7_EnzymeKinetics-Copy1.ipynb

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    "# 7. Enzyme-substrate kinetics\n",
    "\n",
    "## Reminder of concept of Gibbs Free Enery \n",
    "\n",
    "A fundamental task of proteins is to act as catalysts of chemical reactions in the cell. A catalyst is actually a substance or factor to speed up our (chemical or biological) processes/reactions without taking part in them. \n",
    "\n",
    "In biochemical systems, these catalysts are called enzymes, and they increase the rate of virtually all the chemical reactions within cells.\n",
    "\n",
    "To understand how enzymes speed up biochemical reactions we will introduce the concept of __Gibbs Free Energy__: a thermodynamic potential proportional to the maximum of reversible work that may be performed by a thermodynamic system at a constant temperature and pressure \n",
    "                                    (ΔGº = ΔHº-TΔSº)\n",
    "\n",
    "A consequence of the second law of thermodynamics, is that every reaction tends to achieve a minimum in its Gibbs free energy. A quantitative measure of the favorability of a given reaction is the change ΔG caused by the reaction. ∆G can be understood as the amount of \"free\" or \"useful\" energy available to do work.\n",
    "\n",
    "<img src=\"img/endergonic.png\" width=\"400\"/>\n",
    "\n",
    "when the value of ∆G>0 means that the reaction requires free energy, so it cannot take place spontaneously (endergonic).\n",
    "\n",
    "<img src=\"img/exergonic.png\" width=\"400\"/>\n",
    "\n",
    "On the contrary, when the value of ∆G<0 means that the reaction liberates free energy as its proceeds, so its can take place spontaneously (exergonic).\n",
    "\n",
    "### The Transition State and the role of enzymes\n",
    "\n",
    "Between the state where molecules are reactants and the state where molecules are products in biochemical reactions, there is often a \"bump\" in the value of ∆G, which is often refereed as the __transition state__. In the transition state, the reactants are combined in a intermediate species called the activated complex. The stability of this intermediate activated complex affects the rate of the reaction (if it goes towars the producs or it goes towards the reactants). The higher the bump, the more difficult for the reaction to take place, even in conditions of strongly exergonic biochemical reactions (∆G<<0). \n",
    "\n",
    "A way to speed up (i.e., catalyze) a chemical reaction is by stabilization of this intermediate activated complex. A fundamental task of proteins in the cell is to act as these catalysts that increase the rate of virtually all the chemical reactions within cells. These type of proteins are called enzymes. In the absence of enzymatic catalysis, most biochemical reactions are so slow that they would not occur under the mild conditions of temperature and pressure that are compatible with life. Enzymes accelerate the rates of such reactions by well over a million-fold, so reactions that would take years in the absence of catalysis can occur in fractions of seconds if catalyzed by the appropriate enzyme.\n",
    "\n",
    "In brief, the main features of an enzyme are:\n",
    "\n",
    "- Increase the rate of reactions without being consumed or changed by the reaction. \n",
    "- Increase reaction rates without altering the chemical equilibrium between reactants and products (they do not change ∆G).\n",
    "- Enzymes work by lowering the energy of the transition state. They stabilise the intermediate activated complex (they reduce the entropy)\n",
    "\n",
    "<img src=\"img/enzyme.png\" width=\"400\"/>\n",
    "\n",
    "Understanding enzyme kinetics is basic to understand how biological systems work.  In this notebook, we will approach the features of enzymatic reactions using a mathematical perspective, putting to work three of the tools that we just learned so far: \n",
    "- the Mass action Law, \n",
    "- the Mass conservation Law \n",
    "- the ability to solve ODEs numerically  "
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    "## The enzyme-substrate interaction scheme \n",
    "\n",
    "We will now solve the differential equations that result from a generic enzimatic reaction. The scheme of the reaction is as follows:\n",
    "$$ s + e \\overset{k_1}{\\underset{k_2}{\\longleftrightarrow}}  c \\overset{k_3}{\\longrightarrow} p + e \\tag{1}$$\n",
    "\n",
    "where $s$ represents the substrate, $e$ represents the enzyme, $c$ represents the complex, and $p$ the product of the reaction. The reaction can be written as three irreversible reactions: \n",
    "$$\\begin{align*}\n",
    "s + e \\rightarrow  c \\tag{2}\\\\\n",
    "c \\rightarrow s + e \\tag{3}\\\\\n",
    "c \\rightarrow p + e  \\tag{4}\n",
    " \\end{align*}$$\n",
    "\n",
    "\n",
    "with kinetic constants `k1`, `k2` and `k3`, respectively. If we rename variables as $X_1=s,X_2=c, X_3=e, X_4=p$ to use the state vector approach.\n",
    "\n",
    "$$\\begin{align*}\n",
    "X_1 + X_3 \\rightarrow  X_2 \\tag{2}\\\\\n",
    "X_2 \\rightarrow X_1 + X_3 \\tag{3}\\\\\n",
    "X_2 \\rightarrow X_3 +  X_4 \\tag{4}\n",
    " \\end{align*}$$\n",
    "\n",
    "\n",
    "The kinetics of these chemical equations is described by the __Mass action Law__ \n",
    "\n",
    "\n",
    "$$\n",
    "A=\\begin{bmatrix}\n",
    " 1 & 0 & 1 & 0\\\\\n",
    " 0 & 1 & 0 & 0 \\\\\n",
    " 0 & 1 & 0 & 0\\end{bmatrix} ;\n",
    "B=\\begin{bmatrix}\n",
    " 0 & 1 & 0 & 0\\\\\n",
    " 1 & 0 & 1 & 0 \\\\\n",
    " 0 & 0 & 1 & 1\\end{bmatrix} ; (B-A)^T= \\begin{bmatrix}\n",
    "-1 & 1 & 0\\\\\n",
    " 1 & -1  & -1\\\\\n",
    " -1 & 1 & 1 \\\\\n",
    " 0 & 0 & 1\\end{bmatrix} \n",
    "$$\n",
    "\n",
    " in this particular case\n",
    "\n",
    "$$\n",
    "K=\\begin{pmatrix}\n",
    " k_1 & 0 & 0 \\\\\n",
    " 0 & k_2 & 0 \\\\\n",
    " 0 & 0 & k_3 \n",
    "\\end{pmatrix}\n",
    "$$\n",
    "\n",
    "\n",
    "$$X^A=\\begin{pmatrix}\n",
    "X_1^1\\cdot X_2^0 \\cdot X_3^1 \\cdot X_4^0 \\\\\n",
    "X_1^0\\cdot X_2^1 \\cdot X_3^0 \\cdot X_4^0 \\\\\n",
    "X_1^0\\cdot X_2^1 \\cdot X_3^0 \\cdot X_4^0 \\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "X_1 \\cdot X_3\\\\\n",
    " X_2\\\\\n",
    "  X_2\n",
    "\\end{pmatrix}\n",
    "$$\n",
    "\n",
    "so \n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    " \\begin{bmatrix}\n",
    "\\frac{\\mathrm{d} X_1}{\\mathrm{d} t}\\\\ \\frac{\\mathrm{d} X_2}{\\mathrm{d} t} \\\\ \\frac{\\mathrm{d} X_3}{\\mathrm{d} t} \\\\ \\frac{\\mathrm{d} X_4}{\\mathrm{d} t}\\end{bmatrix}= \\begin{bmatrix}\n",
    "-1 & 1 & 0\\\\\n",
    " 1 & -1  & -1\\\\\n",
    " -1 & 1 & 1 \\\\\n",
    " 0 & 0 & 1\\end{bmatrix}\\begin{pmatrix}\n",
    " k_1 & 0 & 0 \\\\\n",
    " 0 & k_2 & 0 \\\\\n",
    " 0 & 0 & k_3 \n",
    "\\end{pmatrix} \\begin{pmatrix}\n",
    "X_1 \\cdot X_3\\\\\n",
    " X_2\\\\\n",
    "  X_2\n",
    "\\end{pmatrix} \n",
    "\\end{align}\n",
    "$$\n",
    "that when we operate the matrices becomes\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    " \\begin{bmatrix}\n",
    "\\frac{\\mathrm{d} X_1}{\\mathrm{d} t}\\\\ \\frac{\\mathrm{d} X_2}{\\mathrm{d} t} \\\\ \\frac{\\mathrm{d} X_3}{\\mathrm{d} t} \\\\ \\frac{\\mathrm{d} X_4}{\\mathrm{d} t}\\end{bmatrix}= \\begin{bmatrix}\n",
    "-1 & 1 & 0\\\\\n",
    " 1 & -1  & -1\\\\\n",
    " -1 & 1 & 1 \\\\\n",
    " 0 & 0 & 1\\end{bmatrix} \\begin{pmatrix}\n",
    "k_1 \\cdot X_1 \\cdot X_3\\\\\n",
    "k_2 \\cdot X_2\\\\\n",
    "k_3 \\cdot  X_2\n",
    "\\end{pmatrix} \n",
    "\\end{align}\n",
    "$$\n",
    "and operating further,\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    " \\begin{bmatrix}\n",
    "\\frac{\\mathrm{d} X_1}{\\mathrm{d} t}\\\\ \\frac{\\mathrm{d} X_2}{\\mathrm{d} t} \\\\ \\frac{\\mathrm{d} X_3}{\\mathrm{d} t} \\\\ \\frac{\\mathrm{d} X_4}{\\mathrm{d} t}\\end{bmatrix}= \\begin{bmatrix}\n",
    "-k_1 \\cdot X_1 \\cdot X_3 + k_2 \\cdot X_2\\\\\n",
    " k_1 \\cdot X_1 \\cdot X_3 - k_2 \\cdot X_2  - k_3 \\cdot  X_2\\\\\n",
    "-k_1 \\cdot X_1 \\cdot X_3 + k_2 \\cdot X_2 + k_3 \\cdot  X_2\\\\\n",
    " k_3 \\cdot  X_2\\end{bmatrix} \n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "which rearranging terms provides the following set of coupled differential equations \n",
    "\n",
    "$$\n",
    "\\begin{align*}\n",
    "  \\dot{X_1}&= k_2 \\cdot X_2 - k_1 X_3 \\cdot X_1 \\tag{8}\\\\ \n",
    "  \\dot{X_2}&= k_1 \\cdot X_3 \\cdot X_1 - (k_2+k_3) \\cdot X_2 \\tag{9}\\\\\n",
    "  \\dot{X_3}&= (k_2+k_3) \\cdot X_2 - k_1 \\cdot X_3(t) \\cdot X_1 \\tag{10}\\\\ \n",
    "  \\dot{X_4}&= k_3 \\cdot X_2 \\tag{11}\n",
    "\\end{align*}\n",
    "$$\n",
    "\n",
    "To solve it numericaly, we define the ODE problem as follows:\n",
    "  "
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   "metadata": {
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    {
     "data": {
      "text/plain": [
       "simpleODEEnzyme! (generic function with 1 method)"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function simpleODEEnzyme!(du,u,p,t)\n",
    "    k1,k2,k3 = p\n",
    "    du[1] = k2*u[2] - k1*u[1]*u[3]\n",
    "    du[2] = k1*u[1]*u[3] - (k2 + k3) * u[2]\n",
    "    du[3] = -k1*u[1]*u[3]+(k2 + k3) * u[2]\n",
    "    du[4] = k3 * u[2] \n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "using DifferentialEquations\n",
    "using ParameterizedFunctions\n",
    "using Plots; gr()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
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   "source": [
    "tspan = (0.0,100)\n",
    "k1=1e3\n",
    "k2=0.1\n",
    "k3=0.05\n",
    "e0=0.002\n",
    "s0=0.002\n",
    "\n",
    "\n",
    "u0=[s0,0,e0,0]\n",
    "p=[k1,k2,k3];"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\u001b[36mODEProblem\u001b[0m with uType \u001b[36mArray{Float64,1}\u001b[0m and tType \u001b[36mFloat64\u001b[0m. In-place: \u001b[36mtrue\u001b[0m\n",
       "timespan: (0.0, 100.0)\n",
       "u0: [0.002, 0.0, 0.002, 0.0]"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "problem = ODEProblem(simpleODEEnzyme!,u0,tspan,p)"
   ]
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   "execution_count": 40,
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       "<g clip-path=\"url(#clip6800)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:66px; text-anchor:middle;\" transform=\"rotate(-90, 89.2861, 773.647)\" x=\"89.2861\" y=\"773.647\">Concentration [M]</text>\n",
       "</g>\n",
       "<polyline clip-path=\"url(#clip6802)\" style=\"stroke:#009af9; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n",
       "  296.041,158.579 298.1,362.84 300.159,507.327 302.217,614.669 304.276,697.396 306.335,762.946 308.394,816.074 310.453,859.932 312.511,896.654 314.57,927.825 \n",
       "  316.629,954.594 318.688,977.788 320.746,998.047 322.805,1015.88 324.864,1031.7 326.923,1045.81 328.981,1058.47 331.04,1069.88 333.099,1080.22 335.158,1089.64 \n",
       "  337.217,1098.25 339.275,1106.16 341.334,1113.45 343.393,1120.19 345.452,1126.44 347.51,1132.27 349.569,1137.72 351.628,1142.83 353.687,1147.64 355.746,1152.17 \n",
       "  357.804,1156.46 359.863,1160.52 361.922,1164.38 363.981,1168.06 366.039,1171.57 368.098,1174.94 370.157,1178.17 372.216,1181.28 374.275,1184.27 376.333,1187.16 \n",
       "  378.392,1189.94 380.451,1192.64 382.51,1195.25 384.568,1197.79 386.627,1200.26 388.686,1202.66 390.745,1205 392.803,1207.29 394.862,1209.52 396.921,1211.7 \n",
       "  398.98,1213.83 401.039,1215.91 403.097,1217.95 405.156,1219.95 407.215,1221.92 409.274,1223.84 411.332,1225.73 413.391,1227.6 415.45,1229.43 417.509,1231.23 \n",
       "  419.568,1233 421.626,1234.74 423.685,1236.45 425.744,1238.14 427.803,1239.8 429.861,1241.43 431.92,1243.04 433.979,1244.63 436.038,1246.19 438.096,1247.73 \n",
       "  440.155,1249.25 442.214,1250.75 444.273,1252.23 446.332,1253.7 448.39,1255.14 450.449,1256.57 452.508,1257.97 454.567,1259.35 456.625,1260.72 458.684,1262.06 \n",
       "  460.743,1263.39 462.802,1264.7 464.861,1265.99 466.919,1267.26 468.978,1268.51 471.037,1269.75 473.096,1270.98 475.154,1272.19 477.213,1273.39 479.272,1274.58 \n",
       "  481.331,1275.75 483.389,1276.91 485.448,1278.05 487.507,1279.18 489.566,1280.29 491.625,1281.38 493.683,1282.46 495.742,1283.53 497.801,1284.57 499.86,1285.61 \n",
       "  501.918,1286.62 503.977,1287.63 506.036,1288.62 508.095,1289.6 510.154,1290.57 512.212,1291.53 514.271,1292.48 516.33,1293.43 518.389,1294.36 520.447,1295.29 \n",
       "  522.506,1296.2 524.565,1297.1 526.624,1297.99 528.683,1298.87 530.741,1299.73 532.8,1300.58 534.859,1301.42 536.918,1302.24 538.976,1303.05 541.035,1303.85 \n",
       "  543.094,1304.63 545.153,1305.41 547.211,1306.17 549.27,1306.93 551.329,1307.68 553.388,1308.43 555.447,1309.17 557.505,1309.91 559.564,1310.64 561.623,1311.37 \n",
       "  563.682,1312.09 565.74,1312.8 567.799,1313.51 569.858,1314.2 571.917,1314.88 573.976,1315.55 576.034,1316.2 578.093,1316.85 580.152,1317.48 582.211,1318.1 \n",
       "  584.269,1318.71 586.328,1319.32 588.387,1319.91 590.446,1320.49 592.504,1321.07 594.563,1321.65 596.622,1322.22 598.681,1322.8 600.74,1323.37 602.798,1323.95 \n",
       "  604.857,1324.52 606.916,1325.09 608.975,1325.66 611.033,1326.22 613.092,1326.77 615.151,1327.31 617.21,1327.84 619.269,1328.36 621.327,1328.87 623.386,1329.36 \n",
       "  625.445,1329.85 627.504,1330.33 629.562,1330.8 631.621,1331.26 633.68,1331.71 635.739,1332.15 637.798,1332.59 639.856,1333.03 641.915,1333.47 643.974,1333.9 \n",
       "  646.033,1334.35 648.091,1334.8 650.15,1335.25 652.209,1335.71 654.268,1336.16 656.326,1336.61 658.385,1337.06 660.444,1337.5 662.503,1337.92 664.562,1338.34 \n",
       "  666.62,1338.75 668.679,1339.15 670.738,1339.54 672.797,1339.91 674.855,1340.28 676.914,1340.64 678.973,1340.98 681.032,1341.32 683.091,1341.66 685.149,1341.99 \n",
       "  687.208,1342.32 689.267,1342.64 691.326,1342.98 693.384,1343.32 695.443,1343.66 697.502,1344.03 699.561,1344.4 701.619,1344.77 703.678,1345.14 705.737,1345.51 \n",
       "  707.796,1345.87 709.855,1346.23 711.913,1346.58 713.972,1346.92 716.031,1347.24 718.09,1347.55 720.148,1347.85 722.207,1348.14 724.266,1348.42 726.325,1348.68 \n",
       "  728.384,1348.94 730.442,1349.19 732.501,1349.43 734.56,1349.67 736.619,1349.91 738.677,1350.15 740.736,1350.4 742.795,1350.66 744.854,1350.94 746.913,1351.23 \n",
       "  748.971,1351.54 751.03,1351.86 753.089,1352.18 755.148,1352.5 757.206,1352.82 759.265,1353.12 761.324,1353.42 763.383,1353.7 765.441,1353.97 767.5,1354.22 \n",
       "  769.559,1354.46 771.618,1354.68 773.677,1354.89 775.735,1355.08 777.794,1355.26 779.853,1355.44 781.912,1355.6 783.97,1355.77 786.029,1355.93 788.088,1356.1 \n",
       "  790.147,1356.28 792.206,1356.48 794.264,1356.7 796.323,1356.95 798.382,1357.22 800.441,1357.51 802.499,1357.8 804.558,1358.1 806.617,1358.38 808.676,1358.66 \n",
       "  810.734,1358.93 812.793,1359.17 814.852,1359.4 816.911,1359.6 818.97,1359.79 821.028,1359.96 823.087,1360.1 825.146,1360.23 827.205,1360.34 829.263,1360.45 \n",
       "  831.322,1360.54 833.381,1360.64 835.44,1360.74 837.499,1360.86 839.557,1360.99 841.616,1361.15 843.675,1361.35 845.734,1361.59 847.792,1361.85 849.851,1362.13 \n",
       "  851.91,1362.41 853.969,1362.69 856.028,1362.95 858.086,1363.2 860.145,1363.43 862.204,1363.63 864.263,1363.81 866.321,1363.97 868.38,1364.09 870.439,1364.2 \n",
       "  872.498,1364.28 874.556,1364.34 876.615,1364.39 878.674,1364.43 880.733,1364.48 882.792,1364.53 884.85,1364.6 886.909,1364.7 888.968,1364.84 891.027,1365.03 \n",
       "  893.085,1365.27 895.144,1365.52 897.203,1365.79 899.262,1366.05 901.321,1366.31 903.379,1366.55 905.438,1366.77 907.497,1366.96 909.556,1367.13 911.614,1367.26 \n",
       "  913.673,1367.37 915.732,1367.45 917.791,1367.5 919.849,1367.53 921.908,1367.54 923.967,1367.55 926.026,1367.56 928.085,1367.58 930.143,1367.62 932.202,1367.7 \n",
       "  934.261,1367.83 936.32,1368.02 938.378,1368.25 940.437,1368.49 942.496,1368.74 944.555,1368.98 946.614,1369.22 948.672,1369.43 950.731,1369.61 952.79,1369.77 \n",
       "  954.849,1369.9 956.907,1369.99 958.966,1370.06 961.025,1370.1 963.084,1370.11 965.143,1370.11 967.201,1370.1 969.26,1370.09 971.319,1370.09 973.378,1370.12 \n",
       "  975.436,1370.18 977.495,1370.3 979.554,1370.47 981.613,1370.68 983.671,1370.91 985.73,1371.13 987.789,1371.35 989.848,1371.56 991.907,1371.74 993.965,1371.9 \n",
       "  996.024,1372.03 998.083,1372.13 1000.14,1372.2 1002.2,1372.24 1004.26,1372.26 1006.32,1372.26 1008.38,1372.24 1010.44,1372.22 1012.49,1372.21 1014.55,1372.22 \n",
       "  1016.61,1372.25 1018.67,1372.33 1020.73,1372.48 1022.79,1372.66 1024.85,1372.86 1026.91,1373.07 1028.96,1373.27 1031.02,1373.45 1033.08,1373.62 1035.14,1373.77 \n",
       "  1037.2,1373.89 1039.26,1373.98 1041.32,1374.04 1043.38,1374.07 1045.43,1374.08 1047.49,1374.08 1049.55,1374.06 1051.61,1374.03 1053.67,1374.01 1055.73,1374.01 \n",
       "  1057.79,1374.05 1059.85,1374.12 1061.9,1374.26 1063.96,1374.43 1066.02,1374.61 1068.08,1374.8 1070.14,1374.98 1072.2,1375.14 1074.26,1375.29 1076.32,1375.42 \n",
       "  1078.38,1375.52 1080.43,1375.59 1082.49,1375.63 1084.55,1375.65 1086.61,1375.65 1088.67,1375.63 1090.73,1375.6 1092.79,1375.58 1094.85,1375.56 1096.9,1375.56 \n",
       "  1098.96,1375.6 1101.02,1375.69 1103.08,1375.8 1105.14,1375.91 1107.2,1376.01 1109.26,1376.11 1111.32,1376.21 1113.37,1376.3 1115.43,1376.4 1117.49,1376.49 \n",
       "  1119.55,1376.59 1121.61,1376.68 1123.67,1376.77 1125.73,1376.85 1127.79,1376.94 1129.84,1377.02 1131.9,1377.1 1133.96,1377.18 1136.02,1377.26 1138.08,1377.34 \n",
       "  1140.14,1377.42 1142.2,1377.49 1144.26,1377.56 1146.31,1377.63 1148.37,1377.7 1150.43,1377.77 1152.49,1377.83 1154.55,1377.9 1156.61,1377.96 1158.67,1378.02 \n",
       "  1160.73,1378.08 1162.78,1378.14 1164.84,1378.19 1166.9,1378.25 1168.96,1378.3 1171.02,1378.35 1173.08,1378.4 1175.14,1378.45 1177.2,1378.49 1179.25,1378.54 \n",
       "  1181.31,1378.58 1183.37,1378.63 1185.43,1378.68 1187.49,1378.73 1189.55,1378.78 1191.61,1378.83 1193.67,1378.87 1195.73,1378.92 1197.78,1378.97 1199.84,1379.02 \n",
       "  1201.9,1379.07 1203.96,1379.12 1206.02,1379.17 1208.08,1379.21 1210.14,1379.26 1212.2,1379.31 1214.25,1379.36 1216.31,1379.41 1218.37,1379.45 1220.43,1379.5 \n",
       "  1222.49,1379.55 1224.55,1379.6 1226.61,1379.65 1228.67,1379.69 1230.72,1379.74 1232.78,1379.79 1234.84,1379.83 1236.9,1379.88 1238.96,1379.93 1241.02,1379.98 \n",
       "  1243.08,1380.02 1245.14,1380.07 1247.19,1380.12 1249.25,1380.16 1251.31,1380.21 1253.37,1380.26 1255.43,1380.3 1257.49,1380.35 1259.55,1380.4 1261.61,1380.44 \n",
       "  1263.66,1380.49 1265.72,1380.53 1267.78,1380.58 1269.84,1380.62 1271.9,1380.67 1273.96,1380.71 1276.02,1380.75 1278.08,1380.8 1280.13,1380.84 1282.19,1380.88 \n",
       "  1284.25,1380.93 1286.31,1380.97 1288.37,1381.01 1290.43,1381.05 1292.49,1381.09 1294.55,1381.14 1296.61,1381.18 1298.66,1381.22 1300.72,1381.26 1302.78,1381.3 \n",
       "  1304.84,1381.34 1306.9,1381.38 1308.96,1381.42 1311.02,1381.45 1313.08,1381.49 1315.13,1381.53 1317.19,1381.57 1319.25,1381.61 1321.31,1381.65 1323.37,1381.68 \n",
       "  1325.43,1381.72 1327.49,1381.76 1329.55,1381.79 1331.6,1381.83 1333.66,1381.87 1335.72,1381.9 1337.78,1381.94 1339.84,1381.97 1341.9,1382.01 1343.96,1382.04 \n",
       "  1346.02,1382.08 1348.07,1382.11 1350.13,1382.14 1352.19,1382.18 1354.25,1382.21 1356.31,1382.24 1358.37,1382.28 1360.43,1382.31 1362.49,1382.34 1364.54,1382.38 \n",
       "  1366.6,1382.41 1368.66,1382.44 1370.72,1382.47 1372.78,1382.51 1374.84,1382.54 1376.9,1382.57 1378.96,1382.6 1381.01,1382.63 1383.07,1382.66 1385.13,1382.7 \n",
       "  1387.19,1382.73 1389.25,1382.76 1391.31,1382.79 1393.37,1382.82 1395.43,1382.85 1397.48,1382.88 1399.54,1382.91 1401.6,1382.94 1403.66,1382.97 1405.72,1383 \n",
       "  1407.78,1383.03 1409.84,1383.06 1411.9,1383.09 1413.96,1383.12 1416.01,1383.15 1418.07,1383.18 1420.13,1383.2 1422.19,1383.23 1424.25,1383.26 1426.31,1383.29 \n",
       "  1428.37,1383.32 1430.43,1383.34 1432.48,1383.37 1434.54,1383.4 1436.6,1383.43 1438.66,1383.45 1440.72,1383.48 1442.78,1383.51 1444.84,1383.53 1446.9,1383.56 \n",
       "  1448.95,1383.59 1451.01,1383.61 1453.07,1383.64 1455.13,1383.67 1457.19,1383.69 1459.25,1383.72 1461.31,1383.74 1463.37,1383.77 1465.42,1383.79 1467.48,1383.82 \n",
       "  1469.54,1383.84 1471.6,1383.87 1473.66,1383.89 1475.72,1383.92 1477.78,1383.94 1479.84,1383.97 1481.89,1383.99 1483.95,1384.01 1486.01,1384.04 1488.07,1384.06 \n",
       "  1490.13,1384.08 1492.19,1384.11 1494.25,1384.13 1496.31,1384.16 1498.36,1384.18 1500.42,1384.2 1502.48,1384.22 1504.54,1384.25 1506.6,1384.27 1508.66,1384.29 \n",
       "  1510.72,1384.32 1512.78,1384.34 1514.84,1384.36 1516.89,1384.38 1518.95,1384.4 1521.01,1384.43 1523.07,1384.45 1525.13,1384.47 1527.19,1384.49 1529.25,1384.51 \n",
       "  1531.31,1384.53 1533.36,1384.56 1535.42,1384.58 1537.48,1384.6 1539.54,1384.62 1541.6,1384.64 1543.66,1384.66 1545.72,1384.68 1547.78,1384.7 1549.83,1384.72 \n",
       "  1551.89,1384.74 1553.95,1384.76 1556.01,1384.78 1558.07,1384.8 1560.13,1384.82 1562.19,1384.84 1564.25,1384.86 1566.3,1384.88 1568.36,1384.9 1570.42,1384.92 \n",
       "  1572.48,1384.94 1574.54,1384.96 1576.6,1384.97 1578.66,1384.99 1580.72,1385.01 1582.77,1385.03 1584.83,1385.05 1586.89,1385.07 1588.95,1385.08 1591.01,1385.1 \n",
       "  1593.07,1385.12 1595.13,1385.14 1597.19,1385.16 1599.24,1385.17 1601.3,1385.19 1603.36,1385.21 1605.42,1385.23 1607.48,1385.24 1609.54,1385.26 1611.6,1385.28 \n",
       "  1613.66,1385.3 1615.71,1385.31 1617.77,1385.33 1619.83,1385.35 1621.89,1385.36 1623.95,1385.38 1626.01,1385.4 1628.07,1385.41 1630.13,1385.43 1632.19,1385.45 \n",
       "  1634.24,1385.46 1636.3,1385.48 1638.36,1385.5 1640.42,1385.51 1642.48,1385.53 1644.54,1385.55 1646.6,1385.56 1648.66,1385.58 1650.71,1385.59 1652.77,1385.61 \n",
       "  1654.83,1385.62 1656.89,1385.64 1658.95,1385.65 1661.01,1385.67 1663.07,1385.69 1665.13,1385.7 1667.18,1385.72 1669.24,1385.73 1671.3,1385.75 1673.36,1385.76 \n",
       "  1675.42,1385.77 1677.48,1385.79 1679.54,1385.8 1681.6,1385.82 1683.65,1385.83 1685.71,1385.85 1687.77,1385.86 1689.83,1385.88 1691.89,1385.89 1693.95,1385.9 \n",
       "  1696.01,1385.92 1698.07,1385.93 1700.12,1385.95 1702.18,1385.96 1704.24,1385.97 1706.3,1385.99 1708.36,1386 1710.42,1386.01 1712.48,1386.03 1714.54,1386.04 \n",
       "  1716.59,1386.05 1718.65,1386.07 1720.71,1386.08 1722.77,1386.09 1724.83,1386.1 1726.89,1386.12 1728.95,1386.13 1731.01,1386.14 1733.07,1386.16 1735.12,1386.17 \n",
       "  1737.18,1386.18 1739.24,1386.19 1741.3,1386.21 1743.36,1386.22 1745.42,1386.23 1747.48,1386.24 1749.54,1386.25 1751.59,1386.27 1753.65,1386.28 1755.71,1386.29 \n",
       "  1757.77,1386.3 1759.83,1386.32 1761.89,1386.33 1763.95,1386.34 1766.01,1386.35 1768.06,1386.36 1770.12,1386.37 1772.18,1386.39 1774.24,1386.4 1776.3,1386.41 \n",
       "  1778.36,1386.42 1780.42,1386.43 1782.48,1386.44 1784.53,1386.45 1786.59,1386.47 1788.65,1386.48 1790.71,1386.49 1792.77,1386.5 1794.83,1386.51 1796.89,1386.52 \n",
       "  1798.95,1386.53 1801,1386.54 1803.06,1386.55 1805.12,1386.56 1807.18,1386.57 1809.24,1386.58 1811.3,1386.6 1813.36,1386.61 1815.42,1386.62 1817.47,1386.63 \n",
       "  1819.53,1386.64 1821.59,1386.65 1823.65,1386.66 1825.71,1386.67 1827.77,1386.68 1829.83,1386.69 1831.89,1386.7 1833.94,1386.71 1836,1386.72 1838.06,1386.73 \n",
       "  1840.12,1386.74 1842.18,1386.75 1844.24,1386.76 1846.3,1386.77 1848.36,1386.77 1850.42,1386.78 1852.47,1386.79 1854.53,1386.8 1856.59,1386.81 1858.65,1386.82 \n",
       "  1860.71,1386.83 1862.77,1386.84 1864.83,1386.85 1866.89,1386.86 1868.94,1386.87 1871,1386.88 1873.06,1386.88 1875.12,1386.89 1877.18,1386.9 1879.24,1386.91 \n",
       "  1881.3,1386.92 1883.36,1386.93 1885.41,1386.94 1887.47,1386.95 1889.53,1386.96 1891.59,1386.96 1893.65,1386.97 1895.71,1386.98 1897.77,1386.99 1899.83,1387 \n",
       "  1901.88,1387.01 1903.94,1387.01 1906,1387.02 1908.06,1387.03 1910.12,1387.04 1912.18,1387.05 1914.24,1387.06 1916.3,1387.06 1918.35,1387.07 1920.41,1387.08 \n",
       "  1922.47,1387.09 1924.53,1387.1 1926.59,1387.1 1928.65,1387.11 1930.71,1387.12 1932.77,1387.13 1934.82,1387.14 1936.88,1387.14 1938.94,1387.15 1941,1387.16 \n",
       "  1943.06,1387.17 1945.12,1387.17 1947.18,1387.18 1949.24,1387.19 1951.3,1387.2 1953.35,1387.2 1955.41,1387.21 1957.47,1387.22 1959.53,1387.23 1961.59,1387.23 \n",
       "  1963.65,1387.24 1965.71,1387.25 1967.77,1387.26 1969.82,1387.26 1971.88,1387.27 1973.94,1387.28 1976,1387.28 1978.06,1387.29 1980.12,1387.3 1982.18,1387.3 \n",
       "  1984.24,1387.31 1986.29,1387.32 1988.35,1387.32 1990.41,1387.33 1992.47,1387.34 1994.53,1387.35 1996.59,1387.35 1998.65,1387.36 2000.71,1387.36 2002.76,1387.37 \n",
       "  2004.82,1387.38 2006.88,1387.38 2008.94,1387.39 2011,1387.4 2013.06,1387.4 2015.12,1387.41 2017.18,1387.42 2019.23,1387.42 2021.29,1387.43 2023.35,1387.43 \n",
       "  2025.41,1387.44 2027.47,1387.45 2029.53,1387.45 2031.59,1387.46 2033.65,1387.47 2035.7,1387.47 2037.76,1387.48 2039.82,1387.48 2041.88,1387.49 2043.94,1387.5 \n",
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       "  2107.76,1387.67 2109.82,1387.67 2111.88,1387.68 2113.94,1387.68 2116,1387.69 2118.06,1387.69 2120.11,1387.7 2122.17,1387.7 2124.23,1387.71 2126.29,1387.71 \n",
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       "  2190.11,1387.85 2192.17,1387.86 2194.23,1387.86 2196.29,1387.86 2198.35,1387.87 2200.41,1387.87 2202.47,1387.88 2204.52,1387.88 2206.58,1387.88 2208.64,1387.89 \n",
       "  2210.7,1387.89 2212.76,1387.9 2214.82,1387.9 2216.88,1387.9 2218.94,1387.91 2220.99,1387.91 2223.05,1387.92 2225.11,1387.92 2227.17,1387.92 2229.23,1387.93 \n",
       "  2231.29,1387.93 2233.35,1387.93 2235.41,1387.94 2237.46,1387.94 2239.52,1387.95 2241.58,1387.95 2243.64,1387.95 2245.7,1387.96 2247.76,1387.96 2249.82,1387.96 \n",
       "  2251.88,1387.97 2253.93,1387.97 2255.99,1387.98 2258.05,1387.98 2260.11,1387.98 2262.17,1387.99 2264.23,1387.99 2266.29,1387.99 2268.35,1388 2270.4,1388 \n",
       "  2272.46,1388 2274.52,1388.01 2276.58,1388.01 2278.64,1388.01 2280.7,1388.02 2282.76,1388.02 2284.82,1388.02 2286.88,1388.03 2288.93,1388.03 2290.99,1388.03 \n",
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       "  2313.64,1388.07 2315.7,1388.07 2317.76,1388.08 2319.82,1388.08 2321.87,1388.08 2323.93,1388.09 2325.99,1388.09 2328.05,1388.09 2330.11,1388.09 2332.17,1388.1 \n",
       "  2334.23,1388.1 2336.29,1388.1 2338.34,1388.11 2340.4,1388.11 2342.46,1388.11 2344.52,1388.11 2346.58,1388.12 2348.64,1388.12 2350.7,1388.12 2352.76,1388.13 \n",
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       "  1634.24,1328.46 1636.3,1328.74 1638.36,1329.03 1640.42,1329.31 1642.48,1329.59 1644.54,1329.87 1646.6,1330.15 1648.66,1330.43 1650.71,1330.71 1652.77,1330.99 \n",
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       "  1675.42,1333.94 1677.48,1334.2 1679.54,1334.46 1681.6,1334.72 1683.65,1334.97 1685.71,1335.23 1687.77,1335.48 1689.83,1335.74 1691.89,1335.99 1693.95,1336.24 \n",
       "  1696.01,1336.49 1698.07,1336.74 1700.12,1336.99 1702.18,1337.23 1704.24,1337.48 1706.3,1337.72 1708.36,1337.96 1710.42,1338.2 1712.48,1338.44 1714.54,1338.68 \n",
       "  1716.59,1338.92 1718.65,1339.15 1720.71,1339.38 1722.77,1339.62 1724.83,1339.85 1726.89,1340.08 1728.95,1340.31 1731.01,1340.53 1733.07,1340.76 1735.12,1340.99 \n",
       "  1737.18,1341.21 1739.24,1341.44 1741.3,1341.66 1743.36,1341.88 1745.42,1342.11 1747.48,1342.33 1749.54,1342.55 1751.59,1342.76 1753.65,1342.98 1755.71,1343.2 \n",
       "  1757.77,1343.42 1759.83,1343.63 1761.89,1343.84 1763.95,1344.06 1766.01,1344.27 1768.06,1344.48 1770.12,1344.69 1772.18,1344.9 1774.24,1345.11 1776.3,1345.32 \n",
       "  1778.36,1345.53 1780.42,1345.73 1782.48,1345.94 1784.53,1346.14 1786.59,1346.34 1788.65,1346.55 1790.71,1346.75 1792.77,1346.95 1794.83,1347.15 1796.89,1347.34 \n",
       "  1798.95,1347.54 1801,1347.74 1803.06,1347.93 1805.12,1348.13 1807.18,1348.32 1809.24,1348.52 1811.3,1348.71 1813.36,1348.9 1815.42,1349.09 1817.47,1349.28 \n",
       "  1819.53,1349.47 1821.59,1349.65 1823.65,1349.84 1825.71,1350.03 1827.77,1350.21 1829.83,1350.39 1831.89,1350.58 1833.94,1350.76 1836,1350.94 1838.06,1351.12 \n",
       "  1840.12,1351.3 1842.18,1351.48 1844.24,1351.65 1846.3,1351.83 1848.36,1352.01 1850.42,1352.18 1852.47,1352.35 1854.53,1352.53 1856.59,1352.7 1858.65,1352.87 \n",
       "  1860.71,1353.04 1862.77,1353.2 1864.83,1353.37 1866.89,1353.54 1868.94,1353.7 1871,1353.87 1873.06,1354.03 1875.12,1354.2 1877.18,1354.36 1879.24,1354.52 \n",
       "  1881.3,1354.68 1883.36,1354.84 1885.41,1355 1887.47,1355.16 1889.53,1355.32 1891.59,1355.48 1893.65,1355.64 1895.71,1355.8 1897.77,1355.95 1899.83,1356.11 \n",
       "  1901.88,1356.26 1903.94,1356.42 1906,1356.57 1908.06,1356.72 1910.12,1356.88 1912.18,1357.03 1914.24,1357.18 1916.3,1357.33 1918.35,1357.48 1920.41,1357.63 \n",
       "  1922.47,1357.78 1924.53,1357.93 1926.59,1358.07 1928.65,1358.22 1930.71,1358.37 1932.77,1358.51 1934.82,1358.65 1936.88,1358.8 1938.94,1358.94 1941,1359.08 \n",
       "  1943.06,1359.23 1945.12,1359.37 1947.18,1359.51 1949.24,1359.65 1951.3,1359.79 1953.35,1359.93 1955.41,1360.06 1957.47,1360.2 1959.53,1360.34 1961.59,1360.47 \n",
       "  1963.65,1360.61 1965.71,1360.74 1967.77,1360.88 1969.82,1361.01 1971.88,1361.14 1973.94,1361.27 1976,1361.41 1978.06,1361.54 1980.12,1361.67 1982.18,1361.8 \n",
       "  1984.24,1361.92 1986.29,1362.05 1988.35,1362.18 1990.41,1362.31 1992.47,1362.43 1994.53,1362.56 1996.59,1362.68 1998.65,1362.81 2000.71,1362.93 2002.76,1363.05 \n",
       "  2004.82,1363.17 2006.88,1363.3 2008.94,1363.42 2011,1363.54 2013.06,1363.66 2015.12,1363.77 2017.18,1363.89 2019.23,1364.01 2021.29,1364.12 2023.35,1364.24 \n",
       "  2025.41,1364.35 2027.47,1364.47 2029.53,1364.58 2031.59,1364.7 2033.65,1364.81 2035.7,1364.92 2037.76,1365.04 2039.82,1365.15 2041.88,1365.26 2043.94,1365.37 \n",
       "  2046,1365.48 2048.06,1365.59 2050.12,1365.7 2052.17,1365.81 2054.23,1365.92 2056.29,1366.03 2058.35,1366.14 2060.41,1366.25 2062.47,1366.35 2064.53,1366.46 \n",
       "  2066.59,1366.57 2068.65,1366.67 2070.7,1366.78 2072.76,1366.88 2074.82,1366.99 2076.88,1367.09 2078.94,1367.19 2081,1367.3 2083.06,1367.4 2085.12,1367.5 \n",
       "  2087.17,1367.6 2089.23,1367.7 2091.29,1367.8 2093.35,1367.9 2095.41,1368 2097.47,1368.1 2099.53,1368.2 2101.59,1368.3 2103.64,1368.4 2105.7,1368.5 \n",
       "  2107.76,1368.59 2109.82,1368.69 2111.88,1368.79 2113.94,1368.88 2116,1368.98 2118.06,1369.07 2120.11,1369.17 2122.17,1369.26 2124.23,1369.35 2126.29,1369.45 \n",
       "  2128.35,1369.54 2130.41,1369.63 2132.47,1369.72 2134.53,1369.81 2136.58,1369.91 2138.64,1370 2140.7,1370.09 2142.76,1370.18 2144.82,1370.26 2146.88,1370.35 \n",
       "  2148.94,1370.44 2151,1370.53 2153.05,1370.61 2155.11,1370.7 2157.17,1370.79 2159.23,1370.87 2161.29,1370.96 2163.35,1371.04 2165.41,1371.13 2167.47,1371.21 \n",
       "  2169.53,1371.3 2171.58,1371.38 2173.64,1371.46 2175.7,1371.54 2177.76,1371.62 2179.82,1371.71 2181.88,1371.79 2183.94,1371.86 2186,1371.94 2188.05,1372.02 \n",
       "  2190.11,1372.1 2192.17,1372.18 2194.23,1372.26 2196.29,1372.34 2198.35,1372.41 2200.41,1372.49 2202.47,1372.57 2204.52,1372.64 2206.58,1372.72 2208.64,1372.79 \n",
       "  2210.7,1372.87 2212.76,1372.95 2214.82,1373.02 2216.88,1373.1 2218.94,1373.17 2220.99,1373.24 2223.05,1373.32 2225.11,1373.39 2227.17,1373.46 2229.23,1373.54 \n",
       "  2231.29,1373.61 2233.35,1373.68 2235.41,1373.75 2237.46,1373.82 2239.52,1373.9 2241.58,1373.97 2243.64,1374.04 2245.7,1374.11 2247.76,1374.18 2249.82,1374.25 \n",
       "  2251.88,1374.32 2253.93,1374.38 2255.99,1374.45 2258.05,1374.52 2260.11,1374.59 2262.17,1374.66 2264.23,1374.73 2266.29,1374.79 2268.35,1374.86 2270.4,1374.93 \n",
       "  2272.46,1374.99 2274.52,1375.06 2276.58,1375.12 2278.64,1375.19 2280.7,1375.25 2282.76,1375.32 2284.82,1375.38 2286.88,1375.45 2288.93,1375.51 2290.99,1375.58 \n",
       "  2293.05,1375.64 2295.11,1375.7 2297.17,1375.76 2299.23,1375.83 2301.29,1375.89 2303.35,1375.95 2305.4,1376.01 2307.46,1376.07 2309.52,1376.13 2311.58,1376.19 \n",
       "  2313.64,1376.25 2315.7,1376.31 2317.76,1376.37 2319.82,1376.43 2321.87,1376.49 2323.93,1376.55 2325.99,1376.61 2328.05,1376.67 2330.11,1376.72 2332.17,1376.78 \n",
       "  2334.23,1376.84 2336.29,1376.89 2338.34,1376.95 2340.4,1377.01 2342.46,1377.06 2344.52,1377.12 2346.58,1377.17 2348.64,1377.23 2350.7,1377.28 2352.76,1377.34 \n",
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       "  1407.78,260.073 1409.84,259.593 1411.9,259.115 1413.96,258.64 1416.01,258.167 1418.07,257.696 1420.13,257.227 1422.19,256.76 1424.25,256.296 1426.31,255.833 \n",
       "  1428.37,255.373 1430.43,254.915 1432.48,254.459 1434.54,254.005 1436.6,253.553 1438.66,253.104 1440.72,252.657 1442.78,252.211 1444.84,251.768 1446.9,251.327 \n",
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       "  1469.54,246.62 1471.6,246.207 1473.66,245.796 1475.72,245.387 1477.78,244.979 1479.84,244.574 1481.89,244.169 1483.95,243.767 1486.01,243.366 1488.07,242.967 \n",
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       "  1510.72,238.686 1512.78,238.307 1514.84,237.93 1516.89,237.554 1518.95,237.18 1521.01,236.807 1523.07,236.437 1525.13,236.067 1527.19,235.7 1529.25,235.334 \n",
       "  1531.31,234.97 1533.36,234.607 1535.42,234.246 1537.48,233.887 1539.54,233.53 1541.6,233.174 1543.66,232.82 1545.72,232.467 1547.78,232.116 1549.83,231.767 \n",
       "  1551.89,231.42 1553.95,231.074 1556.01,230.73 1558.07,230.387 1560.13,230.046 1562.19,229.707 1564.25,229.37 1566.3,229.034 1568.36,228.7 1570.42,228.367 \n",
       "  1572.48,228.036 1574.54,227.707 1576.6,227.379 1578.66,227.053 1580.72,226.729 1582.77,226.407 1584.83,226.086 1586.89,225.767 1588.95,225.449 1591.01,225.135 \n",
       "  1593.07,224.823 1595.13,224.511 1597.19,224.201 1599.24,223.893 1601.3,223.585 1603.36,223.279 1605.42,222.974 1607.48,222.67 1609.54,222.368 1611.6,222.067 \n",
       "  1613.66,221.767 1615.71,221.468 1617.77,221.171 1619.83,220.875 1621.89,220.58 1623.95,220.286 1626.01,219.994 1628.07,219.703 1630.13,219.413 1632.19,219.125 \n",
       "  1634.24,218.837 1636.3,218.551 1638.36,218.266 1640.42,217.983 1642.48,217.701 1644.54,217.42 1646.6,217.14 1648.66,216.861 1650.71,216.584 1652.77,216.308 \n",
       "  1654.83,216.033 1656.89,215.76 1658.95,215.488 1661.01,215.217 1663.07,214.947 1665.13,214.679 1667.18,214.412 1669.24,214.146 1671.3,213.881 1673.36,213.618 \n",
       "  1675.42,213.355 1677.48,213.095 1679.54,212.835 1681.6,212.577 1683.65,212.319 1685.71,212.064 1687.77,211.809 1689.83,211.556 1691.89,211.304 1693.95,211.053 \n",
       "  1696.01,210.803 1698.07,210.555 1700.12,210.308 1702.18,210.062 1704.24,209.818 1706.3,209.574 1708.36,209.332 1710.42,209.092 1712.48,208.852 1714.54,208.614 \n",
       "  1716.59,208.377 1718.65,208.142 1720.71,207.909 1722.77,207.677 1724.83,207.446 1726.89,207.216 1728.95,206.987 1731.01,206.759 1733.07,206.532 1735.12,206.306 \n",
       "  1737.18,206.081 1739.24,205.856 1741.3,205.633 1743.36,205.41 1745.42,205.189 1747.48,204.968 1749.54,204.748 1751.59,204.529 1753.65,204.311 1755.71,204.094 \n",
       "  1757.77,203.878 1759.83,203.663 1761.89,203.449 1763.95,203.236 1766.01,203.023 1768.06,202.812 1770.12,202.601 1772.18,202.392 1774.24,202.183 1776.3,201.976 \n",
       "  1778.36,201.769 1780.42,201.563 1782.48,201.358 1784.53,201.154 1786.59,200.951 1788.65,200.749 1790.71,200.547 1792.77,200.347 1794.83,200.148 1796.89,199.949 \n",
       "  1798.95,199.752 1801,199.555 1803.06,199.359 1805.12,199.165 1807.18,198.971 1809.24,198.778 1811.3,198.586 1813.36,198.395 1815.42,198.205 1817.47,198.015 \n",
       "  1819.53,197.827 1821.59,197.64 1823.65,197.453 1825.71,197.268 1827.77,197.083 1829.83,196.9 1831.89,196.717 1833.94,196.535 1836,196.354 1838.06,196.174 \n",
       "  1840.12,195.995 1842.18,195.817 1844.24,195.64 1846.3,195.464 1848.36,195.288 1850.42,195.114 1852.47,194.94 1854.53,194.768 1856.59,194.596 1858.65,194.425 \n",
       "  1860.71,194.257 1862.77,194.089 1864.83,193.922 1866.89,193.756 1868.94,193.591 1871,193.426 1873.06,193.261 1875.12,193.098 1877.18,192.935 1879.24,192.772 \n",
       "  1881.3,192.611 1883.36,192.45 1885.41,192.289 1887.47,192.129 1889.53,191.97 1891.59,191.812 1893.65,191.654 1895.71,191.497 1897.77,191.341 1899.83,191.185 \n",
       "  1901.88,191.03 1903.94,190.876 1906,190.722 1908.06,190.569 1910.12,190.416 1912.18,190.265 1914.24,190.114 1916.3,189.963 1918.35,189.813 1920.41,189.664 \n",
       "  1922.47,189.516 1924.53,189.368 1926.59,189.221 1928.65,189.074 1930.71,188.929 1932.77,188.783 1934.82,188.639 1936.88,188.495 1938.94,188.352 1941,188.209 \n",
       "  1943.06,188.068 1945.12,187.926 1947.18,187.786 1949.24,187.646 1951.3,187.507 1953.35,187.368 1955.41,187.23 1957.47,187.093 1959.53,186.956 1961.59,186.821 \n",
       "  1963.65,186.685 1965.71,186.551 1967.77,186.417 1969.82,186.284 1971.88,186.151 1973.94,186.019 1976,185.888 1978.06,185.757 1980.12,185.627 1982.18,185.498 \n",
       "  1984.24,185.369 1986.29,185.241 1988.35,185.114 1990.41,184.987 1992.47,184.861 1994.53,184.736 1996.59,184.611 1998.65,184.487 2000.71,184.364 2002.76,184.241 \n",
       "  2004.82,184.119 2006.88,183.998 2008.94,183.877 2011,183.757 2013.06,183.638 2015.12,183.521 2017.18,183.403 2019.23,183.287 2021.29,183.17 2023.35,183.055 \n",
       "  2025.41,182.939 2027.47,182.824 2029.53,182.71 2031.59,182.596 2033.65,182.482 2035.7,182.369 2037.76,182.257 2039.82,182.145 2041.88,182.033 2043.94,181.922 \n",
       "  2046,181.811 2048.06,181.701 2050.12,181.591 2052.17,181.481 2054.23,181.372 2056.29,181.264 2058.35,181.156 2060.41,181.048 2062.47,180.941 2064.53,180.834 \n",
       "  2066.59,180.728 2068.65,180.622 2070.7,180.517 2072.76,180.412 2074.82,180.308 2076.88,180.204 2078.94,180.1 2081,179.997 2083.06,179.895 2085.12,179.793 \n",
       "  2087.17,179.691 2089.23,179.59 2091.29,179.489 2093.35,179.389 2095.41,179.289 2097.47,179.19 2099.53,179.091 2101.59,178.992 2103.64,178.894 2105.7,178.797 \n",
       "  2107.76,178.7 2109.82,178.603 2111.88,178.507 2113.94,178.411 2116,178.316 2118.06,178.221 2120.11,178.127 2122.17,178.033 2124.23,177.939 2126.29,177.846 \n",
       "  2128.35,177.754 2130.41,177.662 2132.47,177.57 2134.53,177.479 2136.58,177.388 2138.64,177.298 2140.7,177.208 2142.76,177.119 2144.82,177.03 2146.88,176.941 \n",
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       "  2169.53,175.999 2171.58,175.916 2173.64,175.833 2175.7,175.751 2177.76,175.67 2179.82,175.588 2181.88,175.508 2183.94,175.429 2186,175.35 2188.05,175.271 \n",
       "  2190.11,175.192 2192.17,175.114 2194.23,175.036 2196.29,174.958 2198.35,174.881 2200.41,174.804 2202.47,174.727 2204.52,174.651 2206.58,174.575 2208.64,174.499 \n",
       "  2210.7,174.423 2212.76,174.348 2214.82,174.273 2216.88,174.199 2218.94,174.124 2220.99,174.05 2223.05,173.977 2225.11,173.903 2227.17,173.83 2229.23,173.758 \n",
       "  2231.29,173.685 2233.35,173.613 2235.41,173.541 2237.46,173.47 2239.52,173.399 2241.58,173.328 2243.64,173.257 2245.7,173.187 2247.76,173.117 2249.82,173.047 \n",
       "  2251.88,172.978 2253.93,172.909 2255.99,172.84 2258.05,172.772 2260.11,172.703 2262.17,172.636 2264.23,172.568 2266.29,172.501 2268.35,172.434 2270.4,172.367 \n",
       "  2272.46,172.301 2274.52,172.235 2276.58,172.17 2278.64,172.104 2280.7,172.039 2282.76,171.974 2284.82,171.91 2286.88,171.846 2288.93,171.782 2290.99,171.718 \n",
       "  2293.05,171.655 2295.11,171.592 2297.17,171.53 2299.23,171.468 2301.29,171.406 2303.35,171.344 2305.4,171.283 2307.46,171.221 2309.52,171.161 2311.58,171.1 \n",
       "  2313.64,171.04 2315.7,170.98 2317.76,170.921 2319.82,170.862 2321.87,170.803 2323.93,170.744 2325.99,170.686 2328.05,170.628 2330.11,170.57 2332.17,170.513 \n",
       "  2334.23,170.456 2336.29,170.399 2338.34,170.343 2340.4,170.286 2342.46,170.231 2344.52,170.175 2346.58,170.12 2348.64,170.065 2350.7,170.01 2352.76,169.956 \n",
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     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sol = solve(problem)\n",
    "plot(sol)\n",
    "title!(\"Enzyme kinetics using ODE and vector matrix notation\")\n",
    "xlabel!(\"Time [s]\")\n",
    "ylabel!(\"Concentration [M]\")\n",
    "#png(\"Michaelis_menten.png\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can also use the simpler DSL direct notation, this is useful because we can use directly the differential equations with the original names of the variables:\n",
    "\n",
    "$$\\begin{align*}\n",
    "  \\dot{s}  &= k_2 \\cdot c - k_1 \\cdot e \\cdot s \\tag{5}\\\\\n",
    "  \\dot{c} &= k_1 \\cdot e\\cdot s - (k_2+k_3) \\cdot c \\tag{6}\\\\\n",
    "  \\dot{e} &= (k_2+k_3) \\cdot c -k_1 \\cdot e \\cdot s \\tag{7}\\\\\n",
    "  \\dot{p} &= k_3 \\cdot c\n",
    "  \\end{align*}$$\n",
    "  \n",
    "(in Julia, we cannot use $e$ for the enzyme, so we use $en$)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(::ab{var\"#3#7\",var\"#4#8\",var\"#5#9\",Nothing,Nothing,var\"#6#10\",Expr,Expr}) (generic function with 2 methods)"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "enzyme_kinetics! = @ode_def ab begin\n",
    "  ds = -k1*en*s+k2*c\n",
    "  dc = k1*en*s-(k2+k3)*c\n",
