Journal of Mathematical Biology

, Volume 71, Issue 2, pp 437–463 | Cite as

Quantifying gene expression variability arising from randomness in cell division times

  • Duarte Antunes
  • Abhyudai SinghEmail author


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.


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

Mathematics Subject Classification

92C37 92C40 37N25 



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

285_2014_811_MOESM1_ESM.pdf (263 kb)
ESM 1 (pdf 264 kb)


  1. Alon U (2006) An introduction to systems biology: design principles of biological circuits. Chapman and Hall/CRC, Boca RatonGoogle Scholar
  2. Antunes D, Hespanha J, Silvestre C (2013) Stability of networked control systems with asynchronous renewal links: An impulsive systems approach. Automatica 49:402–413CrossRefGoogle Scholar
  3. Antunes D, Hespanha JP, Silvestre C (2012) Volterra integral approach to impulsive renewal systems: application to networked control. IEEE Trans Autom Control 57:607–619CrossRefGoogle Scholar
  4. Antunes D, Hespanha JP, Silvestre C (2013) Stochastic hybrid systems with renewal transitions: moment analysis with application to networked control systems with delays. SIAM J Control Optim 51:1481,1499CrossRefGoogle Scholar
  5. Arkin A, Ross J, McAdams HH (1998) Stochastic kinetic analysis of developmental pathway bifurcation in phage \(\lambda \)-infected Escherichia coli cells. Genetics 149:1633–1648Google Scholar
  6. Balaban N, Merrin J, Chait R, Kowalik L, Leibler S (2004) Bacterial persistence as a phenotypic switch. Science 305:1622–1625CrossRefGoogle Scholar
  7. Bar-Even A, Paulsson J, Maheshri N, Carmi M, O’Shea E, Pilpel Y, Barkai N (2006) Noise in protein expression scales with natural protein abundance. Nat Genet 38:636–643CrossRefGoogle Scholar
  8. Berg OG (1978) A model for the statistical fluctuations of protein numbers in a microbial population. J Theor Biol 71:587–603CrossRefGoogle Scholar
  9. Bokes P, King J, Wood A, Loose M (2012) Exact and approximate distributions of protein and mrna levels in the low-copy regime of gene expression. J Math Biol 64:829–854CrossRefGoogle Scholar
  10. Chen J, Lundberg K, Davison D, Bernstein D (2007) The final value theorem revisited: infinite limits and irrational functions. IEEE Control Syst 27:97–99CrossRefGoogle Scholar
  11. Davis MHA (1993) Markov models and optimization. Chapman & Hall, LondonCrossRefGoogle Scholar
  12. Eldar A, Elowitz MB (2010) Functional roles for noise in genetic circuits. Nature 467:167–173CrossRefGoogle Scholar
  13. Elowitz MB, Levine AJ, Siggia ED, Swain PS (2002) Stochastic gene expression in a single cell. Science 297:1183–1186CrossRefGoogle Scholar
  14. Friedman N, Cai L, Xie X (2006) Linking stochastic dynamics to population distribution: an analytical framework of gene expression. Phys Rev Lett 97:168,302CrossRefGoogle Scholar
  15. Golding I, Paulsson J, Zawilski S, Cox E (2005) Real-time kinetics of gene activity in individual bacteria. Cell 123:1025–1036CrossRefGoogle Scholar
  16. Gripenberg G, Londen SO, Staffans O (1990) Volterra Integral and Functional Equations. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  17. Guest PB (1991) Laplace transforms and an introduction to distributions. Ellis Horwood 7, New YorkGoogle Scholar
  18. Hawkins ED, Markham JF, McGuinness LP, Hodgkin P (2009) A single-cell pedigree analysis of alternative stochastic lymphocyte fates. Proc Natl Acad Sci 106(13):457–13, 462Google Scholar
  19. Huh D, Paulsson J (2011) Non-genetic heterogeneity from stochastic partitioning at cell division. Nat Genet 43:95–100CrossRefGoogle Scholar
  20. Innocentini G, Hornos J (2007) Modeling stochastic gene expression under repression. J Math Biol 55:413–431CrossRefGoogle Scholar
  21. Jia T, Kulkarni RV (2011) Intrinsic noise in stochastic models of gene expression with molecular memory and bursting. Phys Rev Lett 106:058,102CrossRefGoogle Scholar
  22. Kaern M, Elston T, Blake W, Collins J (2005) Stochasticity in gene expression: from theories to phenotypes. Nat Rev Genet 6:451–464CrossRefGoogle Scholar
  23. Kuang J, Tang M, Yu J (2013) The mean and noise of protein numbers in stochastic gene expression. J Math Biol 67:261–291CrossRefGoogle Scholar
