Computing mRNA and protein statistical moments for a renewal model of stochastic gene-expression

The level of a given mRNA or protein exhibits significant variations from cell-to-cell across a homogenous population of living cells. Much work has focused on understanding the different sources of noise in the gene-expression process that drive this stochastic variability in gene-expression. Recent experiments tracking growth and division of individual cells reveal that cell division times have considerable intercellular heterogeneity. Here we investigate how randomness in the cell division times can create variability in population counts. We consider a model where mRNA/protein levels evolve according to a linear differential equation with cell divisions times spaced by independent and identically distributed random intervals. Whenever the cell divides the population of mRNA and protein is halved. Considering gamma distributed cell division intervals, we provide a method for computing the mean and variance of mRNA and protein levels and provide exact analytical formulas for the asymptotic values of these statistical moments. Computation of the statistical moments for physiologically relevant parameter values shows that randomness in the cell division process can be a major factor in driving difference in protein levels across a population of cells.