{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# 16b. Profiling code for speed and an application of the Gillespie algorithm\n", "\n", "
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In this appendix, we set up a Gillespie simulation of the [repressilator](../chapters/09_repressilator.ipynb). We will then return to a simple example of gene expression to demonstrate how to assess the speed of the calculation and improve it." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Simulating the repressilator\n", "\n", "We have seen a structure to how we can set up Gillespie simulations; we need to specify the propensities and updates, along with an initial population. It helps clarify the system being simulated and also avoids bugs if we make tables for\n", "\n", "1. The species whose populations we are describing;\n", "2. The update-propensity pairs;\n", "3. The parameter values.\n", "\n", "After constructing the tables, coding up the update and propensity functions is much easier.\n", "\n", "To demonstrate this procedure, we will perform a Gillespie simulation for the repressilator, as described in [Elowitz and Leibler](https://doi.org/10.1038/35002125). We will consider both RNA and DNA in a repressilator where gene 1 represses gene 2, gene 2 represses gene 3, and gene 3 represses gene 1. We explicitly consider mRNA and protein. A repressor binds its operon with chemical rate constant $k_r$, and an operon may have zero, one, or two repressors bound. The unbinding rate of a repressor when one is bound is $k_{u,1}$ and that of a repressor when two are bound if $k_{u,2}$, with $k_{u,2} < k_{u,1}$ to capture cooperativity. Transcription happens when no repressors are bound to a promoter region with rate constant $k_{m,u}$ and happens when one or two repressors are bound with rate $k_{m,o}$.\n", "\n", "As we build the simulation, let's start with a table of the populations we are keeping track of.\n", "\n", "|index|description | variable|\n", "|:----:|:------ |:------: |\n", "|0|gene 1 mRNA copy number | `m1`|\n", "|1|gene 1 protein copy number | `p1`|\n", "|2|gene 2 mRNA copy number | `m2`|\n", "|3|gene 2 protein copy number | `p2`|\n", "|4|gene 3 mRNA copy number | `m3`|\n", "|5|gene 3 protein copy number | `p3`|\n", "|6|Number of repressors bound to promoter of gene 3| `n1`|\n", "|7|Number of repressors bound to promoter of gene 1| `n2`|\n", "|8|Number of repressors bound to promoter of gene 2| `n3`|\n", "\n", "Note that we labeled each species with an index, which corresponds to its position in the array of populations in the simulation.\n", "\n", "Next, we can set up a table of updates and propensities for the moves we allow in the Gillespie simulation. We also assign an index to each entry here, as this helps keep track of everything.\n", "\n", "|index|description | update | propensity|\n", "|:----|:------ |:------ | :----:|\n", "|0|transcription of gene 1 mRNA| `m1 ⟶ m1 + 1`| `kmu*(n3 == 0) + kmo*(n3 > 0)`|\n", "|1|transcription of gene 2 mRNA| `m2 ⟶ m2 + 1`| `kmu*(n1 == 0) + kmo*(n1 > 0)`|\n", "|2|transcription of gene 3 mRNA| `m3 ⟶ m3 + 1`| `kmu*(n2 == 0) + kmo*(n2 > 0)`|\n", "|3|translation of gene 1 protein| `p1 ⟶ p1 + 1`| `kp * m1`|\n", "|4|translation of gene 2 protein| `p2 ⟶ p2 + 1`| `kp * m2`|\n", "|5|translation of gene 3 protein| `p3 ⟶ p3 + 1`| `kp * m3`|\n", "|6|degradation of gene 1 mRNA| `m1 ⟶ m1 - 1`| `gamma_m * m1`|\n", "|7|degradation of gene 2 mRNA| `m2 ⟶ m2 - 1`| `gamma_m * m2`|\n", "|8|degradation of gene 3 mRNA| `m3 ⟶ m3 - 1`| `gamma_m * m3`|\n", "|9|degradation of unbound gene 1 protein| `p1 ⟶ p1 - 1` | `gamma_p * p1`|\n", "|10|degradation of unbound gene 2 protein| `p2 ⟶ p2 - 1` | `gamma_p * p2`|\n", "|11|degradation of unbound gene 3 protein| `p3 ⟶ p3 - 1` | `gamma_p * p3`|\n", "|12|degradation of bound gene 1 protein| `n1 ⟶ n1 - 1` | `gamma_p * n1`|\n", "|13|degradation of bound gene 2 protein| `n2 ⟶ n2 - 1` | `gamma_p * n2`|\n", "|14|degradation of bound gene 3 protein| `n3 ⟶ n3 - 1` | `gamma_p * n3`|\n", "|15|binding of protein to gene 1 operator| `n3 ⟶ n3 + 1`, `p3 ⟶ p3 - 1`| `kr * p3 * (n3 < 2)`|\n", "|16|binding of protein to gene 2 operator| `n1 ⟶ n1 + 1`, `p1 ⟶ p1 - 1`| `kr * p1 * (n1 < 2)`|\n", "|17|binding of protein to gene 3 operator| `n2 ⟶ n2 + 1`, `p2 ⟶ p2 - 1`| `kr * p2 * (n2 < 2)`|\n", "|18|unbinding of protein to gene 1 operator| `n3 ⟶ n3 - 1`, `p3 ⟶ p3 + 1`| `ku1*(n3 == 1) + 2*ku2*(n3 == 2)`|\n", "|19|unbinding of protein to gene 2 operator| `n1 ⟶ n1 - 1`, `p1 ⟶ p1 + 1`| `ku1*(n1 == 1) + 2*ku2*(n1 == 2)`|\n", "|20|unbinding of protein to gene 3 operator| `n2 ⟶ n2 - 1`, `p2 ⟶ p2 + 1`| `ku1*(n2 == 1) + 2*ku2*(n2 == 2)`|\n", "\n", "Finally, we have parameters that were introduced in the propensities, so we should have a table defining them.\n", "\n", "|parameter| value | units |\n", "|:----|:------ |:------: |\n", "|`kmu`| 0.5 | 1/sec|\n", "|`kmo`|5×10$^{-4}$ | 1/sec|\n", "|`kp`| 0.167 | 1/molec-sec|\n", "|`gamma_m`| 0.005776 | 1/sec|\n", "|`gamma_p`| 0.001155 | 1/sec|\n", "| `kr` | 1 | 1/molec-sec|\n", "| `ku1` | 224 | 1/sec|\n", "| `ku2` | 9 | 1/sec|\n", "\n", "We have now clearly defined all of our parameters, updates, and propensities. With the initial population, out simulation is not completely defined. In practice, we recommend constructing tables like this (and including them in your publications!) if you are going to do Gillespie simulations in your work.\n", "\n", "We are now tasked with coding up the propensities and the updates. Starting with the propensities, we recall that the biocircuits module requires a call signature of `propensity(propensities, population, t, *args)`, where the `args` are the parameters. We find that it is clearest to list the arguments one at a time in the function definition. It is also much clearer to unpack the population into individual variables than to use indexing. Finally, when returning the array of propensities, we recommend having one propensity for each line indexed in order." ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "@numba.njit\n", "def repressilator_propensity(\n", " propensities, population, t, kmu, kmo, kp, gamma_m, gamma_p, kr, ku1, ku2\n", "):\n", " m1, p1, m2, p2, m3, p3, n1, n2, n3 = population\n", "\n", " propensities[0] = kmu if n3 == 0 else kmo\n", " propensities[1] = kmu if n1 == 0 else kmo\n", " propensities[2] = kmu if n2 == 0 else kmo\n", " propensities[3] = kp * m1\n", " propensities[4] = kp * m2\n", " propensities[5] = kp * m3\n", " propensities[6] = gamma_m * m1\n", " propensities[7] = gamma_m * m2\n", " propensities[8] = gamma_m * m3\n", " propensities[9] = gamma_p * p1\n", " propensities[10] = gamma_p * p2\n", " propensities[11] = gamma_p * p3\n", " propensities[12] = gamma_p * n1\n", " propensities[13] = gamma_p * n2\n", " propensities[14] = gamma_p * n3\n", " propensities[15] = kr * p3 * (n3 < 2)\n", " propensities[16] = kr * p1 * (n1 < 2)\n", " propensities[17] = kr * p2 * (n2 < 2)\n", " propensities[18] = ku1 * (n3 == 1) + 2 * ku2 * (n3 == 2)\n", " propensities[19] = ku1 * (n1 == 1) + 2 * ku2 * (n1 == 2)\n", " propensities[20] = ku1 * (n2 == 1) + 2 * ku2 * (n2 == 2)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now, we can code up the update. The update is a matrix where entry `i,j` is the change in species `i` due to move `j`. Since we have indexes both the species and the moves (in the update/propensity table), we can include the indices when we define the update for clarity." ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [], "source": [ "repressilator_update = np.array(\n", " [ # 0 1 2 3 4 5 6 7 8\n", " [ 1, 0, 0, 0, 0, 0, 0, 0, 0], # 0\n", " [ 0, 0, 1, 0, 0, 0, 0, 0, 0], # 1\n", " [ 0, 0, 0, 0, 1, 0, 0, 0, 0], # 2\n", " [ 0, 1, 0, 0, 0, 0, 0, 0, 0], # 3\n", " [ 0, 0, 0, 1, 0, 0, 0, 0, 0], # 4\n", " [ 0, 0, 0, 0, 0, 1, 0, 0, 0], # 5\n", " [-1, 0, 0, 0, 0, 0, 0, 0, 0], # 6\n", " [ 0, 0, -1, 0, 0, 0, 0, 0, 0], # 7\n", " [ 0, 0, 0, 0, -1, 0, 0, 0, 0], # 8\n", " [ 0, -1, 0, 0, 0, 0, 0, 0, 0], # 9\n", " [ 0, 0, 0, -1, 0, 0, 0, 0, 0], # 10\n", " [ 0, 0, 0, 0, 0, -1, 0, 0, 0], # 11\n", " [ 0, 0, 0, 0, 0, 0, -1, 0, 0], # 12\n", " [ 0, 0, 0, 0, 0, 0, 0, -1, 0], # 13\n", " [ 0, 0, 0, 0, 0, 0, 0, 0, -1], # 14\n", " [ 0, 0, 0, 0, 0, -1, 0, 0, 1], # 15\n", " [ 0, -1, 0, 0, 0, 0, 1, 0, 0], # 16\n", " [ 0, 0, 0, -1, 0, 0, 0, 1, 0], # 17\n", " [ 0, 0, 0, 0, 0, 1, 0, 0, -1], # 18\n", " [ 0, 1, 0, 0, 0, 0, -1, 0, 0], # 19\n", " [ 0, 0, 0, 1, 0, 0, 0, -1, 0], # 20\n", " ],\n", " dtype=int)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Next, we specify the parameter values according to the parameter table. Remember that we need to package them in a tuple, after defining them." ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [], "source": [ "# Parameter values\n", "kmu = 0.5\n", "kmo = 5e-4\n", "kp = 0.167\n", "gamma_m = 0.005776\n", "gamma_p = 0.001155\n", "kr = 1.0\n", "ku1 = 224.0\n", "ku2 = 9.0\n", "\n", "repressilator_args = (kmu, kmo, kp, gamma_m, gamma_p, kr, ku1, ku2)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Finally, we specify the initial population and the time points we want for the sampling." ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [], "source": [ "# State with 10 copies of everything, nothing bound to operators\n", "repressilator_pop_0 = np.array([10, 10, 10, 10, 10, 10, 0, 0, 0], dtype=np.int64)\n", "\n", "repressilator_time_points = np.linspace(0, 80_000, 4001)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "And we are all set to perform the calculation. We will make a single trace and then plot it." ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "
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\"},\"shape\":[4001],\"dtype\":\"int32\",\"order\":\"little\"}]]}}},\"view\":{\"type\":\"object\",\"name\":\"CDSView\",\"id\":\"p1073\",\"attributes\":{\"filter\":{\"type\":\"object\",\"name\":\"AllIndices\",\"id\":\"p1074\"}}},\"glyph\":{\"type\":\"object\",\"name\":\"Line\",\"id\":\"p1069\",\"attributes\":{\"x\":{\"type\":\"field\",\"field\":\"x\"},\"y\":{\"type\":\"field\",\"field\":\"y\"},\"line_color\":\"#2ca02c\"}},\"nonselection_glyph\":{\"type\":\"object\",\"name\":\"Line\",\"id\":\"p1070\",\"attributes\":{\"x\":{\"type\":\"field\",\"field\":\"x\"},\"y\":{\"type\":\"field\",\"field\":\"y\"},\"line_color\":\"#2ca02c\",\"line_alpha\":0.1}},\"muted_glyph\":{\"type\":\"object\",\"name\":\"Line\",\"id\":\"p1071\",\"attributes\":{\"x\":{\"type\":\"field\",\"field\":\"x\"},\"y\":{\"type\":\"field\",\"field\":\"y\"},\"line_color\":\"#2ca02c\",\"line_alpha\":0.2}}}}],\"toolbar\":{\"type\":\"object\",\"name\":\"Toolbar\",\"id\":\"p1008\",\"attributes\":{\"tools\":[{\"type\":\"object\",\"name\":\"PanTool\",\"id\":\"p1033\"},{\"type\":\"object\",\"name\":\"WheelZoomTool\",\"id\":\"p1034\"},{\"type\":\"object\",\"name\":\"BoxZoomTool\",\"id\":\"p1035\",\"attributes\":{\"overlay\":{\"type\":\"object\",\"name\":\"BoxAnnotation\",\"id\":\"p1036\",\"attributes\":{\"syncable\":false,\"level\":\"overlay\",\"visible\":false,\"left_units\":\"canvas\",\"right_units\":\"canvas\",\"bottom_units\":\"canvas\",\"top_units\":\"canvas\",\"line_color\":\"black\",\"line_alpha\":1.0,\"line_width\":2,\"line_dash\":[4,4],\"fill_color\":\"lightgrey\",\"fill_alpha\":0.5}}}},{\"type\":\"object\",\"name\":\"SaveTool\",\"id\":\"p1037\"},{\"type\":\"object\",\"name\":\"ResetTool\",\"id\":\"p1038\"},{\"type\":\"object\",\"name\":\"HelpTool\",\"id\":\"p1039\"}]}},\"left\":[{\"type\":\"object\",\"name\":\"LinearAxis\",\"id\":\"p1026\",\"attributes\":{\"ticker\":{\"type\":\"object\",\"name\":\"BasicTicker\",\"id\":\"p1028\",\"attributes\":{\"mantissas\":[1,2,5]}},\"formatter\":{\"type\":\"object\",\"name\":\"BasicTickFormatter\",\"id\":\"p1029\"},\"axis_label\":\"protein copy