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

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