The origin of the Poisson distribution in stochastic dynamics of gene expression

TL;DR

This paper clarifies the origin of the Poisson distribution in gene expression dynamics, showing it arises from the count of creation events that survive until a given time, and derives general time-dependent distributions.

Contribution

It introduces a new perspective on the Poisson distribution in gene expression, linking it to creation events that persist, and derives analytic time-dependent distributions for complex models.

Findings

Poisson distribution corresponds to creation events that survive until a given time

Derived time-dependent Poisson distributions as solutions to master equations

Combined Poisson with binomial/multinomial distributions for general distributions

Abstract

The Poisson distribution is the probability distribution of the number of independent events in a given period of time. Although the Poisson distribution appears ubiquitously in various stochastic dynamics of gene expression, both as time-dependent distributions and the stationary distributions, underlying independent events that give rise to such distributions have not been clear, especially in the presence of the degradation of gene products, which is not a Poisson process. I show that, in fact, the variable that follows the Poisson distribution is the number of independent events where biomolecules are created, which are destined to survive until the end of a given time duration. This new viewpoint allows us to derive time-dependent Poisson distributions as solutions of master equations for general class of protein production and degradation dynamics, including models with time-dependent rates and a non-Markovian model with delayed degradation. I then derive analytic forms of general time-dependent probability distributions by combining the Poisson distribution with the binomial or the multinomial distributions.