1 - Constructing simple examples with bond graphs.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Constructing simple models of biochemical networks with BondGraphTools"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In this notebook we introduce the basics of the BondGraphTools package and use it to construct simple bond graph representations of biochemical networks."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Import packages\n",
    "import BondGraphTools as bgt\n",
    "from BondGraphTools.actions import new, add, connect\n",
    "from BondGraphTools import draw, simulate\n",
    "from numpy import log\n",
    "import matplotlib.pyplot as plt\n",
    "import sympy as sp\n",
    "from sympy import init_printing\n",
    "init_printing()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## A simple example\n",
    "In this section we work with the simple biochemical reaction $A + B \\rightleftharpoons C$. The bond graph for this system is shown below, where the bonds have been numbered in blue for ease of reference.\n",
    "\n",
    "![](ABC.svg)\n",
    "\n",
    "We begin by creating each of the components, and adding them to the model."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "OrderedSet([C: A, C: B, C: C, R: R1, 1: AB])"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Create a new model\n",
    "model_ABC = new(name='ABC')\n",
    "\n",
    "# Define the components\n",
    "C_A = new(component='Ce',name='A',library='BioChem') # C:A\n",
    "C_B = new(component='Ce',name='B',library='BioChem') # C:B\n",
    "C_C = new(component='Ce',name='C',library='BioChem') # C:C\n",
    "Re = new(component='Re',name='R1',library='BioChem') # Re:R1\n",
    "flow = new(component='1',name='AB') # 1 junction\n",
    "\n",
    "# Add components to the model\n",
    "add(model_ABC,C_A,C_B,C_C,Re,flow)\n",
    "\n",
    "# List the components to check they have been added\n",
    "model_ABC.components"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Next we define the bonds that link the components together."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[Bond(tail=Port(C: A, 0), head=Port(1: AB, 0)),\n",
       " Bond(tail=Port(C: B, 0), head=Port(1: AB, 1)),\n",
       " Bond(tail=Port(1: AB, 2), head=Port(R: R1, 0)),\n",
       " Bond(tail=Port(R: R1, 1), head=Port(C: C, 0))]"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "connect(C_A,flow) # Bond 1\n",
    "connect(C_B,flow) # Bond 2\n",
    "connect(flow,Re) # Bond 3\n",
    "connect(Re,C_C) # Bond 4\n",
    "\n",
    "# List the bonds to check they have been added\n",
