# Efficient approximations of transcriptional bursting effects on the dynamics of a gene regulatory network

## TL;DR

This paper shows how transcriptional bursting can change gene network dynamics and introduces efficient methods to model it.

## Contribution

A novel extension of the chemical Langevin equation to include transcriptional bursting noise is derived.

## Key findings

- Transcriptional bursting can induce or amplify oscillations in gene regulatory networks.
- The extended Langevin equation reduces computation time while preserving analytical tractability.
- The study provides guidelines for when and how to apply different approximation methods.

## Abstract

Mathematical models of gene regulatory networks are widely used to study cell fate changes and transcriptional regulation. When designing such models, it is important to accurately account for sources of stochasticity. However, doing so can be computationally expensive and analytically untractable, posing limits on the extent of our explorations and on parameter inference. Here, we explore this challenge using the example of a simple auto-negative feedback motif, in which we incorporate stochastic variation due to transcriptional bursting and noise from finite copy numbers. We find that transcriptional bursting may change the qualitative dynamics of the system by inducing oscillations when they would not otherwise be present, or by magnifying existing oscillations. We describe multiple levels of approximation for the model in the form of differential equations, piecewise-deterministic processes and stochastic differential equations. Importantly, we derive how the classical chemical Langevin equation can be extended to include a noise term representing transcriptional bursting. This approximation drastically decreases computation times and allows us to analytically calculate properties of the dynamics, such as their power spectrum. We explore when these approximations break down and provide recommendations for their use. Our analysis illustrates the importance of accounting for transcriptional bursting when simulating gene regulatory network dynamics and provides recommendations to do so with computationally efficient methods.