number\",\"major_label_policy\":{\"type\":\"object\",\"name\":\"AllLabels\",\"id\":\"p1027\"}}}],\"below\":[{\"type\":\"object\",\"name\":\"LinearAxis\",\"id\":\"p1019\",\"attributes\":{\"ticker\":{\"type\":\"object\",\"name\":\"BasicTicker\",\"id\":\"p1021\",\"attributes\":{\"mantissas\":[1,2,5]}},\"formatter\":{\"type\":\"object\",\"name\":\"BasicTickFormatter\",\"id\":\"p1022\"},\"axis_label\":\"time (min)\",\"major_label_policy\":{\"type\":\"object\",\"name\":\"AllLabels\",\"id\":\"p1020\"}}}],\"center\":[{\"type\":\"object\",\"name\":\"Grid\",\"id\":\"p1025\",\"attributes\":{\"axis\":{\"id\":\"p1019\"}}},{\"type\":\"object\",\"name\":\"Grid\",\"id\":\"p1032\",\"attributes\":{\"dimension\":1,\"axis\":{\"id\":\"p1026\"}}}]}}],\"callbacks\":{\"type\":\"map\"}}};\n", " const render_items = [{\"docid\":\"8215192c-d669-4190-b132-d97a4dca7430\",\"roots\":{\"p1002\":\"effeeaba-e023-45d9-90ad-cc83d761ec95\"},\"root_ids\":[\"p1002\"]}];\n", " root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n", " }\n", " if (root.Bokeh !== undefined) {\n", " embed_document(root);\n", " } else {\n", " let attempts = 0;\n", " const timer = setInterval(function(root) {\n", " if (root.Bokeh !== undefined) {\n", " clearInterval(timer);\n", " embed_document(root);\n", " } else {\n", " attempts++;\n", " if (attempts > 100) {\n", " clearInterval(timer);\n", " console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\");\n", " }\n", " }\n", " }, 10, root)\n", " }\n", "})(window);" ], "application/vnd.bokehjs_exec.v0+json": "" }, "metadata": { "application/vnd.bokehjs_exec.v0+json": { "id": "p1002" } }, "output_type": "display_data" } ], "source": [ "# Perform the Gillespie simulation\n", "pop = biocircuits.gillespie_ssa(\n", " repressilator_propensity,\n", " repressilator_update,\n", " repressilator_pop_0,\n", " repressilator_time_points,\n", " args=repressilator_args,\n", ")\n", "\n", "\n", "# Make plot\n", "colors = bokeh.palettes.d3[\"Category10\"][3]\n", "p = bokeh.plotting.figure(\n", " height=250,\n", " width=600,\n", " x_range=[0, 80_000 / 60],\n", " x_axis_label=\"time (min)\",\n", " y_axis_label=\"protein copy number\",\n", ")\n", "for c, i in enumerate([1, 3, 5]):\n", " p.line(repressilator_time_points / 60, pop[0, :, i], color=colors[c])\n", "\n", "bokeh.io.show(p)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We observe the oscillations, but the amplitudes are highly variable." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Increasing speed\n", "\n", "The calculation we implemented without JITting is much slower than the version in the biocircuits package. Still, the algorithm in that package implements a rather naive version of the Gillespie algorithm.\n", "\n", "An obvious speed improvement can be made by only recalculating the propensity for copy numbers we have not visited. For example, for simple gene expression, we do not need to recompute the propensity for mRNA decay if the previous move was a protein decay. The propensity for mRNA decay is the same it was at the previous step. Gibson and Bruck developed the [next reaction method](http://doi.org/10.1021/jp993732q), which makes these improvements, among others, and can result in significant speed-up for complicated sets of reactions. Instead of wading into the algorithmic details, we will instead investigate how we can speed up our implementation of the direct Gillespie algorithm. (These speed boosts are all implemented in the biocircuits package.)\n", "\n", "For simplicity, we will again consider single gene expression,\n", "\n", "\\begin{align}\n", "\\text{DNA} \\rightarrow \\text{mRNA} \\rightarrow \\text{protein}\n", "\\end{align}\n", "\n", "The reactions and propensities are then\n", "\n", "\\begin{align}\n", "\\begin{array}{ll}\n", "\\text{reaction, }r_i & \\text{propensity, } a_i \\\\\n", "m \\rightarrow m+1,\\;\\;\\;\\; & \\beta_m \\\\[0.3em]\n", "m \\rightarrow m-1, \\;\\;\\;\\; & m\\\\[0.3em]\n", "p \\rightarrow p+1, \\;\\;\\;\\; & \\beta_p m \\\\[0.3em]\n", "p \\rightarrow p-1, \\;\\;\\;\\; & \\gamma p.\n", "\\end{array}\n", "\\end{align}\n", "\n", "We will first remake the non-JITted version of the propensity function to test its speed." ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [], "source": [ "def simple_propensity(propensities, population, t, beta_m, beta_p, gamma):\n", " \"\"\"Updates an array of propensities given a set of parameters\n", " and an array of populations.\n", " \"\"\"\n", " # Unpack population\n", " m, p = population\n", "\n", " # Update propensities\n", " propensities[0] = beta_m # Make mRNA transcript\n", " propensities[1] = m # Degrade mRNA\n", " propensities[2] = beta_p * m # Make protein\n", " propensities[3] = gamma * p # Degrade protein" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We also need the updates." ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [], "source": [ "# Column 0 is change in m, column 1 is change in p\n", "simple_update = np.array(\n", " [\n", " [1, 0], # Make mRNA transcript\n", " [-1, 0], # Degrade mRNA\n", " [0, 1], # Make protein\n", " [0, -1], # Degrade protein\n", " ],\n", " dtype=np.int64,\n", ")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We also need functions from [when we introduced the Gillespie algorithm](16a_gillespie.ipynb) to enable non-JITted profiling and to provide a basis of comparison going forward. These function are are in the cell below." ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "tags": [ "hide-input" ] }, "outputs": [], "source": [ "def sample_discrete(probs):\n", " \"\"\"Randomly sample an index with probability given by probs.\"\"\"\n", " # Generate random number\n", " q = np.random.rand()\n", "\n", " # Find index\n", " i = 0\n", " p_sum = 0.0\n", " while p_sum < q:\n", " p_sum += probs[i]\n", " i += 1\n", " \n", " return i - 1\n", "\n", "\n", "def gillespie_draw(propensity_func, propensities, population, t, args=()):\n", " \"\"\"\n", " Draws a reaction and the time it took to do that reaction.