Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Review Article
  • Published:

Stochastic modelling for quantitative description of heterogeneous biological systems

Nature Reviews Genetics volume 10, pages 122–133 (2009)Cite this article

Key Points

  • Cellular dynamics are intrinsically noisy, so mechanistic models must incorporate stochasticity if they are to adequately model experimental observations.

  • As well as intrinsic stochasticity in gene expression, there are other sources of noise and heterogeneity in cells and cell populations.

  • There is a well-developed framework for stochastic modelling, including algorithms for fast, approximate simulation of cellular dynamics.

  • Multiscale models are particularly challenging, and are likely to require the use of fast stochastic emulators.

  • Statistical modelling is concerned with relating models (either stochastic or deterministic) to experimental data, and as such is of key importance in systems biology.

  • Simple statistical models are useful for fitting to high-throughput data such as time course microarray data for uncovering structural relationships between genes.

  • The parameters of complex dynamic models can be estimated from high-resolution dynamic data using sophisticated statistical inference technology.

  • A nonlinear multivariate stochastic differential equation model known as the chemical Langevin equation provides a natural bridge between simple structural statistical models and detailed mechanistic dynamic models.

Abstract

Two related developments are currently changing traditional approaches to computational systems biology modelling. First, stochastic models are being used increasingly in preference to deterministic models to describe biochemical network dynamics at the single-cell level. Second, sophisticated statistical methods and algorithms are being used to fit both deterministic and stochastic models to time course and other experimental data. Both frameworks are needed to adequately describe observed noise, variability and heterogeneity of biological systems over a range of scales of biological organization.

This is a preview of subscription content, access via your institution

Access options

Buy this article

  • Purchase on Springer Link
  • Instant access to full article PDF
Buy now

Prices may be subject to local taxes which are calculated during checkout

Figure 1: Fluctuations in p53 and MDM2 levels in single cells.

Similar content being viewed by others

References

  1. Kitano, H. Computational systems biology. Nature 420, 206–210 (2002).

    Article  CAS  PubMed  Google Scholar 

  2. McAdams, H. H. and Arkin, A. It's a noisy business: genetic regulation at the nanomolecular scale. Trends Genet. 15, 65–69 (1999).

    Article  CAS  PubMed  Google Scholar 

  3. Finch, C. E. & Kirkwood, T. B. L. Chance Development and Aging (Oxford Univ. Press 2000).

    Google Scholar 

  4. Maltzman, W. & Czyzyk, L. UV irradiation stimulates levels of p53 cellular tumor antigen in nontransformed mouse cells. Mol. Cell. Biol. 4, 1689–1694 (1984).

    CAS  PubMed  PubMed Central  Google Scholar 

  5. Lev Bar-Or, R. et al. Generation of oscillations by the p53–mdm2 feedback loop: A theoretical and experimental study. Proc. Natl Acad. Sci. USA 97, 11250–11255 (2000).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  6. Lahav, G. et al. Dynamics of the p53–mdm2 feedback loop in individual cells. Nature Genet. 36, 147–150 (2004).

    Article  CAS  PubMed  Google Scholar 

  7. Geva-Zatorsky, N. et al. Oscillations and variability in the p53 system. Mol. Syst. Biol. 2, 2006.0033 (2006).

  8. Haupt, Y., Maya, R., Kazaz, A. & Oren, M. Mdm2 promotes the rapid degradation of p53. Nature 387, 296–299 (1997).

    Article  CAS  PubMed  Google Scholar 

  9. Clegg, H. V., Itahana, K. & Zhang, Y. Unlocking the mdm2–p53 loop: ubiquitin is the key. Cell Cycle 7, 287–292 (2008).

    Article  CAS  PubMed  Google Scholar 

  10. Hucka, M. et al. The systems biology markup language (SBML): a medium for representation and exchange of biochemical network models. Bioinformatics 19, 524–531 (2003).

