Cyclo-Stationary Distributions of mRNA and Protein Counts for Random Cell Division Times

There is a long history of using experimental and computational approaches to study noise in single-cell levels of mRNA and proteins. The noise originates from myriad factors: intrinsic processes of gene expression, partitioning errors during division, and extrinsic effects, such as random cell-cycle times. Although theoretical methods are well developed to analytically understand the statistics of copy numbers for fixed or Erlang distributed cell cycle times, the general problem of random division times is still open. For any random (but uncorrelated) division time distribution, we present a method to address this challenging problem and obtain exact series representations of the copy number distributions in the cyclo-stationary state. We provide explicit cell age-specific and age-averaged results, and analyze the relative contribution to noise from intrinsic and extrinsic sources. Our analytical approach will aid the analysis of single-cell expression data and help in disentangling the impact of variability in division times.