\n", "\n", " Parameters\n", " ----------\n", " propensity_func : function\n", " Function with call signature propensity_func(population, t, *args)\n", " used for computing propensities. This function must return\n", " an array of propensities.\n", " propensities : ndarray\n", " Propensities for each reaction as a 1D Numpy array.\n", " population : ndarray\n", " Current population of particles\n", " t : float\n", " Value of the current time.\n", " args : tuple, default ()\n", " Arguments to be passed to `propensity_func`.\n", "\n", " Returns\n", " -------\n", " rxn : int\n", " Index of reaction that occured.\n", " time : float\n", " Time it took for the reaction to occur.\n", " \"\"\"\n", " # Compute propensities\n", " propensity_func(propensities, population, t, *args)\n", "\n", " # Sum of propensities\n", " props_sum = propensities.sum()\n", "\n", " # Compute next time\n", " time = np.random.exponential(1.0 / props_sum)\n", "\n", " # Compute discrete probabilities of each reaction\n", " rxn_probs = propensities / props_sum\n", "\n", " # Draw reaction from this distribution\n", " rxn = sample_discrete(rxn_probs)\n", "\n", " return rxn, time\n", "\n", "\n", "def gillespie_ssa(propensity_func, update, population_0, time_points, args=()):\n", " \"\"\"\n", " Uses the Gillespie stochastic simulation algorithm to sample\n", " from probability distribution of particle counts over time.\n", "\n", " Parameters\n", " ----------\n", " propensity_func : function\n", " Function of the form f(params, t, population) that takes the current\n", " population of particle counts and return an array of propensities\n", " for each reaction.\n", " update : ndarray, shape (num_reactions, num_chemical_species)\n", " Entry i, j gives the change in particle counts of species j\n", " for chemical reaction i.\n", " population_0 : array_like, shape (num_chemical_species)\n", " Array of initial populations of all chemical species.\n", " time_points : array_like, shape (num_time_points,)\n", " Array of points in time for which to sample the probability\n", " distribution.\n", " args : tuple, default ()\n", " The set of parameters to be passed to propensity_func.\n", "\n", " Returns\n", " -------\n", " sample : ndarray, shape (num_time_points, num_chemical_species)\n", " Entry i, j is the count of chemical species j at time\n", " time_points[i].\n", " \"\"\"\n", "\n", " # Initialize output\n", " pop_out = np.empty((len(time_points), update.shape[1]), dtype=np.int64)\n", "\n", " # Initialize and perform simulation\n", " i_time = 1\n", " i = 0\n", " t = time_points[0]\n", " population = population_0.copy()\n", " pop_out[0, :] = population\n", " propensities = np.zeros(update.shape[0])\n", " while i < len(time_points):\n", " while t < time_points[i_time]:\n", " # draw the event and time step\n", " event, dt = gillespie_draw(\n", " propensity_func, propensities, population, t, args\n", " )\n", "\n", " # Update the population\n", " population_previous = population.copy()\n", " population += update[event, :]\n", "\n", " # Increment time\n", " t += dt\n", "\n", " # Update the index\n", " i = np.searchsorted(time_points > t, True)\n", "\n", " # Update the population\n", " pop_out[i_time : min(i, len(time_points))] = population_previous\n", "\n", " # Increment index\n", " i_time = i\n", "\n", " return pop_out" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let's do some profiling to see what took so long. We will use the magic function `%lprun` to profile runs of SSAs. The output has line wrapping, so it is kind of hard to read in a Jupyter notebook, unfortunately." ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "\n", "*** Profile printout saved to text file 'lp_results.txt'. \n" ] }, { "data": { "text/plain": [ "Timer unit: 1e-06 s\n", "\n", "Total time: 0.305683 s\n", "File: /var/folders/j_/c5r9ch0913v3h1w4bdwzm0lh0000gn/T/ipykernel_22560/1669721992.py\n", "Function: gillespie_ssa at line 60\n", "\n", "Line # Hits Time Per Hit % Time Line Contents\n", "==============================================================\n", " 60 def gillespie_ssa(propensity_func, update, population_0, time_points, args=()):\n", " 61 \"\"\"\n", " 62 Uses the Gillespie stochastic simulation algorithm to sample\n", " 63 from probability distribution of particle counts over time.\n", " 64 \n", " 65 Parameters\n", " 66 ----------\n", " 67 propensity_func : function\n", " 68 Function of the form f(params, t, population) that takes the current\n", " 69 population of particle counts and return an array of propensities\n", " 70 for each reaction.\n", " 71 update : ndarray, shape (num_reactions, num_chemical_species)\n", " 72 Entry i, j gives the change in particle counts of species j\n", " 73 for chemical reaction i.\n", " 74 population_0 : array_like, shape (num_chemical_species)\n", " 75 Array of initial populations of all chemical species.\n", " 76 time_points : array_like, shape (num_time_points,)\n", " 77 Array of points in time for which to sample the probability\n", " 78 distribution.\n", " 79 args : tuple, default ()\n", " 80 The set of parameters to be passed to propensity_func.\n", " 81 \n", " 82 Returns\n", " 83 -------\n", " 84 sample : ndarray, shape (num_time_points, num_chemical_species)\n", " 85 Entry i, j is the count of chemical species j at time\n", " 86 time_points[i].