12. Stochastic simulation of biological circuits¶
(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. All code contained herein is licensed under an MIT license.
This document was prepared at Caltech with financial support from the Donna and Benjamin M. Rosen Bioengineering Center.
This lesson was generated from a Jupyter notebook. Click the links below for other versions of this lesson.
Concepts¶
- Sampling out of probability distributions is a useful way to computational determine their properties.
- Master equations can equivalently be defined in terms of a set of moves and associated propensities and updates.
- The Gillespie algorithm enables sampling out of probability distributions described by master equations.
Techniques¶
- Application of the Gillespie algorithm.
- Profiling of code for optimization.
import multiprocessing
import tqdm
import numpy as np
import scipy.stats as st
import numba
import biocircuits
# Plotting modules
import bokeh.io
import bokeh.plotting
bokeh.io.output_notebook()
# Line profiler (can install with conda install line_profiler)
%load_ext line_profiler
Sampling out of master equations¶
For some stories, we know the probability distribution. For example, if the story is that we are doing $n$ independent coin flips, each with probability $p$ of landing heads, the number $h$ of heads is Binomially distributed. We could prove that this is the case, or look it up in a book. But sometimes, it is too difficult (or impossible) to prove what the distribution is, so we can numerically compute its properties by sampling out of the distribution. Sampling involved using a random number generator to simulate the story that generates the distribution. So, if you know the story and you have a computer handy, you can sample to be able to get a plot of your distribution, though you will not get the analytical form.
Let's demonstrate this with the Binomial distribution. We will take $n = 25$ and $p = 0.25$ and compute $P(h \mid n, p)$, the probability of getting $h$ heads in $n$ flips, each with probability $p$ of landing heads. We will draw 10, 30, 100, and 300 samples and plot them versus the expected Binomial distribution.
def simulate_coinflips(n, p, size=1):
"""
Simulate n_samples sets of n coin flips with prob. p of heads.
"""
n_heads = np.empty(size, dtype=int)
for i in range(size):
n_heads[i] = np.sum(np.random.random(size=n) < p)
return n_heads
size = (30, 100, 1000, 10000)
n = 25
p = 0.25
h_plot = np.arange(26)
theor_dist = st.binom.pmf(h_plot, n, p)
plots = []
for n_samp in size:
plot = bokeh.plotting.figure(plot_height=200,
plot_width=300,
x_axis_label='h',
y_axis_label='P(h)',
title=f'{n_samp} samples')
h = simulate_coinflips(n, p, size=n_samp)
plot.circle(h_plot, theor_dist)
plot.circle(np.arange(h.max()+1), np.bincount(h)/n_samp, color='orange')
plots.append(plot)
bokeh.io.show(bokeh.layouts.gridplot(plots, ncols=2))
As we can see, if we sample out of the probability distribution, we can approximately calculate the actual distribution. If we sample enough, the approximation is very good.
Sampling is such a powerful strategy that highly efficient algorithms with convenient APIs have been developed to sample out of named probability distributions. For example, we could have used np.random.binom() as a drop-in (and much more efficient) replacement for the simulate_coinflips() function above.
We will use the same strategy for solving master equations. We will find a way to sample out of the distribution that is governed by the master equation. This technique was pioneered by Dan Gillespie in the last 70s. For that reason, these sampling techniques are often called Gillespie simulations. The algorithm is sometimes referred to as a stochastic simulation algorithm, or SSA.
Here, we will explore how this algorithm works by looking at simple production of a protein.
The dynamical equations¶
For simple protein production, we have the following reactions.
\begin{align} \text{DNA} \rightarrow \text{mRNA} \rightarrow \text{protein} \end{align}Macroscale equations¶
As we've seen before, the deterministic dynamics, which describe mean concentrations over a large population of cells, are described by the ODEs
\begin{align} \frac{\mathrm{d}m}{\mathrm{d}t} &= \beta_m - \gamma_m m, \\[1em] \frac{\mathrm{d}p}{\mathrm{d}t} &= \beta_p m - \gamma_p p. \end{align}The same equations should hold if $m$ and $p$ represent the mean numbers of cells; we would just have to appropriately rescale the constants. Assuming the $m$ and $p$ are now numbers (so we are not free to pick their units), we can nondimensionalize using $\gamma_m$ to nondimensionalize time. This leads to redefinition of parameters and variables
\begin{align} &\beta_m/\gamma_m \to \beta_m, \\[1em] &\beta_p/\gamma_m \to \beta_p, \\[1em] &\gamma_m t \to t. \end{align}The dimensionless equations are
\begin{align} \frac{\mathrm{d}m}{\mathrm{d}t} &= \beta_m - m, \\[1em] \frac{\mathrm{d}p}{\mathrm{d}t} &= \beta_p m - \gamma p, \end{align}with $\gamma = \gamma_p/\gamma_m$.