    "  den = -k1*en*s+(k2+k3)*c\n",
    "  dp = k3*c\n",
    "    end k1 k2 k3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
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       "<g clip-path=\"url(#clip7200)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 296.041, 1479.62)\" x=\"296.041\" y=\"1479.62\">0</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip7200)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 707.384, 1479.62)\" x=\"707.384\" y=\"1479.62\">20</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip7200)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 1118.73, 1479.62)\" x=\"1118.73\" y=\"1479.62\">40</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip7200)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 1530.07, 1479.62)\" x=\"1530.07\" y=\"1479.62\">60</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip7200)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 1941.41, 1479.62)\" x=\"1941.41\" y=\"1479.62\">80</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip7200)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 2352.76, 1479.62)\" x=\"2352.76\" y=\"1479.62\">100</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip7200)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 272.041, 1406.21)\" x=\"272.041\" y=\"1406.21\">0.0000</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip7200)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 272.041, 1098.68)\" x=\"272.041\" y=\"1098.68\">0.0005</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip7200)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 272.041, 791.147)\" x=\"272.041\" y=\"791.147\">0.0010</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip7200)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 272.041, 483.613)\" x=\"272.041\" y=\"483.613\">0.0015</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip7200)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 272.041, 176.079)\" x=\"272.041\" y=\"176.079\">0.0020</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip7200)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:84px; text-anchor:middle;\" transform=\"rotate(0, 1324.4, 73.2)\" x=\"1324.4\" y=\"73.2\">Enzyme-substrate kinetics using DSL notation</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip7200)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:66px; text-anchor:middle;\" transform=\"rotate(0, 1324.4, 1559.48)\" x=\"1324.4\" y=\"1559.48\">Time [s]</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip7200)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:66px; text-anchor:middle;\" transform=\"rotate(-90, 89.2861, 773.647)\" x=\"89.2861\" y=\"773.647\">Concentration [M]</text>\n",
       "</g>\n",
       "<polyline clip-path=\"url(#clip7202)\" style=\"stroke:#009af9; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n",
       "  296.041,158.579 298.1,362.84 300.159,507.327 302.217,614.669 304.276,697.396 306.335,762.946 308.394,816.074 310.453,859.932 312.511,896.654 314.57,927.825 \n",
       "  316.629,954.594 318.688,977.788 320.746,998.047 322.805,1015.88 324.864,1031.7 326.923,1045.81 328.981,1058.47 331.04,1069.88 333.099,1080.22 335.158,1089.64 \n",
       "  337.217,1098.25 339.275,1106.16 341.334,1113.45 343.393,1120.19 345.452,1126.44 347.51,1132.27 349.569,1137.72 351.628,1142.83 353.687,1147.64 355.746,1152.17 \n",
       "  357.804,1156.46 359.863,1160.52 361.922,1164.38 363.981,1168.06 366.039,1171.57 368.098,1174.94 370.157,1178.17 372.216,1181.28 374.275,1184.27 376.333,1187.16 \n",
       "  378.392,1189.94 380.451,1192.64 382.51,1195.25 384.568,1197.79 386.627,1200.26 388.686,1202.66 390.745,1205 392.803,1207.29 394.862,1209.52 396.921,1211.7 \n",
       "  398.98,1213.83 401.039,1215.91 403.097,1217.95 405.156,1219.95 407.215,1221.92 409.274,1223.84 411.332,1225.73 413.391,1227.6 415.45,1229.43 417.509,1231.23 \n",
       "  419.568,1233 421.626,1234.74 423.685,1236.45 425.744,1238.14 427.803,1239.8 429.861,1241.43 431.92,1243.04 433.979,1244.63 436.038,1246.19 438.096,1247.73 \n",
       "  440.155,1249.25 442.214,1250.75 444.273,1252.23 446.332,1253.7 448.39,1255.14 450.449,1256.57 452.508,1257.97 454.567,1259.35 456.625,1260.72 458.684,1262.06 \n",
       "  460.743,1263.39 462.802,1264.7 464.861,1265.99 466.919,1267.26 468.978,1268.51 471.037,1269.75 473.096,1270.98 475.154,1272.19 477.213,1273.39 479.272,1274.58 \n",
       "  481.331,1275.75 483.389,1276.91 485.448,1278.05 487.507,1279.18 489.566,1280.29 491.625,1281.38 493.683,1282.46 495.742,1283.53 497.801,1284.57 499.86,1285.61 \n",
       "  501.918,1286.62 503.977,1287.63 506.036,1288.62 508.095,1289.6 510.154,1290.57 512.212,1291.53 514.271,1292.48 516.33,1293.43 518.389,1294.36 520.447,1295.29 \n",
       "  522.506,1296.2 524.565,1297.1 526.624,1297.99 528.683,1298.87 530.741,1299.73 532.8,1300.58 534.859,1301.42 536.918,1302.24 538.976,1303.05 541.035,1303.85 \n",
       "  543.094,1304.63 545.153,1305.41 547.211,1306.17 549.27,1306.93 551.329,1307.68 553.388,1308.43 555.447,1309.17 557.505,1309.91 559.564,1310.64 561.623,1311.37 \n",
       "  563.682,1312.09 565.74,1312.8 567.799,1313.51 569.858,1314.2 571.917,1314.88 573.976,1315.55 576.034,1316.2 578.093,1316.85 580.152,1317.48 582.211,1318.1 \n",
       "  584.269,1318.71 586.328,1319.32 588.387,1319.91 590.446,1320.49 592.504,1321.07 594.563,1321.65 596.622,1322.22 598.681,1322.8 600.74,1323.37 602.798,1323.95 \n",
       "  604.857,1324.52 606.916,1325.09 608.975,1325.66 611.033,1326.22 613.092,1326.77 615.151,1327.31 617.21,1327.84 619.269,1328.36 621.327,1328.87 623.386,1329.36 \n",
       "  625.445,1329.85 627.504,1330.33 629.562,1330.8 631.621,1331.26 633.68,1331.71 635.739,1332.15 637.798,1332.59 639.856,1333.03 641.915,1333.47 643.974,1333.9 \n",
       "  646.033,1334.35 648.091,1334.8 650.15,1335.25 652.209,1335.71 654.268,1336.16 656.326,1336.61 658.385,1337.06 660.444,1337.5 662.503,1337.92 664.562,1338.34 \n",
       "  666.62,1338.75 668.679,1339.15 670.738,1339.54 672.797,1339.91 674.855,1340.28 676.914,1340.64 678.973,1340.98 681.032,1341.32 683.091,1341.66 685.149,1341.99 \n",
       "  687.208,1342.32 689.267,1342.64 691.326,1342.98 693.384,1343.32 695.443,1343.66 697.502,1344.03 699.561,1344.4 701.619,1344.77 703.678,1345.14 705.737,1345.51 \n",
       "  707.796,1345.87 709.855,1346.23 711.913,1346.58 713.972,1346.92 716.031,1347.24 718.09,1347.55 720.148,1347.85 722.207,1348.14 724.266,1348.42 726.325,1348.68 \n",
       "  728.384,1348.94 730.442,1349.19 732.501,1349.43 734.56,1349.67 736.619,1349.91 738.677,1350.15 740.736,1350.4 742.795,1350.66 744.854,1350.94 746.913,1351.23 \n",
       "  748.971,1351.54 751.03,1351.86 753.089,1352.18 755.148,1352.5 757.206,1352.82 759.265,1353.12 761.324,1353.42 763.383,1353.7 765.441,1353.97 767.5,1354.22 \n",
       "  769.559,1354.46 771.618,1354.68 773.677,1354.89 775.735,1355.08 777.794,1355.26 779.853,1355.44 781.912,1355.6 783.97,1355.77 786.029,1355.93 788.088,1356.1 \n",
       "  790.147,1356.28 792.206,1356.48 794.264,1356.7 796.323,1356.95 798.382,1357.22 800.441,1357.51 802.499,1357.8 804.558,1358.1 806.617,1358.38 808.676,1358.66 \n",
       "  810.734,1358.93 812.793,1359.17 814.852,1359.4 816.911,1359.6 818.97,1359.79 821.028,1359.96 823.087,1360.1 825.146,1360.23 827.205,1360.34 829.263,1360.45 \n",
       "  831.322,1360.54 833.381,1360.64 835.44,1360.74 837.499,1360.86 839.557,1360.99 841.616,1361.15 843.675,1361.35 845.734,1361.59 847.792,1361.85 849.851,1362.13 \n",
       "  851.91,1362.41 853.969,1362.69 856.028,1362.95 858.086,1363.2 860.145,1363.43 862.204,1363.63 864.263,1363.81 866.321,1363.97 868.38,1364.09 870.439,1364.2 \n",
       "  872.498,1364.28 874.556,1364.34 876.615,1364.39 878.674,1364.43 880.733,1364.48 882.792,1364.53 884.85,1364.6 886.909,1364.7 888.968,1364.84 891.027,1365.03 \n",
       "  893.085,1365.27 895.144,1365.52 897.203,1365.79 899.262,1366.05 901.321,1366.31 903.379,1366.55 905.438,1366.77 907.497,1366.96 909.556,1367.13 911.614,1367.26 \n",
       "  913.673,1367.37 915.732,1367.45 917.791,1367.5 919.849,1367.53 921.908,1367.54 923.967,1367.55 926.026,1367.56 928.085,1367.58 930.143,1367.62 932.202,1367.7 \n",
       "  934.261,1367.83 936.32,1368.02 938.378,1368.25 940.437,1368.49 942.496,1368.74 944.555,1368.98 946.614,1369.22 948.672,1369.43 950.731,1369.61 952.79,1369.77 \n",
       "  954.849,1369.9 956.907,1369.99 958.966,1370.06 961.025,1370.1 963.084,1370.11 965.143,1370.11 967.201,1370.1 969.26,1370.09 971.319,1370.09 973.378,1370.12 \n",
       "  975.436,1370.18 977.495,1370.3 979.554,1370.47 981.613,1370.68 983.671,1370.91 985.73,1371.13 987.789,1371.35 989.848,1371.56 991.907,1371.74 993.965,1371.9 \n",
       "  996.024,1372.03 998.083,1372.13 1000.14,1372.2 1002.2,1372.24 1004.26,1372.26 1006.32,1372.26 1008.38,1372.24 1010.44,1372.22 1012.49,1372.21 1014.55,1372.22 \n",
       "  1016.61,1372.25 1018.67,1372.33 1020.73,1372.48 1022.79,1372.66 1024.85,1372.86 1026.91,1373.07 1028.96,1373.27 1031.02,1373.45 1033.08,1373.62 1035.14,1373.77 \n",
       "  1037.2,1373.89 1039.26,1373.98 1041.32,1374.04 1043.38,1374.07 1045.43,1374.08 1047.49,1374.08 1049.55,1374.06 1051.61,1374.03 1053.67,1374.01 1055.73,1374.01 \n",
       "  1057.79,1374.05 1059.85,1374.12 1061.9,1374.26 1063.96,1374.43 1066.02,1374.61 1068.08,1374.8 1070.14,1374.98 1072.2,1375.14 1074.26,1375.29 1076.32,1375.42 \n",
       "  1078.38,1375.52 1080.43,1375.59 1082.49,1375.63 1084.55,1375.65 1086.61,1375.65 1088.67,1375.63 1090.73,1375.6 1092.79,1375.58 1094.85,1375.56 1096.9,1375.56 \n",
       "  1098.96,1375.6 1101.02,1375.69 1103.08,1375.8 1105.14,1375.91 1107.2,1376.01 1109.26,1376.11 1111.32,1376.21 1113.37,1376.3 1115.43,1376.4 1117.49,1376.49 \n",
       "  1119.55,1376.59 1121.61,1376.68 1123.67,1376.77 1125.73,1376.85 1127.79,1376.94 1129.84,1377.02 1131.9,1377.1 1133.96,1377.18 1136.02,1377.26 1138.08,1377.34 \n",
       "  1140.14,1377.42 1142.2,1377.49 1144.26,1377.56 1146.31,1377.63 1148.37,1377.7 1150.43,1377.77 1152.49,1377.83 1154.55,1377.9 1156.61,1377.96 1158.67,1378.02 \n",
       "  1160.73,1378.08 1162.78,1378.14 1164.84,1378.19 1166.9,1378.25 1168.96,1378.3 1171.02,1378.35 1173.08,1378.4 1175.14,1378.45 1177.2,1378.49 1179.25,1378.54 \n",
       "  1181.31,1378.58 1183.37,1378.63 1185.43,1378.68 1187.49,1378.73 1189.55,1378.78 1191.61,1378.83 1193.67,1378.87 1195.73,1378.92 1197.78,1378.97 1199.84,1379.02 \n",
       "  1201.9,1379.07 1203.96,1379.12 1206.02,1379.17 1208.08,1379.21 1210.14,1379.26 1212.2,1379.31 1214.25,1379.36 1216.31,1379.41 1218.37,1379.45 1220.43,1379.5 \n",
       "  1222.49,1379.55 1224.55,1379.6 1226.61,1379.65 1228.67,1379.69 1230.72,1379.74 1232.78,1379.79 1234.84,1379.83 1236.9,1379.88 1238.96,1379.93 1241.02,1379.98 \n",
       "  1243.08,1380.02 1245.14,1380.07 1247.19,1380.12 1249.25,1380.16 1251.31,1380.21 1253.37,1380.26 1255.43,1380.3 1257.49,1380.35 1259.55,1380.4 1261.61,1380.44 \n",
       "  1263.66,1380.49 1265.72,1380.53 1267.78,1380.58 1269.84,1380.62 1271.9,1380.67 1273.96,1380.71 1276.02,1380.75 1278.08,1380.8 1280.13,1380.84 1282.19,1380.88 \n",
       "  1284.25,1380.93 1286.31,1380.97 1288.37,1381.01 1290.43,1381.05 1292.49,1381.09 1294.55,1381.14 1296.61,1381.18 1298.66,1381.22 1300.72,1381.26 1302.78,1381.3 \n",
       "  1304.84,1381.34 1306.9,1381.38 1308.96,1381.42 1311.02,1381.45 1313.08,1381.49 1315.13,1381.53 1317.19,1381.57 1319.25,1381.61 1321.31,1381.65 1323.37,1381.68 \n",
       "  1325.43,1381.72 1327.49,1381.76 1329.55,1381.79 1331.6,1381.83 1333.66,1381.87 1335.72,1381.9 1337.78,1381.94 1339.84,1381.97 1341.9,1382.01 1343.96,1382.04 \n",
       "  1346.02,1382.08 1348.07,1382.11 1350.13,1382.14 1352.19,1382.18 1354.25,1382.21 1356.31,1382.24 1358.37,1382.28 1360.43,1382.31 1362.49,1382.34 1364.54,1382.38 \n",
       "  1366.6,1382.41 1368.66,1382.44 1370.72,1382.47 1372.78,1382.51 1374.84,1382.54 1376.9,1382.57 1378.96,1382.6 1381.01,1382.63 1383.07,1382.66 1385.13,1382.7 \n",
       "  1387.19,1382.73 1389.25,1382.76 1391.31,1382.79 1393.37,1382.82 1395.43,1382.85 1397.48,1382.88 1399.54,1382.91 1401.6,1382.94 1403.66,1382.97 1405.72,1383 \n",
       "  1407.78,1383.03 1409.84,1383.06 1411.9,1383.09 1413.96,1383.12 1416.01,1383.15 1418.07,1383.18 1420.13,1383.2 1422.19,1383.23 1424.25,1383.26 1426.31,1383.29 \n",
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       "  1613.66,1385.3 1615.71,1385.31 1617.77,1385.33 1619.83,1385.35 1621.89,1385.36 1623.95,1385.38 1626.01,1385.4 1628.07,1385.41 1630.13,1385.43 1632.19,1385.45 \n",
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       "  1654.83,1385.62 1656.89,1385.64 1658.95,1385.65 1661.01,1385.67 1663.07,1385.69 1665.13,1385.7 1667.18,1385.72 1669.24,1385.73 1671.3,1385.75 1673.36,1385.76 \n",
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       "  1696.01,1385.92 1698.07,1385.93 1700.12,1385.95 1702.18,1385.96 1704.24,1385.97 1706.3,1385.99 1708.36,1386 1710.42,1386.01 1712.48,1386.03 1714.54,1386.04 \n",
       "  1716.59,1386.05 1718.65,1386.07 1720.71,1386.08 1722.77,1386.09 1724.83,1386.1 1726.89,1386.12 1728.95,1386.13 1731.01,1386.14 1733.07,1386.16 1735.12,1386.17 \n",
       "  1737.18,1386.18 1739.24,1386.19 1741.3,1386.21 1743.36,1386.22 1745.42,1386.23 1747.48,1386.24 1749.54,1386.25 1751.59,1386.27 1753.65,1386.28 1755.71,1386.29 \n",
       "  1757.77,1386.3 1759.83,1386.32 1761.89,1386.33 1763.95,1386.34 1766.01,1386.35 1768.06,1386.36 1770.12,1386.37 1772.18,1386.39 1774.24,1386.4 1776.3,1386.41 \n",
       "  1778.36,1386.42 1780.42,1386.43 1782.48,1386.44 1784.53,1386.45 1786.59,1386.47 1788.65,1386.48 1790.71,1386.49 1792.77,1386.5 1794.83,1386.51 1796.89,1386.52 \n",
       "  1798.95,1386.53 1801,1386.54 1803.06,1386.55 1805.12,1386.56 1807.18,1386.57 1809.24,1386.58 1811.3,1386.6 1813.36,1386.61 1815.42,1386.62 1817.47,1386.63 \n",
       "  1819.53,1386.64 1821.59,1386.65 1823.65,1386.66 1825.71,1386.67 1827.77,1386.68 1829.83,1386.69 1831.89,1386.7 1833.94,1386.71 1836,1386.72 1838.06,1386.73 \n",
       "  1840.12,1386.74 1842.18,1386.75 1844.24,1386.76 1846.3,1386.77 1848.36,1386.77 1850.42,1386.78 1852.47,1386.79 1854.53,1386.8 1856.59,1386.81 1858.65,1386.82 \n",
       "  1860.71,1386.83 1862.77,1386.84 1864.83,1386.85 1866.89,1386.86 1868.94,1386.87 1871,1386.88 1873.06,1386.88 1875.12,1386.89 1877.18,1386.9 1879.24,1386.91 \n",
       "  1881.3,1386.92 1883.36,1386.93 1885.41,1386.94 1887.47,1386.95 1889.53,1386.96 1891.59,1386.96 1893.65,1386.97 1895.71,1386.98 1897.77,1386.99 1899.83,1387 \n",
       "  1901.88,1387.01 1903.94,1387.01 1906,1387.02 1908.06,1387.03 1910.12,1387.04 1912.18,1387.05 1914.24,1387.06 1916.3,1387.06 1918.35,1387.07 1920.41,1387.08 \n",
       "  1922.47,1387.09 1924.53,1387.1 1926.59,1387.1 1928.65,1387.11 1930.71,1387.12 1932.77,1387.13 1934.82,1387.14 1936.88,1387.14 1938.94,1387.15 1941,1387.16 \n",
       "  1943.06,1387.17 1945.12,1387.17 1947.18,1387.18 1949.24,1387.19 1951.3,1387.2 1953.35,1387.2 1955.41,1387.21 1957.47,1387.22 1959.53,1387.23 1961.59,1387.23 \n",
       "  1963.65,1387.24 1965.71,1387.25 1967.77,1387.26 1969.82,1387.26 1971.88,1387.27 1973.94,1387.28 1976,1387.28 1978.06,1387.29 1980.12,1387.3 1982.18,1387.3 \n",
       "  1984.24,1387.31 1986.29,1387.32 1988.35,1387.32 1990.41,1387.33 1992.47,1387.34 1994.53,1387.35 1996.59,1387.35 1998.65,1387.36 2000.71,1387.36 2002.76,1387.37 \n",
       "  2004.82,1387.38 2006.88,1387.38 2008.94,1387.39 2011,1387.4 2013.06,1387.4 2015.12,1387.41 2017.18,1387.42 2019.23,1387.42 2021.29,1387.43 2023.35,1387.43 \n",
       "  2025.41,1387.44 2027.47,1387.45 2029.53,1387.45 2031.59,1387.46 2033.65,1387.47 2035.7,1387.47 2037.76,1387.48 2039.82,1387.48 2041.88,1387.49 2043.94,1387.5 \n",
       "  2046,1387.5 2048.06,1387.51 2050.12,1387.51 2052.17,1387.52 2054.23,1387.52 2056.29,1387.53 2058.35,1387.54 2060.41,1387.54 2062.47,1387.55 2064.53,1387.55 \n",
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       "  2087.17,1387.61 2089.23,1387.62 2091.29,1387.62 2093.35,1387.63 2095.41,1387.64 2097.47,1387.64 2099.53,1387.65 2101.59,1387.65 2103.64,1387.66 2105.7,1387.66 \n",
       "  2107.76,1387.67 2109.82,1387.67 2111.88,1387.68 2113.94,1387.68 2116,1387.69 2118.06,1387.69 2120.11,1387.7 2122.17,1387.7 2124.23,1387.71 2126.29,1387.71 \n",
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       "  2190.11,1387.85 2192.17,1387.86 2194.23,1387.86 2196.29,1387.86 2198.35,1387.87 2200.41,1387.87 2202.47,1387.88 2204.52,1387.88 2206.58,1387.88 2208.64,1387.89 \n",
       "  2210.7,1387.89 2212.76,1387.9 2214.82,1387.9 2216.88,1387.9 2218.94,1387.91 2220.99,1387.91 2223.05,1387.92 2225.11,1387.92 2227.17,1387.92 2229.23,1387.93 \n",
       "  2231.29,1387.93 2233.35,1387.93 2235.41,1387.94 2237.46,1387.94 2239.52,1387.95 2241.58,1387.95 2243.64,1387.95 2245.7,1387.96 2247.76,1387.96 2249.82,1387.96 \n",
       "  2251.88,1387.97 2253.93,1387.97 2255.99,1387.98 2258.05,1387.98 2260.11,1387.98 2262.17,1387.99 2264.23,1387.99 2266.29,1387.99 2268.35,1388 2270.4,1388 \n",
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       "  1119.55,1192.83 1121.61,1193.72 1123.67,1194.6 1125.73,1195.48 1127.79,1196.37 1129.84,1197.24 1131.9,1198.12 1133.96,1198.99 1136.02,1199.86 1138.08,1200.73 \n",
       "  1140.14,1201.59 1142.2,1202.46 1144.26,1203.31 1146.31,1204.17 1148.37,1205.03 1150.43,1205.88 1152.49,1206.73 1154.55,1207.57 1156.61,1208.42 1158.67,1209.26 \n",
       "  1160.73,1210.1 1162.78,1210.93 1164.84,1211.77 1166.9,1212.6 1168.96,1213.42 1171.02,1214.25 1173.08,1215.07 1175.14,1215.89 1177.2,1216.71 1179.25,1217.53 \n",
       "  1181.31,1218.34 1183.37,1219.14 1185.43,1219.93 1187.49,1220.73 1189.55,1221.51 1191.61,1222.3 1193.67,1223.08 1195.73,1223.86 1197.78,1224.63 1199.84,1225.41 \n",
       "  1201.9,1226.17 1203.96,1226.94 1206.02,1227.7 1208.08,1228.46 1210.14,1229.21 1212.2,1229.96 1214.25,1230.71 1216.31,1231.45 1218.37,1232.19 1220.43,1232.92 \n",
       "  1222.49,1233.66 1224.55,1234.38 1226.61,1235.11 1228.67,1235.83 1230.72,1236.55 1232.78,1237.26 1234.84,1237.98 1236.9,1238.68 1238.96,1239.39 1241.02,1240.09 \n",
       "  1243.08,1240.78 1245.14,1241.48 1247.19,1242.17 1249.25,1242.85 1251.31,1243.54 1253.37,1244.22 1255.43,1244.89 1257.49,1245.56 1259.55,1246.23 1261.61,1246.9 \n",
       "  1263.66,1247.56 1265.72,1248.22 1267.78,1248.87 1269.84,1249.52 1271.9,1250.17 1273.96,1250.82 1276.02,1251.46 1278.08,1252.11 1280.13,1252.75 1282.19,1253.38 \n",
       "  1284.25,1254.02 1286.31,1254.65 1288.37,1255.28 1290.43,1255.9 1292.49,1256.52 1294.55,1257.15 1296.61,1257.76 1298.66,1258.38 1300.72,1258.99 1302.78,1259.6 \n",
       "  1304.84,1260.21 1306.9,1260.81 1308.96,1261.41 1311.02,1262.01 1313.08,1262.61 1315.13,1263.2 1317.19,1263.79 1319.25,1264.38 1321.31,1264.97 1323.37,1265.55 \n",
       "  1325.43,1266.13 1327.49,1266.71 1329.55,1267.29 1331.6,1267.86 1333.66,1268.43 1335.72,1269 1337.78,1269.56 1339.84,1270.12 1341.9,1270.68 1343.96,1271.24 \n",
       "  1346.02,1271.79 1348.07,1272.34 1350.13,1272.89 1352.19,1273.44 1354.25,1273.98 1356.31,1274.52 1358.37,1275.06 1360.43,1275.59 1362.49,1276.12 1364.54,1276.65 \n",
       "  1366.6,1277.17 1368.66,1277.7 1370.72,1278.22 1372.78,1278.74 1374.84,1279.25 1376.9,1279.77 1378.96,1280.28 1381.01,1280.79 1383.07,1281.3 1385.13,1281.8 \n",
       "  1387.19,1282.31 1389.25,1282.81 1391.31,1283.31 1393.37,1283.8 1395.43,1284.3 1397.48,1284.79 1399.54,1285.28 1401.6,1285.77 1403.66,1286.26 1405.72,1286.74 \n",
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       "  1428.37,1291.92 1430.43,1292.38 1432.48,1292.83 1434.54,1293.29 1436.6,1293.74 1438.66,1294.19 1440.72,1294.64 1442.78,1295.08 1444.84,1295.53 1446.9,1295.97 \n",
       "  1448.95,1296.41 1451.01,1296.84 1453.07,1297.28 1455.13,1297.71 1457.19,1298.14 1459.25,1298.57 1461.31,1298.99 1463.37,1299.42 1465.42,1299.84 1467.48,1300.26 \n",
       "  1469.54,1300.67 1471.6,1301.09 1473.66,1301.5 1475.72,1301.91 1477.78,1302.31 1479.84,1302.72 1481.89,1303.12 1483.95,1303.53 1486.01,1303.93 1488.07,1304.33 \n",
       "  1490.13,1304.72 1492.19,1305.12 1494.25,1305.51 1496.31,1305.91 1498.36,1306.3 1500.42,1306.69 1502.48,1307.07 1504.54,1307.46 1506.6,1307.84 1508.66,1308.23 \n",
       "  1510.72,1308.61 1512.78,1308.99 1514.84,1309.36 1516.89,1309.74 1518.95,1310.11 1521.01,1310.49 1523.07,1310.86 1525.13,1311.23 1527.19,1311.59 1529.25,1311.96 \n",
       "  1531.31,1312.32 1533.36,1312.69 1535.42,1313.05 1537.48,1313.41 1539.54,1313.76 1541.6,1314.12 1543.66,1314.47 1545.72,1314.83 1547.78,1315.18 1549.83,1315.53 \n",
       "  1551.89,1315.87 1553.95,1316.22 1556.01,1316.56 1558.07,1316.91 1560.13,1317.25 1562.19,1317.59 1564.25,1317.92 1566.3,1318.26 1568.36,1318.59 1570.42,1318.93 \n",
       "  1572.48,1319.26 1574.54,1319.59 1576.6,1319.91 1578.66,1320.24 1580.72,1320.56 1582.77,1320.89 1584.83,1321.21 1586.89,1321.53 1588.95,1321.84 1591.01,1322.16 \n",
       "  1593.07,1322.47 1595.13,1322.78 1597.19,1323.09 1599.24,1323.4 1601.3,1323.71 1603.36,1324.01 1605.42,1324.32 1607.48,1324.62 1609.54,1324.93 1611.6,1325.23 \n",
       "  1613.66,1325.53 1615.71,1325.83 1617.77,1326.12 1619.83,1326.42 1621.89,1326.71 1623.95,1327.01 1626.01,1327.3 1628.07,1327.59 1630.13,1327.88 1632.19,1328.17 \n",
       "  1634.24,1328.46 1636.3,1328.74 1638.36,1329.03 1640.42,1329.31 1642.48,1329.59 1644.54,1329.87 1646.6,1330.15 1648.66,1330.43 1650.71,1330.71 1652.77,1330.99 \n",
       "  1654.83,1331.26 1656.89,1331.53 1658.95,1331.81 1661.01,1332.08 1663.07,1332.35 1665.13,1332.62 1667.18,1332.88 1669.24,1333.15 1671.3,1333.41 1673.36,1333.68 \n",
       "  1675.42,1333.94 1677.48,1334.2 1679.54,1334.46 1681.6,1334.72 1683.65,1334.97 1685.71,1335.23 1687.77,1335.48 1689.83,1335.74 1691.89,1335.99 1693.95,1336.24 \n",
       "  1696.01,1336.49 1698.07,1336.74 1700.12,1336.99 1702.18,1337.23 1704.24,1337.48 1706.3,1337.72 1708.36,1337.96 1710.42,1338.2 1712.48,1338.44 1714.54,1338.68 \n",
       "  1716.59,1338.92 1718.65,1339.15 1720.71,1339.38 1722.77,1339.62 1724.83,1339.85 1726.89,1340.08 1728.95,1340.31 1731.01,1340.53 1733.07,1340.76 1735.12,1340.99 \n",
       "  1737.18,1341.21 1739.24,1341.44 1741.3,1341.66 1743.36,1341.88 1745.42,1342.11 1747.48,1342.33 1749.54,1342.55 1751.59,1342.76 1753.65,1342.98 1755.71,1343.2 \n",
       "  1757.77,1343.42 1759.83,1343.63 1761.89,1343.84 1763.95,1344.06 1766.01,1344.27 1768.06,1344.48 1770.12,1344.69 1772.18,1344.9 1774.24,1345.11 1776.3,1345.32 \n",
       "  1778.36,1345.53 1780.42,1345.73 1782.48,1345.94 1784.53,1346.14 1786.59,1346.34 1788.65,1346.55 1790.71,1346.75 1792.77,1346.95 1794.83,1347.15 1796.89,1347.34 \n",
       "  1798.95,1347.54 1801,1347.74 1803.06,1347.93 1805.12,1348.13 1807.18,1348.32 1809.24,1348.52 1811.3,1348.71 1813.36,1348.9 1815.42,1349.09 1817.47,1349.28 \n",
       "  1819.53,1349.47 1821.59,1349.65 1823.65,1349.84 1825.71,1350.03 1827.77,1350.21 1829.83,1350.39 1831.89,1350.58 1833.94,1350.76 1836,1350.94 1838.06,1351.12 \n",
       "  1840.12,1351.3 1842.18,1351.48 1844.24,1351.65 1846.3,1351.83 1848.36,1352.01 1850.42,1352.18 1852.47,1352.35 1854.53,1352.53 1856.59,1352.7 1858.65,1352.87 \n",
       "  1860.71,1353.04 1862.77,1353.2 1864.83,1353.37 1866.89,1353.54 1868.94,1353.7 1871,1353.87 1873.06,1354.03 1875.12,1354.2 1877.18,1354.36 1879.24,1354.52 \n",
       "  1881.3,1354.68 1883.36,1354.84 1885.41,1355 1887.47,1355.16 1889.53,1355.32 1891.59,1355.48 1893.65,1355.64 1895.71,1355.8 1897.77,1355.95 1899.83,1356.11 \n",
       "  1901.88,1356.26 1903.94,1356.42 1906,1356.57 1908.06,1356.72 1910.12,1356.88 1912.18,1357.03 1914.24,1357.18 1916.3,1357.33 1918.35,1357.48 1920.41,1357.63 \n",
       "  1922.47,1357.78 1924.53,1357.93 1926.59,1358.07 1928.65,1358.22 1930.71,1358.37 1932.77,1358.51 1934.82,1358.65 1936.88,1358.8 1938.94,1358.94 1941,1359.08 \n",
       "  1943.06,1359.23 1945.12,1359.37 1947.18,1359.51 1949.24,1359.65 1951.3,1359.79 1953.35,1359.93 1955.41,1360.06 1957.47,1360.2 1959.53,1360.34 1961.59,1360.47 \n",
       "  1963.65,1360.61 1965.71,1360.74 1967.77,1360.88 1969.82,1361.01 1971.88,1361.14 1973.94,1361.27 1976,1361.41 1978.06,1361.54 1980.12,1361.67 1982.18,1361.8 \n",
       "  1984.24,1361.92 1986.29,1362.05 1988.35,1362.18 1990.41,1362.31 1992.47,1362.43 1994.53,1362.56 1996.59,1362.68 1998.65,1362.81 2000.71,1362.93 2002.76,1363.05 \n",
       "  2004.82,1363.17 2006.88,1363.3 2008.94,1363.42 2011,1363.54 2013.06,1363.66 2015.12,1363.77 2017.18,1363.89 2019.23,1364.01 2021.29,1364.12 2023.35,1364.24 \n",
       "  2025.41,1364.35 2027.47,1364.47 2029.53,1364.58 2031.59,1364.7 2033.65,1364.81 2035.7,1364.92 2037.76,1365.04 2039.82,1365.15 2041.88,1365.26 2043.94,1365.37 \n",
       "  2046,1365.48 2048.06,1365.59 2050.12,1365.7 2052.17,1365.81 2054.23,1365.92 2056.29,1366.03 2058.35,1366.14 2060.41,1366.25 2062.47,1366.35 2064.53,1366.46 \n",
       "  2066.59,1366.57 2068.65,1366.67 2070.7,1366.78 2072.76,1366.88 2074.82,1366.99 2076.88,1367.09 2078.94,1367.19 2081,1367.3 2083.06,1367.4 2085.12,1367.5 \n",
       "  2087.17,1367.6 2089.23,1367.7 2091.29,1367.8 2093.35,1367.9 2095.41,1368 2097.47,1368.1 2099.53,1368.2 2101.59,1368.3 2103.64,1368.4 2105.7,1368.5 \n",
       "  2107.76,1368.59 2109.82,1368.69 2111.88,1368.79 2113.94,1368.88 2116,1368.98 2118.06,1369.07 2120.11,1369.17 2122.17,1369.26 2124.23,1369.35 2126.29,1369.45 \n",
       "  2128.35,1369.54 2130.41,1369.63 2132.47,1369.72 2134.53,1369.81 2136.58,1369.91 2138.64,1370 2140.7,1370.09 2142.76,1370.18 2144.82,1370.26 2146.88,1370.35 \n",
       "  2148.94,1370.44 2151,1370.53 2153.05,1370.61 2155.11,1370.7 2157.17,1370.79 2159.23,1370.87 2161.29,1370.96 2163.35,1371.04 2165.41,1371.13 2167.47,1371.21 \n",
       "  2169.53,1371.3 2171.58,1371.38 2173.64,1371.46 2175.7,1371.54 2177.76,1371.62 2179.82,1371.71 2181.88,1371.79 2183.94,1371.86 2186,1371.94 2188.05,1372.02 \n",
       "  2190.11,1372.1 2192.17,1372.18 2194.23,1372.26 2196.29,1372.34 2198.35,1372.41 2200.41,1372.49 2202.47,1372.57 2204.52,1372.64 2206.58,1372.72 2208.64,1372.79 \n",
       "  2210.7,1372.87 2212.76,1372.95 2214.82,1373.02 2216.88,1373.1 2218.94,1373.17 2220.99,1373.24 2223.05,1373.32 2225.11,1373.39 2227.17,1373.46 2229.23,1373.54 \n",
       "  2231.29,1373.61 2233.35,1373.68 2235.41,1373.75 2237.46,1373.82 2239.52,1373.9 2241.58,1373.97 2243.64,1374.04 2245.7,1374.11 2247.76,1374.18 2249.82,1374.25 \n",
       "  2251.88,1374.32 2253.93,1374.38 2255.99,1374.45 2258.05,1374.52 2260.11,1374.59 2262.17,1374.66 2264.23,1374.73 2266.29,1374.79 2268.35,1374.86 2270.4,1374.93 \n",
       "  2272.46,1374.99 2274.52,1375.06 2276.58,1375.12 2278.64,1375.19 2280.7,1375.25 2282.76,1375.32 2284.82,1375.38 2286.88,1375.45 2288.93,1375.51 2290.99,1375.58 \n",
       "  2293.05,1375.64 2295.11,1375.7 2297.17,1375.76 2299.23,1375.83 2301.29,1375.89 2303.35,1375.95 2305.4,1376.01 2307.46,1376.07 2309.52,1376.13 2311.58,1376.19 \n",
       "  2313.64,1376.25 2315.7,1376.31 2317.76,1376.37 2319.82,1376.43 2321.87,1376.49 2323.93,1376.55 2325.99,1376.61 2328.05,1376.67 2330.11,1376.72 2332.17,1376.78 \n",
       "  2334.23,1376.84 2336.29,1376.89 2338.34,1376.95 2340.4,1377.01 2342.46,1377.06 2344.52,1377.12 2346.58,1377.17 2348.64,1377.23 2350.7,1377.28 2352.76,1377.34 \n",
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       "  378.392,1033.31 380.451,1031.63 382.51,1029.88 384.568,1028.06 386.627,1026.18 388.686,1024.24 390.745,1022.26 392.803,1020.23 394.862,1018.15 396.921,1016.03 \n",
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       "  748.971,604.246 751.03,602.339 753.089,600.446 755.148,598.561 757.206,596.681 759.265,594.802 761.324,592.922 763.383,591.038 765.441,589.149 767.5,587.253 \n",
       "  769.559,585.351 771.618,583.442 773.677,581.527 775.735,579.607 777.794,577.685 779.853,575.762 781.912,573.842 783.97,571.929 786.029,570.026 788.088,568.139 \n",
       "  790.147,566.272 792.206,564.432 794.264,562.626 796.323,560.859 798.382,559.124 800.441,557.412 802.499,555.716 804.558,554.028 806.617,552.343 808.676,550.657 \n",
       "  810.734,548.965 812.793,547.265 814.852,545.553 816.911,543.829 818.97,542.093 821.028,540.343 823.087,538.583 825.146,536.813 827.205,535.037 829.263,533.258 \n",
       "  831.322,531.481 833.381,529.711 835.44,527.955 837.499,526.221 839.557,524.515 841.616,522.847 843.675,521.227 845.734,519.656 847.792,518.118 849.851,516.6 \n",
       "  851.91,515.095 853.969,513.594 856.028,512.09 858.086,510.578 860.145,509.052 862.204,507.51 864.263,505.95 866.321,504.371 868.38,502.772 870.439,501.156 \n",
       "  872.498,499.525 874.556,497.884 876.615,496.236 878.674,494.59 880.733,492.951 882.792,491.328 884.85,489.732 886.909,488.174 888.968,486.665 891.027,485.219 \n",
       "  893.085,483.824 895.144,482.458 897.203,481.11 899.262,479.768 901.321,478.425 903.379,477.073 905.438,475.706 907.497,474.319 909.556,472.91 911.614,471.478 \n",
       "  913.673,470.021 915.732,468.543 917.791,467.045 919.849,465.533 921.908,464.013 923.967,462.491 926.026,460.978 928.085,459.483 930.143,458.018 932.202,456.598 \n",
       "  934.261,455.237 936.32,453.946 938.378,452.699 940.437,451.476 942.496,450.265 944.555,449.057 946.614,447.841 948.672,446.612 950.731,445.364 952.79,444.093 \n",
       "  954.849,442.796 956.907,441.474 958.966,440.127 961.025,438.759 963.084,437.374 965.143,435.978 967.201,434.58 969.26,433.187 971.319,431.813 973.378,430.469 \n",
       "  975.436,429.17 977.495,427.932 979.554,426.764 981.613,425.635 983.671,424.527 985.73,423.429 987.789,422.33 989.848,421.222 991.907,420.099 993.965,418.955 \n",
       "  996.024,417.787 998.083,416.594 1000.14,415.377 1002.2,414.136 1004.26,412.875 1006.32,411.601 1008.38,410.319 1010.44,409.039 1012.49,407.772 1014.55,406.528 \n",
       "  1016.61,405.322 1018.67,404.17 1020.73,403.087 1022.79,402.049 1024.85,401.035 1026.91,400.032 1028.96,399.03 1031.02,398.021 1033.08,396.998 1035.14,395.955 \n",
       "  1037.2,394.891 1039.26,393.802 1041.32,392.69 1043.38,391.555 1045.43,390.401 1047.49,389.234 1049.55,388.059 1051.61,386.886 1053.67,385.725 1055.73,384.587 \n",
       "  1057.79,383.486 1059.85,382.438 1061.9,381.455 1063.96,380.511 1066.02,379.587 1068.08,378.673 1070.14,377.757 1072.2,376.834 1074.26,375.895 1076.32,374.938 \n",
       "  1078.38,373.958 1080.43,372.955 1082.49,371.93 1084.55,370.883 1086.61,369.82 1088.67,368.745 1090.73,367.667 1092.79,366.592 1094.85,365.533 1096.9,364.501 \n",
       "  1098.96,363.51 1101.02,362.575 1103.08,361.666 1105.14,360.757 1107.2,359.85 1109.26,358.946 1111.32,358.045 1113.37,357.146 1115.43,356.25 1117.49,355.357 \n",
       "  1119.55,354.466 1121.61,353.577 1123.67,352.692 1125.73,351.809 1127.79,350.928 1129.84,350.051 1131.9,349.175 1133.96,348.303 1136.02,347.433 1138.08,346.566 \n",
       "  1140.14,345.701 1142.2,344.839 1144.26,343.979 1146.31,343.122 1148.37,342.268 1150.43,341.416 1152.49,340.567 1154.55,339.721 1156.61,338.877 1158.67,338.036 \n",
       "  1160.73,337.197 1162.78,336.361 1164.84,335.528 1166.9,334.697 1168.96,333.869 1171.02,333.043 1173.08,332.22 1175.14,331.4 1177.2,330.582 1179.25,329.767 \n",
       "  1181.31,328.957 1183.37,328.157 1185.43,327.361 1187.49,326.569 1189.55,325.78 1191.61,324.994 1193.67,324.213 1195.73,323.434 1197.78,322.659 1199.84,321.888 \n",
       "  1201.9,321.121 1203.96,320.356 1206.02,319.596 1208.08,318.839 1210.14,318.085 1212.2,317.335 1214.25,316.589 1216.31,315.846 1218.37,315.106 1220.43,314.37 \n",
       "  1222.49,313.638 1224.55,312.909 1226.61,312.184 1228.67,311.462 1230.72,310.744 1232.78,310.03 1234.84,309.319 1236.9,308.611 1238.96,307.907 1241.02,307.207 \n",
       "  1243.08,306.51 1245.14,305.816 1247.19,305.126 1249.25,304.44 1251.31,303.757 1253.37,303.078 1255.43,302.402 1257.49,301.73 1259.55,301.062 1261.61,300.397 \n",
       "  1263.66,299.736 1265.72,299.079 1267.78,298.423 1269.84,297.771 1271.9,297.121 1273.96,296.474 1276.02,295.829 1278.08,295.188 1280.13,294.548 1282.19,293.912 \n",
       "  1284.25,293.278 1286.31,292.647 1288.37,292.018 1290.43,291.392 1292.49,290.769 1294.55,290.149 1296.61,289.531 1298.66,288.916 1300.72,288.303 1302.78,287.693 \n",
       "  1304.84,287.086 1306.9,286.482 1308.96,285.88 1311.02,285.281 1313.08,284.684 1315.13,284.09 1317.19,283.499 1319.25,282.911 1321.31,282.325 1323.37,281.742 \n",
       "  1325.43,281.161 1327.49,280.583 1329.55,280.008 1331.6,279.436 1333.66,278.866 1335.72,278.299 1337.78,277.734 1339.84,277.172 1341.9,276.613 1343.96,276.057 \n",
       "  1346.02,275.503 1348.07,274.952 1350.13,274.403 1352.19,273.857 1354.25,273.314 1356.31,272.774 1358.37,272.236 1360.43,271.701 1362.49,271.171 1364.54,270.644 \n",
       "  1366.6,270.119 1368.66,269.596 1370.72,269.076 1372.78,268.557 1374.84,268.041 1376.9,267.527 1378.96,267.015 1381.01,266.505 1383.07,265.997 1385.13,265.492 \n",
       "  1387.19,264.988 1389.25,264.487 1391.31,263.988 1393.37,263.491 1395.43,262.996 1397.48,262.503 1399.54,262.013 1401.6,261.525 1403.66,261.039 1405.72,260.555 \n",
       "  1407.78,260.073 1409.84,259.593 1411.9,259.115 1413.96,258.64 1416.01,258.167 1418.07,257.696 1420.13,257.227 1422.19,256.76 1424.25,256.296 1426.31,255.833 \n",
       "  1428.37,255.373 1430.43,254.915 1432.48,254.459 1434.54,254.005 1436.6,253.553 1438.66,253.104 1440.72,252.657 1442.78,252.211 1444.84,251.768 1446.9,251.327 \n",
       "  1448.95,250.889 1451.01,250.452 1453.07,250.018 1455.13,249.586 1457.19,249.155 1459.25,248.728 1461.31,248.302 1463.37,247.878 1465.42,247.457 1467.48,247.037 \n",
       "  1469.54,246.62 1471.6,246.207 1473.66,245.796 1475.72,245.387 1477.78,244.979 1479.84,244.574 1481.89,244.169 1483.95,243.767 1486.01,243.366 1488.07,242.967 \n",
       "  1490.13,242.57 1492.19,242.174 1494.25,241.78 1496.31,241.387 1498.36,240.996 1500.42,240.607 1502.48,240.22 1504.54,239.834 1506.6,239.45 1508.66,239.067 \n",
       "  1510.72,238.686 1512.78,238.307 1514.84,237.93 1516.89,237.554 1518.95,237.18 1521.01,236.807 1523.07,236.437 1525.13,236.067 1527.19,235.7 1529.25,235.334 \n",
       "  1531.31,234.97 1533.36,234.607 1535.42,234.246 1537.48,233.887 1539.54,233.53 1541.6,233.174 1543.66,232.82 1545.72,232.467 1547.78,232.116 1549.83,231.767 \n",
       "  1551.89,231.42 1553.95,231.074 1556.01,230.73 1558.07,230.387 1560.13,230.046 1562.19,229.707 1564.25,229.37 1566.3,229.034 1568.36,228.7 1570.42,228.367 \n",
       "  1572.48,228.036 1574.54,227.707 1576.6,227.379 1578.66,227.053 1580.72,226.729 1582.77,226.407 1584.83,226.086 1586.89,225.767 1588.95,225.449 1591.01,225.135 \n",
       "  1593.07,224.823 1595.13,224.511 1597.19,224.201 1599.24,223.893 1601.3,223.585 1603.36,223.279 1605.42,222.974 1607.48,222.67 1609.54,222.368 1611.6,222.067 \n",
       "  1613.66,221.767 1615.71,221.468 1617.77,221.171 1619.83,220.875 1621.89,220.58 1623.95,220.286 1626.01,219.994 1628.07,219.703 1630.13,219.413 1632.19,219.125 \n",
       "  1634.24,218.837 1636.3,218.551 1638.36,218.266 1640.42,217.983 1642.48,217.701 1644.54,217.42 1646.6,217.14 1648.66,216.861 1650.71,216.584 1652.77,216.308 \n",
       "  1654.83,216.033 1656.89,215.76 1658.95,215.488 1661.01,215.217 1663.07,214.947 1665.13,214.679 1667.18,214.412 1669.24,214.146 1671.3,213.881 1673.36,213.618 \n",
       "  1675.42,213.355 1677.48,213.095 1679.54,212.835 1681.6,212.577 1683.65,212.319 1685.71,212.064 1687.77,211.809 1689.83,211.556 1691.89,211.304 1693.95,211.053 \n",
       "  1696.01,210.803 1698.07,210.555 1700.12,210.308 1702.18,210.062 1704.24,209.818 1706.3,209.574 1708.36,209.332 1710.42,209.092 1712.48,208.852 1714.54,208.614 \n",
       "  1716.59,208.377 1718.65,208.142 1720.71,207.909 1722.77,207.677 1724.83,207.446 1726.89,207.216 1728.95,206.987 1731.01,206.759 1733.07,206.532 1735.12,206.306 \n",
       "  1737.18,206.081 1739.24,205.856 1741.3,205.633 1743.36,205.41 1745.42,205.189 1747.48,204.968 1749.54,204.748 1751.59,204.529 1753.65,204.311 1755.71,204.094 \n",
       "  1757.77,203.878 1759.83,203.663 1761.89,203.449 1763.95,203.236 1766.01,203.023 1768.06,202.812 1770.12,202.601 1772.18,202.392 1774.24,202.183 1776.3,201.976 \n",
       "  1778.36,201.769 1780.42,201.563 1782.48,201.358 1784.53,201.154 1786.59,200.951 1788.65,200.749 1790.71,200.547 1792.77,200.347 1794.83,200.148 1796.89,199.949 \n",
       "  1798.95,199.752 1801,199.555 1803.06,199.359 1805.12,199.165 1807.18,198.971 1809.24,198.778 1811.3,198.586 1813.36,198.395 1815.42,198.205 1817.47,198.015 \n",
       "  1819.53,197.827 1821.59,197.64 1823.65,197.453 1825.71,197.268 1827.77,197.083 1829.83,196.9 1831.89,196.717 1833.94,196.535 1836,196.354 1838.06,196.174 \n",
       "  1840.12,195.995 1842.18,195.817 1844.24,195.64 1846.3,195.464 1848.36,195.288 1850.42,195.114 1852.47,194.94 1854.53,194.768 1856.59,194.596 1858.65,194.425 \n",
       "  1860.71,194.257 1862.77,194.089 1864.83,193.922 1866.89,193.756 1868.94,193.591 1871,193.426 1873.06,193.261 1875.12,193.098 1877.18,192.935 1879.24,192.772 \n",
       "  1881.3,192.611 1883.36,192.45 1885.41,192.289 1887.47,192.129 1889.53,191.97 1891.59,191.812 1893.65,191.654 1895.71,191.497 1897.77,191.341 1899.83,191.185 \n",
       "  1901.88,191.03 1903.94,190.876 1906,190.722 1908.06,190.569 1910.12,190.416 1912.18,190.265 1914.24,190.114 1916.3,189.963 1918.35,189.813 1920.41,189.664 \n",
       "  1922.47,189.516 1924.53,189.368 1926.59,189.221 1928.65,189.074 1930.71,188.929 1932.77,188.783 1934.82,188.639 1936.88,188.495 1938.94,188.352 1941,188.209 \n",
       "  1943.06,188.068 1945.12,187.926 1947.18,187.786 1949.24,187.646 1951.3,187.507 1953.35,187.368 1955.41,187.23 1957.47,187.093 1959.53,186.956 1961.59,186.821 \n",
       "  1963.65,186.685 1965.71,186.551 1967.77,186.417 1969.82,186.284 1971.88,186.151 1973.94,186.019 1976,185.888 1978.06,185.757 1980.12,185.627 1982.18,185.498 \n",
       "  1984.24,185.369 1986.29,185.241 1988.35,185.114 1990.41,184.987 1992.47,184.861 1994.53,184.736 1996.59,184.611 1998.65,184.487 2000.71,184.364 2002.76,184.241 \n",
       "  2004.82,184.119 2006.88,183.998 2008.94,183.877 2011,183.757 2013.06,183.638 2015.12,183.521 2017.18,183.403 2019.23,183.287 2021.29,183.17 2023.35,183.055 \n",
       "  2025.41,182.939 2027.47,182.824 2029.53,182.71 2031.59,182.596 2033.65,182.482 2035.7,182.369 2037.76,182.257 2039.82,182.145 2041.88,182.033 2043.94,181.922 \n",
       "  2046,181.811 2048.06,181.701 2050.12,181.591 2052.17,181.481 2054.23,181.372 2056.29,181.264 2058.35,181.156 2060.41,181.048 2062.47,180.941 2064.53,180.834 \n",
       "  2066.59,180.728 2068.65,180.622 2070.7,180.517 2072.76,180.412 2074.82,180.308 2076.88,180.204 2078.94,180.1 2081,179.997 2083.06,179.895 2085.12,179.793 \n",
       "  2087.17,179.691 2089.23,179.59 2091.29,179.489 2093.35,179.389 2095.41,179.289 2097.47,179.19 2099.53,179.091 2101.59,178.992 2103.64,178.894 2105.7,178.797 \n",
       "  2107.76,178.7 2109.82,178.603 2111.88,178.507 2113.94,178.411 2116,178.316 2118.06,178.221 2120.11,178.127 2122.17,178.033 2124.23,177.939 2126.29,177.846 \n",
       "  2128.35,177.754 2130.41,177.662 2132.47,177.57 2134.53,177.479 2136.58,177.388 2138.64,177.298 2140.7,177.208 2142.76,177.119 2144.82,177.03 2146.88,176.941 \n",
       "  2148.94,176.853 2151,176.766 2153.05,176.679 2155.11,176.592 2157.17,176.506 2159.23,176.42 2161.29,176.335 2163.35,176.25 2165.41,176.166 2167.47,176.082 \n",
       "  2169.53,175.999 2171.58,175.916 2173.64,175.833 2175.7,175.751 2177.76,175.67 2179.82,175.588 2181.88,175.508 2183.94,175.429 2186,175.35 2188.05,175.271 \n",
       "  2190.11,175.192 2192.17,175.114 2194.23,175.036 2196.29,174.958 2198.35,174.881 2200.41,174.804 2202.47,174.727 2204.52,174.651 2206.58,174.575 2208.64,174.499 \n",
       "  2210.7,174.423 2212.76,174.348 2214.82,174.273 2216.88,174.199 2218.94,174.124 2220.99,174.05 2223.05,173.977 2225.11,173.903 2227.17,173.83 2229.23,173.758 \n",
       "  2231.29,173.685 2233.35,173.613 2235.41,173.541 2237.46,173.47 2239.52,173.399 2241.58,173.328 2243.64,173.257 2245.7,173.187 2247.76,173.117 2249.82,173.047 \n",
       "  2251.88,172.978 2253.93,172.909 2255.99,172.84 2258.05,172.772 2260.11,172.703 2262.17,172.636 2264.23,172.568 2266.29,172.501 2268.35,172.434 2270.4,172.367 \n",
       "  2272.46,172.301 2274.52,172.235 2276.58,172.17 2278.64,172.104 2280.7,172.039 2282.76,171.974 2284.82,171.91 2286.88,171.846 2288.93,171.782 2290.99,171.718 \n",
       "  2293.05,171.655 2295.11,171.592 2297.17,171.53 2299.23,171.468 2301.29,171.406 2303.35,171.344 2305.4,171.283 2307.46,171.221 2309.52,171.161 2311.58,171.1 \n",
       "  2313.64,171.04 2315.7,170.98 2317.76,170.921 2319.82,170.862 2321.87,170.803 2323.93,170.744 2325.99,170.686 2328.05,170.628 2330.11,170.57 2332.17,170.513 \n",
       "  2334.23,170.456 2336.29,170.399 2338.34,170.343 2340.4,170.286 2342.46,170.231 2344.52,170.175 2346.58,170.12 2348.64,170.065 2350.7,170.01 2352.76,169.956 \n",
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       "  316.629,1362.04 318.688,1358.14 320.746,1354.15 322.805,1350.09 324.864,1345.96 326.923,1341.78 328.981,1337.55 331.04,1333.28 333.099,1328.99 335.158,1324.66 \n",
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       "  378.392,1232.08 380.451,1227.71 382.51,1223.34 384.568,1218.98 386.627,1214.64 388.686,1210.3 390.745,1205.97 392.803,1201.65 394.862,1197.35 396.921,1193.05 \n",
       "  398.98,1188.76 401.039,1184.49 403.097,1180.22 405.156,1175.97 407.215,1171.73 409.274,1167.5 411.332,1163.28 413.391,1159.07 415.45,1154.88 417.509,1150.69 \n",
