{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# This notebook shows how to study noise with MuMoT<a class=\"tocSkip\">\n",
    "There are 7 examples worked through below."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import mumot\n",
    "\n",
    "mumot.__version__\n",
    "mumot.about()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Production and decay of protein $P$ and its dimerization into $P_2$ ($P_2$ is called $Q$ below)\n",
    "### (see F. Hayot & C. Jayaprakash (2004), Physical Biology 1, pp. 205-210)<a class=\"tocSkip\">"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "model1 = mumot.parseModel(r\"\"\"\n",
    "\\emptyset -> P : k_3\n",
    "P -> \\emptyset : k\n",
    "P + P -> Q + \\emptyset : k_1\n",
    "Q + \\emptyset -> P + P : k_2\n",
    "\"\"\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "model1.showODEs(method='vanKampen')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "evol1 = model1.integrate(['Q', 'P'], legend_loc='center right')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "evol1.showLogs(tail=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "bookmark_evol1 = model1.integrate(showStateVars=['P', 'Q'], initialState = {'P': 1.0, 'Q': 0.0}, \n",
    "                                   maxTime = 3.0, plotProportions = False, \n",
    "                                   params = [('k_{3}', 2.0), ('k_{2}', 2.0), ('k', 2.0), ('k_{1}', 2.0), \n",
    "                                             ('plotLimits', 1), ('systemSize', 10.0)], \n",
    "                                   legend_loc = 'center right', bookmark = False)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "Ncorr1 = model1.noiseCorrelations(maxTime=120, legend_loc='upper right', legend_fontsize=10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "Ncorr1.showLogs(tail=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "model1.noiseCorrelations()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "bookmark_ncorr1 = model1.noiseCorrelations(initialState = {'P': 1.0, 'Q': 0.0}, params = [('k_{3}', 2.0), ('k_{2}', 2.0), ('k', 2.0), ('k_{1}', 2.0), ('plotLimits', 1), ('systemSize', 10.0)], legend_loc = 'upper right', legend_fontsize = 10, maxTime = 120, bookmark = False)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "model1.showFokkerPlanckEquation()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "model1.showNoiseEquations()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "model1.showNoiseSolutions()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "stream1 = model1.stream('P', 'Q', showFixedPoints=True, showNoise = True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "stream1.showLogs()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "bookmark_stream1 = model1.stream('P', 'Q', params = [('k_{3}', 2.0), ('k_{2}', 2.0), ('k', 2.0), ('k_{1}', 2.0), ('plotLimits', 1.5), ('systemSize', 10.0)], showFixedPoints = True, showNoise = True, bookmark = False)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The 'Brusselator' reaction mechanism (proposed by Prigogene and Lefever in 1968)\n",
    "### (see J.D. Murray (1991), Mathematical Biology, p. 175)<a class=\"tocSkip\">"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "model2 = mumot.parseModel(r\"\"\"\n",
    "(\\alpha) -> X : \\gamma\n",
    "X + X + Y -> X + X + X : \\chi\n",
    "(\\beta) + X -> Y + \\emptyset : \\delta\n",
    "X -> \\emptyset : \\xi\n",
    "\"\"\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "evol2 = model2.integrate(legend_loc='lower right')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "evol2.showLogs()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "evol2b = model2.integrate(['X'], maxTime=50, legend_loc='lower right')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "Ncorr2 = model2.noiseCorrelations(maxTime=20, legend_loc='upper right', legend_fontsize=14)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "Ncorr2.showLogs()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "model2.showFokkerPlanckEquation()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "model2.showNoiseEquations()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "model2.showNoiseSolutions()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "stream2 = model2.stream('X', 'Y', showNoise=True, showFixedPoints=False)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "stream2.showLogs()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Textbook example for multivariate master equation \n",
    "### (see N. van Kampen (1981), Stochastic processes in physics and chemistry, pp. 273-277)<a class=\"tocSkip\">"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "model3 = mumot.parseModel(r\"\"\"\n",
    "(A) -> X : \\alpha\n",
    "X + X -> Y + \\emptyset : \\gamma\n",
    "Y -> (B) : \\beta\n",
