# Effective computational methods for hybrid stochastic gene networks

## TL;DR

This paper introduces improved computational methods for analyzing hybrid stochastic gene networks modeled by PDMPs, enabling more accurate and efficient calculation of their probability distributions.

## Contribution

The authors develop novel algorithms combining simulation and analytic techniques to better compute probability distributions in hybrid gene network models.

## Key findings

- Monte-Carlo method effectively combines simulation with analytic ODE solutions.
- Push-forward method accurately computes probability measures using analytic semigroup expressions.
- Enhanced methods outperform earlier approaches in handling non-linear regulation functions.

## Abstract

At the scale of the individual cell, protein production is a stochastic process with multiple time scales, combining quick and slow random steps with discontinuous and smooth variation. Hybrid stochastic processes, in particular piecewise-deterministic Markov processes (PDMP), are well adapted for describing such situations. PDMPs approximate the jump Markov processes traditionally used as models for stochastic chemical reaction networks. Although hybrid modelling is now well established in biology, these models remain computationally challenging. We propose several improved methods for computing time dependent multivariate probability distributions (MPD) of PDMP models of gene networks. In these models, the promoter dynamics is described by a finite state, continuous time Markov process, whereas the mRNA and protein levels follow ordinary differential equations (ODEs). The Monte-Carlo method combines direct simulation of the PDMP with analytic solutions of the ODEs. The push-forward method numerically computes the probability measure advected by the deterministic ODE flow, through the use of analytic expressions of the corresponding semigroup. Compared to earlier versions of this method, the probability of the promoter states sequence is computed beyond the naive mean field theory and adapted for non-linear regulation functions.