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Stochastic modelling for quantitative description of heterogeneous biological systems

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.

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

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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.

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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.

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

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