It is generally acknowledged that the molecular mechanisms regulating key cellular processes such as gene expression are intrinsically stochastic. The random diffusion of cell signalling molecules and the combinatorial assembly of transcription factor complexes provide extensive opportunities for the action of chance. In recent years stochastic regulatory network models have been developed, based on discrete-event simulation for generating realisations from the complex continuous-time countable-state Markov processes governing the reaction systems. Within the physical sciences, this simulation technique is known as the Gillespie algorithm (Gillespie, 1992). The advantages of discrete stochastic models over their continuous deterministic counterparts are demonstrated by McAdams & Arkin (1997) in the context of an interesting genetic switch. Network models such as this contain many parameters with uncertain values. In addition, the underlying process can only be observed partially, and at discrete time intervals. Inference for such Markov process models is an extremely challenging problem.

This talk will describe the techniques used to model regulatory networks, and the computational tools needed for simulation and analysis. A model based on the Lotka-Volterra system is the simplest exhibiting the non-linear negative feedback mechanisms characteristic of many genetic circuits. Simulating data from this model is straightforward but inference for the parameters based on discrete observation of the process is not. For example, one complicating feature is that the number of reaction events is not determined uniquely by the data. Consequently, MCMC algorithms which attempt to uncover the underlying process and associated parameters are based on a state space of varying dimension. We will explore two approaches, one based on reversible jump MCMC methodology and the other based on more ambitious moves within the state space using a proposal carefully constructed so as to ensure rapid mixing of the resulting chain. The reversible jump moves are similar to those used in Gibson & Renshaw (2001).

An overview will also be given of how such MCMC algorithms generalise to larger and more realistic network models, and the problems associated with applying such techniques in practice.

References:

Gillespie, D.T. (1992). Markov processes: An introduction for physical scientists. Academic Press.

McAdams, H. H. & Arkin, A. (1997). Stochastic mechanisms in gene expression. Proceedings of the National Academy of Sciences, USA, 94: 814--819.

Gibson, G. J. & Renshaw, E. (2001). Likelihood estimation for stochastic compartmental models using Markov chain methods. Statistics and Computing, 11: 347--358.

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