Approximating Solutions of the Chemical Master Equation using Neural Networks

Abstract

The Chemical Master Equation (CME) provides an accurate description of stochastic biochemical reaction networks in well-mixed conditions, but it cannot be solved analytically for most systems of practical interest. While Monte Carlo methods provide a principled means to probe the system dynamics, their high computational cost can render the estimation of molecule number distributions and other numerical tasks infeasible due to the large number of repeated simulations typically required. In this paper we aim to leverage the representational power of neural networks to approximate the solutions of the CME and propose a framework for Neural Estimation of Stochastic Simulations for Inference and Exploration (Nessie). Our approach is based on training a neural network to learn the distributions predicted by the CME from a relatively small number of stochastic simulations, thereby accelerating computationally intensive tasks such as parameter exploration and inference. We show on biologically relevant examples that simple neural networks with one hidden layer are able to capture highly complex distributions across parameter space. We provide a detailed discussion of the neural network implementation and code for easy reproducibility.

Spectral neural approximations for models of transcriptional dynamics

“…To benchmark the algorithm’s performance, we investigated the precision achieved in recovering PMFs on a grid, as well as the runtime, for a series of testing rate vectors (Figure 2a). To quantify precision, we calculate the Hellinger distance [30], a metric of discrepancy between probability distributions: where ß N and ß M are the state space bounds for each θ . Although the system in Figure 1a does not afford a ground truth, we can approximate it by a high-quality generating function approximation.…”

“…To implement this approximation, we need to find values of w k , , and . In principle, this can be accomplished by training a neural network to predict these quantities based on θ ; this is the approach taken in the Nessie framework [30]. However, we find that and .…”

“…Most pertinently, [30] have independently proposed a method for the approximation of CME solutions by a set of negative binomial mixture functions, implemented using the neural Nessie framework. However, the approach is qualitatively different to the one we present.…”

Spectral neural approximations for models of transcriptional dynamics

“…To benchmark the algorithm's performance, we investigated the precision achieved in recovering PMFs on a grid, as well as the runtime, for a series of testing rate vectors (Figure 2a). To quantify precision, we calculate the Hellinger distance [30], a metric of discrepancy between probability distributions:…”

“…Most pertinently, [30] have independently proposed a method for the approximation of CME solutions by a set of negative binomial mixture functions, implemented using the neural Nessie framework. However, the approach is qualitatively di↵erent to the one we present.…”