    "model_ABC.bonds"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We now draw the bond graph to make sure that the structure is the same as the diagram above."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 960x720 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "draw(model_ABC)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The differential equations for the model can be derived using the `constitutive_relations` property."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\left[ dx_{0} + u_{9} e^{\\frac{u_{1} u_{2} \\log{\\left(u_{0} x_{0} \\right)} + u_{4} u_{5} \\log{\\left(u_{3} x_{1} \\right)}}{u_{10} u_{11}}} - u_{9} e^{\\frac{u_{7} u_{8} \\log{\\left(u_{6} x_{2} \\right)}}{u_{10} u_{11}}}, \\  dx_{1} + u_{9} e^{\\frac{u_{1} u_{2} \\log{\\left(u_{0} x_{0} \\right)} + u_{4} u_{5} \\log{\\left(u_{3} x_{1} \\right)}}{u_{10} u_{11}}} - u_{9} e^{\\frac{u_{7} u_{8} \\log{\\left(u_{6} x_{2} \\right)}}{u_{10} u_{11}}}, \\  dx_{2} - u_{9} e^{\\frac{u_{1} u_{2} \\log{\\left(u_{0} x_{0} \\right)} + u_{4} u_{5} \\log{\\left(u_{3} x_{1} \\right)}}{u_{10} u_{11}}} + u_{9} e^{\\frac{u_{7} u_{8} \\log{\\left(u_{6} x_{2} \\right)}}{u_{10} u_{11}}}\\right]$"
      ],
      "text/plain": [
       "⎡          u₁⋅u₂⋅log(u₀⋅x₀) + u₄⋅u₅⋅log(u₃⋅x₁)       u₇⋅u₈⋅log(u₆⋅x₂)         \n",
       "⎢          ───────────────────────────────────       ────────────────         \n",
       "⎢                        u₁₀⋅u₁₁                         u₁₀⋅u₁₁              \n",
       "⎣dx₀ + u₉⋅ℯ                                    - u₉⋅ℯ                , dx₁ + u\n",
       "\n",
       "   u₁⋅u₂⋅log(u₀⋅x₀) + u₄⋅u₅⋅log(u₃⋅x₁)       u₇⋅u₈⋅log(u₆⋅x₂)            u₁⋅u₂\n",
       "   ───────────────────────────────────       ────────────────            ─────\n",
       "                 u₁₀⋅u₁₁                         u₁₀⋅u₁₁                      \n",
       "₉⋅ℯ                                    - u₉⋅ℯ                , dx₂ - u₉⋅ℯ     \n",
       "\n",
       "⋅log(u₀⋅x₀) + u₄⋅u₅⋅log(u₃⋅x₁)       u₇⋅u₈⋅log(u₆⋅x₂)⎤\n",
       "──────────────────────────────       ────────────────⎥\n",
       "         u₁₀⋅u₁₁                         u₁₀⋅u₁₁     ⎥\n",
       "                               + u₉⋅ℯ                ⎦"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "model_ABC.constitutive_relations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As seen above, the differential equations contain a number of unset constants - the control variables $u_n$. We will now set the parameters for each component, and derive the differential equation again."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[ dx_{0} + x_{0} x_{1} - x_{2}, \\  dx_{1} + x_{0} x_{1} - x_{2}, \\  dx_{2} - x_{0} x_{1} + x_{2}\\right]$"
      ],
      "text/plain": [
       "[dx₀ + x₀⋅x₁ - x₂, dx₁ + x₀⋅x₁ - x₂, dx₂ - x₀⋅x₁ + x₂]"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# C:A\n",
    "C_A.set_param('k',1)\n",
    "C_A.set_param('R',1)\n",
    "C_A.set_param('T',1)\n",
    "\n",
    "# C:B\n",
    "C_B.set_param('k',1)\n",
    "C_B.set_param('R',1)\n",
    "C_B.set_param('T',1)\n",
