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.
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
Open Access articles citing this article.
A probabilistic framework for particle-based reaction–diffusion dynamics using classical Fock space representations
Letters in Mathematical Physics Open Access 19 May 2022
BioMedical Engineering OnLine Open Access 11 February 2022
Quantifying the optimal strategy of population control of quorum sensing network in Escherichia coli
npj Systems Biology and Applications Open Access 02 September 2021
Subscribe to this journal
Receive 12 print issues and online access
$189.00 per year
only $15.75 per issue
Rent or buy this article
Prices vary by article type
Prices may be subject to local taxes which are calculated during checkout
Kitano, H. Computational systems biology. Nature 420, 206–210 (2002).
McAdams, H. H. and Arkin, A. It's a noisy business: genetic regulation at the nanomolecular scale. Trends Genet. 15, 65–69 (1999).
Finch, C. E. & Kirkwood, T. B. L. Chance Development and Aging (Oxford Univ. Press 2000).
Maltzman, W. & Czyzyk, L. UV irradiation stimulates levels of p53 cellular tumor antigen in nontransformed mouse cells. Mol. Cell. Biol. 4, 1689–1694 (1984).
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).
Lahav, G. et al. Dynamics of the p53–mdm2 feedback loop in individual cells. Nature Genet. 36, 147–150 (2004).
Geva-Zatorsky, N. et al. Oscillations and variability in the p53 system. Mol. Syst. Biol. 2, 2006.0033 (2006).
Haupt, Y., Maya, R., Kazaz, A. & Oren, M. Mdm2 promotes the rapid degradation of p53. Nature 387, 296–299 (1997).
Clegg, H. V., Itahana, K. & Zhang, Y. Unlocking the mdm2–p53 loop: ubiquitin is the key. Cell Cycle 7, 287–292 (2008).
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).
Cornish-Bowden, A. Fundamentals of Enzyme Kinetics 3rd edn (Portland Press, 2004).
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).
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).
Proctor, C. J. & Gray, D. A. Explaining oscillations and variability in the p53–mdm2 system. BMC Syst. Biol. 2, 75 (2008).
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).
Bahcall, O. G. Single cell resolution in regulation of gene expression. Mol. Syst. Biol. 1, 2005.0015 (2005).
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).
Lehner, B. Selection to minimise noise in living systems and its implications for the evolution of gene expression. Mol. Syst. Biol. 4, 170 (2008).
Ansel, J. Cell-to-cell stochastic variation in gene expression is a complex genetic trait. PLoS Genet. 4, e1000049 (2008).
Raser, J. M. & O'Shea, E. K. Noise in gene expression: origins, consequences, and control. Science 309, 2010–2013 (2005).
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).
Cox, D. R. & Miller, H. D. The Theory of Stochastic Processes (Chapman & Hall, London, 1977).
Gillespie, D. T. Markov Processes: an Introduction for Physical Scientists (Academic, New York, 1992).
Allen, L. J. S. Stochastic Processes with Applications to Biology (Pearson Prentice Hall, Upper Saddle River, 2003).
Wilkinson, D. J. Stochastic Modelling for Systems Biology (Chapman & Hall/CRC, Boca Raton, 2006).
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.
McAdams, H. H. & Arkin, A. Stochastic mechanisms in gene expression. Proc. Natl Acad. Sci. USA 94, 814–819 (1997).
Zlokarnik, G. et al. Quantitation of transcription and clonal selection of single living cells with beta-lactamase as reporter. Science 279, 84–88 (1998).
Renshaw, E. Modelling Biological Populations in Space and Time (Cambridge Univ. Press, 1991).
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).
Higham, D. J. Modeling and simulating chemical reactions. SIAM Rev. 50, 347–368 (2008).
Paulsson, J., Berg, O. & Ehrenberg, M. Stochastic focusing: fluctuation-enhanced sensitivity of intracellular regulation. Proc. Natl. Acad. Sci. USA 97, 7148–7153 (2000).
Dupont, G., Abou-Lovergne, A. & Combettes, L. Stochastic aspects of oscillatory Ca2+ dynamics in hepatocytes. Biophys. J. 95, 2193–2202 (2008).