The Master equation¶
We can write a master equation for these dynamics. In this case, each state is defined by an mRNA copy number $m$ and a protein copy number $p$. So, we will write a master equation for $P(m, p, t)$.
\begin{align} \frac{\mathrm{d}P(m,p,t)}{\mathrm{d}t} &= \beta_m P(m-1,p,t) + (m+1)P(m+1,p,t) - \beta_m P(m,p,t) - mP(m,p,t) \nonumber \\[1em] &+\beta_p mP(m,p-1,t) + \gamma_p (p+1)P(m,p+1,t) - \beta_p mP(m,p,t) - \gamma p P(m,p,t). \end{align}We implicitly define $P(m, p, t) = 0$ if $m < 0$ or $p < 0$. This is the master equation we will sample from using the stochastic simulation algorithm (SSA) or Gillespie algorithm.
The Gillespie algorithm¶
The transition probabilities are also called propensities in the context of stochastic simulation. The propensity for a given transition, say indexed $i$, is denoted as $a_i$. The equivalence to notation we introduced for master equations is that if transition $i$ results in the change of state from $n'$ to $n$, then $a_i = W(n\mid n')$.
To cast this problem for a Gillespie simulation, we can write each change of state (moving either the copy number of mRNA or protein up or down by 1 in this case) and their respective propensities.
\begin{align} \begin{array}{ll} \text{reaction, }r_i & \text{propensity, } a_i \\ m \rightarrow m+1,\;\;\;\; & \beta_m \\[0.3em] m \rightarrow m-1, \;\;\;\; & m\\[0.3em] p \rightarrow p+1, \;\;\;\; & \beta_p m \\[0.3em] p \rightarrow p-1, \;\;\;\; & \gamma p. \end{array} \end{align}We will not carefully prove that the Gillespie algorithm samples from the probability distribution governed by the master equation, but will state the principles behind it. The basic idea is that events (such as those outlined above) are rare, discrete, separate events. I.e., each event is an arrival of a Poisson process. The Gillespie algorithm starts with some state, $(m_0,p_0)$. Then a state change, any state change, will happen in some time $\Delta t$ that has a certain probability distribution (which we will show is Exponential momentarily). The probability that the state change that happens is reaction $j$ is proportional to $a_j$. That is to say, state changes with high propensities are more likely to occur. Thus, choosing which of the $n$ state changes happens in $\Delta t$ is a matter of drawing an integer $j$ in $[1,n]$ where the probability of drawing $j$ is
\begin{align} \frac{a_j}{\sum_i a_i}. \end{align}Now, how do we determine how long the state change took? The probability density function describing that a given state change $i$ takes place in time $t$ is
\begin{align} P(t\mid a_i) = a_i\, \mathrm{e}^{-a_i t}, \end{align}since the time it takes for arrival of a Poisson process is Exponentially distributed. The probability that it has not arrived in time $\Delta t$ is the probability that the arrival time is greater than $\Delta t$, given by the complementary cumulative distribution function for the Exponential distribution.
\begin{align} P(t > \Delta t\mid a_i) = \int_{\Delta t}^\infty \mathrm{d}t\,P(t\mid a_i) = \mathrm{e}^{-a_i \Delta t}. \end{align}Now, say we have $n$ processes that arrive in time $t_1, t_2, \ldots$. The probability that none of them arrive before $\Delta t$ is
\begin{align} P(t_1 > \Delta t, t_2 > \Delta t, \ldots) &= P(t_1 > \Delta t) P(t_2 > \Delta t) \cdots = \prod_i \mathrm{e}^{-a_i \Delta t} = \mathrm{exp}\left[-\Delta t \sum_i a_i\right]. \end{align}This is the same as the probability of a single Poisson process with $a = \sum_i a_i$ not arriving before $\Delta t$. So, the probability that it does arrive in $\Delta t$ is Exponentially distributed with mean $\left(\sum_i a_i\right)^{-1}$.
So, we know how to choose a state change and we also know how long it takes. The Gillespie algorithm then proceeds as follows.
- Choose an initial condition, e.g., $m = p = 0$.
- Calculate the propensity for each of the enumerated state changes. The propensities may be functions of $m$ and $p$, so they need to be recalculated for every $m$ and $p$ we encounter.
- Choose how much time the reaction will take by drawing out of an exponential distribution with a mean equal to $\left(\sum_i a_i\right.)^{-1}$. This means that a change arises from a Poisson process.
- Choose what state change will happen by drawing a sample out of the discrete distribution where $P_i = \left.a_i\middle/\left(\sum_i a_i\right)\right.$. In other words, the probability that a state change will be chosen is proportional to its propensity.
- Increment time by the time step you chose in step 3.
- Update the states according to the state change you choose in step 4.
- If $t$ is less than your pre-determined stopping time, go to step 2. Else stop.
Gillespie proved that this algorithm samples the probability distribution described by the master equation in his seminal papers in 197690041-3) and 1977. (We recommend reading the latter.) You can also read a concise discussion of how the algorithm samples the master equation in section 4.2 of Del Vecchio and Murray.
Coding up the Gillespie simulation¶
To code up the Gillespie simulation, we first make an an array that gives the changes in the counts of $m$ and $p$ for each of the four reactions. This is a way of encoding the updates in the particle counts that we get from choosing the respective state changes.