  24. Kussell E, Leibler S (2005) Phenotypic diversity, population growth, and information in fluctuating environments. Science 309:2075–2078CrossRefGoogle Scholar
  25. Libby E, Perkins TJ, Swain PS (2007) Noisy information processing through transcriptional regulation. Proc Natl Acad Sci 104:7151–7156CrossRefGoogle Scholar
  26. Linz P (1985) Analytical and numerical methods for volterra equations., Chap 7. Studies in Applied and Numerical Mathematics, pp 95–127. ISBN 0-89871-198-3Google Scholar
  27. Losick R, Desplan C (2008) Stochasticity and cell fate. Science 320:65–68CrossRefGoogle Scholar
  28. Meyn S, Tweedie RL (2009) Markov chains and stochastic stability, 2nd edn. Cambridge University Press, New YorkCrossRefGoogle Scholar
  29. Munsky B, Trinh B, Khammash M (2009) Listening to the noise: random fluctuations reveal gene network parameters. Mol Syst Biol 5:318CrossRefGoogle Scholar
  30. Newman JRS, Ghaemmaghami S, Ihmels J, Breslow DK, Noble M, DeRisi JL, Weissman JS (2006) Single-cell proteomic analysis of S. cerevisiae reveals the architecture of biological noise. Nat Genet 441:840–846Google Scholar
  31. Paulsson J (2004) Summing up the noise in gene networks. Nature 427:415–418CrossRefGoogle Scholar
  32. Raj A, van Oudenaarden A (2008) Nature, nurture, or chance: stochastic gene expression and its consequences. Cell 135:216–226CrossRefGoogle Scholar
  33. Raj A, Peskin C, Tranchina D, Vargas D, Tyagi S (2006) Stochastic mRNA synthesis in mammalian cells. PLoS Biol 4:e309CrossRefGoogle Scholar
  34. Resnick SI (1992) Adventures in stochastic processes. Birkhauser, BaselGoogle Scholar
  35. Roeder A, Chickarmane V, Obara B, Manjunath B, Meyerowitz EM (2010) Variability in the control of cell division underlies sepal epidermal patterning in it Arabidopsis thaliana. PLoS Biol 8:e1000,367CrossRefGoogle Scholar
  36. Schwanhausser B, Busse D, Li N, Dittmar G, Schuchhardt J, Wolf J, Chen W, Selbach M (2011) Global quantification of mammalian gene expression control. Nature 473:337–342CrossRefGoogle Scholar
  37. Shahrezaei V, Swain PS (2008) Analytical distributions for stochastic gene expression. Proc Natl Acad Sci 105(17):256–17, 261Google Scholar
  38. Singh A, Dennehy JJ (2014) Stochastic holin expression can account for lysis time variation in the bacteriophage lambda. J Royal Soc Interf 11:20140,140CrossRefGoogle Scholar
  39. Singh A, Razooky B, Cox CD, Simpson ML, Weinberger LS (2010) Transcriptional bursting from the HIV-1 promoter is a significant source of stochastic noise in HIV-1 gene expression. Biophys J 98:L32–L34CrossRefGoogle Scholar
  40. Singh A, Razooky BS, Dar RD, Weinberger LS (2012) Dynamics of protein noise can distinguish between alternate sources of gene-expression variability. Mol Syst Biol 8:607CrossRefGoogle Scholar
  41. Singh A, Soltani M (2013) Quantifying intrinsic and extrinsic variability in stochastic gene expression models. PLoS One 8:e84,301CrossRefGoogle Scholar
  42. Singh A, Weinberger LS (2009) Stochastic gene expression as a molecular switch for viral latency. Curr Opin Microbiol 12:460–466CrossRefGoogle Scholar
  43. Stukalin EB, Aifuwa I, Kim JS, Wirtz D, Sun S (2013) Age-dependent stochastic models for understanding population fluctuations in continuously cultured cells. J Royal Soc Interf 10:20130,325CrossRefGoogle Scholar
  44. Taniguchi Y, Choi P, Li G, Chen H, Babu M, Hearn J, Emili A, Xie X (2010) Quantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells. Science 329:533–538CrossRefGoogle Scholar
  45. Veening JW, Smits WK, Kuipers OP (2008) Bistability, epigenetics, and bet-hedging in bacteria. Annu Rev Microbiol 62:193210CrossRefGoogle Scholar
  46. Weinberger LS, Burnett JC, Toettcher JE, Arkin A, Schaffer D (2005) Stochastic gene expression in a lentiviral positive-feedback loop: HIV-1 Tat fluctuations drive phenotypic diversity. Cell 122:169–182CrossRefGoogle Scholar
  47. Zilman A, Ganusov V, Perelson A (2010) Stochastic models of lymphocyte proliferation and death. PloS One 5:e12,775CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Control Systems Technology, Department of Mechanical EngineeringEindhoven University of TechnologyEindhovenThe Netherlands
  2. 2.Department of Electrical and Computer Engineering, Biomedical Engineering and Mathematical SciencesUniversity of DelawareNewarkUSA

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