Cai, L., Friedman, N. & Xie, X. S. Stochastic protein expression in individual cells at the single molecule level. Nature 440, 358–362 (2006).
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.
Shahrezaei, V., Ollivier, J. and Swain, P. Colored extrinsic fluctuations and stochastic gene expression. Mol. Syst. Biol. 4, 196 (2008).
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).
Gillespie, D. T. Approximate accelerated stochastic simulation of chemically reacting systems. J. Chem. Phys. 115, 1716–1732 (2001).
Gillespie, D. T. & Petzold, L. R. Improved leap-size selection for accelerated stochastic simulation. J. Chem. Phys. 119, 8229–8234 (2003).
Kiehl, T. R., Mattheyses, R. M. & Simmons, M. K. Hybrid simulation of cellular behavior. Bioinformatics 20, 316–322 (2004).
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).
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).
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).
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).
Salis, H. & Kaznessis, Y. Accurate hybrid stochastic simulation of a system of coupled chemical or biochemical reactions. J. Chem. Phys. 122, 054103 (2005).
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).
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).
Weinan, E., Liu, D. & Vanden-Eijnden, E. Nested stochastic simulation algorithm for chemical kinetic systems with disparate rates. J. Chem. Phys. 123, 194107 (2005).
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).
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.
Cyganowski, S., Kloeden, P. & Ombach, J. From Elementary Probability to Stochastic Differential Equations with MAPLE (Springer, New York, 2002).
Kloeden, P. E. & Platen, E. Numerical Solution of Stochastic Differential Equations (Springer, New York, 1992).
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).
Gillespie, C. S. et al. A mathematical model of ageing in yeast. J. Theor. Biol. 44, 493–516 (2004).
Tanase-Nicola, S. & ten Wolde, P. R. Regulatory control and the costs and benefits of biochemical noise. PLoS Comput. Biol. 4, e1000125 (2008).
Speed, T. P. (ed.) Statistical Analysis of Gene Expression Microarray Data (Chapman & Hall/CRC, Boca Raton 2003).
Wit, E. & McClure, J. Statistics for Microarrays: Design, Analysis and Inference (Wiley, New York, 2004).
O'Hagan, A. & Forster, J. J. Kendall's Advanced Theory of Statistics Vol. 2B (Arnold, London, 2004).
Gelman, A., Carlin, J. B., Stern, H. S. & Rubin, D. B. Bayesian Data Analysis 2nd edn (Chapman & Hall/CRC, Boca Raton, 2003).
Vanucci, M., Do, K.-A. & Muller, P. (eds) Bayesian Inference for Gene Expression and Proteomics (Cambridge Univ. Press, New York 2006).
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).
Lewin, A., Richardson, S., Marshall, C., Glazier, A. & Aitman, T. Bayesian modelling of differential gene expression. Biometrics 62, 10–18 (2006).
Friedman, N. Inferring cellular networks using probabilistic graphical models. Science 303, 799–805 (2004).
Pournara, I. & Wernisch, L. Reconstruction of gene networks using Bayesian learning and manipulation experiments. Bioinformatics 20, 2934–2942 (2004).
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).
Schafer, J. & Strimmer, K. An empirical Bayes approach to inferring large-scale gene association networks. Bioinformatics 21, 754–764 (2005).
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).
Dobra, A. et al. Sparse graphical models for exploring gene expression data. J. Multivar. Anal. 90, 196–212 (2004).
Jones, B. et al. Experiments in stochastic computation for high-dimensional graphical models. Stat. Sci. 20, 388–400 (2005).
Husmeier, D. Sensitivity and specificity of inferring genetic regulatory interactions from microarray experiments with dynamic Bayesian networks. Bioinformatics 19, 2271–2282 (2003).
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).
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.
George, E., Sun, D. & Ni, S. Bayesian stochastic search for VAR model restrictions. J. Econom. 142, 553–580 (2008).
Pepperkok, R. & Ellenberg, J. High-throughput fluorescence microscopy for systems biology. Nature Rev. Mol. Cell Biol. 7, 690–696 (2006).
Shen, H. et al. Automated tracking of gene expression profiles in individual cells and cell compartments. J. R. Soc. Interface 3, 787 (2006).
Jaqaman, K. & Danuser, G. Linking data to models: data regression. Nature Rev. Mol. Cell Biol. 7, 813–819 (2006).
