Incorporating age and delay into models for biophysical systems.

In many biological systems, chemical reactions or changes in a physical state are assumed to occur instantaneously. For describing the dynamics of those systems, Markov models that require exponentially distributed inter-event times have been used widely. However, some biophysical processes such as gene transcription and translation are known to have a significant gap between the initiation and the completion of the processes, which renders the usual assumption of exponential distribution untenable. In this paper, we consider relaxing this assumption by incorporating age-dependent random time delays (distributed according to a given probability distribution) into the system dynamics. We do so by constructing a measure-valued Markov process on a more abstract state space, which allows us to keep track of the 'ages' of molecules participating in a chemical reaction. We study the large-volume limit of such age-structured systems. We show that, when appropriately scaled, the stochastic system can be approximated by a system of partial differential equations (PDEs) in the large-volume limit, as opposed to ordinary differential equations (ODEs) in the classical theory. We show how the limiting PDE system can be used for the purpose of further model reductions and for devising efficient simulation algorithms. In order to describe the ideas, we use a simple transcription process as a running example. We, however, note that the methods developed in this paper apply to a wide class of biophysical systems.

Bootstrapping least-squares estimates in biochemical reaction networks.

The paper proposes new computational methods of computing confidence bounds for the least-squares estimates (LSEs) of rate constants in mass action biochemical reaction network and stochastic epidemic models. Such LSEs are obtained by fitting the set of deterministic ordinary differential equations (ODEs), corresponding to the large-volume limit of a reaction network, to network's partially observed trajectory treated as a continuous-time, pure jump Markov process. In the large-volume limit the LSEs are asymptotically Gaussian, but their limiting covariance structure is complicated since it is described by a set of nonlinear ODEs which are often ill-conditioned and numerically unstable. The current paper considers two bootstrap Monte-Carlo procedures, based on the diffusion and linear noise approximations for pure jump processes, which allow one to avoid solving the limiting covariance ODEs. The results are illustrated with both in-silico and real data examples from the LINE 1 gene retrotranscription model and compared with those obtained using other methods.

Model discrimination in dynamic molecular systems: application to parotid de-differentiation network.

In modern systems biology the modeling of longitudinal data, such as changes in mRNA concentrations, is often of interest. Fully parametric, ordinary differential equations (ODE)-based models are typically developed for the purpose, but their lack of fit in some examples indicates that more flexible Bayesian models may be beneficial, particularly when there are relatively few data points available. However, under such sparse data scenarios it is often difficult to identify the most suitable model. The process of falsifying inappropriate candidate models is called model discrimination. We propose here a formal method of discrimination between competing Bayesian mixture-type longitudinal models that is both sensitive and sufficiently flexible to account for the complex variability of the longitudinal molecular data. The ideas from the field of Bayesian analysis of computer model validation are applied, along with modern Markov Chain Monte Carlo (MCMC) algorithms, in order to derive an appropriate Bayes discriminant rule. We restrict attention to the two-model comparison problem and present the application of the proposed rule to the mRNA data in the de-differentiation network of three mRNA concentrations in mammalian salivary glands as well as to a large synthetic dataset derived from the model used in the recent DREAM6 competition.

Inference for discretely observed stochastic kinetic networks with applications to epidemic modeling.

We present a new method for Bayesian Markov Chain Monte Carlo-based inference in certain types of stochastic models, suitable for modeling noisy epidemic data. We apply the so-called uniformization representation of a Markov process, in order to efficiently generate appropriate conditional distributions in the Gibbs sampler algorithm. The approach is shown to work well in various data-poor settings, that is, when only partial information about the epidemic process is available, as illustrated on the synthetic data from SIR-type epidemics and the Center for Disease Control and Prevention data from the onset of the H1N1 pandemic in the United States.

A stochastic model of gene transcription: an application to L1 retrotransposition events.

A simplified mathematical model of gene transcription is presented based on a system of coupled chemical reactions and a corresponding set of stochastic equations similar to those used in enzyme kinetics theory. The quasi-stationary distribution for the model is derived and its usefulness illustrated with an example of model parameters estimation using sparse time course data on L1 retrotransposon expression kinetics. The issue of model validation is also discussed and a simple validation procedure for the estimated model is devised. The procedure compares model predicted values with the laboratory data via the standard Bayesian techniques with the help of modern Markov-Chain Monte-Carlo methodology.