Distribution of population-averaged observables in stochastic gene expression

Observation of phenotypic diversity in a population of genetically identical cells is often linked to the stochastic nature of chemical reactions involved in gene regulatory networks. We investigate the distribution of population-averaged gene expression levels as a function of population, or sample, size for several stochastic gene expression models to find out to what extent population-averaged quantities reflect the underlying mechanism of gene expression. We consider three basic gene regulation networks corresponding to transcription with and without gene state switching and translation. Using analytical expressions for the probability generating function of observables and large deviation theory, we calculate the distribution and first two moments of the population-averaged mRNA and protein levels as a function of model parameters, population size, and number of measurements contained in a data set. We validate our results using stochastic simulations also report exact results on the asymptotic properties of population averages which show qualitative differences among different models.