    Article  CAS  PubMed  Google Scholar 

  11. Cornish-Bowden, A. Fundamentals of Enzyme Kinetics 3rd edn (Portland Press, 2004).

    Google Scholar 

  12. Ma, L. et al. A plausible model for the digital response of p53 to DNA damage. Proc. Natl Acad. Sci. USA 102, 14266–14271 (2005).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  13. Zhang, L. J., Yan, S. W. & Zhuo, Y. Z. A dynamical model of DNA-damage derived p53–mdm2 interaction. Acta Physica Sinica 56, 2442–2447 (2007).

    Article  CAS  Google Scholar 

  14. Proctor, C. J. & Gray, D. A. Explaining oscillations and variability in the p53–mdm2 system. BMC Syst. Biol. 2, 75 (2008).

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  15. Henderson, D. A., Boys, R. J., Proctor, C. J. & Wilkinson, D. J. in Handbook of Applied Bayesian Analysis (eds O'Hagan, A. & West, M.) (Oxford Univ. Press) (in the press).

  16. Bahcall, O. G. Single cell resolution in regulation of gene expression. Mol. Syst. Biol. 1, 2005.0015 (2005).

    Article  PubMed  PubMed Central  Google Scholar 

  17. Maheshri, N. & O'Shea, E. K. Living with noisy genes: how cells function reliably with inherent variability in gene expression. Annu. Rev. Biophys. Biomol. Struct. 36, 413–434 (2007).

    Article  CAS  PubMed  Google Scholar 

  18. Lehner, B. Selection to minimise noise in living systems and its implications for the evolution of gene expression. Mol. Syst. Biol. 4, 170 (2008).

    Article  PubMed  PubMed Central  Google Scholar 

  19. Ansel, J. Cell-to-cell stochastic variation in gene expression is a complex genetic trait. PLoS Genet. 4, e1000049 (2008).

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  20. Raser, J. M. & O'Shea, E. K. Noise in gene expression: origins, consequences, and control. Science 309, 2010–2013 (2005).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  21. Lopez-Maury, L., Marguerat, S. & Bahler, J. Tuning gene expression to changing environments: from rapid responses to evolutionary adaptation. Nature Rev. Genet. 9, 583–593 (2008).

    Article  CAS  PubMed  Google Scholar 

  22. Cox, D. R. & Miller, H. D. The Theory of Stochastic Processes (Chapman & Hall, London, 1977).

    Google Scholar 

  23. Gillespie, D. T. Markov Processes: an Introduction for Physical Scientists (Academic, New York, 1992).

    Google Scholar 

  24. Allen, L. J. S. Stochastic Processes with Applications to Biology (Pearson Prentice Hall, Upper Saddle River, 2003).

    Google Scholar 

  25. Wilkinson, D. J. Stochastic Modelling for Systems Biology (Chapman & Hall/CRC, Boca Raton, 2006).

    Book  Google Scholar 

  26. Gillespie, D. T. Exact stochastic simulation of coupled chemical reactions. J. Phys. Chem. 81, 2340–2361 (1977). The original description of the stochastic simulation algorithm for discrete event simulation of biochemical reaction networks.

    Article  CAS  Google Scholar 

  27. McAdams, H. H. & Arkin, A. Stochastic mechanisms in gene expression. Proc. Natl Acad. Sci. USA 94, 814–819 (1997).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  28. Zlokarnik, G. et al. Quantitation of transcription and clonal selection of single living cells with beta-lactamase as reporter. Science 279, 84–88 (1998).

    Article  CAS  PubMed  Google Scholar 

  29. Renshaw, E. Modelling Biological Populations in Space and Time (Cambridge Univ. Press, 1991).

    Book  Google Scholar 

  30. Li, H., Cao, Y., Petzold, L. R. & Gillespie, D. T. Algorithms and software for stochastic simulation of biochemical reacting systems. Biotechnol. Prog. 24, 56–61 (2007).

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  31. Higham, D. J. Modeling and simulating chemical reactions. SIAM Rev. 50, 347–368 (2008).