\n", " 87 \"\"\"\n", " 88 \n", " 89 # Initialize output\n", " 90 1 18.0 18.0 0.0 pop_out = np.empty((len(time_points), update.shape[1]), dtype=np.int64)\n", " 91 \n", " 92 # Initialize and perform simulation\n", " 93 1 1.0 1.0 0.0 i_time = 1\n", " 94 1 1.0 1.0 0.0 i = 0\n", " 95 1 2.0 2.0 0.0 t = time_points[0]\n", " 96 1 16.0 16.0 0.0 population = population_0.copy()\n", " 97 1 7.0 7.0 0.0 pop_out[0, :] = population\n", " 98 1 3.0 3.0 0.0 propensities = np.zeros(update.shape[0])\n", " 99 101 89.0 0.9 0.0 while i < len(time_points):\n", " 100 9926 8038.0 0.8 2.6 while t < time_points[i_time]:\n", " 101 # draw the event and time step\n", " 102 19652 225373.0 11.5 73.7 event, dt = gillespie_draw(\n", " 103 9826 4122.0 0.4 1.3 propensity_func, propensities, population, t, args\n", " 104 )\n", " 105 \n", " 106 # Update the population\n", " 107 9826 38544.0 3.9 12.6 population_previous = population.copy()\n", " 108 9826 21535.0 2.2 7.0 population += update[event, :]\n", " 109 \n", " 110 # Increment time\n", " 111 9826 5533.0 0.6 1.8 t += dt\n", " 112 \n", " 113 # Update the index\n", " 114 100 1711.0 17.1 0.6 i = np.searchsorted(time_points > t, True)\n", " 115 \n", " 116 # Update the population\n", " 117 100 637.0 6.4 0.2 pop_out[i_time : min(i, len(time_points))] = population_previous\n", " 118 \n", " 119 # Increment index\n", " 120 100 53.0 0.5 0.0 i_time = i\n", " 121 \n", " 122 1 0.0 0.0 0.0 return pop_out" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Specify parameters for calculation\n", "args = (10.0, 10.0, 0.4)\n", "time_points = np.linspace(0, 50, 101)\n", "population_0 = np.array([0, 0], dtype=int)\n", "\n", "%lprun -T lp_results.txt -f gillespie_ssa gillespie_ssa(\\\n", " simple_propensity, simple_update, population_0, time_points, args)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We see that 80% of the time is spent doing draws. Nothing else is really worth looking at. Let's see how we can improve our draw speed." ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "\n", "*** Profile printout saved to text file 'lp_results.txt'. \n" ] }, { "data": { "text/plain": [ "Timer unit: 1e-06 s\n", "\n", "Total time: 0.207239 s\n", "File: /var/folders/j_/c5r9ch0913v3h1w4bdwzm0lh0000gn/T/ipykernel_22560/1669721992.py\n", "Function: gillespie_draw at line 16\n", "\n", "Line # Hits Time Per Hit % Time Line Contents\n", "==============================================================\n", " 16 def gillespie_draw(propensity_func, propensities, population, t, args=()):\n", " 17 \"\"\"\n", " 18 Draws a reaction and the time it took to do that reaction.\n", " 19 \n", " 20 Parameters\n", " 21 ----------\n", " 22 propensity_func : function\n", " 23 Function with call signature propensity_func(population, t, *args)\n", " 24 used for computing propensities. This function must return\n", " 25 an array of propensities.\n", " 26 propensities : ndarray\n", " 27 Propensities for each reaction as a 1D Numpy array.\n", " 28 population : ndarray\n", " 29 Current population of particles\n", " 30 t : float\n", " 31 Value of the current time.\n", " 32 args : tuple, default ()\n", " 33 Arguments to be passed to `propensity_func`.\n", " 34 \n", " 35 Returns\n", " 36 -------\n", " 37 rxn : int\n", " 38 Index of reaction that occured.\n", " 39 time : float\n", " 40 Time it took for the reaction to occur.\n", " 41 \"\"\"\n", " 42 # Compute propensities\n", " 43 10000 80861.0 8.1 39.0 propensity_func(propensities, population, t, *args)\n", " 44 \n", " 45 # Sum of propensities\n", " 46 10000 43009.0 4.3 20.8 props_sum = propensities.sum()\n", " 47 \n", " 48 # Compute next time\n", " 49 10000 21411.0 2.1 10.3 time = np.random.exponential(1.0 / props_sum)\n", " 50 \n", " 51 # Compute discrete probabilities of each reaction\n", " 52 10000 21464.0 2.1 10.4 rxn_probs = propensities / props_sum\n", " 53 \n", " 54 # Draw reaction from this distribution\n", " 55 10000 35997.0 3.6 17.4 rxn = sample_discrete(rxn_probs)\n", " 56 \n", " 57 10000 4497.0 0.4 2.2 return rxn, time" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "propensities = np.ones(4)\n", "\n", "%lprun -T lp_results.txt -f gillespie_draw \\\n", " [gillespie_draw(simple_propensity, propensities, population_0, 0, args) \\\n", " for _ in range(10_000)]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The propensity function is taking the most time. We will focus on improving that." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Speed boost by JIT compilation with Numba\n", "\n", "As we have seen, [Numba](https://numba.pydata.org) is a package that does LLVM optimized just-in-time compilation of Python code. The speed-ups can be substantial. We will use `numba` to compile the parts of the code that we can. For many functions, we just need to decorate the function with\n", "\n", " @numba.jit()\n", "\n", "and that is it. If possible, we can use the `nopython=True` keyword argument to get more speed because the compiler can assume that all of the code is JIT-able. Note that using @numba.njit is equivalent to using @numba.jit(nopython=True).\n", "\n", "We can speed that up by sacrificing some readability and directly accessing the arrays instead of setting up views, as we have been doing." ] }, { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Old propensity function:\n", "4.33 µs ± 17.9 ns per loop (mean ± std. dev. of 7 runs, 100,000 loops each)\n", "\n", "Numba'd propensity function:\n", "576 ns ± 2.73 ns per loop (mean ± std. dev. of 7 runs, 1,000,000 loops each)\n" ] } ], "source": [ "@numba.njit\n", "def simple_propensity_numba(\n", " propensities, population, t, beta_m, beta_p, gamma\n", "):\n", " \"\"\"Updates an array of propensities given a set of parameters\n", " and an array of populations.\n", " \"\"\"\n", " # Unpack population\n", " m, p = population\n", "\n", " # Update propensities\n", " propensities[0] = beta_m # Make mRNA transcript\n", " propensities[1] = m # Degrade mRNA\n", " propensities[2] = beta_p * m # Make protein\n", " propensities[3] = gamma * p # Degrade protein\n", "\n", " \n", "# Check speeds\n", "print('Old propensity function:')\n", "%timeit simple_propensity(propensities, population_0, 0, *args)\n", "\n", "print('\\nNumba\\'d propensity function:')\n", "%timeit simple_propensity_numba(propensities, population_0, 0, *args)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We got some speed up! Nice! \n", "\n", "Now, we also saw that the sums and division of arrays are slow. Let's optimize the sum operation using `numba`." ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "\n", "NumPy sum:\n", "1.8 µs ± 5.97 ns per loop (mean ± std. dev. of 7 runs, 1,000,000 loops each)\n", "\n", "Numba sum:\n", "243 ns ± 0.4 ns per loop (mean ± std. dev. of 7 runs, 1,000,000 loops each)\n" ] } ], "source": [ "@numba.njit\n", "def sum_numba(ar):\n", " return ar.sum()\n", "\n", "\n", "# Make dummy array for testing\n", "ar = np.array([0.3, 0.4, 0.3, 0.2, 0.15])\n", "\n", "print('\\nNumPy sum:')\n", "%timeit ar.sum()\n", "\n", "print('\\nNumba sum:')\n", "%timeit sum_numba(ar)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We get another speed boost, though we should note that this speed boost in the sum is due to `numba` optimizing sums of a certain size. For sums over large numbers of entries, `numba`'s performance will not exceed NumPy's by much at all.\n", "\n", "Finally, we will speed up the sampling of the discrete distribution. We will do this in two ways. First, we notice that the division operation on the propensities took a fair amount of time when we profiled the code. We do not need it; we can instead sample from an unnormalized discrete distribution. Secondly, we can use `numba` to accelerate the while loop in the sampling." ] }, { "cell_type": "code", "execution_count": 14, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Result from non-JITted method:\n", "1.06 µs ± 3.3 ns per loop (mean ± std. dev. of 7 runs, 1,000,000 loops each)\n", "\n", "Results from numba'd version:\n", "298 ns ± 0.296 ns per loop (mean ± std. dev. of 7 runs, 1,000,000 loops each)\n" ] } ], "source": [ "@numba.njit\n", "def sample_discrete_numba(probs, probs_sum):\n", " q = np.random.rand() * probs_sum\n", "\n", " i = 0\n", " p_sum = 0.0\n", " while p_sum < q:\n", " p_sum += probs[i]\n", " i += 1\n", " \n", " return i - 1\n", "\n", "\n", "# Make dummy unnormalized probs\n", "probs = np.array([0.1, 0.3, 0.4, 0.05, 0.15, 0.6])\n", "probs_sum = sum_numba(probs)\n", "\n", "print('Result from non-JITted method:')\n", "%timeit sample_discrete(probs)\n", "\n", "print(\"\\nResults from numba'd version:\")\n", "%timeit sample_discrete_numba(probs, probs_sum)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We get a speed-up of about a factor of three. Let's now make a new `gillespie_draw()` function that is faster. The fast propensity function is just an argument to the fast draw-er." ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Old Gillespie draw:\n", "12.6 µs ± 77.4 ns per loop (mean ± std. dev. of 7 runs, 100,000 loops each)\n", "\n", "Fast Gillespie draw:\n", "1.92 µs ± 11.1 ns per loop (mean ± std. dev. of 7 runs, 1,000,000 loops each)\n" ] } ], "source": [ "def gillespie_draw_fast(propensity_func, propensities, population, t, args):\n", " \"\"\"\n", " Draws a reaction and the time it took to do that reaction.\n", " \"\"\"\n", " # Compute propensities\n", " propensity_func(propensities, population, t, *args)\n", " \n", " # Sum of propensities\n", " props_sum = sum_numba(propensities)\n", " \n", " # Compute time\n", " time = np.random.exponential(1 / props_sum)\n", "\n", " # Draw reaction given propensities\n", " rxn = sample_discrete_numba(propensities, props_sum)\n", " \n", " return rxn, time\n", "\n", "\n", "print('Old Gillespie draw:')\n", "%timeit gillespie_draw(simple_propensity, propensities, population_0, 0.0, args)\n", "\n", "print('\\nFast Gillespie draw:')\n", "%timeit gillespie_draw_fast(simple_propensity_numba, propensities, population_0, 0.0, args)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "So, our optimization got us another speed boost. Let's adjust our SSA function to include the fast draws." ] }, { "cell_type": "code", "execution_count": 16, "metadata": {}, "outputs": [], "source": [ "def gillespie_ssa_fast(\n", " propensity_func, update, population_0, time_points, args=()\n", "):\n", " \"\"\"\n", " Uses the Gillespie stochastic simulation algorithm to sample\n", " from proability distribution of particle counts over time.\n", " \n", " Parameters\n", " ----------\n", " propensity_func : function\n", " Function of the form f(params, population) that takes the current\n", " population of particle counts and return an array of propensities\n", " for each reaction.\n", " update : ndarray, shape (num_reactions, num_chemical_species)\n", " Entry i, j gives the change in particle counts of species j\n", " for chemical reaction i.\n", " population_0 : array_like, shape (num_chemical_species)\n", " Array of initial populations of all chemical species.\n", " time_points : array_like, shape (num_time_points,)\n", " Array of points in time for which to sample the probability\n", " distribution.\n", " args : tuple, default ()\n", " The set of parameters to be passed to propensity_func. \n", " \n", " Returns\n", " -------\n", " rxn : int\n", " Index of reaction that occured.\n", " time : float\n", " Time it took for the reaction to occur. \n", " Returns\n", " -------\n", " sample : ndarray, shape (num_time_points, num_chemical_species)\n", " Entry i, j is the count of chemical species j at time\n", " time_points[i].\n", " \"\"\"\n", "\n", " # Initialize output\n", " pop_out = np.empty((len(time_points), update.shape[1]), dtype=np.int64)\n", "\n", " # Initialize and perform simulation\n", " i_time = 1\n", " i = 0\n", " t = time_points[0]\n", " population = population_0.copy()\n", " pop_out[0, :] = population\n", " propensities = np.zeros(update.shape[0])\n", " while i < len(time_points):\n", " while t < time_points[i_time]:\n", " # draw the event and time step\n", " event, dt = gillespie_draw_fast(\n", " propensity_func, propensities, population, t, args\n", " )\n", "\n", " # Update the population\n", " population_previous = population.copy()\n", " population += update[event, :]\n", "\n", " # Increment time\n", " t += dt\n", "\n", " # Update the index\n", " i = np.searchsorted(time_points > t, True)\n", "\n", " # Update the population\n", " pop_out[i_time : min(i, len(time_points))] = population_previous\n", "\n", " # Increment index\n", " i_time = i\n", "\n", " return