# Markov Jump Process: Reaction Network¶

In the following, we fit stochastic chemical reaction kinetics with pyABC and show how to perform model selection between two competing models.

This notebook can be downloaded here: Markov Jump Process: Reaction Network.

We consider the two Markov jump process models $$m_1$$ and $$m_2$$ for conversion of (chemical) species $$X$$ to species $$Y$$:

$\begin{split}m_1: X + Y \xrightarrow{k_1} 2Y\\ m_2: X \xrightarrow{k_2} Y.\end{split}$

Each model is equipped with a single rate parameter $$k$$. To simulate these models, we define a simple Gillespie simulator:

In [1]:

import scipy as sp

def h(x, pre, c):
return (x**pre).prod(1) * c

def gillespie(x, c, pre, post, max_t):
"""
Gillespie simulation

Parameters
----------

x: 1D array of size n_species
The initial numbers.

c: 1D array of size n_reactions
The reaction rates.

pre: array of size n_reactions x n_species
What is to be consumed.

post: array of size n_reactions x n_species
What is to be produced

max_t: int
Timulate up to time max_t

Returns
-------
t, X: 1d array, 2d array
t: The time points.
X: The history of the species.
X.shape == (t.size, x.size)

"""
t = 0
t_store = [t]
x_store = [x.copy()]
S = post - pre

while t < max_t:
h_vec = h(x, pre, c)
h0 = h_vec.sum()
if h0 == 0:
break
delta_t = sp.random.exponential(1 / h0)
# no reaction can occur any more
if not sp.isfinite(delta_t):
t_store.append(max_t)
x_store.append(x)
break
reaction = sp.random.choice(c.size, p=h_vec/h0)
t = t + delta_t
x = x + S[reaction]

t_store.append(t)
x_store.append(x)

return sp.asarray(t_store), sp.asarray(x_store)


Next, we define the models in terms of ther initial molecule numbers $$x_0$$, an array pre which determines what is to be consumed (the left hand side of the reaction equations) and an array post which determines what is to be produced (the right hand side of the reaction equations). Moreover, we define that the simulation time should not exceed MAX_T seconds.

Model 1 starts with initial concentrations $$X=40$$ and $$Y=3$$. The reaction $$X + Y \rightarrow 2Y$$ is encoded in pre = [[1, 1]] and post = [[0, 2]].

In [2]:

MAX_T = 0.1

class Model1:
__name__ = "Model 1"
x0 = sp.array([40, 3])   # Initial molecule numbers
pre = sp.array([[1, 1]], dtype=int)
post = sp.array([[0, 2]])

def __call__(self, par):
t, X = gillespie(self.x0,
sp.array([float(par["rate"])]),
self.pre, self.post,
MAX_T)
return {"t": t, "X" : X}


Model 2 inherits the initial concentration from model 1. The reaction $$X \rightarrow Y$$ is incoded in pre = [[1, 0]] and post = [[0, 1]].

In [3]:

class Model2(Model1):
__name__ = "Model 2"
pre = sp.array([[1, 0]], dtype=int)
post = sp.array([[0, 1]])


We draw one stochastic simulation from model 1 (the “Observation”) and and one from model 2 (the “Competition”) and visualize both

In [4]:

%matplotlib inline
import matplotlib.pyplot as plt

true_rate = 2.3
observations = [Model1()({"rate": true_rate}),
Model2()({"rate": 30})]
fig, axes = plt.subplots(ncols=2)
fig.set_size_inches((12, 4))
for ax, title, obs in zip(axes, ["Observation", "Competition"],
observations):
ax.step(obs["t"], obs["X"]);
ax.legend(["Species X", "Species Y"]);
ax.set_xlabel("Time");
ax.set_ylabel("Concentration");
ax.set_title(title);


We observe that species $$X$$ is converted into species $$Y$$ in both cases. The difference of the concentrations over time can be quite subtle.

We define a distance function as $$L_1$$ norm of two trajectories, evaluated at 20 time points:

$\mathrm{distance}(X_1, X_2) = \sum_{n=1}^{N} \left |X_1(t_n) -X_2(t_n) \right|, \quad t_n = \frac{n}{N}T, \quad N=20 \,.$

Note that we only consider the concentration of species $$X$$ for distance calculation. And in code:

In [5]:

N_TEST_TIMES = 20

t_test_times = sp.linspace(0, MAX_T, N_TEST_TIMES)
def distance(x, y):
xt_ind = sp.searchsorted(x["t"], t_test_times) - 1
yt_ind = sp.searchsorted(y["t"], t_test_times) - 1
error = (sp.absolute(x["X"][:,1][xt_ind]
- y["X"][:,1][yt_ind]).sum()
/ t_test_times.size)
return error


For ABC, we choose for both models a uniform prior over the interval $$[0, 100]$$ for their single rate parameters:

In [6]:

from pyabc import Distribution, RV

prior = Distribution(rate=RV("uniform", 0, 100))


We initialize the ABCSMC class passing the two models, their priors and the distance function.