    Article  Google Scholar 

  32. Paulsson, J., Berg, O. & Ehrenberg, M. Stochastic focusing: fluctuation-enhanced sensitivity of intracellular regulation. Proc. Natl. Acad. Sci. USA 97, 7148–7153 (2000).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  33. Dupont, G., Abou-Lovergne, A. & Combettes, L. Stochastic aspects of oscillatory Ca2+ dynamics in hepatocytes. Biophys. J. 95, 2193–2202 (2008).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  34. Cai, L., Friedman, N. & Xie, X. S. Stochastic protein expression in individual cells at the single molecule level. Nature 440, 358–362 (2006).

    Article  CAS  PubMed  Google Scholar 

  35. Arkin, A., Ross, J. & McAdams, H. H. Stochastic kinetic analysis of developmental pathway bifurcation in phage λ-infected Escherichia coli cells. Genetics 149, 1633–1648 (1998). An important early example illustrating that stochastic kinetic models can describe important biological phenomena that cannot easily be understood using continuous deterministic models.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  36. Shahrezaei, V., Ollivier, J. and Swain, P. Colored extrinsic fluctuations and stochastic gene expression. Mol. Syst. Biol. 4, 196 (2008).

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  37. Gibson, M. A. & Bruck, J. Efficient exact stochastic simulation of chemical systems with many species and many channels. J. Phys. Chem. A 104, 1876–1889 (2000).

    Article  CAS  Google Scholar 

  38. Gillespie, D. T. Approximate accelerated stochastic simulation of chemically reacting systems. J. Chem. Phys. 115, 1716–1732 (2001).

    Article  CAS  Google Scholar 

  39. Gillespie, D. T. & Petzold, L. R. Improved leap-size selection for accelerated stochastic simulation. J. Chem. Phys. 119, 8229–8234 (2003).

    Article  CAS  Google Scholar 

  40. Kiehl, T. R., Mattheyses, R. M. & Simmons, M. K. Hybrid simulation of cellular behavior. Bioinformatics 20, 316–322 (2004).

    Article  CAS  PubMed  Google Scholar 

  41. Alfonsi, A., Cances, E., Turinici, G., di Ventura, B. & Huisinga, W. Adaptive simulation of hybrid stochastic and deterministic models for biochemical systems. ESAIM: Proc. 14, 1–13 (2005).

    Article  Google Scholar 

  42. Puchalka, J. & Kierzek, A. M. Bridging the gap between stochastic and deterministic regimes in the kinetic simulations of the biochemical reaction networks. Biophys. J. 86, 1357–1372 (2004).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  43. Rao, C. V. & Arkin, A. P. Stochastic chemical kinetics and the quasi-steady-state assumption: application to the Gillespie algorithm. J. Chem. Phys. 118, 4999–5010 (2003).

    Article  CAS  Google Scholar 

  44. Haseltine, E. L. & Rawlings, J. B. Approximate simulation of coupled fast and slow reactions for stochastic chemical kinetics. J. Chem. Phys. 117, 6959–6969 (2002).

    Article  CAS  Google Scholar 

  45. Salis, H. & Kaznessis, Y. Accurate hybrid stochastic simulation of a system of coupled chemical or biochemical reactions. J. Chem. Phys. 122, 054103 (2005).

    Article  CAS  Google Scholar 

  46. Cao, Y., Gillespie, D. T. & Petzold, L. Multiscale stochastic simulation algorithm with stochastic partial equilibrium assumption for chemically reacting systems. J. Comput. Phys. 206, 395–411 (2005).

    Article  CAS  Google Scholar 

  47. Samant, A. & Vlachos, D. G. Overcoming stiffness in stochastic simulation stemming from partial equilibrium: a multiscale Monte Carlo algorithm. J. Chem. Phys. 123, 144114 (2005).

    Article  CAS  PubMed  Google Scholar 

  48. Weinan, E., Liu, D. & Vanden-Eijnden, E. Nested stochastic simulation algorithm for chemical kinetic systems with disparate rates. J. Chem. Phys. 123, 194107 (2005).

    Article  CAS  Google Scholar 

  49. Weinan, E., Liu, D. & Vanden-Eijnden, E. Nested stochastic simulation algorithms for chemical kinetic systems with multiple time scales. J. Comput. Phys. 221, 158–180 (2007).

