Journal of Mathematical Biology

, Volume 64, Issue 5, pp 829–854 | Cite as

Exact and approximate distributions of protein and mRNA levels in the low-copy regime of gene expression

  • Pavol Bokes
  • John R. King
  • Andrew T. A. Wood
  • Matthew Loose


Gene expression at the single-cell level incorporates reaction mechanisms which are intrinsically stochastic as they involve molecular species present at low copy numbers. The dynamics of these mechanisms can be described quantitatively using stochastic master-equation modelling; in this paper we study a generic gene-expression model of this kind which explicitly includes the representations of the processes of transcription and translation. For this model we determine the generating function of the steady-state distribution of mRNA and protein counts and characterise the underlying probability law using a combination of analytic, asymptotic and numerical approaches, finding that the distribution may assume a number of qualitatively distinct forms. The results of the analysis are suitable for comparison with single-molecule resolution gene-expression data emerging from recent experimental studies.


Stochastic modelling Gene expression Master equation Generating function 

Mathematics Subject Classification (2000)

92C40 60J27 


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  1. Abramowitz M, Stegun I (1972) Handbook of mathematical functions with formulas, graphs, and mathematical tables. National Bureau of Standards, Washington, DCzbMATHGoogle Scholar
  2. Ackers G, Johnson A, Shea M (1982) Quantitative model for gene regulation by lambda phage repressor. Proc Natl Acad Sci USA 79: 1129–1133CrossRefGoogle Scholar
  3. Bailey N (1964) The elements of stochastic processes with applications to the natural sciences. Wiley, New YorkzbMATHGoogle Scholar
  4. Belle A, Tanay A, Bitincka L, Shamir R, O’Shea E (2006) Quantification of protein half-lives in the budding yeast proteome. Proc Natl Acad Sci USA 103: 13004–13009CrossRefGoogle Scholar
  5. Berg O (1978) A model for the statistical fluctuations of protein numbers in a microbial population. J Theor Biol 71: 587–603CrossRefGoogle Scholar
  6. Bernstein J, Khodursky A, Lin P, Lin-Chao S, Cohen S (2002) Global analysis of mRNA decay and abundance in Escherichia coli at single-gene resolution using two-color fluorescent DNA microarrays. Proc Natl Acad Sci USA 99: 9697–9702CrossRefGoogle Scholar
  7. Blake W, Kaern M, Cantor C, Collins J (2003) Noise in eukaryotic gene expression. Nature 422: 633–637CrossRefGoogle Scholar
  8. Cai L, Friedman N, Xie X (2006) Stochastic protein expression in individual cells at the single molecule level. Nature 440: 358–362CrossRefGoogle Scholar
  9. Cheong R, Paliwal S, Levchenko A (2010) Models at the single cell level. Wiley Interdiscip Rev Syst Biol Med 2: 34–48CrossRefGoogle Scholar
  10. Coulon A, Gandrillon O, Beslon G (2010) On the spontaneous stochastic dynamics of a single gene: complexity of the molecular interplay at the promoter. BMC Syst Biol 4: 2CrossRefGoogle Scholar
  11. Cox D, Miller H (1977) The theory of stochastic processes. Chapman & Hall/CRC, LondonzbMATHGoogle Scholar
  12. Davidson E, Rast J, Oliveri P, Ransick A, Calestani C, Yuh C, Minokawa T, Amore G, Hinman V, Arenas-Mena C et al (2002) A genomic regulatory network for development. Science 295: 1669–1678CrossRefGoogle Scholar
  13. Elowitz M, Levine A, Siggia E, Swain P (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. Gadgil C, Lee C, Othmer H (2005) A stochastic analysis of first-order reaction networks. B Math Biol 67: 901–946MathSciNetCrossRefGoogle Scholar
