Approximation Techniques for Stochastic Analysis of Biological Systems

There has been an increasing demand for formal methods in the design process of safety-critical synthetic genetic circuits. Probabilistic model checking techniques have demonstrated significant potential in analyzing the intrinsic probabilistic behaviors of complex genetic circuit designs. However, its inability to scale limits its applicability in practice. This chapter addresses the scalability problem by presenting a state-space approximation method to remove unlikely states resulting in a reduced, finite state representation of the infinite-state continuous-time Markov chain that is amenable to probabilistic model checking. The proposed method is evaluated on a design of a genetic toggle switch. Comparisons with another state-of-art tool demonstrates both accuracy and efficiency of the presented method

Bounding Mean First Passage Times in Population Continuous-Time Markov Chains

We consider the problem of bounding mean first passage times and reachability probabilities for the class of population continuous-time Markov chains, which capture stochastic interactions between groups of identical agents. The quantitative analysis of such models is notoriously difficult since typically neither state-based numerical approaches nor methods based on stochastic sampling give efficient and accurate results. Here, we propose a novel approach that leverages techniques from martingale theory and stochastic processes to generate constraints on the statistical moments of first passage time distributions. These constraints induce a semi-definite program that can be used to compute exact bounds on reachability probabilities and mean first passage times without numerically solving the transient probability distribution of the process or sampling from it. We showcase the method on some test examples and tailor it to models exhibiting multimodality, a class of particularly challenging scenarios from biology

Analysis of Timed Properties Using the Jump-Diffusion Approximation

International audienceDensity dependent Markov chains (DDMCs) describe the interaction of groups of identical objects. In case of large numbers of objects a DDMC can be approximated efficiently by means of either a set of ordinary differential equations (ODEs) or by a set of stochastic differential equations (SDEs). While with the ODE approximation the chain stochasticity is not maintained, the SDE approximation, also known as the diffusion approximation, can capture specific stochastic phenomena (e.g., bi-modality) and has also better convergence characteristics. In this paper we introduce a method for assessing temporal properties, specified in terms of a timed automaton, of a DDMC through a jump diffusion approximation. The added value is in terms of runtime: the costly simulation of a very large DDMC model can be replaced through much faster simulation of the corresponding jump diffusion model. We show the efficacy of the framework through the analysis of a biological oscillator