    Article  CAS  Google Scholar 

  50. Gillespie, D. T. The chemical Langevin equation. J. Chem. Phys. 113, 297–306 (2000). A well presented and accessible introduction to the chemical Langevin equation.

    Article  CAS  Google Scholar 

  51. Cyganowski, S., Kloeden, P. & Ombach, J. From Elementary Probability to Stochastic Differential Equations with MAPLE (Springer, New York, 2002).

    Book  Google Scholar 

  52. Kloeden, P. E. & Platen, E. Numerical Solution of Stochastic Differential Equations (Springer, New York, 1992).

    Book  Google Scholar 

  53. Swain, P. S., Elowitz, M. B. & Siggia, E. D. Intrinsic and extrinsic contributions to stochasticity in gene expression. Proc. Natl Acad. Sci. USA 99, 12795–12800 (2002).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  54. Gillespie, C. S. et al. A mathematical model of ageing in yeast. J. Theor. Biol. 44, 493–516 (2004).

    Google Scholar 

  55. Tanase-Nicola, S. & ten Wolde, P. R. Regulatory control and the costs and benefits of biochemical noise. PLoS Comput. Biol. 4, e1000125 (2008).

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  56. Speed, T. P. (ed.) Statistical Analysis of Gene Expression Microarray Data (Chapman & Hall/CRC, Boca Raton 2003).

    Book  Google Scholar 

  57. Wit, E. & McClure, J. Statistics for Microarrays: Design, Analysis and Inference (Wiley, New York, 2004).

    Book  Google Scholar 

  58. O'Hagan, A. & Forster, J. J. Kendall's Advanced Theory of Statistics Vol. 2B (Arnold, London, 2004).

    Google Scholar 

  59. Gelman, A., Carlin, J. B., Stern, H. S. & Rubin, D. B. Bayesian Data Analysis 2nd edn (Chapman & Hall/CRC, Boca Raton, 2003).

    Book  Google Scholar 

  60. Vanucci, M., Do, K.-A. & Muller, P. (eds) Bayesian Inference for Gene Expression and Proteomics (Cambridge Univ. Press, New York 2006).

    Google Scholar 

  61. Hein, A.-M. K., Richardson, S., Causton, H. C., Ambler, G. K. & Green, P. J. BGX: a fully Bayesian integrated approach to the analysis of Affymetrix Gene Chip data. Biostatistics 6, 349–373 (2005).

    Article  PubMed  Google Scholar 

  62. Lewin, A., Richardson, S., Marshall, C., Glazier, A. & Aitman, T. Bayesian modelling of differential gene expression. Biometrics 62, 10–18 (2006).

    Article  Google Scholar 

  63. Friedman, N. Inferring cellular networks using probabilistic graphical models. Science 303, 799–805 (2004).

    Article  CAS  PubMed  Google Scholar 

  64. Pournara, I. & Wernisch, L. Reconstruction of gene networks using Bayesian learning and manipulation experiments. Bioinformatics 20, 2934–2942 (2004).

    Article  CAS  PubMed  Google Scholar 

  65. Beal, M. J., Falciani, F., Ghahramani, Z., Rangel, C. & Wild, D. L. A Bayesian approach to reconstructing genetic regulatory networks with hidden factors. Bioinformatics 21, 349–356, (2005).

    Article  CAS  PubMed  Google Scholar 

  66. Schafer, J. & Strimmer, K. An empirical Bayes approach to inferring large-scale gene association networks. Bioinformatics 21, 754–764 (2005).

    Article  PubMed  Google Scholar 

  67. Werhli, A. V., Grzegorczyk, M. & Husmeier, D. Comparative evaluation of reverse engineering gene regulatory networks with relevance networks, graphical Gaussian models and Bayesian networks. Bioinformatics 22, 2523–2531 (2006).

    Article  CAS  PubMed  Google Scholar 

  68. Dobra, A. et al. Sparse graphical models for exploring gene expression data. J. Multivar. Anal. 90, 196–212 (2004).

    Article  Google Scholar 

  69. Jones, B. et al. Experiments in stochastic computation for high-dimensional graphical models. Stat. Sci. 20, 388–400 (2005).

