{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# 21. Turing patterns\n", "\n", "(c) 2019 Justin Bois and Michael Elowitz. With the exception of pasted graphics, where the source is noted, 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 prepared at [Caltech](http://www.caltech.edu) with financial support from the [Donna and Benjamin M. Rosen Bioengineering Center](http://rosen.caltech.edu).\n", "\n", "\n", "\n", "
\n", "
" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Design principle\n", "- Local activation and long range inhibition can give stable, spatially heterogeneous patterning of morphogens.\n", "\n", "#### Concepts\n", "- Reaction-diffusion equations\n", "\n", "#### Techniques\n", "- Numerical solution of PDEs by the method of lines\n", "- Linear stability analysis of PDEs\n", "\n", "
\n", "\n", "
" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "tags": [ "remove_input" ] }, "outputs": [ { "data": { "text/html": [ "\n", "
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\\n\"+\n \"

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\\n\"+\n \"\\n\"+\n \"\\n\"+\n \"from bokeh.resources import INLINE\\n\"+\n \"output_notebook(resources=INLINE)\\n\"+\n \"\\n\"+\n \"
\"}};\n\n function display_loaded() {\n var el = document.getElementById(\"1073\");\n if (el != null) {\n el.textContent = \"BokehJS is loading...\";\n }\n if (root.Bokeh !== undefined) {\n if (el != null) {\n el.textContent = \"BokehJS \" + root.Bokeh.version + \" successfully loaded.\";\n }\n } else if (Date.now() < root._bokeh_timeout) {\n setTimeout(display_loaded, 100)\n }\n }\n\n\n function run_callbacks() {\n try {\n root._bokeh_onload_callbacks.forEach(function(callback) {\n if (callback != null)\n callback();\n });\n } finally {\n delete root._bokeh_onload_callbacks\n }\n console.debug(\"Bokeh: all callbacks have finished\");\n }\n\n function load_libs(css_urls, js_urls, callback) {\n if (css_urls == null) css_urls = [];\n if (js_urls == null) js_urls = [];\n\n root._bokeh_onload_callbacks.push(callback);\n if (root._bokeh_is_loading > 0) {\n console.debug(\"Bokeh: BokehJS is being loaded, scheduling callback at\", now());\n return null;\n }\n if (js_urls == null || js_urls.length === 0) {\n run_callbacks();\n return null;\n }\n console.debug(\"Bokeh: BokehJS not loaded, scheduling load and callback at\", now());\n root._bokeh_is_loading = css_urls.length + js_urls.length;\n\n function on_load() {\n root._bokeh_is_loading--;\n if (root._bokeh_is_loading === 0) {\n console.debug(\"Bokeh: all BokehJS libraries/stylesheets loaded\");\n run_callbacks()\n }\n }\n\n function on_error() {\n console.error(\"failed to load \" + url);\n }\n\n for (var i = 0; i < css_urls.length; i++) {\n var url = css_urls[i];\n const element = document.createElement(\"link\");\n element.onload = on_load;\n element.onerror = on_error;\n element.rel = \"stylesheet\";\n element.type = \"text/css\";\n element.href = url;\n console.debug(\"Bokeh: injecting link tag for BokehJS stylesheet: \", url);\n document.body.appendChild(element);\n }\n\n for (var i = 0; i < js_urls.length; i++) {\n var url = js_urls[i];\n var element = document.createElement('script');\n element.onload = on_load;\n element.onerror = on_error;\n element.async = false;\n element.src = url;\n console.debug(\"Bokeh: injecting script tag for BokehJS library: \", url);\n document.head.appendChild(element);\n }\n };var element = document.getElementById(\"1073\");\n if (element == null) {\n console.error(\"Bokeh: ERROR: autoload.js configured with elementid '1073' but no matching script tag was found. \")\n return false;\n }\n\n function inject_raw_css(css) {\n const element = document.createElement(\"style\");\n element.appendChild(document.createTextNode(css));\n document.body.appendChild(element);\n }\n\n var js_urls = [\"https://cdn.pydata.org/bokeh/release/bokeh-1.1.0.min.js\", \"https://cdn.pydata.org/bokeh/release/bokeh-widgets-1.1.0.min.js\", \"https://cdn.pydata.org/bokeh/release/bokeh-tables-1.1.0.min.js\", \"https://cdn.pydata.org/bokeh/release/bokeh-gl-1.1.0.min.js\"];\n var css_urls = [\"https://cdn.pydata.org/bokeh/release/bokeh-1.1.0.min.css\", \"https://cdn.pydata.org/bokeh/release/bokeh-widgets-1.1.0.min.css\", \"https://cdn.pydata.org/bokeh/release/bokeh-tables-1.1.0.min.css\"];\n\n var inline_js = [\n function(Bokeh) {\n Bokeh.set_log_level(\"info\");\n },\n \n function(Bokeh) {\n \n },\n function(Bokeh) {} // ensure no trailing comma for IE\n ];\n\n function run_inline_js() {\n \n if ((root.Bokeh !== undefined) || (force === true)) {\n for (var i = 0; i < inline_js.length; i++) {\n inline_js[i].call(root, root.Bokeh);\n }if (force === true) {\n display_loaded();\n }} else if (Date.now() < root._bokeh_timeout) {\n setTimeout(run_inline_js, 100);\n } else if (!root._bokeh_failed_load) {\n console.log(\"Bokeh: BokehJS failed to load within specified timeout.\");\n root._bokeh_failed_load = true;\n } else if (force !== true) {\n var cell = $(document.getElementById(\"1073\")).parents('.cell').data().cell;\n cell.output_area.append_execute_result(NB_LOAD_WARNING)\n }\n\n }\n\n if (root._bokeh_is_loading === 0) {\n console.debug(\"Bokeh: BokehJS loaded, going straight to plotting\");\n run_inline_js();\n } else {\n load_libs(css_urls, js_urls, function() {\n console.debug(\"Bokeh: BokehJS plotting callback run at\", now());\n run_inline_js();\n });\n }\n}(window));" }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import numpy as np\n", "import pandas as pd\n", "import numba\n", "import scipy.integrate\n", "\n", "import biocircuits\n", "\n", "# Plotting modules\n", "import bokeh.io\n", "import bokeh.plotting\n", "\n", "notebook_url = 'localhost:8889'\n", "bokeh.io.output_notebook()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In recent lectures, we have been talking about signaling *between* cells, whether via cytokines regulating homeostasis, via Delta-Notch signaling, or via BMP signaling. The latter two signaling pathways are important in development of an organism, as we saw in lecture. They lead to spatial patterning of cells of different types.