{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Numerical simulatin of β-catenin destruction cycle\n", "\n", "(c) 2016 Justin Bois. This work is licensed under a [Creative Commons Attribution License CC-BY 4.0](https://creativecommons.org/licenses/by/4.0/). All code contained herein is licensed under an [MIT license](https://opensource.org/licenses/MIT).\n", "\n", "*This document was generated from a Jupyter notebook, which may be downloaded [here](betacat_destruction_cycle.ipynb).*" ] }, { "cell_type": "code", "execution_count": 16, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "\n", "
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Most are measured and reported, at least approximately, in the Lee, et al. paper. The parameters $k_8$, $k_{\\text{-}8}$ and $k_{12}$ are unknown. However, $K_\\mathrm{d} = k_{\\text{-}8}/k_8$ is known, so we only have two parameters to set." ] }, { "cell_type": "code", "execution_count": 17, "metadata": { "collapsed": true }, "outputs": [], "source": [ "# Key for names\n", "names = ['Axin complex', 'Axin-βcat', 'Axin-βcat*', 'βcat*', 'βcat']\n", "\n", "# Define known parameters\n", "c_A = 50 # nM (given by fixed GSK-3 concentration)\n", "k9 = 206 # 1/min\n", "k10 = 206 # 1/min\n", "k11 = 0.417 # 1/min\n", "Kd8 = 120 # nM\n", "\n", "# Unknown parameters\n", "km8 = 10 # 1/min\n", "k12 = 100 # nM/min\n", "\n", "# k8 determined form Kd8 and km8\n", "k8 = km8 / Kd8 # 1/nM-min\n", "\n", "# Set up params for ODE\n", "params = np.array([k8, km8, k9, k10, k11, k12])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "For our initial conditions, we will assume that we have no $\\beta$-catenin and that we have a fixed amount of APC/Axin/GKC-3." ] }, { "cell_type": "code", "execution_count": 18, "metadata": { "collapsed": false }, "outputs": [], "source": [ "# Initial conditions\n", "c0 = np.array([c_A, 0, 0, 0, 0])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now we define the time derivative. We use the functional form that is necessary to pass into `scipy.integrate.odeint()`." ] }, { "cell_type": "code", "execution_count": 19, "metadata": { "collapsed": true }, "outputs": [], "source": [ "def dcdt(c, t, k8, km8, k9, k10, k11, k12):\n", " \"\"\"\n", " Time derivative of concentrations.\n", " c = (c3, c8, c9, c10, c11)\n", " \"\"\"\n", " # Unpack concentrations and parameters\n", " c3, c8, c9, c10, c11 = c\n", " k8, km8, k9, k10, k11, k12 = params\n", " \n", " # Build derivatives\n", " deriv = np.empty(5)\n", " deriv[0] = -k8*c3*c11 + km8*c8 + k10*c9\n", " deriv[1] = k8*c3*c11 - (km8 + k9)*c8\n", " deriv[2] = k9*c8 - k10*c9\n", " deriv[3] = k10*c9 - k11*c10\n", " deriv[4] = -k8*c3*c11 + km8*c8 + k12\n", " \n", " return deriv" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now, we just have to specify what time points we want and then run the solver!" ] }, { "cell_type": "code", "execution_count": 22, "metadata": { "collapsed": false }, "outputs": [], "source": [ "# Set up time points and solve\n", "t = np.linspace(0, 15, 300)\n", "c = scipy.integrate.odeint(dcdt, c0, t, args=(k8, km8, k9, k10, k11, k12))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Finally, we can plot the solution." ] }, { "cell_type": "code", "execution_count": 23, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "\n", "\n", "
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