  16. García-Martínez J, González-Candelas F, Pérez-Ortín J (2007) Common gene expression strategies revealed by genome-wide analysis in yeast. Genome Biol 8: R222CrossRefGoogle Scholar
  17. Golding I, Paulsson J, Zawilski S, Cox E (2005) Real-time kinetics of gene activity in individual bacteria. Cell 123: 1025–1036CrossRefGoogle Scholar
  18. Griffith J (1968a) Mathematics of cellular control processes. I. Negative feedback to one gene. J Theor Biol 20: 202–208CrossRefGoogle Scholar
  19. Griffith J (1968b) Mathematics of cellular control processes. II. Positive feedback to one gene. J Theor Biol 20: 209–216CrossRefGoogle Scholar
  20. Gurland J (1958) A generalized class of contagious distributions. Biometrics 14: 229–249zbMATHCrossRefGoogle Scholar
  21. Hornos J, Schultz D, Innocentini G, Wang J, Walczak A, Onuchic J, Wolynes P (2005) Self-regulating gene: an exact solution. Phys Rev E 72: 051,907MathSciNetCrossRefGoogle Scholar
  22. Innocentini G, Hornos J (2007) Modeling stochastic gene expression under repression. J Math Biol 55: 413–431MathSciNetzbMATHCrossRefGoogle Scholar
  23. Iyer-Biswas S, Hayot F, Jayaprakash C (2009) Stochasticity of gene products from transcriptional pulsing. Phys Rev E 79: 031,911CrossRefGoogle Scholar
  24. Johnson N, Kotz S, Kemp A (2005) Univariate discrete distributions, 3rd edn. Wiley-Interscience, LondonzbMATHCrossRefGoogle Scholar
  25. Johnson W (2002) The curious history of Faà di Bruno’s formula. Am Math Mon 109: 217–234zbMATHCrossRefGoogle Scholar
  26. Kendall D (1949) Stochastic processes and population growth. J Roy Stat Soc B 11: 230–282MathSciNetGoogle Scholar
  27. Larson D, Singer R, Zenklusen D (2009) A single molecule view of gene expression. Trends Cell Biol 19: 630–637CrossRefGoogle Scholar
  28. Lee T, Rinaldi N, Robert F, Odom D, Bar-Joseph Z, Gerber G, Hannett N, Harbison C, Thompson C, Simon I et al (2002) Transcriptional regulatory networks in Saccharomyces cerevisiae. Science 298: 799–804CrossRefGoogle Scholar
  29. Lestas I, Paulsson J, Ross N, Vinnicombe G (2008) Noise in gene regulatory networks. IEEE T Circuits-I 53: 189–200Google Scholar
  30. Lewin B (2000) Genes VII. Oxford University Press, OxfordGoogle Scholar
  31. Lu P, Vogel C, Wang R, Yao X, Marcotte E (2007) Absolute protein expression profiling estimates the relative contributions of transcriptional and translational regulation. Nat Biotechnol 25: 117–124CrossRefGoogle Scholar
  32. McAdams H, Arkin A (1997) Stochastic mechanisms in gene expression. Proc Natl Acad Sci USA 94: 814–819CrossRefGoogle Scholar
  33. McAdams H, Arkin A (1999) It is a noisy business! Genetic regulation at the nanomolar scale. Trends Genet 15: 65–69CrossRefGoogle Scholar
  34. Neyman J (1939) On a new class of “contagious” distributions, applicable in entomology and bacteriology. Ann Math Stat 10: 35–57zbMATHCrossRefGoogle Scholar
  35. Ozbudak E, Thattai M, Kurtser I, Grossman A, van Oudenaarden A (2002) Regulation of noise in the expression of a single gene. Nat Genet 31: 69–73CrossRefGoogle Scholar
  36. Paszek P (2007) Modeling stochasticity in gene regulation: characterization in the terms of the underlying distribution function. B Math Biol 69: 1567–1601MathSciNetzbMATHCrossRefGoogle Scholar
  37. Paulsson J (2004) Summing up the noise in gene networks. Nature 427: 415–418CrossRefGoogle Scholar
  38. Paulsson J (2005) Models of stochastic gene expression. Phys Life Rev 2: 157–175CrossRefGoogle Scholar
  39. Paulsson J, Ehrenberg M (2000) Random signal fluctuations can reduce random fluctuations in regulated components of chemical regulatory networks. Phys Rev Lett 84: 5447–5450CrossRefGoogle Scholar
