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);
../_images/examples_chemical_reaction_11_0.png

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,
             population_size=AdaptivePopulationSize(500, 0.15))

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:History:Start <ABCSMC(id=4, start_time=2018-05-08 16:16:00.358367, end_time=None)>
INFO:Epsilon:initial epsilon is 14.074999999999992

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:14.074999999999992
INFO:ABC:t:1 eps:10.6
INFO:Adaptation:Change nr particles 500 -> 101
INFO:ABC:t:2 eps:7.457087679174012
INFO:Adaptation:Change nr particles 101 -> 85
INFO:ABC:t:3 eps:6.270538173626088
INFO:Adaptation:Change nr particles 85 -> 82
INFO:ABC:t:4 eps:5.3
INFO:Adaptation:Change nr particles 82 -> 89
INFO:ABC:t:5 eps:4.85
INFO:Adaptation:Change nr particles 89 -> 97
INFO:ABC:t:6 eps:4.281990310808671
INFO:Adaptation:Change nr particles 97 -> 107
INFO:ABC:t:7 eps:3.336777623025941
INFO:Adaptation:Change nr particles 107 -> 100
INFO:ABC:t:8 eps:2.3319007349977956
INFO:Adaptation:Change nr particles 100 -> 57
INFO:ABC:t:9 eps:1.8334187576797563
INFO:Adaptation:Change nr particles 57 -> 54
INFO:ABC:t:10 eps:1.55
INFO:Adaptation:Change nr particles 54 -> 53
INFO:ABC:t:11 eps:1.35
INFO:Adaptation:Change nr particles 53 -> 56
INFO:ABC:t:12 eps:1.2286362088943308
INFO:Adaptation:Change nr particles 56 -> 60
INFO:ABC:t:13 eps:1.05
INFO:Adaptation:Change nr particles 60 -> 44
INFO:ABC:t:14 eps:0.95
INFO:Adaptation:Change nr particles 44 -> 53
INFO:History:Done <ABCSMC(id=4, start_time=2018-05-08 16:16:00.358367, end_time=2018-05-08 16:30:48.219049)>

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));
../_images/examples_chemical_reaction_24_0.png

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()
../_images/examples_chemical_reaction_26_0.png

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");
../_images/examples_chemical_reaction_28_0.png

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.