Derivation of stationary distributions of biochemical reaction networks via structure transformation

Hyukpyo Hong,M Ali Al-Radhawi,Jae Kyoung Kim,Eduardo D Sontag,Jinsu Kim

doi:10.1038/s42003-021-02117-x

Abstract

Long-term behaviors of biochemical reaction networks (BRNs) are described by steady states in deterministic models and stationary distributions in stochastic models. Unlike deterministic steady states, stationary distributions capturing inherent fluctuations of reactions are extremely difficult to derive analytically due to the curse of dimensionality. Here, we develop a method to derive analytic stationary distributions from deterministic steady states by transforming BRNs to have a special dynamic property, called complex balancing. Specifically, we merge nodes and edges of BRNs to match in- and out-flows of each node. This allows us to derive the stationary distributions of a large class of BRNs, including autophosphorylation networks of EGFR, PAK1, and Aurora B kinase and a genetic toggle switch. This reveals the unique properties of their stochastic dynamics such as robustness, sensitivity, and multi-modality. Importantly, we provide a user-friendly computational package, CASTANET, that automatically derives symbolic expressions of the stationary distributions of BRNs to understand their long-term stochasticity.

Highlights

Long-term behaviors of biochemical reaction networks (BRNs) are described by steady states in deterministic models and stationary distributions in stochastic models
BRNs9 through the connection between Petri nets and BRNs10, shows that for complex balanced networks whose kinetics are described by mass action reactions, stationary distributions can be characterized in terms of jointly distributed Poisson random variables with parameters corresponding to deterministic steady states
As mentioned in the introduction, the stationary distributions of the stochastic mass action models for BRNs can be derived with any choice of rate constants using the previous method[11] if and only if the networks have two structural properties: weak reversibility and zero deficiency

Summary

Introduction

Long-term behaviors of biochemical reaction networks (BRNs) are described by steady states in deterministic models and stationary distributions in stochastic models. We develop a method to derive analytic stationary distributions from deterministic steady states by transforming BRNs to have a special dynamic property, called complex balancing. A standard approach to mathematical modeling of biochemical reaction networks (BRNs) is to use ordinary differential equations (ODEs), whose variables represent concentrations of molecules[1] This deterministic description, while convenient for computation, by its nature cannot capture the inherent randomness of BRNs. In particular, the long-term behavior of ODE systems is characterized by steady states or other attractors, rather than by the stationary distributions statistically observed in real biological systems. Weak reversibility of a network means that the network is a union of closed reaction cycles, and the deficiency of a network is the number of dependent closed reaction cycles, which can be checked Satisfying these two structural properties is a simple condition to derive the stationary distribution of network under mass action reactions with the method in ref.[11].

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