{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "## Enzyme Kinetics (Part 1)\n", "\n", "Implement and simulate a model in which an enzyme E aids a reaction in which a substrate S is consumed to produce a product P.\n", "\n", "\\begin{eqnarray*}\n", "\\frac{dS}{dt} &=& -k_1\\cdot S\\cdot E + k_{-1}\\cdot SE\\\\\n", "\\frac{dE}{dt} &=& -k_1S\\cdot E + k_{-1}SE + k_2\\cdot SE\\\\\n", "\\frac{dP}{dt} &=& k_2\\cdot SE\\\\\n", "\\frac{dSE}{dt} &=& k_1\\cdot S\\cdot E - k_{-1}\\cdot SE - k_2\\cdot SE\\\\\n", "\\end{eqnarray*}\n", "\n", "where SE is the compound of S and E.\n", "\n", "Code it up using names that are more appropriate for programming ($k_1$ becomes k1f for \"forward\", $k_{-1}$ becomes k1b for \"backward\", $k_2$ becomes k2)." ] }, { "cell_type": "code", "execution_count": 66, "metadata": {}, "outputs": [], "source": [ "import numpy as np\n", "import scipy.integrate\n", "\n", "def enzyme_model( t, y, params ):\n", " S = y[0]\n", " E = y[1]\n", " P = y[2]\n", " SE = y[3]\n", "\n", " k1f = params[0]\n", " k1b = params[1]\n", " k2 = params[2]\n", "\n", " dydt = np.zeros( (4,) )\n", " dydt[0] = -k1f*S*E + k1b*SE # S\n", " dydt[1] = -k1f*S*E + k1b*SE + k2*SE #E\n", " dydt[2] = k2*SE #P\n", " dydt[3] = k1f*S*E - k1b*SE - k2*SE #SE\n", " return dydt" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Simulate the model in the not-very-common situation in which there are lots of enzymes, but not a lot of substrate.\n", "\n", "Use k1f = 2, k1b=2, k2 = 3, S(0) = 4, E(0) = 100, SE(0) = 0, P(0) = 0.\n", "\n", "Simulate with from t=0 to t=100 with a timestep of 0.1" ] }, { "cell_type": "code", "execution_count": 67, "metadata": {}, "outputs": [], "source": [ "k1f = 2\n", "k1b = 1\n", "k2 = 3\n", "params = (k1f, k1b, k2)\n", "\n", "# Simulate the situation where there is very little\n", "# substrate and lots of enzymes.\n", "# This is not common in biological systems.