\n", "\n", "As we saw in the discussion of BMP, the expression of other genes in the cells of interest depend on the concentration of the ligands and receptors involved in signaling. It seems that in many contexts, biochemical cues that influence cell fate do so in a *concentration-dependent manner*; higher concentrations result in\n", "stronger signals than lower concentrations. Such chemical species,\n", "which determine cell fate in a developmental context in a\n", "concentration-dependent way, are called **morphogens**.\n", "\n", "For this lecture, we will discuss **reaction-diffusion**\n", "mechanisms for generating spatial distributions of morphogens. This is\n", "important both practically, and historically." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Turing's thoughts on reaction-diffusion mechanisms for morphogenesis\n", "In his landmark paper ([Turing, 1952](https://doi.org/10.1098%2Frstb.1952.0012)), Alan Turing laid out a prescription for morphogenesis. He described what\n", "should be considered when studying the \"changes of state\" of a\n", "developing organism. Turing said,\n", ">In determining the changes of state one should take into account:\n", ">\n", ">(i) the changes of position and velocity as given by Newton’s\n", " laws of motion;\n", ">\n", ">(ii) the stresses as given by the elasticities and motions,\n", " also taking into account the osmotic pressures as given from the\n", " chemical data;\n", "> \n", ">(iii) the chemical reactions;\n", ">\n", ">(iv) the diffusion of the chemical substances (the region in which this diffusion is possible is given from the mechanical data).\n", "\n", "This makes perfect sense; it we are shaping an organism, we have to think about what changes the shape and also how the little molecules that govern the shape move about and react. That said, a few lines later in the paper, Turing wrote, \"The interdependence of the chemical and mechanical data adds enormously to the difficulty, and attention will therefore be confined, so far as is possible, to cases where these can be separated.\"\n", "\n", "Turing went on to consider only the chemical reactions coupled with diffusion. His claim, which we will explore mathematically in this lecture, was that when diffusion was considered, spatial patterns in chemical species could form. Stop and think about that for a moment. Diffusion, a *dissipative* process, can help give spatial patterns in chemical systems that otherwise have no spatial information. In all of your life experience, I bet diffusion has served to flatten out patterns, like when you put food coloring in a glass of water. Let's explore Turing's counterintuitive proposal.\n", "\n", "We will specifically explore a type of patterning that now bears the name \"Turing patterns.\" A Turing pattern is formed *spontaneously* due to diffusion from a system of two chemical species that have a stable fixed point in the absence of diffusion." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Reaction-diffusion equations for two components\n", "\n", "Consider two chemical components, U and V, that can react with each other. The dynamical equations we have considered thus far are of the form\n", "\n", "\\begin{align}\n", " \\frac{\\mathrm{d}u}{\\mathrm{d}t} &= f(u, v)\\\\[1em]\n", " \\frac{\\mathrm{d}v}{\\mathrm{d}t} &= g(u, v).\n", "\\end{align}\n", "\n", "If the species can undergo diffusion, we need to include diffusive terms. We use Fick's second law, which says that the rate of change of concentration of a diffusing species if\n", "\n", "\\begin{align}\n", " \\frac{\\partial c}{\\partial t} = D\\nabla^2 c,\n", "\\end{align}\n", "\n", "to update our dynamical equations.\n", "\n", "\\begin{align}\n", " \\frac{\\partial u}{\\partial t} &= D_u\\nabla^2 u + f(u, v)\\\\[1em]\n", " \\frac{\\partial v}{\\partial t} &= D_v\\nabla^2 v + g(u, v).\n", "\\end{align}\n", "\n", "Our goal is to determine what type of regulation we need in the chemical circuit (i.e., to decide what type of arrowheads, activating or repressing, to use in the figure below, and what values of $D_u$ and $D_v$ are necessary to give spontaneous patterning of U and V.\n", "\n", "

" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Linear stability analysis of a PDE\n", "\n", "Earlier in the class, we learned how to do linear stability analysis of a fixed point of a system of ordinary differential equations. For this system, we also have spatial derivatives, so we need to perform linear stability analysis on a system of _partial_ differential equations. Specifically, we will consider a chemical reaction system that has a stable fixed point in the absence of diffusion. The idea is that the chemical reaction system itself will not be oscillatory and run away to some other state. We will test to see if the inclusion of spatial information, by explicitly considering diffusion, can make the entire system unstable, thereby giving rise to patterns.\n", "\n", "Let us assume that $(u_0, v_0)$ is a stable fixed point of the dynamical system in the absence of diffusion. That is, $\"f(u_0, v_0) = g(u_0, v_0) = 0$. Therefore, we can construct a _homogeneous steady state_ where $a(\\mathbf{x}) = u_0$ and $b(\\mathbf{x}) = v_0$, where $\\mathbf{x}$ denotes spatial variables. This is to say that the concentrations of species $u$ and $v$ are everywhere equal to their value at the stable fixed point. This is the unpatterned state, which is stable in the absence of diffusion.