Hawkes Process Modelling for Chemical Reaction Networks in a Random Environment

SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 2444-2488, September 2024.
Abstract.Cellular processes are open systems, situated in a heterogeneous context, rather than operating in isolation. Chemical reaction networks (CRNs) whose reaction rates are modelled as external stochastic processes account for the heterogeneous environment when describing the embedded process. A marginal description of the embedded process is of interest for (i) marginal simulations that bypass the co-simulation of the environment, (ii) obtaining new process equations from which moment equations can be derived, (iii) the computation of information-theoretic quantities, and (iv) state estimation. It is known since Snyder’s and related works that marginalization over a stochastic intensity turns point processes into self-exciting ones. While the Snyder filter specifies the exact history-dependent propensities in the framework of CRNs in a Markov environment, it was recently suggested to use approximate filters for the marginal description. By regarding the chemical reactions as events, we establish a link between CRNs in a linear random environment and Hawkes processes, a class of self-exciting counting processes widely used in event analysis. The Hawkes approximation can be obtained via a moment closure scheme or as the optimal linear approximation under the quadratic criterion. We show the equivalence of both approaches. Furthermore, we use martingale techniques to provide results on the agreement of the Hawkes process and the exact marginal process in their second-order statistics, i.e., covariance, auto/cross-correlation. We introduce an approximate marginal simulation algorithm and illustrate it in case studies.