This paper is concerned with the Bayesian estimation of stochastic rate constants in the context of dynamic models of intra-cellular processes. The underlying discrete stochastic kinetic model is replaced by a di#usion approximation (or stochastic di#erential equation approach) where a white noise term models stochastic behaviour and the model is identified using equispaced time course data. The estimation framework involves the introduction of m-1 latent data points between every pair of observations. MCMC methods are then used to sample the posterior distribution of the latent process and the model parameters

The purpose of time series analysis via mechanistic models is to reconcile the known or hypothesized structure of a dynamical system with observations collected over time. We develop a framework for constructing nonlinear mechanistic models and carrying out inference. Our framework permits the consideration of implicit dynamic models, meaning statistical models for stochastic dynamical systems which are specified by a simulation algorithm to generate sample paths. Inference procedures that operate on implicit models are said to have the plug-and-play property. Our work builds on recently developed plug-and-play inference methodology for partially observed Markov models. We introduce a class of implicitly specified Markov chains with stochastic transition rates, and we demonstrate its applicability to open problems in statistical inference for biological systems. As one example, these models are shown to give a fresh perspective on measles transmission dynamics. As a second example, we present a mechanistic analysis of cholera incidence data, involving interaction between two competing strains of the pathogen Vibrio cholerae. 1. Introduction. A

We consider analysis of complex stochastic models based upon partial information. MCMC and reversible jump MCMC are often the methods of choice for such problems, but in some situations they can be difficult to implement; and suffer from problems such as poor mixing, and the difficulty of diagnosing convergence. Here we review three alternatives to MCMC methods: importance sampling, the forward-backward algorithm, and sequential Monte Carlo (SMC). We discuss how to design good proposal densities for importance sampling, show some of the range of models for which the forward-backward algorithm can be applied, and show how resampling ideas from SMC can be used to improve the efficiency of the other two methods. We demonstrate these methods on a range of examples, including estimating the transition density of a diffusion and of a discrete-state continuous-time Markov chain; inferring structure in population genetics; and segmenting genetic divergence data.

This chapter considers the assessment and refinement of a dynamic stochastic process model of the cellular response to DNA damage. The proposed model is a complex nonlinear continuous time latent stochastic process. It is compared to time course data on the levels of two key proteins involved in this response, captured at the level of individual cells in a human cancer cell line. The primary goal of this study is to “calibrate ” the model by finding parameters of the model (kinetic rate constants) that are most consistent with the experimental data. Significant amounts of prior information are available for the model parameters. It is therefore most natural to consider a Bayesian analysis of the problem, using sophisticated MCMC methods to overcome the formidable computational challenges.

This chapter considers stochastic differential equations for Systems Biology models derived from the Chemical Langevin Equation (CLE). After outlining the derivation of such models, Bayesian inference for the parameters is considered, based on state-of-the-art Markov chain Monte Carlo algorithms. Starting with a basic scheme for models observed perfectly, but discretely in time, problems with standard schemes and their solutions are discussed. Extensions of these schemes to partial observation and observations subject to measurement error are also considered. Finally, the techniques are demonstrated in the context of a simple stochastic kinetic model of a genetic regulatory network. 1

Top-down modelling

Gillespie’s algorithm serves to simulate a network of stochastic reactions with given initial quantities and kinetic rate constants. In this paper we consider the estimation of the kinetic rate constants of the reactions based on a set of discrete observations generated by Gillespie’s algorithm. In particular, we present an Expectation Maximisation (EM) method to perform maximum likelihood estimation of the rate constants. Applicability of the method is tested on a simple reaction network. 1.

Dynamic simulation modelling of complex biological processes forms the backbone of systems biology. Discrete stochastic models are particularly appropriate for describing sub-cellular molecular interactions, especially when critical molecular species are thought to be present at low copy-numbers. For example, these stochastic effects play an important role in models of human ageing, where ageing results from the long-term accumulation of random damage at various biological scales. Unfortunately, realistic stochastic simulation of discrete biological processes is highly computationally intensive, requiring specialist hardware, and can benefit greatly from parallel and distributed approaches to computation and analysis. For these reasons, we have developed the BASIS system for the simulation and storage of stochastic SBML models together with associated simulation results. This system is exposed as a set of web services to allow users to incorporate its simulation tools into their workflows. Parameter inference for stochastic models is also difficult and computationally expensive. The CaliBayes system provides a set of web services (together with an R package for consuming these and formatting data) which addresses this problem for SBML models. It uses a sequential Bayesian MCMC method which is powerful and flexible, providing very rich information. However this approach is exceptionally computationally intensive and requires the use of a carefully designed architecture. Again, these tools are exposed as web services to allow users to take advantage of this system. In this paper we describe these two systems and demonstrate their integrated use with an example workflow to estimate the parameters of a simple model of S. Cerevisiae growth on agar plates.

Modelling and inference for biochemical network dynamicsOverview Stochastic modelling Bayesian parameter inference Dynamic Bayesian network inference Summary and conclusion

Summary and conclusion Bayesian inference for Markov process models, with applications to systems biology