{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Enzyme Kinetics (Part 1)\n",
    "\n",
    "Implement and simulate a model in which an enzyme E in necessary for 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": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import scipy.integrate\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "\n",
    "## write definition here"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Simulating when Enzyme concentration is low\n",
    "A 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": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# There is lots of substrate, but not lots of enzymes.\n",
    " \n",
    "# Write code here."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Plot the simulation over time."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Write code here\n"
   ]
  },
  {
   "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": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Write code here"
   ]
  },
  {
   "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": null,
   "metadata": {
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "# Write code here\n"
   ]
  }
 ],
 "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
}