Oscillatory Regulation of Hes1: Discrete Stochastic Delay Modelling and Simulation

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Abstract
Discrete stochastic simulations are a powerful tool for understanding the dynamics of chemical kinetics when there are small-to-moderate numbers of certain molecular species. In this paper we introduce delays into the stochastic simulation algorithm, thus mimicking delays associated with transcription and translation. We then show that this process may well explain more faithfully than continuous deterministic models the observed sustained oscillations in expression levels of hes1 mRNA and Hes1 protein. Delay processes are ubiquitous in the biological sciences but are not always well-represented in mathematical models attempting to describe these biological processes. Additional issues arise when attempting to capture the uncertainty (intrinsic noise) associated with chemical kinetics in dealing with when and in what order reactions take place. Complicating the situation further are important instances when certain key molecules occur only in small numbers, so that it is not meaningful to talk about concentrations. In this paper Barrio et al. show how to incorporate delay, intrinsic noise, and discreteness associated with chemical kinetic systems into a very simple algorithm called the delay stochastic simulation algorithm (DSSA). This algorithm very naturally generalises the stochastic simulation algorithm that does not treat delays. The authors then apply the DSSA to a specific set of experiments performed by Hirata et al. who showed, amongst other things, that serum treatment of cultured cells induces cyclic expression of both mRNA and protein of the Notch effector Hes1 with a two-hour period. The authors show how this approach can explain additional experiments performed by Hirata et al., and, because this approach is very general, suggest that it can provide deep insights into the relationship between delayed processes, intrinsic noise, and small numbers of molecules in many biological systems.