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       "  646.033,766.036 648.091,763.274 650.15,760.523 652.209,757.785 654.268,755.058 656.326,752.342 658.385,749.638 660.444,746.945 662.503,744.264 664.562,741.593 \n",
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       "  872.498,523.962 874.556,522.257 876.615,520.56 878.674,518.87 880.733,517.188 882.792,515.514 884.85,513.848 886.909,512.191 888.968,510.542 891.027,508.903 \n",
       "  893.085,507.273 895.144,505.651 897.203,504.036 899.262,502.429 901.321,500.829 903.379,499.236 905.438,497.649 907.497,496.069 909.556,494.495 911.614,492.927 \n",
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       "  1037.2,409.717 1039.26,408.539 1041.32,407.365 1043.38,406.196 1045.43,405.032 1047.49,403.872 1049.55,402.718 1051.61,401.569 1053.67,400.425 1055.73,399.287 \n",
       "  1057.79,398.155 1059.85,397.03 1061.9,395.912 1063.96,394.8 1066.02,393.693 1068.08,392.592 1070.14,391.496 1072.2,390.404 1074.26,389.317 1076.32,388.235 \n",
       "  1078.38,387.157 1080.43,386.083 1082.49,385.013 1084.55,383.947 1086.61,382.886 1088.67,381.829 1090.73,380.777 1092.79,379.729 1094.85,378.687 1096.9,377.65 \n",
       "  1098.96,376.62 1101.02,375.595 1103.08,374.577 1105.14,373.565 1107.2,372.556 1109.26,371.552 1111.32,370.552 1113.37,369.556 1115.43,368.564 1117.49,367.577 \n",
       "  1119.55,366.594 1121.61,365.615 1123.67,364.641 1125.73,363.671 1127.79,362.705 1129.84,361.743 1131.9,360.786 1133.96,359.833 1136.02,358.884 1138.08,357.939 \n",
       "  1140.14,356.999 1142.2,356.063 1144.26,355.131 1146.31,354.204 1148.37,353.281 1150.43,352.362 1152.49,351.447 1154.55,350.537 1156.61,349.631 1158.67,348.729 \n",
       "  1160.73,347.831 1162.78,346.938 1164.84,346.049 1166.9,345.164 1168.96,344.284 1171.02,343.408 1173.08,342.536 1175.14,341.668 1177.2,340.805 1179.25,339.946 \n",
       "  1181.31,339.091 1183.37,338.243 1185.43,337.398 1187.49,336.556 1189.55,335.718 1191.61,334.884 1193.67,334.053 1195.73,333.226 1197.78,332.403 1199.84,331.583 \n",
       "  1201.9,330.767 1203.96,329.954 1206.02,329.145 1208.08,328.34 1210.14,327.538 1212.2,326.74 1214.25,325.945 1216.31,325.154 1218.37,324.367 1220.43,323.583 \n",
       "  1222.49,322.803 1224.55,322.027 1226.61,321.254 1228.67,320.484 1230.72,319.719 1232.78,318.957 1234.84,318.198 1236.9,317.443 1238.96,316.692 1241.02,315.945 \n",
       "  1243.08,315.201 1245.14,314.46 1247.19,313.723 1249.25,312.99 1251.31,312.261 1253.37,311.535 1255.43,310.812 1257.49,310.094 1259.55,309.379 1261.61,308.668 \n",
       "  1263.66,307.962 1265.72,307.259 1267.78,306.559 1269.84,305.862 1271.9,305.168 1273.96,304.477 1276.02,303.789 1278.08,303.104 1280.13,302.422 1282.19,301.742 \n",
       "  1284.25,301.066 1286.31,300.393 1288.37,299.722 1290.43,299.055 1292.49,298.39 1294.55,297.728 1296.61,297.069 1298.66,296.414 1300.72,295.761 1302.78,295.111 \n",
       "  1304.84,294.464 1306.9,293.82 1308.96,293.179 1311.02,292.54 1313.08,291.905 1315.13,291.273 1317.19,290.643 1319.25,290.017 1321.31,289.394 1323.37,288.773 \n",
       "  1325.43,288.155 1327.49,287.541 1329.55,286.929 1331.6,286.32 1333.66,285.714 1335.72,285.111 1337.78,284.511 1339.84,283.914 1341.9,283.32 1343.96,282.729 \n",
       "  1346.02,282.141 1348.07,281.555 1350.13,280.973 1352.19,280.394 1354.25,279.817 1356.31,279.244 1358.37,278.673 1360.43,278.105 1362.49,277.543 1364.54,276.983 \n",
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       "  1387.19,270.975 1389.25,270.443 1391.31,269.913 1393.37,269.385 1395.43,268.86 1397.48,268.337 1399.54,267.817 1401.6,267.298 1403.66,266.782 1405.72,266.268 \n",
       "  1407.78,265.757 1409.84,265.248 1411.9,264.741 1413.96,264.237 1416.01,263.735 1418.07,263.235 1420.13,262.737 1422.19,262.242 1424.25,261.749 1426.31,261.259 \n",
       "  1428.37,260.771 1430.43,260.285 1432.48,259.801 1434.54,259.32 1436.6,258.841 1438.66,258.364 1440.72,257.89 1442.78,257.418 1444.84,256.948 1446.9,256.481 \n",
       "  1448.95,256.016 1451.01,255.553 1453.07,255.093 1455.13,254.634 1457.19,254.179 1459.25,253.725 1461.31,253.274 1463.37,252.825 1465.42,252.379 1467.48,251.934 \n",
       "  1469.54,251.493 1471.6,251.054 1473.66,250.619 1475.72,250.185 1477.78,249.753 1479.84,249.323 1481.89,248.895 1483.95,248.468 1486.01,248.043 1488.07,247.62 \n",
       "  1490.13,247.199 1492.19,246.78 1494.25,246.362 1496.31,245.946 1498.36,245.532 1500.42,245.12 1502.48,244.709 1504.54,244.301 1506.6,243.894 1508.66,243.489 \n",
       "  1510.72,243.085 1512.78,242.684 1514.84,242.284 1516.89,241.886 1518.95,241.49 1521.01,241.095 1523.07,240.703 1525.13,240.312 1527.19,239.923 1529.25,239.535 \n",
       "  1531.31,239.15 1533.36,238.766 1535.42,238.384 1537.48,238.004 1539.54,237.625 1541.6,237.249 1543.66,236.874 1545.72,236.501 1547.78,236.13 1549.83,235.76 \n",
       "  1551.89,235.392 1553.95,235.027 1556.01,234.662 1558.07,234.3 1560.13,233.939 1562.19,233.581 1564.25,233.224 1566.3,232.868 1568.36,232.515 1570.42,232.163 \n",
       "  1572.48,231.814 1574.54,231.465 1576.6,231.119 1578.66,230.775 1580.72,230.432 1582.77,230.091 1584.83,229.752 1586.89,229.414 1588.95,229.079 1591.01,228.747 \n",
       "  1593.07,228.416 1595.13,228.087 1597.19,227.759 1599.24,227.433 1601.3,227.108 1603.36,226.784 1605.42,226.461 1607.48,226.14 1609.54,225.82 1611.6,225.502 \n",
       "  1613.66,225.185 1615.71,224.869 1617.77,224.555 1619.83,224.242 1621.89,223.93 1623.95,223.619 1626.01,223.31 1628.07,223.003 1630.13,222.696 1632.19,222.391 \n",
       "  1634.24,222.088 1636.3,221.785 1638.36,221.484 1640.42,221.184 1642.48,220.886 1644.54,220.589 1646.6,220.294 1648.66,219.999 1650.71,219.706 1652.77,219.415 \n",
       "  1654.83,219.124 1656.89,218.835 1658.95,218.548 1661.01,218.262 1663.07,217.977 1665.13,217.693 1667.18,217.411 1669.24,217.13 1671.3,216.85 1673.36,216.572 \n",
       "  1675.42,216.295 1677.48,216.02 1679.54,215.745 1681.6,215.473 1683.65,215.201 1685.71,214.931 1687.77,214.662 1689.83,214.395 1691.89,214.129 1693.95,213.864 \n",
       "  1696.01,213.6 1698.07,213.338 1700.12,213.077 1702.18,212.818 1704.24,212.56 1706.3,212.303 1708.36,212.048 1710.42,211.794 1712.48,211.541 1714.54,211.289 \n",
       "  1716.59,211.039 1718.65,210.791 1720.71,210.545 1722.77,210.3 1724.83,210.057 1726.89,209.814 1728.95,209.572 1731.01,209.331 1733.07,209.091 1735.12,208.852 \n",
       "  1737.18,208.615 1739.24,208.378 1741.3,208.142 1743.36,207.907 1745.42,207.673 1747.48,207.44 1749.54,207.208 1751.59,206.977 1753.65,206.747 1755.71,206.518 \n",
       "  1757.77,206.29 1759.83,206.063 1761.89,205.837 1763.95,205.611 1766.01,205.387 1768.06,205.164 1770.12,204.942 1772.18,204.721 1774.24,204.501 1776.3,204.281 \n",
       "  1778.36,204.063 1780.42,203.846 1782.48,203.63 1784.53,203.414 1786.59,203.2 1788.65,202.987 1790.71,202.774 1792.77,202.563 1794.83,202.353 1796.89,202.143 \n",
       "  1798.95,201.935 1801,201.728 1803.06,201.521 1805.12,201.316 1807.18,201.111 1809.24,200.908 1811.3,200.705 1813.36,200.504 1815.42,200.303 1817.47,200.104 \n",
       "  1819.53,199.905 1821.59,199.707 1823.65,199.511 1825.71,199.315 1827.77,199.12 1829.83,198.927 1831.89,198.734 1833.94,198.542 1836,198.352 1838.06,198.162 \n",
       "  1840.12,197.973 1842.18,197.785 1844.24,197.599 1846.3,197.413 1848.36,197.228 1850.42,197.044 1852.47,196.861 1854.53,196.679 1856.59,196.499 1858.65,196.319 \n",
       "  1860.71,196.141 1862.77,195.964 1864.83,195.788 1866.89,195.613 1868.94,195.438 1871,195.264 1873.06,195.091 1875.12,194.918 1877.18,194.746 1879.24,194.575 \n",
       "  1881.3,194.405 1883.36,194.235 1885.41,194.066 1887.47,193.897 1889.53,193.73 1891.59,193.563 1893.65,193.396 1895.71,193.231 1897.77,193.066 1899.83,192.902 \n",
       "  1901.88,192.738 1903.94,192.576 1906,192.413 1908.06,192.252 1910.12,192.091 1912.18,191.931 1914.24,191.772 1916.3,191.614 1918.35,191.456 1920.41,191.298 \n",
       "  1922.47,191.142 1924.53,190.986 1926.59,190.831 1928.65,190.677 1930.71,190.523 1932.77,190.37 1934.82,190.218 1936.88,190.066 1938.94,189.915 1941,189.765 \n",
       "  1943.06,189.615 1945.12,189.467 1947.18,189.319 1949.24,189.171 1951.3,189.024 1953.35,188.878 1955.41,188.733 1957.47,188.589 1959.53,188.445 1961.59,188.301 \n",
       "  1963.65,188.159 1965.71,188.017 1967.77,187.876 1969.82,187.736 1971.88,187.596 1973.94,187.457 1976,187.319 1978.06,187.181 1980.12,187.044 1982.18,186.908 \n",
       "  1984.24,186.772 1986.29,186.638 1988.35,186.503 1990.41,186.37 1992.47,186.237 1994.53,186.105 1996.59,185.974 1998.65,185.843 2000.71,185.713 2002.76,185.584 \n",
       "  2004.82,185.456 2006.88,185.328 2008.94,185.201 2011,185.074 2013.06,184.949 2015.12,184.825 2017.18,184.702 2019.23,184.579 2021.29,184.456 2023.35,184.334 \n",
       "  2025.41,184.213 2027.47,184.092 2029.53,183.971 2031.59,183.851 2033.65,183.732 2035.7,183.613 2037.76,183.494 2039.82,183.376 2041.88,183.258 2043.94,183.141 \n",
       "  2046,183.024 2048.06,182.908 2050.12,182.792 2052.17,182.677 2054.23,182.562 2056.29,182.448 2058.35,182.334 2060.41,182.221 2062.47,182.108 2064.53,181.996 \n",
       "  2066.59,181.884 2068.65,181.772 2070.7,181.661 2072.76,181.551 2074.82,181.441 2076.88,181.332 2078.94,181.223 2081,181.114 2083.06,181.006 2085.12,180.899 \n",
       "  2087.17,180.791 2089.23,180.685 2091.29,180.579 2093.35,180.473 2095.41,180.368 2097.47,180.263 2099.53,180.159 2101.59,180.055 2103.64,179.952 2105.7,179.85 \n",
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   ],
   "source": [
    "prob = ODEProblem(enzyme_kinetics!,u0,tspan,p)\n",
    "sol = solve(prob)\n",
    "plot(sol)\n",
    "title!(\"Enzyme-substrate kinetics using DSL notation\")\n",
    "xlabel!(\"Time [s]\")\n",
    "ylabel!(\"Concentration [M]\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The next step is to take advantage of the __Mass Conservation Law__ to reduce the number of independent variables (i.e, the number of differential equations to solve). \n",
    "\n",
    "\n",
    "\n",
    "$$ \\begin{align*}\n",
    "C_1 \\cdot X_1 + C_2 \\cdot X_2 + C_3 \\cdot X_3 + C_4 \\cdot X_4 & = cte \\\\\n",
    "\\end{align*}$$\n",
    "\n",
    "so\n",
    "$$ \\begin{align*}\n",
    "C \\cdot (B-A)^T &=0\\\\ \n",
    "\\begin{pmatrix}C_1 & C_2 & C_3 & C_4 \\end{pmatrix} \\begin{bmatrix}\n",
    "-1 & 1 & 0\\\\\n",
    " 1 & -1  & -1\\\\\n",
    " -1 & 1 & 1 \\\\\n",
    " 0 & 0 & 1\\end{bmatrix} &=0\n",
    " \\end{align*}$$\n",
    " \n",
    "the solution of the system of equations is: \n",
    " $$ \\begin{align*}\n",
    " -C_1 + C_2 - C_3= 0 &\\implies  C_2 = C_3 + C_1\\\\\n",
    " -C_2 + C_3 + C_4= 0 &\\implies  C_2 = C_3 + C_4 \n",
    "\\end{align*}$$\n",
    "\n",
    "Therefore, from the two equations above, $C_1 = C_4$, and the conservation is:\n",
    "\n",
    "$$ \\begin{align*}\n",
    "C_1 \\cdot X_1(t) + (C_3 + C_1) \\cdot X_2(t) + C_3 \\cdot X_3(t) + C_1 \\cdot X_4(t) &= cte\\\\\n",
    "C_1 \\cdot X_1(t) + C_3 \\cdot X_2(t) + C_1 \\cdot X_2(t)  + C_3 \\cdot X_3(t) + C_1 \\cdot X_4(t) &= cte\\\\\n",
    "\\end{align*}$$\n",
    "\n",
    "which is valid for any value of $C_1$ and $C_3$, so, separating these two dependencies\n",
    "\n",
    "$$ \\begin{align*}\n",
    "C_1 \\cdot X_1(t) + C_1 \\cdot X_2(t)  + C_1 \\cdot X_4(t) &= cte\\\\\n",
    "\\end{align*}$$\n",
    "\n",
    "which makes sense, since the $s$, $c$ and $p$ are three forms of the same protein, and their sum has to be constant,a nd equal to the initial value of substrate used. Using the value $C_1=1$ we obtain:\n",
    "\n",
    "$$ \\begin{align*}\n",
    "X_1(t) + X_2(t) + X_4(t) &= cte = X_1(0) + X_2(0) + X_4(0) =  X_1(0) \\\\\n",
    "\\end{align*}$$\n",
    "\n",
    "is our first conservation law (typically we start a biochemical reaction adding only $e$ and $s$). For the second, we use the parameters that depend on the other constant $C_2$ \n",
    "\n",
    "$$ \\begin{align*}\n",
    "C_3 \\cdot X_2(t) + C_3 \\cdot X_3(t) &= cte\n",
    "\\end{align*}$$\n",
    "\n",
    "which also makes sense, because the amoun of enzyme is not consumed and is only in teh form of the reactans $e$ and $c$. So, giving the value $C_3=1$, we obtain the conservation law, \n",
    "$$ \\begin{align*}\n",
    "X_2(t) + X_3(t) & = cte &= X_2(0) + X_3(0) = X_3 (0) \n",
    "\\end{align*}$$\n",
    "\n",
    "which is true because the enzyme is a catalyst that facilitates the reaction but does not react itself, and if we assume that there is no complex $c(0)=0$ before the reacton starts, we can assume ($e(0)+c(0)=e_0$). This conservation law allows us to reduce the four differential equations into the following three coupled ordinary differential equations:\n",
    "\n",
    "$$\\begin{align*}\n",
    "  \\dot{s}  &= k_2 \\cdot c - k_1 \\cdot (e_0-c) \\cdot s \\tag{12}\\\\\n",
    "  \\dot{c} &= k_1 \\cdot (e_0-c)\\cdot s - (k_2+k_3) \\cdot c \\tag{13}\\\\\n",
    "  \\dot{p} &= k_3 \\cdot c \\tag{14}\n",
    "  \\end{align*}$$\n",
    "  \n",
    "which rearranging terms become:\n",
    "\n",
    "$$\\begin{align*}\n",
    "  \\dot{s}  &= (k_2+k_1 \\cdot s) \\cdot c -k_1 \\cdot e_0 \\cdot s \\tag{15}\\\\ \n",
    "  \\dot{c} &= k_1\\cdot e_0\\cdot s-(k_3+k_2+k_1 \\cdot s) \\cdot c \\tag{16}\\\\\n",
    "  \\dot{p} &= k_3 \\cdot c \\tag{17}\n",
    "  \\end{align*}$$\n",
    "  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(::ab{var\"#67#71\",var\"#68#72\",var\"#69#73\",Nothing,Nothing,var\"#70#74\",Expr,Expr}) (generic function with 2 methods)"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "enzyme_kinetics2! = @ode_def ab begin\n",
    "  ds = (k2+s*k1)*c-k1*e0*s\n",
    "  dc = k1*e0*s-(k2+k3+k1*s)*c\n",
    "  dp = k3*c\n",
    "    end k1 k2 k3 e0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\u001b[36mODEProblem\u001b[0m with uType \u001b[36mArray{Float64,1}\u001b[0m and tType \u001b[36mFloat64\u001b[0m. In-place: \u001b[36mtrue\u001b[0m\n",
       "timespan: (0.0, 100.0)\n",
       "u0: [0.002, 0.0, 0.0]"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tspan = (0.0,100.0)\n",
    "u0=[s0,0,0]\n",
    "p=[k1,k2,k3,e0];\n",
    "\n",
    "prob = ODEProblem(enzyme_kinetics2!,u0,tspan,p)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
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       "  460.743,1263.39 462.802,1264.7 464.861,1265.99 466.919,1267.27 468.978,1268.52 471.037,1269.76 473.096,1270.98 475.154,1272.19 477.213,1273.38 479.272,1274.57 \n",
       "  481.331,1275.74 483.389,1276.9 485.448,1278.04 487.507,1279.17 489.566,1280.28 491.625,1281.38 493.683,1282.46 495.742,1283.53 497.801,1284.58 499.86,1285.62 \n",
       "  501.918,1286.64 503.977,1287.64 506.036,1288.63 508.095,1289.61 510.154,1290.58 512.212,1291.53 514.271,1292.48 516.33,1293.42 518.389,1294.35 520.447,1295.27 \n",
       "  522.506,1296.18 524.565,1297.08 526.624,1297.98 528.683,1298.86 530.741,1299.73 532.8,1300.58 534.859,1301.42 536.918,1302.25 538.976,1303.07 541.035,1303.87 \n",
       "  543.094,1304.66 545.153,1305.43 547.211,1306.2 549.27,1306.95 551.329,1307.7 553.388,1308.44 555.447,1309.17 557.505,1309.9 559.564,1310.62 561.623,1311.34 \n",
       "  563.682,1312.06 565.74,1312.77 567.799,1313.48 569.858,1314.17 571.917,1314.86 573.976,1315.53 576.034,1316.2 578.093,1316.85 580.152,1317.49 582.211,1318.12 \n",
       "  584.269,1318.74 586.328,1319.35 588.387,1319.94 590.446,1320.53 592.504,1321.11 594.563,1321.68 596.622,1322.25 598.681,1322.81 600.74,1323.37 602.798,1323.92 \n",
       "  604.857,1324.49 606.916,1325.05 608.975,1325.61 611.033,1326.17 613.092,1326.72 615.151,1327.27 617.21,1327.81 619.269,1328.34 621.327,1328.86 623.386,1329.37 \n",
       "  625.445,1329.87 627.504,1330.35 629.562,1330.83 631.621,1331.3 633.68,1331.75 635.739,1332.2 637.798,1332.64 639.856,1333.08 641.915,1333.51 643.974,1333.93 \n",
       "  646.033,1334.36 648.091,1334.79 650.15,1335.22 652.209,1335.66 654.268,1336.1 656.326,1336.55 658.385,1336.99 660.444,1337.43 662.503,1337.87 664.562,1338.3 \n",
       "  666.62,1338.72 668.679,1339.13 670.738,1339.54 672.797,1339.93 674.855,1340.31 676.914,1340.68 678.973,1341.04 681.032,1341.38 683.091,1341.72 685.149,1342.06 \n",
       "  687.208,1342.38 689.267,1342.71 691.326,1343.03 693.384,1343.35 695.443,1343.67 697.502,1344 699.561,1344.34 701.619,1344.7 703.678,1345.06 705.737,1345.42 \n",
       "  707.796,1345.79 709.855,1346.15 711.913,1346.51 713.972,1346.86 716.031,1347.2 718.09,1347.53 720.148,1347.85 722.207,1348.16 724.266,1348.45 726.325,1348.74 \n",
       "  728.384,1349.01 730.442,1349.27 732.501,1349.52 734.56,1349.76 736.619,1350 738.677,1350.24 740.736,1350.47 742.795,1350.71 744.854,1350.95 746.913,1351.21 \n",
       "  748.971,1351.48 751.03,1351.77 753.089,1352.07 755.148,1352.38 757.206,1352.7 759.265,1353.01 761.324,1353.32 763.383,1353.62 765.441,1353.91 767.5,1354.19 \n",
       "  769.559,1354.45 771.618,1354.7 773.677,1354.93 775.735,1355.15 777.794,1355.35 779.853,1355.54 781.912,1355.72 783.97,1355.89 786.029,1356.05 788.088,1356.21 \n",
       "  790.147,1356.38 792.206,1356.55 794.264,1356.73 796.323,1356.93 798.382,1357.15 800.441,1357.4 802.499,1357.67 804.558,1357.95 806.617,1358.23 808.676,1358.52 \n",
       "  810.734,1358.8 812.793,1359.06 814.852,1359.32 816.911,1359.56 818.97,1359.77 821.028,1359.97 823.087,1360.15 825.146,1360.31 827.205,1360.46 829.263,1360.58 \n",
       "  831.322,1360.69 833.381,1360.8 835.44,1360.9 837.499,1361 839.557,1361.1 841.616,1361.22 843.675,1361.36 845.734,1361.53 847.792,1361.74 849.851,1361.98 \n",
       "  851.91,1362.24 853.969,1362.51 856.028,1362.78 858.086,1363.04 860.145,1363.29 862.204,1363.53 864.263,1363.74 866.321,1363.93 868.38,1364.1 870.439,1364.25 \n",
       "  872.498,1364.36 874.556,1364.46 876.615,1364.54 878.674,1364.6 880.733,1364.66 882.792,1364.71 884.85,1364.76 886.909,1364.82 888.968,1364.91 891.027,1365.02 \n",
       "  893.085,1365.18 895.144,1365.39 897.203,1365.63 899.262,1365.88 901.321,1366.13 903.379,1366.38 905.438,1366.62 907.497,1366.84 909.556,1367.04 911.614,1367.22 \n",
       "  913.673,1367.37 915.732,1367.49 917.791,1367.58 919.849,1367.65 921.908,1367.69 923.967,1367.72 926.026,1367.74 928.085,1367.76 930.143,1367.78 932.202,1367.82 \n",
       "  934.261,1367.89 936.32,1367.99 938.378,1368.15 940.437,1368.36 942.496,1368.58 944.555,1368.82 946.614,1369.05 948.672,1369.28 950.731,1369.49 952.79,1369.68 \n",
       "  954.849,1369.84 956.907,1369.98 958.966,1370.08 961.025,1370.16 963.084,1370.22 965.143,1370.25 967.201,1370.26 969.26,1370.27 971.319,1370.26 973.378,1370.27 \n",
       "  975.436,1370.29 977.495,1370.34 979.554,1370.44 981.613,1370.59 983.671,1370.78 985.73,1370.99 987.789,1371.2 989.848,1371.42 991.907,1371.62 993.965,1371.8 \n",
       "  996.024,1371.96 998.083,1372.09 1000.14,1372.2 1002.2,1372.28 1004.26,1372.34 1006.32,1372.37 1008.38,1372.38 1010.44,1372.38 1012.49,1372.37 1014.55,1372.36 \n",
       "  1016.61,1372.37 1018.67,1372.4 1020.73,1372.47 1022.79,1372.59 1024.85,1372.75 1026.91,1372.94 1028.96,1373.13 1031.02,1373.32 1033.08,1373.51 1035.14,1373.67 \n",
       "  1037.2,1373.82 1039.26,1373.94 1041.32,1374.04 1043.38,1374.11 1045.43,1374.15 1047.49,1374.17 1049.55,1374.18 1051.61,1374.17 1053.67,1374.15 1055.73,1374.14 \n",
       "  1057.79,1374.15 1059.85,1374.17 1061.9,1374.24 1063.96,1374.35 1066.02,1374.51 1068.08,1374.68 1070.14,1374.85 1072.2,1375.03 1074.26,1375.19 1076.32,1375.33 \n",
       "  1078.38,1375.46 1080.43,1375.56 1082.49,1375.64 1084.55,1375.69 1086.61,1375.72 1088.67,1375.73 1090.73,1375.72 1092.79,1375.7 1094.85,1375.69 1096.9,1375.67 \n",
       "  1098.96,1375.68 1101.02,1375.71 1103.08,1375.79 1105.14,1375.92 1107.2,1376.07 1109.26,1376.23 1111.32,1376.39 1113.37,1376.54 1115.43,1376.68 1117.49,1376.81 \n",
       "  1119.55,1376.91 1121.61,1376.98 1123.67,1377.04 1125.73,1377.07 1127.79,1377.08 1129.84,1377.07 1131.9,1377.05 1133.96,1377.03 1136.02,1377.01 1138.08,1377 \n",
       "  1140.14,1377.02 1142.2,1377.07 1144.26,1377.17 1146.31,1377.26 1148.37,1377.35 1150.43,1377.44 1152.49,1377.52 1154.55,1377.61 1156.61,1377.69 1158.67,1377.77 \n",
       "  1160.73,1377.86 1162.78,1377.94 1164.84,1378.01 1166.9,1378.09 1168.96,1378.17 1171.02,1378.24 1173.08,1378.31 1175.14,1378.38 1177.2,1378.45 1179.25,1378.52 \n",
       "  1181.31,1378.59 1183.37,1378.65 1185.43,1378.72 1187.49,1378.78 1189.55,1378.84 1191.61,1378.9 1193.67,1378.96 1195.73,1379.01 1197.78,1379.07 1199.84,1379.12 \n",
       "  1201.9,1379.18 1203.96,1379.23 1206.02,1379.28 1208.08,1379.32 1210.14,1379.37 1212.2,1379.42 1214.25,1379.46 1216.31,1379.5 1218.37,1379.54 1220.43,1379.58 \n",
       "  1222.49,1379.62 1224.55,1379.66 1226.61,1379.71 1228.67,1379.75 1230.72,1379.8 1232.78,1379.84 1234.84,1379.88 1236.9,1379.93 1238.96,1379.97 1241.02,1380.01 \n",
       "  1243.08,1380.05 1245.14,1380.1 1247.19,1380.14 1249.25,1380.18 1251.31,1380.23 1253.37,1380.27 1255.43,1380.31 1257.49,1380.35 1259.55,1380.4 1261.61,1380.44 \n",
       "  1263.66,1380.48 1265.72,1380.52 1267.78,1380.57 1269.84,1380.61 1271.9,1380.65 1273.96,1380.69 1276.02,1380.74 1278.08,1380.78 1280.13,1380.82 1282.19,1380.86 \n",
       "  1284.25,1380.9 1286.31,1380.95 1288.37,1380.99 1290.43,1381.03 1292.49,1381.07 1294.55,1381.11 1296.61,1381.15 1298.66,1381.2 1300.72,1381.24 1302.78,1381.28 \n",
       "  1304.84,1381.32 1306.9,1381.36 1308.96,1381.4 1311.02,1381.44 1313.08,1381.48 1315.13,1381.52 1317.19,1381.55 1319.25,1381.59 1321.31,1381.63 1323.37,1381.67 \n",
       "  1325.43,1381.71 1327.49,1381.74 1329.55,1381.78 1331.6,1381.82 1333.66,1381.86 1335.72,1381.89 1337.78,1381.93 1339.84,1381.96 1341.9,1382 1343.96,1382.04 \n",
       "  1346.02,1382.07 1348.07,1382.11 1350.13,1382.14 1352.19,1382.18 1354.25,1382.21 1356.31,1382.25 1358.37,1382.28 1360.43,1382.31 1362.49,1382.35 1364.54,1382.38 \n",
       "  1366.6,1382.41 1368.66,1382.45 1370.72,1382.48 1372.78,1382.51 1374.84,1382.54 1376.9,1382.58 1378.96,1382.61 1381.01,1382.64 1383.07,1382.67 1385.13,1382.7 \n",
       "  1387.19,1382.73 1389.25,1382.76 1391.31,1382.79 1393.37,1382.82 1395.43,1382.85 1397.48,1382.88 1399.54,1382.91 1401.6,1382.94 1403.66,1382.97 1405.72,1383 \n",
       "  1407.78,1383.03 1409.84,1383.06 1411.9,1383.09 1413.96,1383.12 1416.01,1383.14 1418.07,1383.17 1420.13,1383.2 1422.19,1383.23 1424.25,1383.26 1426.31,1383.29 \n",
       "  1428.37,1383.31 1430.43,1383.34 1432.48,1383.37 1434.54,1383.4 1436.6,1383.42 1438.66,1383.45 1440.72,1383.48 1442.78,1383.51 1444.84,1383.53 1446.9,1383.56 \n",
       "  1448.95,1383.59 1451.01,1383.61 1453.07,1383.64 1455.13,1383.66 1457.19,1383.69 1459.25,1383.72 1461.31,1383.74 1463.37,1383.77 1465.42,1383.79 1467.48,1383.82 \n",
       "  1469.54,1383.84 1471.6,1383.87 1473.66,1383.89 1475.72,1383.92 1477.78,1383.94 1479.84,1383.96 1481.89,1383.99 1483.95,1384.01 1486.01,1384.04 1488.07,1384.06 \n",
       "  1490.13,1384.08 1492.19,1384.11 1494.25,1384.13 1496.31,1384.15 1498.36,1384.18 1500.42,1384.2 1502.48,1384.22 1504.54,1384.25 1506.6,1384.27 1508.66,1384.29 \n",
       "  1510.72,1384.31 1512.78,1384.34 1514.84,1384.36 1516.89,1384.38 1518.95,1384.4 1521.01,1384.42 1523.07,1384.45 1525.13,1384.47 1527.19,1384.49 1529.25,1384.51 \n",
       "  1531.31,1384.53 1533.36,1384.55 1535.42,1384.57 1537.48,1384.59 1539.54,1384.62 1541.6,1384.64 1543.66,1384.66 1545.72,1384.68 1547.78,1384.7 1549.83,1384.72 \n",
       "  1551.89,1384.74 1553.95,1384.76 1556.01,1384.78 1558.07,1384.8 1560.13,1384.82 1562.19,1384.84 1564.25,1384.86 1566.3,1384.88 1568.36,1384.9 1570.42,1384.92 \n",
       "  1572.48,1384.94 1574.54,1384.95 1576.6,1384.97 1578.66,1384.99 1580.72,1385.01 1582.77,1385.03 1584.83,1385.05 1586.89,1385.07 1588.95,1385.09 1591.01,1385.1 \n",
       "  1593.07,1385.12 1595.13,1385.14 1597.19,1385.16 1599.24,1385.18 1601.3,1385.19 1603.36,1385.21 1605.42,1385.23 1607.48,1385.25 1609.54,1385.26 1611.6,1385.28 \n",
       "  1613.66,1385.3 1615.71,1385.31 1617.77,1385.33 1619.83,1385.35 1621.89,1385.36 1623.95,1385.38 1626.01,1385.4 1628.07,1385.41 1630.13,1385.43 1632.19,1385.45 \n",
       "  1634.24,1385.46 1636.3,1385.48 1638.36,1385.49 1640.42,1385.51 1642.48,1385.53 1644.54,1385.54 1646.6,1385.56 1648.66,1385.57 1650.71,1385.59 1652.77,1385.61 \n",
       "  1654.83,1385.62 1656.89,1385.64 1658.95,1385.65 1661.01,1385.67 1663.07,1385.68 1665.13,1385.7 1667.18,1385.71 1669.24,1385.73 1671.3,1385.74 1673.36,1385.76 \n",
       "  1675.42,1385.77 1677.48,1385.79 1679.54,1385.8 1681.6,1385.82 1683.65,1385.83 1685.71,1385.84 1687.77,1385.86 1689.83,1385.87 1691.89,1385.89 1693.95,1385.9 \n",
       "  1696.01,1385.92 1698.07,1385.93 1700.12,1385.94 1702.18,1385.96 1704.24,1385.97 1706.3,1385.98 1708.36,1386 1710.42,1386.01 1712.48,1386.03 1714.54,1386.04 \n",
       "  1716.59,1386.05 1718.65,1386.07 1720.71,1386.08 1722.77,1386.09 1724.83,1386.1 1726.89,1386.12 1728.95,1386.13 1731.01,1386.14 1733.07,1386.16 1735.12,1386.17 \n",
       "  1737.18,1386.18 1739.24,1386.19 1741.3,1386.21 1743.36,1386.22 1745.42,1386.23 1747.48,1386.24 1749.54,1386.25 1751.59,1386.27 1753.65,1386.28 1755.71,1386.29 \n",
       "  1757.77,1386.3 1759.83,1386.31 1761.89,1386.33 1763.95,1386.34 1766.01,1386.35 1768.06,1386.36 1770.12,1386.37 1772.18,1386.38 1774.24,1386.4 1776.3,1386.41 \n",
       "  1778.36,1386.42 1780.42,1386.43 1782.48,1386.44 1784.53,1386.45 1786.59,1386.46 1788.65,1386.47 1790.71,1386.49 1792.77,1386.5 1794.83,1386.51 1796.89,1386.52 \n",
       "  1798.95,1386.53 1801,1386.54 1803.06,1386.55 1805.12,1386.56 1807.18,1386.57 1809.24,1386.58 1811.3,1386.59 1813.36,1386.6 1815.42,1386.61 1817.47,1386.62 \n",
       "  1819.53,1386.63 1821.59,1386.65 1823.65,1386.66 1825.71,1386.67 1827.77,1386.68 1829.83,1386.69 1831.89,1386.7 1833.94,1386.71 1836,1386.72 1838.06,1386.73 \n",
       "  1840.12,1386.74 1842.18,1386.74 1844.24,1386.75 1846.3,1386.76 1848.36,1386.77 1850.42,1386.78 1852.47,1386.79 1854.53,1386.8 1856.59,1386.81 1858.65,1386.82 \n",
       "  1860.71,1386.83 1862.77,1386.84 1864.83,1386.85 1866.89,1386.86 1868.94,1386.87 1871,1386.88 1873.06,1386.89 1875.12,1386.89 1877.18,1386.9 1879.24,1386.91 \n",
       "  1881.3,1386.92 1883.36,1386.93 1885.41,1386.94 1887.47,1386.95 1889.53,1386.96 1891.59,1386.96 1893.65,1386.97 1895.71,1386.98 1897.77,1386.99 1899.83,1387 \n",
       "  1901.88,1387.01 1903.94,1387.01 1906,1387.02 1908.06,1387.03 1910.12,1387.04 1912.18,1387.05 1914.24,1387.05 1916.3,1387.06 1918.35,1387.07 1920.41,1387.08 \n",
       "  1922.47,1387.09 1924.53,1387.1 1926.59,1387.1 1928.65,1387.11 1930.71,1387.12 1932.77,1387.13 1934.82,1387.13 1936.88,1387.14 1938.94,1387.15 1941,1387.16 \n",
       "  1943.06,1387.16 1945.12,1387.17 1947.18,1387.18 1949.24,1387.19 1951.3,1387.2 1953.35,1387.2 1955.41,1387.21 1957.47,1387.22 1959.53,1387.22 1961.59,1387.23 \n",
       "  1963.65,1387.24 1965.71,1387.25 1967.77,1387.25 1969.82,1387.26 1971.88,1387.27 1973.94,1387.28 1976,1387.28 1978.06,1387.29 1980.12,1387.3 1982.18,1387.3 \n",
       "  1984.24,1387.31 1986.29,1387.32 1988.35,1387.32 1990.41,1387.33 1992.47,1387.34 1994.53,1387.34 1996.59,1387.35 1998.65,1387.36 2000.71,1387.36 2002.76,1387.37 \n",
       "  2004.82,1387.38 2006.88,1387.38 2008.94,1387.39 2011,1387.4 2013.06,1387.4 2015.12,1387.41 2017.18,1387.42 2019.23,1387.42 2021.29,1387.43 2023.35,1387.43 \n",
       "  2025.41,1387.44 2027.47,1387.45 2029.53,1387.45 2031.59,1387.46 2033.65,1387.47 2035.7,1387.47 2037.76,1387.48 2039.82,1387.48 2041.88,1387.49 2043.94,1387.5 \n",
       "  2046,1387.5 2048.06,1387.51 2050.12,1387.51 2052.17,1387.52 2054.23,1387.52 2056.29,1387.53 2058.35,1387.54 2060.41,1387.54 2062.47,1387.55 2064.53,1387.55 \n",
       "  2066.59,1387.56 2068.65,1387.56 2070.7,1387.57 2072.76,1387.57 2074.82,1387.58 2076.88,1387.59 2078.94,1387.59 2081,1387.6 2083.06,1387.6 2085.12,1387.61 \n",
       "  2087.17,1387.61 2089.23,1387.62 2091.29,1387.62 2093.35,1387.63 2095.41,1387.63 2097.47,1387.64 2099.53,1387.64 2101.59,1387.65 2103.64,1387.66 2105.7,1387.66 \n",
       "  2107.76,1387.67 2109.82,1387.67 2111.88,1387.68 2113.94,1387.68 2116,1387.69 2118.06,1387.69 2120.11,1387.7 2122.17,1387.7 2124.23,1387.71 2126.29,1387.71 \n",
       "  2128.35,1387.72 2130.41,1387.72 2132.47,1387.73 2134.53,1387.73 2136.58,1387.73 2138.64,1387.74 2140.7,1387.74 2142.76,1387.75 2144.82,1387.75 2146.88,1387.76 \n",
       "  2148.94,1387.76 2151,1387.77 2153.05,1387.77 2155.11,1387.78 2157.17,1387.78 2159.23,1387.79 2161.29,1387.79 2163.35,1387.8 2165.41,1387.8 2167.47,1387.8 \n",
       "  2169.53,1387.81 2171.58,1387.81 2173.64,1387.82 2175.7,1387.82 2177.76,1387.83 2179.82,1387.83 2181.88,1387.83 2183.94,1387.84 2186,1387.84 2188.05,1387.85 \n",
       "  2190.11,1387.85 2192.17,1387.86 2194.23,1387.86 2196.29,1387.86 2198.35,1387.87 2200.41,1387.87 2202.47,1387.88 2204.52,1387.88 2206.58,1387.88 2208.64,1387.89 \n",
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       "  1819.53,1349.43 1821.59,1349.62 1823.65,1349.81 1825.71,1349.99 1827.77,1350.18 1829.83,1350.36 1831.89,1350.55 1833.94,1350.73 1836,1350.91 1838.06,1351.09 \n",
       "  1840.12,1351.27 1842.18,1351.45 1844.24,1351.63 1846.3,1351.8 1848.36,1351.98 1850.42,1352.16 1852.47,1352.33 1854.53,1352.5 1856.59,1352.68 1858.65,1352.85 \n",
       "  1860.71,1353.02 1862.77,1353.19 1864.83,1353.36 1866.89,1353.53 1868.94,1353.7 1871,1353.86 1873.06,1354.03 1875.12,1354.19 1877.18,1354.36 1879.24,1354.52 \n",
       "  1881.3,1354.68 1883.36,1354.85 1885.41,1355.01 1887.47,1355.17 1889.53,1355.33 1891.59,1355.48 1893.65,1355.64 1895.71,1355.8 1897.77,1355.95 1899.83,1356.11 \n",
       "  1901.88,1356.26 1903.94,1356.41 1906,1356.57 1908.06,1356.72 1910.12,1356.87 1912.18,1357.02 1914.24,1357.17 1916.3,1357.32 1918.35,1357.47 1920.41,1357.61 \n",
       "  1922.47,1357.76 1924.53,1357.91 1926.59,1358.05 1928.65,1358.2 1930.71,1358.34 1932.77,1358.49 1934.82,1358.63 1936.88,1358.78 1938.94,1358.92 1941,1359.06 \n",
       "  1943.06,1359.2 1945.12,1359.34 1947.18,1359.48 1949.24,1359.62 1951.3,1359.76 1953.35,1359.9 1955.41,1360.04 1957.47,1360.17 1959.53,1360.31 1961.59,1360.45 \n",
       "  1963.65,1360.58 1965.71,1360.72 1967.77,1360.85 1969.82,1360.98 1971.88,1361.12 1973.94,1361.25 1976,1361.38 1978.06,1361.51 1980.12,1361.64 1982.18,1361.77 \n",
       "  1984.24,1361.9 1986.29,1362.03 1988.35,1362.16 1990.41,1362.28 1992.47,1362.41 1994.53,1362.54 1996.59,1362.66 1998.65,1362.79 2000.71,1362.91 2002.76,1363.03 \n",
       "  2004.82,1363.16 2006.88,1363.28 2008.94,1363.4 2011,1363.52 2013.06,1363.64 2015.12,1363.76 2017.18,1363.88 2019.23,1364 2021.29,1364.12 2023.35,1364.23 \n",
       "  2025.41,1364.35 2027.47,1364.47 2029.53,1364.58 2031.59,1364.7 2033.65,1364.81 2035.7,1364.93 2037.76,1365.04 2039.82,1365.15 2041.88,1365.26 2043.94,1365.38 \n",
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       "  2066.59,1366.56 2068.65,1366.66 2070.7,1366.77 2072.76,1366.87 2074.82,1366.97 2076.88,1367.08 2078.94,1367.18 2081,1367.28 2083.06,1367.38 2085.12,1367.49 \n",
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       "  2190.11,1372.1 2192.17,1372.18 2194.23,1372.26 2196.29,1372.33 2198.35,1372.41 2200.41,1372.49 2202.47,1372.57 2204.52,1372.64 2206.58,1372.72 2208.64,1372.8 \n",
       "  2210.7,1372.87 2212.76,1372.95 2214.82,1373.02 2216.88,1373.09 2218.94,1373.17 2220.99,1373.24 2223.05,1373.32 2225.11,1373.39 2227.17,1373.46 2229.23,1373.53 \n",
       "  2231.29,1373.6 2233.35,1373.68 2235.41,1373.75 2237.46,1373.82 2239.52,1373.89 2241.58,1373.96 2243.64,1374.03 2245.7,1374.1 2247.76,1374.17 2249.82,1374.24 \n",
       "  2251.88,1374.31 2253.93,1374.38 2255.99,1374.44 2258.05,1374.51 2260.11,1374.58 2262.17,1374.65 2264.23,1374.71 2266.29,1374.78 2268.35,1374.85 2270.4,1374.91 \n",
       "  2272.46,1374.98 2274.52,1375.05 2276.58,1375.11 2278.64,1375.18 2280.7,1375.24 2282.76,1375.31 2284.82,1375.37 2286.88,1375.43 2288.93,1375.5 2290.99,1375.56 \n",
       "  2293.05,1375.62 2295.11,1375.69 2297.17,1375.75 2299.23,1375.81 2301.29,1375.87 2303.35,1375.93 2305.4,1376 2307.46,1376.06 2309.52,1376.12 2311.58,1376.18 \n",
       "  2313.64,1376.24 2315.7,1376.3 2317.76,1376.36 2319.82,1376.42 2321.87,1376.47 2323.93,1376.53 2325.99,1376.59 2328.05,1376.65 2330.11,1376.71 2332.17,1376.76 \n",
       "  2334.23,1376.82 2336.29,1376.88 2338.34,1376.93 2340.4,1376.99 2342.46,1377.05 2344.52,1377.1 2346.58,1377.16 2348.64,1377.21 2350.7,1377.27 2352.76,1377.32 \n",
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     "execution_count": 44,
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   "source": [
    "sol_ = solve(prob)\n",
    "plot(sol_)\n",
    "e=[p[4]-u[2] for (u,t) in tuples(sol_)]\n",
    "plot!(sol_.t,e,\n",
    "        label=\"e\",\n",
    "        linealpha = 1,\n",
    "        linewidth = 3,\n",
    "        linecolor = :purple)\n",
    "title!(\"Enzyme-substrate kinetics simplified using mass action\")\n",
    "xlabel!(\"Time [s]\")\n",
    "ylabel!(\"Concentration [M]\")"
   ]
  },
  {
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   "source": [
    "## Quasi steady state approximation:\n",
    "\n",
    "The other conservation law could be used to reduce teh system to just two equations, but instead, we will use another approiximation that allows us to go further and reduce the system to a single differential equation. Thsi aproximation si swidely used in all biochemistry books and has been famously proposed by Miachelis and Menten. In briefm it assumes that the formation and degradation of the intermediate complex is often very fast and reaches equilibrium early in the reaction. \n",
    "\n",
    "$$k_1\\cdot x_1(0) ≈ k_2 >> k_3 \\tag{18}$$ \n",
    "\n",
    "\n",
    "The substrate-enzyme binding occurs at much faster time scales than the turnover into product (often the case in biologically relevant biochemical reactions)\n",
    "\n",
    "Under these circumstances one expects that after an initial short transient period there will be a balance between the formation of the enzyme-substrate complex and the breaking apart of complex (either to enzyme and substrate, or to enzyme and product) \n",
    "\n",
    "$$ \\frac{d c}{dt}= \\frac{d e}{dt}=0 \\tag{19}$$\n",
    "\n",
    "In these conditions, we can calculate the equilibrium for $c$ from eq. 2.16 as \n",
    "\n",
    "$$c_{eq}=\\frac{k_1\\cdot e_0\\cdot s}{k_3+k_2+k_1 \\cdot s} \\tag{20}$$\n",
    "\n",
    "This is called the __pseudo-steady state aproximation__ for biochemical systems"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
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       "<g clip-path=\"url(#clip8000)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 1530.07, 1479.62)\" x=\"1530.07\" y=\"1479.62\">60</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip8000)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 1941.41, 1479.62)\" x=\"1941.41\" y=\"1479.62\">80</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip8000)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:middle;\" transform=\"rotate(0, 2352.76, 1479.62)\" x=\"2352.76\" y=\"1479.62\">100</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip8000)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 272.041, 1406.21)\" x=\"272.041\" y=\"1406.21\">0.0000</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip8000)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 272.041, 1098.68)\" x=\"272.041\" y=\"1098.68\">0.0005</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip8000)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 272.041, 791.147)\" x=\"272.041\" y=\"791.147\">0.0010</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip8000)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 272.041, 483.613)\" x=\"272.041\" y=\"483.613\">0.0015</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip8000)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 272.041, 176.079)\" x=\"272.041\" y=\"176.079\">0.0020</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip8000)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:84px; text-anchor:middle;\" transform=\"rotate(0, 1324.4, 73.2)\" x=\"1324.4\" y=\"73.2\">Enzyme-substrate kinetics simplified using mass action</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip8000)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:66px; text-anchor:middle;\" transform=\"rotate(0, 1324.4, 1559.48)\" x=\"1324.4\" y=\"1559.48\">Time [s]</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip8000)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:66px; text-anchor:middle;\" transform=\"rotate(-90, 89.2861, 773.647)\" x=\"89.2861\" y=\"773.647\">Concentration [M]</text>\n",
       "</g>\n",
       "<polyline clip-path=\"url(#clip8002)\" style=\"stroke:#009af9; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n",
       "  296.041,158.579 298.1,362.839 300.159,507.327 302.217,614.669 304.276,697.399 306.335,762.944 308.394,816.074 310.453,859.93 312.511,896.654 314.57,927.831 \n",
       "  316.629,954.6 318.688,977.793 320.746,998.047 322.805,1015.88 324.864,1031.7 326.923,1045.81 328.981,1058.47 331.04,1069.88 333.099,1080.22 335.158,1089.63 \n",
       "  337.217,1098.25 339.275,1106.16 341.334,1113.44 343.393,1120.19 345.452,1126.44 347.51,1132.27 349.569,1137.72 351.628,1142.83 353.687,1147.64 355.746,1152.17 \n",
       "  357.804,1156.46 359.863,1160.52 361.922,1164.38 363.981,1168.06 366.039,1171.57 368.098,1174.94 370.157,1178.17 372.216,1181.28 374.275,1184.27 376.333,1187.16 \n",
       "  378.392,1189.94 380.451,1192.64 382.51,1195.25 384.568,1197.79 386.627,1200.26 388.686,1202.66 390.745,1205 392.803,1207.28 394.862,1209.52 396.921,1211.7 \n",
       "  398.98,1213.83 401.039,1215.91 403.097,1217.96 405.156,1219.96 407.215,1221.92 409.274,1223.84 411.332,1225.73 413.391,1227.59 415.45,1229.42 417.509,1231.22 \n",
       "  419.568,1232.99 421.626,1234.74 423.685,1236.45 425.744,1238.14 427.803,1239.8 429.861,1241.44 431.92,1243.05 433.979,1244.63 436.038,1246.19 438.096,1247.73 \n",
       "  440.155,1249.25 442.214,1250.75 444.273,1252.23 446.332,1253.69 448.39,1255.14 450.449,1256.56 452.508,1257.97 454.567,1259.35 456.625,1260.72 458.684,1262.07 \n",
       "  460.743,1263.39 462.802,1264.7 464.861,1265.99 466.919,1267.27 468.978,1268.52 471.037,1269.76 473.096,1270.98 475.154,1272.19 477.213,1273.38 479.272,1274.57 \n",
       "  481.331,1275.74 483.389,1276.9 485.448,1278.04 487.507,1279.17 489.566,1280.28 491.625,1281.38 493.683,1282.46 495.742,1283.53 497.801,1284.58 499.86,1285.62 \n",
       "  501.918,1286.64 503.977,1287.64 506.036,1288.63 508.095,1289.61 510.154,1290.58 512.212,1291.53 514.271,1292.48 516.33,1293.42 518.389,1294.35 520.447,1295.27 \n",
       "  522.506,1296.18 524.565,1297.08 526.624,1297.98 528.683,1298.86 530.741,1299.73 532.8,1300.58 534.859,1301.42 536.918,1302.25 538.976,1303.07 541.035,1303.87 \n",
       "  543.094,1304.66 545.153,1305.43 547.211,1306.2 549.27,1306.95 551.329,1307.7 553.388,1308.44 555.447,1309.17 557.505,1309.9 559.564,1310.62 561.623,1311.34 \n",
       "  563.682,1312.06 565.74,1312.77 567.799,1313.48 569.858,1314.17 571.917,1314.86 573.976,1315.53 576.034,1316.2 578.093,1316.85 580.152,1317.49 582.211,1318.12 \n",
       "  584.269,1318.74 586.328,1319.35 588.387,1319.94 590.446,1320.53 592.504,1321.11 594.563,1321.68 596.622,1322.25 598.681,1322.81 600.74,1323.37 602.798,1323.92 \n",
       "  604.857,1324.49 606.916,1325.05 608.975,1325.61 611.033,1326.17 613.092,1326.72 615.151,1327.27 617.21,1327.81 619.269,1328.34 621.327,1328.86 623.386,1329.37 \n",
       "  625.445,1329.87 627.504,1330.35 629.562,1330.83 631.621,1331.3 633.68,1331.75 635.739,1332.2 637.798,1332.64 639.856,1333.08 641.915,1333.51 643.974,1333.93 \n",
       "  646.033,1334.36 648.091,1334.79 650.15,1335.22 652.209,1335.66 654.268,1336.1 656.326,1336.55 658.385,1336.99 660.444,1337.43 662.503,1337.87 664.562,1338.3 \n",
       "  666.62,1338.72 668.679,1339.13 670.738,1339.54 672.797,1339.93 674.855,1340.31 676.914,1340.68 678.973,1341.04 681.032,1341.38 683.091,1341.72 685.149,1342.06 \n",
       "  687.208,1342.38 689.267,1342.71 691.326,1343.03 693.384,1343.35 695.443,1343.67 697.502,1344 699.561,1344.34 701.619,1344.7 703.678,1345.06 705.737,1345.42 \n",
       "  707.796,1345.79 709.855,1346.15 711.913,1346.51 713.972,1346.86 716.031,1347.2 718.09,1347.53 720.148,1347.85 722.207,1348.16 724.266,1348.45 726.325,1348.74 \n",
       "  728.384,1349.01 730.442,1349.27 732.501,1349.52 734.56,1349.76 736.619,1350 738.677,1350.24 740.736,1350.47 742.795,1350.71 744.854,1350.95 746.913,1351.21 \n",
       "  748.971,1351.48 751.03,1351.77 753.089,1352.07 755.148,1352.38 757.206,1352.7 759.265,1353.01 761.324,1353.32 763.383,1353.62 765.441,1353.91 767.5,1354.19 \n",
       "  769.559,1354.45 771.618,1354.7 773.677,1354.93 775.735,1355.15 777.794,1355.35 779.853,1355.54 781.912,1355.72 783.97,1355.89 786.029,1356.05 788.088,1356.21 \n",
       "  790.147,1356.38 792.206,1356.55 794.264,1356.73 796.323,1356.93 798.382,1357.15 800.441,1357.4 802.499,1357.67 804.558,1357.95 806.617,1358.23 808.676,1358.52 \n",
       "  810.734,1358.8 812.793,1359.06 814.852,1359.32 816.911,1359.56 818.97,1359.77 821.028,1359.97 823.087,1360.15 825.146,1360.31 827.205,1360.46 829.263,1360.58 \n",
       "  831.322,1360.69 833.381,1360.8 835.44,1360.9 837.499,1361 839.557,1361.1 841.616,1361.22 843.675,1361.36 845.734,1361.53 847.792,1361.74 849.851,1361.98 \n",