\n", "\n", "The question now is _is the homogeneous steady state stable in the presence of diffusion_? To answer this question, we will perform a linear stability analysis. We linearize $f(u, v)$ and $g(u, v)$ about $(u_0, v_0)$.\n", "\n", "\\begin{align}\n", " f(u, v) \\approx f(u_0, v_0) + f_u \\delta u + f_v\\delta v,\\\\[1em]\n", " g(u, v) \\approx g(u_0, v_0) + g_u\\delta u + g_v\\delta v,\n", "\\end{align}\n", "\n", "where\n", "\n", "\\begin{align}\n", " f_u &= \\left.\\frac{\\partial f}{\\partial u}\\right|_{(u_0, v_0)}, \\;\\;\n", " f_v = \\left.\\frac{\\partial f}{\\partial v}\\right|_{(u_0, v_0)}, \\\\[1em]\n", " g_u &= \\left.\\frac{\\partial g}{\\partial u}\\right|_{(u_0, v_0)}, \\;\\;\n", " g_v = \\left.\\frac{\\partial g}{\\partial v}\\right|_{(u_0, v_0)},\n", "\\end{align}\n", "\n", "and $\\delta u = u - u_0$ and $\\delta\\!v = v - v_0$ represent the perturbation from the homogeneous steady state. Note that $\\delta u$ and $\\delta v$ are functions of position $\\mathbf{x}$ and time $t$. Thus, our linearized system is\n", "\n", "\\begin{align}\n", " \\frac{\\partial \\delta u}{\\partial t} &= D_u\\nabla^2 \\delta u + f_u\\delta u + f_v\\delta v, \\\\[1em]\n", " \\frac{\\partial \\delta v}{\\partial t} &= D_v\\nabla^2 \\delta v + g_u\\delta u + g_v\\delta v. \n", "\\end{align}\n", "\n", "Note that the terms $f_u$ and $g_v$ describe the nature of the autoregulation of the genetic circuit\n", "\n", "Taking the Fourier transform of the equations gives\n", "\n", "\\begin{align}\n", " \\frac{\\partial \\widehat{\\delta u}}{\\partial t} &= -k^2D_u \\widehat{\\delta u} + f_u\\widehat{\\delta u} + f_v\\widehat{\\delta v} \\\\\n", " \\frac{\\partial \\widehat{\\delta v}}{\\partial t} &= -k^2D_v \\widehat{\\delta v} + g_u\\widehat{\\delta u} + g_v\\widehat{\\delta v},\n", "\\end{align}\n", "\n", "where $\\mathbf{k}$ is the vector of wave numbers (e.g., $\\mathbf{k} = (k_x, k_y)$ for a two-dimensional system) and $k^2 \\equiv \\mathbf{k}\\cdot \\mathbf{k}$. This is now a system of linear _ordinary_ differential equations in $\\widehat{\\delta u}$ and $\\widehat{\\delta v}$, which we can approach in our usual way of doing linear stability analysis.\n", "\n", "To avoid the cumbersome hats on the Fourier-transformed variables, we note that solving the linear system above is the same as searching for harmonic solutions to equations to the linearized PDEs describing the perturbations $\\delta u$ and $\\delta v$ of the form\n", "\n", "\\begin{align}\n", " \\delta u &= \\delta u_0 \\, \\mathrm{e}^{-i\\mathbf{k}\\cdot\\mathbf{x} + \\lambda t},\\\\[1em]\n", " \\delta v &= \\delta v_0 \\, \\mathrm{e}^{-i\\mathbf{k}\\cdot\\mathbf{x} + \\lambda t}.\n", "\\end{align}\n", "\n", "We can then write the resulting system of linear ODEs as\n", "\n", "\\begin{align}\n", "\\lambda \\begin{pmatrix}\n", "\\delta u\\\\\n", "\\delta v\n", "\\end{pmatrix}\n", "= \\mathsf{A}\\,\\begin{pmatrix}\n", "\\delta u\\\\\n", "\\delta v\n", "\\end{pmatrix},\n", "\\end{align}\n", "\n", "where the linear stability matrix is given by\n", "\n", "\\begin{align}\n", " \\mathsf{A} = \\begin{pmatrix}\n", " -k^2D_u + f_u & f_v \\\\\n", " g_u & -k^2D_v + g_v\n", "\\end{pmatrix}.\n", "\\end{align}\n", "\n", "As before, our goal is to compute the eigenvalues $\\lambda$ of this matrix. For a 2$\\times$2 matrix, the eigenvalues are\n", "\n", "\\begin{align}\n", " \\lambda = \\frac{1}{2}\\left(\\mathrm{tr}(\\mathsf{A}) \\pm \\sqrt{\\mathrm{tr}^2(\\mathsf{A}) - 4\\mathrm{det}(\\mathsf{A})}\\right),\n", "\\end{align}\n", "\n", "where $\\mathrm{tr}(\\mathsf{A})$ is the trace of $\\mathsf{A}$ and $\\mathrm{det}(\\mathsf{A})$ is its determinant.\n", "\n", "In the absence of diffusion, $\\mathrm{tr}(\\mathsf{A}) = f_u + g_v$. Because we have specified that in the absence of diffusion, the chemical reaction system has a stable fixed point, at least one of $f_u$ or $g_v$ must be negative. Interestingly, the trace of the linear stability matrix with diffusion included is maximal for the zeroth mode ($k = 0$), which means that an instability arising from the trace becoming positive has the zeroth mode as its fastest growing. If the determinant is positive at the onset of the instability (when the trace crosses zero), the eigenvalues are imaginary, which means that the zeroth mode is oscillatory. This is called a Hopf bifurcation.\n", "\n", "We will stipulate that $g_v < 0$, leaving for now the possibility that $f_u$ may be negative as well. In the presence of diffusion, the trace is\n", " \n", "\\begin{align}\n", " \\mathrm{tr}(\\mathsf{A}) = f_u + g_v - \\left(D_u + D_v\\right)k^2.\n", "\\end{align}\n", "\n", "Since the second term is always negative, and we have decreed that $f_u + g_v < 0$ by virtue of the chemical reaction system being stable in the absence of diffusion, the trace is always negative. As such, we can only get an eigenvalue with a positive real part (an instability) if the determinant is negative.\n", "\n", "The determinant of the linear stability matrix is\n", "\n", "\\begin{align}\n", " \\mathrm{det}(\\mathsf{A}) = D_uD_v\\,k^4 - (D_v\\,f_u + D_u\\, g_v)k^2 + f_u \\,g_v - f_v\\, g_u.\n", "\\end{align}\n", "\n", "This is quadratic in $k^2$, and since $D_u D_v > 0$, this is a convex parabola. Therefore, we can calculate the wave number where the parabola is a minimum.\n", "\n", "\\begin{align}\n", " \\frac{\\partial}{\\partial k^2} \\,\\mathrm{det}(\\mathsf{A})\n", " = 2D_u D_v \\,k^2 - D_v f_u - D_u g_v = 0.\n", "\\end{align}\n", "\n", "Thus, we have\n", "\n", "\\begin{align}\n", " k^2_\\mathrm{min} = \\frac{D_u\\, g_v + D_v \\,f_u}{2D_u D_v}.\n", "\\end{align}\n", "\n", "Inserting this expression into the expression for the determinant gives the minimal value of the determinant.