# Column 0 is change in m, column 1 is change in p
simple_update = np.array([[1, 0], # Make mRNA transcript
[-1, 0], # Degrade mRNA
[0, 1], # Make protein
[0, -1]], # Degrade protein
dtype=np.int)
Next, we make a function that updates the array of propensities for each of the four reactions. We update the propensities (which are passed into the function as an argument) instead of instantiating them and returning them to save on memory allocation while running the code. It has the added benefit that it forces you to keep track of the indices corresponding to the update matrix. This helps prevent bugs. It will naturally be a function of the current population of molecules. It may in general also be a function of time, so we explicitly allow for time dependence (even though we will not use it in this simple example) as well.
def simple_propensity(propensities, population, t, beta_m, beta_p, gamma):
"""Updates an array of propensities given a set of parameters
and an array of populations.
"""
# Unpack population
m, p = population
# Update propensities
propensities[0] = beta_m # Make mRNA transcript
propensities[1] = m # Degrade mRNA
propensities[2] = beta_p * m # Make protein
propensities[3] = gamma * p # Degrade protein
Making a draw¶
Finally, we write a general function that draws a choice of reaction and the time interval for that reaction. This is the heart of the Gillespie algorithm, so we will take some time to discuss speed. First, to get the time interval, we sample a random number from an exponential distribution with mean $\left(\sum_i a_i\right)^{-1}$. This is easily done using the np.random.exponential() function.
Next, we have to select which reaction will take place. This amounts to drawing a sample over the discrete distribution where $P_i = a_i\left(\sum_i a_i\right)^{-1}$, or the probability of each reaction is proportional to its propensity. This can be done using scipy.stats.rv_discrete, which allows specification of an arbitrary discrete distribution. We will write a function to do this.
def sample_discrete_scipy(probs):
"""Randomly sample an index with probability given by probs."""
return st.rv_discrete(values=(range(len(probs)), probs)).rvs()
This is a nice one-liner, but is it fast? There may be significant overhead in setting up the scipy.stats discrete random variable object to sample from each time. Remember, we can't just do this once because the array probs changes with each step in the SSA because the propensities change. We will therefore write a less elegant, but maybe faster way of doing it.
Another way to sample the distribution is to generate a uniformly distributed random number $q$, with $0 < q < 1$ and return the value $j$ such that
\begin{align} \sum_{i=0}^{j-1} p_i < q < \sum_{i=0}^{j}p_i. \end{align}We'll code this up.
def sample_discrete(probs):
"""Randomly sample an index with probability given by probs."""
# Generate random number
q = np.random.rand()
# Find index
i = 0
p_sum = 0.0
while p_sum < q:
p_sum += probs[i]
i += 1
return i - 1
Now let's compare the speeds using the %timeit magic function. This is a useful tool to help diagnose slow spots in your code.
# Make dummy probs
probs = np.array([0.1, 0.3, 0.4, 0.05, 0.15])
print('Result from scipy.stats:')
%timeit sample_discrete_scipy(probs)
print('\nResult from hand-coded method:')
%timeit sample_discrete(probs)
Wow! The less concise method is a couple of orders of magnitude faster! So, we will ditch using scipy.stats, and use our hand-built sampler instead.
Now we can write a function to do our draws.
def gillespie_draw(propensity_func, propensities, population, t, args=()):
"""
Draws a reaction and the time it took to do that reaction.
Parameters
----------
propensity_func : function
Function with call signature propensity_func(population, t, *args)
used for computing propensities. This function must return
an array of propensities.
population : ndarray
Current population of particles
t : float
Value of the current time.
args : tuple, default ()
Arguments to be passed to `propensity_func`.
Returns
-------
rxn : int
Index of reaction that occured.
time : float
Time it took for the reaction to occur.
"""
# Compute propensities
propensity_func(propensities, population, t, *args)
# Sum of propensities
props_sum = propensities.sum()
# Compute next time
time = np.random.exponential(1.0 / props_sum)
# Compute discrete probabilities of each reaction
rxn_probs = propensities / props_sum
# Draw reaction from this distribution
rxn = sample_discrete(rxn_probs)
return rxn, time
SSA time stepping¶
Now we are ready to write our main SSA loop. We will only keep the counts at pre-specified time points. This saves on RAM, and we really only care about the values at given time points anyhow.
Note that this function is generic. All we need to specify our system are:
- A function to compute the propensities
- How the updates for a given reaction are made
- Initial population
Additionally, we specify necessary parameters, an initial condition, and the time points at which we want to store our samples. So, providing the propensity function and update are analogous to providing the time derivatives when using scipy.integrate.odeint().
def gillespie_ssa(propensity_func, update, population_0, time_points, args=()):
"""
Uses the Gillespie stochastic simulation algorithm to sample
from probability distribution of particle counts over time.
Parameters
----------
propensity_func : function
Function of the form f(params, t, population) that takes the current
population of particle counts and return an array of propensities
for each reaction.
update : ndarray, shape (num_reactions, num_chemical_species)
Entry i, j gives the change in particle counts of species j
for chemical reaction i.
population_0 : array_like, shape (num_chemical_species)
Array of initial populations of all chemical species.
time_points : array_like, shape (num_time_points,)
Array of points in time for which to sample the probability
distribution.
args : tuple, default ()
The set of parameters to be passed to propensity_func.
Returns
-------
sample : ndarray, shape (num_time_points, num_chemical_species)
Entry i, j is the count of chemical species j at time
time_points[i].