    "\n",
    "# C:C\n",
    "C_C.set_param('k',1)\n",
    "C_C.set_param('R',1)\n",
    "C_C.set_param('T',1)\n",
    "\n",
    "# Re:R1\n",
    "Re.set_param('r',1)\n",
    "Re.set_param('R',1)\n",
    "Re.set_param('T',1)\n",
    "\n",
    "# Derive the differential equations\n",
    "model_ABC.constitutive_relations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It is easily seen that the differential equations follow the law of mass action. The definition of each state can be found by using the `state_vars` method. The states $x_0$, $x_1$ and $x_2$ correspond to A, B and C respectively."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "x_0 -> C: A\n",
      "x_1 -> C: B\n",
      "x_2 -> C: C\n"
     ]
    }
   ],
   "source": [
    "for k,v in model_ABC.state_vars.items(): print(f'{k} -> {v[0]}')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Therefore the differential equations can be interpreted as\n",
    "$$ \n",
    "\\begin{align*}\n",
    "\\frac{dx_A}{dt} &= -x_A x_B + x_C \\\\\n",
    "\\frac{dx_B}{dt} &= -x_A x_B + x_C \\\\\n",
    "\\frac{dx_C}{dt} &= x_A x_B - x_C \n",
    "\\end{align*}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We will now attempt to simulate the model using the `simulate` method in BondGraphTools. We set $x_0 = 1$, $x_1 = 2$ and $x_2 = 3$ as the initial conditions, and simulate the system from $t=0$ to $t=10$. The results are plotted below."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "t,x = simulate(model_ABC, timespan=(0.,10.), x0=[1,2,3])\n",
    "plt.plot(t,x)\n",
    "plt.xlabel('Time')\n",
    "plt.ylabel('Amount')\n",
    "plt.legend(['A','B','C'])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can run a different simulation for the same system using the initial conditions $x_0 = 3$, $x_1 = 1$ and $x_2 = 2$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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F6b2ZmQ1y5QTB2cBBEbEl62LMzGz7Vq1axcUXX8ycOXMYNmwYtbW1/OQnP2HSpEm9brOcO4ufBfbudQ9mZtYvIoKzzz6bqVOn8te//pXnn3+e7373u7z22mt9arecPYJ9gCWS5vD2OYKICA9FbWY2gGbNmkWhUOCiiy7atmzrncZ9UU4QXNFpXsApFIeZMDNL0g/++AOWrF3Sr20eNvowvnHcN7a7zsKFCzn22GP7tV8o49BQRDwBrAM+DNwOnAbc2O+VmJlZLnrcI5A0CTiPtweZ+y9AEfH+AarNzGyXtKPf3LMyefJk7r333n5vd3t7BEso/vb/0Yg4JSKuA7IZ8cjMzHbo1FNPZfPmzdx8883bls2ZM6fPzy3eXhCcA6wCZkm6WdJpFM8RmJlZDiQxY8YMZs6cycEHH8zkyZO58sormTBhQp/a7fHQUETMAGZIGgF8HPg6sI+knwEzIuK3ferZzMx22oQJE/jlL3/Zr22Wc7L4zYi4OyI+AtQA84HL+rUKMzPLTTk3lG0TEWsj4ucR4eElzMyGiJ0KAjOzlEVE3iXsUG9qdBCYmZWhurqa5ubmXToMIoLm5maqq6t3arty7iw2M0teTU0NjY2NNDU15V3KdlVXV1NTU7NT2zgIzMzKUCgUqKury7uMTPjQkJlZ4hwEZmaJcxCYmSXOQWBmljgHgZlZ4hwEZmaJyywIJN0qabWkhT18PlXSOknzS9O/Z1WLmZn1LMv7CG4Hrgfu3M46/10azM7MzHKS2R5BRDwJrM2qfTMz6x95nyM4UdKzkh6WNLmnlSRNk9QgqWFXv73bzGywyTMI5gEHRsRRwHXAfT2tGBE3RUR9RNSPGzduoOozM0tCbkEQES0RsaE0/xBQkDQ2r3rMzFKVWxBI2leSSvPHlWppzqseM7NUZXbVkKTpwFRgrKRG4AqgABARNwLnAl+U1AZsBM6LXXmgbzOzISqzIIiIT+/g8+spXl5qZmY5yvuqITMzy5mDwMwscQ4CM7PEOQjMzBLnIDAzS5yDwMwscQ4CM7PEOQjMzBLnIDAzS5yDwMwscQ4CM7PEOQjMzBLnIDAzS5yDwMwscQ4CM7PEOQjMzBLnIDAzS5yDwMwscQ4CM7PEOQjMzBLnIDAzS5yDwMwscQ4CM7PEOQjMzBLnIDAzS5yDwMwscQ4CM7PEOQjMzBLnIDAzS5yDwMwscQ4CM7PEOQjMzBLnIDAzS5yDwMwscQ4CM7PEOQjMzBKXWRBIulXSakkLe/hckq6VtFTSc5KOyaoWMzPrWZZ7BLcDZ2zn8zOBQ0rTNOBnGdZiZmY9yCwIIuJJYO12VjkLuDOKngb2lrRfVvXQ9Gd46FJo25JZF2Zmg1Ge5wj2B1Z0et9YWvYukqZJapDU0NTU1Lve3lgOf/w5LH6gd9ubmQ1ReQaBulkW3a0YETdFRH1E1I8bN653vR18Gow+GJ65sXfbm5kNUXkGQSNwQKf3NcCrmfVWUQHHfwEa50Dj3My6MTMbbPIMggeAC0pXD50ArIuIlZn2OOUzsNtI7xWYmXWS5eWj04E/AIdKapT0z5IuknRRaZWHgBeBpcDNwL9mVcs2w0bC0efDohmwflXm3ZmZDQZVWTUcEZ/ewecBfCmr/nt03OeLewQNt8L7vzng3ZuZ7WrSu7N4zMEw6R+KQdC2Oe9qzMxyl14QAJz4JXizCZ74Qd6VmJnlLs0gqPs7mHI+zP4xvPx03tWYmeUqzSAAOPP7sPdE+PU02NSSdzVmZrlJNwiGjYSzb4J1K+CRy/OuxswsN+kGAcDE4+GU/wXz74Inr4bo9sZmM7MhLbPLRweNqZcVxyF67DvQ8ip86CqoqMy7KjOzAeMgqCwUDxHtOQGeuqZ4o9nHfwrD9867MjOzAZH2oaGtKirgA/8bzrwKXngIrp0CT9/oIavNLAkOgs6OnwZfeAL2PRIe+Qb89Hj4ww3Qku0QSGZmeVIMshOk9fX10dDQkG0nEfCXmfD4d+HVPwGC2lPgoKlQ8z7Y/5jiVUdmZoOEpLkRUd/dZz5H0B0JJn2wOK35Cyz8FSy6r3hCubgC7HUAjK6FUXXF8wsjxsKI8cVzC8NGwrA9obA7FKqhajhU7lY8BGVmtovxHsHO2Pg6vDK3+DyD5qXw+kuw9kV4q7m87VVZCoSq4pVJW19V8faESo/sUTGQ3vVa+mxbm90936fLOjurxzbNLFdHfxZO+nKvNvUeQX8ZPgrec3px6qy9tRgGG1bDpnWweX1xan0TWjdB28biOu1bilNHO3S0FafoKE4dpVeidD9DN6/Q5V6HHkK8T+E+uH4xMEvKHuMzadZB0B8qCzBy3+JkZjbI+KC1mVniHARmZolzEJiZJc5BYGaWOAeBmVniHARmZolzEJiZJc5BYGaWuEE3xISkJmB5LzcfC6zpx3IGA3/nNPg7p6Ev3/nAiBjX3QeDLgj6QlJDT2NtDFX+zmnwd05DVt/Zh4bMzBLnIDAzS1xqQXBT3gXkwN85Df7OacjkOyd1jsDMzN4ttT0CMzPrwkFgZpa4ZIJA0hmSXpC0VNJledeTNUkHSJolabGkRZK+lndNA0FSpaQ/SfpN3rUMFEl7S7pX0pLSn/eJedeUJUlfL/2dXihpuqTqvGvKgqRbJa2WtLDTstGSZkr6S+l1VH/0lUQQSKoEbgDOBI4APi3piHyrylwbcElEHA6cAHwpge8M8DVgcd5FDLBrgEci4jDgKIbw95e0P/BVoD4i3gtUAuflW1VmbgfO6LLsMuB3EXEI8LvS+z5LIgiA44ClEfFiRGwBfgGclXNNmYqIlRExrzS/nuIPh/3zrSpbkmqADwP/mXctA0XSnsDfAbcARMSWiHgj16KyVwUMl1QF7A68mnM9mYiIJ4G1XRafBdxRmr8D+Hh/9JVKEOwPrOj0vpEh/kOxM0m1wNHAMzmXkrWfAJcCHTnXMZAOApqA20qHxP5T0oi8i8pKRLwCXA28DKwE1kXEb/OtakDtExErofjLHtAvT7NPJQjUzbIkrpuVtAfwK+DiiGjJu56sSPoIsDoi5uZdywCrAo4BfhYRRwNv0k+HC3ZFpWPiZwF1wARghKTz861q8EslCBqBAzq9r2GI7k52JqlAMQTujohf511Pxk4GPiZpGcVDf6dKuivfkgZEI9AYEVv39u6lGAxD1enASxHRFBGtwK+Bk3KuaSC9Jmk/gNLr6v5oNJUgmAMcIqlO0m4UTy49kHNNmZIkiseNF0fEj/KuJ2sRcXlE1ERELcU/38ciYsj/phgRq4AVkg4tLToNeD7HkrL2MnCCpN1Lf8dPYwifHO/GA8CFpfkLgfv7o9Gq/mhkVxcRbZK+DDxK8SqDWyNiUc5lZe1k4LPAAknzS8u+GREP5VeSZeQrwN2lX3JeBP4p53oyExHPSLoXmEfxyrg/MUSHmpA0HZgKjJXUCFwBfB/4paR/phiKn+yXvjzEhJlZ2lI5NGRmZj1wEJiZJc5BYGaWOAeBmVniHARmZolzEJj1QNIYSfNL0ypJr5TmN0j6ad71mfUXXz5qVgZJVwIbIuLqvGsx62/eIzDbSZKmbn3egaQrJd0h6beSlkn6hKQfSlog6ZHSMB9IOlbSE5LmSnp06zABZrsCB4FZ3x1Mcfjrs4C7gFkRcSSwEfhwKQyuA86NiGOBW4H/k1exZl0lMcSEWcYejohWSQsoDmHySGn5AqAWOBR4LzCzODwOlRSHUDbbJTgIzPpuM0BEdEhqjbdPvHVQ/DcmYFFEDOlHSNrg5UNDZtl7ARi39VnCkgqSJudck9k2DgKzjJUej3ou8ANJzwLzSWsMfdvF+fJRM7PEeY/AzCxxDgIzs8Q5CMzMEucgMDNLnIPAzCxxDgIzs8Q5CMzMEvf/AdyiZ/RBnj+PAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "t,x = simulate(model_ABC, timespan=(0.,10.), x0=[3,1,2])\n",