    Article  Google Scholar 

  70. Husmeier, D. Sensitivity and specificity of inferring genetic regulatory interactions from microarray experiments with dynamic Bayesian networks. Bioinformatics 19, 2271–2282 (2003).

    Article  CAS  PubMed  Google Scholar 

  71. Yu, J., Smith, V. A., Wang, P. P., Hartemink, A. J. & Jarvis, E. D. Advances to Bayesian network inference for generating causal networks from observational data. Bioinformatics 20, 3594–3603 (2004).

    Article  CAS  PubMed  Google Scholar 

  72. Opgen-Rhein, R. & Strimmer, K. Learning causal networks from systems biology time course data: an effective model selection procedure for the vector autoregressive process. BMC Bioinformatics 8 (Suppl. 2), S3 (2007). The first paper to explore the use of sparse vector autoregressive models for inferring causal genetic regulatory relationships.

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  73. George, E., Sun, D. & Ni, S. Bayesian stochastic search for VAR model restrictions. J. Econom. 142, 553–580 (2008).

    Article  Google Scholar 

  74. Pepperkok, R. & Ellenberg, J. High-throughput fluorescence microscopy for systems biology. Nature Rev. Mol. Cell Biol. 7, 690–696 (2006).

    Article  CAS  Google Scholar 

  75. Shen, H. et al. Automated tracking of gene expression profiles in individual cells and cell compartments. J. R. Soc. Interface 3, 787 (2006).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  76. Jaqaman, K. & Danuser, G. Linking data to models: data regression. Nature Rev. Mol. Cell Biol. 7, 813–819 (2006).

    Article  CAS  Google Scholar 

  77. Moles, C. G., Mendes, P. & Banga, J. R. Parameter estimation in biochemical pathways: a comparison of global optimization methods. Genome Res. 13, 2467–2474 (2003).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  78. Brown, K. S. & Sethna, J. P. Statistical mechanical approaches to models with many poorly known parameters. Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 68, 021904 (2003). An early example of applying MCMC methods for inferring parameters of continuous deterministic models.

    Article  PubMed  CAS  Google Scholar 

  79. Barenco, M. et al. Ranked prediction of p53 targets using hidden variable dynamic modeling. Genome Biol. 7, R25 (2006).

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  80. Vyshemirsky, V. & Girolami, M. Bayesian ranking of biochemical system models. Bioinformatics 24, 833 (2008). Describes the use of MCMC for parameter inference and model selection using deterministic models.

    Article  CAS  PubMed  Google Scholar 

  81. Liebermeister, W. & Klipp, E. Biochemical networks with uncertain parameters. IEE Syst. Biol. 152, 97–107 (2005).

    Article  CAS  Google Scholar 

  82. Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H. & Teller, E. Equations of state calculations by fast computing machines. J. Chem. Phys. 21, 1087–1092 (1953).

    Article  CAS  Google Scholar 

  83. Hastings, W. K. Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57, 97–109 (1970).

    Article  Google Scholar 

  84. Gamerman, D. Markov Chain Monte Carlo (Texts in Statistical Science) (Chapman & Hall, New York, 1997).

    Google Scholar 

  85. Vyshemirsky, V. & Girolami, M. BioBayes: a software package for Bayesian inference in systems biology. Bioinformatics 24, 1933–1934 (2008).

    Article  CAS  PubMed  Google Scholar 

  86. Reinker, S., Altman, R. M. & Timmer, J. Parameter estimation in stochastic biochemical reactions. IEE Syst. Biol. 153, 168–178 (2006).

    Article  CAS  Google Scholar 

  87. Tian, T., Xu, S., Gao, J. & Burrage, K. Simulated maximum likelihood method for estimating kinetic rates in gene expression. Bioinformatics 23, 84–91 (2007).

    Article  CAS  PubMed  Google Scholar 

  88. Boys, R. J., Wilkinson, D. J. & Kirkwood, T. B. L. Bayesian inference for a discretely observed stochastic kinetic model. Stat. Comput. 18, 125–135 (2008). The first paper to demonstrate the possibility of conducting fully Bayesian inference for the parameters of stochastic kinetic models.