       "  851.91,1362.24 853.969,1362.51 856.028,1362.78 858.086,1363.04 860.145,1363.29 862.204,1363.53 864.263,1363.74 866.321,1363.93 868.38,1364.1 870.439,1364.25 \n",
       "  872.498,1364.36 874.556,1364.46 876.615,1364.54 878.674,1364.6 880.733,1364.66 882.792,1364.71 884.85,1364.76 886.909,1364.82 888.968,1364.91 891.027,1365.02 \n",
       "  893.085,1365.18 895.144,1365.39 897.203,1365.63 899.262,1365.88 901.321,1366.13 903.379,1366.38 905.438,1366.62 907.497,1366.84 909.556,1367.04 911.614,1367.22 \n",
       "  913.673,1367.37 915.732,1367.49 917.791,1367.58 919.849,1367.65 921.908,1367.69 923.967,1367.72 926.026,1367.74 928.085,1367.76 930.143,1367.78 932.202,1367.82 \n",
       "  934.261,1367.89 936.32,1367.99 938.378,1368.15 940.437,1368.36 942.496,1368.58 944.555,1368.82 946.614,1369.05 948.672,1369.28 950.731,1369.49 952.79,1369.68 \n",
       "  954.849,1369.84 956.907,1369.98 958.966,1370.08 961.025,1370.16 963.084,1370.22 965.143,1370.25 967.201,1370.26 969.26,1370.27 971.319,1370.26 973.378,1370.27 \n",
       "  975.436,1370.29 977.495,1370.34 979.554,1370.44 981.613,1370.59 983.671,1370.78 985.73,1370.99 987.789,1371.2 989.848,1371.42 991.907,1371.62 993.965,1371.8 \n",
       "  996.024,1371.96 998.083,1372.09 1000.14,1372.2 1002.2,1372.28 1004.26,1372.34 1006.32,1372.37 1008.38,1372.38 1010.44,1372.38 1012.49,1372.37 1014.55,1372.36 \n",
       "  1016.61,1372.37 1018.67,1372.4 1020.73,1372.47 1022.79,1372.59 1024.85,1372.75 1026.91,1372.94 1028.96,1373.13 1031.02,1373.32 1033.08,1373.51 1035.14,1373.67 \n",
       "  1037.2,1373.82 1039.26,1373.94 1041.32,1374.04 1043.38,1374.11 1045.43,1374.15 1047.49,1374.17 1049.55,1374.18 1051.61,1374.17 1053.67,1374.15 1055.73,1374.14 \n",
       "  1057.79,1374.15 1059.85,1374.17 1061.9,1374.24 1063.96,1374.35 1066.02,1374.51 1068.08,1374.68 1070.14,1374.85 1072.2,1375.03 1074.26,1375.19 1076.32,1375.33 \n",
       "  1078.38,1375.46 1080.43,1375.56 1082.49,1375.64 1084.55,1375.69 1086.61,1375.72 1088.67,1375.73 1090.73,1375.72 1092.79,1375.7 1094.85,1375.69 1096.9,1375.67 \n",
       "  1098.96,1375.68 1101.02,1375.71 1103.08,1375.79 1105.14,1375.92 1107.2,1376.07 1109.26,1376.23 1111.32,1376.39 1113.37,1376.54 1115.43,1376.68 1117.49,1376.81 \n",
       "  1119.55,1376.91 1121.61,1376.98 1123.67,1377.04 1125.73,1377.07 1127.79,1377.08 1129.84,1377.07 1131.9,1377.05 1133.96,1377.03 1136.02,1377.01 1138.08,1377 \n",
       "  1140.14,1377.02 1142.2,1377.07 1144.26,1377.17 1146.31,1377.26 1148.37,1377.35 1150.43,1377.44 1152.49,1377.52 1154.55,1377.61 1156.61,1377.69 1158.67,1377.77 \n",
       "  1160.73,1377.86 1162.78,1377.94 1164.84,1378.01 1166.9,1378.09 1168.96,1378.17 1171.02,1378.24 1173.08,1378.31 1175.14,1378.38 1177.2,1378.45 1179.25,1378.52 \n",
       "  1181.31,1378.59 1183.37,1378.65 1185.43,1378.72 1187.49,1378.78 1189.55,1378.84 1191.61,1378.9 1193.67,1378.96 1195.73,1379.01 1197.78,1379.07 1199.84,1379.12 \n",
       "  1201.9,1379.18 1203.96,1379.23 1206.02,1379.28 1208.08,1379.32 1210.14,1379.37 1212.2,1379.42 1214.25,1379.46 1216.31,1379.5 1218.37,1379.54 1220.43,1379.58 \n",
       "  1222.49,1379.62 1224.55,1379.66 1226.61,1379.71 1228.67,1379.75 1230.72,1379.8 1232.78,1379.84 1234.84,1379.88 1236.9,1379.93 1238.96,1379.97 1241.02,1380.01 \n",
       "  1243.08,1380.05 1245.14,1380.1 1247.19,1380.14 1249.25,1380.18 1251.31,1380.23 1253.37,1380.27 1255.43,1380.31 1257.49,1380.35 1259.55,1380.4 1261.61,1380.44 \n",
       "  1263.66,1380.48 1265.72,1380.52 1267.78,1380.57 1269.84,1380.61 1271.9,1380.65 1273.96,1380.69 1276.02,1380.74 1278.08,1380.78 1280.13,1380.82 1282.19,1380.86 \n",
       "  1284.25,1380.9 1286.31,1380.95 1288.37,1380.99 1290.43,1381.03 1292.49,1381.07 1294.55,1381.11 1296.61,1381.15 1298.66,1381.2 1300.72,1381.24 1302.78,1381.28 \n",
       "  1304.84,1381.32 1306.9,1381.36 1308.96,1381.4 1311.02,1381.44 1313.08,1381.48 1315.13,1381.52 1317.19,1381.55 1319.25,1381.59 1321.31,1381.63 1323.37,1381.67 \n",
       "  1325.43,1381.71 1327.49,1381.74 1329.55,1381.78 1331.6,1381.82 1333.66,1381.86 1335.72,1381.89 1337.78,1381.93 1339.84,1381.96 1341.9,1382 1343.96,1382.04 \n",
       "  1346.02,1382.07 1348.07,1382.11 1350.13,1382.14 1352.19,1382.18 1354.25,1382.21 1356.31,1382.25 1358.37,1382.28 1360.43,1382.31 1362.49,1382.35 1364.54,1382.38 \n",
       "  1366.6,1382.41 1368.66,1382.45 1370.72,1382.48 1372.78,1382.51 1374.84,1382.54 1376.9,1382.58 1378.96,1382.61 1381.01,1382.64 1383.07,1382.67 1385.13,1382.7 \n",
       "  1387.19,1382.73 1389.25,1382.76 1391.31,1382.79 1393.37,1382.82 1395.43,1382.85 1397.48,1382.88 1399.54,1382.91 1401.6,1382.94 1403.66,1382.97 1405.72,1383 \n",
       "  1407.78,1383.03 1409.84,1383.06 1411.9,1383.09 1413.96,1383.12 1416.01,1383.14 1418.07,1383.17 1420.13,1383.2 1422.19,1383.23 1424.25,1383.26 1426.31,1383.29 \n",
       "  1428.37,1383.31 1430.43,1383.34 1432.48,1383.37 1434.54,1383.4 1436.6,1383.42 1438.66,1383.45 1440.72,1383.48 1442.78,1383.51 1444.84,1383.53 1446.9,1383.56 \n",
       "  1448.95,1383.59 1451.01,1383.61 1453.07,1383.64 1455.13,1383.66 1457.19,1383.69 1459.25,1383.72 1461.31,1383.74 1463.37,1383.77 1465.42,1383.79 1467.48,1383.82 \n",
       "  1469.54,1383.84 1471.6,1383.87 1473.66,1383.89 1475.72,1383.92 1477.78,1383.94 1479.84,1383.96 1481.89,1383.99 1483.95,1384.01 1486.01,1384.04 1488.07,1384.06 \n",
       "  1490.13,1384.08 1492.19,1384.11 1494.25,1384.13 1496.31,1384.15 1498.36,1384.18 1500.42,1384.2 1502.48,1384.22 1504.54,1384.25 1506.6,1384.27 1508.66,1384.29 \n",
       "  1510.72,1384.31 1512.78,1384.34 1514.84,1384.36 1516.89,1384.38 1518.95,1384.4 1521.01,1384.42 1523.07,1384.45 1525.13,1384.47 1527.19,1384.49 1529.25,1384.51 \n",
       "  1531.31,1384.53 1533.36,1384.55 1535.42,1384.57 1537.48,1384.59 1539.54,1384.62 1541.6,1384.64 1543.66,1384.66 1545.72,1384.68 1547.78,1384.7 1549.83,1384.72 \n",
       "  1551.89,1384.74 1553.95,1384.76 1556.01,1384.78 1558.07,1384.8 1560.13,1384.82 1562.19,1384.84 1564.25,1384.86 1566.3,1384.88 1568.36,1384.9 1570.42,1384.92 \n",
       "  1572.48,1384.94 1574.54,1384.95 1576.6,1384.97 1578.66,1384.99 1580.72,1385.01 1582.77,1385.03 1584.83,1385.05 1586.89,1385.07 1588.95,1385.09 1591.01,1385.1 \n",
       "  1593.07,1385.12 1595.13,1385.14 1597.19,1385.16 1599.24,1385.18 1601.3,1385.19 1603.36,1385.21 1605.42,1385.23 1607.48,1385.25 1609.54,1385.26 1611.6,1385.28 \n",
       "  1613.66,1385.3 1615.71,1385.31 1617.77,1385.33 1619.83,1385.35 1621.89,1385.36 1623.95,1385.38 1626.01,1385.4 1628.07,1385.41 1630.13,1385.43 1632.19,1385.45 \n",
       "  1634.24,1385.46 1636.3,1385.48 1638.36,1385.49 1640.42,1385.51 1642.48,1385.53 1644.54,1385.54 1646.6,1385.56 1648.66,1385.57 1650.71,1385.59 1652.77,1385.61 \n",
       "  1654.83,1385.62 1656.89,1385.64 1658.95,1385.65 1661.01,1385.67 1663.07,1385.68 1665.13,1385.7 1667.18,1385.71 1669.24,1385.73 1671.3,1385.74 1673.36,1385.76 \n",
       "  1675.42,1385.77 1677.48,1385.79 1679.54,1385.8 1681.6,1385.82 1683.65,1385.83 1685.71,1385.84 1687.77,1385.86 1689.83,1385.87 1691.89,1385.89 1693.95,1385.9 \n",
       "  1696.01,1385.92 1698.07,1385.93 1700.12,1385.94 1702.18,1385.96 1704.24,1385.97 1706.3,1385.98 1708.36,1386 1710.42,1386.01 1712.48,1386.03 1714.54,1386.04 \n",
       "  1716.59,1386.05 1718.65,1386.07 1720.71,1386.08 1722.77,1386.09 1724.83,1386.1 1726.89,1386.12 1728.95,1386.13 1731.01,1386.14 1733.07,1386.16 1735.12,1386.17 \n",
       "  1737.18,1386.18 1739.24,1386.19 1741.3,1386.21 1743.36,1386.22 1745.42,1386.23 1747.48,1386.24 1749.54,1386.25 1751.59,1386.27 1753.65,1386.28 1755.71,1386.29 \n",
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       "  1778.36,1386.42 1780.42,1386.43 1782.48,1386.44 1784.53,1386.45 1786.59,1386.46 1788.65,1386.47 1790.71,1386.49 1792.77,1386.5 1794.83,1386.51 1796.89,1386.52 \n",
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       "  1819.53,1386.63 1821.59,1386.65 1823.65,1386.66 1825.71,1386.67 1827.77,1386.68 1829.83,1386.69 1831.89,1386.7 1833.94,1386.71 1836,1386.72 1838.06,1386.73 \n",
       "  1840.12,1386.74 1842.18,1386.74 1844.24,1386.75 1846.3,1386.76 1848.36,1386.77 1850.42,1386.78 1852.47,1386.79 1854.53,1386.8 1856.59,1386.81 1858.65,1386.82 \n",
       "  1860.71,1386.83 1862.77,1386.84 1864.83,1386.85 1866.89,1386.86 1868.94,1386.87 1871,1386.88 1873.06,1386.89 1875.12,1386.89 1877.18,1386.9 1879.24,1386.91 \n",
       "  1881.3,1386.92 1883.36,1386.93 1885.41,1386.94 1887.47,1386.95 1889.53,1386.96 1891.59,1386.96 1893.65,1386.97 1895.71,1386.98 1897.77,1386.99 1899.83,1387 \n",
       "  1901.88,1387.01 1903.94,1387.01 1906,1387.02 1908.06,1387.03 1910.12,1387.04 1912.18,1387.05 1914.24,1387.05 1916.3,1387.06 1918.35,1387.07 1920.41,1387.08 \n",
       "  1922.47,1387.09 1924.53,1387.1 1926.59,1387.1 1928.65,1387.11 1930.71,1387.12 1932.77,1387.13 1934.82,1387.13 1936.88,1387.14 1938.94,1387.15 1941,1387.16 \n",
       "  1943.06,1387.16 1945.12,1387.17 1947.18,1387.18 1949.24,1387.19 1951.3,1387.2 1953.35,1387.2 1955.41,1387.21 1957.47,1387.22 1959.53,1387.22 1961.59,1387.23 \n",
       "  1963.65,1387.24 1965.71,1387.25 1967.77,1387.25 1969.82,1387.26 1971.88,1387.27 1973.94,1387.28 1976,1387.28 1978.06,1387.29 1980.12,1387.3 1982.18,1387.3 \n",
       "  1984.24,1387.31 1986.29,1387.32 1988.35,1387.32 1990.41,1387.33 1992.47,1387.34 1994.53,1387.34 1996.59,1387.35 1998.65,1387.36 2000.71,1387.36 2002.76,1387.37 \n",
       "  2004.82,1387.38 2006.88,1387.38 2008.94,1387.39 2011,1387.4 2013.06,1387.4 2015.12,1387.41 2017.18,1387.42 2019.23,1387.42 2021.29,1387.43 2023.35,1387.43 \n",
       "  2025.41,1387.44 2027.47,1387.45 2029.53,1387.45 2031.59,1387.46 2033.65,1387.47 2035.7,1387.47 2037.76,1387.48 2039.82,1387.48 2041.88,1387.49 2043.94,1387.5 \n",
       "  2046,1387.5 2048.06,1387.51 2050.12,1387.51 2052.17,1387.52 2054.23,1387.52 2056.29,1387.53 2058.35,1387.54 2060.41,1387.54 2062.47,1387.55 2064.53,1387.55 \n",
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       "  2087.17,1387.61 2089.23,1387.62 2091.29,1387.62 2093.35,1387.63 2095.41,1387.63 2097.47,1387.64 2099.53,1387.64 2101.59,1387.65 2103.64,1387.66 2105.7,1387.66 \n",
       "  2107.76,1387.67 2109.82,1387.67 2111.88,1387.68 2113.94,1387.68 2116,1387.69 2118.06,1387.69 2120.11,1387.7 2122.17,1387.7 2124.23,1387.71 2126.29,1387.71 \n",
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       "  2190.11,1387.85 2192.17,1387.86 2194.23,1387.86 2196.29,1387.86 2198.35,1387.87 2200.41,1387.87 2202.47,1387.88 2204.52,1387.88 2206.58,1387.88 2208.64,1387.89 \n",
       "  2210.7,1387.89 2212.76,1387.9 2214.82,1387.9 2216.88,1387.9 2218.94,1387.91 2220.99,1387.91 2223.05,1387.92 2225.11,1387.92 2227.17,1387.92 2229.23,1387.93 \n",
       "  2231.29,1387.93 2233.35,1387.93 2235.41,1387.94 2237.46,1387.94 2239.52,1387.95 2241.58,1387.95 2243.64,1387.95 2245.7,1387.96 2247.76,1387.96 2249.82,1387.96 \n",
       "  2251.88,1387.97 2253.93,1387.97 2255.99,1387.97 2258.05,1387.98 2260.11,1387.98 2262.17,1387.99 2264.23,1387.99 2266.29,1387.99 2268.35,1388 2270.4,1388 \n",
       "  2272.46,1388 2274.52,1388.01 2276.58,1388.01 2278.64,1388.01 2280.7,1388.02 2282.76,1388.02 2284.82,1388.02 2286.88,1388.03 2288.93,1388.03 2290.99,1388.03 \n",
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       "  2313.64,1388.07 2315.7,1388.07 2317.76,1388.07 2319.82,1388.08 2321.87,1388.08 2323.93,1388.08 2325.99,1388.09 2328.05,1388.09 2330.11,1388.09 2332.17,1388.1 \n",
       "  2334.23,1388.1 2336.29,1388.1 2338.34,1388.11 2340.4,1388.11 2342.46,1388.11 2344.52,1388.11 2346.58,1388.12 2348.64,1388.12 2350.7,1388.12 2352.76,1388.13 \n",
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       "  1593.07,1322.45 1595.13,1322.76 1597.19,1323.08 1599.24,1323.39 1601.3,1323.7 1603.36,1324.01 1605.42,1324.32 1607.48,1324.62 1609.54,1324.92 1611.6,1325.23 \n",
       "  1613.66,1325.53 1615.71,1325.83 1617.77,1326.13 1619.83,1326.42 1621.89,1326.72 1623.95,1327.01 1626.01,1327.3 1628.07,1327.59 1630.13,1327.88 1632.19,1328.16 \n",
       "  1634.24,1328.45 1636.3,1328.73 1638.36,1329.01 1640.42,1329.3 1642.48,1329.58 1644.54,1329.86 1646.6,1330.13 1648.66,1330.41 1650.71,1330.69 1652.77,1330.96 \n",
       "  1654.83,1331.23 1656.89,1331.51 1658.95,1331.78 1661.01,1332.05 1663.07,1332.31 1665.13,1332.58 1667.18,1332.85 1669.24,1333.11 1671.3,1333.38 1673.36,1333.64 \n",
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       "  1696.01,1336.45 1698.07,1336.7 1700.12,1336.95 1702.18,1337.2 1704.24,1337.44 1706.3,1337.69 1708.36,1337.93 1710.42,1338.17 1712.48,1338.41 1714.54,1338.65 \n",
       "  1716.59,1338.89 1718.65,1339.13 1720.71,1339.36 1722.77,1339.6 1724.83,1339.83 1726.89,1340.07 1728.95,1340.3 1731.01,1340.53 1733.07,1340.76 1735.12,1340.99 \n",
       "  1737.18,1341.21 1739.24,1341.44 1741.3,1341.66 1743.36,1341.89 1745.42,1342.11 1747.48,1342.33 1749.54,1342.55 1751.59,1342.77 1753.65,1342.99 1755.71,1343.2 \n",
       "  1757.77,1343.41 1759.83,1343.63 1761.89,1343.84 1763.95,1344.05 1766.01,1344.26 1768.06,1344.47 1770.12,1344.68 1772.18,1344.89 1774.24,1345.09 1776.3,1345.3 \n",
       "  1778.36,1345.51 1780.42,1345.71 1782.48,1345.91 1784.53,1346.12 1786.59,1346.32 1788.65,1346.52 1790.71,1346.72 1792.77,1346.92 1794.83,1347.12 1796.89,1347.32 \n",
       "  1798.95,1347.51 1801,1347.71 1803.06,1347.9 1805.12,1348.1 1807.18,1348.29 1809.24,1348.48 1811.3,1348.68 1813.36,1348.87 1815.42,1349.06 1817.47,1349.25 \n",
       "  1819.53,1349.43 1821.59,1349.62 1823.65,1349.81 1825.71,1349.99 1827.77,1350.18 1829.83,1350.36 1831.89,1350.55 1833.94,1350.73 1836,1350.91 1838.06,1351.09 \n",
       "  1840.12,1351.27 1842.18,1351.45 1844.24,1351.63 1846.3,1351.8 1848.36,1351.98 1850.42,1352.16 1852.47,1352.33 1854.53,1352.5 1856.59,1352.68 1858.65,1352.85 \n",
       "  1860.71,1353.02 1862.77,1353.19 1864.83,1353.36 1866.89,1353.53 1868.94,1353.7 1871,1353.86 1873.06,1354.03 1875.12,1354.19 1877.18,1354.36 1879.24,1354.52 \n",
       "  1881.3,1354.68 1883.36,1354.85 1885.41,1355.01 1887.47,1355.17 1889.53,1355.33 1891.59,1355.48 1893.65,1355.64 1895.71,1355.8 1897.77,1355.95 1899.83,1356.11 \n",
       "  1901.88,1356.26 1903.94,1356.41 1906,1356.57 1908.06,1356.72 1910.12,1356.87 1912.18,1357.02 1914.24,1357.17 1916.3,1357.32 1918.35,1357.47 1920.41,1357.61 \n",
       "  1922.47,1357.76 1924.53,1357.91 1926.59,1358.05 1928.65,1358.2 1930.71,1358.34 1932.77,1358.49 1934.82,1358.63 1936.88,1358.78 1938.94,1358.92 1941,1359.06 \n",
       "  1943.06,1359.2 1945.12,1359.34 1947.18,1359.48 1949.24,1359.62 1951.3,1359.76 1953.35,1359.9 1955.41,1360.04 1957.47,1360.17 1959.53,1360.31 1961.59,1360.45 \n",
       "  1963.65,1360.58 1965.71,1360.72 1967.77,1360.85 1969.82,1360.98 1971.88,1361.12 1973.94,1361.25 1976,1361.38 1978.06,1361.51 1980.12,1361.64 1982.18,1361.77 \n",
       "  1984.24,1361.9 1986.29,1362.03 1988.35,1362.16 1990.41,1362.28 1992.47,1362.41 1994.53,1362.54 1996.59,1362.66 1998.65,1362.79 2000.71,1362.91 2002.76,1363.03 \n",
       "  2004.82,1363.16 2006.88,1363.28 2008.94,1363.4 2011,1363.52 2013.06,1363.64 2015.12,1363.76 2017.18,1363.88 2019.23,1364 2021.29,1364.12 2023.35,1364.23 \n",
       "  2025.41,1364.35 2027.47,1364.47 2029.53,1364.58 2031.59,1364.7 2033.65,1364.81 2035.7,1364.93 2037.76,1365.04 2039.82,1365.15 2041.88,1365.26 2043.94,1365.38 \n",
       "  2046,1365.49 2048.06,1365.59 2050.12,1365.7 2052.17,1365.81 2054.23,1365.92 2056.29,1366.03 2058.35,1366.13 2060.41,1366.24 2062.47,1366.35 2064.53,1366.45 \n",
       "  2066.59,1366.56 2068.65,1366.66 2070.7,1366.77 2072.76,1366.87 2074.82,1366.97 2076.88,1367.08 2078.94,1367.18 2081,1367.28 2083.06,1367.38 2085.12,1367.49 \n",
       "  2087.17,1367.59 2089.23,1367.69 2091.29,1367.79 2093.35,1367.89 2095.41,1367.99 2097.47,1368.08 2099.53,1368.18 2101.59,1368.28 2103.64,1368.38 2105.7,1368.48 \n",
       "  2107.76,1368.57 2109.82,1368.67 2111.88,1368.77 2113.94,1368.86 2116,1368.96 2118.06,1369.05 2120.11,1369.14 2122.17,1369.24 2124.23,1369.33 2126.29,1369.43 \n",
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       "  2169.53,1371.28 2171.58,1371.36 2173.64,1371.45 2175.7,1371.53 2177.76,1371.61 2179.82,1371.69 2181.88,1371.77 2183.94,1371.86 2186,1371.94 2188.05,1372.02 \n",
       "  2190.11,1372.1 2192.17,1372.18 2194.23,1372.26 2196.29,1372.33 2198.35,1372.41 2200.41,1372.49 2202.47,1372.57 2204.52,1372.64 2206.58,1372.72 2208.64,1372.8 \n",
       "  2210.7,1372.87 2212.76,1372.95 2214.82,1373.02 2216.88,1373.09 2218.94,1373.17 2220.99,1373.24 2223.05,1373.32 2225.11,1373.39 2227.17,1373.46 2229.23,1373.53 \n",
       "  2231.29,1373.6 2233.35,1373.68 2235.41,1373.75 2237.46,1373.82 2239.52,1373.89 2241.58,1373.96 2243.64,1374.03 2245.7,1374.1 2247.76,1374.17 2249.82,1374.24 \n",
       "  2251.88,1374.31 2253.93,1374.38 2255.99,1374.44 2258.05,1374.51 2260.11,1374.58 2262.17,1374.65 2264.23,1374.71 2266.29,1374.78 2268.35,1374.85 2270.4,1374.91 \n",
       "  2272.46,1374.98 2274.52,1375.05 2276.58,1375.11 2278.64,1375.18 2280.7,1375.24 2282.76,1375.31 2284.82,1375.37 2286.88,1375.43 2288.93,1375.5 2290.99,1375.56 \n",
       "  2293.05,1375.62 2295.11,1375.69 2297.17,1375.75 2299.23,1375.81 2301.29,1375.87 2303.35,1375.93 2305.4,1376 2307.46,1376.06 2309.52,1376.12 2311.58,1376.18 \n",
       "  2313.64,1376.24 2315.7,1376.3 2317.76,1376.36 2319.82,1376.42 2321.87,1376.47 2323.93,1376.53 2325.99,1376.59 2328.05,1376.65 2330.11,1376.71 2332.17,1376.76 \n",
       "  2334.23,1376.82 2336.29,1376.88 2338.34,1376.93 2340.4,1376.99 2342.46,1377.05 2344.52,1377.1 2346.58,1377.16 2348.64,1377.21 2350.7,1377.27 2352.76,1377.32 \n",
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       "  1551.89,235.436 1553.95,235.07 1556.01,234.707 1558.07,234.344 1560.13,233.984 1562.19,233.625 1564.25,233.268 1566.3,232.912 1568.36,232.558 1570.42,232.205 \n",
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       "  1654.83,219.154 1656.89,218.867 1658.95,218.58 1661.01,218.295 1663.07,218.011 1665.13,217.729 1667.18,217.447 1669.24,217.167 1671.3,216.888 1673.36,216.611 \n",
       "  1675.42,216.334 1677.48,216.059 1679.54,215.785 1681.6,215.512 1683.65,215.241 1685.71,214.971 1687.77,214.702 1689.83,214.434 1691.89,214.167 1693.95,213.902 \n",
       "  1696.01,213.638 1698.07,213.375 1700.12,213.113 1702.18,212.853 1704.24,212.594 1706.3,212.336 1708.36,212.079 1710.42,211.824 1712.48,211.57 1714.54,211.317 \n",
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       "  1778.36,204.084 1780.42,203.868 1782.48,203.653 1784.53,203.439 1786.59,203.226 1788.65,203.014 1790.71,202.802 1792.77,202.592 1794.83,202.382 1796.89,202.174 \n",
       "  1798.95,201.966 1801,201.759 1803.06,201.554 1805.12,201.349 1807.18,201.145 1809.24,200.941 1811.3,200.739 1813.36,200.538 1815.42,200.337 1817.47,200.138 \n",
       "  1819.53,199.939 1821.59,199.742 1823.65,199.545 1825.71,199.349 1827.77,199.154 1829.83,198.96 1831.89,198.767 1833.94,198.575 1836,198.383 1838.06,198.193 \n",
       "  1840.12,198.003 1842.18,197.815 1844.24,197.627 1846.3,197.44 1848.36,197.254 1850.42,197.069 1852.47,196.885 1854.53,196.702 1856.59,196.52 1858.65,196.338 \n",
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       "  1881.3,194.403 1883.36,194.233 1885.41,194.063 1887.47,193.894 1889.53,193.727 1891.59,193.56 1893.65,193.394 1895.71,193.23 1897.77,193.067 1899.83,192.905 \n",
       "  1901.88,192.743 1903.94,192.582 1906,192.421 1908.06,192.261 1910.12,192.102 1912.18,191.943 1914.24,191.785 1916.3,191.628 1918.35,191.471 1920.41,191.315 \n",
       "  1922.47,191.16 1924.53,191.005 1926.59,190.851 1928.65,190.698 1930.71,190.545 1932.77,190.393 1934.82,190.241 1936.88,190.09 1938.94,189.94 1941,189.79 \n",
       "  1943.06,189.641 1945.12,189.493 1947.18,189.346 1949.24,189.199 1951.3,189.052 1953.35,188.907 1955.41,188.762 1957.47,188.617 1959.53,188.473 1961.59,188.33 \n",
       "  1963.65,188.188 1965.71,188.046 1967.77,187.905 1969.82,187.765 1971.88,187.625 1973.94,187.485 1976,187.347 1978.06,187.209 1980.12,187.072 1982.18,186.935 \n",
       "  1984.24,186.799 1986.29,186.664 1988.35,186.529 1990.41,186.395 1992.47,186.262 1994.53,186.129 1996.59,185.997 1998.65,185.866 2000.71,185.735 2002.76,185.605 \n",
       "  2004.82,185.475 2006.88,185.346 2008.94,185.218 2011,185.091 2013.06,184.964 2015.12,184.837 2017.18,184.712 2019.23,184.587 2021.29,184.463 2023.35,184.339 \n",
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      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c_eq=[(p[1] * p[4] * u[1])/(p[3]+p[2]+p[1] * u[1]) for (u,t) in tuples(sol)]\n",
    "plot!(sol.t,c_eq,\n",
    "        label=\"c_eq\",        \n",
    "        linealpha = 0.5,\n",
    "        linewidth = 3,\n",
    "        linestyle= :dash,\n",
    "        linecolor = :red)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As we can see, the equilibrium solution $c_{eq}$ is quite close to the concentration of $c$, in this conditions we can simplify all equations as:\n",
    "  \n",
    "$$\\begin{align*}\n",
    "  \\dot{s}  &= (k_2+k_1 \\cdot s) \\cdot \\frac{k_1\\cdot e_0\\cdot s}{k_3+k_2+k_1 \\cdot s} -k_1 \\cdot e_0 \\cdot s \\tag{21}\\\\ \n",
    "  \\dot{p}  &= k_3 \\frac{k_1\\cdot e_0\\cdot s}{k_3+k_2+k_1 \\cdot s} \\tag{22}\n",
    "\\end{align*}$$\n",
    "\n",
    "Eq. 2.20 can be rewritten in a more simplified form, after some some algebraic manipulation:\n",
    "\n",
    "$$\\begin{align*}\n",
    "  \\dot{s}  &= (k_2+k_1 \\cdot s) \\cdot \\frac{k_1\\cdot e_0\\cdot s}{k_3+k_2+k_1 \\cdot s} -k_1 \\cdot e_0 \\cdot s \\tag{23}\\\\ \n",
    "   \\dot{s}  &= k_1 \\cdot e_0 \\cdot s  \\left [ \\frac{k_2+k_1 \\cdot s}{k_3+k_2+k_1 \\cdot s} -1 \\right ]  \\tag{24}\\\\ \n",
    "\\dot{s}  &= k_1 \\cdot e_0 \\cdot s  \\left [ \\frac{k_2+k_1 \\cdot s - k_3-k_2-k_1 \\cdot s}{k_3+k_2+k_1 \\cdot s} \\right ]  \\tag{25}\\\\ \n",
    "\\dot{s}  &=   \\left [ \\frac{- k_3\\cdot k_1 \\cdot e_0  \\cdot s}{k_3+k_2+k_1 \\cdot s} \\right ]  \\tag{26}\\\\ \n",
    "\\end{align*}$$\n",
    "\n",
    "So the final set of equations takes the form:\n",
    "\n",
    "$$\\begin{align*}\n",
    "  \\dot{s}  &=- \\frac{k_1 \\cdot k_3 \\cdot e_0 \\cdot s}{k_3+k_2+k_1 \\cdot s} \\tag{27}\\\\ \n",
    "  \\dot{p}  &=  \\frac{ k_3 \\cdot k_1\\cdot e_0\\cdot s}{k_3+k_2+k_1 \\cdot s} \\tag{28}\n",
    "\\end{align*}$$\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(::ab{var\"#79#83\",var\"#80#84\",var\"#81#85\",Nothing,Nothing,var\"#82#86\",Expr,Expr}) (generic function with 2 methods)"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "enzyme_kinetics10! = @ode_def ab begin\n",
    "     ds = - (k3 * e0 * s)/((k3+k2)/k1 +  s)\n",
    "     dp = (k3 * e0 * s)/((k3+k2)/k1 +  s)\n",
    "    end k1 k2 k3 e0\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\u001b[36mODEProblem\u001b[0m with uType \u001b[36mArray{Float64,1}\u001b[0m and tType \u001b[36mFloat64\u001b[0m. In-place: \u001b[36mtrue\u001b[0m\n",
       "timespan: (0.0, 100.0)\n",
       "u0: [0.002, 1.0e-5]"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tspan = (0.0,100.0)\n",
    "\n",
    "k1=1e3\n",
    "k2=0.1\n",
    "k3=0.05\n",
    "e0=0.002\n",
    "s0=0.002\n",
    "\n",
    "u0=[s0,0.00001]\n",
    "p=[k1,k2,k3,e0];\n",
    "prob10 = ODEProblem(enzyme_kinetics10!,u0,tspan,p)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
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       "  976.427,1385.12 978.548,1385.13 980.668,1385.13 982.789,1385.14 984.91,1385.14 987.031,1385.14 989.151,1385.14 991.272,1385.15 993.393,1385.15 995.513,1385.15 \n",
       "  997.634,1385.15 999.755,1385.15 1001.88,1385.15 1004,1385.16 1006.12,1385.16 1008.24,1385.16 1010.36,1385.16 1012.48,1385.16 1014.6,1385.16 1016.72,1385.16 \n",
       "  1018.84,1385.16 1020.96,1385.16 1023.08,1385.16 1025.2,1385.16 1027.32,1385.16 1029.44,1385.16 1031.57,1385.16 1033.69,1385.16 1035.81,1385.17 1037.93,1385.17 \n",
       "  1040.05,1385.17 1042.17,1385.17 1044.29,1385.17 1046.41,1385.17 1048.53,1385.17 1050.65,1385.17 1052.77,1385.18 1054.89,1385.18 1057.01,1385.18 1059.13,1385.18 \n",
       "  1061.25,1385.18 1063.38,1385.18 1065.5,1385.18 1067.62,1385.18 1069.74,1385.18 1071.86,1385.18 1073.98,1385.18 1076.1,1385.18 1078.22,1385.18 1080.34,1385.18 \n",
       "  1082.46,1385.18 1084.58,1385.18 1086.7,1385.18 1088.82,1385.18 1090.94,1385.18 1093.07,1385.18 1095.19,1385.18 1097.31,1385.18 1099.43,1385.17 1101.55,1385.17 \n",
       "  1103.67,1385.17 1105.79,1385.17 1107.91,1385.17 1110.03,1385.17 1112.15,1385.17 1114.27,1385.17 1116.39,1385.17 1118.51,1385.17 1120.63,1385.17 1122.75,1385.17 \n",
       "  1124.88,1385.18 1127,1385.18 1129.12,1385.18 1131.24,1385.18 1133.36,1385.18 1135.48,1385.18 1137.6,1385.18 1139.72,1385.18 1141.84,1385.18 1143.96,1385.18 \n",
       "  1146.08,1385.18 1148.2,1385.18 1150.32,1385.18 1152.44,1385.19 1154.57,1385.19 1156.69,1385.19 1158.81,1385.19 1160.93,1385.19 1163.05,1385.19 1165.17,1385.19 \n",
       "  1167.29,1385.19 1169.41,1385.19 1171.53,1385.19 1173.65,1385.19 1175.77,1385.19 1177.89,1385.19 1180.01,1385.19 1182.13,1385.18 1184.25,1385.18 1186.38,1385.18 \n",
       "  1188.5,1385.18 1190.62,1385.18 1192.74,1385.18 1194.86,1385.18 1196.98,1385.18 1199.1,1385.18 1201.22,1385.18 1203.34,1385.18 1205.46,1385.18 1207.58,1385.18 \n",
       "  1209.7,1385.17 1211.82,1385.17 1213.94,1385.17 1216.07,1385.17 1218.19,1385.17 1220.31,1385.17 1222.43,1385.17 1224.55,1385.17 1226.67,1385.17 1228.79,1385.17 \n",
       "  1230.91,1385.17 1233.03,1385.17 1235.15,1385.17 1237.27,1385.17 1239.39,1385.17 1241.51,1385.18 1243.63,1385.18 1245.75,1385.18 1247.88,1385.18 1250,1385.18 \n",
       "  1252.12,1385.19 1254.24,1385.19 1256.36,1385.19 1258.48,1385.19 1260.6,1385.2 1262.72,1385.2 1264.84,1385.2 1266.96,1385.2 1269.08,1385.21 1271.2,1385.21 \n",
       "  1273.32,1385.21 1275.44,1385.21 1277.57,1385.21 1279.69,1385.21 1281.81,1385.21 1283.93,1385.22 1286.05,1385.22 1288.17,1385.22 1290.29,1385.22 1292.41,1385.22 \n",
       "  1294.53,1385.22 1296.65,1385.22 1298.77,1385.22 1300.89,1385.21 1303.01,1385.21 1305.13,1385.21 1307.25,1385.21 1309.38,1385.21 1311.5,1385.21 1313.62,1385.2 \n",
       "  1315.74,1385.2 1317.86,1385.2 1319.98,1385.2 1322.1,1385.19 1324.22,1385.19 1326.34,1385.19 1328.46,1385.18 1330.58,1385.18 1332.7,1385.18 1334.82,1385.17 \n",
       "  1336.94,1385.17 1339.07,1385.17 1341.19,1385.16 1343.31,1385.16 1345.43,1385.16 1347.55,1385.15 1349.67,1385.15 1351.79,1385.15 1353.91,1385.15 1356.03,1385.14 \n",
       "  1358.15,1385.14 1360.27,1385.14 1362.39,1385.14 1364.51,1385.14 1366.63,1385.14 1368.76,1385.14 1370.88,1385.14 1373,1385.14 1375.12,1385.15 1377.24,1385.15 \n",
       "  1379.36,1385.16 1381.48,1385.17 1383.6,1385.18 1385.72,1385.19 1387.84,1385.2 1389.96,1385.21 1392.08,1385.22 1394.2,1385.23 1396.32,1385.25 1398.44,1385.26 \n",
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       "  1421.77,1385.36 1423.89,1385.37 1426.01,1385.37 1428.13,1385.37 1430.26,1385.37 1432.38,1385.37 1434.5,1385.37 1436.62,1385.37 1438.74,1385.36 1440.86,1385.36 \n",
       "  1442.98,1385.35 1445.1,1385.35 1447.22,1385.34 1449.34,1385.33 1451.46,1385.32 1453.58,1385.3 1455.7,1385.29 1457.82,1385.28 1459.94,1385.26 1462.07,1385.25 \n",
       "  1464.19,1385.23 1466.31,1385.22 1468.43,1385.2 1470.55,1385.18 1472.67,1385.16 1474.79,1385.15 1476.91,1385.13 1479.03,1385.11 1481.15,1385.09 1483.27,1385.08 \n",
       "  1485.39,1385.06 1487.51,1385.05 1489.63,1385.03 1491.76,1385.02 1493.88,1385.01 1496,1385 1498.12,1384.99 1500.24,1384.98 1502.36,1384.98 1504.48,1384.97 \n",
       "  1506.6,1384.97 1508.72,1384.98 1510.84,1384.98 1512.96,1384.99 1515.08,1385.01 1517.2,1385.03 1519.32,1385.06 1521.44,1385.09 1523.57,1385.13 1525.69,1385.17 \n",
       "  1527.81,1385.22 1529.93,1385.27 1532.05,1385.32 1534.17,1385.37 1536.29,1385.42 1538.41,1385.47 1540.53,1385.52 1542.65,1385.56 1544.77,1385.61 1546.89,1385.65 \n",
       "  1549.01,1385.69 1551.13,1385.73 1553.26,1385.76 1555.38,1385.79 1557.5,1385.81 1559.62,1385.83 1561.74,1385.84 1563.86,1385.85 1565.98,1385.86 1568.1,1385.85 \n",
       "  1570.22,1385.85 1572.34,1385.83 1574.46,1385.82 1576.58,1385.79 1578.7,1385.77 1580.82,1385.73 1582.94,1385.69 1585.07,1385.65 1587.19,1385.6 1589.31,1385.55 \n",
       "  1591.43,1385.5 1593.55,1385.44 1595.67,1385.38 1597.79,1385.31 1599.91,1385.25 1602.03,1385.18 1604.15,1385.11 1606.27,1385.04 1608.39,1384.97 1610.51,1384.9 \n",
       "  1612.63,1384.83 1614.76,1384.77 1616.88,1384.71 1619,1384.65 1621.12,1384.6 1623.24,1384.55 1625.36,1384.51 1627.48,1384.48 1629.6,1384.45 1631.72,1384.44 \n",
       "  1633.84,1384.44 1635.96,1384.45 1638.08,1384.47 1640.2,1384.5 1642.32,1384.56 1644.44,1384.64 1646.57,1384.73 1648.69,1384.84 1650.81,1384.96 1652.93,1385.09 \n",
       "  1655.05,1385.23 1657.17,1385.37 1659.29,1385.51 1661.41,1385.65 1663.53,1385.78 1665.65,1385.91 1667.77,1386.04 1669.89,1386.15 1672.01,1386.25 1674.13,1386.35 \n",
       "  1676.26,1386.43 1678.38,1386.49 1680.5,1386.55 1682.62,1386.58 1684.74,1386.6 1686.86,1386.61 1688.98,1386.6 1691.1,1386.57 1693.22,1386.52 1695.34,1386.46 \n",
       "  1697.46,1386.38 1699.58,1386.29 1701.7,1386.19 1703.82,1386.07 1705.94,1385.93 1708.07,1385.79 1710.19,1385.63 1712.31,1385.47 1714.43,1385.3 1716.55,1385.12 \n",
       "  1718.67,1384.94 1720.79,1384.76 1722.91,1384.58 1725.03,1384.41 1727.15,1384.24 1729.27,1384.08 1731.39,1383.93 1733.51,1383.79 1735.63,1383.68 1737.76,1383.58 \n",
       "  1739.88,1383.51 1742,1383.47 1744.12,1383.46 1746.24,1383.49 1748.36,1383.56 1750.48,1383.66 1752.6,1383.78 1754.72,1383.9 1756.84,1384.03 1758.96,1384.16 \n",
       "  1761.08,1384.3 1763.2,1384.43 1765.32,1384.56 1767.45,1384.68 1769.57,1384.79 1771.69,1384.9 1773.81,1385 1775.93,1385.09 1778.05,1385.16 1780.17,1385.23 \n",
       "  1782.29,1385.28 1784.41,1385.32 1786.53,1385.34 1788.65,1385.36 1790.77,1385.36 1792.89,1385.34 1795.01,1385.32 1797.13,1385.28 1799.26,1385.24 1801.38,1385.18 \n",
       "  1803.5,1385.12 1805.62,1385.05 1807.74,1384.98 1809.86,1384.9 1811.98,1384.82 1814.1,1384.74 1816.22,1384.66 1818.34,1384.59 1820.46,1384.53 1822.58,1384.48 \n",
       "  1824.7,1384.44 1826.82,1384.42 1828.95,1384.42 1831.07,1384.45 1833.19,1384.49 1835.31,1384.54 1837.43,1384.59 1839.55,1384.65 1841.67,1384.7 1843.79,1384.75 \n",
       "  1845.91,1384.8 1848.03,1384.85 1850.15,1384.9 1852.27,1384.94 1854.39,1384.98 1856.51,1385.02 1858.63,1385.06 1860.76,1385.09 1862.88,1385.11 1865,1385.13 \n",
       "  1867.12,1385.15 1869.24,1385.17 1871.36,1385.18 1873.48,1385.18 1875.6,1385.18 1877.72,1385.18 1879.84,1385.17 1881.96,1385.17 1884.08,1385.15 1886.2,1385.14 \n",
       "  1888.32,1385.12 1890.45,1385.1 1892.57,1385.08 1894.69,1385.06 1896.81,1385.03 1898.93,1385.01 1901.05,1384.99 1903.17,1384.97 1905.29,1384.96 1907.41,1384.95 \n",
       "  1909.53,1384.94 1911.65,1384.94 1913.77,1384.94 1915.89,1384.96 1918.01,1384.97 1920.13,1384.99 1922.26,1385.01 1924.38,1385.02 1926.5,1385.04 1928.62,1385.06 \n",
       "  1930.74,1385.08 1932.86,1385.09 1934.98,1385.11 1937.1,1385.13 1939.22,1385.14 1941.34,1385.15 1943.46,1385.17 1945.58,1385.18 1947.7,1385.19 1949.82,1385.19 \n",
       "  1951.95,1385.2 1954.07,1385.21 1956.19,1385.21 1958.31,1385.21 1960.43,1385.21 1962.55,1385.21 1964.67,1385.21 1966.79,1385.21 1968.91,1385.2 1971.03,1385.2 \n",
       "  1973.15,1385.19 1975.27,1385.18 1977.39,1385.17 1979.51,1385.17 1981.63,1385.16 1983.76,1385.15 1985.88,1385.14 1988,1385.13 1990.12,1385.12 1992.24,1385.11 \n",
       "  1994.36,1385.1 1996.48,1385.09 1998.6,1385.09 2000.72,1385.08 2002.84,1385.08 2004.96,1385.08 2007.08,1385.09 2009.2,1385.09 2011.32,1385.1 2013.45,1385.11 \n",
       "  2015.57,1385.12 2017.69,1385.13 2019.81,1385.14 2021.93,1385.14 2024.05,1385.15 2026.17,1385.16 2028.29,1385.17 2030.41,1385.18 2032.53,1385.19 2034.65,1385.2 \n",
       "  2036.77,1385.2 2038.89,1385.21 2041.01,1385.21 2043.13,1385.22 2045.26,1385.22 2047.38,1385.23 2049.5,1385.23 2051.62,1385.23 2053.74,1385.23 2055.86,1385.23 \n",
       "  2057.98,1385.23 2060.1,1385.23 2062.22,1385.23 2064.34,1385.22 2066.46,1385.22 2068.58,1385.21 2070.7,1385.21 2072.82,1385.2 2074.95,1385.2 2077.07,1385.19 \n",
       "  2079.19,1385.18 2081.31,1385.17 2083.43,1385.17 2085.55,1385.16 2087.67,1385.15 2089.79,1385.14 2091.91,1385.14 2094.03,1385.13 2096.15,1385.12 2098.27,1385.12 \n",
       "  2100.39,1385.11 2102.51,1385.11 2104.63,1385.11 2106.76,1385.11 2108.88,1385.11 2111,1385.11 2113.12,1385.12 2115.24,1385.12 2117.36,1385.13 2119.48,1385.14 \n",
       "  2121.6,1385.15 2123.72,1385.15 2125.84,1385.16 2127.96,1385.17 2130.08,1385.18 2132.2,1385.19 2134.32,1385.2 2136.45,1385.21 2138.57,1385.22 2140.69,1385.23 \n",
       "  2142.81,1385.23 2144.93,1385.24 2147.05,1385.25 2149.17,1385.25 2151.29,1385.26 2153.41,1385.26 2155.53,1385.26 2157.65,1385.26 2159.77,1385.27 2161.89,1385.27 \n",
       "  2164.01,1385.26 2166.14,1385.26 2168.26,1385.26 2170.38,1385.26 2172.5,1385.25 2174.62,1385.25 2176.74,1385.24 2178.86,1385.23 2180.98,1385.23 2183.1,1385.22 \n",
       "  2185.22,1385.21 2187.34,1385.2 2189.46,1385.19 2191.58,1385.18 2193.7,1385.17 2195.82,1385.16 2197.95,1385.15 2200.07,1385.14 2202.19,1385.13 2204.31,1385.12 \n",
       "  2206.43,1385.11 2208.55,1385.1 2210.67,1385.1 2212.79,1385.09 2214.91,1385.09 2217.03,1385.08 2219.15,1385.08 2221.27,1385.08 2223.39,1385.08 2225.51,1385.09 \n",
       "  2227.64,1385.1 2229.76,1385.1 2231.88,1385.11 2234,1385.13 2236.12,1385.14 2238.24,1385.15 2240.36,1385.17 2242.48,1385.18 2244.6,1385.2 2246.72,1385.21 \n",
       "  2248.84,1385.23 2250.96,1385.24 2253.08,1385.26 2255.2,1385.27 2257.32,1385.28 2259.45,1385.3 2261.57,1385.31 2263.69,1385.32 2265.81,1385.32 2267.93,1385.33 \n",
       "  2270.05,1385.34 2272.17,1385.34 2274.29,1385.34 2276.41,1385.34 2278.53,1385.34 2280.65,1385.34 2282.77,1385.34 2284.89,1385.33 2287.01,1385.33 2289.14,1385.32 \n",
       "  2291.26,1385.31 2293.38,1385.3 2295.5,1385.28 2297.62,1385.27 2299.74,1385.26 2301.86,1385.24 2303.98,1385.22 2306.1,1385.21 2308.22,1385.19 2310.34,1385.17 \n",
       "  2312.46,1385.15 2314.58,1385.13 2316.7,1385.12 2318.82,1385.1 2320.95,1385.08 2323.07,1385.07 2325.19,1385.05 2327.31,1385.04 2329.43,1385.02 2331.55,1385.01 \n",
       "  2333.67,1385.01 2335.79,1385 2337.91,1385 2340.03,1385 2342.15,1385 2344.27,1385.01 2346.39,1385.02 2348.51,1385.03 2350.64,1385.04 2352.76,1385.05 \n",
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       "  255.392,1318.72 257.513,1312.73 259.633,1306.74 261.754,1300.75 263.875,1294.77 265.995,1288.79 268.116,1282.81 270.237,1276.84 272.357,1270.86 274.478,1264.89 \n",
       "  276.599,1258.92 278.72,1252.95 280.84,1246.99 282.961,1241.03 285.082,1235.07 287.202,1229.11 289.323,1223.15 291.444,1217.2 293.564,1211.25 295.685,1205.3 \n",
       "  297.806,1199.36 299.926,1193.42 302.047,1187.48 304.168,1181.54 306.289,1175.6 308.409,1169.67 310.53,1163.74 312.651,1157.81 314.771,1151.89 316.892,1145.97 \n",
       "  319.013,1140.05 321.133,1134.13 323.254,1128.22 325.375,1122.31 327.495,1116.4 329.616,1110.5 331.737,1104.6 333.858,1098.7 335.978,1092.8 338.099,1086.91 \n",
       "  340.22,1081.02 342.34,1075.14 344.461,1069.25 346.582,1063.37 348.702,1057.5 350.823,1051.62 352.944,1045.75 355.064,1039.89 357.185,1034.02 359.306,1028.16 \n",
       "  361.426,1022.3 363.547,1016.45 365.668,1010.6 367.789,1004.75 369.909,998.912 372.03,993.073 374.151,987.237 376.271,981.405 378.392,975.577 380.513,969.753 \n",
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       "  446.254,791.334 448.375,785.657 450.496,779.986 452.616,774.32 454.737,768.659 456.858,763.005 458.978,757.356 461.099,751.713 463.22,746.076 465.34,740.445 \n",
       "  467.461,734.82 469.582,729.202 471.702,723.589 473.823,717.983 475.944,712.384 478.065,706.791 480.185,701.205 482.306,695.625 484.427,690.053 486.547,684.488 \n",
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       "  1082.46,87.6837 1084.58,87.6842 1086.7,87.6846 1088.82,87.6852 1090.94,87.6857 1093.07,87.6863 1095.19,87.6868 1097.31,87.6874 1099.43,87.688 1101.55,87.6885 \n",
       "  1103.67,87.6889 1105.79,87.6893 1107.91,87.6896 1110.03,87.6898 1112.15,87.6898 1114.27,87.6897 1116.39,87.6893 1118.51,87.6889 1120.63,87.6884 1122.75,87.6877 \n",
       "  1124.88,87.6871 1127,87.6863 1129.12,87.6856 1131.24,87.6848 1133.36,87.684 1135.48,87.6832 1137.6,87.6824 1139.72,87.6816 1141.84,87.6808 1143.96,87.6801 \n",
       "  1146.08,87.6794 1148.2,87.6787 1150.32,87.6781 1152.44,87.6776 1154.57,87.6771 1156.69,87.6767 1158.81,87.6764 1160.93,87.6761 1163.05,87.676 1165.17,87.6759 \n",
       "  1167.29,87.6758 1169.41,87.6759 1171.53,87.676 1173.65,87.6762 1175.77,87.6765 1177.89,87.6769 1180.01,87.6773 1182.13,87.6779 1184.25,87.6784 1186.38,87.6791 \n",
       "  1188.5,87.6798 1190.62,87.6805 1192.74,87.6813 1194.86,87.6821 1196.98,87.683 1199.1,87.6838 1201.22,87.6847 1203.34,87.6856 1205.46,87.6865 1207.58,87.6874 \n",
       "  1209.7,87.6882 1211.82,87.689 1213.94,87.6897 1216.07,87.6904 1218.19,87.691 1220.31,87.6914 1222.43,87.6918 1224.55,87.6921 1226.67,87.6922 1228.79,87.6921 \n",
       "  1230.91,87.6919 1233.03,87.6914 1235.15,87.6907 1237.27,87.6896 1239.39,87.6883 1241.51,87.6867 1243.63,87.6848 1245.75,87.6828 1247.88,87.6807 1250,87.6784 \n",
       "  1252.12,87.676 1254.24,87.6736 1256.36,87.6711 1258.48,87.6687 1260.6,87.6663 1262.72,87.6639 1264.84,87.6616 1266.96,87.6594 1269.08,87.6573 1271.2,87.6554 \n",
       "  1273.32,87.6536 1275.44,87.6519 1277.57,87.6505 1279.69,87.6492 1281.81,87.6481 1283.93,87.6473 1286.05,87.6466 1288.17,87.6462 1290.29,87.646 1292.41,87.6461 \n",
       "  1294.53,87.6464 1296.65,87.6469 1298.77,87.6477 1300.89,87.6487 1303.01,87.65 1305.13,87.6514 1307.25,87.6531 1309.38,87.6551 1311.5,87.6572 1313.62,87.6595 \n",
       "  1315.74,87.662 1317.86,87.6647 1319.98,87.6676 1322.1,87.6705 1324.22,87.6737 1326.34,87.6769 1328.46,87.6802 1330.58,87.6835 1332.7,87.6869 1334.82,87.6904 \n",
       "  1336.94,87.6938 1339.07,87.6972 1341.19,87.7005 1343.31,87.7037 1345.43,87.7068 1347.55,87.7097 1349.67,87.7124 1351.79,87.7149 1353.91,87.7171 1356.03,87.719 \n",
       "  1358.15,87.7206 1360.27,87.7218 1362.39,87.7225 1364.51,87.7227 1366.63,87.7225 1368.76,87.7216 1370.88,87.7202 1373,87.718 1375.12,87.7144 1377.24,87.709 \n",
       "  1379.36,87.7022 1381.48,87.6941 1383.6,87.6849 1385.72,87.6748 1387.84,87.6639 1389.96,87.6525 1392.08,87.6406 1394.2,87.6284 1396.32,87.6161 1398.44,87.6038 \n",
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       "  1421.77,87.5 1423.89,87.4956 1426.01,87.4924 1428.13,87.4903 1430.26,87.4894 1432.38,87.4897 1434.5,87.4912 1436.62,87.4939 1438.74,87.4979 1440.86,87.503 \n",
       "  1442.98,87.5093 1445.1,87.5168 1447.22,87.5255 1449.34,87.5353 1451.46,87.5461 1453.58,87.5579 1455.7,87.5707 1457.82,87.5844 1459.94,87.5989 1462.07,87.6141 \n",
       "  1464.19,87.63 1466.31,87.6465 1468.43,87.6634 1470.55,87.6806 1472.67,87.6981 1474.79,87.7157 1476.91,87.7333 1479.03,87.7507 1481.15,87.7679 1483.27,87.7845 \n",
       "  1485.39,87.8006 1487.51,87.8159 1489.63,87.8303 1491.76,87.8435 1493.88,87.8554 1496,87.8659 1498.12,87.8746 1500.24,87.8814 1502.36,87.886 1504.48,87.8884 \n",
       "  1506.6,87.8881 1508.72,87.885 1510.84,87.8789 1512.96,87.8694 1515.08,87.8552 1517.2,87.8336 1519.32,87.8053 1521.44,87.7714 1523.57,87.7327 1525.69,87.69 \n",
       "  1527.81,87.6443 1529.93,87.5964 1532.05,87.5468 1534.17,87.4965 1536.29,87.4459 1538.41,87.3959 1540.53,87.3469 1542.65,87.2995 1544.77,87.2543 1546.89,87.2118 \n",
       "  1549.01,87.1723 1551.13,87.1362 1553.26,87.104 1555.38,87.076 1557.5,87.0524 1559.62,87.0335 1561.74,87.0195 1563.86,87.0106 1565.98,87.007 1568.1,87.0086 \n",
       "  1570.22,87.0156 1572.34,87.028 1574.46,87.0457 1576.58,87.0688 1578.7,87.097 1580.82,87.1304 1582.94,87.1686 1585.07,87.2115 1587.19,87.2588 1589.31,87.3103 \n",
       "  1591.43,87.3656 1593.55,87.4244 1595.67,87.4862 1597.79,87.5505 1599.91,87.617 1602.03,87.6851 1604.15,87.7543 1606.27,87.8238 1608.39,87.8932 1610.51,87.9617 \n",
       "  1612.63,88.0286 1614.76,88.0931 1616.88,88.1545 1619,88.212 1621.12,88.2646 1623.24,88.3115 1625.36,88.3516 1627.48,88.3842 1629.6,88.408 1631.72,88.4221 \n",
       "  1633.84,88.4254 1635.96,88.4166 1638.08,88.3947 1640.2,88.3584 1642.32,88.3031 1644.44,88.2266 1646.57,88.132 1648.69,88.0227 1650.81,87.9018 1652.93,87.772 \n",
       "  1655.05,87.6362 1657.17,87.4969 1659.29,87.3566 1661.41,87.2175 1663.53,87.0817 1665.65,86.9511 1667.77,86.8277 1669.89,86.713 1672.01,86.6085 1674.13,86.5155 \n",
       "  1676.26,86.4353 1678.38,86.3687 1680.5,86.3168 1682.62,86.2801 1684.74,86.2593 1686.86,86.2547 1688.98,86.2666 1691.1,86.2949 1693.22,86.3398 1695.34,86.4009 \n",
       "  1697.46,86.4778 1699.58,86.57 1701.7,86.6767 1703.82,86.7972 1705.94,86.9303 1708.07,87.075 1710.19,87.2299 1712.31,87.3935 1714.43,87.5641 1716.55,87.74 \n",
       "  1718.67,87.9192 1720.79,88.0996 1722.91,88.2789 1725.03,88.4547 1727.15,88.6244 1729.27,88.7853 1731.39,88.9344 1733.51,89.0688 1735.63,89.1852 1737.76,89.2802 \n",
       "  1739.88,89.3504 1742,89.392 1744.12,89.4012 1746.24,89.3741 1748.36,89.3064 1750.48,89.2018 1752.6,89.085 1754.72,88.9604 1756.84,88.8307 1758.96,88.6983 \n",
       "  1761.08,88.5654 1763.2,88.4342 1765.32,88.3065 1767.45,88.1842 1769.57,88.0686 1771.69,87.9612 1773.81,87.8633 1775.93,87.7758 1778.05,87.6996 1780.17,87.6354 \n",