    "\"\"\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "model3.showODEs(method='vanKampen')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "evol3 = model3.integrate(legend_loc='center right')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "evol3.showLogs()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "Ncorr3 = model3.noiseCorrelations(maxTime=20, legend_loc='upper right', legend_fontsize=12)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "Ncorr3.showLogs()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "model3.showFokkerPlanckEquation()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "model3.showNoiseEquations()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "model3.showNoiseSolutions()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "stream3 = model3.stream('X', 'Y', showNoise=True, showFixedPoints=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "stream3.showLogs()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Pitchfork bifurcation: a prototypical example "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "model4 = mumot.parseModel(r\"\"\"\n",
    "\\emptyset + X -> X + X : \\mu\n",
    "X + X + X -> X + X + \\emptyset : \\alpha\n",
    "Y -> \\emptyset : \\beta\n",
    "(A) -> Y : \\kappa\n",
    "X -> \\emptyset : \\gamma\n",
    "\"\"\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "model4.showODEs(method='vanKampen')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "evol4 = model4.integrate(['X', 'Y'], legend_loc='center right')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "evol4.showLogs()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "Ncorr4 = model4.noiseCorrelations(maxTime=20, legend_loc='upper right', legend_fontsize=12)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "Ncorr4.showLogs()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "model4.showFokkerPlanckEquation()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "model4.showNoiseEquations()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "model4.showNoiseSolutions()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "stream4 = model4.stream('X', 'Y', showNoise=True, showFixedPoints=False)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "stream4.showLogs()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Time evolution of stop-signal model"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "model5 = mumot.parseModel(r\"\"\"\n",
    "U -> A : g_A\n",
    "U -> B : g_B\n",
    "A -> U : a_A\n",
    "B -> U : a_B\n",
    "A + U -> A + A : r_A\n",
    "B + U -> B + B : r_B\n",
    "A + B -> A + U : s\n",
    "A + B -> B + U : s\n",
    "\"\"\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "evol5 = model5.integrate(['A', 'B', 'U'], maxTime=50, legend_loc='center right', legend_fontsize=13)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "evol5.showLogs()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "Ncorr5 = model5.noiseCorrelations(maxTime=50, legend_loc='upper right', legend_fontsize=12)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "Ncorr5.showLogs()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "model5.showNoiseEquations()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Stop-signal model with 3 options"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "model6 = mumot.parseModel(r\"\"\"\n",
    "U -> A : g_1\n",
    "U -> B : g_2\n",
    "U -> C : g_3\n",
    "A -> U : a_1\n",
    "B -> U : a_2\n",
    "C -> U : a_3\n",
    "A + U -> A + A : r_1\n",
    "B + U -> B + B : r_2\n",
    "C + U -> C + C : r_3\n",
    "A + B -> A + U : s\n",
    "A + B -> B + U : s\n",
    "A + C -> A + U : s\n",
    "A + C -> C + U : s\n",
    "B + C -> B + U : s\n",
    "B + C -> C + U : s\n",
    "\"\"\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "evol6 = model6.integrate(['A', 'B', 'C'], maxTime=50, legend_loc='center right')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "evol6.showLogs()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Population control of *E. coli* cells\n",
    "### (see You et al. (2004), Nature 428, pp. 868-871)<a class=\"tocSkip\">"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "model7 = mumot.parseModel(r\"\"\"\n",
    "A -> E : k_E\n",
    "\\emptyset + X -> X + X : k\n",
    "\\emptyset + X -> X + X : v_A\n",
    "X + X -> X + \\emptyset : k_m\n",
    "E -> \\emptyset : d_E\n",
    "X -> A : v_A\n",
    "A -> \\emptyset : d_A\n",
    "E + X -> E + \\emptyset : d_N\n",
    "A + \\emptyset -> A + A : k_E\n",
    "\"\"\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "model7.showODEs(method='vanKampen')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "evol7 = model7.integrate(['A', 'E', 'X'], maxTime=50, legend_loc='center right')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "evol7.showLogs()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "Ncorr7 = model7.noiseCorrelations(maxTimeDS=100, tstep=0.02, maxTime=20, \n",
    "                                  legend_loc='upper right', legend_fontsize=16)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "Ncorr7.showLogs()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "model7.showFokkerPlanckEquation()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "model7.showNoiseEquations()"
   ]
  }
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