\n", "\n", "\\begin{align}\n", " \\mathrm{det}(\\mathsf{A}(k_\\mathrm{min})) = f_u g_v - f_v g_u - \\frac{(D_u\\,g_v + D_v\\,f_u)^2}{4D_u D_v}.\n", "\\end{align}\n", "\n", "Therefore, the determinant is negative when\n", "\n", "\\begin{align}\n", " D_u \\,g_v + D_v\\,f_u > 2\\sqrt{D_uD_v(f_u g_v - f_v g_u)}.\n", "\\end{align}\n", "\n", "Because we have decreed that the chemical system in the absence of diffusion is stable, the determinant of the linear stability matrix of the diffusion-less system, $f_u g_v - f_v g_u$, must be positive, so the term in parentheses in the square root must be positive. Thus, we must have\n", "\n", "\\begin{align}\n", " D_u\\,g_v + D_v\\,f_u > 0\n", "\\end{align}\n", "\n", "in order to have a negative determinant and therefore an instability. Since we already showed that at least one of $f_u$ or $g_v$ must be negative, we must have *exactly* one be negative. We will take $f_u > 0$, as we already decreed that $g_v < 0$. Note that having $g_v < 0$ does not necessarily mean that we have autoinhibition, since we could also have $g_v < 0$ by degradation or dilution. However, the signs of $f_u$ and $g_v$ indicate that U is autoactivating and is in a negative feedback loop with V.\n", "\n", "Finally, let's look again at our expression for $k_\\mathrm{min}^2$. For $k_\\mathrm{min}$ to be real, we must have\n", "\n", "\\begin{align}\n", " D_v \\,f_u + D_u\\,g_v > 0.\n", "\\end{align}\n", "\n", "We also have $f_u + g_v < 0$ from the condition that the trace of the linear stability matrix is negative. So, we have four inequalities,\n", "\n", "\\begin{align}\n", " &f_u > 0, \\\\[1em]\n", " &g_v < 0, \\\\[1em]\n", " &f_u + g_v < 0,\\\\[1em]\n", " &\\frac{D_v}{D_u}\\,f_u + g_v > 0.\n", "\\end{align}\n", "\n", "A requirement for all of these inequalities can only hold is that $D_v > D_u$. Thus, we have arrived at our conditions for formation of Turing patterns.\n", "\n", "1. We need an autocatalytic species (an \"activator\").\n", "2. The activator must have a negative feedback loop with the other species, the \"inhibitor.\"\n", "3. The inhibitor must diffuse more rapidly than the activator.\n", "\n", "The intuition here is that the activator starts producing more of itself locally. The local peak starts to spread, but the inhibitor diffuses more quickly so that it pins the peak of activator in so that it cannot spread. This gives a set wavelength of the pattern of peaks. The circuits below can lead to Turing patterns. The dashed line indicates that autorepression is not necessary, only that $g_v < 0$.\n", "\n", "

" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Wavelength of a Turing pattern\n", "\n", "The wavelength of a Turing pattern is set, at least as the instability begins to grow, by the _fastest growing mode_, $k$. That is, for the value of $k$ for which the eigenvalue $\\lambda$ is largest. We have already shown that $k_\\mathrm{min}$, the $k$ for which the determinant of the linear stability matrix is minimal, is the $k$ corresponding to the fastest growing mode.\n", "\n", "We can derive a **dispersion relation**, which relates the eigenvalues of the linear stability matrix $\\mathsf{A}$ to the wave number $k$. Specifically, is $\\lambda_\\mathrm{max}$ is the eigenvalue with the largest real part, the dispersion relation is the function $\\mathrm{Re}[\\lambda_\\mathrm{max}(k)]$. The dispersion relation describes what modes contribute to the instability; these set the wavelength of the emergent pattern. This is perhaps best understood by example, which we consider below with the **activator substrate depletion model**." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Turing patterns do not scale\n", "\n", "Turing patterns do not scale with the size of the organism because the wavelength of the pattern, given by the fastest growing mode, $k_\\mathrm{min}$, is independent of system size. So, if a system is twice as large, it would have twice as many peaks and valleys in the pattern. In the next lesson, we will address this problem of **scaling** and talk about reaction-diffusion systems that can give patterns that scale with organism size." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Example: The Activator-Substrate Depletion Model\n", "\n", "As an example of a system that undergoes a Turing pattern, let us consider now the activator-substrate depletion model, or ASDM. This is a simple model in which there are two species, the activator and substrate. The activator is autocatalytic, but consumes substrate in the process. It also has our standard degradation. The substrate is constitutively produced, and for the sake of simplicity, we assume it is very stable such that its degradation is all by consumption by the activator. This corresponds to the right circuit in our figure of circuits that can give Turing patterns, above. The dynamical equations are\n", "\n", "\\begin{align}\n", " \\frac{\\partial a}{\\partial t} &= D_a\\,\\nabla^2 a + \\rho a^2s - \\gamma a,\\\\[1em]\n", " \\frac{\\partial s}{\\partial t} &= D_s\\,\\nabla^2 s + \\beta - \\rho a^2s.\n", "\\end{align}\n", "\n", "We nondimensionalize by choosing $t \\leftarrow t\\gamma$, $(x,y,z) \\leftarrow \\sqrt{D_s/\\gamma}(x, y, z)$, $a \\leftarrow \\gamma a/\\beta$, $s \\leftarrow \\beta\\rho s/\\gamma^2$, $d = D_a/D_s$ and $\\mu = \\beta^2\\rho/\\gamma^3$. Then, the dimensionless equations are\n", "\n", "\\begin{align}\n", " \\frac{\\partial a}{\\partial t} &= d \\nabla^2 a + a^2s - a,\\\\[1em]\n", " \\frac{\\partial s}{\\partial t} &= \\nabla^2 s + \\mu(1-a^2s).\n", "\\end{align}\n", "\n", "This system has a unique homogeneous steady state of $a_0 = s_0 = 1$.