       "  1782.29,87.5838 1784.41,87.5449 1786.53,87.5191 1788.65,87.5062 1790.77,87.5062 1792.89,87.5185 1795.01,87.5428 1797.13,87.5784 1799.26,87.6242 1801.38,87.6793 \n",
       "  1803.5,87.7424 1805.62,87.8121 1807.74,87.8869 1809.86,87.965 1811.98,88.0444 1814.1,88.1231 1816.22,88.1988 1818.34,88.2691 1820.46,88.3312 1822.58,88.3825 \n",
       "  1824.7,88.4199 1826.82,88.4402 1828.95,88.4403 1831.07,88.4166 1833.19,88.3706 1835.31,88.3205 1837.43,88.2689 1839.55,88.2165 1841.67,88.1641 1843.79,88.1122 \n",
       "  1845.91,88.0615 1848.03,88.0124 1850.15,87.9655 1852.27,87.9212 1854.39,87.8798 1856.51,87.8418 1858.63,87.8074 1860.76,87.7768 1862.88,87.7502 1865,87.7278 \n",
       "  1867.12,87.7096 1869.24,87.6957 1871.36,87.6861 1873.48,87.6806 1875.6,87.6792 1877.72,87.6817 1879.84,87.6879 1881.96,87.6976 1884.08,87.7103 1886.2,87.7258 \n",
       "  1888.32,87.7436 1890.45,87.7633 1892.57,87.7843 1894.69,87.8061 1896.81,87.828 1898.93,87.8495 1901.05,87.8697 1903.17,87.8879 1905.29,87.9033 1907.41,87.915 \n",
       "  1909.53,87.9222 1911.65,87.9238 1913.77,87.9187 1915.89,87.9063 1918.01,87.8904 1920.13,87.8738 1922.26,87.8564 1924.38,87.8388 1926.5,87.821 1928.62,87.8033 \n",
       "  1930.74,87.7858 1932.86,87.7689 1934.98,87.7526 1937.1,87.7371 1939.22,87.7225 1941.34,87.7091 1943.46,87.6968 1945.58,87.6857 1947.7,87.676 1949.82,87.6677 \n",
       "  1951.95,87.6609 1954.07,87.6556 1956.19,87.6517 1958.31,87.6494 1960.43,87.6485 1962.55,87.6491 1964.67,87.6511 1966.79,87.6545 1968.91,87.6591 1971.03,87.6648 \n",
       "  1973.15,87.6717 1975.27,87.6794 1977.39,87.688 1979.51,87.6972 1981.63,87.7069 1983.76,87.7168 1985.88,87.7268 1988,87.7367 1990.12,87.7462 1992.24,87.755 \n",
       "  1994.36,87.763 1996.48,87.7698 1998.6,87.7751 2000.72,87.7786 2002.84,87.78 2004.96,87.779 2007.08,87.7751 2009.2,87.7689 2011.32,87.7617 2013.45,87.7539 \n",
       "  2015.57,87.7455 2017.69,87.7367 2019.81,87.7277 2021.93,87.7186 2024.05,87.7094 2026.17,87.7004 2028.29,87.6916 2030.41,87.6831 2032.53,87.675 2034.65,87.6675 \n",
       "  2036.77,87.6604 2038.89,87.654 2041.01,87.6483 2043.13,87.6433 2045.26,87.6391 2047.38,87.6358 2049.5,87.6332 2051.62,87.6315 2053.74,87.6307 2055.86,87.6307 \n",
       "  2057.98,87.6316 2060.1,87.6333 2062.22,87.6359 2064.34,87.6392 2066.46,87.6433 2068.58,87.6481 2070.7,87.6536 2072.82,87.6596 2074.95,87.6662 2077.07,87.6732 \n",
       "  2079.19,87.6806 2081.31,87.6882 2083.43,87.696 2085.55,87.7039 2087.67,87.7116 2089.79,87.7192 2091.91,87.7265 2094.03,87.7333 2096.15,87.7394 2098.27,87.7448 \n",
       "  2100.39,87.7492 2102.51,87.7526 2104.63,87.7546 2106.76,87.7551 2108.88,87.754 2111,87.7509 2113.12,87.7461 2115.24,87.7401 2117.36,87.7332 2119.48,87.7256 \n",
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   "source": [
    "sol10 = solve(prob10)\n",
    "plot(sol10)"
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   "source": [
    "\n",
    "Now, the conservation of mass $s+p=s_0$ allows us to reduce everything to a single ODE:\n",
    "$$\n",
    "  \\dot{p}  =  \\frac{ k_3 \\cdot k_1\\cdot e_0\\cdot (s_0-p)}{k_3+k_2+k_1 \\cdot (s_0-p)} \\tag{29}\n",
    "$$\n",
    "\n",
    "Or, if the divide every term by $k_1$\n",
    "$$\n",
    "  \\dot{p}  =  \\frac{ k_3 \\cdot e_0\\cdot (s_0-p)}{\\frac{k_3+k_2}{k_1}+(s_0-p)} \\tag{30}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
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       "  1061.25,89.0366 1063.38,89.0378 1065.5,89.039 1067.62,89.0404 1069.74,89.0417 1071.86,89.0431 1073.98,89.0444 1076.1,89.0457 1078.22,89.0468 1080.34,89.0478 \n",
       "  1082.46,89.0485 1084.58,89.0491 1086.7,89.0493 1088.82,89.0492 1090.94,89.0487 1093.07,89.0479 1095.19,89.0467 1097.31,89.0454 1099.43,89.0439 1101.55,89.0423 \n",
       "  1103.67,89.0406 1105.79,89.0388 1107.91,89.037 1110.03,89.0351 1112.15,89.0333 1114.27,89.0314 1116.39,89.0297 1118.51,89.0279 1120.63,89.0263 1122.75,89.0248 \n",
       "  1124.88,89.0233 1127,89.022 1129.12,89.0209 1131.24,89.0199 1133.36,89.019 1135.48,89.0183 1137.6,89.0178 1139.72,89.0175 1141.84,89.0174 1143.96,89.0174 \n",
       "  1146.08,89.0176 1148.2,89.018 1150.32,89.0186 1152.44,89.0194 1154.57,89.0203 1156.69,89.0214 1158.81,89.0226 1160.93,89.024 1163.05,89.0256 1165.17,89.0272 \n",
       "  1167.29,89.029 1169.41,89.0309 1171.53,89.0328 1173.65,89.0348 1175.77,89.0368 1177.89,89.0389 1180.01,89.0409 1182.13,89.0429 1184.25,89.0449 1186.38,89.0467 \n",
       "  1188.5,89.0485 1190.62,89.0501 1192.74,89.0516 1194.86,89.0528 1196.98,89.0538 1199.1,89.0545 1201.22,89.0549 1203.34,89.055 1205.46,89.0546 1207.58,89.0539 \n",
       "  1209.7,89.0526 1211.82,89.0505 1213.94,89.0476 1216.07,89.0441 1218.19,89.0401 1220.31,89.0355 1222.43,89.0306 1224.55,89.0254 1226.67,89.0199 1228.79,89.0143 \n",
       "  1230.91,89.0085 1233.03,89.0028 1235.15,88.997 1237.27,88.9914 1239.39,88.9859 1241.51,88.9806 1243.63,88.9756 1245.75,88.9708 1247.88,88.9664 1250,88.9623 \n",
       "  1252.12,88.9587 1254.24,88.9555 1256.36,88.9527 1258.48,88.9505 1260.6,88.9487 1262.72,88.9475 1264.84,88.9469 1266.96,88.9467 1269.08,88.9472 1271.2,88.9482 \n",
       "  1273.32,88.9498 1275.44,88.9519 1277.57,88.9547 1279.69,88.9579 1281.81,88.9617 1283.93,88.966 1286.05,88.9708 1288.17,88.9761 1290.29,88.9819 1292.41,88.9881 \n",
       "  1294.53,88.9946 1296.65,89.0016 1298.77,89.0088 1300.89,89.0163 1303.01,89.024 1305.13,89.0319 1307.25,89.0399 1309.38,89.048 1311.5,89.0561 1313.62,89.0641 \n",
       "  1315.74,89.072 1317.86,89.0797 1319.98,89.0871 1322.1,89.0942 1324.22,89.1008 1326.34,89.107 1328.46,89.1125 1330.58,89.1173 1332.7,89.1214 1334.82,89.1245 \n",
       "  1336.94,89.1267 1339.07,89.1278 1341.19,89.1277 1343.31,89.1262 1345.43,89.1234 1347.55,89.119 1349.67,89.1119 1351.79,89.1 1353.91,89.084 1356.03,89.0643 \n",
       "  1358.15,89.0414 1360.27,89.0159 1362.39,88.9882 1364.51,88.9587 1366.63,88.9279 1368.76,88.8962 1370.88,88.864 1373,88.8317 1375.12,88.7995 1377.24,88.7679 \n",
       "  1379.36,88.7372 1381.48,88.7076 1383.6,88.6795 1385.72,88.653 1387.84,88.6285 1389.96,88.6061 1392.08,88.5861 1394.2,88.5686 1396.32,88.5537 1398.44,88.5417 \n",
       "  1400.57,88.5326 1402.69,88.5266 1404.81,88.5237 1406.93,88.5239 1409.05,88.5274 1411.17,88.5342 1413.29,88.5442 1415.41,88.5574 1417.53,88.5738 1419.65,88.5934 \n",
       "  1421.77,88.6161 1423.89,88.6417 1426.01,88.6702 1428.13,88.7014 1430.26,88.7352 1432.38,88.7714 1434.5,88.8099 1436.62,88.8503 1438.74,88.8926 1440.86,88.9363 \n",
       "  1442.98,88.9813 1445.1,89.0273 1447.22,89.074 1449.34,89.121 1451.46,89.168 1453.58,89.2146 1455.7,89.2605 1457.82,89.3052 1459.94,89.3483 1462.07,89.3894 \n",
       "  1464.19,89.428 1466.31,89.4636 1468.43,89.4957 1470.55,89.5239 1472.67,89.5475 1474.79,89.566 1476.91,89.5788 1479.03,89.5854 1481.15,89.585 1483.27,89.5771 \n",
       "  1485.39,89.561 1487.51,89.5361 1489.63,89.4954 1491.76,89.4222 1493.88,89.3195 1496,89.1916 1498.12,89.0424 1500.24,88.8759 1502.36,88.6956 1504.48,88.5052 \n",
       "  1506.6,88.308 1508.72,88.107 1510.84,87.9053 1512.96,87.7057 1515.08,87.5108 1517.2,87.3231 1519.32,87.1448 1521.44,86.9781 1523.57,86.8248 1525.69,86.6868 \n",
       "  1527.81,86.5656 1529.93,86.4626 1532.05,86.3791 1534.17,86.316 1536.29,86.2743 1538.41,86.2547 1540.53,86.2577 1542.65,86.2836 1544.77,86.3326 1546.89,86.4047 \n",
       "  1549.01,86.4998 1551.13,86.6175 1553.26,86.7572 1555.38,86.9183 1557.5,87.0999 1559.62,87.3009 1561.74,87.5201 1563.86,87.7561 1565.98,88.0073 1568.1,88.272 \n",
       "  1570.22,88.5483 1572.34,88.8339 1574.46,89.1267 1576.58,89.4242 1578.7,89.7238 1580.82,90.0226 1582.94,90.3178 1585.07,90.606 1587.19,90.884 1589.31,91.1483 \n",
       "  1591.43,91.3952 1593.55,91.6209 1595.67,91.8212 1597.79,91.9921 1599.91,92.129 1602.03,92.2275 1604.15,92.2829 1606.27,92.2901 1608.39,92.2442 1610.51,92.1398 \n",
       "  1612.63,91.9722 1614.76,91.7728 1616.88,91.565 1619,91.3527 1621.12,91.1392 1623.24,90.9278 1625.36,90.7213 1627.48,90.5225 1629.6,90.3336 1631.72,90.1568 \n",
       "  1633.84,89.9938 1635.96,89.8462 1638.08,89.7152 1640.2,89.6017 1642.32,89.5065 1644.44,89.4299 1646.57,89.3721 1648.69,89.3328 1650.81,89.3116 1652.93,89.3079 \n",
       "  1655.05,89.3205 1657.17,89.3483 1659.29,89.3896 1661.41,89.4427 1663.53,89.5053 1665.65,89.575 1667.77,89.6493 1669.89,89.725 1672.01,89.7991 1674.13,89.8679 \n",
       "  1676.26,89.9276 1678.38,89.9741 1680.5,90.0032 1682.62,90.01 1684.74,89.9898 1686.86,89.9384 1688.98,89.8756 1691.1,89.8109 1693.22,89.7453 1695.34,89.6795 \n",
       "  1697.46,89.6145 1699.58,89.5509 1701.7,89.4893 1703.82,89.4305 1705.94,89.3749 1708.07,89.323 1710.19,89.2752 1712.31,89.2318 1714.43,89.1932 1716.55,89.1596 \n",
       "  1718.67,89.1311 1720.79,89.1078 1722.91,89.0898 1725.03,89.0769 1727.15,89.0692 1729.27,89.0665 1731.39,89.0685 1733.51,89.0749 1735.63,89.0855 1737.76,89.0997 \n",
       "  1739.88,89.1171 1742,89.1373 1744.12,89.1594 1746.24,89.183 1748.36,89.2073 1750.48,89.2314 1752.6,89.2546 1754.72,89.2759 1756.84,89.2942 1758.96,89.3087 \n",
       "  1761.08,89.3181 1763.2,89.3213 1765.32,89.3171 1767.45,89.3041 1769.57,89.2854 1771.69,89.2655 1773.81,89.2447 1775.93,89.2232 1778.05,89.2014 1780.17,89.1796 \n",
       "  1782.29,89.1581 1784.41,89.137 1786.53,89.1167 1788.65,89.0972 1790.77,89.0789 1792.89,89.0619 1795.01,89.0463 1797.13,89.0322 1799.26,89.0198 1801.38,89.0091 \n",
       "  1803.5,89.0002 1805.62,88.9932 1807.74,88.9881 1809.86,88.9849 1811.98,88.9836 1814.1,88.9841 1816.22,88.9864 1818.34,88.9905 1820.46,88.9962 1822.58,89.0034 \n",
       "  1824.7,89.012 1826.82,89.0218 1828.95,89.0328 1831.07,89.0446 1833.19,89.0571 1835.31,89.07 1837.43,89.0832 1839.55,89.0962 1841.67,89.1089 1843.79,89.121 \n",
       "  1845.91,89.132 1848.03,89.1417 1850.15,89.1497 1852.27,89.1555 1854.39,89.1589 1856.51,89.1593 1858.63,89.1563 1860.76,89.1496 1862.88,89.1406 1865,89.1304 \n",
       "  1867.12,89.1191 1869.24,89.1069 1871.36,89.0942 1873.48,89.0811 1875.6,89.0677 1877.72,89.0543 1879.84,89.0411 1881.96,89.0282 1884.08,89.0157 1886.2,89.0039 \n",
       "  1888.32,88.9927 1890.45,88.9824 1892.57,88.973 1894.69,88.9647 1896.81,88.9574 1898.93,88.9513 1901.05,88.9465 1903.17,88.9429 1905.29,88.9406 1907.41,88.9396 \n",
       "  1909.53,88.9399 1911.65,88.9415 1913.77,88.9444 1915.89,88.9486 1918.01,88.9541 1920.13,88.9607 1922.26,88.9684 1924.38,88.9771 1926.5,88.9868 1928.62,88.9973 \n",
       "  1930.74,89.0086 1932.86,89.0205 1934.98,89.0328 1937.1,89.0455 1939.22,89.0584 1941.34,89.0712 1943.46,89.0839 1945.58,89.0962 1947.7,89.1079 1949.82,89.1188 \n",
       "  1951.95,89.1286 1954.07,89.1372 1956.19,89.1443 1958.31,89.1496 1960.43,89.1527 1962.55,89.1536 1964.67,89.1518 1966.79,89.147 1968.91,89.1391 1971.03,89.1286 \n",
       "  1973.15,89.116 1975.27,89.1016 1977.39,89.0858 1979.51,89.0687 1981.63,89.0508 1983.76,89.0323 1985.88,89.0136 1988,88.9947 1990.12,88.9761 1992.24,88.9579 \n",
       "  1994.36,88.9403 1996.48,88.9235 1998.6,88.9077 2000.72,88.8931 2002.84,88.8798 2004.96,88.868 2007.08,88.8577 2009.2,88.849 2011.32,88.8421 2013.45,88.8371 \n",
       "  2015.57,88.8339 2017.69,88.8326 2019.81,88.8332 2021.93,88.8358 2024.05,88.8404 2026.17,88.8469 2028.29,88.8552 2030.41,88.8654 2032.53,88.8774 2034.65,88.891 \n",
       "  2036.77,88.9063 2038.89,88.923 2041.01,88.941 2043.13,88.9603 2045.26,88.9806 2047.38,89.0017 2049.5,89.0236 2051.62,89.0459 2053.74,89.0685 2055.86,89.0911 \n",
       "  2057.98,89.1134 2060.1,89.1353 2062.22,89.1564 2064.34,89.1764 2066.46,89.195 2068.58,89.2118 2070.7,89.2267 2072.82,89.239 2074.95,89.2486 2077.07,89.255 \n",
       "  2079.19,89.2579 2081.31,89.2567 2083.43,89.251 2085.55,89.2405 2087.67,89.2241 2089.79,89.2014 2091.91,89.1735 2094.03,89.1411 2096.15,89.105 2098.27,89.066 \n",
       "  2100.39,89.0247 2102.51,88.982 2104.63,88.9384 2106.76,88.8946 2108.88,88.8512 2111,88.8086 2113.12,88.7675 2115.24,88.7282 2117.36,88.6913 2119.48,88.657 \n",
       "  2121.6,88.6258 2123.72,88.5981 2125.84,88.574 2127.96,88.5539 2130.08,88.5379 2132.2,88.5263 2134.32,88.5191 2136.45,88.5166 2138.57,88.5187 2140.69,88.5255 \n",
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       "  2164.01,88.8832 2166.14,88.9356 2168.26,88.9899 2170.38,89.0455 2172.5,89.1019 2174.62,89.1586 2176.74,89.2149 2178.86,89.2703 2180.98,89.3241 2183.1,89.3756 \n",
       "  2185.22,89.424 2187.34,89.4686 2189.46,89.5086 2191.58,89.5432 2193.7,89.5714 2195.82,89.5924 2197.95,89.6051 2200.07,89.6087 2202.19,89.602 2204.31,89.584 \n",
       "  2206.43,89.5536 2208.55,89.5057 2210.67,89.4401 2212.79,89.3594 2214.91,89.2663 2217.03,89.1633 2219.15,89.0527 2221.27,88.9367 2223.39,88.8175 2225.51,88.697 \n",
       "  2227.64,88.5772 2229.76,88.4596 2231.88,88.3461 2234,88.238 2236.12,88.1368 2238.24,88.0437 2240.36,87.9598 2242.48,87.8861 2244.6,87.8236 2246.72,87.7729 \n",
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       "  2270.05,88.0966 2272.17,88.2037 2274.29,88.3215 2276.41,88.4491 2278.53,88.5853 2280.65,88.729 2282.77,88.8788 2284.89,89.0334 2287.01,89.1911 2289.14,89.3503 \n",
       "  2291.26,89.5092 2293.38,89.6659 2295.5,89.8182 2297.62,89.9641 2299.74,90.1011 2301.86,90.2271 2303.98,90.3392 2306.1,90.435 2308.22,90.5116 2310.34,90.5662 \n",
       "  2312.46,90.5956 2314.58,90.5968 2316.7,90.5664 2318.82,90.5012 2320.95,90.4095 2323.07,90.3217 2325.19,90.2395 2327.31,90.1624 2329.43,90.0902 2331.55,90.0226 \n",
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       "  \n",
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      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "enzyme_kinetics3! = @ode_def ab begin\n",
    "  dp = (k3 * e0 * (s0 - p))/((k3+k2)/k1 +  (s0 - p))\n",
    "    end k1 k2 k3 e0 s0\n",
    "tspan = (0.0,100.0)\n",
    "\n",
    "u0=[0.00001]\n",
    "p=[k1,k2,k3,e0,s0];\n",
    "prob2 = ODEProblem(enzyme_kinetics3!,u0,tspan,p)\n",
    "sol2 = solve(prob2)\n",
    "plot(sol2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can compare directly this steady state aproximation wit the mass action solution for the product"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### <font color=red>Computer Exercise:</font>\n",
    "\n",
    "Solve numerically the enzyme-substrate system simplified with the previous pseudo-steady state approximation and compare its performance and accuracy by proving the value of p(t) of the simplified system with the prediction of $p(t)$ using the full system. Find parameters where the pseudo-steady state aproximation is a better approximation to the the full system. \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.002"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tspan = (0.0,100.0)\n",
    "k1=1e3\n",
    "k2=0.1\n",
    "k3=0.05\n",
    "e0=0.002\n",
    "s0=0.002"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Michaelis-Menten equation: \n",
    "\n",
    "The previous equation \n",
    "\n",
    "$$\n",
    "  \\dot{p}  =  \\frac{ k_3 \\cdot e_0\\cdot (s_0-p)}{\\frac{k_3+k_2}{k_1}+(s_0-p)} \\tag{31}\n",
    "$$\n",
    "\n",
    "leads to the traditional Michaelis-Menten equation, which predicts the initial turnover rate of the enzymatic reaction $V_0$ as a function of initial substrate concentration $s_0$. So at the initial state of the reaction ($p=0$):\n",
    "\n",
    "$$\n",
    "  V_0  =  \\frac{ k_3 \\cdot e_0\\cdot s_0}{\\frac{k_3+k_2}{k_1}+s_0}=\\frac{ v_{max} \\cdot s_0}{K_M+s_0} \\tag{32}\n",
    "$$\n",
    "\n",
    "\n",
    "where the constant $K_M = \\frac{k_3+k_2}{k_1}$ is called the Michaelis-Menten constant and $v_{max}= k_3 \\cdot e_0$ is the maximum turn-over rate. \n",
    "The $K_M$  reflects the affinity of the reaction. Strong affinity means small $K_M$. At $s_0=K_M$ the turn-over rate is half maximal, i.e., $\\dot{p}=\\frac{v_{max}}{2}$\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Michaelis-Menten curve\n",
    "\n",
    "The famous Michaelis-Menten plot is a culve that shows the dependence of the velocity of the reaction in terms of production of product (i.e., $\\dot p$) on teh concetration of substrate. It showns three regimes:\n",
    "- __linear__: there is a lot of enzyme available to bind the substrate, so the velocity of teh reaction increases linearly with the concetration of subtrate, as it ocurs in noncatalized reactions. \n",
    "- __Constrained__: the intermediate regime \n",
    "- __Saturated__: there is a lot of substrate available, and the enzyme is the limitant factor for the catalisis, so the speed of teh reactioon does not increase if the increase further the concetration of substrate."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "K_m = 0.00015000000000000001\n"
     ]
    },
    {
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     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "s_vector= s0*LinRange(0,1,100)\n",
    "plot(s_vector,k3 * e0 * s_vector ./(s_vector.+(k3+k2)/k1),label=\"Michaelis Menten\",)\n",
    "println(\"K_m = \",(k3+k2)/k1)\n",
    "title!(\"Michaelis-Menten Plot\")\n",
    "xlabel!(\"concentration of substrate\")\n",
    "ylabel!(\"speed of the reaction\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### How good is the Michaelis-Menten Approach"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can compare this analytical equation with the result of the numerical simulation of the full ODE system. To do that we solve the numerical system, and compare the rates of the reactions predicted by the Mass action and the Michaelis-Menten approach."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
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     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\"k1=1e2\n",
    "k2=1e0\n",
    "k3=0.01\n",
    "e0=1e-2\n",
    "s0=1e-2\"\n",
    "\n",
    "s_vector= s0*LinRange(0,1,100)*5\n",
    "p=[k1,k2,k3,e0]\n",
    "plot(s_vector,k3 * e0 * s_vector ./(s_vector.+(k3+k2)/k1),label=\"Michaelis Menten\",)\n",
    "title!(\"Michaelis-Menten Plot\")\n",
    "xlabel!(\"Concentration of substrate\")\n",
    "ylabel!(\"Speed of the reaction\")\n",
    "\n",
    "tspan = (0.0,10)\n",
    "#p=[1e3,0.1,0.05,0.001]\n",
    "speed_vector = similar(s_vector)\n",
    "for i in 1:100 \n",
    "u0=[s_vector[i],0,0]\n",
    "    prob = ODEProblem(enzyme_kinetics2!,u0,tspan,p)\n",
    "    sol_ = solve(prob)\n",
    "    # we shoudl compute the speed at the maximum formation of Complex. \n",
    "    speed=[k3*u[2] for (u,t) in tuples(sol_)]\n",
    " speed_vector[i]=maximum(speed)\n",
    "end\n",
    "plot!(s_vector,speed_vector,label=\"rate full mass action\")\n",
    "title!(\"Michaelis-Menten vs. Mass action\")\n",
    "xlabel!(\"Concentration of substrate\")\n",
    "ylabel!(\"Speed of the reaction\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This analysis shows that the Michaelis-Menten approach is valid when the amount of $s$ is higher that amount of $e$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Kinase-Phosphatase systems\n",
    "\n",
    "If we want to compute the equilibrium concentration for the product `p`, we have to make the last eq equal to `0` :, \n",
    "$$\n",
    "\\dot{p}=\\frac{k_3 \\cdot e_0 (s_0-p)}{K_M + (s_0-p)} =0  \\tag{33}\n",
    "$$\n",
    "\n",
    "so we simply obtain:\n",
    " \n",
    " $$\\begin{align*}\n",
    "k_3 \\cdot e_0 (s_0-p_{eq})=0\\tag{34}\\\\\n",
    "p_{eq}=s_0\\tag{35}\n",
    " \\end{align*}$$\n",
    "\n",
    "which means that all substrate is converted into product eventually. This means that, once all substrate is consumed the reaction is finished, and there is no way to reverse it. Of course, living systems do not work like that, i.e., there is allways a balance in the form of a dynamical equilibrium between reactions, (such as in reversible chemical reactions). This equilibrium, in enzyme-catalized reactions, is achieved by combining pairs of enzymes that perform opposite tasks, i.e., that catalize opposite reactions. \n",
    "\n",
    "Quite often, one enzyme catalizes the formation from `s` to `p`, there is another enzyme that catalizes the transformation of `p` back to `s`.\n",
    "\n",
    "$$\\begin{align*}\n",
    "s + e1 \\leftrightarrow  e1s \\rightarrow p + e1 \\tag{36}\\\\\n",
    "p + e2 \\leftrightarrow  e2p \\rightarrow s + e2  \\tag{37}\n",
    "\\end{align*}$$\n",
    "\n",
    "whith kinetic rate constants $k_1$, $k_2$ and $k_3$, for the first reaction, and $k_4$, $k_5$ and $k_6$, for the second reaction. If we write the sistem as irreversible reactions, we obatin 6 reactions:\n",
    "\n",
    "$$\\begin{align*}\n",
    "s + e1 \\rightarrow  e1s \\tag{38}\\\\\n",
    "e1s \\rightarrow s + e1 \\tag{39}\\\\\n",
    "e1s \\rightarrow p + e1 \\tag{40}\\\\\n",
    "p + e2 \\rightarrow e2p \\tag{41}\\\\\n",
    "e2p \\rightarrow p + e2 \\tag{42}\\\\\n",
    "e2p \\rightarrow s + e2 \\tag{43}\n",
    " \\end{align*}$$\n",
    " \n",
    "Lets compute the matrices form the Mass action law. If we rename variables as $X_1=s, X_2=e1s, X_3=e1, X_4=p, X_5=e2p, X_6=e2$ to use the state vector approach.\n",
    "\n",
    "$$\\begin{align*}\n",
    "X_1 + X_3 \\rightarrow  X_2 \\\\\n",
    "X_2 \\rightarrow X_1 + X_3 \\\\\n",
    "X_2 \\rightarrow X_3 +  X_4 \\\\\n",
    "X_4 + X_6 \\rightarrow X_5\\\\\n",
    "X_5 \\rightarrow X_4 + X_6 \\\\\n",
    "X_5 \\rightarrow X_1 + X_6 \n",
    " \\end{align*}$$\n",
    "\n",
    "\n",
    "The kinetics of these chemical equations is described by the __Mass action Law__ \n",
    "\n",
    "\n",
    "$$\n",
    "A=\\begin{bmatrix}\n",
    " 1 & 0 & 1 & 0 & 0 & 0 \\\\\n",
    " 0 & 1 & 0 & 0 & 0 & 0\\\\\n",
    " 0 & 1 & 0 & 0 & 0 & 0 \\\\\n",
    " 0 & 0 & 0 & 1 & 0 & 1\\\\\n",
    "  0 & 0 & 0 & 0 & 1 & 0\\\\\n",
    "   0 & 0 & 0 & 0 & 1 & 0\\end{bmatrix} ;\n",
    "B=\\begin{bmatrix}\n",
    " 0 & 1 & 0 & 0 & 0 & 0 \\\\\n",
    " 1 & 0 & 1 & 0 & 0 & 0 \\\\\n",
    " 0 & 0 & 1 & 1 & 0 & 0 \\\\\n",
    " 0 & 0 & 0 & 0 & 1 & 0 \\\\\n",
    " 0 & 0 & 0 & 1 & 0 & 1 \\\\\n",
    " 1 & 0 & 0 & 0 & 0 & 1\n",
    " \\end{bmatrix} ; \n",
    " (B-A)^T= \\begin{bmatrix}\n",
    " -1 &  1 &  0 &  0 &  0 &  1\\\\\n",
    "  1 & -1 & -1 &  0 &  0 &  0\\\\\n",
    " -1 &  1 &  1 &  0 &  0 &  0 \\\\\n",
    "  0 &  0 &  1 & -1 &  1 &  0 \\\\\n",
    "  0 &  0 &  0 &  1 & -1 & -1 \\\\\n",
    "  0 &  0 &  0 & -1 &  1 &  1\\end{bmatrix} \n",
    "$$\n",
    " in this particular case\n",
    "\n",
    "$$\n",
    "K=\\begin{pmatrix}\n",
    " k_1 & 0 & 0 & 0 & 0 & 0\\\\\n",
    " 0 & k_2 & 0 & 0 & 0  & 0\\\\\n",
    " 0 & 0 & k_3 & 0 & 0 & 0\\\\\n",
    " 0 & 0 & 0 & k_4 & 0  &0\\\\\n",
    " 0 & 0 & 0 & 0 & k_5 & 0 \\\\\n",
    "  0 & 0 & 0 & 0 & 0 & k_6 \n",
    "\\end{pmatrix}\n",
    "$$\n",
    "\n",
    "\n",
    "$$X^A=\\begin{pmatrix}\n",
    "X_1^1\\cdot X_2^0 \\cdot X_3^1 \\cdot X_4^0 \\cdot X_5^0 \\cdot X_6^0\\\\\n",
    "X_1^0\\cdot X_2^1 \\cdot X_3^0 \\cdot X_4^0 \\cdot X_5^0 \\cdot X_6^0\\\\\n",
    "X_1^0\\cdot X_2^1 \\cdot X_3^0 \\cdot X_4^0 \\cdot X_5^0 \\cdot X_6^0\\\\\n",
    "X_1^0\\cdot X_2^0 \\cdot X_3^0 \\cdot X_4^1 \\cdot X_5^0 \\cdot X_6^1\\\\\n",
    "X_1^0\\cdot X_2^0 \\cdot X_3^0 \\cdot X_4^0 \\cdot X_5^1 \\cdot X_6^0\\\\\n",
    "X_1^0\\cdot X_2^0 \\cdot X_3^0 \\cdot X_4^0 \\cdot X_5^1 \\cdot X_6^0\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "X_1 \\cdot X_3\\\\\n",
    " X_2\\\\\n",
    "  X_2\\\\\n",
    "  X_4 \\cdot X_6\\\\\n",
    "  X_5\\\\\n",
    "  X_5\n",
    "\\end{pmatrix}\n",
    "$$\n",
    "so \n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    " \\begin{bmatrix}\n",
    "\\frac{\\mathrm{d} X_1}{\\mathrm{d} t}\\\\ \\frac{\\mathrm{d} X_2}{\\mathrm{d} t} \\\\ \\frac{\\mathrm{d} X_3}{\\mathrm{d} t} \\\\ \\frac{\\mathrm{d} X_4}{\\mathrm{d} t} \\\\ \\frac{\\mathrm{d} X_5}{\\mathrm{d} t} \\\\ \\frac{\\mathrm{d} X_6}{\\mathrm{d} t}\\end{bmatrix}= \\begin{bmatrix}\n",
    " -1 &  1 &  0 &  0 &  0 &  1\\\\\n",
    "  1 & -1 & -1 &  0 &  0 &  0\\\\\n",
    " -1 &  1 &  1 &  0 &  0 &  0 \\\\\n",
    "  0 &  0 &  1 & -1 &  1 &  0 \\\\\n",
    "  0 &  0 &  0 &  1 & -1 & -1 \\\\\n",
    "  0 &  0 &  0 & -1 &  1 &  1\\end{bmatrix}\\begin{pmatrix}\n",
    " k_1 & 0 & 0 & 0 & 0 & 0\\\\\n",
    " 0 & k_2 & 0 & 0 & 0  & 0\\\\\n",
    " 0 & 0 & k_3 & 0 & 0 & 0\\\\\n",
    " 0 & 0 & 0 & k_4 & 0  &0\\\\\n",
    " 0 & 0 & 0 & 0 & k_5 & 0 \\\\\n",
    "  0 & 0 & 0 & 0 & 0 & k_6 \n",
    "\\end{pmatrix} \\begin{pmatrix}\n",
    "X_1 \\cdot X_3\\\\\n",
    " X_2\\\\\n",
    "  X_2\\\\\n",
    "  X_4 \\cdot X_6\\\\\n",
    "  X_5\\\\\n",
    "  X_5\n",
    "\\end{pmatrix}\n",
    "\\end{align}\n",
    "$$\n",
    "that when we operate the matrices becomes\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    " \\begin{bmatrix}\n",
    "\\frac{\\mathrm{d} X_1}{\\mathrm{d} t}\\\\ \\frac{\\mathrm{d} X_2}{\\mathrm{d} t} \\\\ \\frac{\\mathrm{d} X_3}{\\mathrm{d} t} \\\\ \\frac{\\mathrm{d} X_4}{\\mathrm{d} t} \\\\ \\frac{\\mathrm{d} X_5}{\\mathrm{d} t} \\\\ \\frac{\\mathrm{d} X_6}{\\mathrm{d} t}\\end{bmatrix}= \\begin{bmatrix}\n",
    " -1 &  1 &  0 &  0 &  0 &  1\\\\\n",
    "  1 & -1 & -1 &  0 &  0 &  0\\\\\n",
    " -1 &  1 &  1 &  0 &  0 &  0 \\\\\n",
    "  0 &  0 &  1 & -1 &  1 &  0 \\\\\n",
    "  0 &  0 &  0 &  1 & -1 & -1 \\\\\n",
    "  0 &  0 &  0 & -1 &  1 &  1\\end{bmatrix} \\begin{pmatrix}\n",
    "k_1 \\cdot X_1 \\cdot X_3\\\\\n",
    " k_2 \\cdot X_2\\\\\n",
    "  k_3 \\cdot X_2\\\\\n",
    "  k_4 \\cdot X_4 \\cdot X_6\\\\\n",
    "  k_5 \\cdot X_5\\\\\n",
    "  k_6 \\cdot X_5\n",
    "\\end{pmatrix}\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "and operating further,\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    " \\begin{bmatrix}\n",
    "\\frac{\\mathrm{d} X_1}{\\mathrm{d} t}\\\\ \\frac{\\mathrm{d} X_2}{\\mathrm{d} t} \\\\ \\frac{\\mathrm{d} X_3}{\\mathrm{d} t} \\\\ \\frac{\\mathrm{d} X_4}{\\mathrm{d} t} \\\\ \\frac{\\mathrm{d} X_5}{\\mathrm{d} t} \\\\ \\frac{\\mathrm{d} X_6}{\\mathrm{d} t}\\end{bmatrix}=  \\begin{bmatrix}\n",
    "-k_1 \\cdot X_1 \\cdot X_3 + k_2 \\cdot X_2 + k_6 \\cdot X_5\\\\\n",
    " k_1 \\cdot X_1 \\cdot X_3 - k_2 \\cdot X_2  - k_3 \\cdot  X_2\\\\\n",
    "-k_1 \\cdot X_1 \\cdot X_3 + k_2 \\cdot X_2 + k_3 \\cdot  X_2\\\\\n",
    "k_3 \\cdot X_2 -  k_4 \\cdot X_4 \\cdot X_6 + k_5 \\cdot X_5\\\\\n",
    "  k_4 \\cdot X_4 \\cdot X_6 - k_5 \\cdot X_5 - k_6 \\cdot X_5\\\\\n",
    "-  k_4 \\cdot X_4 \\cdot X_6 + k_5 \\cdot X_5 + k_6 \\cdot X_5\n",
    "\\end{bmatrix} \n",
    "\\end{align}\n",
    "$$\n",
    " \n",
    " which,operating further and substituting for the original names of the variables, it gives us a set of six differential equations:\n",
    " \n",
    " $$\\begin{align*}\n",
    "  \\dot{s}   &= -k_1 \\cdot s \\cdot e1 + k_2 \\cdot e1s + k_6 \\cdot e2p \\\\\n",
    "  \\dot{e1s} &= k_1 \\cdot e1 \\cdot s - (k_2 + k_3) \\cdot e1s \\\\\n",
    "  \\dot{e1}  &= -k_1 \\cdot s \\cdot e1 + (k_2 + k_3) \\cdot e1s \\\\\n",
    "  \\dot{p}   &= -k_4 \\cdot p \\cdot e2  + k_5 \\cdot e2p + k_3 \\cdot e1s \\\\\n",
    "   \\dot{e2p} &= k_4 \\cdot e2 \\cdot p - (k_5 + k_6) \\cdot e2p \\\\\n",
    "  \\dot{e2}  &= -k_4 \\cdot p \\cdot e2 + (k_5 + k_6) \\cdot e2p \n",
    "  \\end{align*}$$\n",
    "\n",
    "With these equations we define an ODE vector function using the DSL notation:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(::ab{var\"#43#47\",var\"#44#48\",var\"#45#49\",Nothing,Nothing,var\"#46#50\",Expr,Expr}) (generic function with 2 methods)"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "enzyme_kinetics4! = @ode_def ab begin\n",
    "  ds   = -k1 * s * e1 + k2 * e1s + k6 * e2p\n",
    "  de1  = -k1 * s * e1 + (k2 + k3) * e1s\n",
    "  de2  = -k4 * p * e2 + (k5 + k6) * e2p\n",
    "  de1s =  k1 * e1 * s - (k2 + k3) * e1s\n",
    "  de2p =  k4 * e2 * p - (k5 + k6) * e2p\n",
    "  dp   = -k4 * p* e2  + k5 * e2p + k3 * e1s \n",
    "    end k1 k2 k3 k4 k5 k6"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [],
   "source": [
    "tspan = (0.0,100)\n",
    "k1=0.5e3\n",
    "k2=0.15\n",
    "k3=0.03\n",
    "k4=1e3\n",
    "k5=0.1\n",
    "k6=0.01\n",
    "\n",
    "e10=0.002\n",
    "e20=0.002\n",
    "s0=0.002\n",
    "p0=0.001\n",
    "\n",
    "u0=[s0,e10,e20,0,0,p0]\n",
    "p=[k1,k2,k3,k4,k5,k6];"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\u001b[36mODEProblem\u001b[0m with uType \u001b[36mArray{Float64,1}\u001b[0m and tType \u001b[36mFloat64\u001b[0m. In-place: \u001b[36mtrue\u001b[0m\n",
       "timespan: (0.0, 100.0)\n",
       "u0: [0.002, 0.002, 0.002, 0.0, 0.0, 0.001]"
      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "prob4 = ODEProblem(enzyme_kinetics4!,u0,tspan,p)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
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       "  728.384,1138.38 730.442,1138.93 732.501,1139.47 734.56,1140.01 736.619,1140.55 738.677,1141.08 740.736,1141.62 742.795,1142.15 744.854,1142.68 746.913,1143.21 \n",
       "  748.971,1143.73 751.03,1144.26 753.089,1144.79 755.148,1145.31 757.206,1145.83 759.265,1146.36 761.324,1146.88 763.383,1147.4 765.441,1147.92 767.5,1148.43 \n",
       "  769.559,1148.95 771.618,1149.46 773.677,1149.97 775.735,1150.48 777.794,1150.98 779.853,1151.49 781.912,1151.99 783.97,1152.48 786.029,1152.98 788.088,1153.47 \n",
       "  790.147,1153.96 792.206,1154.45 794.264,1154.94 796.323,1155.42 798.382,1155.9 800.441,1156.38 802.499,1156.85 804.558,1157.33 806.617,1157.8 808.676,1158.27 \n",
       "  810.734,1158.74 812.793,1159.2 814.852,1159.67 816.911,1160.14 818.97,1160.6 821.028,1161.06 823.087,1161.53 825.146,1161.99 827.205,1162.45 829.263,1162.91 \n",
       "  831.322,1163.37 833.381,1163.83 835.44,1164.29 837.499,1164.75 839.557,1165.2 841.616,1165.65 843.675,1166.1 845.734,1166.55 847.792,1167 849.851,1167.44 \n",
       "  851.91,1167.88 853.969,1168.32 856.028,1168.76 858.086,1169.19 860.145,1169.62 862.204,1170.05 864.263,1170.48 866.321,1170.91 868.38,1171.33 870.439,1171.75 \n",
       "  872.498,1172.17 874.556,1172.58 876.615,1173 878.674,1173.41 880.733,1173.82 882.792,1174.23 884.85,1174.64 886.909,1175.05 888.968,1175.46 891.027,1175.87 \n",
       "  893.085,1176.27 895.144,1176.68 897.203,1177.09 899.262,1177.5 901.321,1177.9 903.379,1178.31 905.438,1178.72 907.497,1179.12 909.556,1179.52 911.614,1179.92 \n",
       "  913.673,1180.32 915.732,1180.72 917.791,1181.11 919.849,1181.51 921.908,1181.9 923.967,1182.28 926.026,1182.67 928.085,1183.05 930.143,1183.44 932.202,1183.82 \n",
       "  934.261,1184.19 936.32,1184.57 938.378,1184.94 940.437,1185.31 942.496,1185.68 944.555,1186.05 946.614,1186.41 948.672,1186.77 950.731,1187.14 952.79,1187.5 \n",
       "  954.849,1187.86 956.907,1188.22 958.966,1188.57 961.025,1188.93 963.084,1189.29 965.143,1189.65 967.201,1190.01 969.26,1190.37 971.319,1190.73 973.378,1191.08 \n",
       "  975.436,1191.44 977.495,1191.8 979.554,1192.16 981.613,1192.51 983.671,1192.87 985.73,1193.22 987.789,1193.57 989.848,1193.92 991.907,1194.27 993.965,1194.61 \n",
       "  996.024,1194.96 998.083,1195.3 1000.14,1195.64 1002.2,1195.97 1004.26,1196.31 1006.32,1196.64 1008.38,1196.97 1010.44,1197.3 1012.49,1197.62 1014.55,1197.95 \n",
       "  1016.61,1198.27 1018.67,1198.59 1020.73,1198.91 1022.79,1199.23 1024.85,1199.54 1026.91,1199.86 1028.96,1200.17 1031.02,1200.49 1033.08,1200.8 1035.14,1201.12 \n",
       "  1037.2,1201.43 1039.26,1201.74 1041.32,1202.06 1043.38,1202.38 1045.43,1202.69 1047.49,1203.01 1049.55,1203.33 1051.61,1203.64 1053.67,1203.96 1055.73,1204.27 \n",
       "  1057.79,1204.58 1059.85,1204.9 1061.9,1205.21 1063.96,1205.51 1066.02,1205.82 1068.08,1206.12 1070.14,1206.43 1072.2,1206.73 1074.26,1207.03 1076.32,1207.32 \n",
       "  1078.38,1207.62 1080.43,1207.91 1082.49,1208.2 1084.55,1208.49 1086.61,1208.77 1088.67,1209.06 1090.73,1209.34 1092.79,1209.62 1094.85,1209.9 1096.9,1210.18 \n",
       "  1098.96,1210.46 1101.02,1210.73 1103.08,1211.01 1105.14,1211.28 1107.2,1211.56 1109.26,1211.83 1111.32,1212.11 1113.37,1212.38 1115.43,1212.66 1117.49,1212.94 \n",
       "  1119.55,1213.22 1121.61,1213.5 1123.67,1213.78 1125.73,1214.06 1127.79,1214.34 1129.84,1214.62 1131.9,1214.89 1133.96,1215.17 1136.02,1215.44 1138.08,1215.72 \n",
       "  1140.14,1215.99 1142.2,1216.26 1144.26,1216.52 1146.31,1216.79 1148.37,1217.05 1150.43,1217.32 1152.49,1217.58 1154.55,1217.83 1156.61,1218.09 1158.67,1218.34 \n",
       "  1160.73,1218.6 1162.78,1218.85 1164.84,1219.09 1166.9,1219.34 1168.96,1219.59 1171.02,1219.83 1173.08,1220.07 1175.14,1220.32 1177.2,1220.56 1179.25,1220.8 \n",
       "  1181.31,1221.04 1183.37,1221.28 1185.43,1221.52 1187.49,1221.77 1189.55,1222.01 1191.61,1222.25 1193.67,1222.5 1195.73,1222.75 1197.78,1222.99 1199.84,1223.24 \n",
       "  1201.9,1223.49 1203.96,1223.74 1206.02,1223.98 1208.08,1224.23 1210.14,1224.47 1212.2,1224.71 1214.25,1224.96 1216.31,1225.19 1218.37,1225.43 1220.43,1225.67 \n",
       "  1222.49,1225.9 1224.55,1226.14 1226.61,1226.37 1228.67,1226.6 1230.72,1226.82 1232.78,1227.05 1234.84,1227.27 1236.9,1227.49 1238.96,1227.71 1241.02,1227.93 \n",
       "  1243.08,1228.15 1245.14,1228.36 1247.19,1228.58 1249.25,1228.79 1251.31,1229 1253.37,1229.21 1255.43,1229.42 1257.49,1229.63 1259.55,1229.85 1261.61,1230.06 \n",
       "  1263.66,1230.27 1265.72,1230.48 1267.78,1230.7 1269.84,1230.92 1271.9,1231.13 1273.96,1231.35 1276.02,1231.57 1278.08,1231.79 1280.13,1232.01 1282.19,1232.23 \n",
       "  1284.25,1232.45 1286.31,1232.67 1288.37,1232.88 1290.43,1233.1 1292.49,1233.31 1294.55,1233.52 1296.61,1233.73 1298.66,1233.94 1300.72,1234.14 1302.78,1234.35 \n",
       "  1304.84,1234.55 1306.9,1234.75 1308.96,1234.95 1311.02,1235.15 1313.08,1235.34 1315.13,1235.53 1317.19,1235.72 1319.25,1235.91 1321.31,1236.1 1323.37,1236.29 \n",
       "  1325.43,1236.48 1327.49,1236.66 1329.55,1236.85 1331.6,1237.03 1333.66,1237.22 1335.72,1237.4 1337.78,1237.59 1339.84,1237.78 1341.9,1237.96 1343.96,1238.15 \n",
       "  1346.02,1238.35 1348.07,1238.54 1350.13,1238.73 1352.19,1238.93 1354.25,1239.13 1356.31,1239.32 1358.37,1239.52 1360.43,1239.71 1362.49,1239.9 1364.54,1240.1 \n",
       "  1366.6,1240.29 1368.66,1240.48 1370.72,1240.66 1372.78,1240.85 1374.84,1241.04 1376.9,1241.22 1378.96,1241.4 1381.01,1241.58 1383.07,1241.76 1385.13,1241.93 \n",
       "  1387.19,1242.11 1389.25,1242.28 1391.31,1242.45 1393.37,1242.62 1395.43,1242.78 1397.48,1242.95 1399.54,1243.12 1401.6,1243.28 1403.66,1243.44 1405.72,1243.6 \n",
       "  1407.78,1243.77 1409.84,1243.93 1411.9,1244.09 1413.96,1244.25 1416.01,1244.42 1418.07,1244.58 1420.13,1244.75 1422.19,1244.92 1424.25,1245.09 1426.31,1245.26 \n",
       "  1428.37,1245.43 1430.43,1245.6 1432.48,1245.77 1434.54,1245.94 1436.6,1246.11 1438.66,1246.28 1440.72,1246.45 1442.78,1246.61 1444.84,1246.78 1446.9,1246.94 \n",
       "  1448.95,1247.11 1451.01,1247.27 1453.07,1247.44 1455.13,1247.6 1457.19,1247.76 1459.25,1247.92 1461.31,1248.09 1463.37,1248.25 1465.42,1248.41 1467.48,1248.57 \n",
       "  1469.54,1248.73 1471.6,1248.88 1473.66,1249.04 1475.72,1249.2 1477.78,1249.36 1479.84,1249.51 1481.89,1249.67 1483.95,1249.82 1486.01,1249.98 1488.07,1250.13 \n",
       "  1490.13,1250.29 1492.19,1250.44 1494.25,1250.59 1496.31,1250.74 1498.36,1250.89 1500.42,1251.04 1502.48,1251.19 1504.54,1251.34 1506.6,1251.49 1508.66,1251.64 \n",
       "  1510.72,1251.79 1512.78,1251.93 1514.84,1252.08 1516.89,1252.23 1518.95,1252.37 1521.01,1252.51 1523.07,1252.66 1525.13,1252.8 1527.19,1252.94 1529.25,1253.09 \n",
       "  1531.31,1253.23 1533.36,1253.37 1535.42,1253.51 1537.48,1253.65 1539.54,1253.79 1541.6,1253.93 1543.66,1254.07 1545.72,1254.2 1547.78,1254.34 1549.83,1254.48 \n",
       "  1551.89,1254.61 1553.95,1254.75 1556.01,1254.88 1558.07,1255.02 1560.13,1255.15 1562.19,1255.28 1564.25,1255.42 1566.3,1255.55 1568.36,1255.68 1570.42,1255.81 \n",
       "  1572.48,1255.94 1574.54,1256.07 1576.6,1256.2 1578.66,1256.33 1580.72,1256.46 1582.77,1256.59 1584.83,1256.71 1586.89,1256.84 1588.95,1256.97 1591.01,1257.1 \n",
       "  1593.07,1257.23 1595.13,1257.35 1597.19,1257.48 1599.24,1257.6 1601.3,1257.73 1603.36,1257.86 1605.42,1257.98 1607.48,1258.11 1609.54,1258.23 1611.6,1258.36 \n",
       "  1613.66,1258.48 1615.71,1258.6 1617.77,1258.73 1619.83,1258.85 1621.89,1258.97 1623.95,1259.09 1626.01,1259.21 1628.07,1259.34 1630.13,1259.46 1632.19,1259.58 \n",
       "  1634.24,1259.7 1636.3,1259.82 1638.36,1259.94 1640.42,1260.06 1642.48,1260.18 1644.54,1260.3 1646.6,1260.41 1648.66,1260.53 1650.71,1260.65 1652.77,1260.77 \n",
       "  1654.83,1260.88 1656.89,1261 1658.95,1261.12 1661.01,1261.23 1663.07,1261.35 1665.13,1261.46 1667.18,1261.58 1669.24,1261.69 1671.3,1261.81 1673.36,1261.92 \n",
       "  1675.42,1262.03 1677.48,1262.15 1679.54,1262.26 1681.6,1262.37 1683.65,1262.48 1685.71,1262.6 1687.77,1262.71 1689.83,1262.82 1691.89,1262.93 1693.95,1263.04 \n",
       "  1696.01,1263.15 1698.07,1263.26 1700.12,1263.37 1702.18,1263.48 1704.24,1263.59 1706.3,1263.69 1708.36,1263.8 1710.42,1263.91 1712.48,1264.02 1714.54,1264.12 \n",
       "  1716.59,1264.23 1718.65,1264.34 1720.71,1264.44 1722.77,1264.55 1724.83,1264.66 1726.89,1264.76 1728.95,1264.87 1731.01,1264.97 1733.07,1265.08 1735.12,1265.18 \n",
       "  1737.18,1265.28 1739.24,1265.39 1741.3,1265.49 1743.36,1265.59 1745.42,1265.7 1747.48,1265.8 1749.54,1265.9 1751.59,1266 1753.65,1266.11 1755.71,1266.21 \n",
       "  1757.77,1266.31 1759.83,1266.41 1761.89,1266.51 1763.95,1266.61 1766.01,1266.71 1768.06,1266.81 1770.12,1266.91 1772.18,1267.01 1774.24,1267.1 1776.3,1267.2 \n",
       "  1778.36,1267.3 1780.42,1267.4 1782.48,1267.5 1784.53,1267.59 1786.59,1267.69 1788.65,1267.79 1790.71,1267.88 1792.77,1267.98 1794.83,1268.07 1796.89,1268.17 \n",
       "  1798.95,1268.26 1801,1268.36 1803.06,1268.45 1805.12,1268.55 1807.18,1268.64 1809.24,1268.74 1811.3,1268.83 1813.36,1268.92 1815.42,1269.01 1817.47,1269.11 \n",
       "  1819.53,1269.2 1821.59,1269.29 1823.65,1269.38 1825.71,1269.47 1827.77,1269.56 1829.83,1269.65 1831.89,1269.75 1833.94,1269.84 1836,1269.92 1838.06,1270.01 \n",
       "  1840.12,1270.1 1842.18,1270.19 1844.24,1270.28 1846.3,1270.37 1848.36,1270.46 1850.42,1270.54 1852.47,1270.63 1854.53,1270.72 1856.59,1270.81 1858.65,1270.89 \n",
       "  1860.71,1270.98 1862.77,1271.06 1864.83,1271.15 1866.89,1271.23 1868.94,1271.32 1871,1271.4 1873.06,1271.49 1875.12,1271.57 1877.18,1271.66 1879.24,1271.74 \n",
       "  1881.3,1271.82 1883.36,1271.91 1885.41,1271.99 1887.47,1272.07 1889.53,1272.15 1891.59,1272.23 1893.65,1272.32 1895.71,1272.4 1897.77,1272.48 1899.83,1272.56 \n",
       "  1901.88,1272.64 1903.94,1272.72 1906,1272.8 1908.06,1272.88 1910.12,1272.96 1912.18,1273.04 1914.24,1273.12 1916.3,1273.2 1918.35,1273.28 1920.41,1273.36 \n",
       "  1922.47,1273.44 1924.53,1273.52 1926.59,1273.6 1928.65,1273.68 1930.71,1273.76 1932.77,1273.83 1934.82,1273.91 1936.88,1273.99 1938.94,1274.07 1941,1274.14 \n",
       "  1943.06,1274.22 1945.12,1274.3 1947.18,1274.37 1949.24,1274.45 1951.3,1274.52 1953.35,1274.6 1955.41,1274.68 1957.47,1274.75 1959.53,1274.83 1961.59,1274.9 \n",
       "  1963.65,1274.98 1965.71,1275.05 1967.77,1275.12 1969.82,1275.2 1971.88,1275.27 1973.94,1275.35 1976,1275.42 1978.06,1275.49 1980.12,1275.57 1982.18,1275.64 \n",
       "  1984.24,1275.71 1986.29,1275.78 1988.35,1275.86 1990.41,1275.93 1992.47,1276 1994.53,1276.07 1996.59,1276.14 1998.65,1276.21 2000.71,1276.28 2002.76,1276.35 \n",
       "  2004.82,1276.42 2006.88,1276.49 2008.94,1276.56 2011,1276.63 2013.06,1276.7 2015.12,1276.77 2017.18,1276.84 2019.23,1276.91 2021.29,1276.98 2023.35,1277.05 \n",
       "  2025.41,1277.12 2027.47,1277.18 2029.53,1277.25 2031.59,1277.32 2033.65,1277.39 2035.7,1277.45 2037.76,1277.52 2039.82,1277.59 2041.88,1277.65 2043.94,1277.72 \n",
       "  2046,1277.79 2048.06,1277.85 2050.12,1277.92 2052.17,1277.98 2054.23,1278.05 2056.29,1278.11 2058.35,1278.18 2060.41,1278.24 2062.47,1278.31 2064.53,1278.37 \n",
       "  2066.59,1278.44 2068.65,1278.5 2070.7,1278.56 2072.76,1278.63 2074.82,1278.69 2076.88,1278.75 2078.94,1278.82 2081,1278.88 2083.06,1278.94 2085.12,1279.01 \n",
       "  2087.17,1279.07 2089.23,1279.13 2091.29,1279.19 2093.35,1279.26 2095.41,1279.32 2097.47,1279.38 2099.53,1279.44 2101.59,1279.5 2103.64,1279.56 2105.7,1279.63 \n",
       "  2107.76,1279.69 2109.82,1279.75 2111.88,1279.81 2113.94,1279.87 2116,1279.93 2118.06,1279.99 2120.11,1280.05 2122.17,1280.11 2124.23,1280.17 2126.29,1280.23 \n",
       "  2128.35,1280.29 2130.41,1280.35 2132.47,1280.41 2134.53,1280.46 2136.58,1280.52 2138.64,1280.58 2140.7,1280.64 2142.76,1280.7 2144.82,1280.76 2146.88,1280.81 \n",
       "  2148.94,1280.87 2151,1280.93 2153.05,1280.99 2155.11,1281.04 2157.17,1281.1 2159.23,1281.16 2161.29,1281.21 2163.35,1281.27 2165.41,1281.33 2167.47,1281.38 \n",
       "  2169.53,1281.44 2171.58,1281.49 2173.64,1281.55 2175.7,1281.6 2177.76,1281.66 2179.82,1281.71 2181.88,1281.77 2183.94,1281.82 2186,1281.88 2188.05,1281.93 \n",
       "  2190.11,1281.99 2192.17,1282.04 2194.23,1282.1 2196.29,1282.15 2198.35,1282.2 2200.41,1282.26 2202.47,1282.31 2204.52,1282.36 2206.58,1282.42 2208.64,1282.47 \n",
       "  2210.7,1282.52 2212.76,1282.57 2214.82,1282.63 2216.88,1282.68 2218.94,1282.73 2220.99,1282.78 2223.05,1282.83 2225.11,1282.88 2227.17,1282.94 2229.23,1282.99 \n",
       "  2231.29,1283.04 2233.35,1283.09 2235.41,1283.14 2237.46,1283.19 2239.52,1283.24 2241.58,1283.29 2243.64,1283.34 2245.7,1283.39 2247.76,1283.44 2249.82,1283.49 \n",
       "  2251.88,1283.54 2253.93,1283.59 2255.99,1283.64 2258.05,1283.68 2260.11,1283.73 2262.17,1283.78 2264.23,1283.83 2266.29,1283.88 2268.35,1283.93 2270.4,1283.98 \n",