    Article  Google Scholar 

  89. Rempala, G. A., Ramos, K. S. & Kalbfleisch, T. A stochastic model of gene transcription: an application to L1 retrotransposition events. J. Theor. Biol. 242, 101–116 (2006).

    Article  CAS  PubMed  Google Scholar 

  90. Iacus, S. M. Simulation and Inference for Stochastic Differential Equations — with R Examples (Springer, New York, 2008).

    Book  Google Scholar 

  91. Golightly, A. & Wilkinson, D. J. Bayesian inference for stochastic kinetic models using a diffusion approximation. Biometrics 61, 781–788 (2005).

    Article  CAS  PubMed  Google Scholar 

  92. Heron, E. A., Finkenstadt, B. & Rand, D. A. Bayesian inference for dynamic transcriptional regulation; the Hes1 system as a case study. Bioinformatics 23, 2596–2603 (2007).

    Article  CAS  PubMed  Google Scholar 

  93. Golightly, A. & Wilkinson, D. J. Bayesian sequential inference for stochastic kinetic biochemical network models. J. Comput. Biol. 13, 838–851 (2006). Describes using Bayesian inference for stochastic kinetic models using multiple, partial and noisy experimental data sets.

    Article  CAS  PubMed  Google Scholar 

  94. Golightly, A. & Wilkinson, D. J. Bayesian inference for nonlinear multivariate diffusion models observed with error. Comput. Stat. Data Anal. 52, 1674–1693 (2008).

    Article  Google Scholar 

  95. Kennedy, M. C. & O'Hagan, A. Bayesian calibration of computer models. J. R. Stat. Soc. Ser. B 63, 425–464 (2001).

    Article  Google Scholar 

  96. Goldstein, M. & Rougier, J. Bayes linear calibrated prediction for complex systems. J. Am. Stat. Assoc. 101, 1132–114 (2006).

    Article  CAS  Google Scholar 

  97. Challenor, P. G., Hankin, R. K. S. & Marsh, R. in Avoiding Dangerous Climate Change (Schellnhuber, H. J., Cramer, W., Nakicenovic, N., Wigley, T. & Yohe, G. eds) 53–63 (Cambridge Univ. Press, 2006).

    Google Scholar 

  98. Henderson, D. A., Boys, R. J., Krishnan, K. J., Lawless, C. & Wilkinson, D. J. Bayesian emulation and calibration of a stochastic computer model of mitochondrial DNA deletions in substantia nigra neurons. J. Am. Stat. Assoc. (in the press). The first example of using inference for a single-cell model based on cell population data and a statistical emulator of a stochastic cell population model.

  99. Uhlenbeck, G. E. & Ornstein, L. S. On the theory of Brownian motion. Phys. Rev. 36, 823–841 (1930).

    Article  CAS  Google Scholar 

  100. Orlando, D. et al. A probabilistic model for cell cycle distributions in synchrony experiments. Cell Cycle 6, 478–488 (2007).

    Article  CAS  PubMed  Google Scholar 

  101. Orlando, D. et al. Global control of cell-cycle transcription by coupled CDK and network oscillators. Nature 453, 944–947 (2008).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  102. Beaumont, M. A. & Rannala, B. The Bayesian revolution in genetics. Nature Rev. Genet. 5, 251–261 (2004).

    Article  CAS  PubMed  Google Scholar 

  103. Wilkinson, D. J. Bayesian methods in bioinformatics and computational systems biology. Brief. Bioinformatics 8, 109–116 (2007).

    Article  CAS  PubMed  Google Scholar 

  104. Schultz, D., Jacob, E. B., Onuchic, J. N. & Wolynes, P. G. Molecular level stochastic model for competence cycles in Bacillus subtilis. Proc. Natl Acad. Sci. USA 104, 17582–17587 (2007).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  105. Smits, W. K. et al. Stripping Bacillus: ComK auto-stimulation is responsible for the bistable response in competence development. Mol. Microbiol. 56, 604–614 (2005).