\n", "In the language of our general Turing system, here we have $f_a = f_s = 1$, $g_a = -2\\mu$, and $g_s = -\\mu$, so we have a linear stability matrix\n", "\n", "\\begin{align}\n", " \\mathsf{A} = \\begin{pmatrix}\n", " 1-dk^2 & 1 \\\\\n", " -2\\mu & -\\mu-k^2\n", "\\end{pmatrix}.\n", "\\end{align}" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Dispersion relation\n", "\n", "With the linear stability matrix in hand, we can compute and plot the dispersion relation. We will allow for various values of the two parameters $d$ and $\\mu$." ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "application/vnd.bokehjs_exec.v0+json": "", "text/html": [ "\n", "" ] }, "metadata": { "application/vnd.bokehjs_exec.v0+json": { "server_id": "4c567d7a28f54df3a247567475b5baa2" } }, "output_type": "display_data" } ], "source": [ "slider_params = (biocircuits.AttributeContainer(title='d', start=0.01, \n", " end=2, value=0.25, step=0.01),\n", " biocircuits.AttributeContainer(title='μ', start=0.01, \n", " end=2, value=1.5, step=0.005))\n", "\n", "def dispersion_relation(k_vals, d, mu):\n", " lam = np.empty_like(k_vals)\n", " for i, k in enumerate(k_vals):\n", " A = np.array([[1-d*k**2, 1],\n", " [-2*mu, -mu - k**2]])\n", " lam[i] = np.linalg.eigvals(A).real.max()\n", " \n", " return lam\n", "\n", "def dispersion_callback(source, k_range, lam_range, sliders, toggles, n_points):\n", " # Set up k values, keeping minimum at zero\n", " k = np.linspace(0, k_range.end, n_points)\n", " \n", " # Pull out slider values\n", " d = sliders[0].value\n", " mu = sliders[1].value\n", " \n", " # Compute dispersion relation\n", " source.data['k'] = k\n", " source.data['lam'] = dispersion_relation(k, d, mu)\n", "\n", "\n", "def dispersion_plot(callback, sliders, toggles, extra_args):\n", " # Set up plot\n", " p = bokeh.plotting.figure(plot_width=500,\n", " plot_height=300,\n", " x_axis_label='k',\n", " y_axis_label='Re[λ-max]')\n", "\n", " # Set up empty data source\n", " source = bokeh.models.ColumnDataSource()\n", "\n", " # Populate glyphs\n", " p.line('k', 'lam', source=source, line_width=2)\n", "\n", " # Update data according to callback\n", " callback(source, bokeh.models.Range1d(0, 10), None, \n", " sliders, toggles, *extra_args)\n", "\n", " return p, source\n", "\n", "dispersion_app = biocircuits.interactive_xy_plot(dispersion_plot, dispersion_callback, \n", " slider_params, (), (400,))\n", "bokeh.io.show(dispersion_app, notebook_url=notebook_url)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "By playing with the sliders, we can see that the maximal real part of the eigenvalues is always below zero for large $d$. For $\\mu < 1$, we For small $d$, and for $\\mu > 1$, we can get a positive real part of an eigenvalue, with the maximum occurring at $k = 0$. This implies that the zero mode, or uniform concentration, grows fastest. However, for small $d$ with $\\mu > 1$, we get a fastest growing mode for finite positive $k$. In this regimes, we can form patterns." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Linear stability diagram\n", "\n", "We have just explored the parameter values that give patterns with interactive plots of the dispersion relation. We can take a similar approach to construct a **linear stability diagram**. There are two parameters, $\\mu$ and $d$, and we can imagine a map of the $d$-$\\mu$ plane identifying which regions of parameter space that have different linear stability properties. We have five possibilities in this system. Denoting by $k_\\mathrm{max}$ a wavenumber for which the largest real part of the eigenvalues if (locally) maximal, we have the following.\n", "\n", "|Eigenvalues | Description| color |\n", "|:----:|:----|:----:|\n", "|Negative real parts | Homogeneous steady state stable | green |\n", "|Real, positive, $k_\\mathrm{max}>0$ | Pattern forming | purple|\n", "|Real, positive, $k_\\mathrm{max} = 0$| Homogeneous conc. grows without bound | blue|\n", "|Real, positive, $k_\\mathrm{max} = 0$ and $k_\\mathrm{max} > 0$ | Possible patterns, depending on initial condition | yellow |\n", "|Imaginary, $k_mathrm{max} > 0$ | Oscillating homogeneous patterns | orange |\n", "\n", "\n", "In some cases, we can conveniently compute the boundaries between the different regions of parameter spaces, but it is often easier to take a \"brute force\" approach, wherein we compute the eigenvalues for each set of parameter values for a range of $k$-values, and then display the results as an image colored according to which class of eigenvalues we get. This is implemented below." ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "tags": [ "remove_input" ] }, "outputs": [ { "data": { "text/html": [ "\n", "\n", "\n", "\n", "\n", "\n", "
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Application\",\"version\":\"1.1.0\"}};\n", " var render_items = [{\"docid\":\"2d08efb0-b8ce-4839-ba30-ef8c126fdc2d\",\"roots\":{\"2550\":\"3e3f29d4-e203-4e31-a95f-264dc0b632c5\"}}];\n", " root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n", "\n", " }\n", " if (root.Bokeh !== undefined) {\n", " embed_document(root);\n", " } else {\n", " var attempts = 0;\n", " var timer = setInterval(function(root) {\n", " if (root.Bokeh !== undefined) {\n", " embed_document(root);\n", " clearInterval(timer);\n", " }\n", " attempts++;\n", " if (attempts > 100) {\n", " console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\");\n", " clearInterval(timer);\n", " }\n", " }, 10, root)\n", " }\n", "})(window);" ], "application/vnd.bokehjs_exec.v0+json": "" }, "metadata": { "application/vnd.bokehjs_exec.v0+json": { "id": "2550" } }, "output_type": "display_data" } ], "source": [ "# Values for the plot\n", "mu_vals = np.linspace(0.001, 2, 400)\n", "d_vals = np.linspace(0.001, 1, 400)\n", "\n", "@numba.njit\n", "def region(d, mu):\n", " \"\"\"Returns which region the eigenvalue lives in.