In [7]:

from pyabc import ABCSMC
from pyabc.populationstrategy import AdaptivePopulationSize

abc = ABCSMC([Model1(),
Model2()],
[prior, prior],
distance,


We initialize a new ABC run, taking as observed data the one generated by model 1. The ABC run is to be stored in the sqlite database located at /tmp/mjp.db.

In [8]:

abc_id = abc.new("sqlite:////tmp/mjp.db", observations[0])

INFO:Epsilon:initial epsilon is 11.9
INFO:History:Start <ABCSMC(id=2, start_time=2018-04-11 08:15:33.836708, end_time=None)>


We start pyABC which automatically parallelizes across all available cores.

In [9]:

history = abc.run(minimum_epsilon=0.7, max_nr_populations=15)

INFO:ABC:t:0 eps:11.9
INFO:ABC:t:1 eps:8.074999999999992
INFO:Adaptation:Change nr particles 500 -> 103
INFO:ABC:t:2 eps:6.7
INFO:Adaptation:Change nr particles 103 -> 84
INFO:ABC:t:3 eps:5.548103189731676
INFO:Adaptation:Change nr particles 84 -> 81
INFO:ABC:t:4 eps:4.829384979405069
INFO:Adaptation:Change nr particles 81 -> 96
INFO:ABC:t:5 eps:4.160126641930985
INFO:Adaptation:Change nr particles 96 -> 83
INFO:ABC:t:6 eps:3.15
INFO:Adaptation:Change nr particles 83 -> 111
INFO:ABC:t:7 eps:2.352496741582326
INFO:Adaptation:Change nr particles 111 -> 63
INFO:ABC:t:8 eps:1.75
INFO:Adaptation:Change nr particles 63 -> 60
INFO:ABC:t:9 eps:1.4527327452006253
INFO:Adaptation:Change nr particles 60 -> 41
INFO:ABC:t:10 eps:1.3
INFO:Adaptation:Change nr particles 41 -> 71
INFO:ABC:t:11 eps:1.1
INFO:Adaptation:Change nr particles 71 -> 57
INFO:ABC:t:12 eps:1.0
INFO:Adaptation:Change nr particles 57 -> 61
INFO:ABC:t:13 eps:0.9
INFO:Adaptation:Change nr particles 61 -> 42
INFO:ABC:t:14 eps:0.7938654326356155
INFO:Adaptation:Change nr particles 42 -> 56
INFO:History:Done <ABCSMC(id=2, start_time=2018-04-11 08:15:33.836708, end_time=2018-04-11 08:36:46.517874)>


We first inspect the model probabilities.

In [10]:

ax = history.get_model_probabilities().plot.bar();
ax.set_ylabel("Probability");
ax.set_xlabel("Generation");
ax.legend([1, 2], title="Model", ncol=2,
loc="lower center", bbox_to_anchor=(.5, 1));


The mass at model 2 decreased, the mass at model 1 increased slowly. The correct model 1 is detected towards the later generations. We then inspect the distribution of the rate parameters:

In [11]:

from pyabc.visualization import plot_kde_1d
fig, axes = plt.subplots(2)
fig.set_size_inches((6, 6))
axes = axes.flatten()
axes[0].axvline(true_rate, color="black", linestyle="dotted")
for m, ax in enumerate(axes):
for t in range(0, history.n_populations, 2):
df, w = history.get_distribution(m=m, t=t)
if len(w) > 0:  # Particles in a model might die out
plot_kde_1d(df, w, "rate", ax=ax, label=f"t={t}",
xmin=0, xmax=20 if m == 0 else 100,
numx=200)
ax.set_title(f"Model {m+1}")
axes[0].legend(title="Generation",
loc="upper left", bbox_to_anchor=(1, 1));

fig.tight_layout()


The true rate is closely approximated by the posterior over the rate of model 1. It is a little harder to interpret the posterior over model 2. Apparently a rate between 20 and 40 yields data most similar to the observed data.

Lastly, we visualize the evolution of the population sizes. The population sizes were automatically selected by pyABC and varied over the course of the generations. (We do not plot the size of th first generation, which was set to 500)

In [12]:

populations = history.get_all_populations()
ax = populations[populations.t >= 1].plot("t", "particles",
style= "o-")
ax.set_xlabel("Generation");


The initially chosen population size was adapted to the desired target accuracy. A larger population size was automatically selected by pyABC while both models were still alive. The population size decreased during the later populations thereby saving computational time.