Efficient parametric inference for stochastic biological systems with measured variability

Iain G. Johnston 1
  • 1 Department of Mathematics, Imperial College London, London SW7 2AZ, UK
Iain G. Johnston


Stochastic systems in biology often exhibit substantial variability within and between cells. This variability, as well as having dramatic functional consequences, provides information about the underlying details of the system’s behavior. It is often desirable to infer properties of the parameters governing such systems given experimental observations of the mean and variance of observed quantities. In some circumstances, analytic forms for the likelihood of these observations allow very efficient inference: we present these forms and demonstrate their usage. When likelihood functions are unavailable or difficult to calculate, we show that an implementation of approximate Bayesian computation (ABC) is a powerful tool for parametric inference in these systems. However, the calculations required to apply ABC to these systems can also be computationally expensive, relying on repeated stochastic simulations. We propose an ABC approach that cheaply eliminates unimportant regions of parameter space, by addressing computationally simple mean behavior before explicitly simulating the more computationally demanding variance behavior. We show that this approach leads to a substantial increase in speed when applied to synthetic and experimental datasets.

  • Bailey, N. T. J. (1964): The elements of stochastic processes with applications to the natural sciences, Wiley: New York.

  • Beaumont, M. A., W. Zhang and D. J. Balding (2002): “Approximate bayesian computation in population genetics,” Genetics, 162, 2025–2035.

  • Blake, W. J., M. Kærn, C. R. Cantor and J. J. Collins (2003): “Noise in eukaryotic gene expression,” Nature, 422, 633–637.

    • Crossref
  • Brock, A., H. Chang and S. Huang (2009): “Non-genetic heterogeneity – a mutation-independent driving force for the somatic evolution of tumours,” Nat. Rev. Genet., 10, 336–342.

    • PubMed
  • Chang, H. H., M. Hemberg, M. Barahona, D. E. Ingber and S. Huang (2008): “Transcriptome-wide noise controls lineage choice in mammalian progenitor cells,” Nature, 453, 544–547.

    • Crossref
  • das Neves, R. P., N. S. Jones, L. Andreu, R. Gupta, T. Enver and F. J. Iborra (2010): “Connecting variability in global transcription rate to mitochondrial variability,” PLoS Biol., 8, 451–464.

  • Ding, H., G. Trajcevski, P. Scheuermann, X. Wang and E. Keogh (2008): “Querying and mining of time series data: experimental comparison of representations and distance measures,” Proceedings of the VLDB Endowment, 1, 1542–1552.

    • Crossref
  • Elowitz, M. B., A. J. Levine, E. D. Siggia and P. S. Swain (2002): “Stochastic gene expression in a single cell,” Science, 297, 1183–1186.

    • Crossref
  • Enver, T., M. Pera, C. Peterson and P. W. Andrews (2009): “Stem cell states, fates, and the rules of attraction,” Cell Stem Cell, 4, 387–397.

    • Crossref
    • PubMed
  • Gillespie, D. T. (1977) “Exact stochastic simulation of coupled chemical reactions,” J. Phys. Chem., 81, 2340–2361.

    • Crossref
  • Golding, I., J. Paulsson, S. M. Zawilski and E. C. Cox (2005): “Real-time kinetics of gene activity in individual bacteria,” Cell, 123, 1025–1036.

    • Crossref
  • Graf, T. and M. Stadtfeld (2008): “Heterogeneity of embryonic and adult stem cells,” Cell Stem Cell, 3, 480–483.

    • Crossref
    • PubMed
  • Johnston, I. G., B. Gaal, R. P. das Neves, T. Enver, F. J. Iborra and N. S. Jones (2012): “Mitochondrial variability as a source of extrinsic cellular noise,” PLoS Comput. Biol., 8, e1002416.

  • Kærn, M., T. C. Elston, W. J. Blake and J. J. Collins (2005): “Stochasticity in gene expression: from theories to phenotypes,” Nat. Rev. Genet., 6, 451–464.

    • PubMed
  • Knight, K. (2000): Mathematical statistics. Chapman & Hall: New York, NY, USA.

    • Crossref
  • Kussell, E., R. Kishony, N. Q. Balaban and S. Leibler (2005): “Bacterial persistence: a model of survival in changing environments,” Genetics, 169, 1807.

    • Crossref
  • Marin, J.-M., P. Pudlo, C. P. Robert and R. J. Ryder (2012): “Approximate bayesian computational methods,” Stat. Comput., 22, 1167–1180.

  • Marjoram, P., J. Molitor, V. Plagnol and S. Tavaré (2003): “Markov chain monte carlo without likelihoods,” Proc. Natl. Acad. Sci., 100, 15324–15328.

  • Paulsson, J. (2005): “Models of stochastic gene expression,” Phys. Life Rev., 2, 157–175.

  • Raser, J. M. and E. K. O’Shea (2004): “Control of stochasticity in eukaryotic gene expression,” Science, 304, 1811–1814.

    • Crossref
  • Rausenberger, J. and M. Kollmann (2008): “Quantifying origins of cell-to-cell variations in gene expression,” Biophys. J., 95, 4523–4528.

  • Sisson, S. A., Y. Fan and M. M. Tanaka (2007): “Sequential monte carlo without likelihoods,” Proc. Natl. Acad. Sci., 104, 1760–1765.

  • Spencer, S. L., S. Gaudet, J. G. Albeck, J. M. Burke and P. K. Sorger (2009): “Non-genetic origins of cell-to-cell variability in TRAIL-induced apoptosis,” Nature, 459, 428–432.

    • Crossref
  • Sunnåker, Mikael, Alberto Giovanni Busetto, Elina Numminen, Jukka Corander, Matthieu Foll and Christophe Dessimoz (2013): “Approximate bayesian computation,” PLoS Comput. Biol., 9, e1002803.

  • Toni, T., D. Welch, N. Strelkowa, A. Ipsen and M. P. H. Stumpf (2009): “Approximate bayesian computation scheme for parameter inference and model selection in dynamical systems,” J. Roy. Soc. Interface, 6, 187–202.

  • Wilkinson, D. J. (2012): Stochastic modelling for systems biology. CRC press: Boca Raton, FL, USA.

Purchase article
Get instant unlimited access to the article.
Log in
Already have access? Please log in.

Log in with your institution

Journal + Issues

SAGMB publishes significant research on the application of statistical ideas to problems arising from computational biology. The range of topics includes linkage mapping, association studies, gene finding and sequence alignment, protein structure prediction, design and analysis of microarrary data, molecular evolution and phylogenetic trees, DNA topology, and data base search strategies.