       "  2272.46,1284.02 2274.52,1284.07 2276.58,1284.12 2278.64,1284.17 2280.7,1284.22 2282.76,1284.26 2284.82,1284.31 2286.88,1284.36 2288.93,1284.4 2290.99,1284.45 \n",
       "  2293.05,1284.5 2295.11,1284.55 2297.17,1284.59 2299.23,1284.64 2301.29,1284.68 2303.35,1284.73 2305.4,1284.78 2307.46,1284.82 2309.52,1284.87 2311.58,1284.91 \n",
       "  2313.64,1284.96 2315.7,1285.01 2317.76,1285.05 2319.82,1285.1 2321.87,1285.14 2323.93,1285.19 2325.99,1285.23 2328.05,1285.28 2330.11,1285.32 2332.17,1285.37 \n",
       "  2334.23,1285.41 2336.29,1285.45 2338.34,1285.5 2340.4,1285.54 2342.46,1285.59 2344.52,1285.63 2346.58,1285.67 2348.64,1285.72 2350.7,1285.76 2352.76,1285.8 \n",
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       "  316.629,729.242 318.688,752.441 320.746,772.932 322.805,791.086 324.864,807.216 326.923,821.578 328.981,834.391 331.04,845.837 333.099,856.078 335.158,865.252 \n",
       "  337.217,873.477 339.275,880.856 341.334,887.476 343.393,893.415 345.452,898.745 347.51,903.528 349.569,907.816 351.628,911.66 353.687,915.1 355.746,918.173 \n",
       "  357.804,920.914 359.863,923.353 361.922,925.519 363.981,927.437 366.039,929.13 368.098,930.618 370.157,931.918 372.216,933.046 374.275,934.016 376.333,934.842 \n",
       "  378.392,935.537 380.451,936.112 382.51,936.579 384.568,936.946 386.627,937.222 388.686,937.416 390.745,937.532 392.803,937.577 394.862,937.557 396.921,937.476 \n",
       "  398.98,937.341 401.039,937.155 403.097,936.923 405.156,936.65 407.215,936.339 409.274,935.994 411.332,935.617 413.391,935.21 415.45,934.776 417.509,934.317 \n",
       "  419.568,933.834 421.626,933.328 423.685,932.803 425.744,932.259 427.803,931.698 429.861,931.122 431.92,930.533 433.979,929.931 436.038,929.317 438.096,928.695 \n",
       "  440.155,928.063 442.214,927.423 444.273,926.774 446.332,926.118 448.39,925.454 450.449,924.784 452.508,924.107 454.567,923.424 456.625,922.736 458.684,922.043 \n",
       "  460.743,921.345 462.802,920.643 464.861,919.938 466.919,919.23 468.978,918.519 471.037,917.806 473.096,917.092 475.154,916.377 477.213,915.659 479.272,914.941 \n",
       "  481.331,914.221 483.389,913.499 485.448,912.776 487.507,912.051 489.566,911.325 491.625,910.598 493.683,909.869 495.742,909.139 497.801,908.409 499.86,907.677 \n",
       "  501.918,906.945 503.977,906.212 506.036,905.478 508.095,904.744 510.154,904.011 512.212,903.277 514.271,902.544 516.33,901.811 518.389,901.079 520.447,900.348 \n",
       "  522.506,899.617 524.565,898.886 526.624,898.155 528.683,897.425 530.741,896.694 532.8,895.964 534.859,895.234 536.918,894.503 538.976,893.772 541.035,893.042 \n",
       "  543.094,892.311 545.153,891.58 547.211,890.849 549.27,890.118 551.329,889.388 553.388,888.657 555.447,887.927 557.505,887.197 559.564,886.468 561.623,885.739 \n",
       "  563.682,885.011 565.74,884.284 567.799,883.558 569.858,882.833 571.917,882.109 573.976,881.387 576.034,880.665 578.093,879.943 580.152,879.223 582.211,878.503 \n",
       "  584.269,877.783 586.328,877.063 588.387,876.344 590.446,875.625 592.504,874.906 594.563,874.188 596.622,873.469 598.681,872.751 600.74,872.033 602.798,871.315 \n",
       "  604.857,870.597 606.916,869.88 608.975,869.163 611.033,868.447 613.092,867.731 615.151,867.015 617.21,866.301 619.269,865.587 621.327,864.875 623.386,864.163 \n",
       "  625.445,863.453 627.504,862.744 629.562,862.037 631.621,861.331 633.68,860.626 635.739,859.922 637.798,859.219 639.856,858.516 641.915,857.814 643.974,857.113 \n",
       "  646.033,856.412 648.091,855.711 650.15,855.011 652.209,854.311 654.268,853.611 656.326,852.911 658.385,852.211 660.444,851.512 662.503,850.813 664.562,850.114 \n",
       "  666.62,849.415 668.679,848.717 670.738,848.02 672.797,847.323 674.855,846.626 676.914,845.931 678.973,845.236 681.032,844.543 683.091,843.85 685.149,843.159 \n",
       "  687.208,842.47 689.267,841.782 691.326,841.097 693.384,840.412 695.443,839.73 697.502,839.048 699.561,838.367 701.619,837.687 703.678,837.008 705.737,836.33 \n",
       "  707.796,835.652 709.855,834.975 711.913,834.298 713.972,833.621 716.031,832.944 718.09,832.268 720.148,831.591 722.207,830.915 724.266,830.24 726.325,829.564 \n",
       "  728.384,828.889 730.442,828.214 732.501,827.539 734.56,826.865 736.619,826.192 738.677,825.519 740.736,824.847 742.795,824.176 744.854,823.506 746.913,822.837 \n",
       "  748.971,822.17 751.03,821.504 753.089,820.841 755.148,820.179 757.206,819.52 759.265,818.862 761.324,818.206 763.383,817.551 765.441,816.898 767.5,816.245 \n",
       "  769.559,815.594 771.618,814.943 773.677,814.292 775.735,813.642 777.794,812.993 779.853,812.343 781.912,811.694 783.97,811.046 786.029,810.397 788.088,809.749 \n",
       "  790.147,809.1 792.206,808.452 794.264,807.805 796.323,807.157 798.382,806.51 800.441,805.863 802.499,805.217 804.558,804.571 806.617,803.926 808.676,803.282 \n",
       "  810.734,802.639 812.793,801.997 814.852,801.356 816.911,800.718 818.97,800.08 821.028,799.445 823.087,798.813 825.146,798.182 827.205,797.554 829.263,796.928 \n",
       "  831.322,796.303 833.381,795.68 835.44,795.058 837.499,794.436 839.557,793.816 841.616,793.196 843.675,792.577 845.734,791.958 847.792,791.34 849.851,790.722 \n",
       "  851.91,790.104 853.969,789.486 856.028,788.869 858.086,788.251 860.145,787.634 862.204,787.017 864.263,786.4 866.321,785.783 868.38,785.167 870.439,784.551 \n",
       "  872.498,783.935 874.556,783.32 876.615,782.706 878.674,782.093 880.733,781.481 882.792,780.87 884.85,780.26 886.909,779.653 888.968,779.047 891.027,778.443 \n",
       "  893.085,777.841 895.144,777.242 897.203,776.646 899.262,776.052 901.321,775.46 903.379,774.869 905.438,774.279 907.497,773.691 909.556,773.103 911.614,772.517 \n",
       "  913.673,771.931 915.732,771.345 917.791,770.76 919.849,770.175 921.908,769.591 923.967,769.006 926.026,768.422 928.085,767.838 930.143,767.254 932.202,766.67 \n",
       "  934.261,766.086 936.32,765.502 938.378,764.918 940.437,764.335 942.496,763.752 944.555,763.17 946.614,762.588 948.672,762.007 950.731,761.427 952.79,760.848 \n",
       "  954.849,760.27 956.907,759.694 958.966,759.119 961.025,758.546 963.084,757.976 965.143,757.408 967.201,756.843 969.26,756.28 971.319,755.72 973.378,755.162 \n",
       "  975.436,754.606 977.495,754.051 979.554,753.497 981.613,752.944 983.671,752.392 985.73,751.841 987.789,751.29 989.848,750.74 991.907,750.19 993.965,749.64 \n",
       "  996.024,749.09 998.083,748.541 1000.14,747.991 1002.2,747.442 1004.26,746.892 1006.32,746.343 1008.38,745.794 1010.44,745.245 1012.49,744.696 1014.55,744.147 \n",
       "  1016.61,743.598 1018.67,743.051 1020.73,742.503 1022.79,741.957 1024.85,741.411 1026.91,740.867 1028.96,740.324 1031.02,739.782 1033.08,739.243 1035.14,738.705 \n",
       "  1037.2,738.17 1039.26,737.637 1041.32,737.107 1043.38,736.581 1045.43,736.056 1047.49,735.534 1049.55,735.013 1051.61,734.494 1053.67,733.976 1055.73,733.459 \n",
       "  1057.79,732.943 1059.85,732.427 1061.9,731.912 1063.96,731.397 1066.02,730.883 1068.08,730.369 1070.14,729.855 1072.2,729.34 1074.26,728.826 1076.32,728.312 \n",
       "  1078.38,727.798 1080.43,727.284 1082.49,726.77 1084.55,726.255 1086.61,725.741 1088.67,725.227 1090.73,724.714 1092.79,724.201 1094.85,723.688 1096.9,723.176 \n",
       "  1098.96,722.665 1101.02,722.155 1103.08,721.646 1105.14,721.139 1107.2,720.634 1109.26,720.13 1111.32,719.629 1113.37,719.13 1115.43,718.635 1117.49,718.142 \n",
       "  1119.55,717.652 1121.61,717.165 1123.67,716.679 1125.73,716.195 1127.79,715.712 1129.84,715.231 1131.9,714.75 1133.96,714.27 1136.02,713.79 1138.08,713.311 \n",
       "  1140.14,712.832 1142.2,712.353 1144.26,711.874 1146.31,711.396 1148.37,710.917 1150.43,710.438 1152.49,709.959 1154.55,709.48 1156.61,709.001 1158.67,708.521 \n",
       "  1160.73,708.042 1162.78,707.563 1164.84,707.083 1166.9,706.605 1168.96,706.126 1171.02,705.648 1173.08,705.171 1175.14,704.695 1177.2,704.219 1179.25,703.746 \n",
       "  1181.31,703.273 1183.37,702.803 1185.43,702.334 1187.49,701.868 1189.55,701.405 1191.61,700.945 1193.67,700.488 1195.73,700.034 1197.78,699.582 1199.84,699.132 \n",
       "  1201.9,698.683 1203.96,698.236 1206.02,697.789 1208.08,697.344 1210.14,696.899 1212.2,696.454 1214.25,696.01 1216.31,695.566 1218.37,695.122 1220.43,694.679 \n",
       "  1222.49,694.234 1224.55,693.79 1226.61,693.346 1228.67,692.901 1230.72,692.456 1232.78,692.011 1234.84,691.566 1236.9,691.121 1238.96,690.676 1241.02,690.23 \n",
       "  1243.08,689.785 1245.14,689.341 1247.19,688.896 1249.25,688.453 1251.31,688.01 1253.37,687.568 1255.43,687.128 1257.49,686.689 1259.55,686.252 1261.61,685.817 \n",
       "  1263.66,685.385 1265.72,684.955 1267.78,684.529 1269.84,684.105 1271.9,683.685 1273.96,683.267 1276.02,682.851 1278.08,682.436 1280.13,682.023 1282.19,681.611 \n",
       "  1284.25,681.199 1286.31,680.788 1288.37,680.378 1290.43,679.968 1292.49,679.558 1294.55,679.147 1296.61,678.737 1298.66,678.327 1300.72,677.916 1302.78,677.505 \n",
       "  1304.84,677.094 1306.9,676.682 1308.96,676.27 1311.02,675.858 1313.08,675.446 1315.13,675.033 1317.19,674.621 1319.25,674.208 1321.31,673.796 1323.37,673.384 \n",
       "  1325.43,672.973 1327.49,672.562 1329.55,672.153 1331.6,671.744 1333.66,671.337 1335.72,670.932 1337.78,670.529 1339.84,670.128 1341.9,669.73 1343.96,669.335 \n",
       "  1346.02,668.944 1348.07,668.556 1350.13,668.17 1352.19,667.786 1354.25,667.404 1356.31,667.023 1358.37,666.643 1360.43,666.264 1362.49,665.886 1364.54,665.508 \n",
       "  1366.6,665.13 1368.66,664.752 1370.72,664.374 1372.78,663.996 1374.84,663.618 1376.9,663.24 1378.96,662.861 1381.01,662.481 1383.07,662.102 1385.13,661.721 \n",
       "  1387.19,661.341 1389.25,660.96 1391.31,660.579 1393.37,660.197 1395.43,659.816 1397.48,659.435 1399.54,659.054 1401.6,658.673 1403.66,658.293 1405.72,657.914 \n",
       "  1407.78,657.536 1409.84,657.159 1411.9,656.784 1413.96,656.411 1416.01,656.04 1418.07,655.672 1420.13,655.306 1422.19,654.944 1424.25,654.586 1426.31,654.231 \n",
       "  1428.37,653.876 1430.43,653.521 1432.48,653.167 1434.54,652.814 1436.6,652.461 1438.66,652.109 1440.72,651.758 1442.78,651.407 1444.84,651.057 1446.9,650.707 \n",
       "  1448.95,650.359 1451.01,650.01 1453.07,649.663 1455.13,649.316 1457.19,648.969 1459.25,648.624 1461.31,648.278 1463.37,647.934 1465.42,647.59 1467.48,647.247 \n",
       "  1469.54,646.904 1471.6,646.562 1473.66,646.221 1475.72,645.88 1477.78,645.54 1479.84,645.201 1481.89,644.862 1483.95,644.524 1486.01,644.186 1488.07,643.849 \n",
       "  1490.13,643.513 1492.19,643.177 1494.25,642.842 1496.31,642.508 1498.36,642.174 1500.42,641.841 1502.48,641.508 1504.54,641.176 1506.6,640.845 1508.66,640.515 \n",
       "  1510.72,640.185 1512.78,639.855 1514.84,639.526 1516.89,639.198 1518.95,638.871 1521.01,638.544 1523.07,638.218 1525.13,637.892 1527.19,637.567 1529.25,637.243 \n",
       "  1531.31,636.919 1533.36,636.596 1535.42,636.273 1537.48,635.952 1539.54,635.63 1541.6,635.31 1543.66,634.99 1545.72,634.67 1547.78,634.352 1549.83,634.034 \n",
       "  1551.89,633.716 1553.95,633.399 1556.01,633.083 1558.07,632.768 1560.13,632.453 1562.19,632.138 1564.25,631.825 1566.3,631.512 1568.36,631.199 1570.42,630.887 \n",
       "  1572.48,630.576 1574.54,630.266 1576.6,629.956 1578.66,629.648 1580.72,629.342 1582.77,629.036 1584.83,628.731 1586.89,628.427 1588.95,628.124 1591.01,627.821 \n",
       "  1593.07,627.519 1595.13,627.217 1597.19,626.916 1599.24,626.616 1601.3,626.317 1603.36,626.018 1605.42,625.72 1607.48,625.422 1609.54,625.125 1611.6,624.829 \n",
       "  1613.66,624.534 1615.71,624.239 1617.77,623.945 1619.83,623.651 1621.89,623.358 1623.95,623.066 1626.01,622.775 1628.07,622.484 1630.13,622.194 1632.19,621.904 \n",
       "  1634.24,621.615 1636.3,621.327 1638.36,621.04 1640.42,620.753 1642.48,620.467 1644.54,620.181 1646.6,619.896 1648.66,619.612 1650.71,619.329 1652.77,619.046 \n",
       "  1654.83,618.764 1656.89,618.482 1658.95,618.201 1661.01,617.921 1663.07,617.642 1665.13,617.363 1667.18,617.085 1669.24,616.807 1671.3,616.531 1673.36,616.255 \n",
       "  1675.42,615.979 1677.48,615.704 1679.54,615.43 1681.6,615.157 1683.65,614.884 1685.71,614.612 1687.77,614.34 1689.83,614.069 1691.89,613.799 1693.95,613.53 \n",
       "  1696.01,613.261 1698.07,612.993 1700.12,612.725 1702.18,612.459 1704.24,612.192 1706.3,611.927 1708.36,611.662 1710.42,611.398 1712.48,611.135 1714.54,610.872 \n",
       "  1716.59,610.61 1718.65,610.349 1720.71,610.089 1722.77,609.831 1724.83,609.572 1726.89,609.315 1728.95,609.058 1731.01,608.801 1733.07,608.545 1735.12,608.29 \n",
       "  1737.18,608.035 1739.24,607.78 1741.3,607.526 1743.36,607.273 1745.42,607.02 1747.48,606.768 1749.54,606.516 1751.59,606.265 1753.65,606.014 1755.71,605.764 \n",
       "  1757.77,605.515 1759.83,605.266 1761.89,605.017 1763.95,604.769 1766.01,604.522 1768.06,604.275 1770.12,604.029 1772.18,603.783 1774.24,603.537 1776.3,603.293 \n",
       "  1778.36,603.049 1780.42,602.805 1782.48,602.562 1784.53,602.319 1786.59,602.077 1788.65,601.836 1790.71,601.595 1792.77,601.354 1794.83,601.114 1796.89,600.875 \n",
       "  1798.95,600.636 1801,600.398 1803.06,600.16 1805.12,599.923 1807.18,599.686 1809.24,599.45 1811.3,599.215 1813.36,598.979 1815.42,598.745 1817.47,598.511 \n",
       "  1819.53,598.278 1821.59,598.045 1823.65,597.812 1825.71,597.58 1827.77,597.349 1829.83,597.118 1831.89,596.888 1833.94,596.659 1836,596.429 1838.06,596.201 \n",
       "  1840.12,595.973 1842.18,595.745 1844.24,595.518 1846.3,595.292 1848.36,595.066 1850.42,594.84 1852.47,594.616 1854.53,594.391 1856.59,594.168 1858.65,593.944 \n",
       "  1860.71,593.722 1862.77,593.5 1864.83,593.278 1866.89,593.057 1868.94,592.836 1871,592.616 1873.06,592.397 1875.12,592.178 1877.18,591.96 1879.24,591.742 \n",
       "  1881.3,591.524 1883.36,591.308 1885.41,591.091 1887.47,590.876 1889.53,590.661 1891.59,590.448 1893.65,590.235 1895.71,590.023 1897.77,589.811 1899.83,589.599 \n",
       "  1901.88,589.388 1903.94,589.178 1906,588.968 1908.06,588.758 1910.12,588.549 1912.18,588.341 1914.24,588.133 1916.3,587.925 1918.35,587.718 1920.41,587.511 \n",
       "  1922.47,587.304 1924.53,587.099 1926.59,586.893 1928.65,586.688 1930.71,586.484 1932.77,586.28 1934.82,586.076 1936.88,585.873 1938.94,585.671 1941,585.468 \n",
       "  1943.06,585.267 1945.12,585.066 1947.18,584.865 1949.24,584.664 1951.3,584.465 1953.35,584.265 1955.41,584.066 1957.47,583.868 1959.53,583.67 1961.59,583.472 \n",
       "  1963.65,583.275 1965.71,583.079 1967.77,582.883 1969.82,582.687 1971.88,582.492 1973.94,582.297 1976,582.103 1978.06,581.909 1980.12,581.716 1982.18,581.523 \n",
       "  1984.24,581.33 1986.29,581.138 1988.35,580.947 1990.41,580.756 1992.47,580.565 1994.53,580.375 1996.59,580.185 1998.65,579.996 2000.71,579.807 2002.76,579.619 \n",
       "  2004.82,579.431 2006.88,579.244 2008.94,579.057 2011,578.871 2013.06,578.685 2015.12,578.499 2017.18,578.314 2019.23,578.13 2021.29,577.946 2023.35,577.762 \n",
       "  2025.41,577.579 2027.47,577.396 2029.53,577.214 2031.59,577.032 2033.65,576.851 2035.7,576.67 2037.76,576.489 2039.82,576.309 2041.88,576.13 2043.94,575.951 \n",
       "  2046,575.772 2048.06,575.594 2050.12,575.417 2052.17,575.24 2054.23,575.063 2056.29,574.887 2058.35,574.712 2060.41,574.538 2062.47,574.364 2064.53,574.191 \n",
       "  2066.59,574.018 2068.65,573.845 2070.7,573.672 2072.76,573.5 2074.82,573.329 2076.88,573.157 2078.94,572.987 2081,572.816 2083.06,572.646 2085.12,572.476 \n",
       "  2087.17,572.307 2089.23,572.138 2091.29,571.969 2093.35,571.801 2095.41,571.633 2097.47,571.465 2099.53,571.298 2101.59,571.132 2103.64,570.965 2105.7,570.799 \n",
       "  2107.76,570.634 2109.82,570.468 2111.88,570.303 2113.94,570.139 2116,569.975 2118.06,569.811 2120.11,569.648 2122.17,569.485 2124.23,569.322 2126.29,569.16 \n",
       "  2128.35,568.998 2130.41,568.836 2132.47,568.675 2134.53,568.515 2136.58,568.354 2138.64,568.194 2140.7,568.035 2142.76,567.875 2144.82,567.717 2146.88,567.558 \n",
       "  2148.94,567.4 2151,567.242 2153.05,567.085 2155.11,566.928 2157.17,566.771 2159.23,566.615 2161.29,566.459 2163.35,566.304 2165.41,566.149 2167.47,565.994 \n",
       "  2169.53,565.839 2171.58,565.686 2173.64,565.532 2175.7,565.379 2177.76,565.226 2179.82,565.073 2181.88,564.921 2183.94,564.77 2186,564.618 2188.05,564.467 \n",
       "  2190.11,564.317 2192.17,564.166 2194.23,564.017 2196.29,563.867 2198.35,563.718 2200.41,563.569 2202.47,563.421 2204.52,563.273 2206.58,563.125 2208.64,562.978 \n",
       "  2210.7,562.831 2212.76,562.685 2214.82,562.539 2216.88,562.393 2218.94,562.248 2220.99,562.103 2223.05,561.958 2225.11,561.814 2227.17,561.67 2229.23,561.527 \n",
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       "  1263.66,1095.08 1265.72,1095.41 1267.78,1095.75 1269.84,1096.1 1271.9,1096.47 1273.96,1096.84 1276.02,1097.21 1278.08,1097.59 1280.13,1097.97 1282.19,1098.35 \n",
       "  1284.25,1098.72 1286.31,1099.1 1288.37,1099.47 1290.43,1099.83 1292.49,1100.19 1294.55,1100.54 1296.61,1100.89 1298.66,1101.23 1300.72,1101.56 1302.78,1101.88 \n",
       "  1304.84,1102.19 1306.9,1102.5 1308.96,1102.79 1311.02,1103.08 1313.08,1103.36 1315.13,1103.64 1317.19,1103.91 1319.25,1104.17 1321.31,1104.42 1323.37,1104.68 \n",
       "  1325.43,1104.93 1327.49,1105.18 1329.55,1105.43 1331.6,1105.68 1333.66,1105.93 1335.72,1106.19 1337.78,1106.45 1339.84,1106.72 1341.9,1107 1343.96,1107.29 \n",
       "  1346.02,1107.6 1348.07,1107.92 1350.13,1108.25 1352.19,1108.59 1354.25,1108.93 1356.31,1109.27 1358.37,1109.61 1360.43,1109.95 1362.49,1110.28 1364.54,1110.61 \n",
       "  1366.6,1110.94 1368.66,1111.26 1370.72,1111.58 1372.78,1111.89 1374.84,1112.19 1376.9,1112.49 1378.96,1112.77 1381.01,1113.05 1383.07,1113.32 1385.13,1113.58 \n",
       "  1387.19,1113.83 1389.25,1114.07 1391.31,1114.31 1393.37,1114.54 1395.43,1114.76 1397.48,1114.98 1399.54,1115.2 1401.6,1115.41 1403.66,1115.62 1405.72,1115.83 \n",
       "  1407.78,1116.04 1409.84,1116.25 1411.9,1116.47 1413.96,1116.69 1416.01,1116.92 1418.07,1117.16 1420.13,1117.41 1422.19,1117.67 1424.25,1117.95 1426.31,1118.23 \n",
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       "  1448.95,1121.18 1451.01,1121.44 1453.07,1121.7 1455.13,1121.96 1457.19,1122.21 1459.25,1122.47 1461.31,1122.73 1463.37,1122.98 1465.42,1123.23 1467.48,1123.48 \n",
       "  1469.54,1123.73 1471.6,1123.98 1473.66,1124.23 1475.72,1124.47 1477.78,1124.72 1479.84,1124.96 1481.89,1125.21 1483.95,1125.45 1486.01,1125.69 1488.07,1125.93 \n",
       "  1490.13,1126.16 1492.19,1126.4 1494.25,1126.64 1496.31,1126.87 1498.36,1127.1 1500.42,1127.34 1502.48,1127.57 1504.54,1127.79 1506.6,1128.02 1508.66,1128.25 \n",
       "  1510.72,1128.48 1512.78,1128.7 1514.84,1128.92 1516.89,1129.14 1518.95,1129.37 1521.01,1129.59 1523.07,1129.8 1525.13,1130.02 1527.19,1130.24 1529.25,1130.45 \n",
       "  1531.31,1130.66 1533.36,1130.88 1535.42,1131.09 1537.48,1131.3 1539.54,1131.51 1541.6,1131.71 1543.66,1131.92 1545.72,1132.12 1547.78,1132.33 1549.83,1132.53 \n",
       "  1551.89,1132.73 1553.95,1132.93 1556.01,1133.13 1558.07,1133.33 1560.13,1133.53 1562.19,1133.72 1564.25,1133.91 1566.3,1134.11 1568.36,1134.3 1570.42,1134.49 \n",
       "  1572.48,1134.68 1574.54,1134.87 1576.6,1135.05 1578.66,1135.24 1580.72,1135.43 1582.77,1135.62 1584.83,1135.81 1586.89,1136 1588.95,1136.19 1591.01,1136.37 \n",
       "  1593.07,1136.56 1595.13,1136.75 1597.19,1136.93 1599.24,1137.12 1601.3,1137.3 1603.36,1137.49 1605.42,1137.67 1607.48,1137.85 1609.54,1138.04 1611.6,1138.22 \n",
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       "  1634.24,1140.2 1636.3,1140.37 1638.36,1140.55 1640.42,1140.72 1642.48,1140.9 1644.54,1141.07 1646.6,1141.25 1648.66,1141.42 1650.71,1141.59 1652.77,1141.77 \n",
       "  1654.83,1141.94 1656.89,1142.11 1658.95,1142.28 1661.01,1142.45 1663.07,1142.62 1665.13,1142.79 1667.18,1142.96 1669.24,1143.13 1671.3,1143.3 1673.36,1143.46 \n",
       "  1675.42,1143.63 1677.48,1143.8 1679.54,1143.96 1681.6,1144.13 1683.65,1144.29 1685.71,1144.46 1687.77,1144.62 1689.83,1144.79 1691.89,1144.95 1693.95,1145.11 \n",
       "  1696.01,1145.27 1698.07,1145.44 1700.12,1145.6 1702.18,1145.76 1704.24,1145.92 1706.3,1146.08 1708.36,1146.24 1710.42,1146.4 1712.48,1146.55 1714.54,1146.71 \n",
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       "  1737.18,1148.42 1739.24,1148.57 1741.3,1148.72 1743.36,1148.88 1745.42,1149.03 1747.48,1149.18 1749.54,1149.33 1751.59,1149.48 1753.65,1149.63 1755.71,1149.77 \n",
       "  1757.77,1149.92 1759.83,1150.07 1761.89,1150.22 1763.95,1150.36 1766.01,1150.51 1768.06,1150.66 1770.12,1150.8 1772.18,1150.95 1774.24,1151.09 1776.3,1151.23 \n",
       "  1778.36,1151.38 1780.42,1151.52 1782.48,1151.66 1784.53,1151.8 1786.59,1151.95 1788.65,1152.09 1790.71,1152.23 1792.77,1152.37 1794.83,1152.51 1796.89,1152.64 \n",
       "  1798.95,1152.78 1801,1152.92 1803.06,1153.06 1805.12,1153.19 1807.18,1153.33 1809.24,1153.47 1811.3,1153.6 1813.36,1153.74 1815.42,1153.87 1817.47,1154.01 \n",
       "  1819.53,1154.14 1821.59,1154.27 1823.65,1154.4 1825.71,1154.54 1827.77,1154.67 1829.83,1154.8 1831.89,1154.93 1833.94,1155.06 1836,1155.19 1838.06,1155.32 \n",
       "  1840.12,1155.45 1842.18,1155.58 1844.24,1155.7 1846.3,1155.83 1848.36,1155.96 1850.42,1156.08 1852.47,1156.21 1854.53,1156.33 1856.59,1156.46 1858.65,1156.58 \n",
       "  1860.71,1156.71 1862.77,1156.83 1864.83,1156.95 1866.89,1157.07 1868.94,1157.2 1871,1157.32 1873.06,1157.44 1875.12,1157.56 1877.18,1157.68 1879.24,1157.8 \n",
       "  1881.3,1157.92 1883.36,1158.04 1885.41,1158.15 1887.47,1158.27 1889.53,1158.39 1891.59,1158.51 1893.65,1158.62 1895.71,1158.74 1897.77,1158.86 1899.83,1158.97 \n",
       "  1901.88,1159.09 1903.94,1159.21 1906,1159.32 1908.06,1159.44 1910.12,1159.55 1912.18,1159.67 1914.24,1159.78 1916.3,1159.9 1918.35,1160.01 1920.41,1160.12 \n",
       "  1922.47,1160.24 1924.53,1160.35 1926.59,1160.46 1928.65,1160.57 1930.71,1160.68 1932.77,1160.8 1934.82,1160.91 1936.88,1161.02 1938.94,1161.13 1941,1161.24 \n",
       "  1943.06,1161.35 1945.12,1161.46 1947.18,1161.57 1949.24,1161.67 1951.3,1161.78 1953.35,1161.89 1955.41,1162 1957.47,1162.11 1959.53,1162.21 1961.59,1162.32 \n",
       "  1963.65,1162.43 1965.71,1162.53 1967.77,1162.64 1969.82,1162.74 1971.88,1162.85 1973.94,1162.95 1976,1163.06 1978.06,1163.16 1980.12,1163.26 1982.18,1163.37 \n",
       "  1984.24,1163.47 1986.29,1163.57 1988.35,1163.68 1990.41,1163.78 1992.47,1163.88 1994.53,1163.98 1996.59,1164.08 1998.65,1164.18 2000.71,1164.28 2002.76,1164.38 \n",
       "  2004.82,1164.48 2006.88,1164.58 2008.94,1164.68 2011,1164.78 2013.06,1164.88 2015.12,1164.98 2017.18,1165.08 2019.23,1165.17 2021.29,1165.27 2023.35,1165.37 \n",
       "  2025.41,1165.46 2027.47,1165.56 2029.53,1165.66 2031.59,1165.75 2033.65,1165.85 2035.7,1165.94 2037.76,1166.04 2039.82,1166.13 2041.88,1166.22 2043.94,1166.32 \n",
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       "  2066.59,1167.33 2068.65,1167.42 2070.7,1167.51 2072.76,1167.6 2074.82,1167.69 2076.88,1167.78 2078.94,1167.87 2081,1167.96 2083.06,1168.05 2085.12,1168.13 \n",
       "  2087.17,1168.22 2089.23,1168.31 2091.29,1168.4 2093.35,1168.49 2095.41,1168.57 2097.47,1168.66 2099.53,1168.75 2101.59,1168.83 2103.64,1168.92 2105.7,1169.01 \n",
       "  2107.76,1169.09 2109.82,1169.18 2111.88,1169.26 2113.94,1169.35 2116,1169.43 2118.06,1169.52 2120.11,1169.6 2122.17,1169.69 2124.23,1169.77 2126.29,1169.85 \n",
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       "  2190.11,1172.32 2192.17,1172.39 2194.23,1172.47 2196.29,1172.54 2198.35,1172.62 2200.41,1172.69 2202.47,1172.77 2204.52,1172.84 2206.58,1172.91 2208.64,1172.99 \n",
       "  2210.7,1173.06 2212.76,1173.13 2214.82,1173.2 2216.88,1173.28 2218.94,1173.35 2220.99,1173.42 2223.05,1173.49 2225.11,1173.56 2227.17,1173.63 2229.23,1173.7 \n",
       "  2231.29,1173.78 2233.35,1173.85 2235.41,1173.92 2237.46,1173.99 2239.52,1174.05 2241.58,1174.12 2243.64,1174.19 2245.7,1174.26 2247.76,1174.33 2249.82,1174.4 \n",
       "  2251.88,1174.47 2253.93,1174.54 2255.99,1174.6 2258.05,1174.67 2260.11,1174.74 2262.17,1174.81 2264.23,1174.87 2266.29,1174.94 2268.35,1175.01 2270.4,1175.07 \n",
       "  2272.46,1175.14 2274.52,1175.21 2276.58,1175.27 2278.64,1175.34 2280.7,1175.4 2282.76,1175.47 2284.82,1175.54 2286.88,1175.6 2288.93,1175.67 2290.99,1175.73 \n",
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       "  1696.01,934.033 1698.07,934.301 1700.12,934.568 1702.18,934.835 1704.24,935.101 1706.3,935.367 1708.36,935.632 1710.42,935.896 1712.48,936.159 1714.54,936.422 \n",
       "  1716.59,936.684 1718.65,936.945 1720.71,937.204 1722.77,937.463 1724.83,937.721 1726.89,937.979 1728.95,938.236 1731.01,938.493 1733.07,938.749 1735.12,939.004 \n",
       "  1737.18,939.259 1739.24,939.513 1741.3,939.767 1743.36,940.021 1745.42,940.273 1747.48,940.526 1749.54,940.777 1751.59,941.029 1753.65,941.279 1755.71,941.529 \n",
       "  1757.77,941.779 1759.83,942.028 1761.89,942.277 1763.95,942.525 1766.01,942.772 1768.06,943.019 1770.12,943.265 1772.18,943.511 1774.24,943.756 1776.3,944.001 \n",
       "  1778.36,944.245 1780.42,944.489 1782.48,944.732 1784.53,944.975 1786.59,945.217 1788.65,945.458 1790.71,945.699 1792.77,945.94 1794.83,946.179 1796.89,946.419 \n",
       "  1798.95,946.658 1801,946.896 1803.06,947.134 1805.12,947.371 1807.18,947.607 1809.24,947.844 1811.3,948.079 1813.36,948.314 1815.42,948.549 1817.47,948.783 \n",
       "  1819.53,949.016 1821.59,949.249 1823.65,949.481 1825.71,949.713 1827.77,949.945 1829.83,950.175 1831.89,950.406 1833.94,950.635 1836,950.864 1838.06,951.093 \n",
       "  1840.12,951.321 1842.18,951.549 1844.24,951.776 1846.3,952.002 1848.36,952.228 1850.42,952.453 1852.47,952.678 1854.53,952.902 1856.59,953.126 1858.65,953.349 \n",
       "  1860.71,953.572 1862.77,953.794 1864.83,954.016 1866.89,954.237 1868.94,954.457 1871,954.677 1873.06,954.897 1875.12,955.116 1877.18,955.334 1879.24,955.552 \n",
       "  1881.3,955.769 1883.36,955.986 1885.41,956.202 1887.47,956.418 1889.53,956.633 1891.59,956.846 1893.65,957.059 1895.71,957.271 1897.77,957.483 1899.83,957.694 \n",
       "  1901.88,957.905 1903.94,958.116 1906,958.326 1908.06,958.535 1910.12,958.744 1912.18,958.953 1914.24,959.161 1916.3,959.369 1918.35,959.576 1920.41,959.783 \n",
       "  1922.47,959.989 1924.53,960.195 1926.59,960.401 1928.65,960.605 1930.71,960.81 1932.77,961.014 1934.82,961.217 1936.88,961.421 1938.94,961.623 1941,961.825 \n",
       "  1943.06,962.027 1945.12,962.228 1947.18,962.429 1949.24,962.629 1951.3,962.829 1953.35,963.028 1955.41,963.227 1957.47,963.426 1959.53,963.624 1961.59,963.821 \n",
       "  1963.65,964.018 1965.71,964.215 1967.77,964.411 1969.82,964.607 1971.88,964.802 1973.94,964.997 1976,965.191 1978.06,965.385 1980.12,965.578 1982.18,965.771 \n",
       "  1984.24,965.964 1986.29,966.156 1988.35,966.347 1990.41,966.538 1992.47,966.729 1994.53,966.919 1996.59,967.108 1998.65,967.298 2000.71,967.486 2002.76,967.675 \n",
       "  2004.82,967.862 2006.88,968.05 2008.94,968.237 2011,968.423 2013.06,968.609 2015.12,968.795 2017.18,968.98 2019.23,969.164 2021.29,969.348 2023.35,969.532 \n",
       "  2025.41,969.715 2027.47,969.898 2029.53,970.08 2031.59,970.262 2033.65,970.443 2035.7,970.624 2037.76,970.804 2039.82,970.984 2041.88,971.164 2043.94,971.343 \n",
       "  2046,971.521 2048.06,971.699 2050.12,971.877 2052.17,972.054 2054.23,972.231 2056.29,972.407 2058.35,972.581 2060.41,972.756 2062.47,972.929 2064.53,973.103 \n",
       "  2066.59,973.276 2068.65,973.449 2070.7,973.621 2072.76,973.793 2074.82,973.965 2076.88,974.136 2078.94,974.307 2081,974.478 2083.06,974.648 2085.12,974.818 \n",
       "  2087.17,974.987 2089.23,975.156 2091.29,975.325 2093.35,975.493 2095.41,975.661 2097.47,975.828 2099.53,975.995 2101.59,976.162 2103.64,976.329 2105.7,976.495 \n",
       "  2107.76,976.66 2109.82,976.825 2111.88,976.99 2113.94,977.155 2116,977.319 2118.06,977.483 2120.11,977.646 2122.17,977.809 2124.23,977.972 2126.29,978.134 \n",
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       "  2169.53,981.454 2171.58,981.608 2173.64,981.762 2175.7,981.915 2177.76,982.068 2179.82,982.22 2181.88,982.372 2183.94,982.524 2186,982.676 2188.05,982.827 \n",
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       "  2210.7,984.462 2212.76,984.609 2214.82,984.755 2216.88,984.901 2218.94,985.046 2220.99,985.191 2223.05,985.335 2225.11,985.48 2227.17,985.623 2229.23,985.767 \n",
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       "  1531.31,416.629 1533.36,416.417 1535.42,416.206 1537.48,415.996 1539.54,415.787 1541.6,415.58 1543.66,415.374 1545.72,415.169 1547.78,414.965 1549.83,414.763 \n",
       "  1551.89,414.561 1553.95,414.361 1556.01,414.163 1558.07,413.965 1560.13,413.768 1562.19,413.573 1564.25,413.379 1566.3,413.186 1568.36,412.995 1570.42,412.805 \n",
       "  1572.48,412.615 1574.54,412.428 1576.6,412.241 1578.66,412.053 1580.72,411.863 1582.77,411.674 1584.83,411.485 1586.89,411.296 1588.95,411.108 1591.01,410.921 \n",
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       "  1613.66,408.891 1615.71,408.71 1617.77,408.529 1619.83,408.348 1621.89,408.168 1623.95,407.989 1626.01,407.81 1628.07,407.631 1630.13,407.453 1632.19,407.276 \n",
       "  1634.24,407.099 1636.3,406.922 1638.36,406.746 1640.42,406.571 1642.48,406.395 1644.54,406.221 1646.6,406.047 1648.66,405.873 1650.71,405.7 1652.77,405.528 \n",
       "  1654.83,405.356 1656.89,405.184 1658.95,405.013 1661.01,404.843 1663.07,404.673 1665.13,404.503 1667.18,404.334 1669.24,404.165 1671.3,403.997 1673.36,403.83 \n",
       "  1675.42,403.663 1677.48,403.496 1679.54,403.33 1681.6,403.164 1683.65,402.999 1685.71,402.835 1687.77,402.671 1689.83,402.507 1691.89,402.344 1693.95,402.181 \n",
       "  1696.01,402.019 1698.07,401.858 1700.12,401.697 1702.18,401.536 1704.24,401.376 1706.3,401.216 1708.36,401.057 1710.42,400.899 1712.48,400.74 1714.54,400.583 \n",
       "  1716.59,400.426 1718.65,400.269 1720.71,400.112 1722.77,399.956 1724.83,399.8 1726.89,399.644 1728.95,399.489 1731.01,399.335 1733.07,399.181 1735.12,399.027 \n",
       "  1737.18,398.874 1739.24,398.722 1741.3,398.57 1743.36,398.418 1745.42,398.267 1747.48,398.117 1749.54,397.967 1751.59,397.817 1753.65,397.668 1755.71,397.519 \n",
       "  1757.77,397.371 1759.83,397.224 1761.89,397.076 1763.95,396.93 1766.01,396.784 1768.06,396.638 1770.12,396.493 1772.18,396.348 1774.24,396.204 1776.3,396.06 \n",
       "  1778.36,395.917 1780.42,395.774 1782.48,395.632 1784.53,395.49 1786.59,395.349 1788.65,395.208 1790.71,395.068 1792.77,394.928 1794.83,394.788 1796.89,394.65 \n",
       "  1798.95,394.511 1801,394.373 1803.06,394.236 1805.12,394.099 1807.18,393.963 1809.24,393.827 1811.3,393.691 1813.36,393.556 1815.42,393.422 1817.47,393.288 \n",
       "  1819.53,393.154 1821.59,393.021 1823.65,392.889 1825.71,392.757 1827.77,392.625 1829.83,392.494 1831.89,392.364 1833.94,392.234 1836,392.104 1838.06,391.975 \n",
       "  1840.12,391.846 1842.18,391.718 1844.24,391.591 1846.3,391.463 1848.36,391.337 1850.42,391.211 1852.47,391.085 1854.53,390.96 1856.59,390.835 1858.65,390.711 \n",
       "  1860.71,390.587 1862.77,390.464 1864.83,390.341 1866.89,390.219 1868.94,390.097 1871,389.976 1873.06,389.855 1875.12,389.735 1877.18,389.615 1879.24,389.496 \n",
       "  1881.3,389.377 1883.36,389.258 1885.41,389.14 1887.47,389.023 1889.53,388.906 1891.59,388.788 1893.65,388.67 1895.71,388.553 1897.77,388.436 1899.83,388.319 \n",
       "  1901.88,388.203 1903.94,388.087 1906,387.971 1908.06,387.856 1910.12,387.741 1912.18,387.626 1914.24,387.512 1916.3,387.398 1918.35,387.285 1920.41,387.171 \n",
       "  1922.47,387.058 1924.53,386.946 1926.59,386.833 1928.65,386.721 1930.71,386.61 1932.77,386.498 1934.82,386.387 1936.88,386.277 1938.94,386.166 1941,386.056 \n",
       "  1943.06,385.947 1945.12,385.837 1947.18,385.728 1949.24,385.62 1951.3,385.511 1953.35,385.403 1955.41,385.295 1957.47,385.188 1959.53,385.081 1961.59,384.974 \n",
       "  1963.65,384.868 1965.71,384.762 1967.77,384.656 1969.82,384.551 1971.88,384.446 1973.94,384.341 1976,384.237 1978.06,384.133 1980.12,384.029 1982.18,383.926 \n",
       "  1984.24,383.822 1986.29,383.72 1988.35,383.617 1990.41,383.515 1992.47,383.413 1994.53,383.312 1996.59,383.211 1998.65,383.11 2000.71,383.01 2002.76,382.91 \n",
       "  2004.82,382.81 2006.88,382.711 2008.94,382.611 2011,382.513 2013.06,382.414 2015.12,382.316 2017.18,382.218 2019.23,382.121 2021.29,382.024 2023.35,381.927 \n",
       "  2025.41,381.83 2027.47,381.734 2029.53,381.639 2031.59,381.543 2033.65,381.448 2035.7,381.353 2037.76,381.259 2039.82,381.165 2041.88,381.071 2043.94,380.977 \n",
       "  2046,380.884 2048.06,380.791 2050.12,380.699 2052.17,380.607 2054.23,380.515 2056.29,380.423 2058.35,380.331 2060.41,380.239 2062.47,380.148 2064.53,380.057 \n",
       "  2066.59,379.966 2068.65,379.875 2070.7,379.785 2072.76,379.695 2074.82,379.605 2076.88,379.515 2078.94,379.426 2081,379.336 2083.06,379.248 2085.12,379.159 \n",
       "  2087.17,379.071 2089.23,378.982 2091.29,378.895 2093.35,378.807 2095.41,378.72 2097.47,378.632 2099.53,378.546 2101.59,378.459 2103.64,378.373 2105.7,378.287 \n",
       "  2107.76,378.201 2109.82,378.115 2111.88,378.03 2113.94,377.945 2116,377.86 2118.06,377.775 2120.11,377.691 2122.17,377.607 2124.23,377.523 2126.29,377.44 \n",
       "  2128.35,377.357 2130.41,377.274 2132.47,377.191 2134.53,377.108 2136.58,377.026 2138.64,376.944 2140.7,376.862 2142.76,376.781 2144.82,376.7 2146.88,376.619 \n",
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       "  2169.53,375.745 2171.58,375.667 2173.64,375.589 2175.7,375.512 2177.76,375.435 2179.82,375.358 2181.88,375.281 2183.94,375.205 2186,375.129 2188.05,375.053 \n",
       "  2190.11,374.977 2192.17,374.902 2194.23,374.827 2196.29,374.752 2198.35,374.677 2200.41,374.603 2202.47,374.529 2204.52,374.455 2206.58,374.381 2208.64,374.308 \n",
       "  2210.7,374.235 2212.76,374.162 2214.82,374.09 2216.88,374.017 2218.94,373.945 2220.99,373.873 2223.05,373.802 2225.11,373.731 2227.17,373.66 2229.23,373.589 \n",
       "  2231.29,373.518 2233.35,373.448 2235.41,373.378 2237.46,373.308 2239.52,373.239 2241.58,373.17 2243.64,373.101 2245.7,373.032 2247.76,372.963 2249.82,372.895 \n",
       "  2251.88,372.826 2253.93,372.758 2255.99,372.69 2258.05,372.623 2260.11,372.555 2262.17,372.488 2264.23,372.42 2266.29,372.353 2268.35,372.286 2270.4,372.22 \n",
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       "  2334.23,370.239 2336.29,370.178 2338.34,370.117 2340.4,370.056 2342.46,369.996 2344.52,369.935 2346.58,369.875 2348.64,369.815 2350.7,369.755 2352.76,369.695 \n",
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       "  1098.96,1191.35 1101.02,1190.97 1103.08,1190.59 1105.14,1190.22 1107.2,1189.85 1109.26,1189.49 1111.32,1189.13 1113.37,1188.79 1115.43,1188.46 1117.49,1188.14 \n",
       "  1119.55,1187.84 1121.61,1187.54 1123.67,1187.24 1125.73,1186.96 1127.79,1186.67 1129.84,1186.38 1131.9,1186.1 1133.96,1185.81 1136.02,1185.52 1138.08,1185.23 \n",
       "  1140.14,1184.93 1142.2,1184.63 1144.26,1184.32 1146.31,1184.01 1148.37,1183.69 1150.43,1183.37 1152.49,1183.04 1154.55,1182.7 1156.61,1182.36 1158.67,1182.01 \n",
       "  1160.73,1181.66 1162.78,1181.3 1164.84,1180.94 1166.9,1180.57 1168.96,1180.21 1171.02,1179.83 1173.08,1179.46 1175.14,1179.09 1177.2,1178.72 1179.25,1178.36 \n",
       "  1181.31,1177.99 1183.37,1177.64 1185.43,1177.29 1187.49,1176.95 1189.55,1176.62 1191.61,1176.3 1193.67,1176 1195.73,1175.71 1197.78,1175.43 1199.84,1175.15 \n",
       "  1201.9,1174.88 1203.96,1174.61 1206.02,1174.34 1208.08,1174.07 1210.14,1173.8 1212.2,1173.52 1214.25,1173.24 1216.31,1172.96 1218.37,1172.68 1220.43,1172.38 \n",
       "  1222.49,1172.09 1224.55,1171.78 1226.61,1171.47 1228.67,1171.16 1230.72,1170.83 1232.78,1170.5 1234.84,1170.17 1236.9,1169.83 1238.96,1169.48 1241.02,1169.13 \n",
       "  1243.08,1168.78 1245.14,1168.42 1247.19,1168.06 1249.25,1167.7 1251.31,1167.34 1253.37,1166.98 1255.43,1166.63 1257.49,1166.28 1259.55,1165.93 1261.61,1165.59 \n",
       "  1263.66,1165.26 1265.72,1164.95 1267.78,1164.65 1269.84,1164.36 1271.9,1164.08 1273.96,1163.82 1276.02,1163.56 1278.08,1163.3 1280.13,1163.05 1282.19,1162.79 \n",
       "  1284.25,1162.54 1286.31,1162.29 1288.37,1162.03 1290.43,1161.77 1292.49,1161.51 1294.55,1161.24 1296.61,1160.96 1298.66,1160.68 1300.72,1160.4 1302.78,1160.1 \n",
       "  1304.84,1159.8 1306.9,1159.49 1308.96,1159.18 1311.02,1158.86 1313.08,1158.54 1315.13,1158.2 1317.19,1157.87 1319.25,1157.53 1321.31,1157.19 1323.37,1156.84 \n",
       "  1325.43,1156.49 1327.49,1156.14 1329.55,1155.8 1331.6,1155.45 1333.66,1155.12 1335.72,1154.78 1337.78,1154.46 1339.84,1154.14 1341.9,1153.83 1343.96,1153.54 \n",
       "  1346.02,1153.27 1348.07,1153.01 1350.13,1152.76 1352.19,1152.51 1354.25,1152.27 1356.31,1152.04 1358.37,1151.8 1360.43,1151.57 1362.49,1151.33 1364.54,1151.09 \n",
       "  1366.6,1150.85 1368.66,1150.61 1370.72,1150.36 1372.78,1150.1 1374.84,1149.84 1376.9,1149.57 1378.96,1149.3 1381.01,1149.02 1383.07,1148.73 1385.13,1148.43 \n",
       "  1387.19,1148.13 1389.25,1147.82 1391.31,1147.51 1393.37,1147.19 1395.43,1146.86 1397.48,1146.54 1399.54,1146.2 1401.6,1145.87 1403.66,1145.54 1405.72,1145.21 \n",
       "  1407.78,1144.87 1409.84,1144.55 1411.9,1144.23 1413.96,1143.91 1416.01,1143.61 1418.07,1143.31 1420.13,1143.03 1422.19,1142.77 1424.25,1142.52 1426.31,1142.26 \n",
       "  1428.37,1142.01 1430.43,1141.76 1432.48,1141.51 1434.54,1141.26 1436.6,1141.01 1438.66,1140.75 1440.72,1140.5 1442.78,1140.25 1444.84,1140 1446.9,1139.75 \n",
       "  1448.95,1139.5 1451.01,1139.24 1453.07,1138.99 1455.13,1138.74 1457.19,1138.49 1459.25,1138.24 1461.31,1137.99 1463.37,1137.73 1465.42,1137.48 1467.48,1137.23 \n",
       "  1469.54,1136.98 1471.6,1136.73 1473.66,1136.47 1475.72,1136.22 1477.78,1135.97 1479.84,1135.72 1481.89,1135.47 1483.95,1135.22 1486.01,1134.96 1488.07,1134.71 \n",
       "  1490.13,1134.46 1492.19,1134.21 1494.25,1133.96 1496.31,1133.7 1498.36,1133.45 1500.42,1133.2 1502.48,1132.95 1504.54,1132.7 1506.6,1132.45 1508.66,1132.19 \n",
       "  1510.72,1131.94 1512.78,1131.69 1514.84,1131.44 1516.89,1131.19 1518.95,1130.93 1521.01,1130.68 1523.07,1130.43 1525.13,1130.18 1527.19,1129.93 1529.25,1129.67 \n",
       "  1531.31,1129.42 1533.36,1129.17 1535.42,1128.92 1537.48,1128.67 1539.54,1128.41 1541.6,1128.16 1543.66,1127.91 1545.72,1127.66 1547.78,1127.41 1549.83,1127.15 \n",
       "  1551.89,1126.9 1553.95,1126.65 1556.01,1126.4 1558.07,1126.15 1560.13,1125.89 1562.19,1125.64 1564.25,1125.39 1566.3,1125.14 1568.36,1124.89 1570.42,1124.63 \n",
       "  1572.48,1124.38 1574.54,1124.13 1576.6,1123.88 1578.66,1123.63 1580.72,1123.38 1582.77,1123.14 1584.83,1122.89 1586.89,1122.65 1588.95,1122.41 1591.01,1122.16 \n",
       "  1593.07,1121.92 1595.13,1121.68 1597.19,1121.44 1599.24,1121.2 1601.3,1120.96 1603.36,1120.72 1605.42,1120.48 1607.48,1120.24 1609.54,1120 1611.6,1119.76 \n",
       "  1613.66,1119.52 1615.71,1119.29 1617.77,1119.05 1619.83,1118.82 1621.89,1118.58 1623.95,1118.35 1626.01,1118.11 1628.07,1117.88 1630.13,1117.64 1632.19,1117.41 \n",
       "  1634.24,1117.18 1636.3,1116.95 1638.36,1116.72 1640.42,1116.49 1642.48,1116.26 1644.54,1116.03 1646.6,1115.8 1648.66,1115.57 1650.71,1115.34 1652.77,1115.11 \n",
       "  1654.83,1114.89 1656.89,1114.66 1658.95,1114.43 1661.01,1114.21 1663.07,1113.98 1665.13,1113.76 1667.18,1113.53 1669.24,1113.31 1671.3,1113.09 1673.36,1112.87 \n",
       "  1675.42,1112.64 1677.48,1112.42 1679.54,1112.2 1681.6,1111.98 1683.65,1111.76 1685.71,1111.54 1687.77,1111.32 1689.83,1111.11 1691.89,1110.89 1693.95,1110.67 \n",
       "  1696.01,1110.45 1698.07,1110.24 1700.12,1110.02 1702.18,1109.81 1704.24,1109.59 1706.3,1109.38 1708.36,1109.16 1710.42,1108.95 1712.48,1108.74 1714.54,1108.53 \n",
       "  1716.59,1108.31 1718.65,1108.1 1720.71,1107.89 1722.77,1107.69 1724.83,1107.48 1726.89,1107.27 1728.95,1107.06 1731.01,1106.86 1733.07,1106.65 1735.12,1106.44 \n",
       "  1737.18,1106.24 1739.24,1106.03 1741.3,1105.83 1743.36,1105.62 1745.42,1105.42 1747.48,1105.21 1749.54,1105.01 1751.59,1104.81 1753.65,1104.6 1755.71,1104.4 \n",
       "  1757.77,1104.2 1759.83,1104 1761.89,1103.79 1763.95,1103.59 1766.01,1103.39 1768.06,1103.19 1770.12,1102.99 1772.18,1102.79 1774.24,1102.59 1776.3,1102.39 \n",
       "  1778.36,1102.19 1780.42,1101.99 1782.48,1101.8 1784.53,1101.6 1786.59,1101.4 1788.65,1101.2 1790.71,1101.01 1792.77,1100.81 1794.83,1100.61 1796.89,1100.42 \n",
       "  1798.95,1100.22 1801,1100.03 1803.06,1099.83 1805.12,1099.64 1807.18,1099.44 1809.24,1099.25 1811.3,1099.06 1813.36,1098.86 1815.42,1098.67 1817.47,1098.48 \n",
       "  1819.53,1098.29 1821.59,1098.09 1823.65,1097.9 1825.71,1097.71 1827.77,1097.52 1829.83,1097.33 1831.89,1097.14 1833.94,1096.95 1836,1096.76 1838.06,1096.57 \n",
       "  1840.12,1096.38 1842.18,1096.2 1844.24,1096.01 1846.3,1095.82 1848.36,1095.63 1850.42,1095.45 1852.47,1095.26 1854.53,1095.07 1856.59,1094.89 1858.65,1094.7 \n",
       "  1860.71,1094.52 1862.77,1094.33 1864.83,1094.15 1866.89,1093.96 1868.94,1093.78 1871,1093.6 1873.06,1093.41 1875.12,1093.23 1877.18,1093.05 1879.24,1092.87 \n",
       "  1881.3,1092.69 1883.36,1092.51 1885.41,1092.32 1887.47,1092.14 1889.53,1091.96 1891.59,1091.79 1893.65,1091.61 1895.71,1091.43 1897.77,1091.26 1899.83,1091.08 \n",