"""
# Initialize output
pop_out = np.empty((len(time_points), update.shape[1]), dtype=np.int)
# Initialize and perform simulation
i_time = 1
i = 0
t = time_points[0]
population = population_0.copy()
pop_out[0,:] = population
propensities = np.zeros(update.shape[0])
while i < len(time_points):
while t < time_points[i_time]:
# draw the event and time step
event, dt = gillespie_draw(propensity_func, propensities, population, t, args)
# Update the population
population_previous = population.copy()
population += update[event,:]
# Increment time
t += dt
# Update the index
i = np.searchsorted(time_points > t, True)
# Update the population
pop_out[i_time:min(i,len(time_points))] = population_previous
# Increment index
i_time = i
return pop_out
Running and parsing results¶
We can now run a set of SSA simulations and plot the results. We will run 100 trajectories and store them, using $\beta_p = \beta_m = 10$ and $\gamma = 0.4$. We will also use the nifty package tqdm to give a progress bar so we know how long it is taking.
# Specify parameters for calculation
args = (10.0, 10.0, 0.4)
time_points = np.linspace(0, 50, 101)
population_0 = np.array([0, 0], dtype=int)
size = 100
# Seed random number generator for reproducibility
np.random.seed(42)
# Initialize output array
samples = np.empty((size, len(time_points), 2), dtype=int)
# Run the calculations
for i in tqdm.tqdm_notebook(range(size)):
samples[i,:,:] = gillespie_ssa(simple_propensity, simple_update,
population_0, time_points, args=args)
We now have our samples, so we can plot the trajectories. For visualization, we will plot every trajectory as a thin blue line, and then the average of the trajectories as a thick orange line.
# Set up plots
plots = [bokeh.plotting.figure(plot_width=300,
plot_height=200,
x_axis_label='dimensionless time',
y_axis_label='number of mRNAs'),
bokeh.plotting.figure(plot_width=300,
plot_height=200,
x_axis_label='dimensionless time',
y_axis_label='number of proteins')]
# Plot trajectories and mean
for i in [0, 1]:
for x in samples[:,:,i]:
plots[i].line(time_points, x, line_width=0.3,
alpha=0.2, line_join='bevel')
plots[i].line(time_points, samples[:,:,i].mean(axis=0),
line_width=6, color='orange', line_join='bevel')
# Link axes
plots[0].x_range = plots[1].x_range
bokeh.io.show(bokeh.layouts.gridplot(plots, ncols=2))
We can also compute the steady state properties by considering the end of the simulation. The last 50 time points are at steady state, so we will average over them.
print('mRNA mean copy number =', samples[:,-50:,0].mean())
print('protein mean copy number =', samples[:,-50:,1].mean())
print('\nmRNA variance =', samples[:,-50:,0].std()**2)
print('protein variance =', samples[:,-50:,1].std()**2)
print('\nmRNA noise =', samples[:,-50:,0].std() / samples[:,-50:,0].mean())
print('protein noise =', samples[:,-50:,1].std() / samples[:,-50:,1].mean())
Finally, we can compute the steady state probability distributions. To plot them, we plot the empirical cumulative distribution function (ECDF) from the sampling. The theoretical distribution for mRNA is Poisson, which is overlayed in orange.
# mRNA ECDF
p_m = biocircuits.viz.ecdf(samples[:,-50:,0].flatten(),
plot_width=300,
plot_height=200,
x_axis_label='mRNA copy number')
# Theoretial mRNA CDF (Poisson)
p_m.circle(np.arange(25), st.poisson.cdf(np.arange(25), args[0]), color='orange')
# protein ECDF
p_p = biocircuits.viz.ecdf(samples[:,-50:,1].flatten(),
plot_width=300,
plot_height=200,
x_axis_label='protein copy number')
bokeh.io.show(bokeh.layouts.gridplot([p_m, p_p], ncols=2))
As we expect, the mRNA copy number matches the Poisson distribution. We also managed to get a distribution for the protein copy number that we could plot. So, you now have the basic tools for doing Gillespie simulations. You just need to code the propensity function and the update, and you're on your way! (Note, though, that this only works for Gillespie simulations where the states are defined by particle counts, which should suffice for this course.)
Implementation in the biocircuits package¶
This algorithm is implemented in the biocircuits package. A significant speed boost is achieved by just-in-time compliation using Numba. To utilize this feature, you need to just-in-time compile (JIT) your propensity function. You can insist that everything is compiled (and therefore skips the comparably slow Python interpreter) by using the @numba.njit decorator. In many cases, like in this one for simple gene expression, that is all you have to do.
@numba.njit
def simple_propensity(propensities, population, t, beta_m, beta_p, gamma):
"""Updates an array of propensities given a set of parameters
and an array of populations.