\n", "S0 = 4\n", "E0 = 100\n", "SE0 = 0\n", "P0 = 0\n", "yinit = (S0,E0,P0,SE0)\n", "\n", "em = lambda t,y: enzyme_model(t,y,params)\n", "sol = scipy.integrate.solve_ivp( fun=em, t_span=(0,100), y0=yinit, \n", " t_eval=np.arange(0,100,1) )" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Plot the simulation output." ] }, { "cell_type": "code", "execution_count": 68, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "import matplotlib.pyplot as plt\n", "\n", "plt.plot( sol.t, sol.y.T )\n", "plt.xlabel( \"Time\")\n", "plt.ylabel( \"Concentration\" )\n", "plt.legend( [\"S\",\"E\",\"P\",\"SE\"])\n", "plt.title( \"Enzyme-catalyzed reaction\");" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Since we are interested in finding the appropriate mathematical expression for dP/dt as a function of S, determine what it is for this model, using simulation output.\n", "\n", "Approximate dP/dt by calling np.diff on the simulation output for P." ] }, { "cell_type": "code", "execution_count": 69, "metadata": {}, "outputs": [ { "data": { "image/png": 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/vQeNGtQPO5okEBWBSEi+yN3F2GkZLF6/m9N6teX2EamktG0WdixJQCoCkRjbU1DMn+cs5cmP1tCueSMeuOJELj6uk2YCJDQqApEYcXde+WIjt89cwva9hVw9LIWbzu3NUY2Two4mCU5FIBIDq7btY8KMDN5fvo3jurTksatPYmAXzQRIfKhSEZjZU+5+1aEeE5GvKiwp5aF3V/LXd3NoVL8ek0YM4PtDu2kmQOJKVd8RDCh/x8zqA4NrPo5I3TFv+TbGz8hg1bZ9XHL8MYy/qB/tj2ocdiyRrzloEZjZzUQuL9nEzL680LwBRcDUQ3xvV+BJoAPgwFR3v6/COgbcB1wI7AdGu/uiavx3iMSNLXkF3PlqFjM+20BKm6Y8de0QTuvVLuxYIgd00CJw98nAZDOb7O43H+bPLgF+5e6LzKwFsNDM3nD3JeXWuQDoFf0aCvwt+qdIrVNa5jwzfw13z1lKYXEZvzi7FzeccSyNkzQTIPHtUO8IBkVvvlju9n8c7F/v7r4R2Bi9nWdmWUBnoHwRjACejF6w/iMza2VmnaLfK1JrZKzfzdhpi/k8dzen9mzLpBED6NGuedixRKrkUPsI/hz9szGQBnxO5KOh44AFwLCqPImZpQAnAvMrLOoMrCt3Pzf62FeKwMzGAGMAkpOTq/KUIjGRV1DMPW8s44kPVtO6WSPuu/wEhh9/jGYCpFY51EdDZwKY2cvAIHdfHL2fCkysyhOYWXPgX8Av3X3PodY/QI6pRPdJpKWleXV+hkhNcndmLd7EpJmZbMkr5AdDu/Hr8/rQsolmAqT2qepRQ32+LAEAd88ws36H+iYzSyJSAk+7+8uVrLIe6FrufpfoYyJxa832fUyYkcl7y7Yy4JijmHpVGsd3bRV2LJFqq2oRfGFmjwD/jN7/PvDFwb4hekTQo0CWu99zgNXSgRvN7DkiO4l3a/+AxKvCklKmvreSKe/kkFS/Hrde0p+rTu5Gg/q6aLzUblUtgmuAG4BfRO/PJXKEz8GcAlwFLDazz6KP3QIkA7j7Q8AsIoeO5hA5fPSaqgYXiaUPVmxj3PQMVm7dx0UDOzH+4v50bKmZAKkbDnXU0FRgNvCmu98L3FvVH+zu84jsWD7YOg78tKo/UyTWtu0t5PevZvHyp+tJbt2Ux685iTP6tA87lkiNOtQ7gkeJHOt/k5kVAa8Dr7n754EnEwlRWZnz7CdruWt2NvnFpfzsrJ789MyemgmQOulQRw3NJ3LI50QzawOcC/zKzAYCnxIphReCjykSO0s27GHs9MV8unYXw3q04faRqfRsr5kAqbuqfPZRd98OPBv9wswGA+cHlEsk5vYWlnDvG8t4/IPVtGqSxL3fO56RJ3TWTIDUeYfaRzCUyPH7xwKLgWu/PEWEuy8EFgaeUCRg7s6czE1MTF/C5rwCrhySzG/O60vLppoJkMRwqHcEfwV+TeQooeFEdhafF3QokVhZt2M/t6Zn8nb2Fvp1OooHfzCIQclHhx1LJKYOVQT13P2N6O0Xo2cjFan1ikrKePj9lTzw9nLqmzHuon6M/kaKZgIkIR2qCFqZ2bcPdP8A08IicW3+yu2MnZ5Bzpa9XJDakQmX9KdTyyZhxxIJzaGK4D3g4uhtK3ffiFxjQEUgtcb2vYVMnp3NSwtz6XJ0Ex4bncZZfTuEHUskdIcqgoxyt7882ZuVuy0S98rKnBcWrGPy7Gz2F5Xw0zOP5cYze9GkoWYCRODQRfDlwdN9gJOAGUSK4BLg4wBzidSI7E17GDstg4VrdjKke2vuHJlKrw4two4lElcONVB2G4CZzSVyGuq86P2JwKuBpxOppn2FJdz31nIenbeKlk2S+NN3jueyQZoJEKlMVQfKOhC5TvGXiqKPicSd1zM3MTE9kw27C7hiSFd+c15fjm7WMOxYInGrqkXwJPCxmU2L3h8JPB5EIJHqyt25n4npS3gzazN9O7bg/itOJC2lddixROJelYrA3e80s9nAadGHrnH3T4OLJVJ1xaVlPDpvFfe9uRyAWy7syzWndCdJMwEiVXI45xpaBBzwYvUiYfhk9Q7GTctg6eY8zu3fgVuHD6BzK80EiByOKheBSDzZsa+IP8zO4oUFuXRu1YSHf5jGOf2120qkOlQEUqu4Oy8uzGXyrCzyCkr4yenH8vOze9K0of4qi1RXYK8eM3uMyBTyFndPrWT5GUTmElZFH3rZ3ScFlUdqv2Wb8xg3LYOPV+/gpJSjuWPkQPp01EyAyJEK8p9RjwNTiBxxdCDvu/vFB1kuwv6iEu5/K4dH3l9J88YNuPuy4xg1uAv16mkmQKQmBFYE7j7XzFKC+vmSGN7K2syEGZms35XPd9O68LsL+tFaMwEiNSrsD1aHmdnnwAbg1+6eWdlKZjYGGAOQnJwcw3gSlg278rntlUzmZG6md4fmvHD9MIZ010yASBDCLIJFQDd332tmFwLTgV6VrejuU4lcKY20tDSd8K4OKy4t4/F/r+beN5dR5s5vz+/Ltad2p2EDzQSIBCW0InD3PeVuzzKzB82srbtvCyuThGvhmp2MnbaY7E15nN23PROHD6Br66ZhxxKp80IrAjPrCGx2dzezIUA9YHtYeSQ8u/YXcddr2Tz78To6tWzM368azLn9O+gEcSIxEuTho88CZwBtzSwXuBVIAnD3h4BRwA1mVgLkA5e7uz72SSDuzsuL1vP7WVnsyi9mzDd78Iuze9GsUdi7rkQSS5BHDV1xiOVTiBxeKgkoZ0seY6dlMH/VDgYlt+Kflw6kX6ejwo4lkpD0Ty+JqfyiUqa8s5ypc1fStGEDJn97IN9L66qZAJEQqQgkZt7J3sKE9AzW7cjnskFduPnCvrRt3ijsWCIJT0Uggdu0u4BJMzOZtXgTx7ZrxrPXncywY9uEHUtEolQEEpiS0jKe+HAN97y+lJIy5//O68N1p/XQTIBInFERSCA+XbuTsdMyWLJxD2f0acek4akkt9FMgEg8UhFIjdq9v5i752TzzMdr6dCiMX/7/iDOT+2omQCROKYikBrh7sz4bAN3vLqEHfuK+NEp3fnfc3rTXDMBInFPr1I5Yiu27mX89Aw+WLGd47u24vFrhpDauWXYsUSkilQEUm0FxaU8+E4OD723ksZJ9bhjZCpXDEmmvmYCRGoVFYFUy3vLtjJhRgZrtu/n0hM7c8uF/WjXQjMBIrWRikAOy+Y9BUyauYRXv9hIj7bNeObHQ/lGz7ZhxxKRI6AikCopLXOe+nA1f3p9GUWlZfzqnN6MOb0HjRrUDzuaiBwhFYEc0he5uxg7LYPF63dzWq+23D4ilZS2zcKOJSI1REUgB7SnoJg/zVnKUx+toV3zRky58kQuGthJMwEidYyKQL7G3Xnli43cPnMJ2/cWcvWwFH51bm9aNE4KO5qIBEBFIF+xats+xk/PYF7ONo7r0pLHrj6JgV00EyBSl6kIBIjMBDz03goefHcFjerX4/YRA7hyaDfNBIgkgCAvVfkYcDGwxd1TK1luwH3AhcB+YLS7LwoqjxzYvOXbGD8jg1Xb9jH8+GMYd1E/2h/VOOxYIhIjQb4jeJzIpSifPMDyC4Be0a+hwN+if0qMbMkr4I6ZWaR/voHubZvxz2uHcmovzQSIJJogr1k818xSDrLKCODJ6AXrPzKzVmbWyd03BpVJIkrLnGfmr+HuOUspLC7jl9/qxU9OP5bGSZoJEElEYe4j6AysK3c/N/rY14rAzMYAYwCSk5NjEq6uyli/m7HTFvN57m5O7dmWSSMG0KNd87BjiUiIasXOYnefCkwFSEtL85Dj1Ep5BcX8+fVlPPnhalo3a8T9V5zIJcdpJkBEwi2C9UDXcve7RB+TGuTuvLp4I5NeWcLWvYVcdXI3fnVuH1o20UyAiESEWQTpwI1m9hyRncS7tX+gZq3Zvo/xMzKZu2wrqZ2P4uEfpnF811ZhxxKROBPk4aPPAmcAbc0sF7gVSAJw94eAWUQOHc0hcvjoNUFlSTSFJaVMfW8lU97JIal+PSZe0p+rhqVoJkBEKhXkUUNXHGK5Az8N6vkT1QcrtjFuegYrt+7jouM6MeHi/nTQTICIHESt2Fksh7Y1r5Dfz8pi2qfrSW7dlCd+NITTe7cLO5aI1AIqglqurMx59pO13DU7m/ziUn5+Vk/+58yemgkQkSpTEdRimRt2M3ZaBp+t28WwHm24fWQqPdtrJkBEDo+KoBbaW1jCvW8s4x//XkXrZg35y/dOYMQJx2gmQESqRUVQi7g7r2Vs4rZXlrA5r4ArhyTzm/P60rKpZgJEpPpUBLXEuh37mTAjg3eWbqVfp6N48AeDGJR8dNixRKQOUBHEuaKSMh5+fyUPvL2c+maMv7g/Vw/rRoP69cKOJiJ1hIogjn20cjvjpmeQs2UvF6R2ZMIl/enUsknYsUSkjlERxKHtewv5/axs/rUol66tm/CP0SdxZt/2YccSkTpKRRBHysqcFxasY/LsbPYXlfDTM4/lxjN70aShZgJEJDgqgjiRtXEPY6ctZtHaXQzp3po7R6bSq0OLsGOJSAJQEYRsX2EJ9721nEfnraJlkyT+9J3juWxQZ80EiEjMqAhC9HrmJiamZ7JhdwFXDOnKb8/vS6umDcOOJSIJRkUQgtyd+5mYnsmbWVvo27EFD1x5IoO7tQ47logkKBVBDBWXlvHovFXc9+ZyzGDshf0YfUoKSZoJEJEQqQhi5JPVOxg7bTHLNu/l3P4duHX4ADq30kyAiIRPRRCwHfuK+MPsLF5YkEvnVk145IdpfKt/h7BjiYj8R6BFYGbnA/cB9YFH3P0PFZaPBv7Ify9aP8XdHwkyU6yUlTkvLcpl8qws8gpK+Mnpx/Lzs3vStKG6V0TiS5DXLK4P/BU4B8gFPjGzdHdfUmHV5939xqByhGHppjzGTV/MJ6t3clLK0dwxciB9OmomQETiU5D/PB0C5Lj7SgAzew4YAVQsgjpjf1EJ97+VwyPvr6RF4wbcPeo4Rg3qQj1dNF5E4liQRdAZWFfufi4wtJL1LjOzbwLLgP9193UVVzCzMcAYgOTk5ACiHrm3sjYzYUYm63fl8920Lvzugn60bqaZABGJf2F/YP0K8Ky7F5rZ9cATwFkVV3L3qcBUgLS0NI9txINbvyuf29IzeX3JZnp3aM6LPxnGSSmaCRCR2iPIIlgPdC13vwv/3SkMgLtvL3f3EeDuAPPUqOLSMh7/92rufXMZZe787oK+XHtqd80EiEitE2QRfAL0MrPuRArgcuDK8iuYWSd33xi9OxzICjBPjVm4Zgdjp2WQvSmPb/Vrz62XDKBr66ZhxxIRqZbAisDdS8zsRmAOkcNHH3P3TDObBCxw93Tg52Y2HCgBdgCjg8pTE3