  40. Peccoud J, Ycart B (1995) Markovian modeling of gene-product synthesis. Theor Popul Biol 48: 222–234zbMATHCrossRefGoogle Scholar
  41. Press W, Teukolsky S, Vetterling W, Flannery B (2007) Numerical recipes: the art of scientific computing. Cambridge university press, CambridgezbMATHGoogle Scholar
  42. Raj A, van Oudenaarden A (2009) Single-molecule approaches to stochastic gene expression. Annu Rev Biophys 38: 255–270CrossRefGoogle Scholar
  43. Raj A, Peskin C, Tranchina D, Vargas D, Tyagi S (2006) Stochastic mRNA synthesis in mammalian cells. PLoS Biol 4: e309CrossRefGoogle Scholar
  44. Raser J, O’Shea E (2004) Control of stochasticity in eukaryotic gene expression. Science 304: 1811–1814CrossRefGoogle Scholar
  45. Shahrezaei V, Swain P (2008a) Analytical distributions for stochastic gene expression. Proc Natl Acad Sci USA 105: 17,256CrossRefGoogle Scholar
  46. Shahrezaei V, Swain P (2008b) The stochastic nature of biochemical networks. Curr Opin Biotechnol 19: 369–374CrossRefGoogle Scholar
  47. Shea M, Ackers G (1985) The OR control system of bacteriophage lambda. A physical–chemical model for gene regulation. J Mol Biol 181: 211–230CrossRefGoogle Scholar
  48. Shen-Orr S, Milo R, Mangan S, Alon U (2002) Network motifs in the transcriptional regulation network of Escherichia coli. Nat Genet 31: 64–68CrossRefGoogle Scholar
  49. Singh A, Hespanha J (2007) Stochastic analysis of gene regulatory networks using moment closure. In: Proceedings of the American control conferenceGoogle Scholar
  50. Swiers G, Patient R, Loose M (2006) Genetic regulatory networks programming hematopoietic stem cells and erythroid lineage specification. Dev Biol 294: 525–540CrossRefGoogle Scholar
  51. 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
  52. Thattai M, van Oudenaarden A (2001) Intrinsic noise in gene regulatory networks. Proc Natl Acad Sci USA 98: 151588,598CrossRefGoogle Scholar
  53. Tomioka R, Kimura H, J Kobayashi T, Aihara K (2004) Multivariate analysis of noise in genetic regulatory networks. J Theor Biol 229: 501–521CrossRefGoogle Scholar
  54. Tomlin C, Axelrod J (2007) Biology by numbers: mathematical modelling in developmental biology. Nat Rev Genet 8: 331–340CrossRefGoogle Scholar
  55. Tyson J, Chen K, Novak B (2003) Sniffers, buzzers, toggles and blinkers: dynamics of regulatory and signaling pathways in the cell. Curr Opin Cell Biol 15: 221–231CrossRefGoogle Scholar
  56. van Kampen N (2006) Stochastic processes in physics and chemistry. Elsevier, New YorkGoogle Scholar
  57. Wang Y, Liu C, Storey J, Tibshirani R, Herschlag D, Brown P (2002) Precision and functional specificity in mRNA decay. Proc Natl Acad Sci USA 99: 5860–5865CrossRefGoogle Scholar
  58. Xie X, Choi P, Li G, Lee N, Lia G (2008) Single-molecule approach to molecular biology in living bacterial cells. Annu Rev Biophys 37: 417–444CrossRefGoogle Scholar
  59. Yu J, Xiao J, Ren X, Lao K, Xie X (2006) Probing gene expression in live cells, one protein molecule at a time. Science 311: 1600–1603CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • Pavol Bokes
    • 1
    • 2
  • John R. King
    • 1
  • Andrew T. A. Wood
    • 1
  • Matthew Loose
    • 3
  1. 1.Centre for Mathematical Medicine and Biology, School of Mathematical SciencesUniversity of NottinghamNottinghamUK
  2. 2.Department of Applied Mathematics and StatisticsComenius UniversityBratislavaSlovakia
  3. 3.Institute of Genetics, Queen’s Medical CentreUniversity of NottinghamNottinghamUK

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