\"\"\"\n", " osc = False\n", " unstable = False\n", " turing = False\n", " \n", " if 3-2*np.sqrt(2) < mu < 1:\n", " osc = True\n", "\n", " k2 = np.linspace(0, 10, 200)\n", " det = (1 - d*k2) * (-mu - k2) + 2*mu\n", " tr = 1 - (1+d)*k2 - mu\n", " \n", " real_lam = np.empty_like(k2)\n", " for i, (t, d) in enumerate(zip(tr, det)):\n", " discriminant = t**2 - 4 * d\n", " if discriminant < 0:\n", " real_lam[i] = t / 2\n", " else:\n", " real_lam[i] = (t + np.sqrt(discriminant)) / 2\n", "\n", " if (real_lam > 0).any():\n", " unstable = True\n", "\n", " if np.argmax(real_lam) > 0:\n", " turing = True\n", "\n", " if osc:\n", " if turing:\n", " return 3\n", " return 2\n", " \n", " if unstable:\n", " if turing:\n", " return 1\n", " return 4\n", " \n", " return 0\n", " \n", "# Compute linear stability diagram\n", "d, mu = np.meshgrid(d_vals, mu_vals)\n", "im = np.empty_like(d)\n", "for i in range(len(mu_vals)):\n", " for j in range(len(d_vals)):\n", " im[i, j] = region(d_vals[j], mu_vals[i])\n", " \n", "# Set up the coloring\n", "color_mapper = bokeh.models.LinearColorMapper(\n", " ['#7fc97f','#beaed4','#fdc086','#ffff99','#386cb0'])\n", "\n", "# Make plot and display\n", "p = biocircuits.imshow(im, \n", " color_mapper=color_mapper,\n", " x_range=[d.min(), d.max()], \n", " y_range=[mu.min(), mu.max()],\n", " x_axis_label='d',\n", " y_axis_label='µ',\n", " flip=False,\n", " display_clicks=True)\n", "bokeh.io.show(p)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Tissue size\n", "\n", "We will solve the PDEs on the domain $0 \\le x \\le L$, where $L$ is the total length of the growing tissue. We did not consider a finite domain size when we were doing the linear stability analysis. If all wave numbers $k$ for which an eigenvalue has positive real part are larger than the domain size $L$, the domain cannot accommodate the nascent pattern. In this case the homogeneous steady state remains stable.\n", "\n", "This is another mechanism by which a reaction-diffusion system can give patterns. As a tissue grows, when it passes a length that can accommodate the pattern, the pattern can spontaneously form." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Boundary conditions\n", "\n", "For any set of PDEs, we need to also specify boundary and initial conditions. We can choose periodic boundary conditions, $c(0,t) = c(L,t)$, where $c$ is the concentration of a morphogen. This is appropriate, for example, in examining patterning around a cylinder, which was the case in the Turing paper in looking at the pattering of developing [whorl](https://en.wikipedia.org/wiki/Whorl_(botany)) around a stem.\n", "\n", "We may also want to consider **no-flux boundary conditions**, in which the flux of morphogen at the boundaries is zero. This is the case for a tissue of finite size; the morphogen cannot diffuse off of the end of the tissue. The diffusive flux, as given by [Fick's Law](https://en.wikipedia.org/wiki/Fick%27s_laws_of_diffusion) is\n", "\n", "\\begin{align}\n", "j(x) = -D\\,\\frac{\\partial c}{\\partial x}.\n", "\\end{align}\n", "\n", "So, a no-flux condition means that we set the first derivative of the concentration profile to zero at the boundaries. In what follows, we will use no-flux conditions." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Numerical solution of PDEs\n", "\n", "To investigate the dynamics of spontaneously forming patterns, we need to solve the governing PDEs numerically. We will use a technique known as the [**method of lines**](https://en.wikipedia.org/wiki/Method_of_lines) to do so. The idea is that we discretize the spatial coordinate $x$ in evenly spaced points with spacing $\\Delta x$ between each point. To compute the second derivatives that come about in the diffusion terms, we use [**central differencing**](https://en.wikipedia.org/wiki/Central_differencing_scheme). In the central differencing scheme, we approximate the second derivative of a concentration evaluated at point $x_i$ (which is one of our discrete grid points) as\n", "\n", "\\begin{align}\n", "\\left.\\frac{\\partial^2 c}{\\partial x^2}\\right|_{x=x_i} \\approx \\frac{c(x_{i+1}) + c(x_{i-1}) - 2c(x_i)}{2\\Delta x}.\n", "\\end{align}\n", "\n", "By discretizing the spatial variable, we now have a system of *ordinary* differential equations. In this case, where we have two species, we have two times the number of grid points ordinary differential equations. We can then use `scipy.integrate.odeint()` to solve this system.\n", "\n", "This is implemented in the functions below, which also allow for nonconstant diffusion coefficients." ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [], "source": [ "def dc_dt(c, t, x, derivs_0, derivs_L, diff_coeff_fun, diff_coeff_params, \n", " rxn_fun, rxn_params, n_species, h):\n", " \"\"\"\n", " Time derivative of concentrations in an R-D system \n", " for constant flux BCs.\n", " \n", " Parameters\n", " ----------\n", " c : ndarray, shape (n_species * n_gridpoints)\n", " The concentration of the chemical species interleaved in a\n", " a NumPy array. The interleaving allows us to take advantage \n", " of the banded structure of the Jacobian when using the \n", " Hindmarsh algorithm for integrating in time.\n", " t : float\n", " Time.\n", " derivs_0 : ndarray, shape (n_species)\n", " derivs_0[i] is the value of the diffusive flux,\n", " D dc_i/dx, at x = 0, the leftmost boundary of the domain of x.\n", " derivs_L : ndarray, shape (n_species)\n", " derivs_0[i] is the value of the diffusive flux,\n", " D dc_i/dx, at x = L, the rightmost boundary of the domain of x.\n", " diff_coeff_fun : function\n", " Function of the form diff_coeff_fun(c_tuple, t, x, *diff_coeff_params).