    Article  CAS  PubMed  Google Scholar 

  106. Veening, J.-W., Hamoen, L. W. & Kuipers, O. P. Phosphatases modulate the bistable sporulation gene expression pattern in Bacillus subtilis. Mol. Microbiol. 56, 1481–1494 (2005).

    Article  CAS  PubMed  Google Scholar 

  107. Veening, J.-W. et al. Transient heterogeneity in extracellular protease production by Bacillus subtilis. Mol. Syst. Biol. 4, 184 (2008).

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  108. Shimizu, T. S., Aksenov, S. V. & Bray, D. A spatially extended stochastic model of the bacterial chemotaxis signalling pathway. J. Mol. Biol. 329, 291–309 (2003).

    Article  CAS  PubMed  Google Scholar 

  109. Fraser, H. B., Hirsh, A. E., Giaever, G., Kumm, J. & Eisen M. B. Noise minimization in eukaryotic gene expression. PLoS Biol. 2, e137 (2004).

    Article  PubMed  PubMed Central  Google Scholar 

  110. Bar-Even, A. et al. Noise in protein expression scales with natural protein abundance. Nature Genet. 38, 636–643 (2006).

    Article  CAS  PubMed  Google Scholar 

  111. Newman, J. et al. Single-cell proteomic analysis of S. cerevisiae reveals the architecture of biological noise. Nature 441, 840–846 (2006).

    Article  CAS  PubMed  Google Scholar 

  112. Kirkwood, T. B. L. et al. Towards an e-biology of ageing: integrating theory and data. Nature Rev. Mol. Cell Biol. 4, 243–249 (2003).

    Article  CAS  Google Scholar 

  113. Kirkwood, T. B. L. et al. in Handbook of the Biology of Aging 6th edn (eds Masoro, E. J. & Austad, S. N.) 334–357 (Academic, New York, 2005).

    Book  Google Scholar 

  114. Proctor, C. J. et al. Modelling the checkpoint response to telomere uncapping in budding yeast. J. R. Soc. Interface 4, 73–90 (2007).

    Article  CAS  PubMed  Google Scholar 

  115. Proctor, C. J. et al. Modelling the action of chaperones and their role in ageing. Mech. Ageing Dev. 126, 119–131 (2005).

    Article  CAS  PubMed  Google Scholar 

  116. Kowald, A. & Kirkwood, T. B. Towards a network theory of ageing: a model combining the free radical theory and the protein error theory. J. Theor. Biol. 168, 75–94 (1994).

    Article  CAS  PubMed  Google Scholar 

  117. de Sozou, P. & Kirkwood, T. B. L. A stochastic model of cell replicative senescence based on telomere shortening, oxidative stress, and somatic mutations in nuclear and mitochondrial DNA. J. Theor. Biol. 213, 573 (2001).

    Article  CAS  Google Scholar 

Download references

Acknowledgements

The author would like to thank three anonymous referees for numerous suggestions that have helped to improve this article. This work was funded by the Biotechnology and Biological Sciences Research Council through grants BBF0235451, BBSB16550 and BBC0082001.

Author information

Authors and Affiliations

  1. School of Mathematics & Statistics and the Centre for Integrated Systems Biology of Ageing and Nutrition (CISBAN), Newcastle University, Newcastle upon Tyne, NE1 7RU, Tyne and Wear, UK

    Darren J. Wilkinson

Authors
  1. Darren J. Wilkinson
    View author publications

    You can also search for this author in PubMed Google Scholar

Related links

Related links

FURTHER INFORMATION

Darren J. Wilkinson's homepage

BioBayes

Biology of ageing e-science integration and simulation system (BASIS)

BioModels database

CaliBayes

FERN — A Java Framework for Stochastic Simulation and Evaluation of Reaction Networks

Systems Biology Markup Language (SBML)

SloppyCell

Glossary

Continuous deterministic mathematical model

A model that does not contain any element of unpredictability, and that describes the smooth and gradual change of model elements (such as biochemical substances) according to pre-determined mathematical rules. The precise behaviour of the model is entirely pre-determined (and hence, in principle, predictable) from the form of the equations and the starting conditions.

