# Quantifying gene expression variability arising from randomness in cell division times

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## Abstract

The level of a given mRNA or protein exhibits significant variations from cell-to-cell across a homogeneous 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 inter-cellular heterogeneity. Here we investigate how randomness in the cell division times can create variability in population counts. We consider a model by which mRNA/protein levels in a given cell evolve according to a linear differential equation and cell divisions occur at times spaced by independent and identically distributed random intervals. Whenever the cell divides the levels of mRNA and protein are halved. For this model, we provide a method for computing any statistical moment (mean, variance, skewness, etcetera) of the mRNA and protein levels. The key to our approach is to establish that the time evolution of the mRNA and protein statistical moments is described by an upper triangular system of Volterra equations. 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.

## Keywords

Stochastic gene expression Non-genetic heterogeneity cell division times Asymptotic levels Volterra equations Statistical moments## Mathematics Subject Classification

92C37 92C40 37N25## Notes

### Acknowledgments

Duarte Antunes was supported by the Dutch Science Foundation (STW) and the Dutch Organization for Scientific Research (NWO) under the VICI Grant No. 11382, and by the European 7th Framework Network of Excellence by the project HYCON2-257462. Abhyudai Singh was supported by the National Science Foundation Grant DMS-1312926, University of Delaware Research Foundation (UDRF) and Oak Ridge Associated Universities (ORAU).

## Supplementary material

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