Marginal process framework: A model reduction tool for Markov jump processes

Markov jump process models have many applications across science. Often these models are defined on a state space of product form and only one of the components of the process is of direct interest. In this paper we extend the marginal process framework, which provides a marginal description of the component of interest, to the case of fully coupled processes. We use entropic matching to obtain a finite-dimensional approximation of the filtering equation, which governs the transition rates of the marginal process. The resulting equations can be seen as a combination of two projection operations applied to the full master equation so that we obtain a principled model reduction framework. We demonstrate the resulting reduced description on the totally asymmetric exclusion process. An important class of Markov jump processes are stochastic reaction networks, which have applications in chemical and biomolecular kinetics, ecological models, and models of social networks. We obtain a particularly simple instantiation of the marginal process framework for mass-action systems by using product Poisson distributions for the approximate solution of the filtering equation. We investigate the resulting approximate marginal process analytically and numerically.

Figure 1

Schematic illustration of the concepts involved in the construction of the marginal process, based on the example network (5). Bottom: Bold curve curve and shaded area show mean and plus or minus one standard deviation of the filtering distribution. The marginal process trajectory (top) drives the evolution of the filtering distribution and the filtering distribution mean determines the transition rates of the marginal process $X$. Note that jumps of the filtering distribution occur only if the marginal process increases by a jump.

Figure 2

Monte Carlo evaluation of the approximation quality of the Poisson marginal process and of timescale separation. Distributions of protein abundance at stationarity from 50 000 Monte Carlo runs for each case. The parameters are $c_1=\gamma$, $c_2=\gamma /2$, $c_3=1$, and $c_4=0.1$. Rows correspond to system sizes $\mathrm{\Omega }=0.1$ for (a–d), $\mathrm{\Omega }=1$ for (e–h) and $\mathrm{\Omega }=10$ for (i–l). Columns correspond to mRNA process speeds of (a), (e), and (i) $\gamma =0.5$, (b), (f), and (j) $\gamma =2$, (c), (g), and (k) $\gamma =5$, and (d), (h), and (l) $\gamma =25$. Note that the Poisson marginal process has a somewhat heavier right tail than the exact marginal process, especially at low system size and low value of $\gamma$. The algorithm used for stochastic simulation of the marginal process is explained in Appendix pp1.

Figure 3

Example of a sampled trajectory of the process $(X,\mathrm{\Theta })$. The solid red curve shows the abundance of gene product $X$ over time. The dashed blue curve shows the state of the filtering distribution mean $\mathrm{\Theta }$. Note that, as can be seen from (25), jumps in $\mathrm{\Theta }$ occurring when $\mathrm{\Theta }=1$ always lead to the same value of $\mathrm{\Theta }=c_2/(c_2+c_3)$, indicated by the dotted line. Whether $\mathrm{\Theta }$ increases or decreases after the jump depends on the value of $X$. The parameters are $c_1=1$, $c_2=0.25$, $c_3=0.5$, and $c_4=2$. The initial state was $X\left(0\right)=0$ and $\mathrm{\Theta }\left(0\right)=1$.

Figure 4

Numerical evaluation of the accuracy of the Bernoulli marginal TASEP approximation on $N=10$ sites at stationarity. Waiting-time distributions are shown for a particle to enter site $X_N$ after the previous particle left $X_N$. The parameters are $\alpha =\beta =1$ and (a) $c=0.01$, (b) $c=0.1$, (c) $c=1$, and (d) $c=2$. Distributions are from 100 000 samples.

Figure 5

Monte Carlo evaluation of the accuracy of the Poisson marginal process for the Lotka-Volterra system at system size (a), (d), and (g) $\mathrm{\Omega }=0.5$, (b), (e), and (h) $\mathrm{\Omega }=1$, and (c), (f), and (i) $\mathrm{\Omega }=5$. The mean of the Poisson marginal process approaches the true mean as the system size increases. In all three cases, the Poisson marginal process has a smaller standard deviation than the exact process. (g)–(i) show the full distributions at time $t=100$. The parameters are $c_1=1$, $c_2=5$, $c_3=0.5$, $c_4=0.003$, and $c_5=0.3$. Initial conditions were of product Poisson form with means $\langle X\rangle =\langle \widehat{X}\rangle =75\mathrm{\Omega }$. The number of simulated trajectories was 100 000 for $\mathrm{\Omega }=0.5$, 20 000 for $\mathrm{\Omega }=1$, and 10 000 for $\mathrm{\Omega }=5$.