Comparing hitting time behaviour of Markov jump processes and their diffusion approximations

Markov jump processes can provide accurate models in many applications, notably chemical and biochemical kinetics, and population dynamics. Stochastic differential equations offer a computationally efficient way to approximate these processes. It is therefore of interest to establish results that shed light on the extent to which the jump and diffusion models agree. In this work we focus on mean hitting time behavior in a thermodynamic limit. We study three simple types of reactions where analytical results can be derived, and we find that the match between mean hitting time behavior of the two models is vastly different in each case. In particular, for a degradation reaction we find that the relative discrepancy decays extremely slowly, namely, as the inverse of the logarithm of the system size. After giving some further computational results, we conclude by pointing out that studying hitting times allows the Markov jump and stochastic differential equation regimes to be compared in a manner that avoids pitfalls that may invalidate other approaches

Uncoupled Analysis of Stochastic Reaction Networks in Fluctuating Environments

The dynamics of stochastic reaction networks within cells are inevitably modulated by factors considered extrinsic to the network such as for instance the fluctuations in ribsome copy numbers for a gene regulatory network. While several recent studies demonstrate the importance of accounting for such extrinsic components, the resulting models are typically hard to analyze. In this work we develop a general mathematical framework that allows to uncouple the network from its dynamic environment by incorporating only the environment's effect onto the network into a new model. More technically, we show how such fluctuating extrinsic components (e.g., chemical species) can be marginalized in order to obtain this decoupled model. We derive its corresponding process- and master equations and show how stochastic simulations can be performed. Using several case studies, we demonstrate the significance of the approach. For instance, we exemplarily formulate and solve a marginal master equation describing the protein translation and degradation in a fluctuating environment.Comment: 7 pages, 4 figures, Appendix attached as SI.pdf, under submissio

Chemical master equation and Langevin regimes for a gene transcription model

Gene transcription models must take account of intrinsic stochasticity. The Chemical Master Equation framework is based on modelling assumptions that are highly appropriate for this context, and the Stochastic Simulation Algorithm (also known as Gillespie's algorithm) allows for practical simulations to be performed. However, for large networks and/or fast reactions, such computations can be prohibitatively expensive. The Chemical Langevin regime replaces the massive ordinary dierential equation system with a small stochastic dierential equation system that is more amenable to computation. Although the transition from Chemical Master Equation to Chemical Langevin Equation can be justied rigorously in the large system size limit, there is very little guidance available about how closely the two models match for a xed system. Here, we consider a transcription model from the recent literature and show that it is possible to compare rst and second moments in the two stochastic settings. To analyse the Chemical Master Equation we use some recent work of Gadgil, Lee and Othmer, and to analyse the Chemical Langevin Equation we use Ito's Lemma. We nd that there is a perfect match|both modelling regimes give the same means, variances and correlations for all components in the system. The model that we analyse involves 'unimolecular reactions', and we nish with some numerical simulations involving dimerization to show that the means and variances in the two regimes can also be close when more general 'bimolecular reactions' are involved

Stationary distributions and condensation in autocatalytic CRN

We investigate a broad family of non weakly reversible stochastically modeled reaction networks (CRN), by looking at their steady-state distributions. Most known results on stationary distributions assume weak reversibility and zero deficiency. We first give explicitly product-form steady-state distributions for a class of non weakly reversible autocatalytic CRN of arbitrary deficiency. Examples of interest in statistical mechanics (inclusion process), life sciences and robotics (collective decision making in ant and robot swarms) are provided. The product-form nature of the steady-state then enables the study of condensation in particle systems that are generalizations of the inclusion process.Comment: 25 pages. Some typos corrected, shortened some part