pop_out" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now we can test the speed of the two SSAs." ] }, { "cell_type": "code", "execution_count": 17, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Gillespie SSA:\n", "198 ms ± 9.57 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)\n", "\n", "Fast Gillespie SSA:\n", "41.8 ms ± 985 µs per loop (mean ± std. dev. of 7 runs, 10 loops each)\n" ] } ], "source": [ "print('Gillespie SSA:')\n", "%timeit gillespie_ssa(simple_propensity, simple_update, \\\n", " population_0, time_points, args)\n", "\n", "print('\\nFast Gillespie SSA:')\n", "%timeit gillespie_ssa_fast(simple_propensity_numba, simple_update,\\\n", " population_0, time_points, args)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "So, we are now faster with not too much work. This is still a general solver that you can use with any propensity function and update.\n", "\n", "We have constructed a generic tool for doing Gillespie simulations. Specifically, we pass a propensity function into the algorithm. Passing a function as an argument precludes use of `numba`. This means that we cannot just-in-time compile the entire Gillespie simulation. We could insist that our propensity function be encoded in a global function `prop_func()`, then we can fully JIT compile the entire simulation. (Note that there are ways to get around this insistence on a global function, and the `biocircuits.gillespie_ssa()` function does not rely on such a global function, but for the purposes of this demonstration, it is convenient.)" ] }, { "cell_type": "code", "execution_count": 18, "metadata": {}, "outputs": [], "source": [ "@numba.njit\n", "def prop_func(propensities, population, t, beta_m, beta_p, gamma):\n", " \"\"\"Updates an array of propensities given a set of parameters\n", " and an array of populations.\n", " \"\"\"\n", " # Unpack population\n", " m, p = population\n", " \n", " # Update propensities\n", " propensities[0] = beta_m # Make mRNA transcript\n", " propensities[1] = m # Degrade mRNA\n", " propensities[2] = beta_p * m # Make protein\n", " propensities[3] = gamma * p # Degrade protein\n", "\n", "\n", "@numba.njit\n", "def gillespie_draw_numba(propensities, population, t, args):\n", " \"\"\"\n", " Draws a reaction and the time it took to do that reaction.\n", " \n", " Assumes that there is a globally scoped function\n", " `prop_func` that is Numba'd with nopython=True.\n", " \"\"\"\n", " # Compute propensities\n", " prop_func(propensities, population, t, *args)\n", "\n", " # Sum of propensities\n", " props_sum = np.sum(propensities)\n", " \n", " # Compute time\n", " time = np.random.exponential(1 / props_sum)\n", " \n", " # Draw reaction given propensities\n", " rxn = sample_discrete_numba(propensities, props_sum)\n", "\n", " return rxn, time\n", "\n", "\n", "@numba.njit\n", "def gillespie_ssa_numba(update, population_0, time_points, args):\n", " \"\"\"\n", " Uses the Gillespie stochastic simulation algorithm to sample\n", " from proability distribution of particle counts over time.\n", " \n", " Parameters\n", " ----------\n", " update : ndarray, shape (num_reactions, num_chemical_species)\n", " Entry i, j gives the change in particle counts of species j\n", " for chemical reaction i.\n", " population_0 : array_like, shape (num_chemical_species)\n", " Array of initial populations of all chemical species.\n", " time_points : array_like, shape (num_time_points,)\n", " Array of points in time for which to sample the probability\n", " distribution.\n", " args : tuple, default ()\n", " The set of parameters to be passed to propensity_func.\n", " \n", " Returns\n", " -------\n", " sample : ndarray, shape (num_time_points, num_chemical_species)\n", " Entry i, j is the count of chemical species j at time\n", " time_points[i].\n", " \n", " Notes\n", " -----\n", " .. Assumes that there is a globally scoped function\n", " `propensity_func` that is Numba'd with nopython=True.\n", " \"\"\"\n", " # Initialize output\n", " pop_out = np.empty((len(time_points), update.shape[1]), dtype=np.int64)\n", "\n", " # Initialize and perform simulation\n", " i_time = 1\n", " i = 0\n", " t = time_points[0]\n", " population = population_0.copy()\n", " pop_out[0,:] = population\n", " propensities = np.zeros(update.shape[0])\n", " while i < len(time_points):\n", " while t < time_points[i_time]:\n", " # draw the event and time step\n", " event, dt = gillespie_draw_numba(propensities, population, t, args)\n", " \n", " # Update the population\n", " population_previous = population.copy()\n", " population += update[event,:]\n", " \n", " # Increment time\n", " t += dt\n", "\n", " # Update the index (Have to be careful about types for Numba)\n", " i = np.searchsorted((time_points > t).astype(np.int64), 1)\n", "\n", " # Update the population\n", " for j in np.arange(i_time, min(i, len(time_points))):\n", " pop_out[j,:] = population_previous\n", " \n", " # Increment index\n", " i_time = i\n", " \n", " return pop_out" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now let's test the speed of all three of our functions." ] }, { "cell_type": "code", "execution_count": 19, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Gillespie SSA:\n", "165 ms ± 4.43 ms per loop (mean ± std. dev. of 7 runs, 10 loops each)\n", "\n", "Fast Gillespie SSA:\n", "41.5 ms ± 501 µs per loop (mean ± std. dev. of 7 runs, 10 loops each)\n", "\n", "Totally numba'd Gillespie SSA:\n", "1.9 ms ± 75.9 µs per loop (mean ± std. dev. of 7 runs, 1 loop each)\n" ] } ], "source": [ "print('Gillespie SSA:')\n", "%timeit gillespie_ssa(simple_propensity, simple_update, \\\n", " population_0, time_points, args)\n", "\n", "print('\\nFast Gillespie SSA:')\n", "%timeit gillespie_ssa_fast(simple_propensity_numba, simple_update,\\\n", " population_0, time_points, args)\n", "\n", "print('\\nTotally numba\\'d Gillespie SSA:')\n", "%timeit gillespie_ssa_numba(simple_update, population_0, time_points, args)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We got an extra order of magnitude speed boost by totally JIT compiling everything. The speed up is significant, so we should probably use Numba'd code, which is what the `biocircuits.gillespie_ssa()` function uses when possible." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Parallel Gillespie simulations\n", "\n", "Sampling by the Gillespie algorithm is trivially parallelizable. We can use the `multiprocessing` module to parallelize the computation. Syntactically, we need to specify a function that takes a single argument. Below, we set up a parallel calculation of Gillespie simulations for our specific example.\n", "\n", "\n", "
