Mathematical Models of Gene Expression

TL;DR

This paper analyzes stochastic models of gene expression in bacteria, proving convergence to equilibrium, deriving moments, and comparing theoretical results with biological approximations.

Contribution

It introduces an extended model including elongation phases and provides new convergence results and explicit moment formulas for gene expression.

Findings

Proves convergence to equilibrium for the extended gene expression model

Derives explicit formulas for the first two moments of mRNA and protein numbers

Analyzes the validity of biological approximations for equilibrium distributions

Abstract

In this paper we analyze the equilibrium properties of a large class of stochastic processes describing the fundamental biological process within bacterial cells, {\em the production process of proteins}. Stochastic models classically used in this context to describe the time evolution of the numbers of mRNAs and proteins are presented and discussed. An extension of these models, which includes elongation phases of mRNAs and proteins, is introduced. A convergence result to equilibrium for the process associated to the number of proteins and mRNAs is proved and a representation of this equilibrium as a functional of a Poisson process in an extended state space is obtained. Explicit expressions for the first two moments of the number of mRNAs and proteins at equilibrium are derived, generalizing some classical formulas. Approximations used in the biological literature for the equilibrium distribution of the number of proteins are discussed and investigated in the light of these results. Several convergence results for the distribution of the number of proteins at equilibrium are in particular obtained under different scaling assumptions.