btL+Ku17J59uN1HNOyMVOvGsy5AzqGHUtE5IiYe1x95H5IaWlpvmDBgpg+p7vz8qL13Dkri935xVx7and+cXYvmjUKexeLiEjVmNlCd0+rbJl+kx1CzpY8xk7LYP6qHQzudjR3jEylX6ejwo4lIlJjVAQHkF9UypR3ljN17kqaNmzAH749kO+mddVMgIjUOSqCSryTvYUJ6Rms25HPqMFduPmCvrRp3ijsWCIigVARlLNxdz6TXlnC7IxN9GzfnOfGnMzJPdqEHUtEJFAqAqCktIzHP1jNvW8so6TM+b/z+nDdaT1o2EAzASJS9yV8EXy6die3TMsga+MezuzTjkkjUjUTICIJJWGLYPf+Yu6ek80zH6+lQ4vGPPSDQZw3oKMuGi8iCSfhisDdmf7Zeu58NYud+4u59pTu/PKc3jTXTICIJKiE+u23Yutexk3L4MOV2zmhayue+FEqA45pGXYsEZFQJUwR5O7czwX3vU/jBvW489JUrjgpWTMBIiIkUBGs2b6fopIyHrv6JE7t1TbsOCIicSNhjo8si55TqXFSwvwni4hUScL8ViyLnltPRwWJiHxVAhVBpAm0W0BE5KsSpgj8P0WgJhARKS9hiqCsLPKnikBE5KsSpwii7wjUAyIiXxVoEZjZ+Wa21MxyzOx3lSxvZGbPR5fPN7OUoLJ8ubNY7whERL4qsCIws/rAX4ELgP7AFWbWv8Jq1wI73b0ncC9wV1B5/rOPIGHeA4mIVE2QvxaHADnuvtLdi4DngBEV1hkBPBG9/RJwtgV0fOcNTy8C4Py/vB/EjxcRqbWCLILOwLpy93Ojj1W6jruXALuBr10JxszGmNkCM1uwdevWIwr1o1O6H9H3i4jUNbXiFBPuPhWYCpCWlubV+Rmr/3BRjWYSEakrgnxHsB7oWu5+l+hjla5jZg2AlsD2ADOJiEgFQRbBJ0AvM+tuZg2By4H0CuukA1dHb48C3vYv9+qKiEhMBPbRkLuXmNmNwBygPvCYu2ea2SRggbunA48CT5lZDrCDSFmIiEgMBbqPwN1nAbMqPDah3O0C4DtBZhARkYPTUfUiIglORSAikuBUBCIiCU5FICKS4Ky2Ha1pZluBNdX89rbAthqMU1PiNRfEbzblOjzKdXjqYq5u7t6usgW1rgiOhJktcPe0sHNUFK+5IH6zKdfhUa7Dk2i59NGQiEiCUxGIiCS4RCuCqWEHOIB4zQXxm025Do9yHZ6EypVQ+whEROTrEu0dgYiIVKAiEBFJcHWyCMzsfDNbamY5Zva7SpY3MrPno8vnm1lKnOQabWZbzeyz6NePY5TrMTPbYmYZB1huZnZ/NPcXZjYoTnKdYWa7y22vCZWtV8OZuprZO2a2xMwyzewXlawT8+1VxVwx317R521sZh+b2efRbLdVsk7MX5NVzBXWa7K+mX1qZjMrWVbz28rd69QXkVNerwB6AA2Bz4H+Fdb5H+Ch6O3LgefjJNdoYEoI2+ybwCAg4wDLLwRmAwacDMyPk1xnADNjvK06AYOit1sAyyr5/xjz7VXFXDHfXtHnNaB59HYSMB84ucI6Ybwmq5IrrNfkTcAzlf3/CmJb1cV3BEOAHHdf6e5FwHPAiArrjACeiN5+CTjbzCwOcoXC3ecSuR7EgYwAnvSIj4BWZtYpDnLFnLtvdPdF0dt5QBZfvxZ3zLdXFXOFIrod9kbvJkW/Kh6lEvPXZBVzxZyZdQEuAh45wCo1vq3qYhF0BtaVu5/L118Q/1nH3UuA3UCbOMgFcFn044SXzKxrJcvDUNXsYRgWfWs/28wGxPKJo2/JTyTyL8nyQt1eB8kFIW2v6EcdnwFbgDfc/YDbLIavyarkgti/Jv8C/AYoO8DyGt9WdbEIarNXgBR3Pw54g/+2vlRuEZHzpxwPPABMj9UTm1lz4F/AL919T6ye91AOkSu07eXupe5+ApFrlw8xs9RYPffBVCFXTF+TZnYxsMXdFwb5PBXVxSJYD5Rv7S7Rxypdx8waAC2B7WHncvft7l4YvfsIMDjgTFVVlW0ac+6+58u39h65Gl6SmbUN+nnNLInIL9un3f3lSlYJZXsdKldY26tChl3AO8D5