"""
# Unpack population
m, p = population
# Update propensities
propensities[0] = beta_m # Make mRNA transcript
propensities[1] = m # Degrade mRNA
propensities[2] = beta_p * m # Make protein
propensities[3] = gamma * p # Degrade protein
You can also utilize more threads and do the Gillespie simulations in parallel if you like. Because of the speed boosts, we will take more samples using the biocircuits implementation.
samples = biocircuits.gillespie_ssa(simple_propensity,
simple_update,
population_0,
time_points,
size=1000,
args=args,
n_threads=4,
progress_bar=False)
We now have 4000 trajectories, and we can again make the plots as we did before. With so many trajectories, though, we should not show them all, since there would be too many glyphs on the plot. We therefore thin out the samples for plotting, taking only every 100th trajectory.
# Set up plots
plots = [bokeh.plotting.figure(plot_width=300,
plot_height=200,
x_axis_label='dimensionless time',
y_axis_label='number of mRNAs'),
bokeh.plotting.figure(plot_width=300,
plot_height=200,
x_axis_label='dimensionless time',
y_axis_label='number of proteins')]
# Plot trajectories and mean
for i in [0, 1]:
for x in samples[::100,:,i]:
plots[i].line(time_points, x, line_width=0.3,
alpha=0.2, line_join='bevel')
plots[i].line(time_points, samples[:,:,i].mean(axis=0),
line_width=6, color='orange', line_join='bevel')
# Link axes
plots[0].x_range = plots[1].x_range
bokeh.io.show(bokeh.layouts.gridplot(plots, ncols=2))
Simulating the repressilator¶
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
- The species whose populations we are describing;
- The update-propensity pairs;
- The parameter values.
After constructing the tables, coding up the update and propensity functions is much easier.
To demonstrate this procedure, we will perform a Gillespie simulation for the repressilator, as described in Elowitz and Leibler. We will consider both RNA and DNA in a repressilator where gene 1 represses gene 2, gene 2 repressed gene 3, and gene 3 repressed 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 $k_{m,o}$.
As we build the simulation, let's start with a table of the populations we are keeping track of.
| index | description | variable |
|---|---|---|
| 0 | gene 1 mRNA copy number | m1 |
| 1 | gene 1 protein copy number | p1 |
| 2 | gene 2 mRNA copy number | m2 |
| 3 | gene 2 protein copy number | p2 |
| 4 | gene 3 mRNA copy number | m3 |
| 5 | gene 3 protein copy number | p3 |
| 6 | Number of repressors bound to promoter of gene 3 | n1 |
| 7 | Number of repressors bound to promoter of gene 1 | n2 |
| 8 | Number of repressors bound to promoter of gene 2 | n3 |
Note that we labeled each species with an index, which corresponds to its position in the array of populations in the simulation.
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.
| index | description | update | propensity |
|---|---|---|---|
| 0 | transcription of gene 1 mRNA | m1 ⟶ m1 + 1 |
kmu*(n3 == 0) + kmo*(n3 > 0) |
| 1 | transcription of gene 2 mRNA | m2 ⟶ m2 + 1 |
kmu*(n1 == 0) + kmo*(n1 > 0) |
| 2 | transcription of gene 3 mRNA | m3 ⟶ m3 + 1 |
kmu*(n2 == 0) + kmo*(n2 > 0) |
| 3 | translation of gene 1 protein | p1 ⟶ p1 + 1 |
kp * m1 |
| 4 | translation of gene 1 protein | p2 ⟶ p2 + 1 |
kp * m2 |
| 5 | translation of gene 1 protein | p3 ⟶ p3 + 1 |
kp * m3 |
| 6 | degradation of gene 1 mRNA | m1 ⟶ m1 - 1 |
gamma_m * m1 |
| 7 | degradation of gene 2 mRNA | m2 ⟶ m2 - 1 |
gamma_m * m2 |
| 8 | degradation of gene 3 mRNA | m3 ⟶ m3 - 1 |
gamma_m * m3 |
| 9 | degradation of unbound gene 1 protein | p1 ⟶ p1 - 1 |
gamma_p * p1 |
| 10 | degradation of unbound gene 2 protein | p2 ⟶ p2 - 1 |
gamma_p * p2 |
| 11 | degradation of unbound gene 3 protein | p3 ⟶ p3 - 1 |
gamma_p * p3 |
| 12 | degradation of bound gene 1 protein | n1 ⟶ n1 - 1 |
gamma_p * n1 |
| 13 | degradation of bound gene 2 protein | n2 ⟶ n2 - 1 |
gamma_p * n2 |
| 14 | degradation of bound gene 3 protein | n3 ⟶ n3 - 1 |
gamma_p * n3 |
| 15 | binding of protein to gene 1 operator | n3 ⟶ n3 + 1, p3 ⟶ p3 - 1 |
kr * p3 * (n3 < 2) |
| 16 | binding of protein to gene 2 operator | n1 ⟶ n1 + 1, p1 ⟶ p1 - 1 |
kr * p1 * (n1 < 2) |
| 17 | binding of protein to gene 3 operator | n2 ⟶ n2 + 1, p2 ⟶ p2 - 1 |
kr * p2 * (n2 < 2) |
| 18 | unbinding of protein to gene 1 operator | n3 ⟶ n3 - 1, p3 ⟶ p3 + 1 |
ku1*(n3 == 1) + 2*ku2*(n3 == 2) |
| 19 | unbinding of protein to gene 2 operator | n1 ⟶ n1 - 1, p1 ⟶ p1 + 1 |
ku1*(n1 == 1) + 2*ku2*(n1 == 2) |
| 20 | unbinding of protein to gene 3 operator | n2 ⟶ n2 - 1, p2 ⟶ p2 + 1 |
ku1*(n2 == 1) + 2*ku2*(n2 == 2) |
Finally, we have parameters that were introduced in the propensities, so we should have a table defining them.
| parameter | value | units |
|---|---|---|
kmu |
0.5 | 1/sec |
kmo |
5×10$^{-4}$ | 1/sec |
kp |
0.167 | 1/molec-sec |
gamma_m |
0.005776 | 1/sec |
gamma_p |
0.001155 | 1/sec |
kr |
1 | 1/molec-sec |
ku1 |
224 | 1/sec |
ku2 |
9 | 1/sec |
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.
