{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#The response of an I1-FFL"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here, we make the plot of the response of an incoherent fead-forward loop (I1-FFL).  The IPython notebook may be downloaded [here](i1-ffl.ipynb).\n",
    "\n",
    "First thing's first.  Let's import the necessary modules to do the computation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Import necessary modules\n",
    "import numpy as np\n",
    "from scipy.integrate import odeint\n",
    "import matplotlib.pyplot as plt\n",
    "import seaborn as sns\n",
    "sns.set_style('ticks')\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The incoherent feed-forward loop is shown below.\n",
    "\n",
    "<img src=i1-ffl.png width=\"25%\">\n",
    "\n",
    "The input, X, activates Y and Z, and Y represses Z.  A dimensionless set of differential equations describing an I1-FFL network motif is\n",
    "\n",
    "\\begin{align}\n",
    "\\dot{y}&=F-y,\\\\[1em]\n",
    "\\dot{z} &= \\frac{1}{r}\\left(\\frac{F}{y}-z\\right),\n",
    "\\end{align}\n",
    "\n",
    "where $F$ is the fold change in the input X and $r$ is the ratio of the degradation rates of species Y and Z.  The variables $y$ and $z$ represent the dimensionless concentrations of species Y and Z.\n",
    "\n",
    "Assume that for $t < 0$, we have $F = 2$.  For $t \\ge 0$, we have $F = 4$.  Assuming steady state for $t < 0$, we have $y_\\mathrm{ss} = 2$ and $z_\\mathrm{ss} = 1$.  We can solve the first ODE analytically for $y(t)$, but we can easily solve these coupled equations using `odeint`, which we will do for illustrative purposes."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def rhs(yz, t, F, r):\n",
    "    \"\"\"\n",
    "    Right hand side of system of ODEs describing I1-FFL.\n",
    "    \"\"\"\n",
    "    y, z = yz\n",
    "    return np.array([F - y, (F / y - z) / r])\n",
    "\n",
    "# Time points we want for the solution\n",
    "t = np.linspace(0, 4, 1000)\n",
    "\n",
    "# Initial condition\n",
    "yz_0 = np.array([2.0, 1.0])\n",
    "\n",
    "# Parameters\n",
    "F = 4.0\n",
    "r = 0.1\n",
    "\n",
    "# Integrate ODES\n",
    "yz = odeint(rhs, yz_0, t, args=(F, r))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "With the results in hand, we can generate our plots."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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       "AElFTkSuQmCC\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x115814f90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Pluck out y and z\n",
    "y, z = yz.transpose()\n",
    "\n",
    "# Append the pre-fold change steady state\n",
    "t = np.insert(t, 0, -1.0)\n",
    "y = np.insert(y, 0, 2.0)\n",
    "z = np.insert(z, 0, 1.0)\n",
    "\n",
    "# Generate x\n",
    "x = 2.0 + 2.0 * (t > 0.0).astype(np.float)\n",
    "\n",
    "# Plot the results\n",
    "fig, ax = plt.subplots(1, 1)\n",
    "ax.plot(t, x, '-')\n",
    "ax.plot(t, y, '-')\n",
    "ax.plot(t, z, '-')\n",
    "ax.set_xticks([])\n",
    "ax.set_yticks([])\n",
    "ax.margins(0.02)\n",
    "ax.set_xlabel('time', x=1, fontsize=14)\n",
    "ax.set_ylabel('level', rotation=0, fontsize=14)\n",
    "ax.yaxis.set_label_coords(-0.05, 0.95) \n",
    "sns.despine(offset=0.1)\n",
    "plt.savefig('fig.pdf', bbox_inches='tight', transparent=True)"
   ]
  }
 ],
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