\n", " Returns an tuple where entry i is a NumPy array containing\n", " the diffusion coefficient of species i at the grid points.\n", " c_tuple[i] is a NumPy array containing the concentrations of\n", " species i at the grid poitns.\n", " diff_coeff_params : arbitrary\n", " Tuple of parameters to be passed into diff_coeff_fun.\n", " rxn_fun : function\n", " Function of the form rxn_fun(c_tuple, t, *rxn_params).\n", " Returns an tuple where entry i is a NumPy array containing\n", " the net rate of production of species i by chemical reaction\n", " at the grid points. c_tuple[i] is a NumPy array containing \n", " the concentrations of species i at the grid poitns.\n", " rxn_params : arbitrary\n", " Tuple of parameters to be passed into rxn_fun.\n", " n_species : int\n", " Number of chemical species.\n", " h : float\n", " Grid spacing (assumed to be constant)\n", " \n", " Returns\n", " -------\n", " dc_dt : ndarray, shape (n_species * n_gridpoints)\n", " The time derivatives of the concentrations of the chemical\n", " species at the grid points interleaved in a NumPy array.\n", " \"\"\"\n", " # Tuple of concentrations\n", " c_tuple = tuple([c[i::n_species] for i in range(n_species)])\n", " \n", " # Compute diffusion coefficients\n", " D_tuple = diff_coeff_fun(c_tuple, t, x, *diff_coeff_params)\n", " \n", " # Compute reaction terms\n", " rxn_tuple = rxn_fun(c_tuple, t, *rxn_params)\n", "\n", " # Return array\n", " conc_deriv = np.empty_like(c)\n", " \n", " # Convenient array for storing concentrations\n", " da_dt = np.empty(len(c_tuple[0]))\n", "\n", " # Useful to have square of grid spacing around\n", " h2 = h**2\n", " \n", " # Compute diffusion terms (central differencing w/ Neumann BCs)\n", " for i in range(n_species):\n", " # View of concentrations and diffusion coeff. for convenience\n", " a = np.copy(c_tuple[i])\n", " D = np.copy(D_tuple[i])\n", " \n", " # Time derivative at left boundary\n", " da_dt[0] = D[0] / h2 * 2 * (a[1] - a[0] - h * derivs_0[i])\n", " \n", " # First derivatives of D and a\n", " dD_dx = (D[2:] - D[:-2]) / (2*h)\n", " da_dx = (a[2:] - a[:-2]) / (2*h)\n", "\n", " # Time derivative for middle grid points\n", " da_dt[1:-1] = D[1:-1] * np.diff(a, 2) / h2 + dD_dx*da_dx\n", " \n", " # Time derivative at left boundary\n", " da_dt[-1] = D[-1] / h2 * 2 * (a[-2] - a[-1] + h * derivs_L[i])\n", " \n", " # Store in output array with reaction terms\n", " conc_deriv[i::n_species] = da_dt + rxn_tuple[i]\n", " \n", " return conc_deriv\n", "\n", "\n", "def rd_solve(c_0_tuple, t, L=1, derivs_0=0, derivs_L=0, diff_coeff_fun=None, \n", " diff_coeff_params=(), rxn_fun=None, rxn_params=(),\n", " rtol=1.49012e-8, atol=1.49012e-8):\n", " \"\"\"\n", " Parameters\n", " ----------\n", " c_0_tuple : tuple\n", " c_0_tuple[i] is a NumPy array of length n_gridpoints with the \n", " initial concentrations of chemical species i at the grid points.\n", " t : ndarray\n", " An array of time points for which the solution is desired.\n", " L : float\n", " Total length of the x-domain.\n", " derivs_0 : ndarray, shape (n_species)\n", " derivs_0[i] is the value of dc_i/dx at x = 0.\n", " derivs_L : ndarray, shape (n_species)\n", " derivs_L[i] is the value of dc_i/dx at x = L, the rightmost\n", " boundary of the domain of x.\n", " diff_coeff_fun : function\n", " Function of the form diff_coeff_fun(c_tuple, x, t, *diff_coeff_params).\n", " Returns an tuple where entry i is a NumPy array containing\n", " the diffusion coefficient of species i at the grid points.\n", " c_tuple[i] is a NumPy array containing the concentrations of\n", " species i at the grid poitns.\n", " diff_coeff_params : arbitrary\n", " Tuple of parameters to be passed into diff_coeff_fun.\n", " rxn_fun : function\n", " Function of the form rxn_fun(c_tuple, t, *rxn_params).\n", " Returns an tuple where entry i is a NumPy array containing\n", " the net rate of production of species i by chemical reaction\n", " at the grid points. c_tuple[i] is a NumPy array containing \n", " the concentrations of species i at the grid poitns.\n", " rxn_params : arbitrary\n", " Tuple of parameters to be passed into rxn_fun.\n", " rtol : float\n", " Relative tolerance for solver. Default os odeint's default.\n", " atol : float\n", " Absolute tolerance for solver. Default os odeint's default.\n", " \n", " Returns\n", " -------\n", " c_tuple : tuple\n", " c_tuple[i] is a NumPy array of shape (len(t), n_gridpoints)\n", " with the initial concentrations of chemical species i at \n", " the grid points over time.\n", " \n", " Notes\n", " -----\n", " .. When intergrating for long times near a steady state, you\n", " may need to lower the absolute tolerance (atol) because the\n", " solution does not change much over time and it may be difficult\n", " for the solver to maintain tight tolerances.\n", " \"\"\"\n", " # Number of grid points\n", " n_gridpoints = len(c_0_tuple[0])\n", " \n", " # Number of chemical species\n", " n_species = len(c_0_tuple)\n", "\n", " # Grid spacing\n", " h = (L / (n_gridpoints - 1))\n", " \n", " # Grid points\n", " x = np.linspace(0, L, n_gridpoints)\n", " \n", " # Set up boundary conditions\n", " if np.isscalar(derivs_0):\n", " derivs_0 = np.array(n_species * [derivs_0])\n", " if np.isscalar(derivs_L):\n", " derivs_L = np.array(n_species * [derivs_L])\n", "\n", " # Set up parameters to be passed in to dc_dt\n", " params = (x, derivs_0, derivs_L, diff_coeff_fun, diff_coeff_params, \n", " rxn_fun, rxn_params, n_species, h)\n", "\n", " # Set up initial condition\n", " c0 = np.empty(n_species * n_gridpoints)\n", " for i in range(n_species):\n", " c0[i::n_species] = c_0_tuple[i]\n", "\n", " # Solve using odeint, taking advantage of banded structure\n", " c = scipy.integrate.odeint(\n", " dc_dt, c0, t, args=params, ml=n_species, mu=n_species,\n", " rtol=rtol, atol=atol)\n", " \n", " return tuple([c[:,i::n_species] for i in range(n_species)])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The `rd_solve()` function is included in the `biocircuits` package.