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.
@numba.njit
def repressilator_propensity(propensities, population, t,
kmu,
kmo,
kp,
gamma_m,
gamma_p,
kr,
ku1,
ku2):
m1, p1, m2, p2, m3, p3, n1, n2, n3 = population
propensities[0] = kmu if n3 == 0 else kmo
propensities[1] = kmu if n1 == 0 else kmo
propensities[2] = kmu if n2 == 0 else kmo
propensities[3] = kp * m1
propensities[4] = kp * m2
propensities[5] = kp * m3
propensities[6] = gamma_m * m1
propensities[7] = gamma_m * m2
propensities[8] = gamma_m * m3
propensities[9] = gamma_p * p1
propensities[10] = gamma_p * p2
propensities[11] = gamma_p * p3
propensities[12] = gamma_p * n1
propensities[13] = gamma_p * n2
propensities[14] = gamma_p * n3
propensities[15] = kr * p3 * (n3 < 2)
propensities[16] = kr * p1 * (n1 < 2)
propensities[17] = kr * p2 * (n2 < 2)
propensities[18] = ku1*(n3==1) + 2*ku2*(n3==2)
propensities[19] = ku1*(n1==1) + 2*ku2*(n1==2)
propensities[20] = ku1*(n2==1) + 2*ku2*(n2==2)
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.
repressilator_update = np.array([
# 0 1 2 3 4 5 6 7 8
[ 1, 0, 0, 0, 0, 0, 0, 0, 0], # 0
[ 0, 0, 1, 0, 0, 0, 0, 0, 0], # 1
[ 0, 0, 0, 0, 1, 0, 0, 0, 0], # 2
[ 0, 1, 0, 0, 0, 0, 0, 0, 0], # 3
[ 0, 0, 0, 1, 0, 0, 0, 0, 0], # 4
[ 0, 0, 0, 0, 0, 1, 0, 0, 0], # 5
[-1, 0, 0, 0, 0, 0, 0, 0, 0], # 6
[ 0, 0, -1, 0, 0, 0, 0, 0, 0], # 7
[ 0, 0, 0, 0, -1, 0, 0, 0, 0], # 8
[ 0, -1, 0, 0, 0, 0, 0, 0, 0], # 9
[ 0, 0, 0, -1, 0, 0, 0, 0, 0], # 10
[ 0, 0, 0, 0, 0, -1, 0, 0, 0], # 11
[ 0, 0, 0, 0, 0, 0, -1, 0, 0], # 12
[ 0, 0, 0, 0, 0, 0, 0, -1, 0], # 13
[ 0, 0, 0, 0, 0, 0, 0, 0, -1], # 14
[ 0, 0, 0, 0, 0, -1, 0, 0, 1], # 15
[ 0, -1, 0, 0, 0, 0, 1, 0, 0], # 16
[ 0, 0, 0, -1, 0, 0, 0, 1, 0], # 17
[ 0, 0, 0, 0, 0, 1, 0, 0, -1], # 18
[ 0, 1, 0, 0, 0, 0, -1, 0, 0], # 19
[ 0, 0, 0, 1, 0, 0, 0, -1, 0], # 20
], dtype=int)
Next, we specify the parameter values according to the parameter table. Remember that we need to package them in a tuple, after defining them.
# Parameter values
kmu = 0.5
kmo = 5e-4
kp = 0.167
gamma_m = 0.005776
gamma_p = 0.001155
kr = 1.0
ku1 = 224.0
ku2 = 9.0
repressilator_args = (kmu,
kmo,
kp,
gamma_m,
gamma_p,
kr,
ku1,
ku2)
Finally, we specify the initial population and the time points we want for the sampling.
# State with 10 copies of everything, nothing bound to operators
repressilator_pop_0 = np.array([10, 10, 10, 10, 10, 10, 0, 0, 0], dtype=int)
repressilator_time_points = np.linspace(0, 80000, 4001)
And we are all set to perform the calculation. We will make a single trace and then plot it.
# Perform the Gillespie simulation
pop = biocircuits.gillespie_ssa(repressilator_propensity,
repressilator_update,
repressilator_pop_0,
repressilator_time_points,
args=repressilator_args)
# Make plot
colors = bokeh.palettes.d3['Category10'][3]
p = bokeh.plotting.figure(height=250, width=600,
x_axis_label='time (min)',
y_axis_label='protein copy number')
for c, i in enumerate([1, 3, 5]):
p.line(repressilator_time_points/60,
pop[0,:,i], color=colors[c])
bokeh.io.show(p)
We observe the oscillations, but the amplitudes are highly variable.
