Stochastic Hybrid Models of Gene Regulatory Networks - A PDE Approach

TL;DR

This paper introduces a hybrid PDE-based model for gene regulatory networks that simplifies the analysis of complex stochastic systems by approximating protein counts continuously, providing accurate results with reduced computational effort.

Contribution

It presents a novel hybrid PDE approach combining stochastic and deterministic modeling for gene networks, improving efficiency and accuracy over traditional methods.

Findings

Hybrid PDE model matches stochastic results for large molecule counts.

Reduces computational complexity from many ODEs to fewer PDEs.

Provides analytical steady-state solution for self-regulatory gene case.

Abstract

A widely used approach to describe the dynamics of gene regulatory networks is based on the chemical master equation, which considers probability distributions over all possible combinations of molecular counts. The analysis of such models is extremely challenging due to their large discrete state space. We therefore propose a hybrid approximation approach based on a system of partial differential equations, where we assume a continuous-deterministic evolution for the protein counts. We discuss efficient analysis methods for both modeling approaches and compare their performance. We show that the hybrid approach yields accurate results for sufficiently large molecule counts, while reducing the computational effort from one ordinary differential equation for each state to one partial differential equation for each mode of the system. Furthermore, we give an analytical steady-state solution of the hybrid model for the case of a self-regulatory gene.