Stochastic reaction network models are a popular tool for studying the effects of dynamical randomness in biological systems. Such models are typically analysed by estimating the solution of Kolmogorov's forward equation, called the chemical master equation (CME), which describes the evolution of the probability distribution of the random state-vector representing molecular counts of the reacting species. The size of the CME system is typically very large or even infinite, and due to this high-dimensional nature, accurate numerical solutions of the CME are very difficult to obtain. In this talk we will present a novel deep learning approach for computing solution statistics of high-dimensional CMEs by reformulating the stochastic dynamics using Kolmogorov's backward equation. The proposed method leverages superior approximation properties of Deep Neural Networks (DNNs) to reliably estimate expectations under the CME solution for several user-defined functions of the state-vector. Our method only requires a handful of stochastic simulations and it allows not just the numerical approximation of various expectations for the CME solution but also of its sensitivities with respect to all the reaction network parameters (e.g. rate constants). We illustrate the method with a number of examples and discuss possible extensions and improvements. 

This is joint work with Prof. Christoph Schwab (Seminar for Applied Mathematics, ETH Zürich) and Prof. Mustafa Khammash (Department of Biosystems Science and Engineering, ETH Zürich)

Reference:  Gupta A, Schwab C, Khammash M (2021) DeepCME: A deep learning framework for computing solution statistics of the chemical master equation. PLoS Comput Biol 17(12): e1009623. https://doi.org/10.1371/journal.pcbi.1009623