Increasing speed¶
In this last section of the tutorial, we briefly discuss some strategies for boosting the speed of your Gillespie simulation. There are no major conceptual topics here, and this section may be skipped.
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.
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, 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.)
We will first remake the non-JITted version of the propensity function to test its speed.
def simple_propensity(propensities, population, t, beta_m, beta_p, gamma):
"""Updates an array of propensities given a set of parameters
and an array of populations.
"""
# Unpack population
m, p = population
# Update propensities
propensities[0] = beta_m # Make mRNA transcript
propensities[1] = m # Degrade mRNA
propensities[2] = beta_p * m # Make protein
propensities[3] = gamma * p # Degrade protein
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.
%lprun -T lp_results.txt -f gillespie_ssa gillespie_ssa(\
simple_propensity, simple_update, population_0, time_points, args)
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.
propensities = np.ones(4)
%lprun -T lp_results.txt -f gillespie_draw \
[gillespie_draw(simple_propensity, propensities, population_0, 0, args) \
for _ in range(10000)]
The propensity function is taking the most time. We will focus on improving that.
Speed boost by JIT compilation with Numba¶
As we have seen Numba 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
@numba.jit()
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).
We can speed that up by sacrificing some readability and directly accessing the arrays instead of setting up views, as we have been doing.
@numba.njit
def simple_propensity_numba(propensities, population, t, beta_m, beta_p, gamma):
"""Updates an array of propensities given a set of parameters
and an array of populations.
"""
# Unpack population
m, p = population
# Update propensities
propensities[0] = beta_m # Make mRNA transcript
propensities[1] = m # Degrade mRNA
propensities[2] = beta_p * m # Make protein
propensities[3] = gamma * p # Degrade protein
# Check speeds
print('Old propensity function:')
%timeit simple_propensity(propensities, population_0, 0, *args)
print('\nNumba\'d propensity function:')
%timeit simple_propensity_numba(propensities, population_0, 0, *args)
We got some speed up! Nice!
Now, we also saw that the sums and division of arrays are slow. Let's optimize the sum operation using numba.
@numba.jit(nopython=True)
def sum_numba(ar):
return ar.sum()
# Make dummy array for testing
ar = np.array([0.3, 0.4, 0.3, 0.2, 0.15])
print('\nNumPy sum:')
%timeit ar.sum()
print('\nNumba sum:')
%timeit sum_numba(ar)
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.
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.
@numba.njit
def sample_discrete_numba(probs, probs_sum):
q = np.random.rand() * probs_sum
i = 0
p_sum = 0.0
while p_sum < q:
p_sum += probs[i]
i += 1
return i - 1
# Make dummy unnormalized probs
probs = np.array([0.1, 0.3, 0.4, 0.05, 0.15, 0.6])
probs_sum = sum_numba(probs)
print('Result from hand-coded method:')
%timeit sample_discrete(probs)
print("\nResults from numba'd version:")
%timeit sample_discrete_numba(probs, probs_sum)
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.
def gillespie_draw_fast(propensity_func, propensities, population, t, args):
"""
Draws a reaction and the time it took to do that reaction.
"""
# Compute propensities
propensity_func(propensities, population, t, *args)
# Sum of propensities
props_sum = sum_numba(propensities)
# Compute time
time = np.random.exponential(1 / props_sum)
# Draw reaction given propensities
rxn = sample_discrete_numba(propensities, props_sum)
return rxn, time
print('Old Gillespie draw:')
%timeit gillespie_draw(simple_propensity, propensities, population_0, 0.0, args)
print('\nFast Gillespie draw:')
%timeit gillespie_draw_fast(simple_propensity_numba, propensities, population_0, 0.0, args)
So, our optimization got us another speed boost. Let's adjust our SSA function to include the fast draws.
def gillespie_ssa_fast(propensity_func, update, population_0,
time_points, args=()):
"""
Uses the Gillespie stochastic simulation algorithm to sample
from proability distribution of particle counts over time.
Parameters
----------
propensity_func : function
Function of the form f(params, population) that takes the current
population of particle counts and return an array of propensities
for each reaction.
update : ndarray, shape (num_reactions, num_chemical_species)
Entry i, j gives the change in particle counts of species j
for chemical reaction i.
population_0 : array_like, shape (num_chemical_species)
Array of initial populations of all chemical species.
time_points : array_like, shape (num_time_points,)
Array of points in time for which to sample the probability
distribution.
args : tuple, default ()
The set of parameters to be passed to propensity_func.
Returns
-------
rxn : int
Index of reaction that occured.
time : float
Time it took for the reaction to occur.
Returns
-------
sample : ndarray, shape (num_time_points, num_chemical_species)
Entry i, j is the count of chemical species j at time
time_points[i].