\n", "\n", "Let us proceed to set up and solve for the ASDM model. We will specify an initial condition that is a tiny perturbation from the homogeneous steady state of $a = s = 1$." ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [], "source": [ "def constant_diff_coeffs(c_tuple, t, x, diff_coeffs):\n", " n = len(c_tuple[0])\n", " return tuple([diff_coeffs[i] * np.ones(n) for i in range(len(c_tuple))])\n", "\n", "\n", "def asdm_rxn(as_tuple, t, mu):\n", " \"\"\"\n", " Reaction expression for activator-substrate depletion model.\n", "\n", " Returns the rate of production of activator and substrate, respectively.\n", "\n", " r_a = a**2 * s - a\n", " r_s = mu * (1 - a**2 * s)\n", " \"\"\"\n", " # Unpack concentrations\n", " a, s = as_tuple\n", " \n", " # Compute and return reaction rates\n", " a2s = a**2 * s\n", " return (a2s - a, mu * (1.0 - a2s))\n", "\n", "\n", "# Set up steady state (using 500 grid points)\n", "a_0 = np.ones(500)\n", "s_0 = np.ones(500)\n", "\n", "# Make a small perturbation to a_0\n", "a_0 += 0.001 * np.random.rand(len(a_0))\n", "\n", "# Time points\n", "t = np.linspace(0.0, 100.0, 100)\n", "\n", "# Physical length of system\n", "L = 20.0\n", "\n", "# x-coordinates for plotting\n", "x = np.linspace(0, L, len(a_0))\n", "\n", "# Diffusion coefficients\n", "diff_coeffs = (0.05, 1.0)\n", "\n", "# Reaction parameter (must be a tuple of params, even though only 1 for ASDM)\n", "rxn_params = (1.5,)\n", "\n", "# Solve\n", "conc = rd_solve((a_0, s_0), t, L=L, derivs_0=0, derivs_L=0, \n", " diff_coeff_fun=constant_diff_coeffs, \n", " diff_coeff_params=(diff_coeffs,), rxn_fun=asdm_rxn, \n", " rxn_params=rxn_params)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can now make an interactive plot with a time slider to investigate the dynamics." ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "data": { "application/vnd.bokehjs_exec.v0+json": "", "text/html": [ "\n", "" ] }, "metadata": { "application/vnd.bokehjs_exec.v0+json": { "server_id": "111e8552b2b0491cb3d3112b21d4ba17" } }, "output_type": "display_data" } ], "source": [ "slider_params = (biocircuits.AttributeContainer(title='t', start=t[0], \n", " end=t[-1], value=t[0], \n", " step=np.diff(t).mean()),)\n", "\n", "extra_args = (x, conc)\n", "\n", "def turing_callback(source, x_range, y_range, sliders, toggles, x, conc):\n", " # Get index of time point\n", " t_point = sliders[0].value\n", " i = np.searchsorted(t, t_point)\n", "\n", " # Update source\n", " source.data['x'] = x\n", " source.data['conc_0'] = conc[0][i, :]\n", " source.data['conc_1'] = conc[1][i, :]\n", "\n", "\n", "def turing_plot(callback, sliders, toggles, extra_args):\n", " # Determine y-range\n", " y_range = bokeh.models.Range1d(0, np.concatenate(conc).max()*1.02)\n", " \n", " # Set up plot\n", " p = bokeh.plotting.figure(plot_width=500,\n", " plot_height=300,\n", " x_axis_label='x',\n", " y_axis_label='a, s',\n", " x_range=[0, L],\n", " y_range=y_range)\n", "\n", " # Set up empty data source\n", " source = bokeh.models.ColumnDataSource()\n", "\n", " # Populate glyphs\n", " p.line('x', 'conc_0', source=source, line_width=2, legend='activator')\n", " p.line('x', 'conc_1', source=source, line_width=2, color='orange', legend='substrate')\n", " \n", " # Update data according to callback\n", " callback(source, None, None, \n", " sliders, toggles, *extra_args)\n", "\n", " return p, source\n", "\n", "turing_app = biocircuits.interactive_xy_plot(turing_plot, turing_callback, \n", " slider_params, (), extra_args)\n", "bokeh.io.show(turing_app, notebook_url=notebook_url)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Two-dimensional solutions\n", "\n", "Using more sophisticated numerical integration techniques, we can solve the ADSM in two dimensions. This is implemented in the [rdsolver package](https://github.com/justinbois/rdsolver), which solves two-dimensional reaction-diffusion problems using [spectral methods](https://en.wikipedia.org/wiki/Spectral_method) with implicit-explicit (IMEX) time stepping. For fun, we demonstrate it here." ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "name": "stderr", "output_type": "stream", "text": [ "100%|██████████| 100/100 [00:11<00:00, 11.05it/s]\n" ] }, { "data": { "application/vnd.bokehjs_exec.v0+json": "", "text/html": [ "\n", "" ] }, "metadata": { "application/vnd.bokehjs_exec.v0+json": { "server_id": "7a93f964087f4b83867dc91bcb4ffb01" } }, "output_type": "display_data" } ], "source": [ "import rdsolver\n", "\n", "D, beta, gamma, f, f_args, homo_ss = rdsolver.models.asdm()\n", "\n", "n = (32, 32)\n", "L = (50, 50)\n", "t = np.linspace(0, 100000, 100)\n", "c0 = rdsolver.initial_condition(uniform_conc=homo_ss, n=n, L=L)\n", "\n", "c = rdsolver.solve(c0, t, D=D, beta=beta, gamma=gamma, f=f, f_args=f_args, L=L)\n", "c_interp = rdsolver.viz.interpolate_concs(c)\n", "\n", "bokeh.io.show(rdsolver.viz.display_notebook(t, c_interp), notebook_url=notebook_url)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Computing environment" ] }, { "cell_type": "code", "execution_count": 15, "metadata": { "tags": [ "remove_input" ] }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "CPython 3.7.3\n", "IPython 7.5.0\n", "\n", "numpy 1.16.3\n", "scipy 1.2.1\n", "numba 0.43.1\n", "bokeh 1.1.0\n", "jupyterlab 0.35.5\n", "rdsolver 0.1.3\n", "biocircuits 0.0.11\n" ] } ], "source": [ "%load_ext watermark\n", "%watermark -v -p numpy,scipy,numba,bokeh,jupyterlab,rdsolver,biocircuits" ] } ], "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": "3.7.3" } }, "nbformat": 4, "nbformat_minor": 2 }