# Stochastic modeling of gene expression: application of ensembles of trajectories

## TL;DR

This paper demonstrates how ensembles of trajectories can be used to analyze stochastic gene expression models, providing insights into dynamics, fluctuations, and rare events beyond static distributions.

## Contribution

It applies the ensemble of trajectories approach to various gene expression models, including complex ones with bursts and switches, to analyze dynamics and rare events.

## Key findings

- Generation of cumulant generating functions for protein numbers
- Analysis of rare event statistics in gene expression models
- Application to simple and complex stochastic models

## Abstract

It is well established that gene expression can be modeled as a Markovian stochastic process and hence proper observables might be subjected to large fluctuations and rare events. Since dynamics is often more than statics, one can work with ensembles of trajectories for long but fixed times, instead of states or configurations, to study dynamics of these Markovian stochastic processes and glean more information. In this paper we aim to show that the concept of ensemble of trajectories can be applied to a variety of stochastic models of gene expression ranging from a simple birth-death process to a more sophisticate model containing burst and switch. By considering the protein numbers as a relevant dynamical observable, apart from asymptotic behavior of remote tails of probability distribution, generating function for the cumulants of this observable can also be obtained. We discuss the unconditional stochastic Markov processes which generate the statistics of rare events in these models.