Hybrid Master Equation for Jump-Diffusion Approximation of Biomolecular Reaction Networks

TL;DR

This paper develops a hybrid jump-diffusion model for biomolecular reaction networks, proving the associated hybrid master equation and proposing an efficient solution algorithm, demonstrated on gene regulation models.

Contribution

It introduces a hybrid master equation for jump-diffusion approximations and an algorithm to solve it using moments and maximum entropy, advancing multi-scale biomolecular modeling.

Findings

The hybrid master equation combines chemical master and Fokker-Planck equations.

The proposed algorithm efficiently solves the hybrid master equation.

Application to gene regulation shows the method's effectiveness.

Abstract

Cellular reactions have multi-scale nature in the sense that the abundance of molecular species and the magnitude of reaction rates can vary in a wide range. This diversity leads to hybrid models that combine deterministic and stochastic modeling approaches. To reveal this multi-scale nature, we proposed jump-diffusion approximation in a previous study. The key idea behind the model was to partition reactions into fast and slow groups, and then to combine Markov chain updating scheme for the slow set with diffusion (Langevin) approach updating scheme for the fast set. Then, the state vector of the model was defined as the summation of the random time change model and the solution of the Langevin equation. In this study, we have proved that the joint probability density function of the jump-diffusion approximation over the reaction counting process satisfies the hybrid master equation, which is the summation of the chemical master equation and the Fokker-Planck equation. To solve the hybrid master equation, we propose an algorithm using the moments of reaction counters of fast reactions given the reaction counters of slow reactions. Then, we solve a constrained optimization problem for each conditional probability density at the time point of interest utilizing the maximum entropy approach. Based on the multiplication rule for joint probability density functions, we construct the solution of the hybrid master equation. To show the efficiency of the method, we implement it to a canonical model of gene regulation.