Stochastic model

A model that contains an element of unpredictability or randomness specified in a precise mathematical way. Each run of a given model will produce different results, but the statistical properties of the results of many such runs are pre-determined by the mathematical formulation of the model.

Michaelis–Menten

A simple kinetic law that modifies the rate of conversion from substrate to product based on enzyme concentration.

Hill kinetics

A more complex enzyme kinetic law than simple Michaelis–Menten kinetics.

Ordinary differential equation

A mathematical equation involving differential calculus. In simple cases, explicit formulas can be derived for their solution, but typically they must be numerically integrated on a computer.

Probability theory

The mathematical theory of chance, randomness, uncertainty and stochasticity.

Markov jump process

A class of stochastic processes that is well studied in probability theory and that includes the class of processes described by stochastic chemical kinetics.

Stochastic chemical kinetics

A chemical kinetic theory which recognizes that molecules are discrete entities, and that reaction events occur at random when particular combinations of molecules interact.

Probability distribution

A precise mathematical description of a stochastic quantity.

Stochastic simulation algorithm

In the context of stochastic chemical kinetics, this refers to an exact discrete event simulation algorithm for generating time course trajectories of chemical reaction network models.

Monte Carlo error

The unavoidable error associated with estimating a population quantity from a finite number of stochastic samples from the population. It can often be reduced by averaging large numbers of samples.

Intrinsic noise

A crude categorization of stochasticity in biological systems that loosely corresponds to noise that cannot be controlled for.

Diffusion process

A stochastic process continuous in both time and space and that can be described by a stochastic differential equation.

Next reaction method

An alternative exact simulation algorithm to the stochastic simulation algorithm, which in certain situations can be faster.

Diffusion approximation

A diffusion process that approximates a Markov jump process.

Chemical Langevin equation

(CLE). A diffusion approximation to a stochastic chemical kinetic model.

Stochastic differential equation

(SDE). A mathematical equation involving both differential calculus and a stochastic process (typically Brownian motion). Simple cases can be 'solved' exactly, but typically solutions must be generated using a stochastic form of numerical integration.

Numerical integration

An algorithm (typically implemented on a computer) for generating approximate solutions to ordinary differential equations.

Multiscale model

A model that spans multiple scales in space and/or time. Such models generally require approximate algorithmic solutions, and are often computationally intensive.

Extrinsic noise

A crude categorization of stochasticity in biological systems that loosely corresponds to noise that can be controlled for.

Fluorescence-activated cell sorting

(FACS). An experimental technology that can be used to make quantitative measurements on a cell population with single-cell resolution. It is particularly useful for quantifying heterogeneity in cell populations.

Bayesian methods

Fully probabilistic methods for describing models, parameters and data. So called because extensive use is made of Bayes theorem to compute the probability distribution of model parameters given the experimental data.

Likelihood

The probability of the data given the statistical model and its parameters. In classical statistics it is often regarded as a function of the model parameters for given fixed experimental data.

Identifiability

The extent to which it is possible to accurately estimate model parameters given sufficient experimental data.

Confounded

Describes a problematic situation that arises when only a subset of a given set of model parameters is identifiable.

Model selection

The assessment of which model among a class of models has the most support on the basis of the available experimental data.

Markov chain Monte Carlo

(MCMC). A powerful class of algorithms that can be used to provide numerical solutions to most problems in Bayesian analysis. For complex problems they are notoriously computationally expensive, and many obscure techniques exist to increase the rate of convergence.

Posterior distribution

A probability distribution describing information about model parameters having taken into account all available information in the experimental data. From this it is possible to extract parameter estimates, together with associated levels of uncertainty.

Emulator

A fast surrogate for a more complex, and hence slower, computational model. Emulators are often used in place of the original model in iterative algorithms that require many model evaluations.

Time-discretized

The conversion of a continuous time model to a discrete time model, formed by considering the states of the continuous time model only at given discrete times.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wilkinson, D. Stochastic modelling for quantitative description of heterogeneous biological systems. Nat Rev Genet 10, 122–133 (2009). https://doi.org/10.1038/nrg2509

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1038/nrg2509

This article is cited by

Search

Advanced search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing