Probability Distributions for Stochastic Gene Expression with Multiple Binding Sites

Abstract

While regulation of intrinsic stochasticity in gene expression is essential for many cellular functions, there is considerable interest in understanding how different molecular mechanisms of gene expression impact variations in protein levels across a population of cells. Stochastic properties of the corresponding gene systems are usually examined by directly finding the probability distributions of mRNA or protein governed by chemical master equations. Here, we analyze four stochastic models of gene expression, each describing a representative topology of binding sites. The total mRNA or protein probability distributions are derived by the weight assignment method that we propose here. These distributions indicate that the total mRNA or protein number obeys a beta distribution or a linear superposition of two beta distributions in spite of differences in the local mRNA or protein numbers distributed in individual binding sites among the models. Our results reveal essential mechanisms of gene expression in terms of the probability distribution rather than finite-order moments, as was done in previous studies.