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).
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.
Barenco, M. et al. Ranked prediction of p53 targets using hidden variable dynamic modeling. Genome Biol. 7, R25 (2006).
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.
Liebermeister, W. & Klipp, E. Biochemical networks with uncertain parameters. IEE Syst. Biol. 152, 97–107 (2005).
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).
Hastings, W. K. Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57, 97–109 (1970).
Gamerman, D. Markov Chain Monte Carlo (Texts in Statistical Science) (Chapman & Hall, New York, 1997).
Vyshemirsky, V. & Girolami, M. BioBayes: a software package for Bayesian inference in systems biology. Bioinformatics 24, 1933–1934 (2008).
Reinker, S., Altman, R. M. & Timmer, J. Parameter estimation in stochastic biochemical reactions. IEE Syst. Biol. 153, 168–178 (2006).
Tian, T., Xu, S., Gao, J. & Burrage, K. Simulated maximum likelihood method for estimating kinetic rates in gene expression. Bioinformatics 23, 84–91 (2007).
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.
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).
Iacus, S. M. Simulation and Inference for Stochastic Differential Equations — with R Examples (Springer, New York, 2008).
Golightly, A. & Wilkinson, D. J. Bayesian inference for stochastic kinetic models using a diffusion approximation. Biometrics 61, 781–788 (2005).
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).
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.
Golightly, A. & Wilkinson, D. J. Bayesian inference for nonlinear multivariate diffusion models observed with error. Comput. Stat. Data Anal. 52, 1674–1693 (2008).
Kennedy, M. C. & O'Hagan, A. Bayesian calibration of computer models. J. R. Stat. Soc. Ser. B 63, 425–464 (2001).
Goldstein, M. & Rougier, J. Bayes linear calibrated prediction for complex systems. J. Am. Stat. Assoc. 101, 1132–114 (2006).
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).
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.
Uhlenbeck, G. E. & Ornstein, L. S. On the theory of Brownian motion. Phys. Rev. 36, 823–841 (1930).
Orlando, D. et al. A probabilistic model for cell cycle distributions in synchrony experiments. Cell Cycle 6, 478–488 (2007).
Orlando, D. et al. Global control of cell-cycle transcription by coupled CDK and network oscillators. Nature 453, 944–947 (2008).
Beaumont, M. A. & Rannala, B. The Bayesian revolution in genetics. Nature Rev. Genet. 5, 251–261 (2004).
Wilkinson, D. J. Bayesian methods in bioinformatics and computational systems biology. Brief. Bioinformatics 8, 109–116 (2007).
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).
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).
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).
Veening, J.-W. et al. Transient heterogeneity in extracellular protease production by Bacillus subtilis. Mol. Syst. Biol. 4, 184 (2008).
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).
Fraser, H. B., Hirsh, A. E., Giaever, G., Kumm, J. & Eisen M. B. Noise minimization in eukaryotic gene expression. PLoS Biol. 2, e137 (2004).
Bar-Even, A. et al. Noise in protein expression scales with natural protein abundance. Nature Genet. 38, 636–643 (2006).
Newman, J. et al. Single-cell proteomic analysis of S. cerevisiae reveals the architecture of biological noise. Nature 441, 840–846 (2006).
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).
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).
Proctor, C. J. et al. Modelling the checkpoint response to telomere uncapping in budding yeast. J. R. Soc. Interface 4, 73–90 (2007).
Proctor, C. J. et al. Modelling the action of chaperones and their role in ageing. Mech. Ageing Dev. 126, 119–131 (2005).
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).
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).
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.
- 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.
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.
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.
The extent to which it is possible to accurately estimate model parameters given sufficient experimental data.
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.
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.
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.
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
This article is cited by
BioMedical Engineering OnLine (2022)
A probabilistic framework for particle-based reaction–diffusion dynamics using classical Fock space representations
Letters in Mathematical Physics (2022)
Quantifying the optimal strategy of population control of quorum sensing network in Escherichia coli
npj Systems Biology and Applications (2021)
Stochastic Modelling of Respiratory System Elastance for Mechanically Ventilated Respiratory Failure Patients
Annals of Biomedical Engineering (2021)
Scientific Reports (2020)