FRaF8Zo8ZK4QXpOnAMPNbDWRj4/PMrN/VlinxrdVXSyCT4BeZtbdzBoS2ZmSXmGddODq6O1RwNse3fMSZq4KnyMPJ/I5bzxIB34YPRrmZGC3u28MO5SZdfzys1EzG0Lk73Ogvzyiz/cokOXu9xxgtZhvr6rkCmN7RZ+rnZm1it5uApwDZFdYLeavyarkivVr0t1vdvcu7p5C5HfE2+7+gwqr1fi2CvSaxWFw9xIzuxGYQ+RIncfcPdPMJgEL3D2dyAvmKTPLIbIz8vI4yfVzMxsOlERzjQ46F4CZPUvkiJK2ZpYL3Epkxxnu/hCR605fCOQA+4Fr4iTXKOAGMysB8oHLY1DopwBXAYujny0D3AIkl8sVxvaqSq4wthdEjmh6wszqEymfF9x9ZtivySrmCuU1WVHQ20qnmBARSXB18aMhERE5DCoCEZEEpyIQEUlwKgIRkQSnIhARSXAqApEaYGZjo2ew/CJ6lsqhYWcSqao6N0cgEmtmNgy4mMjZPwuj07oNQ44lUmUqApEj1wnY9uWpCNx9W8h5RA6LBspEjlD0RG/zgKbAm0TOD/9euKlEqk77CESOUPREboOBMcBW4HkzGx1qKJHDoHcEIjXMzEYBV7v7JWFnEakKvSMQOUJm1sfMepV76ARgTUhxRA6bdhaLHLnmwAPRUxqXEDnr6JhQE4kcBn00JCKS4PTRkIhIglMRiIgkOBWBiEiCUxGIiCQ4FYGISIJTEYiIJDgVgYhIgvt/UXgfPBXmtJYAAAAASUVORK5CYII=\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "# If we plot dP/dt against S, we see that it looks like\n", "# mass action kinetics that don't involve an enzyme.\n", "# i.e. we see that dP/dt is linearly proportional to S.\n", "P = sol.y[2,:].T\n", "S = sol.y[0,:-1]\n", "velocity_of_P = np.divide(np.diff(P), (sol.t[1]-sol.t[0]))\n", "plt.plot( S, velocity_of_P );\n", "plt.xlabel( 'S' );\n", "plt.ylabel( 'dP/dt' );\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In the rare circumstance above, the reaction may be enzyme catalyzed, but there are so many enzymes that the rate of accumulation of P depends on S (not E).\n", "\n", "### Simulating when Enzyme concentration is low\n", "A more realistic scenario is one in which the enzyme concentration is low.\n", "\n", "Use k1f = 2, k1b=2, k2 = 3, S(0) = 100, E(0) = 1, SE(0) = 0, P(0) = 0.\n", "\n", "Simulate with from t=0 to t=100 with a timestep of 0.1" ] }, { "cell_type": "code", "execution_count": 70, "metadata": {}, "outputs": [], "source": [ "# On the other extreme (and more common in biology), there is lots of substrate\n", "# but not lots of enzymes. So let's model that\n", "k1f = 2\n", "k1b = 1\n", "k2 = 3\n", "params = (k1f, k1b, k2)\n", "\n", "S0 = 100\n", "E0 = 1\n", "SE0 = 0\n", "P0 = 0\n", "yinit = (S0,E0,P0,SE0)\n", "\n", "em = lambda t,y: enzyme_model(t,y,params)\n", "sol = scipy.integrate.solve_ivp( fun=em, t_span=(0,100), y0=yinit, \n", " t_eval=np.arange(0,100,1) )" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Plot the simulation over time." ] }, { "cell_type": "code", "execution_count": 71, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "plt.plot( sol.t, sol.y.T )\n", "plt.xlabel( \"Time\")\n", "plt.ylabel( \"Concentration\" )\n", "plt.legend( [\"S\",\"E\",\"P\",\"SE\"])\n", "plt.title( \"Enzyme-catalyzed reaction\");" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "And plot dP/dt vs S, computing dP/dt from P as above." ] }, { "cell_type": "code", "execution_count": 72, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "# And here, we see that dP/dt depends on S in a nonlinear manner.\n", "# In particular, when S is high, dP/dt does not depend on S at all.\n", "# and when S is smaller, we have a curve. \n", "P = sol.y[2,:].T\n", "S = sol.y[0,:-1].T\n", "velocity_of_P = np.divide(np.diff(P), (sol.t[1]-sol.t[0]))\n", "plt.plot( S, velocity_of_P );\n", "plt.xlabel( 'S' );\n", "plt.ylabel( 'dP/dt' );" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The curve dP/dt vs S can be expressed as\n", "\n", "$v_{max}\\frac{S}{K_m+S}$\n", "\n", "where\n", "\n", "$v_{max} = E(0)\\cdot k_2$\n", "\n", "and \n", "\n", "$K_m = \\frac{k_{-1}+k_2}{k_1}$\n", "\n", "Plot the analytic curve (from the Michaelis-Mention model for enzyme kinetics) on the same plot as the approximated dP/dt (you can just replot the curve from above and the plot the analytic curve as well)." ] }, { "cell_type": "code", "execution_count": 73, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "# The curve dP/dt vs S can be expressed as\n", "Km = (k1b + k2)/k1f\n", "vmax = E0*k2\n", "analytic_dPdt = vmax * np.divide( S, (S+Km) )\n", "plt.plot(S, analytic_dPdt)\n", "plt.plot( S, velocity_of_P)\n", "plt.xlabel( \"S\")\n", "plt.ylabel(\"dP/dt\")\n", "plt.legend( [\"M-M expression\",\"Simulation\"]);" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.8.5" } }, "nbformat": 4, "nbformat_minor": 4 }