    "plt.plot(t,x)\n",
    "plt.xlabel('Time')\n",
    "plt.ylabel('Amount')\n",
    "plt.legend(['A','B','C'])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The reversible Michaelis-Menten model\n",
    "Here we will create a bond graph of the two-state enzyme model used for the Michaelis-Menten equations. The chemical reactions in this system are \n",
    "$$\n",
    "E + S \\rightleftharpoons C \\\\ \n",
    "C \\rightleftharpoons E + P\n",
    "$$\n",
    "As in the typical formulation, we assume that S and P are chemostats, i.e. their amounts are fixed at constant values. The corresponding bond graph for this system is shown below. The bonds have been labelled in blue.\n",
    "\n",
    "![](enzyme_cycle.svg)\n",
    "\n",
    "We begin by creating each of the components, and placing them in a list.\n",
    "\n",
    "While we chose to initialise components without their parameters in the previous example, we use this example to demonstrate how to initialise components with their parameters. All the bond graph parameters have been set to 1, and we assume that S is fixed at an amount of 2, and P is fixed at an amount of 1."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "OrderedSet([C: E, C: C, SS: S, SS: P, R: R1, R: R2, 0: E, 0: C, 1: R1, 1: R2])"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "model_MM = new(name='Michaelis-Menten')\n",
    "\n",
    "R = 8.3144598\n",
    "T = 310\n",
    "K_S = 1\n",
    "K_P = 1\n",
    "x_S = 2\n",
    "x_P = 1\n",
    "\n",
    "# Chemical species\n",
    "C_E = new(component='Ce',name='E',library='BioChem',value={'k':1,'R':R,'T':T})\n",
    "C_C = new(component='Ce',name='C',library='BioChem',value={'k':1,'R':R,'T':T})\n",
    "# Chemostats\n",
    "Se_S = new(component='Se',name='S',value={'e':R*T*log(K_S*x_S)})\n",
    "Se_P = new(component='Se',name='P',value={'e':R*T*log(K_P*x_P)})\n",
    "# Reactions\n",
    "R1 = new(component='Re',name='R1',library='BioChem',value={'r':1,'R':R,'T':T})\n",
    "R2 = new(component='Re',name='R2',library='BioChem',value={'r':1,'R':R,'T':T})\n",
    "# 0 junctions\n",
    "ej_E = new(component='0',name='E')\n",
    "ej_C = new(component='0',name='C')\n",
    "# 1 junctions\n",
    "fj_1 = new(component='1',name='R1')\n",
    "fj_2 = new(component='1',name='R2')\n",
    "\n",
    "# Place components in a list and add to model\n",
    "components = [C_E,C_C,Se_S,Se_P,R1,R2,ej_E,ej_C,fj_1,fj_2]\n",
    "add(model_MM, *components)\n",
    "\n",
    "# Check that the components are in the model\n",