"""
# Initialize output
pop_out = np.empty((len(time_points), update.shape[1]), dtype=np.int)
# Initialize and perform simulation
i_time = 1
i = 0
t = time_points[0]
population = population_0.copy()
pop_out[0,:] = population
propensities = np.zeros(update.shape[0])
while i < len(time_points):
while t < time_points[i_time]:
# draw the event and time step
event, dt = gillespie_draw_fast(propensity_func,
propensities, population, t, args)
# Update the population
population_previous = population.copy()
population += update[event,:]
# Increment time
t += dt
# Update the index
i = np.searchsorted(time_points > t, True)
# Update the population
pop_out[i_time:min(i,len(time_points))] = population_previous
# Increment index
i_time = i
return pop_out
Now we can test the speed of the two SSAs.
print('Gillespie SSA:')
%timeit gillespie_ssa(simple_propensity, simple_update, \
population_0, time_points, args)
print('\nFast Gillespie SSA:')
%timeit gillespie_ssa_fast(simple_propensity_numba, simple_update,\
population_0, time_points, args)
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.
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 package does not rely on such a global function, but for the purposes of this demonstration, it is convenient.)
@numba.njit
def prop_func(propensities, population, t, beta_m, beta_p, gamma):
"""Updates an array of propensities given a set of parameters
and an array of populations.
"""
# Unpack population
m, p = population
# Update propensities
propensities[0] = beta_m # Make mRNA transcript
propensities[1] = m # Degrade mRNA
propensities[2] = beta_p * m # Make protein
propensities[3] = gamma * p # Degrade protein
@numba.njit
def gillespie_draw_numba(propensities, population, t, args):
"""
Draws a reaction and the time it took to do that reaction.
Assumes that there is a globally scoped function
`prop_func` that is Numba'd with nopython=True.
"""
# Compute propensities
prop_func(propensities, population, t, *args)
# Sum of propensities
props_sum = np.sum(propensities)
# Compute time
time = np.random.exponential(1 / props_sum)
# Draw reaction given propensities
rxn = sample_discrete_numba(propensities, props_sum)
return rxn, time
@numba.njit
def gillespie_ssa_numba(update, population_0, time_points, args):
"""
Uses the Gillespie stochastic simulation algorithm to sample
from proability distribution of particle counts over time.
Parameters
----------
update : ndarray, shape (num_reactions, num_chemical_species)
Entry i, j gives the change in particle counts of species j
for chemical reaction i.
population_0 : array_like, shape (num_chemical_species)
Array of initial populations of all chemical species.
time_points : array_like, shape (num_time_points,)
Array of points in time for which to sample the probability
distribution.
args : tuple, default ()
The set of parameters to be passed to propensity_func.
Returns
-------
sample : ndarray, shape (num_time_points, num_chemical_species)
Entry i, j is the count of chemical species j at time
time_points[i].
Notes
-----
.. Assumes that there is a globally scoped function
`propensity_func` that is Numba'd with nopython=True.
"""
# Initialize output
pop_out = np.empty((len(time_points), update.shape[1]), dtype=np.int64)
# Initialize and perform simulation
i_time = 1
i = 0
t = time_points[0]
population = population_0.copy()
pop_out[0,:] = population
propensities = np.zeros(update.shape[0])
while i < len(time_points):
while t < time_points[i_time]:
# draw the event and time step
event, dt = gillespie_draw_numba(propensities, population, t, args)
# Update the population
population_previous = population.copy()
population += update[event,:]
# Increment time
t += dt
# Update the index (Have to be careful about types for Numba)
i = np.searchsorted((time_points > t).astype(np.int64), 1)
# Update the population
for j in np.arange(i_time, min(i, len(time_points))):
pop_out[j,:] = population_previous
# Increment index
i_time = i
return pop_out
Now let's test the speed of all three of our functions.
print('Gillespie SSA:')
%timeit gillespie_ssa(simple_propensity, simple_update, \
population_0, time_points, args)
print('\nFast Gillespie SSA:')
%timeit gillespie_ssa_fast(simple_propensity_numba, simple_update,\
population_0, time_points, args)
print('\nTotally numba\'d Gillespie SSA:')
%timeit gillespie_ssa_numba(simple_update, population_0, time_points, args)
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 module uses.
Parallel Gillespie simulations¶
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.
def gillespie_fn(args):
return gillespie_ssa_numba(*args)
def gillespie_parallel(fn, update, population_0, time_points, args,
n_simulations, n_threads):
"""
Convenience function to do parallel Gillespie simulations.
"""
input_args = (update, population_0, time_points, args)
with multiprocessing.Pool(n_threads) as p:
populations = p.map(fn, [input_args]*n_simulations)
return np.array(populations)
We are paying some overhead in setting up the parallelization. Let's time it to see how we do with four threads. Note: Your machine have have fewer than four cores, which means that you cannot parallelize the code more than this.
n_simulations = 1000
print('\nnumba\'d Gillespie SSA:')
%timeit [gillespie_ssa_numba(simple_update, population_0, time_points, args) \
for _ in range(n_simulations)]
print('\nParallel numba\'d Gillespie SSA:')
%timeit gillespie_parallel(gillespie_fn, simple_update, population_0, time_points,\
args, n_simulations, 2)
We get a factor of two speed boost from two cores. This brings our total speed boost from the optimization to near two orders of magnitude.
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. If we are willing to approximately sample out of the probability distribution, there are many fast, approximate methods (e.g., Salis and Kaznessis) available.