\n", "\n", "Note \n", "\n", "The `multiprocessing` module is part of the standard Python distribution. As of April 2022, [multiprocessing hangs in IPython and Jupyter environments](https://github.com/ipython/ipython/issues/12396). As an alternative, we will use the [multiprocess](https://pypi.org/project/multiprocess/) package. It can be installed by running `pip install multiprocess` on the command line. In the contexts in which we are doing parallel calculation, the `multiprocess` package can be used as a drop-in replacement for `multiprocessing`.\n", " \n", "
" ] }, { "cell_type": "code", "execution_count": 20, "metadata": {}, "outputs": [], "source": [ "def gillespie_fn(args):\n", " return gillespie_ssa_numba(*args)\n", " \n", " \n", "def gillespie_parallel(fn, update, population_0, time_points, args,\n", " n_simulations, n_threads):\n", " \"\"\"\n", " Convenience function to do parallel Gillespie simulations.\n", " \"\"\"\n", " input_args = (update, population_0, time_points, args)\n", "\n", " with multiprocess.Pool(n_threads) as p:\n", " populations = p.map(fn, [input_args]*n_simulations)\n", " \n", " return np.array(populations)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We are paying some overhead in setting up the parallelization. Let's time it to see how we do with two threads." ] }, { "cell_type": "code", "execution_count": 21, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "\n", "numba'd Gillespie SSA:\n", "1.96 s ± 3.29 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)\n", "\n", "Parallel numba'd Gillespie SSA:\n", "1.23 s ± 11.3 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)\n" ] } ], "source": [ "n_simulations = 1000\n", "\n", "print('\\nnumba\\'d Gillespie SSA:')\n", "%timeit [gillespie_ssa_numba(simple_update, population_0, time_points, args) \\\n", " for _ in range(n_simulations)]\n", "\n", "print('\\nParallel numba\\'d Gillespie SSA:')\n", "%timeit gillespie_parallel(gillespie_fn, simple_update, population_0, time_points,\\\n", " args, n_simulations, 2)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We get a factor of two speed boost from two cores. This brings our total speed boost from the optimization to nearly two orders of magnitude.\n", "\n", "If we insist on *exact* sampling out of a probability distribution defined by a master question, we can get significant speed boosts by switching to the [Gibson and Bruck algorithm](http://doi.org/10.1021/jp993732q), especially for more complicated systems and propensity functions. If we are willing to *approximately* sample out of the probability distribution, there are many fast, approximate methods (e.g., [Salis and Kaznessis](https://doi.org/10.1063%2F1.1835951)) available.\n", "\n", "The function `gillespie_ssa()` in the biocircuits package uses all of the speed boosts (and a few extra) that we discussed here. You will get the best performance if you pass a Numba'd propensity function." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "\n", "## References\n", "\n", "- Elowitz, M. B. and Leibler, S., A synthetic oscillatory network of transcriptional regulators, _Science_, 403, 355–338, 2000. ([link](https://doi.org/10.1038/35002125))\n", "- Gibson, M. A. and Bruck, J., Efficient exact stochastic simulation of chemical systems with many species and many channels, *J. Phys. Chem. A*, 104, 1876–1889, 2000. ([link](https://doi.org/10.1021/jp993732q))\n", "- Salis, H. and Kaznessis, Y., Accurate hybrid stochastic simulation of a system of coupled chemical or biochemical reactions, *J. Chem. Phys.*, 122, 054103, 2005. ([link](https://doi.org/10.1063/1.1835951))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Computing environment" ] }, { "cell_type": "code", "execution_count": 22, "metadata": { "tags": [ "hide-input" ] }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Python implementation: CPython\n", "Python version : 3.10.10\n", "IPython version : 8.12.0\n", "\n", "numpy : 1.23.5\n", "scipy : 1.10.1\n", "numba : 0.56.4\n", "tqdm : 4.65.0\n", "line_profiler: 3.3.1\n", "multiprocess : 0.70.14\n", "biocircuits : 0.1.11\n", "bokeh : 3.1.0\n", "jupyterlab : 3.5.3\n", "\n" ] } ], "source": [ "%load_ext watermark\n", "%watermark -v -p numpy,scipy,numba,tqdm,line_profiler,multiprocess,biocircuits,bokeh,jupyterlab" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3 (ipykernel)", "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": "3.10.10" }, "toc": { "base_numbering": 1, "nav_menu": {}, "number_sections": true, "sideBar": true, "skip_h1_title": false, "title_cell": "Table of Contents", "title_sidebar": "Contents", "toc_cell": false, "toc_position": {}, "toc_section_display": true, "toc_window_display": false }, "varInspector": { "cols": { "lenName": 16, "lenType": 16, "lenVar": 40 }, "kernels_config": { "python": { "delete_cmd_postfix": "", "delete_cmd_prefix": "del ", "library": "var_list.py", "varRefreshCmd": "print(var_dic_list())" }, "r": { "delete_cmd_postfix": ") ", "delete_cmd_prefix": "rm(", "library": "var_list.r", "varRefreshCmd": "cat(var_dic_list()) " } }, "types_to_exclude": [ "module", "function", "builtin_function_or_method", "instance", "_Feature" ], "window_display": false } }, "nbformat": 4, "nbformat_minor": 4 }