Approximate analysis of biological systems by hybrid switching jump diffusion

Abstract

In this paper we consider large state space continuous time Markov chains arising in the field of systems biology. For a class of such models, namely, for density dependent families of Markov chains that represent the interaction of large groups of identical objects, Kurtz has proposed two kinds of approximations. One is based on ordinary differential equations and provides a deterministic approximation, while the other uses a diffusion process with which the resulting approximation is stochastic. The computational cost of the deterministic approximation is significantly lower, but the diffusion approximation retains stochasticity and is able to reproduce relevant random features like variance, bimodality, and tail behavior that cannot be captured by a single deterministic quantity.In a recent paper, for particular stochastic Petri net models, we proposed a jump diffusion approximation that aims at being applicable beyond the limits of Kurtz's diffusion approximation in order to cover the case when the process reaches the boundary with non-negligible probability. In this paper we generalize the method so that it can be applied to any density dependent Markov chains. Other limitations of the diffusion approximation in its original form are that it can provide inaccurate results when the number of objects in some groups is often or constantly low and that it can be applied only to pure density dependent Markov chains. In order to overcome these drawbacks, in this paper we propose to apply the jump-diffusion approximation only to those components of the model that are in density dependent form and are associated with high population levels. The remaining components are treated as discrete quantities. The resulting process is a hybrid switching jump diffusion, i.e., a diffusion with hybrid state space and jumps where the discrete state changes can be seen as switches that take the diffusion from one condition to another. We show that the stochastic differential equations that characterize this process can be derived automatically both from the description of the original Markov chains or starting from a higher level description language, like stochastic Petri nets. The proposed approach is illustrated on three models: one modeling the so-called crazy clock reaction, one describing viral infection kinetics and the last considering transcription regulation.