# Analytical results for a stochastic model of gene expression with arbitrary partitioning of proteins

## TL;DR

This paper provides an exact analytical solution for a stochastic gene expression model with arbitrary protein partitioning, utilizing the PPA-mapping to transform the problem into a hierarchy of simpler models, enabling explicit calculations of key statistics.

## Contribution

It introduces a novel application of the PPA-mapping to a variation of the two-stage gene expression model with time-dependent rates, deriving explicit solutions and extending to models with parameter fluctuations.

## Key findings

- Derived an integral expression for the time-dependent generating function.
- Obtained explicit formulas for mean, variance, and correlation functions.
- Extended results to models with stochastic parameter fluctuations.

## Abstract

In biophysics, the search for analytical solutions of stochastic models of cellular processes is often a challenging task. In recent work on models of gene expression, it was shown that a mapping based on partitioning of Poisson arrivals (PPA-mapping) can lead to exact solutions for previously unsolved problems. While the approach can be used in general when the model involves Poisson processes corresponding to creation or degradation, current applications of the method and new results derived using it have been limited to date. In this paper, we present the exact solution of a variation of the two-stage model of gene expression (with time dependent transition rates) describing the arbitrary partitioning of proteins. The methodology proposed makes full use of the the PPA-mapping by transforming the original problem into a new process describing the evolution of three biological switches. Based on a succession of transformations, the method leads to a hierarchy of reduced models. We give an integral expression of the time dependent generating function as well as explicit results for the mean, variance, and correlation function. Finally, we discuss how results for time dependent parameters can be extended to the three-stage model and used to make inferences about models with parameter fluctuations induced by hidden stochastic variables.