       "  1901.88,1090.9 1903.94,1090.73 1906,1090.55 1908.06,1090.38 1910.12,1090.2 1912.18,1090.03 1914.24,1089.86 1916.3,1089.68 1918.35,1089.51 1920.41,1089.34 \n",
       "  1922.47,1089.17 1924.53,1088.99 1926.59,1088.82 1928.65,1088.65 1930.71,1088.48 1932.77,1088.31 1934.82,1088.14 1936.88,1087.97 1938.94,1087.8 1941,1087.63 \n",
       "  1943.06,1087.46 1945.12,1087.29 1947.18,1087.13 1949.24,1086.96 1951.3,1086.79 1953.35,1086.62 1955.41,1086.46 1957.47,1086.29 1959.53,1086.12 1961.59,1085.96 \n",
       "  1963.65,1085.79 1965.71,1085.63 1967.77,1085.46 1969.82,1085.3 1971.88,1085.13 1973.94,1084.97 1976,1084.81 1978.06,1084.65 1980.12,1084.48 1982.18,1084.32 \n",
       "  1984.24,1084.16 1986.29,1084 1988.35,1083.84 1990.41,1083.68 1992.47,1083.51 1994.53,1083.35 1996.59,1083.19 1998.65,1083.04 2000.71,1082.88 2002.76,1082.72 \n",
       "  2004.82,1082.56 2006.88,1082.4 2008.94,1082.24 2011,1082.09 2013.06,1081.93 2015.12,1081.77 2017.18,1081.62 2019.23,1081.46 2021.29,1081.3 2023.35,1081.15 \n",
       "  2025.41,1080.99 2027.47,1080.84 2029.53,1080.69 2031.59,1080.53 2033.65,1080.38 2035.7,1080.23 2037.76,1080.07 2039.82,1079.92 2041.88,1079.77 2043.94,1079.62 \n",
       "  2046,1079.46 2048.06,1079.31 2050.12,1079.16 2052.17,1079.01 2054.23,1078.86 2056.29,1078.71 2058.35,1078.57 2060.41,1078.42 2062.47,1078.27 2064.53,1078.12 \n",
       "  2066.59,1077.98 2068.65,1077.83 2070.7,1077.69 2072.76,1077.54 2074.82,1077.39 2076.88,1077.25 2078.94,1077.1 2081,1076.96 2083.06,1076.82 2085.12,1076.67 \n",
       "  2087.17,1076.53 2089.23,1076.38 2091.29,1076.24 2093.35,1076.1 2095.41,1075.96 2097.47,1075.81 2099.53,1075.67 2101.59,1075.53 2103.64,1075.39 2105.7,1075.25 \n",
       "  2107.76,1075.11 2109.82,1074.97 2111.88,1074.83 2113.94,1074.69 2116,1074.55 2118.06,1074.41 2120.11,1074.27 2122.17,1074.13 2124.23,1073.99 2126.29,1073.85 \n",
       "  2128.35,1073.72 2130.41,1073.58 2132.47,1073.44 2134.53,1073.3 2136.58,1073.17 2138.64,1073.03 2140.7,1072.89 2142.76,1072.76 2144.82,1072.62 2146.88,1072.49 \n",
       "  2148.94,1072.35 2151,1072.22 2153.05,1072.08 2155.11,1071.95 2157.17,1071.82 2159.23,1071.68 2161.29,1071.55 2163.35,1071.42 2165.41,1071.28 2167.47,1071.15 \n",
       "  2169.53,1071.02 2171.58,1070.89 2173.64,1070.76 2175.7,1070.62 2177.76,1070.49 2179.82,1070.36 2181.88,1070.23 2183.94,1070.1 2186,1069.97 2188.05,1069.84 \n",
       "  2190.11,1069.71 2192.17,1069.58 2194.23,1069.46 2196.29,1069.33 2198.35,1069.2 2200.41,1069.07 2202.47,1068.94 2204.52,1068.82 2206.58,1068.69 2208.64,1068.56 \n",
       "  2210.7,1068.44 2212.76,1068.31 2214.82,1068.19 2216.88,1068.06 2218.94,1067.93 2220.99,1067.81 2223.05,1067.68 2225.11,1067.56 2227.17,1067.44 2229.23,1067.31 \n",
       "  2231.29,1067.19 2233.35,1067.07 2235.41,1066.94 2237.46,1066.82 2239.52,1066.7 2241.58,1066.58 2243.64,1066.45 2245.7,1066.33 2247.76,1066.21 2249.82,1066.09 \n",
       "  2251.88,1065.97 2253.93,1065.85 2255.99,1065.73 2258.05,1065.61 2260.11,1065.49 2262.17,1065.37 2264.23,1065.26 2266.29,1065.14 2268.35,1065.02 2270.4,1064.9 \n",
       "  2272.46,1064.78 2274.52,1064.67 2276.58,1064.55 2278.64,1064.43 2280.7,1064.32 2282.76,1064.2 2284.82,1064.09 2286.88,1063.97 2288.93,1063.85 2290.99,1063.74 \n",
       "  2293.05,1063.62 2295.11,1063.51 2297.17,1063.4 2299.23,1063.28 2301.29,1063.17 2303.35,1063.06 2305.4,1062.94 2307.46,1062.83 2309.52,1062.72 2311.58,1062.6 \n",
       "  2313.64,1062.49 2315.7,1062.38 2317.76,1062.27 2319.82,1062.16 2321.87,1062.05 2323.93,1061.94 2325.99,1061.83 2328.05,1061.72 2330.11,1061.61 2332.17,1061.5 \n",
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      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sol4 = solve(prob4)\n",
    "plot(sol4)\n",
    "title!(\"kinase-phosphatase\")\n",
    "xlabel!(\"Time [s]\")\n",
    "ylabel!(\"Concentration [M]\")\n",
    "#png(\"Michaelis_menten.png\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can take advantage of the conservation of mass (as in systems with one enzyme) to go from six to four differential equations. \n",
    "\n",
    "$$ \\begin{align*}\n",
    "C_1 \\cdot X_1 + C_2 \\cdot X_2 + C_3 \\cdot X_3 + C_4 \\cdot X_4 + C_5 \\cdot X_5 + C_6 \\cdot X_6 & = cte \\\\\n",
    "\\end{align*}$$\n",
    "\n",
    "so\n",
    "$$ \\begin{align*}\n",
    "C \\cdot (B-A)^T &=0\\\\ \n",
    "\\begin{pmatrix}C_1 & C_2 & C_3 & C_4 & C_5 & C_6\\end{pmatrix} \\begin{bmatrix}\n",
    " -1 &  1 &  0 &  0 &  0 &  1\\\\\n",
    "  1 & -1 & -1 &  0 &  0 &  0\\\\\n",
    " -1 &  1 &  1 &  0 &  0 &  0 \\\\\n",
    "  0 &  0 &  1 & -1 &  1 &  0 \\\\\n",
    "  0 &  0 &  0 &  1 & -1 & -1 \\\\\n",
    "  0 &  0 &  0 & -1 &  1 &  1\\end{bmatrix}  &=0\n",
    " \\end{align*}$$\n",
    " \n",
    " \n",
    " the solution of the system of equations is: \n",
    " $$ \\begin{align*}\n",
    " -C_1 + C_2 - C_3= 0 &\\implies  C_2 = C_3 + C_1\\\\\n",
    " -C_2 + C_3 + C_4= 0 &\\implies  C_2 = C_3 + C_4\\\\\n",
    " - C_4 + C_5 - C_6 = 0 &\\implies  C_5=C_4 + C_6\\\\\n",
    " C_1 - C_5 + C_6 =0 &\\implies  C_5= C_1 + C_6\n",
    "\\end{align*}$$\n",
    "\n",
    "Therefore, from the set of equations above, we obatin $C_1 = C_4$ (as in the case of one enzyem and one substrate. Therefore, the conservation of mass is:\n",
    "\n",
    "$$ \\begin{align*}\n",
    "C_1 \\cdot X_1(t) + (C_3 + C_1) \\cdot X_2(t) + C_3 \\cdot X_3(t) + C_1 \\cdot X_4(t) +  (C_6 + C_1) \\cdot X_5(t) + C_6 \\cdot X_6(t)&= cte\\\\\n",
    "C_1 \\cdot X_1(t) + C_3 \\cdot X_2(t) + C_1 \\cdot X_2(t)  + C_3 \\cdot X_3(t) + C_1 \\cdot X_4(t) + C_6 \\cdot X_5(t) + C_1 \\cdot X_5(t) + C_6 \\cdot X_6(t)&= cte\\\\\n",
    "\\end{align*}$$\n",
    "which is valid for any value of $C_1$, $C_3$ and $C_6$, so, separating these dependencies\n",
    "\n",
    "\n",
    "$$ \\begin{align*}\n",
    "C_1 \\cdot X_1(t) + C_1 \\cdot X_2(t) + C_1 \\cdot X_4(t) + C_1 \\cdot X_5 (t)&= cte\\\\\n",
    "\\end{align*}$$\n",
    "\n",
    "which makes sense, since the $s$, $e1s$, $e2p$ and $p$ are four forms of the same protein, and their sum has to be constant, and equal to the initial value of substrate used. Using the value $C_1=1$ we obtain:\n",
    "\n",
    "$$ \\begin{align*}\n",
    "X_1(t) + X_2(t) + X_4(t) + X_5 (t) &= cte = X_1(0) + X_2(0) + X_4(0) + X_5(0)=  X_1(0) + X_4(0)\\\\\n",
    "\\end{align*}$$\n",
    "\n",
    "is our first conservation law (typically we start a biochemical reaction adding only the two substrates $s$ and $p$ and the two enzymes $e1$ and $e2$). For the second, we use the parameters that depend on the other constant $C_2$ \n",
    " \n",
    "$$ \\begin{align*}\n",
    "C_3 \\cdot X_2(t) + C_3 \\cdot X_3(t) &= cte\n",
    "\\end{align*}$$\n",
    "\n",
    "This is the same conservation law that we obtained for the simple single enzyme-substrate system studiend previously. So, giving the value $C_3=1$, we obtain the conservation law, \n",
    "\n",
    "$$ \\begin{align*}\n",
    "X_2(t) + X_3(t) & = cte &= X_2(0) + X_3(0) = X_3 (0) \n",
    "\\end{align*}$$\n",
    "\n",
    "For the other constant $C_6$: \n",
    "$$ \\begin{align*}\n",
    "C_6 \\cdot X_5(t) + C_6 \\cdot X_6(t) &= cte\n",
    "\\end{align*}$$\n",
    "\n",
    "which is true because the enzyme $e2$ is also a catalyst that facilitates the reaction but does not react itself, and if we assume that there is no complex $e2p(0)=0$ before the reacton starts, we can assume ($e2(0)+e2p(0)=e2_0$). \n",
    "\n",
    "This conservation law together with the previous one, allows us to reduce the system of 6 differential equations to just four  coupled ordinary differential equations:\n",
    " \n",
    " $$\\begin{align*}\n",
    "  \\dot{s}   &= -k_1 \\cdot s \\cdot e1_0 + (k_2 + k_1 \\cdot s) \\cdot e1s + k_6 \\cdot e2p \\tag{52}\\\\\n",
    "  \\dot{e1s} &= k_1 \\cdot e1_0 \\cdot s - (k_2 + k_3 + k_1 \\cdot s) \\cdot e1s \\tag{53}\\\\\n",
    "  \\dot{e2p} &= k_4 \\cdot e2_0 \\cdot p - (k_5 + k_6 + k_4 \\cdot p) \\cdot e2p \\tag{54}\\\\\n",
    "  \\dot{p}   &= -k_4 \\cdot p \\cdot e2_0  + (k_5 + k_4 \\cdot p) \\cdot e2p + k_3 \\cdot e1s \\tag{55}\n",
    "  \\end{align*}$$"
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  {
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    {
     "data": {
      "text/plain": [
       "(::ab{var\"#51#55\",var\"#52#56\",var\"#53#57\",Nothing,Nothing,var\"#54#58\",Expr,Expr}) (generic function with 2 methods)"
      ]
     },
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   "source": [
    "enzyme_kinetics5! = @ode_def ab begin\n",
    "  ds   = -k1 * s * e10 + (k2 + k1 * s) * e1s + k6 * e2p\n",
    "  de1s =  k1 * e10 * s - (k2 + k3 + k1 * s) * e1s\n",
    "  de2p =  k4 * e20 * p - (k5 + k6 + k4 * p) * e2p\n",
    "  dp   = -k4 * p * e20  + (k5 + k4 * p) * e2p + k3 * e1s\n",
    "    end k1 k2 k3 k4 k5 k6 e10 e20"
   ]
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       "  853.236,1272.57 855.322,1272.86 857.407,1273.16 859.493,1273.45 861.579,1273.75 863.664,1274.04 865.75,1274.34 867.835,1274.63 869.921,1274.93 872.006,1275.22 \n",
       "  874.092,1275.51 876.178,1275.81 878.263,1276.1 880.349,1276.39 882.434,1276.67 884.52,1276.96 886.605,1277.24 888.691,1277.53 890.776,1277.81 892.862,1278.09 \n",
       "  894.948,1278.37 897.033,1278.64 899.119,1278.92 901.204,1279.19 903.29,1279.46 905.375,1279.74 907.461,1280 909.547,1280.27 911.632,1280.54 913.718,1280.8 \n",
       "  915.803,1281.07 917.889,1281.33 919.974,1281.59 922.06,1281.85 924.145,1282.11 926.231,1282.37 928.317,1282.63 930.402,1282.89 932.488,1283.15 934.573,1283.41 \n",
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       "  957.514,1286.24 959.6,1286.5 961.686,1286.75 963.771,1286.99 965.857,1287.24 967.942,1287.49 970.028,1287.73 972.113,1287.97 974.199,1288.21 976.285,1288.45 \n",
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       "  999.226,1291.01 1001.31,1291.23 1003.4,1291.46 1005.48,1291.69 1007.57,1291.92 1009.65,1292.14 1011.74,1292.37 1013.82,1292.6 1015.91,1292.83 1018,1293.06 \n",
       "  1020.08,1293.29 1022.17,1293.52 1024.25,1293.75 1026.34,1293.97 1028.42,1294.2 1030.51,1294.42 1032.59,1294.64 1034.68,1294.87 1036.77,1295.09 1038.85,1295.31 \n",
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       "  1103.5,1301.64 1105.59,1301.84 1107.67,1302.03 1109.76,1302.23 1111.85,1302.43 1113.93,1302.62 1116.02,1302.81 1118.1,1303 1120.19,1303.19 1122.27,1303.38 \n",
       "  1124.36,1303.57 1126.45,1303.75 1128.53,1303.94 1130.62,1304.12 1132.7,1304.3 1134.79,1304.48 1136.87,1304.66 1138.96,1304.84 1141.04,1305.02 1143.13,1305.19 \n",
       "  1145.22,1305.37 1147.3,1305.54 1149.39,1305.72 1151.47,1305.89 1153.56,1306.07 1155.64,1306.24 1157.73,1306.42 1159.81,1306.59 1161.9,1306.77 1163.99,1306.95 \n",
       "  1166.07,1307.12 1168.16,1307.3 1170.24,1307.48 1172.33,1307.66 1174.41,1307.84 1176.5,1308.02 1178.58,1308.2 1180.67,1308.37 1182.76,1308.55 1184.84,1308.72 \n",
       "  1186.93,1308.9 1189.01,1309.07 1191.1,1309.24 1193.18,1309.41 1195.27,1309.58 1197.35,1309.75 1199.44,1309.91 1201.53,1310.08 1203.61,1310.24 1205.7,1310.4 \n",
       "  1207.78,1310.56 1209.87,1310.72 1211.95,1310.88 1214.04,1311.03 1216.12,1311.19 1218.21,1311.34 1220.3,1311.5 1222.38,1311.65 1224.47,1311.81 1226.55,1311.96 \n",
       "  1228.64,1312.11 1230.72,1312.26 1232.81,1312.42 1234.89,1312.57 1236.98,1312.72 1239.07,1312.88 1241.15,1313.04 1243.24,1313.2 1245.32,1313.35 1247.41,1313.51 \n",
       "  1249.49,1313.67 1251.58,1313.83 1253.66,1313.99 1255.75,1314.15 1257.84,1314.3 1259.92,1314.46 1262.01,1314.61 1264.09,1314.77 1266.18,1314.92 1268.26,1315.07 \n",
       "  1270.35,1315.22 1272.43,1315.37 1274.52,1315.52 1276.61,1315.67 1278.69,1315.81 1280.78,1315.96 1282.86,1316.1 1284.95,1316.24 1287.03,1316.38 1289.12,1316.52 \n",
       "  1291.2,1316.66 1293.29,1316.79 1295.38,1316.93 1297.46,1317.06 1299.55,1317.2 1301.63,1317.33 1303.72,1317.46 1305.8,1317.6 1307.89,1317.73 1309.97,1317.87 \n",
       "  1312.06,1318 1314.15,1318.14 1316.23,1318.27 1318.32,1318.41 1320.4,1318.55 1322.49,1318.69 1324.57,1318.83 1326.66,1318.97 1328.74,1319.12 1330.83,1319.26 \n",
       "  1332.92,1319.4 1335,1319.54 1337.09,1319.68 1339.17,1319.81 1341.26,1319.95 1343.34,1320.09 1345.43,1320.22 1347.51,1320.36 1349.6,1320.49 1351.69,1320.62 \n",
       "  1353.77,1320.75 1355.86,1320.88 1357.94,1321.01 1360.03,1321.13 1362.11,1321.26 1364.2,1321.38 1366.28,1321.5 1368.37,1321.62 1370.46,1321.74 1372.54,1321.86 \n",
       "  1374.63,1321.98 1376.71,1322.1 1378.8,1322.22 1380.88,1322.33 1382.97,1322.45 1385.05,1322.57 1387.14,1322.69 1389.23,1322.8 1391.31,1322.92 1393.4,1323.04 \n",
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       "  1458.05,1326.71 1460.14,1326.82 1462.22,1326.93 1464.31,1327.04 1466.39,1327.15 1468.48,1327.26 1470.56,1327.37 1472.65,1327.48 1474.73,1327.59 1476.82,1327.7 \n",
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       "  1499.76,1328.86 1501.85,1328.97 1503.93,1329.07 1506.02,1329.17 1508.1,1329.27 1510.19,1329.38 1512.27,1329.48 1514.36,1329.58 1516.45,1329.68 1518.53,1329.78 \n",
       "  1520.62,1329.88 1522.7,1329.98 1524.79,1330.08 1526.87,1330.17 1528.96,1330.27 1531.04,1330.37 1533.13,1330.47 1535.22,1330.56 1537.3,1330.66 1539.39,1330.75 \n",
       "  1541.47,1330.85 1543.56,1330.94 1545.64,1331.04 1547.73,1331.13 1549.81,1331.23 1551.9,1331.32 1553.99,1331.41 1556.07,1331.5 1558.16,1331.6 1560.24,1331.69 \n",
       "  1562.33,1331.78 1564.41,1331.88 1566.5,1331.97 1568.58,1332.06 1570.67,1332.15 1572.76,1332.24 1574.84,1332.33 1576.93,1332.42 1579.01,1332.52 1581.1,1332.61 \n",
       "  1583.18,1332.7 1585.27,1332.79 1587.35,1332.88 1589.44,1332.96 1591.53,1333.05 1593.61,1333.14 1595.7,1333.23 1597.78,1333.32 1599.87,1333.41 1601.95,1333.5 \n",
       "  1604.04,1333.58 1606.12,1333.67 1608.21,1333.76 1610.3,1333.85 1612.38,1333.93 1614.47,1334.02 1616.55,1334.1 1618.64,1334.19 1620.72,1334.28 1622.81,1334.36 \n",
       "  1624.89,1334.45 1626.98,1334.53 1629.07,1334.62 1631.15,1334.7 1633.24,1334.79 1635.32,1334.87 1637.41,1334.95 1639.49,1335.04 1641.58,1335.12 1643.66,1335.2 \n",
       "  1645.75,1335.29 1647.84,1335.37 1649.92,1335.45 1652.01,1335.53 1654.09,1335.61 1656.18,1335.69 1658.26,1335.78 1660.35,1335.86 1662.43,1335.94 1664.52,1336.02 \n",
       "  1666.61,1336.1 1668.69,1336.18 1670.78,1336.26 1672.86,1336.34 1674.95,1336.42 1677.03,1336.5 1679.12,1336.57 1681.2,1336.65 1683.29,1336.73 1685.38,1336.81 \n",
       "  1687.46,1336.89 1689.55,1336.96 1691.63,1337.04 1693.72,1337.12 1695.8,1337.2 1697.89,1337.27 1699.97,1337.35 1702.06,1337.42 1704.15,1337.5 1706.23,1337.58 \n",
       "  1708.32,1337.65 1710.4,1337.73 1712.49,1337.8 1714.57,1337.88 1716.66,1337.95 1718.74,1338.03 1720.83,1338.1 1722.92,1338.18 1725,1338.25 1727.09,1338.32 \n",
       "  1729.17,1338.4 1731.26,1338.47 1733.34,1338.54 1735.43,1338.62 1737.51,1338.69 1739.6,1338.76 1741.69,1338.83 1743.77,1338.91 1745.86,1338.98 1747.94,1339.05 \n",
       "  1750.03,1339.12 1752.11,1339.19 1754.2,1339.26 1756.28,1339.33 1758.37,1339.4 1760.46,1339.47 1762.54,1339.55 1764.63,1339.61 1766.71,1339.68 1768.8,1339.75 \n",
       "  1770.88,1339.82 1772.97,1339.89 1775.06,1339.96 1777.14,1340.03 1779.23,1340.1 1781.31,1340.17 1783.4,1340.23 1785.48,1340.3 1787.57,1340.37 1789.65,1340.44 \n",
       "  1791.74,1340.5 1793.83,1340.57 1795.91,1340.64 1798,1340.7 1800.08,1340.77 1802.17,1340.84 1804.25,1340.9 1806.34,1340.97 1808.42,1341.03 1810.51,1341.1 \n",
       "  1812.6,1341.16 1814.68,1341.23 1816.77,1341.29 1818.85,1341.36 1820.94,1341.42 1823.02,1341.49 1825.11,1341.55 1827.19,1341.61 1829.28,1341.68 1831.37,1341.74 \n",
       "  1833.45,1341.8 1835.54,1341.86 1837.62,1341.93 1839.71,1341.99 1841.79,1342.05 1843.88,1342.11 1845.96,1342.17 1848.05,1342.24 1850.14,1342.3 1852.22,1342.36 \n",
       "  1854.31,1342.42 1856.39,1342.48 1858.48,1342.54 1860.56,1342.6 1862.65,1342.66 1864.73,1342.72 1866.82,1342.78 1868.91,1342.84 1870.99,1342.9 1873.08,1342.96 \n",
       "  1875.16,1343.01 1877.25,1343.07 1879.33,1343.13 1881.42,1343.19 1883.5,1343.25 1885.59,1343.31 1887.68,1343.36 1889.76,1343.42 1891.85,1343.48 1893.93,1343.54 \n",
       "  1896.02,1343.6 1898.1,1343.65 1900.19,1343.71 1902.27,1343.77 1904.36,1343.82 1906.45,1343.88 1908.53,1343.94 1910.62,1343.99 1912.7,1344.05 1914.79,1344.1 \n",
       "  1916.87,1344.16 1918.96,1344.22 1921.04,1344.27 1923.13,1344.33 1925.22,1344.38 1927.3,1344.44 1929.39,1344.49 1931.47,1344.55 1933.56,1344.6 1935.64,1344.66 \n",
       "  1937.73,1344.71 1939.81,1344.76 1941.9,1344.82 1943.99,1344.87 1946.07,1344.93 1948.16,1344.98 1950.24,1345.03 1952.33,1345.09 1954.41,1345.14 1956.5,1345.19 \n",
       "  1958.58,1345.25 1960.67,1345.3 1962.76,1345.35 1964.84,1345.4 1966.93,1345.45 1969.01,1345.51 1971.1,1345.56 1973.18,1345.61 1975.27,1345.66 1977.35,1345.71 \n",
       "  1979.44,1345.76 1981.53,1345.82 1983.61,1345.87 1985.7,1345.92 1987.78,1345.97 1989.87,1346.02 1991.95,1346.07 1994.04,1346.12 1996.12,1346.17 1998.21,1346.22 \n",
       "  2000.3,1346.27 2002.38,1346.32 2004.47,1346.37 2006.55,1346.42 2008.64,1346.47 2010.72,1346.51 2012.81,1346.56 2014.89,1346.61 2016.98,1346.66 2019.07,1346.71 \n",
       "  2021.15,1346.76 2023.24,1346.8 2025.32,1346.85 2027.41,1346.9 2029.49,1346.95 2031.58,1346.99 2033.66,1347.04 2035.75,1347.09 2037.84,1347.14 2039.92,1347.18 \n",
       "  2042.01,1347.23 2044.09,1347.28 2046.18,1347.32 2048.26,1347.37 2050.35,1347.41 2052.43,1347.46 2054.52,1347.51 2056.61,1347.55 2058.69,1347.6 2060.78,1347.64 \n",
       "  2062.86,1347.69 2064.95,1347.73 2067.03,1347.78 2069.12,1347.83 2071.2,1347.87 2073.29,1347.92 2075.38,1347.96 2077.46,1348 2079.55,1348.05 2081.63,1348.09 \n",
       "  2083.72,1348.14 2085.8,1348.18 2087.89,1348.23 2089.98,1348.27 2092.06,1348.31 2094.15,1348.36 2096.23,1348.4 2098.32,1348.44 2100.4,1348.49 2102.49,1348.53 \n",
       "  2104.57,1348.57 2106.66,1348.62 2108.75,1348.66 2110.83,1348.7 2112.92,1348.75 2115,1348.79 2117.09,1348.83 2119.17,1348.87 2121.26,1348.91 2123.34,1348.96 \n",
       "  2125.43,1349 2127.52,1349.04 2129.6,1349.08 2131.69,1349.12 2133.77,1349.17 2135.86,1349.21 2137.94,1349.25 2140.03,1349.29 2142.11,1349.33 2144.2,1349.37 \n",
       "  2146.29,1349.41 2148.37,1349.45 2150.46,1349.49 2152.54,1349.53 2154.63,1349.57 2156.71,1349.61 2158.8,1349.65 2160.88,1349.69 2162.97,1349.73 2165.06,1349.77 \n",
       "  2167.14,1349.81 2169.23,1349.85 2171.31,1349.89 2173.4,1349.93 2175.48,1349.97 2177.57,1350.01 2179.65,1350.05 2181.74,1350.08 2183.83,1350.12 2185.91,1350.16 \n",
       "  2188,1350.2 2190.08,1350.24 2192.17,1350.28 2194.25,1350.31 2196.34,1350.35 2198.42,1350.39 2200.51,1350.43 2202.6,1350.47 2204.68,1350.5 2206.77,1350.54 \n",
       "  2208.85,1350.58 2210.94,1350.61 2213.02,1350.65 2215.11,1350.69 2217.19,1350.72 2219.28,1350.76 2221.37,1350.8 2223.45,1350.83 2225.54,1350.87 2227.62,1350.91 \n",
       "  2229.71,1350.94 2231.79,1350.98 2233.88,1351.01 2235.96,1351.05 2238.05,1351.08 2240.14,1351.12 2242.22,1351.15 2244.31,1351.19 2246.39,1351.23 2248.48,1351.26 \n",
       "  2250.56,1351.29 2252.65,1351.33 2254.73,1351.36 2256.82,1351.4 2258.91,1351.43 2260.99,1351.47 2263.08,1351.5 2265.16,1351.54 2267.25,1351.57 2269.33,1351.6 \n",
       "  2271.42,1351.64 2273.5,1351.67 2275.59,1351.71 2277.68,1351.74 2279.76,1351.77 2281.85,1351.81 2283.93,1351.84 2286.02,1351.87 2288.1,1351.91 2290.19,1351.94 \n",
       "  2292.27,1351.97 2294.36,1352.01 2296.45,1352.04 2298.53,1352.07 2300.62,1352.11 2302.7,1352.14 2304.79,1352.17 2306.87,1352.2 2308.96,1352.24 2311.04,1352.27 \n",
       "  2313.13,1352.3 2315.22,1352.33 2317.3,1352.36 2319.39,1352.4 2321.47,1352.43 2323.56,1352.46 2325.64,1352.49 2327.73,1352.52 2329.81,1352.56 2331.9,1352.59 \n",
       "  2333.99,1352.62 2336.07,1352.65 2338.16,1352.68 2340.24,1352.71 2342.33,1352.74 2344.41,1352.77 2346.5,1352.8 2348.58,1352.84 2350.67,1352.87 2352.76,1352.9 \n",
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       "  290.134,1022.35 292.22,1005.96 294.306,991.477 296.391,978.647 298.477,967.248 300.562,957.099 302.648,948.045 304.733,939.957 306.819,932.72 308.904,926.236 \n",
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       "  456.979,891.629 459.065,892.14 461.151,892.651 463.236,893.164 465.322,893.679 467.407,894.194 469.493,894.709 471.578,895.226 473.664,895.743 475.749,896.26 \n",
       "  477.835,896.778 479.921,897.296 482.006,897.813 484.092,898.331 486.177,898.848 488.263,899.365 490.348,899.881 492.434,900.398 494.52,900.914 496.605,901.431 \n",
       "  498.691,901.948 500.776,902.464 502.862,902.981 504.947,903.498 507.033,904.015 509.118,904.532 511.204,905.049 513.29,905.567 515.375,906.084 517.461,906.601 \n",
       "  519.546,907.119 521.632,907.636 523.717,908.153 525.803,908.67 527.889,909.186 529.974,909.702 532.06,910.217 534.145,910.732 536.231,911.246 538.316,911.76 \n",
       "  540.402,912.272 542.487,912.784 544.573,913.296 546.659,913.807 548.744,914.318 550.83,914.828 552.915,915.339 555.001,915.849 557.086,916.359 559.172,916.869 \n",
       "  561.258,917.378 563.343,917.888 565.429,918.398 567.514,918.907 569.6,919.416 571.685,919.925 573.771,920.434 575.856,920.943 577.942,921.451 580.028,921.959 \n",
       "  582.113,922.466 584.199,922.973 586.284,923.48 588.37,923.985 590.455,924.49 592.541,924.994 594.627,925.497 596.712,925.999 598.798,926.501 600.883,927.001 \n",
       "  602.969,927.501 605.054,928.001 607.14,928.499 609.225,928.998 611.311,929.496 613.397,929.994 615.482,930.492 617.568,930.99 619.653,931.487 621.739,931.984 \n",
       "  623.824,932.481 625.91,932.978 627.996,933.475 630.081,933.972 632.167,934.468 634.252,934.964 636.338,935.46 638.423,935.955 640.509,936.45 642.594,936.945 \n",
       "  644.68,937.439 646.766,937.932 648.851,938.424 650.937,938.916 653.022,939.406 655.108,939.896 657.193,940.384 659.279,940.871 661.365,941.357 663.45,941.842 \n",
       "  665.536,942.326 667.621,942.81 669.707,943.293 671.792,943.776 673.878,944.259 675.963,944.741 678.049,945.223 680.135,945.705 682.22,946.186 684.306,946.667 \n",
       "  686.391,947.149 688.477,947.63 690.562,948.111 692.648,948.591 694.734,949.072 696.819,949.552 698.905,950.032 700.99,950.512 703.076,950.991 705.161,951.47 \n",
       "  707.247,951.948 709.332,952.425 711.418,952.902 713.504,953.378 715.589,953.853 717.675,954.327 719.76,954.799 721.846,955.27 723.931,955.74 726.017,956.208 \n",
       "  728.103,956.675 730.188,957.142 732.274,957.607 734.359,958.072 736.445,958.537 738.53,959.001 740.616,959.464 742.701,959.927 744.787,960.39 746.873,960.853 \n",
       "  748.958,961.315 751.044,961.778 753.129,962.24 755.215,962.702 757.3,963.164 759.386,963.626 761.472,964.087 763.557,964.549 765.643,965.01 767.728,965.471 \n",
       "  769.814,965.931 771.899,966.391 773.985,966.851 776.071,967.309 778.156,967.767 780.242,968.224 782.327,968.681 784.413,969.136 786.498,969.589 788.584,970.041 \n",
       "  790.669,970.492 792.755,970.941 794.841,971.388 796.926,971.835 799.012,972.28 801.097,972.725 803.183,973.168 805.268,973.612 807.354,974.054 809.44,974.496 \n",
       "  811.525,974.938 813.611,975.38 815.696,975.821 817.782,976.262 819.867,976.703 821.953,977.144 824.038,977.584 826.124,978.025 828.21,978.465 830.295,978.906 \n",
       "  832.381,979.346 834.466,979.786 836.552,980.226 838.637,980.665 840.723,981.104 842.809,981.542 844.894,981.98 846.98,982.417 849.065,982.854 851.151,983.289 \n",
       "  853.236,983.723 855.322,984.157 857.407,984.588 859.493,985.018 861.579,985.447 863.664,985.873 865.75,986.298 867.835,986.722 869.921,987.144 872.006,987.566 \n",
       "  874.092,987.986 876.178,988.406 878.263,988.826 880.349,989.245 882.434,989.663 884.52,990.082 886.605,990.5 888.691,990.917 890.776,991.335 892.862,991.753 \n",
       "  894.948,992.17 897.033,992.588 899.119,993.005 901.204,993.422 903.29,993.839 905.375,994.256 907.461,994.673 909.547,995.09 911.632,995.506 913.718,995.922 \n",
       "  915.803,996.337 917.889,996.751 919.974,997.165 922.06,997.578 924.145,997.99 926.231,998.401 928.317,998.811 930.402,999.219 932.488,999.625 934.573,1000.03 \n",
       "  936.659,1000.43 938.744,1000.83 940.83,1001.23 942.916,1001.63 945.001,1002.03 947.087,1002.42 949.172,1002.82 951.258,1003.21 953.343,1003.61 955.429,1004 \n",
       "  957.514,1004.4 959.6,1004.79 961.686,1005.18 963.771,1005.58 965.857,1005.97 967.942,1006.36 970.028,1006.76 972.113,1007.15 974.199,1007.54 976.285,1007.94 \n",
       "  978.37,1008.33 980.456,1008.72 982.541,1009.11 984.627,1009.51 986.712,1009.9 988.798,1010.29 990.883,1010.68 992.969,1011.07 995.055,1011.46 997.14,1011.85 \n",
       "  999.226,1012.24 1001.31,1012.62 1003.4,1013.01 1005.48,1013.39 1007.57,1013.77 1009.65,1014.15 1011.74,1014.53 1013.82,1014.9 1015.91,1015.28 1018,1015.65 \n",
       "  1020.08,1016.02 1022.17,1016.39 1024.25,1016.76 1026.34,1017.13 1028.42,1017.5 1030.51,1017.87 1032.59,1018.24 1034.68,1018.61 1036.77,1018.98 1038.85,1019.34 \n",
       "  1040.94,1019.71 1043.02,1020.08 1045.11,1020.45 1047.19,1020.82 1049.28,1021.18 1051.36,1021.55 1053.45,1021.92 1055.54,1022.29 1057.62,1022.66 1059.71,1023.02 \n",
       "  1061.79,1023.39 1063.88,1023.76 1065.96,1024.13 1068.05,1024.49 1070.13,1024.86 1072.22,1025.22 1074.31,1025.58 1076.39,1025.95 1078.48,1026.31 1080.56,1026.67 \n",
       "  1082.65,1027.02 1084.73,1027.38 1086.82,1027.73 1088.9,1028.08 1090.99,1028.43 1093.08,1028.78 1095.16,1029.13 1097.25,1029.47 1099.33,1029.82 1101.42,1030.16 \n",
       "  1103.5,1030.51 1105.59,1030.85 1107.67,1031.19 1109.76,1031.54 1111.85,1031.88 1113.93,1032.22 1116.02,1032.56 1118.1,1032.91 1120.19,1033.25 1122.27,1033.59 \n",
       "  1124.36,1033.94 1126.45,1034.28 1128.53,1034.62 1130.62,1034.97 1132.7,1035.31 1134.79,1035.65 1136.87,1036 1138.96,1036.34 1141.04,1036.68 1143.13,1037.02 \n",
       "  1145.22,1037.36 1147.3,1037.71 1149.39,1038.05 1151.47,1038.38 1153.56,1038.72 1155.64,1039.06 1157.73,1039.39 1159.81,1039.72 1161.9,1040.05 1163.99,1040.38 \n",
       "  1166.07,1040.71 1168.16,1041.03 1170.24,1041.35 1172.33,1041.67 1174.41,1041.99 1176.5,1042.31 1178.58,1042.63 1180.67,1042.95 1182.76,1043.27 1184.84,1043.59 \n",
       "  1186.93,1043.91 1189.01,1044.23 1191.1,1044.54 1193.18,1044.86 1195.27,1045.18 1197.35,1045.5 1199.44,1045.82 1201.53,1046.14 1203.61,1046.46 1205.7,1046.77 \n",
       "  1207.78,1047.09 1209.87,1047.41 1211.95,1047.73 1214.04,1048.05 1216.12,1048.37 1218.21,1048.69 1220.3,1049.01 1222.38,1049.32 1224.47,1049.64 1226.55,1049.96 \n",
       "  1228.64,1050.27 1230.72,1050.59 1232.81,1050.9 1234.89,1051.21 1236.98,1051.52 1239.07,1051.82 1241.15,1052.13 1243.24,1052.43 1245.32,1052.73 1247.41,1053.03 \n",
       "  1249.49,1053.32 1251.58,1053.62 1253.66,1053.91 1255.75,1054.21 1257.84,1054.5 1259.92,1054.8 1262.01,1055.09 1264.09,1055.39 1266.18,1055.68 1268.26,1055.97 \n",
       "  1270.35,1056.27 1272.43,1056.56 1274.52,1056.86 1276.61,1057.15 1278.69,1057.45 1280.78,1057.74 1282.86,1058.04 1284.95,1058.33 1287.03,1058.63 1289.12,1058.93 \n",
       "  1291.2,1059.22 1293.29,1059.52 1295.38,1059.81 1297.46,1060.11 1299.55,1060.4 1301.63,1060.7 1303.72,1060.99 1305.8,1061.28 1307.89,1061.57 1309.97,1061.86 \n",
       "  1312.06,1062.15 1314.15,1062.44 1316.23,1062.72 1318.32,1063 1320.4,1063.28 1322.49,1063.56 1324.57,1063.83 1326.66,1064.11 1328.74,1064.38 1330.83,1064.65 \n",
       "  1332.92,1064.92 1335,1065.19 1337.09,1065.47 1339.17,1065.74 1341.26,1066.01 1343.34,1066.28 1345.43,1066.55 1347.51,1066.82 1349.6,1067.09 1351.69,1067.36 \n",
       "  1353.77,1067.63 1355.86,1067.91 1357.94,1068.18 1360.03,1068.45 1362.11,1068.73 1364.2,1069 1366.28,1069.27 1368.37,1069.55 1370.46,1069.82 1372.54,1070.09 \n",
       "  1374.63,1070.37 1376.71,1070.64 1378.8,1070.91 1380.88,1071.18 1382.97,1071.45 1385.05,1071.72 1387.14,1071.99 1389.23,1072.26 1391.31,1072.52 1393.4,1072.79 \n",
       "  1395.48,1073.05 1397.57,1073.3 1399.65,1073.56 1401.74,1073.81 1403.82,1074.07 1405.91,1074.32 1408,1074.58 1410.08,1074.83 1412.17,1075.08 1414.25,1075.33 \n",
       "  1416.34,1075.58 1418.42,1075.84 1420.51,1076.09 1422.59,1076.34 1424.68,1076.59 1426.77,1076.84 1428.85,1077.09 1430.94,1077.33 1433.02,1077.58 1435.11,1077.83 \n",
       "  1437.19,1078.08 1439.28,1078.32 1441.37,1078.57 1443.45,1078.82 1445.54,1079.06 1447.62,1079.31 1449.71,1079.55 1451.79,1079.79 1453.88,1080.04 1455.96,1080.28 \n",
       "  1458.05,1080.52 1460.14,1080.77 1462.22,1081.01 1464.31,1081.25 1466.39,1081.49 1468.48,1081.73 1470.56,1081.97 1472.65,1082.21 1474.73,1082.45 1476.82,1082.69 \n",
       "  1478.91,1082.93 1480.99,1083.16 1483.08,1083.4 1485.16,1083.64 1487.25,1083.87 1489.33,1084.11 1491.42,1084.35 1493.5,1084.58 1495.59,1084.81 1497.68,1085.05 \n",
       "  1499.76,1085.28 1501.85,1085.52 1503.93,1085.75 1506.02,1085.98 1508.1,1086.21 1510.19,1086.44 1512.27,1086.68 1514.36,1086.91 1516.45,1087.14 1518.53,1087.37 \n",
       "  1520.62,1087.6 1522.7,1087.82 1524.79,1088.05 1526.87,1088.28 1528.96,1088.51 1531.04,1088.74 1533.13,1088.96 1535.22,1089.19 1537.3,1089.41 1539.39,1089.64 \n",
       "  1541.47,1089.86 1543.56,1090.09 1545.64,1090.31 1547.73,1090.54 1549.81,1090.76 1551.9,1090.98 1553.99,1091.2 1556.07,1091.42 1558.16,1091.64 1560.24,1091.86 \n",
       "  1562.33,1092.08 1564.41,1092.3 1566.5,1092.52 1568.58,1092.73 1570.67,1092.95 1572.76,1093.17 1574.84,1093.38 1576.93,1093.6 1579.01,1093.81 1581.1,1094.02 \n",
       "  1583.18,1094.24 1585.27,1094.45 1587.35,1094.66 1589.44,1094.88 1591.53,1095.09 1593.61,1095.3 1595.7,1095.51 1597.78,1095.72 1599.87,1095.93 1601.95,1096.14 \n",
       "  1604.04,1096.35 1606.12,1096.56 1608.21,1096.76 1610.3,1096.97 1612.38,1097.18 1614.47,1097.39 1616.55,1097.59 1618.64,1097.8 1620.72,1098 1622.81,1098.21 \n",
       "  1624.89,1098.41 1626.98,1098.61 1629.07,1098.82 1631.15,1099.02 1633.24,1099.22 1635.32,1099.42 1637.41,1099.63 1639.49,1099.83 1641.58,1100.03 1643.66,1100.23 \n",
       "  1645.75,1100.43 1647.84,1100.62 1649.92,1100.82 1652.01,1101.02 1654.09,1101.22 1656.18,1101.42 1658.26,1101.61 1660.35,1101.81 1662.43,1102 1664.52,1102.2 \n",
       "  1666.61,1102.39 1668.69,1102.59 1670.78,1102.78 1672.86,1102.97 1674.95,1103.17 1677.03,1103.36 1679.12,1103.55 1681.2,1103.74 1683.29,1103.93 1685.38,1104.12 \n",
       "  1687.46,1104.31 1689.55,1104.5 1691.63,1104.69 1693.72,1104.88 1695.8,1105.07 1697.89,1105.25 1699.97,1105.44 1702.06,1105.63 1704.15,1105.81 1706.23,1105.99 \n",
       "  1708.32,1106.18 1710.4,1106.36 1712.49,1106.55 1714.57,1106.73 1716.66,1106.91 1718.74,1107.09 1720.83,1107.28 1722.92,1107.46 1725,1107.64 1727.09,1107.82 \n",
       "  1729.17,1108 1731.26,1108.18 1733.34,1108.36 1735.43,1108.54 1737.51,1108.72 1739.6,1108.9 1741.69,1109.07 1743.77,1109.25 1745.86,1109.43 1747.94,1109.61 \n",
       "  1750.03,1109.78 1752.11,1109.96 1754.2,1110.14 1756.28,1110.31 1758.37,1110.49 1760.46,1110.66 1762.54,1110.83 1764.63,1111.01 1766.71,1111.18 1768.8,1111.36 \n",
       "  1770.88,1111.53 1772.97,1111.7 1775.06,1111.87 1777.14,1112.04 1779.23,1112.21 1781.31,1112.39 1783.4,1112.56 1785.48,1112.73 1787.57,1112.9 1789.65,1113.06 \n",
       "  1791.74,1113.23 1793.83,1113.4 1795.91,1113.57 1798,1113.74 1800.08,1113.91 1802.17,1114.07 1804.25,1114.24 1806.34,1114.41 1808.42,1114.57 1810.51,1114.74 \n",
       "  1812.6,1114.9 1814.68,1115.07 1816.77,1115.23 1818.85,1115.39 1820.94,1115.56 1823.02,1115.72 1825.11,1115.88 1827.19,1116.05 1829.28,1116.21 1831.37,1116.37 \n",
       "  1833.45,1116.53 1835.54,1116.69 1837.62,1116.85 1839.71,1117.01 1841.79,1117.17 1843.88,1117.33 1845.96,1117.49 1848.05,1117.65 1850.14,1117.81 1852.22,1117.96 \n",
       "  1854.31,1118.12 1856.39,1118.28 1858.48,1118.44 1860.56,1118.59 1862.65,1118.75 1864.73,1118.9 1866.82,1119.06 1868.91,1119.21 1870.99,1119.37 1873.08,1119.52 \n",
       "  1875.16,1119.67 1877.25,1119.83 1879.33,1119.98 1881.42,1120.13 1883.5,1120.28 1885.59,1120.43 1887.68,1120.58 1889.76,1120.73 1891.85,1120.88 1893.93,1121.03 \n",
       "  1896.02,1121.18 1898.1,1121.33 1900.19,1121.48 1902.27,1121.63 1904.36,1121.77 1906.45,1121.92 1908.53,1122.07 1910.62,1122.22 1912.7,1122.36 1914.79,1122.51 \n",
       "  1916.87,1122.65 1918.96,1122.8 1921.04,1122.94 1923.13,1123.09 1925.22,1123.23 1927.3,1123.38 1929.39,1123.52 1931.47,1123.67 1933.56,1123.81 1935.64,1123.95 \n",
       "  1937.73,1124.09 1939.81,1124.24 1941.9,1124.38 1943.99,1124.52 1946.07,1124.66 1948.16,1124.8 1950.24,1124.94 1952.33,1125.08 1954.41,1125.22 1956.5,1125.36 \n",
       "  1958.58,1125.5 1960.67,1125.64 1962.76,1125.78 1964.84,1125.92 1966.93,1126.06 1969.01,1126.19 1971.1,1126.33 1973.18,1126.47 1975.27,1126.61 1977.35,1126.74 \n",
       "  1979.44,1126.88 1981.53,1127.01 1983.61,1127.15 1985.7,1127.28 1987.78,1127.42 1989.87,1127.55 1991.95,1127.69 1994.04,1127.82 1996.12,1127.95 1998.21,1128.09 \n",
       "  2000.3,1128.22 2002.38,1128.35 2004.47,1128.48 2006.55,1128.62 2008.64,1128.75 2010.72,1128.88 2012.81,1129.01 2014.89,1129.14 2016.98,1129.27 2019.07,1129.4 \n",
       "  2021.15,1129.53 2023.24,1129.66 2025.32,1129.79 2027.41,1129.92 2029.49,1130.04 2031.58,1130.17 2033.66,1130.3 2035.75,1130.43 2037.84,1130.55 2039.92,1130.68 \n",
       "  2042.01,1130.8 2044.09,1130.93 2046.18,1131.06 2048.26,1131.18 2050.35,1131.3 2052.43,1131.43 2054.52,1131.55 2056.61,1131.67 2058.69,1131.8 2060.78,1131.92 \n",
       "  2062.86,1132.04 2064.95,1132.16 2067.03,1132.29 2069.12,1132.41 2071.2,1132.53 2073.29,1132.65 2075.38,1132.77 2077.46,1132.89 2079.55,1133.01 2081.63,1133.13 \n",
       "  2083.72,1133.25 2085.8,1133.37 2087.89,1133.49 2089.98,1133.61 2092.06,1133.73 2094.15,1133.85 2096.23,1133.97 2098.32,1134.08 2100.4,1134.2 2102.49,1134.32 \n",
       "  2104.57,1134.44 2106.66,1134.55 2108.75,1134.67 2110.83,1134.78 2112.92,1134.9 2115,1135.02 2117.09,1135.13 2119.17,1135.25 2121.26,1135.36 2123.34,1135.48 \n",
       "  2125.43,1135.59 2127.52,1135.71 2129.6,1135.82 2131.69,1135.93 2133.77,1136.05 2135.86,1136.16 2137.94,1136.27 2140.03,1136.39 2142.11,1136.5 2144.2,1136.61 \n",
       "  2146.29,1136.72 2148.37,1136.83 2150.46,1136.94 2152.54,1137.05 2154.63,1137.17 2156.71,1137.28 2158.8,1137.39 2160.88,1137.5 2162.97,1137.61 2165.06,1137.72 \n",
       "  2167.14,1137.82 2169.23,1137.93 2171.31,1138.04 2173.4,1138.15 2175.48,1138.26 2177.57,1138.37 2179.65,1138.47 2181.74,1138.58 2183.83,1138.69 2185.91,1138.79 \n",
       "  2188,1138.9 2190.08,1139.01 2192.17,1139.11 2194.25,1139.22 2196.34,1139.32 2198.42,1139.43 2200.51,1139.53 2202.6,1139.64 2204.68,1139.74 2206.77,1139.85 \n",
       "  2208.85,1139.95 2210.94,1140.05 2213.02,1140.16 2215.11,1140.26 2217.19,1140.36 2219.28,1140.47 2221.37,1140.57 2223.45,1140.67 2225.54,1140.77 2227.62,1140.87 \n",
       "  2229.71,1140.97 2231.79,1141.07 2233.88,1141.17 2235.96,1141.28 2238.05,1141.38 2240.14,1141.48 2242.22,1141.57 2244.31,1141.67 2246.39,1141.77 2248.48,1141.87 \n",
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       "  1374.63,750.77 1376.71,750.619 1378.8,750.469 1380.88,750.319 1382.97,750.167 1385.05,750.013 1387.14,749.854 1389.23,749.691 1391.31,749.521 1393.4,749.342 \n",
       "  1395.48,749.154 1397.57,748.953 1399.65,748.753 1401.74,748.554 1403.82,748.356 1405.91,748.158 1408,747.962 1410.08,747.766 1412.17,747.572 1414.25,747.378 \n",
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       "  1437.19,745.306 1439.28,745.122 1441.37,744.94 1443.45,744.759 1445.54,744.578 1447.62,744.399 1449.71,744.22 1451.79,744.042 1453.88,743.866 1455.96,743.69 \n",
       "  1458.05,743.514 1460.14,743.34 1462.22,743.167 1464.31,742.994 1466.39,742.823 1468.48,742.652 1470.56,742.482 1472.65,742.313 1474.73,742.145 1476.82,741.978 \n",
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       "  1520.62,738.673 1522.7,738.525 1524.79,738.379 1526.87,738.233 1528.96,738.088 1531.04,737.944 1533.13,737.801 1535.22,737.658 1537.3,737.517 1539.39,737.376 \n",
       "  1541.47,737.237 1543.56,737.098 1545.64,736.96 1547.73,736.823 1549.81,736.687 1551.9,736.552 1553.99,736.417 1556.07,736.28 1558.16,736.143 1560.24,736.007 \n",
       "  1562.33,735.871 1564.41,735.735 1566.5,735.599 1568.58,735.464 1570.67,735.329 1572.76,735.195 1574.84,735.06 1576.93,734.927 1579.01,734.793 1581.1,734.66 \n",
       "  1583.18,734.527 1585.27,734.395 1587.35,734.263 1589.44,734.132 1591.53,734 1593.61,733.869 1595.7,733.739 1597.78,733.609 1599.87,733.479 1601.95,733.349 \n",
       "  1604.04,733.22 1606.12,733.092 1608.21,732.963 1610.3,732.835 1612.38,732.708 1614.47,732.58 1616.55,732.453 1618.64,732.327 1620.72,732.201 1622.81,732.075 \n",
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       "  1645.75,730.714 1647.84,730.593 1649.92,730.471 1652.01,730.351 1654.09,730.23 1656.18,730.11 1658.26,729.99 1660.35,729.871 1662.43,729.752 1664.52,729.633 \n",
       "  1666.61,729.515 1668.69,729.397 1670.78,729.279 1672.86,729.162 1674.95,729.045 1677.03,728.929 1679.12,728.813 1681.2,728.697 1683.29,728.581 1685.38,728.466 \n",
       "  1687.46,728.352 1689.55,728.237 1691.63,728.123 1693.72,728.01 1695.8,727.896 1697.89,727.783 1699.97,727.67 1702.06,727.558 1704.15,727.445 1706.23,727.333 \n",
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       "  1729.17,726.123 1731.26,726.015 1733.34,725.908 1735.43,725.801 1737.51,725.694 1739.6,725.588 1741.69,725.481 1743.77,725.376 1745.86,725.27 1747.94,725.165 \n",
       "  1750.03,725.061 1752.11,724.956 1754.2,724.852 1756.28,724.749 1758.37,724.645 1760.46,724.542 1762.54,724.44 1764.63,724.338 1766.71,724.236 1768.8,724.134 \n",
       "  1770.88,724.033 1772.97,723.932 1775.06,723.832 1777.14,723.732 1779.23,723.632 1781.31,723.533 1783.4,723.434 1785.48,723.335 1787.57,723.237 1789.65,723.139 \n",
       "  1791.74,723.041 1793.83,722.944 1795.91,722.847 1798,722.75 1800.08,722.654 1802.17,722.558 1804.25,722.463 1806.34,722.367 1808.42,722.273 1810.51,722.178 \n",
       "  1812.6,722.084 1814.68,721.99 1816.77,721.897 1818.85,721.804 1820.94,721.711 1823.02,721.619 1825.11,721.527 1827.19,721.435 1829.28,721.344 1831.37,721.253 \n",
       "  1833.45,721.162 1835.54,721.072 1837.62,720.982 1839.71,720.893 1841.79,720.803 1843.88,720.715 1845.96,720.626 1848.05,720.538 1850.14,720.45 1852.22,720.363 \n",
       "  1854.31,720.276 1856.39,720.189 1858.48,720.103 1860.56,720.017 1862.65,719.931 1864.73,719.846 1866.82,719.761 1868.91,719.676 1870.99,719.592 1873.08,719.508 \n",
       "  1875.16,719.423 1877.25,719.339 1879.33,719.255 1881.42,719.171 1883.5,719.087 1885.59,719.003 1887.68,718.92 1889.76,718.837 1891.85,718.754 1893.93,718.672 \n",
       "  1896.02,718.589 1898.1,718.507 1900.19,718.426 1902.27,718.344 1904.36,718.263 1906.45,718.181 1908.53,718.101 1910.62,718.02 1912.7,717.94 1914.79,717.859 \n",
       "  1916.87,717.779 1918.96,717.7 1921.04,717.62 1923.13,717.541 1925.22,717.462 1927.3,717.383 1929.39,717.305 1931.47,717.227 1933.56,717.148 1935.64,717.071 \n",
       "  1937.73,716.993 1939.81,716.916 1941.9,716.839 1943.99,716.762 1946.07,716.685 1948.16,716.609 1950.24,716.533 1952.33,716.457 1954.41,716.381 1956.5,716.306 \n",
       "  1958.58,716.231 1960.67,716.156 1962.76,716.081 1964.84,716.006 1966.93,715.932 1969.01,715.858 1971.1,715.784 1973.18,715.711 1975.27,715.638 1977.35,715.565 \n",
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     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u0=[s0,0,0,p0]\n",
    "p=[k1,k2,k3,k4,k5,k6,e10,e20];\n",
    "prob5 = ODEProblem(enzyme_kinetics5!,u0,tspan,p)\n",
    "sol5 = solve(prob5)\n",
    "plot(sol5,ylims = (0,s0+p0))\n",
    "title!(\"kinase-phosphatase\")\n",
    "xlabel!(\"Time [s]\")\n",
    "ylabel!(\"Concentration [M]\")\n",
    "#png(\"Michaelis_menten.png\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The concentration of `e1` and `e2` can be also plotted by simply using the mass conservation:;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {},
   "outputs": [
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     "execution_count": 63,
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   ],
   "source": [
    "plot(sol5.t,[e10.-sol5[2,:],e20.-sol5[3,:]],label = [\"e1\" \"e2\"],ylims = (0,s0+p0))"
   ]
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  {
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   "source": [
    "## Quasi steady state approximation:\n",
    "\n",
    "If we consider the pseudo-steady state approximation typical for biochemical systems (i.e., dynamic of intermediate complexes very fast compared with the other reactions), \n",
    "\n",
    "$$ \n",
    "\\begin{align*}\n",
    "k_1 \\cdot s(0) ≈ k_2 >> k_3 \\tag{56}\\\\\n",
    "k_4 \\cdot p(0) ≈ k_5 >> k_6 \\tag{57}\n",
    " \\end{align*}$$\n",
    " \n",
    "if this is true, we wan simplify the system by using the following rules:\n",
    "\n",
    "$$ \\frac{d e1s}{dt}= \\frac{d e1}{dt}=\\frac{d e2p}{dt}=\\frac{d e2}{dt}=0 \\tag{58}$$\n",
    "\n",
    "so, the equilibrium concentrations for `e1s`and `e2p` are simply:\n",
    "\n",
    " $$\\begin{align*}\n",
    "  e1s &= \\frac{k_1 \\cdot e1_0 \\cdot s}{k_2 + k_3 + k_1 \\cdot s} \\tag{59}\\\\\n",
    "  e2p &= \\frac{k_4 \\cdot e2_0 \\cdot p }{k_5 + k_6 + k_4 \\cdot p} \\tag{60}\n",
    "  \\end{align*}$$"
   ]
  },
  {
   "cell_type": "code",