    "model_MM.components"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We next connect the components together, as in the diagram above."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[Bond(tail=Port(SS: S, 0), head=Port(1: R1, 0)),\n",
       " Bond(tail=Port(1: R1, 1), head=Port(R: R1, 0)),\n",
       " Bond(tail=Port(R: R1, 1), head=Port(0: C, 0)),\n",
       " Bond(tail=Port(0: C, 1), head=Port(C: C, 0)),\n",
       " Bond(tail=Port(0: C, 2), head=Port(R: R2, 0)),\n",
       " Bond(tail=Port(R: R2, 1), head=Port(1: R2, 0)),\n",
       " Bond(tail=Port(1: R2, 1), head=Port(SS: P, 0)),\n",
       " Bond(tail=Port(1: R2, 2), head=Port(0: E, 0)),\n",
       " Bond(tail=Port(0: E, 1), head=Port(C: E, 0)),\n",
       " Bond(tail=Port(0: E, 2), head=Port(1: R1, 2))]"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "connect(Se_S,fj_1) # Bond 1\n",
    "connect(fj_1,(R1,0)) # Bond 2\n",
    "connect((R1,1),ej_C) # Bond 3\n",
    "connect(ej_C,C_C) # Bond 4\n",
    "connect(ej_C,(R2,0)) # Bond 5\n",
    "connect((R2,1),fj_2) # Bond 6\n",
    "connect(fj_2,Se_P) # Bond 7\n",
    "connect(fj_2,ej_E) # Bond 8\n",
    "connect(ej_E,C_E) # Bond 9\n",
    "connect(ej_E,fj_1) # Bond 10\n",
    "\n",
    "# Check that the bonds have been added\n",
    "model_MM.bonds"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The bond graph drawn by the software verifies these connections."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 960x720 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "draw(model_MM)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For this model, the constitutive equations reduce to a linear system"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAATwAAAAVCAYAAADb0VJMAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAGi0lEQVR4Ae2c7XXVRhCGhY8LMKYD0wEfFUA6iJMKYjowh1/2vxzoAKgggQ5MKjDQQejA5naQPI/urizJujeSkHQlZ+ecubvaL807Ozs7Wsm+d3Z2dppl2QtYOj4/P/+6zqbfpIGkgaSBZWsAf1bxb/dweK+BdEHFp2VDS9InDSQNJA00awD/lvu5/ebq5lI6PaLmPXwE/8l1jAybO8y8FPnFETEckPf6NeV3xvkHjC/DVDwhvYZfUr7YSB7Zkx2GCV1KMhc77OrwXCSPEf4f0oulKLtJTjDo4Fz40eFl5H8WF6mP9h+b+i2pDAw68LekP0W5ybvTfbEMXqRjR+5kh3FCF5AyX7Oxw72u+kL456HPzheLssAnXTGE9vY7ob9OLlLE9CoWLDzVuRUOXSzgNdpbwR+8XiqBYzZ2+IM6XIQdqm+471qbjR12dnhMrtHCN8C7aHZNRmlyHzJKEEOBYyaY+mDZ1Een8De46jrSsR9QfrSp4wLK52SHP6KupdihNlS3o7a4Z2OH+20lLrVT+BgJlYqXlWWxi+F+WWrKYrT3tly+4LwYH4GrcOo1LH0NuDbMTi6THe5E7b1uOhs73OrwQgRgOPoNvoJzwUl/hwuinQsnhrtPyf8GGz38CkuXtJn1mRjyuYDy0Jv8u1zq8LNUfMh9XMZRynvon1FvdJHTnDEim7aU7HCh64z5m40dbnykRUgdwBfYg335TTA6klsRnm8234Q2l9T7Jtdnfs+LdIYa6ywJGY2A4rc6OoDPDYIuFl8di3gp04HEN7exySwxIm+ywzhD6y8IFrnObiCsc7uyw8YID2F0Uh5q6+iM7iKZ/0rZKhaQN7IrOzTrfDQ0ypMO4XJ9Rp94bdT4gOv64rPfJMS9dXJ5pENeuX2DWbylJb9ofA1KdF4/gssNLKeeGLWRX+AX9H+8HmnYX8Yd2w5Hx9BWI2BNdthvrXWaw0aHxyTpkDzUrjzaUeZuW380/Uy7slPU+AunSF0lnOVaB3JFmi84UiNBPwXxELqRqPNMzXvXSWeaUV95ExkaKUPl3vXO9Wva6whWlH8gvR/yo+NTDu7lxP0Fm7alTn8Zwz3Uoy+c6nrpitEoMc5HF3nb4ortxrTDzhimmCOBc59d2uHoaw18O7PDTQ7PndvzuoLCZPsoVPn+jvI8Oioarnf9yhlfqc6sRvwsltH/E6zDO4LLjjM2yShvcmiWG5HZr4hWik7/kaGPBu8Ydfl9pHUxyxpevV7dDIqP8ZRDRztKpBTGd6M55D63NpauGEN7N5T4ksdbjEGj2WEfDPQZfI4Yc252OPhaKxsGeHdqh3tlYcwjkDu2XHFsXGt81lccoWWRqNNJ2LeIAh1Ptg2pDtN83bGtKMsnnnQq8nzSx9dctjY3pe2S8BWQkFvH9JC0iOzIu1E4HxWibCvGSuMRL5DDeZEHt8MRxe4zdLLDCe1wf8sM1Z2SkUEe7WCMLiDz17BnQvHPsdwdVtSX+77iOp7RbXIujnMIT0krbmZ0aVqmJ+HCOuVdKr4cBhjcSJ6W5iDAy5zDdz0wxv6tUseH6zpu1Tc0KtuSRUPYYZf7j9022eEEdhgnsSnCyyeABsXuj8G6OFw48Q2mf5akIRoNyNcatilcEGXWXRYFmzM6O/tPSTrhSvQQcCqHB/HqYcn4MjA4hzpsnY5/YlbwCBgZskrcT11+JzWK6UT0Uf8+TUxph51kHKhxssNh19rWadnfUOujz3uM7pT0AfwH7M5qJGeZh46SBumLDR1DRp2OwvM46zVyHWHxeMu1RtxELoz6Tt7UbrAy5DK6eR5kjeO6uMp/Y7pYfAGQDl1MnpvUyQhd6opx3avFL7qN0b4Ot0+kN5YdtpB+miboJdnhWtWj2WF5Jif/91BM8HcEeEYaF1xG3n9G4BlTJ6dHeyPPXi8tykoYMo9Mg+EbUq4xxgr6dxN8uG380K7p+GBbt0nq2mKYRJgBbwKuQe0w6GlWay2qq80c0saXpRd7sdOEqW8484jQeyKIed+GdnJ29oWMGDdFjXmDHfwMiW8H4o9yS88Q5zZPowCd0aBD2+Ec11pndU8e4Skhxq+3la5gowM/cL4zC+J/gM/HZF9QuVl5tutnQcW3leQLQhcHXJyQdv50qBhkhAzytMYwwu0nGTLZ4Y2agy4uduLwbsRIubuuAQztFJ6Vs7vrOk/4bmug7vA8C5M6fbm/7pJ+kwaSBpIG5qkBN1wk82nEL0GO/wWlyM/D/m7SfQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$\\displaystyle \\left[ dx_{0} + 3 x_{0} - 2 x_{1}, \\  dx_{1} - 3 x_{0} + 2 x_{1}\\right]$"
      ],
      "text/plain": [
       "[dx₀ + 3⋅x₀ - 2⋅x₁, dx₁ - 3⋅x₀ + 2⋅x₁]"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "model_MM.constitutive_relations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "with the state variables defined as"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "x_0 -> C: E\n",
      "x_1 -> C: C\n"
     ]
    }
   ],
   "source": [
    "for k,v in model_MM.state_vars.items(): print(f'{k} -> {v[0]}')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We may now simulate this model using the initial conditions $x_E = 1$ and $x_C = 2$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "t,x = simulate(model_MM, timespan=(0.,3.), x0=[1,2])\n",
    "plt.plot(t,x)\n",
    "plt.xlabel('Time')\n",
    "plt.ylabel('Amount')\n",
    "plt.legend(['E','C'])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We may also run a simulation where the amount of substrate is gradually increased, i.e. $x_S = 1+t$. We turn the **Se:S** component back into a controlled variable, and set the effort to be $e = RT\\ln(K_S (1+t))$. As the amount of substrate increases more of the enzyme is driven to the complex state."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "Se_S.set_param('e',None)\n",
    "t,x = simulate(model_MM, timespan=(0.,10.), x0=[1,2], control_vars={'u_0':f'{R}*{T}*log({K_S}*(1+t))'})\n",
    "plt.plot(t,x)\n",
    "plt.xlabel('Time')\n",
    "plt.ylabel('Amount')\n",
    "plt.legend(['E','C'])\n",
    "plt.show()"
   ]
  }
 ],
 "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": "3.8.8"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}