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       "  811.525,1266.4 813.611,1266.72 815.696,1267.05 817.782,1267.37 819.867,1267.68 821.953,1268 824.038,1268.32 826.124,1268.63 828.21,1268.94 830.295,1269.25 \n",
       "  832.381,1269.56 834.466,1269.87 836.552,1270.17 838.637,1270.48 840.723,1270.78 842.809,1271.08 844.894,1271.38 846.98,1271.68 849.065,1271.97 851.151,1272.27 \n",
       "  853.236,1272.57 855.322,1272.86 857.407,1273.16 859.493,1273.45 861.579,1273.75 863.664,1274.04 865.75,1274.34 867.835,1274.63 869.921,1274.93 872.006,1275.22 \n",
       "  874.092,1275.51 876.178,1275.81 878.263,1276.1 880.349,1276.39 882.434,1276.67 884.52,1276.96 886.605,1277.24 888.691,1277.53 890.776,1277.81 892.862,1278.09 \n",
       "  894.948,1278.37 897.033,1278.64 899.119,1278.92 901.204,1279.19 903.29,1279.46 905.375,1279.74 907.461,1280 909.547,1280.27 911.632,1280.54 913.718,1280.8 \n",
       "  915.803,1281.07 917.889,1281.33 919.974,1281.59 922.06,1281.85 924.145,1282.11 926.231,1282.37 928.317,1282.63 930.402,1282.89 932.488,1283.15 934.573,1283.41 \n",
       "  936.659,1283.67 938.744,1283.93 940.83,1284.19 942.916,1284.45 945.001,1284.71 947.087,1284.97 949.172,1285.22 951.258,1285.48 953.343,1285.74 955.429,1285.99 \n",
       "  957.514,1286.24 959.6,1286.5 961.686,1286.75 963.771,1286.99 965.857,1287.24 967.942,1287.49 970.028,1287.73 972.113,1287.97 974.199,1288.21 976.285,1288.45 \n",
       "  978.37,1288.69 980.456,1288.93 982.541,1289.16 984.627,1289.4 986.712,1289.63 988.798,1289.86 990.883,1290.09 992.969,1290.32 995.055,1290.55 997.14,1290.78 \n",
       "  999.226,1291.01 1001.31,1291.23 1003.4,1291.46 1005.48,1291.69 1007.57,1291.92 1009.65,1292.14 1011.74,1292.37 1013.82,1292.6 1015.91,1292.83 1018,1293.06 \n",
       "  1020.08,1293.29 1022.17,1293.52 1024.25,1293.75 1026.34,1293.97 1028.42,1294.2 1030.51,1294.42 1032.59,1294.64 1034.68,1294.87 1036.77,1295.09 1038.85,1295.31 \n",
       "  1040.94,1295.52 1043.02,1295.74 1045.11,1295.95 1047.19,1296.17 1049.28,1296.38 1051.36,1296.59 1053.45,1296.8 1055.54,1297.01 1057.62,1297.21 1059.71,1297.42 \n",
       "  1061.79,1297.62 1063.88,1297.82 1065.96,1298.02 1068.05,1298.23 1070.13,1298.43 1072.22,1298.63 1074.31,1298.83 1076.39,1299.02 1078.48,1299.22 1080.56,1299.42 \n",
       "  1082.65,1299.62 1084.73,1299.82 1086.82,1300.03 1088.9,1300.23 1090.99,1300.43 1093.08,1300.63 1095.16,1300.83 1097.25,1301.04 1099.33,1301.24 1101.42,1301.44 \n",
       "  1103.5,1301.64 1105.59,1301.84 1107.67,1302.03 1109.76,1302.23 1111.85,1302.43 1113.93,1302.62 1116.02,1302.81 1118.1,1303 1120.19,1303.19 1122.27,1303.38 \n",
       "  1124.36,1303.57 1126.45,1303.75 1128.53,1303.94 1130.62,1304.12 1132.7,1304.3 1134.79,1304.48 1136.87,1304.66 1138.96,1304.84 1141.04,1305.02 1143.13,1305.19 \n",
       "  1145.22,1305.37 1147.3,1305.54 1149.39,1305.72 1151.47,1305.89 1153.56,1306.07 1155.64,1306.24 1157.73,1306.42 1159.81,1306.59 1161.9,1306.77 1163.99,1306.95 \n",
       "  1166.07,1307.12 1168.16,1307.3 1170.24,1307.48 1172.33,1307.66 1174.41,1307.84 1176.5,1308.02 1178.58,1308.2 1180.67,1308.37 1182.76,1308.55 1184.84,1308.72 \n",
       "  1186.93,1308.9 1189.01,1309.07 1191.1,1309.24 1193.18,1309.41 1195.27,1309.58 1197.35,1309.75 1199.44,1309.91 1201.53,1310.08 1203.61,1310.24 1205.7,1310.4 \n",
       "  1207.78,1310.56 1209.87,1310.72 1211.95,1310.88 1214.04,1311.03 1216.12,1311.19 1218.21,1311.34 1220.3,1311.5 1222.38,1311.65 1224.47,1311.81 1226.55,1311.96 \n",
       "  1228.64,1312.11 1230.72,1312.26 1232.81,1312.42 1234.89,1312.57 1236.98,1312.72 1239.07,1312.88 1241.15,1313.04 1243.24,1313.2 1245.32,1313.35 1247.41,1313.51 \n",
       "  1249.49,1313.67 1251.58,1313.83 1253.66,1313.99 1255.75,1314.15 1257.84,1314.3 1259.92,1314.46 1262.01,1314.61 1264.09,1314.77 1266.18,1314.92 1268.26,1315.07 \n",
       "  1270.35,1315.22 1272.43,1315.37 1274.52,1315.52 1276.61,1315.67 1278.69,1315.81 1280.78,1315.96 1282.86,1316.1 1284.95,1316.24 1287.03,1316.38 1289.12,1316.52 \n",
       "  1291.2,1316.66 1293.29,1316.79 1295.38,1316.93 1297.46,1317.06 1299.55,1317.2 1301.63,1317.33 1303.72,1317.46 1305.8,1317.6 1307.89,1317.73 1309.97,1317.87 \n",
       "  1312.06,1318 1314.15,1318.14 1316.23,1318.27 1318.32,1318.41 1320.4,1318.55 1322.49,1318.69 1324.57,1318.83 1326.66,1318.97 1328.74,1319.12 1330.83,1319.26 \n",
       "  1332.92,1319.4 1335,1319.54 1337.09,1319.68 1339.17,1319.81 1341.26,1319.95 1343.34,1320.09 1345.43,1320.22 1347.51,1320.36 1349.6,1320.49 1351.69,1320.62 \n",
       "  1353.77,1320.75 1355.86,1320.88 1357.94,1321.01 1360.03,1321.13 1362.11,1321.26 1364.2,1321.38 1366.28,1321.5 1368.37,1321.62 1370.46,1321.74 1372.54,1321.86 \n",
       "  1374.63,1321.98 1376.71,1322.1 1378.8,1322.22 1380.88,1322.33 1382.97,1322.45 1385.05,1322.57 1387.14,1322.69 1389.23,1322.8 1391.31,1322.92 1393.4,1323.04 \n",
       "  1395.48,1323.16 1397.57,1323.29 1399.65,1323.41 1401.74,1323.53 1403.82,1323.66 1405.91,1323.78 1408,1323.9 1410.08,1324.02 1412.17,1324.15 1414.25,1324.27 \n",
       "  1416.34,1324.39 1418.42,1324.51 1420.51,1324.63 1422.59,1324.75 1424.68,1324.87 1426.77,1324.98 1428.85,1325.1 1430.94,1325.22 1433.02,1325.34 1435.11,1325.45 \n",
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       "  1458.05,1326.71 1460.14,1326.82 1462.22,1326.93 1464.31,1327.04 1466.39,1327.15 1468.48,1327.26 1470.56,1327.37 1472.65,1327.48 1474.73,1327.59 1476.82,1327.7 \n",
       "  1478.91,1327.81 1480.99,1327.92 1483.08,1328.02 1485.16,1328.13 1487.25,1328.24 1489.33,1328.34 1491.42,1328.45 1493.5,1328.55 1495.59,1328.66 1497.68,1328.76 \n",
       "  1499.76,1328.86 1501.85,1328.97 1503.93,1329.07 1506.02,1329.17 1508.1,1329.27 1510.19,1329.38 1512.27,1329.48 1514.36,1329.58 1516.45,1329.68 1518.53,1329.78 \n",
       "  1520.62,1329.88 1522.7,1329.98 1524.79,1330.08 1526.87,1330.17 1528.96,1330.27 1531.04,1330.37 1533.13,1330.47 1535.22,1330.56 1537.3,1330.66 1539.39,1330.75 \n",
       "  1541.47,1330.85 1543.56,1330.94 1545.64,1331.04 1547.73,1331.13 1549.81,1331.23 1551.9,1331.32 1553.99,1331.41 1556.07,1331.5 1558.16,1331.6 1560.24,1331.69 \n",
       "  1562.33,1331.78 1564.41,1331.88 1566.5,1331.97 1568.58,1332.06 1570.67,1332.15 1572.76,1332.24 1574.84,1332.33 1576.93,1332.42 1579.01,1332.52 1581.1,1332.61 \n",
       "  1583.18,1332.7 1585.27,1332.79 1587.35,1332.88 1589.44,1332.96 1591.53,1333.05 1593.61,1333.14 1595.7,1333.23 1597.78,1333.32 1599.87,1333.41 1601.95,1333.5 \n",
       "  1604.04,1333.58 1606.12,1333.67 1608.21,1333.76 1610.3,1333.85 1612.38,1333.93 1614.47,1334.02 1616.55,1334.1 1618.64,1334.19 1620.72,1334.28 1622.81,1334.36 \n",
       "  1624.89,1334.45 1626.98,1334.53 1629.07,1334.62 1631.15,1334.7 1633.24,1334.79 1635.32,1334.87 1637.41,1334.95 1639.49,1335.04 1641.58,1335.12 1643.66,1335.2 \n",
       "  1645.75,1335.29 1647.84,1335.37 1649.92,1335.45 1652.01,1335.53 1654.09,1335.61 1656.18,1335.69 1658.26,1335.78 1660.35,1335.86 1662.43,1335.94 1664.52,1336.02 \n",
       "  1666.61,1336.1 1668.69,1336.18 1670.78,1336.26 1672.86,1336.34 1674.95,1336.42 1677.03,1336.5 1679.12,1336.57 1681.2,1336.65 1683.29,1336.73 1685.38,1336.81 \n",
       "  1687.46,1336.89 1689.55,1336.96 1691.63,1337.04 1693.72,1337.12 1695.8,1337.2 1697.89,1337.27 1699.97,1337.35 1702.06,1337.42 1704.15,1337.5 1706.23,1337.58 \n",
       "  1708.32,1337.65 1710.4,1337.73 1712.49,1337.8 1714.57,1337.88 1716.66,1337.95 1718.74,1338.03 1720.83,1338.1 1722.92,1338.18 1725,1338.25 1727.09,1338.32 \n",
       "  1729.17,1338.4 1731.26,1338.47 1733.34,1338.54 1735.43,1338.62 1737.51,1338.69 1739.6,1338.76 1741.69,1338.83 1743.77,1338.91 1745.86,1338.98 1747.94,1339.05 \n",
       "  1750.03,1339.12 1752.11,1339.19 1754.2,1339.26 1756.28,1339.33 1758.37,1339.4 1760.46,1339.47 1762.54,1339.55 1764.63,1339.61 1766.71,1339.68 1768.8,1339.75 \n",
       "  1770.88,1339.82 1772.97,1339.89 1775.06,1339.96 1777.14,1340.03 1779.23,1340.1 1781.31,1340.17 1783.4,1340.23 1785.48,1340.3 1787.57,1340.37 1789.65,1340.44 \n",
       "  1791.74,1340.5 1793.83,1340.57 1795.91,1340.64 1798,1340.7 1800.08,1340.77 1802.17,1340.84 1804.25,1340.9 1806.34,1340.97 1808.42,1341.03 1810.51,1341.1 \n",
       "  1812.6,1341.16 1814.68,1341.23 1816.77,1341.29 1818.85,1341.36 1820.94,1341.42 1823.02,1341.49 1825.11,1341.55 1827.19,1341.61 1829.28,1341.68 1831.37,1341.74 \n",
       "  1833.45,1341.8 1835.54,1341.86 1837.62,1341.93 1839.71,1341.99 1841.79,1342.05 1843.88,1342.11 1845.96,1342.17 1848.05,1342.24 1850.14,1342.3 1852.22,1342.36 \n",
       "  1854.31,1342.42 1856.39,1342.48 1858.48,1342.54 1860.56,1342.6 1862.65,1342.66 1864.73,1342.72 1866.82,1342.78 1868.91,1342.84 1870.99,1342.9 1873.08,1342.96 \n",
       "  1875.16,1343.01 1877.25,1343.07 1879.33,1343.13 1881.42,1343.19 1883.5,1343.25 1885.59,1343.31 1887.68,1343.36 1889.76,1343.42 1891.85,1343.48 1893.93,1343.54 \n",
       "  1896.02,1343.6 1898.1,1343.65 1900.19,1343.71 1902.27,1343.77 1904.36,1343.82 1906.45,1343.88 1908.53,1343.94 1910.62,1343.99 1912.7,1344.05 1914.79,1344.1 \n",
       "  1916.87,1344.16 1918.96,1344.22 1921.04,1344.27 1923.13,1344.33 1925.22,1344.38 1927.3,1344.44 1929.39,1344.49 1931.47,1344.55 1933.56,1344.6 1935.64,1344.66 \n",
       "  1937.73,1344.71 1939.81,1344.76 1941.9,1344.82 1943.99,1344.87 1946.07,1344.93 1948.16,1344.98 1950.24,1345.03 1952.33,1345.09 1954.41,1345.14 1956.5,1345.19 \n",
       "  1958.58,1345.25 1960.67,1345.3 1962.76,1345.35 1964.84,1345.4 1966.93,1345.45 1969.01,1345.51 1971.1,1345.56 1973.18,1345.61 1975.27,1345.66 1977.35,1345.71 \n",
       "  1979.44,1345.76 1981.53,1345.82 1983.61,1345.87 1985.7,1345.92 1987.78,1345.97 1989.87,1346.02 1991.95,1346.07 1994.04,1346.12 1996.12,1346.17 1998.21,1346.22 \n",
       "  2000.3,1346.27 2002.38,1346.32 2004.47,1346.37 2006.55,1346.42 2008.64,1346.47 2010.72,1346.51 2012.81,1346.56 2014.89,1346.61 2016.98,1346.66 2019.07,1346.71 \n",
       "  2021.15,1346.76 2023.24,1346.8 2025.32,1346.85 2027.41,1346.9 2029.49,1346.95 2031.58,1346.99 2033.66,1347.04 2035.75,1347.09 2037.84,1347.14 2039.92,1347.18 \n",
       "  2042.01,1347.23 2044.09,1347.28 2046.18,1347.32 2048.26,1347.37 2050.35,1347.41 2052.43,1347.46 2054.52,1347.51 2056.61,1347.55 2058.69,1347.6 2060.78,1347.64 \n",
       "  2062.86,1347.69 2064.95,1347.73 2067.03,1347.78 2069.12,1347.83 2071.2,1347.87 2073.29,1347.92 2075.38,1347.96 2077.46,1348 2079.55,1348.05 2081.63,1348.09 \n",
       "  2083.72,1348.14 2085.8,1348.18 2087.89,1348.23 2089.98,1348.27 2092.06,1348.31 2094.15,1348.36 2096.23,1348.4 2098.32,1348.44 2100.4,1348.49 2102.49,1348.53 \n",
       "  2104.57,1348.57 2106.66,1348.62 2108.75,1348.66 2110.83,1348.7 2112.92,1348.75 2115,1348.79 2117.09,1348.83 2119.17,1348.87 2121.26,1348.91 2123.34,1348.96 \n",
       "  2125.43,1349 2127.52,1349.04 2129.6,1349.08 2131.69,1349.12 2133.77,1349.17 2135.86,1349.21 2137.94,1349.25 2140.03,1349.29 2142.11,1349.33 2144.2,1349.37 \n",
       "  2146.29,1349.41 2148.37,1349.45 2150.46,1349.49 2152.54,1349.53 2154.63,1349.57 2156.71,1349.61 2158.8,1349.65 2160.88,1349.69 2162.97,1349.73 2165.06,1349.77 \n",
       "  2167.14,1349.81 2169.23,1349.85 2171.31,1349.89 2173.4,1349.93 2175.48,1349.97 2177.57,1350.01 2179.65,1350.05 2181.74,1350.08 2183.83,1350.12 2185.91,1350.16 \n",
       "  2188,1350.2 2190.08,1350.24 2192.17,1350.28 2194.25,1350.31 2196.34,1350.35 2198.42,1350.39 2200.51,1350.43 2202.6,1350.47 2204.68,1350.5 2206.77,1350.54 \n",
       "  2208.85,1350.58 2210.94,1350.61 2213.02,1350.65 2215.11,1350.69 2217.19,1350.72 2219.28,1350.76 2221.37,1350.8 2223.45,1350.83 2225.54,1350.87 2227.62,1350.91 \n",
       "  2229.71,1350.94 2231.79,1350.98 2233.88,1351.01 2235.96,1351.05 2238.05,1351.08 2240.14,1351.12 2242.22,1351.15 2244.31,1351.19 2246.39,1351.23 2248.48,1351.26 \n",
       "  2250.56,1351.29 2252.65,1351.33 2254.73,1351.36 2256.82,1351.4 2258.91,1351.43 2260.99,1351.47 2263.08,1351.5 2265.16,1351.54 2267.25,1351.57 2269.33,1351.6 \n",
       "  2271.42,1351.64 2273.5,1351.67 2275.59,1351.71 2277.68,1351.74 2279.76,1351.77 2281.85,1351.81 2283.93,1351.84 2286.02,1351.87 2288.1,1351.91 2290.19,1351.94 \n",
       "  2292.27,1351.97 2294.36,1352.01 2296.45,1352.04 2298.53,1352.07 2300.62,1352.11 2302.7,1352.14 2304.79,1352.17 2306.87,1352.2 2308.96,1352.24 2311.04,1352.27 \n",
       "  2313.13,1352.3 2315.22,1352.33 2317.3,1352.36 2319.39,1352.4 2321.47,1352.43 2323.56,1352.46 2325.64,1352.49 2327.73,1352.52 2329.81,1352.56 2331.9,1352.59 \n",
       "  2333.99,1352.62 2336.07,1352.65 2338.16,1352.68 2340.24,1352.71 2342.33,1352.74 2344.41,1352.77 2346.5,1352.8 2348.58,1352.84 2350.67,1352.87 2352.76,1352.9 \n",
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       "  290.134,1022.35 292.22,1005.96 294.306,991.477 296.391,978.647 298.477,967.248 300.562,957.099 302.648,948.045 304.733,939.957 306.819,932.72 308.904,926.236 \n",
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       "  352.701,876.57 354.787,876.163 356.872,875.832 358.958,875.571 361.044,875.375 363.129,875.239 365.215,875.158 367.3,875.127 369.386,875.142 371.471,875.2 \n",
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       "  415.268,881.847 417.354,882.299 419.439,882.758 421.525,883.223 423.61,883.693 425.696,884.168 427.782,884.647 429.867,885.13 431.953,885.618 434.038,886.108 \n",
       "  436.124,886.601 438.209,887.097 440.295,887.594 442.38,888.093 444.466,888.594 446.552,889.096 448.637,889.6 450.723,890.105 452.808,890.612 454.894,891.12 \n",
       "  456.979,891.629 459.065,892.14 461.151,892.651 463.236,893.164 465.322,893.679 467.407,894.194 469.493,894.709 471.578,895.226 473.664,895.743 475.749,896.26 \n",
       "  477.835,896.778 479.921,897.296 482.006,897.813 484.092,898.331 486.177,898.848 488.263,899.365 490.348,899.881 492.434,900.398 494.52,900.914 496.605,901.431 \n",
       "  498.691,901.948 500.776,902.464 502.862,902.981 504.947,903.498 507.033,904.015 509.118,904.532 511.204,905.049 513.29,905.567 515.375,906.084 517.461,906.601 \n",
       "  519.546,907.119 521.632,907.636 523.717,908.153 525.803,908.67 527.889,909.186 529.974,909.702 532.06,910.217 534.145,910.732 536.231,911.246 538.316,911.76 \n",
       "  540.402,912.272 542.487,912.784 544.573,913.296 546.659,913.807 548.744,914.318 550.83,914.828 552.915,915.339 555.001,915.849 557.086,916.359 559.172,916.869 \n",
       "  561.258,917.378 563.343,917.888 565.429,918.398 567.514,918.907 569.6,919.416 571.685,919.925 573.771,920.434 575.856,920.943 577.942,921.451 580.028,921.959 \n",
       "  582.113,922.466 584.199,922.973 586.284,923.48 588.37,923.985 590.455,924.49 592.541,924.994 594.627,925.497 596.712,925.999 598.798,926.501 600.883,927.001 \n",
       "  602.969,927.501 605.054,928.001 607.14,928.499 609.225,928.998 611.311,929.496 613.397,929.994 615.482,930.492 617.568,930.99 619.653,931.487 621.739,931.984 \n",
       "  623.824,932.481 625.91,932.978 627.996,933.475 630.081,933.972 632.167,934.468 634.252,934.964 636.338,935.46 638.423,935.955 640.509,936.45 642.594,936.945 \n",
       "  644.68,937.439 646.766,937.932 648.851,938.424 650.937,938.916 653.022,939.406 655.108,939.896 657.193,940.384 659.279,940.871 661.365,941.357 663.45,941.842 \n",
       "  665.536,942.326 667.621,942.81 669.707,943.293 671.792,943.776 673.878,944.259 675.963,944.741 678.049,945.223 680.135,945.705 682.22,946.186 684.306,946.667 \n",
       "  686.391,947.149 688.477,947.63 690.562,948.111 692.648,948.591 694.734,949.072 696.819,949.552 698.905,950.032 700.99,950.512 703.076,950.991 705.161,951.47 \n",
       "  707.247,951.948 709.332,952.425 711.418,952.902 713.504,953.378 715.589,953.853 717.675,954.327 719.76,954.799 721.846,955.27 723.931,955.74 726.017,956.208 \n",
       "  728.103,956.675 730.188,957.142 732.274,957.607 734.359,958.072 736.445,958.537 738.53,959.001 740.616,959.464 742.701,959.927 744.787,960.39 746.873,960.853 \n",
       "  748.958,961.315 751.044,961.778 753.129,962.24 755.215,962.702 757.3,963.164 759.386,963.626 761.472,964.087 763.557,964.549 765.643,965.01 767.728,965.471 \n",
       "  769.814,965.931 771.899,966.391 773.985,966.851 776.071,967.309 778.156,967.767 780.242,968.224 782.327,968.681 784.413,969.136 786.498,969.589 788.584,970.041 \n",
       "  790.669,970.492 792.755,970.941 794.841,971.388 796.926,971.835 799.012,972.28 801.097,972.725 803.183,973.168 805.268,973.612 807.354,974.054 809.44,974.496 \n",
       "  811.525,974.938 813.611,975.38 815.696,975.821 817.782,976.262 819.867,976.703 821.953,977.144 824.038,977.584 826.124,978.025 828.21,978.465 830.295,978.906 \n",
       "  832.381,979.346 834.466,979.786 836.552,980.226 838.637,980.665 840.723,981.104 842.809,981.542 844.894,981.98 846.98,982.417 849.065,982.854 851.151,983.289 \n",
       "  853.236,983.723 855.322,984.157 857.407,984.588 859.493,985.018 861.579,985.447 863.664,985.873 865.75,986.298 867.835,986.722 869.921,987.144 872.006,987.566 \n",
       "  874.092,987.986 876.178,988.406 878.263,988.826 880.349,989.245 882.434,989.663 884.52,990.082 886.605,990.5 888.691,990.917 890.776,991.335 892.862,991.753 \n",
       "  894.948,992.17 897.033,992.588 899.119,993.005 901.204,993.422 903.29,993.839 905.375,994.256 907.461,994.673 909.547,995.09 911.632,995.506 913.718,995.922 \n",
       "  915.803,996.337 917.889,996.751 919.974,997.165 922.06,997.578 924.145,997.99 926.231,998.401 928.317,998.811 930.402,999.219 932.488,999.625 934.573,1000.03 \n",
       "  936.659,1000.43 938.744,1000.83 940.83,1001.23 942.916,1001.63 945.001,1002.03 947.087,1002.42 949.172,1002.82 951.258,1003.21 953.343,1003.61 955.429,1004 \n",
       "  957.514,1004.4 959.6,1004.79 961.686,1005.18 963.771,1005.58 965.857,1005.97 967.942,1006.36 970.028,1006.76 972.113,1007.15 974.199,1007.54 976.285,1007.94 \n",
       "  978.37,1008.33 980.456,1008.72 982.541,1009.11 984.627,1009.51 986.712,1009.9 988.798,1010.29 990.883,1010.68 992.969,1011.07 995.055,1011.46 997.14,1011.85 \n",
       "  999.226,1012.24 1001.31,1012.62 1003.4,1013.01 1005.48,1013.39 1007.57,1013.77 1009.65,1014.15 1011.74,1014.53 1013.82,1014.9 1015.91,1015.28 1018,1015.65 \n",
       "  1020.08,1016.02 1022.17,1016.39 1024.25,1016.76 1026.34,1017.13 1028.42,1017.5 1030.51,1017.87 1032.59,1018.24 1034.68,1018.61 1036.77,1018.98 1038.85,1019.34 \n",
       "  1040.94,1019.71 1043.02,1020.08 1045.11,1020.45 1047.19,1020.82 1049.28,1021.18 1051.36,1021.55 1053.45,1021.92 1055.54,1022.29 1057.62,1022.66 1059.71,1023.02 \n",
       "  1061.79,1023.39 1063.88,1023.76 1065.96,1024.13 1068.05,1024.49 1070.13,1024.86 1072.22,1025.22 1074.31,1025.58 1076.39,1025.95 1078.48,1026.31 1080.56,1026.67 \n",
       "  1082.65,1027.02 1084.73,1027.38 1086.82,1027.73 1088.9,1028.08 1090.99,1028.43 1093.08,1028.78 1095.16,1029.13 1097.25,1029.47 1099.33,1029.82 1101.42,1030.16 \n",
       "  1103.5,1030.51 1105.59,1030.85 1107.67,1031.19 1109.76,1031.54 1111.85,1031.88 1113.93,1032.22 1116.02,1032.56 1118.1,1032.91 1120.19,1033.25 1122.27,1033.59 \n",
       "  1124.36,1033.94 1126.45,1034.28 1128.53,1034.62 1130.62,1034.97 1132.7,1035.31 1134.79,1035.65 1136.87,1036 1138.96,1036.34 1141.04,1036.68 1143.13,1037.02 \n",
       "  1145.22,1037.36 1147.3,1037.71 1149.39,1038.05 1151.47,1038.38 1153.56,1038.72 1155.64,1039.06 1157.73,1039.39 1159.81,1039.72 1161.9,1040.05 1163.99,1040.38 \n",
       "  1166.07,1040.71 1168.16,1041.03 1170.24,1041.35 1172.33,1041.67 1174.41,1041.99 1176.5,1042.31 1178.58,1042.63 1180.67,1042.95 1182.76,1043.27 1184.84,1043.59 \n",
       "  1186.93,1043.91 1189.01,1044.23 1191.1,1044.54 1193.18,1044.86 1195.27,1045.18 1197.35,1045.5 1199.44,1045.82 1201.53,1046.14 1203.61,1046.46 1205.7,1046.77 \n",
       "  1207.78,1047.09 1209.87,1047.41 1211.95,1047.73 1214.04,1048.05 1216.12,1048.37 1218.21,1048.69 1220.3,1049.01 1222.38,1049.32 1224.47,1049.64 1226.55,1049.96 \n",
       "  1228.64,1050.27 1230.72,1050.59 1232.81,1050.9 1234.89,1051.21 1236.98,1051.52 1239.07,1051.82 1241.15,1052.13 1243.24,1052.43 1245.32,1052.73 1247.41,1053.03 \n",
       "  1249.49,1053.32 1251.58,1053.62 1253.66,1053.91 1255.75,1054.21 1257.84,1054.5 1259.92,1054.8 1262.01,1055.09 1264.09,1055.39 1266.18,1055.68 1268.26,1055.97 \n",
       "  1270.35,1056.27 1272.43,1056.56 1274.52,1056.86 1276.61,1057.15 1278.69,1057.45 1280.78,1057.74 1282.86,1058.04 1284.95,1058.33 1287.03,1058.63 1289.12,1058.93 \n",
       "  1291.2,1059.22 1293.29,1059.52 1295.38,1059.81 1297.46,1060.11 1299.55,1060.4 1301.63,1060.7 1303.72,1060.99 1305.8,1061.28 1307.89,1061.57 1309.97,1061.86 \n",
       "  1312.06,1062.15 1314.15,1062.44 1316.23,1062.72 1318.32,1063 1320.4,1063.28 1322.49,1063.56 1324.57,1063.83 1326.66,1064.11 1328.74,1064.38 1330.83,1064.65 \n",
       "  1332.92,1064.92 1335,1065.19 1337.09,1065.47 1339.17,1065.74 1341.26,1066.01 1343.34,1066.28 1345.43,1066.55 1347.51,1066.82 1349.6,1067.09 1351.69,1067.36 \n",
       "  1353.77,1067.63 1355.86,1067.91 1357.94,1068.18 1360.03,1068.45 1362.11,1068.73 1364.2,1069 1366.28,1069.27 1368.37,1069.55 1370.46,1069.82 1372.54,1070.09 \n",
       "  1374.63,1070.37 1376.71,1070.64 1378.8,1070.91 1380.88,1071.18 1382.97,1071.45 1385.05,1071.72 1387.14,1071.99 1389.23,1072.26 1391.31,1072.52 1393.4,1072.79 \n",
       "  1395.48,1073.05 1397.57,1073.3 1399.65,1073.56 1401.74,1073.81 1403.82,1074.07 1405.91,1074.32 1408,1074.58 1410.08,1074.83 1412.17,1075.08 1414.25,1075.33 \n",
       "  1416.34,1075.58 1418.42,1075.84 1420.51,1076.09 1422.59,1076.34 1424.68,1076.59 1426.77,1076.84 1428.85,1077.09 1430.94,1077.33 1433.02,1077.58 1435.11,1077.83 \n",
       "  1437.19,1078.08 1439.28,1078.32 1441.37,1078.57 1443.45,1078.82 1445.54,1079.06 1447.62,1079.31 1449.71,1079.55 1451.79,1079.79 1453.88,1080.04 1455.96,1080.28 \n",
       "  1458.05,1080.52 1460.14,1080.77 1462.22,1081.01 1464.31,1081.25 1466.39,1081.49 1468.48,1081.73 1470.56,1081.97 1472.65,1082.21 1474.73,1082.45 1476.82,1082.69 \n",
       "  1478.91,1082.93 1480.99,1083.16 1483.08,1083.4 1485.16,1083.64 1487.25,1083.87 1489.33,1084.11 1491.42,1084.35 1493.5,1084.58 1495.59,1084.81 1497.68,1085.05 \n",
       "  1499.76,1085.28 1501.85,1085.52 1503.93,1085.75 1506.02,1085.98 1508.1,1086.21 1510.19,1086.44 1512.27,1086.68 1514.36,1086.91 1516.45,1087.14 1518.53,1087.37 \n",
       "  1520.62,1087.6 1522.7,1087.82 1524.79,1088.05 1526.87,1088.28 1528.96,1088.51 1531.04,1088.74 1533.13,1088.96 1535.22,1089.19 1537.3,1089.41 1539.39,1089.64 \n",
       "  1541.47,1089.86 1543.56,1090.09 1545.64,1090.31 1547.73,1090.54 1549.81,1090.76 1551.9,1090.98 1553.99,1091.2 1556.07,1091.42 1558.16,1091.64 1560.24,1091.86 \n",
       "  1562.33,1092.08 1564.41,1092.3 1566.5,1092.52 1568.58,1092.73 1570.67,1092.95 1572.76,1093.17 1574.84,1093.38 1576.93,1093.6 1579.01,1093.81 1581.1,1094.02 \n",
       "  1583.18,1094.24 1585.27,1094.45 1587.35,1094.66 1589.44,1094.88 1591.53,1095.09 1593.61,1095.3 1595.7,1095.51 1597.78,1095.72 1599.87,1095.93 1601.95,1096.14 \n",
       "  1604.04,1096.35 1606.12,1096.56 1608.21,1096.76 1610.3,1096.97 1612.38,1097.18 1614.47,1097.39 1616.55,1097.59 1618.64,1097.8 1620.72,1098 1622.81,1098.21 \n",
       "  1624.89,1098.41 1626.98,1098.61 1629.07,1098.82 1631.15,1099.02 1633.24,1099.22 1635.32,1099.42 1637.41,1099.63 1639.49,1099.83 1641.58,1100.03 1643.66,1100.23 \n",
       "  1645.75,1100.43 1647.84,1100.62 1649.92,1100.82 1652.01,1101.02 1654.09,1101.22 1656.18,1101.42 1658.26,1101.61 1660.35,1101.81 1662.43,1102 1664.52,1102.2 \n",
       "  1666.61,1102.39 1668.69,1102.59 1670.78,1102.78 1672.86,1102.97 1674.95,1103.17 1677.03,1103.36 1679.12,1103.55 1681.2,1103.74 1683.29,1103.93 1685.38,1104.12 \n",
       "  1687.46,1104.31 1689.55,1104.5 1691.63,1104.69 1693.72,1104.88 1695.8,1105.07 1697.89,1105.25 1699.97,1105.44 1702.06,1105.63 1704.15,1105.81 1706.23,1105.99 \n",
       "  1708.32,1106.18 1710.4,1106.36 1712.49,1106.55 1714.57,1106.73 1716.66,1106.91 1718.74,1107.09 1720.83,1107.28 1722.92,1107.46 1725,1107.64 1727.09,1107.82 \n",
       "  1729.17,1108 1731.26,1108.18 1733.34,1108.36 1735.43,1108.54 1737.51,1108.72 1739.6,1108.9 1741.69,1109.07 1743.77,1109.25 1745.86,1109.43 1747.94,1109.61 \n",
       "  1750.03,1109.78 1752.11,1109.96 1754.2,1110.14 1756.28,1110.31 1758.37,1110.49 1760.46,1110.66 1762.54,1110.83 1764.63,1111.01 1766.71,1111.18 1768.8,1111.36 \n",
       "  1770.88,1111.53 1772.97,1111.7 1775.06,1111.87 1777.14,1112.04 1779.23,1112.21 1781.31,1112.39 1783.4,1112.56 1785.48,1112.73 1787.57,1112.9 1789.65,1113.06 \n",
       "  1791.74,1113.23 1793.83,1113.4 1795.91,1113.57 1798,1113.74 1800.08,1113.91 1802.17,1114.07 1804.25,1114.24 1806.34,1114.41 1808.42,1114.57 1810.51,1114.74 \n",
       "  1812.6,1114.9 1814.68,1115.07 1816.77,1115.23 1818.85,1115.39 1820.94,1115.56 1823.02,1115.72 1825.11,1115.88 1827.19,1116.05 1829.28,1116.21 1831.37,1116.37 \n",
       "  1833.45,1116.53 1835.54,1116.69 1837.62,1116.85 1839.71,1117.01 1841.79,1117.17 1843.88,1117.33 1845.96,1117.49 1848.05,1117.65 1850.14,1117.81 1852.22,1117.96 \n",
       "  1854.31,1118.12 1856.39,1118.28 1858.48,1118.44 1860.56,1118.59 1862.65,1118.75 1864.73,1118.9 1866.82,1119.06 1868.91,1119.21 1870.99,1119.37 1873.08,1119.52 \n",
       "  1875.16,1119.67 1877.25,1119.83 1879.33,1119.98 1881.42,1120.13 1883.5,1120.28 1885.59,1120.43 1887.68,1120.58 1889.76,1120.73 1891.85,1120.88 1893.93,1121.03 \n",
       "  1896.02,1121.18 1898.1,1121.33 1900.19,1121.48 1902.27,1121.63 1904.36,1121.77 1906.45,1121.92 1908.53,1122.07 1910.62,1122.22 1912.7,1122.36 1914.79,1122.51 \n",
       "  1916.87,1122.65 1918.96,1122.8 1921.04,1122.94 1923.13,1123.09 1925.22,1123.23 1927.3,1123.38 1929.39,1123.52 1931.47,1123.67 1933.56,1123.81 1935.64,1123.95 \n",
       "  1937.73,1124.09 1939.81,1124.24 1941.9,1124.38 1943.99,1124.52 1946.07,1124.66 1948.16,1124.8 1950.24,1124.94 1952.33,1125.08 1954.41,1125.22 1956.5,1125.36 \n",
       "  1958.58,1125.5 1960.67,1125.64 1962.76,1125.78 1964.84,1125.92 1966.93,1126.06 1969.01,1126.19 1971.1,1126.33 1973.18,1126.47 1975.27,1126.61 1977.35,1126.74 \n",
       "  1979.44,1126.88 1981.53,1127.01 1983.61,1127.15 1985.7,1127.28 1987.78,1127.42 1989.87,1127.55 1991.95,1127.69 1994.04,1127.82 1996.12,1127.95 1998.21,1128.09 \n",
       "  2000.3,1128.22 2002.38,1128.35 2004.47,1128.48 2006.55,1128.62 2008.64,1128.75 2010.72,1128.88 2012.81,1129.01 2014.89,1129.14 2016.98,1129.27 2019.07,1129.4 \n",
       "  2021.15,1129.53 2023.24,1129.66 2025.32,1129.79 2027.41,1129.92 2029.49,1130.04 2031.58,1130.17 2033.66,1130.3 2035.75,1130.43 2037.84,1130.55 2039.92,1130.68 \n",
       "  2042.01,1130.8 2044.09,1130.93 2046.18,1131.06 2048.26,1131.18 2050.35,1131.3 2052.43,1131.43 2054.52,1131.55 2056.61,1131.67 2058.69,1131.8 2060.78,1131.92 \n",
       "  2062.86,1132.04 2064.95,1132.16 2067.03,1132.29 2069.12,1132.41 2071.2,1132.53 2073.29,1132.65 2075.38,1132.77 2077.46,1132.89 2079.55,1133.01 2081.63,1133.13 \n",
       "  2083.72,1133.25 2085.8,1133.37 2087.89,1133.49 2089.98,1133.61 2092.06,1133.73 2094.15,1133.85 2096.23,1133.97 2098.32,1134.08 2100.4,1134.2 2102.49,1134.32 \n",
       "  2104.57,1134.44 2106.66,1134.55 2108.75,1134.67 2110.83,1134.78 2112.92,1134.9 2115,1135.02 2117.09,1135.13 2119.17,1135.25 2121.26,1135.36 2123.34,1135.48 \n",
       "  2125.43,1135.59 2127.52,1135.71 2129.6,1135.82 2131.69,1135.93 2133.77,1136.05 2135.86,1136.16 2137.94,1136.27 2140.03,1136.39 2142.11,1136.5 2144.2,1136.61 \n",
       "  2146.29,1136.72 2148.37,1136.83 2150.46,1136.94 2152.54,1137.05 2154.63,1137.17 2156.71,1137.28 2158.8,1137.39 2160.88,1137.5 2162.97,1137.61 2165.06,1137.72 \n",
       "  2167.14,1137.82 2169.23,1137.93 2171.31,1138.04 2173.4,1138.15 2175.48,1138.26 2177.57,1138.37 2179.65,1138.47 2181.74,1138.58 2183.83,1138.69 2185.91,1138.79 \n",
       "  2188,1138.9 2190.08,1139.01 2192.17,1139.11 2194.25,1139.22 2196.34,1139.32 2198.42,1139.43 2200.51,1139.53 2202.6,1139.64 2204.68,1139.74 2206.77,1139.85 \n",
       "  2208.85,1139.95 2210.94,1140.05 2213.02,1140.16 2215.11,1140.26 2217.19,1140.36 2219.28,1140.47 2221.37,1140.57 2223.45,1140.67 2225.54,1140.77 2227.62,1140.87 \n",
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       "  1583.18,734.527 1585.27,734.395 1587.35,734.263 1589.44,734.132 1591.53,734 1593.61,733.869 1595.7,733.739 1597.78,733.609 1599.87,733.479 1601.95,733.349 \n",
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       "  1666.61,729.515 1668.69,729.397 1670.78,729.279 1672.86,729.162 1674.95,729.045 1677.03,728.929 1679.12,728.813 1681.2,728.697 1683.29,728.581 1685.38,728.466 \n",
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       "  1750.03,725.061 1752.11,724.956 1754.2,724.852 1756.28,724.749 1758.37,724.645 1760.46,724.542 1762.54,724.44 1764.63,724.338 1766.71,724.236 1768.8,724.134 \n",
       "  1770.88,724.033 1772.97,723.932 1775.06,723.832 1777.14,723.732 1779.23,723.632 1781.31,723.533 1783.4,723.434 1785.48,723.335 1787.57,723.237 1789.65,723.139 \n",
       "  1791.74,723.041 1793.83,722.944 1795.91,722.847 1798,722.75 1800.08,722.654 1802.17,722.558 1804.25,722.463 1806.34,722.367 1808.42,722.273 1810.51,722.178 \n",
       "  1812.6,722.084 1814.68,721.99 1816.77,721.897 1818.85,721.804 1820.94,721.711 1823.02,721.619 1825.11,721.527 1827.19,721.435 1829.28,721.344 1831.37,721.253 \n",
       "  1833.45,721.162 1835.54,721.072 1837.62,720.982 1839.71,720.893 1841.79,720.803 1843.88,720.715 1845.96,720.626 1848.05,720.538 1850.14,720.45 1852.22,720.363 \n",
       "  1854.31,720.276 1856.39,720.189 1858.48,720.103 1860.56,720.017 1862.65,719.931 1864.73,719.846 1866.82,719.761 1868.91,719.676 1870.99,719.592 1873.08,719.508 \n",
       "  1875.16,719.423 1877.25,719.339 1879.33,719.255 1881.42,719.171 1883.5,719.087 1885.59,719.003 1887.68,718.92 1889.76,718.837 1891.85,718.754 1893.93,718.672 \n",
       "  1896.02,718.589 1898.1,718.507 1900.19,718.426 1902.27,718.344 1904.36,718.263 1906.45,718.181 1908.53,718.101 1910.62,718.02 1912.7,717.94 1914.79,717.859 \n",
       "  1916.87,717.779 1918.96,717.7 1921.04,717.62 1923.13,717.541 1925.22,717.462 1927.3,717.383 1929.39,717.305 1931.47,717.227 1933.56,717.148 1935.64,717.071 \n",
       "  1937.73,716.993 1939.81,716.916 1941.9,716.839 1943.99,716.762 1946.07,716.685 1948.16,716.609 1950.24,716.533 1952.33,716.457 1954.41,716.381 1956.5,716.306 \n",
       "  1958.58,716.231 1960.67,716.156 1962.76,716.081 1964.84,716.006 1966.93,715.932 1969.01,715.858 1971.1,715.784 1973.18,715.711 1975.27,715.638 1977.35,715.565 \n",
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       "  1604.04,1235.38 1606.12,1235.21 1608.21,1235.04 1610.3,1234.88 1612.38,1234.71 1614.47,1234.55 1616.55,1234.38 1618.64,1234.22 1620.72,1234.05 1622.81,1233.89 \n",
       "  1624.89,1233.72 1626.98,1233.56 1629.07,1233.4 1631.15,1233.23 1633.24,1233.07 1635.32,1232.91 1637.41,1232.75 1639.49,1232.59 1641.58,1232.43 1643.66,1232.27 \n",
       "  1645.75,1232.11 1647.84,1231.95 1649.92,1231.79 1652.01,1231.63 1654.09,1231.47 1656.18,1231.31 1658.26,1231.15 1660.35,1231 1662.43,1230.84 1664.52,1230.68 \n",
       "  1666.61,1230.52 1668.69,1230.37 1670.78,1230.21 1672.86,1230.06 1674.95,1229.9 1677.03,1229.75 1679.12,1229.59 1681.2,1229.44 1683.29,1229.29 1685.38,1229.13 \n",
       "  1687.46,1228.98 1689.55,1228.83 1691.63,1228.68 1693.72,1228.52 1695.8,1228.37 1697.89,1228.22 1699.97,1228.07 1702.06,1227.92 1704.15,1227.77 1706.23,1227.63 \n",
       "  1708.32,1227.48 1710.4,1227.33 1712.49,1227.18 1714.57,1227.04 1716.66,1226.89 1718.74,1226.74 1720.83,1226.59 1722.92,1226.45 1725,1226.3 1727.09,1226.16 \n",
       "  1729.17,1226.01 1731.26,1225.86 1733.34,1225.72 1735.43,1225.57 1737.51,1225.43 1739.6,1225.28 1741.69,1225.14 1743.77,1225 1745.86,1224.85 1747.94,1224.71 \n",
       "  1750.03,1224.57 1752.11,1224.42 1754.2,1224.28 1756.28,1224.14 1758.37,1224 1760.46,1223.85 1762.54,1223.71 1764.63,1223.57 1766.71,1223.43 1768.8,1223.29 \n",
       "  1770.88,1223.15 1772.97,1223.01 1775.06,1222.87 1777.14,1222.73 1779.23,1222.59 1781.31,1222.45 1783.4,1222.31 1785.48,1222.17 1787.57,1222.03 1789.65,1221.89 \n",
       "  1791.74,1221.75 1793.83,1221.61 1795.91,1221.48 1798,1221.34 1800.08,1221.2 1802.17,1221.06 1804.25,1220.93 1806.34,1220.79 1808.42,1220.65 1810.51,1220.52 \n",
       "  1812.6,1220.38 1814.68,1220.25 1816.77,1220.11 1818.85,1219.98 1820.94,1219.84 1823.02,1219.71 1825.11,1219.57 1827.19,1219.44 1829.28,1219.3 1831.37,1219.17 \n",
       "  1833.45,1219.04 1835.54,1218.9 1837.62,1218.77 1839.71,1218.64 1841.79,1218.5 1843.88,1218.37 1845.96,1218.24 1848.05,1218.11 1850.14,1217.98 1852.22,1217.85 \n",
       "  1854.31,1217.71 1856.39,1217.58 1858.48,1217.45 1860.56,1217.32 1862.65,1217.19 1864.73,1217.06 1866.82,1216.93 1868.91,1216.8 1870.99,1216.68 1873.08,1216.55 \n",
       "  1875.16,1216.42 1877.25,1216.29 1879.33,1216.17 1881.42,1216.04 1883.5,1215.92 1885.59,1215.79 1887.68,1215.67 1889.76,1215.54 1891.85,1215.42 1893.93,1215.29 \n",
       "  1896.02,1215.17 1898.1,1215.04 1900.19,1214.92 1902.27,1214.79 1904.36,1214.67 1906.45,1214.55 1908.53,1214.43 1910.62,1214.3 1912.7,1214.18 1914.79,1214.06 \n",
       "  1916.87,1213.94 1918.96,1213.82 1921.04,1213.69 1923.13,1213.57 1925.22,1213.45 1927.3,1213.33 1929.39,1213.21 1931.47,1213.09 1933.56,1212.97 1935.64,1212.85 \n",
       "  1937.73,1212.73 1939.81,1212.61 1941.9,1212.49 1943.99,1212.38 1946.07,1212.26 1948.16,1212.14 1950.24,1212.02 1952.33,1211.9 1954.41,1211.79 1956.5,1211.67 \n",
       "  1958.58,1211.55 1960.67,1211.44 1962.76,1211.32 1964.84,1211.2 1966.93,1211.09 1969.01,1210.97 1971.1,1210.86 1973.18,1210.74 1975.27,1210.63 1977.35,1210.51 \n",
       "  1979.44,1210.4 1981.53,1210.28 1983.61,1210.17 1985.7,1210.06 1987.78,1209.94 1989.87,1209.83 1991.95,1209.72 1994.04,1209.6 1996.12,1209.49 1998.21,1209.38 \n",
       "  2000.3,1209.27 2002.38,1209.15 2004.47,1209.04 2006.55,1208.93 2008.64,1208.82 2010.72,1208.71 2012.81,1208.6 2014.89,1208.49 2016.98,1208.38 2019.07,1208.27 \n",
       "  2021.15,1208.16 2023.24,1208.05 2025.32,1207.94 2027.41,1207.83 2029.49,1207.72 2031.58,1207.62 2033.66,1207.51 2035.75,1207.4 2037.84,1207.29 2039.92,1207.19 \n",
       "  2042.01,1207.08 2044.09,1206.97 2046.18,1206.87 2048.26,1206.76 2050.35,1206.66 2052.43,1206.55 2054.52,1206.45 2056.61,1206.34 2058.69,1206.24 2060.78,1206.14 \n",
       "  2062.86,1206.03 2064.95,1205.93 2067.03,1205.83 2069.12,1205.72 2071.2,1205.62 2073.29,1205.52 2075.38,1205.41 2077.46,1205.31 2079.55,1205.21 2081.63,1205.11 \n",
       "  2083.72,1205.01 2085.8,1204.91 2087.89,1204.8 2089.98,1204.7 2092.06,1204.6 2094.15,1204.5 2096.23,1204.4 2098.32,1204.3 2100.4,1204.2 2102.49,1204.1 \n",
       "  2104.57,1204 2106.66,1203.9 2108.75,1203.8 2110.83,1203.71 2112.92,1203.61 2115,1203.51 2117.09,1203.41 2119.17,1203.31 2121.26,1203.21 2123.34,1203.12 \n",
       "  2125.43,1203.02 2127.52,1202.92 2129.6,1202.82 2131.69,1202.73 2133.77,1202.63 2135.86,1202.53 2137.94,1202.44 2140.03,1202.34 2142.11,1202.25 2144.2,1202.15 \n",
       "  2146.29,1202.06 2148.37,1201.96 2150.46,1201.86 2152.54,1201.77 2154.63,1201.68 2156.71,1201.58 2158.8,1201.49 2160.88,1201.39 2162.97,1201.3 2165.06,1201.21 \n",
       "  2167.14,1201.11 2169.23,1201.02 2171.31,1200.93 2173.4,1200.83 2175.48,1200.74 2177.57,1200.65 2179.65,1200.56 2181.74,1200.46 2183.83,1200.37 2185.91,1200.28 \n",
       "  2188,1200.19 2190.08,1200.1 2192.17,1200.01 2194.25,1199.92 2196.34,1199.83 2198.42,1199.74 2200.51,1199.65 2202.6,1199.56 2204.68,1199.47 2206.77,1199.38 \n",
       "  2208.85,1199.29 2210.94,1199.2 2213.02,1199.11 2215.11,1199.02 2217.19,1198.93 2219.28,1198.84 2221.37,1198.76 2223.45,1198.67 2225.54,1198.58 2227.62,1198.49 \n",
       "  2229.71,1198.4 2231.79,1198.32 2233.88,1198.23 2235.96,1198.14 2238.05,1198.06 2240.14,1197.97 2242.22,1197.89 2244.31,1197.8 2246.39,1197.71 2248.48,1197.63 \n",
       "  2250.56,1197.54 2252.65,1197.46 2254.73,1197.37 2256.82,1197.29 2258.91,1197.21 2260.99,1197.12 2263.08,1197.04 2265.16,1196.95 2267.25,1196.87 2269.33,1196.79 \n",
       "  2271.42,1196.71 2273.5,1196.62 2275.59,1196.54 2277.68,1196.46 2279.76,1196.37 2281.85,1196.29 2283.93,1196.21 2286.02,1196.13 2288.1,1196.05 2290.19,1195.97 \n",
       "  2292.27,1195.89 2294.36,1195.8 2296.45,1195.72 2298.53,1195.64 2300.62,1195.56 2302.7,1195.48 2304.79,1195.4 2306.87,1195.32 2308.96,1195.24 2311.04,1195.16 \n",
       "  2313.13,1195.09 2315.22,1195.01 2317.3,1194.93 2319.39,1194.85 2321.47,1194.77 2323.56,1194.69 2325.64,1194.62 2327.73,1194.54 2329.81,1194.46 2331.9,1194.38 \n",
       "  2333.99,1194.31 2336.07,1194.23 2338.16,1194.15 2340.24,1194.08 2342.33,1194 2344.41,1193.92 2346.5,1193.85 2348.58,1193.77 2350.67,1193.7 2352.76,1193.62 \n",
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      ]
     },
     "execution_count": 64,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e1s_eq=(p[1] .* p[7] .* sol5[1,:])./(p[2].+p[3].+p[1] .* sol5[1,:])\n",
    "e2sp_eq=(p[4] .* p[8] .* sol5[4,:])./(p[5].+p[6].+p[4] .* sol5[4,:])\n",
    "\n",
    "plot(sol5,label=[\"s\" \"e1s\" \"e2p\" \"p\"])\n",
    "title!(\"kinase-phosphatase\")\n",
    "xlabel!(\"Time [s]\")\n",
    "ylabel!(\"Concentration [M]\")\n",
    "#png(\"Michaelis_menten.png\")\n",
    "\n",
    "plot!(sol5.t,e1s_eq,\n",
    "        ylims = (0,s0+p0),\n",
    "        label=\"e1s_eq\",        \n",
    "        linealpha = 0.5,\n",
    "        linewidth = 3,\n",
    "        linestyle= :dash,\n",
    "        linecolor = :red)\n",
    "\n",
    "\n",
    "plot!(sol5.t,e2sp_eq,\n",
    "        label=\"e1s_eq\",        \n",
    "        linealpha = 0.5,\n",
    "        linewidth = 3,\n",
    "        linestyle= :dash,\n",
    "        linecolor = :green)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As we can see, the two equilibrium solutions are again quite close to the concentration of `e1s` and `e2p`, in this conditions we can simplify all equations as:\n",
    "  \n",
    "$$\\begin{align*}\n",
    "  \\dot{s}   &= -k_1 \\cdot s \\cdot e1_0 + (k_2 + k_1 \\cdot s) \\cdot \\frac{k_1 \\cdot e1_0 \\cdot s}{k_2 + k_3 + k_1 \\cdot s} +  \\frac{k_6 \\cdot k_4 \\cdot e2_0 \\cdot p }{k_5 + k_6 + k_4 \\cdot p} \\tag{61}\\\\\n",
    "  \\dot{p}   &= -k_4 \\cdot p \\cdot e2_0  + (k_5 + k_4 \\cdot p) \\cdot \\frac{k_4 \\cdot e2_0 \\cdot p }{k_5 + k_6 + k_4 \\cdot p} +  \\frac{ k_3 \\cdot k_1 \\cdot e1_0 \\cdot s}{k_2 + k_3 + k_1 \\cdot s} \\tag{62}\n",
    "\\end{align*}$$\n",
    "\n",
    "which, after some manipulation become\n",
    "\n",
    "$$\\begin{align*}\n",
    "  \\dot{s}   &= k_1 \\cdot e1_0 \\cdot  s \\left [  \\frac{k_2 + k_1 \\cdot s}{k_2 + k_3 + k_1 \\cdot s} -1\\right ]  +  \\frac{k_6 \\cdot k_4 \\cdot e2_0 \\cdot p }{k_5 + k_6 + k_4 \\cdot p} \\tag{63}\\\\\n",
    "  \\dot{p}   &= k_4 \\cdot e2_0 \\cdot p   \\left [  \\frac{k_5 + k_4 \\cdot p }{k_5 + k_6 + k_4 \\cdot p} -1 \\right ] +  \\frac{ k_3 \\cdot k_1 \\cdot e1_0 \\cdot s}{k_2 + k_3 + k_1 \\cdot s} \\tag{64}\n",
    "\\end{align*}$$\n",
    "\n",
    "\n",
    "$$\\begin{align*}\n",
    "  \\dot{s}   &= k_1 \\cdot e1_0 \\cdot  s \\left [  \\frac{k_2 + k_1 \\cdot s- k_2 -  k_3 - k_1 \\cdot s}{k_2 + k_3 + k_1 \\cdot s} \\right ]  +  \\frac{k_6 \\cdot k_4 \\cdot e2_0 \\cdot p }{k_5 + k_6 + k_4 \\cdot p} \\tag{65}\\\\\n",
    "  \\dot{p}   &= k_4 \\cdot e2_0 \\cdot p   \\left [  \\frac{k_5 + k_4 \\cdot p -k_5 - k_6-  k_4 \\cdot p}{k_5 + k_6 + k_4 \\cdot p}  \\right ] +  \\frac{ k_3 \\cdot k_1 \\cdot e1_0 \\cdot s}{k_2 + k_3 + k_1 \\cdot s} \\tag{66}\n",
    "\\end{align*}$$\n",
    "\n",
    "$$\\begin{align*}\n",
    "  \\dot{s}   &=  \\frac{k_6 \\cdot k_4 \\cdot e2_0 \\cdot p }{k_5 + k_6 + k_4 \\cdot p}  - \\frac{ k_3 \\cdot k_1 \\cdot e1_0 \\cdot  s }{k_2 + k_3 + k_1 \\cdot s}  \\tag{67}  \\\\\n",
    "  \\dot{p}   &=   \\frac{ k_3 \\cdot k_1 \\cdot e1_0 \\cdot s}{k_2 + k_3 + k_1 \\cdot s} -   \\frac{k_6 \\cdot k_4 \\cdot e2_0 \\cdot p}{k_5 + k_6 + k_4 \\cdot p}   \\tag{68}\n",
    "\\end{align*}$$\n",
    "\n",
    "Finally, using the definition of the Michaelis-Menten constant for each enzyme-substrate pair, $K_{1M} = \\frac{k_3+k_2}{k_1}$, $K_{2M} = \\frac{k_5+k_6}{k_4}$, the previous equations can be simplified as: \n",
    "\n",
    "$$\\begin{align*}\n",
    "  \\dot{s}   &=   \\frac{k_6 \\cdot e2_0 \\cdot p}{K_{2M} + p}   -  \\frac{ k_3  \\cdot e1_0 \\cdot s}{K_{1M}+  s}    \\tag{69}\\\\\n",
    "  \\dot{p}   &=   \\frac{ k_3  \\cdot e1_0 \\cdot s}{K_{1M}+  s} -   \\frac{k_6 \\cdot e2_0 \\cdot p}{K_{2M} + p}   \\tag{70}\n",
    "\\end{align*}$$\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "\n",
    "## Conclusion \n",
    "\n",
    "Enzyme kinetics is one of the most basics sistems where Mass Action combined with mathematical modeling allows us to extract relevant information of the dynamics of the interactions that shape the responses of biological systems "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### <font color=red>Computer Exercise:</font>\n",
    "\n",
    "Solve numerically the simplified kinase-phosphatase system and compare its accuracy in the prediction of the value of $s(t)$ and $p(t)$ with the values predicted using the full system.  Find parameters where the pseudo-steady state aproximation is a better approximation to the the full system. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "tspan = (0.0,100)\n",
    "k1=0.5e3\n",
    "k2=0.15\n",
    "k3=0.03\n",
    "k4=1e3\n",
    "k5=0.1\n",
    "k6=0.01\n",
    "\n",
    "e10=0.002\n",
    "e20=0.002\n",
    "s0=0.002\n",
    "p0=0.001;"
   ]
  }
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