Abstract

Intrinsic noise is a common phenomenon in biochemical reaction networks and may affect the occurence and amplitude of sustained oscillations in the states of the network. To evaluate properties of such oscillations in the time domain, it is usually required to conduct long-term stochastic simulations, using for example the Gillespie algorithm. In this paper, we present a new method to compute the amplitude distribution of the oscillations without the need for long-term stochastic simulations. By the derivation of the method, we also gain insight into the structural features underlying the stochastic oscillations. The method is applicable to a wide class of non-linear stochastic differential equations that exhibit stochastic oscillations. The application is exemplified for the MAPK cascade, a fundamental element of several biochemical signalling pathways. This example shows that the proposed method can accurately predict the amplitude distribution for the stochastic oscillations even when using further computational approximations.PACS Codes: 87.10.Mn, 87.18.Tt, 87.18.VfMSC Codes: 92B05, 60G10, 65C30

Highlights

  • Oscillations are a widely occurring phenomenon in the dynamics of biological systems

  • In this article we focus on biochemical systems that can be modeled as a set of interconnected, possibly nonlinear, stochastic differential equations (SDEs)

  • The method is applicable to systems where a stationary density distributions exists and can be computed either analytically or numerically

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Summary

Introduction

Oscillations are a widely occurring phenomenon in the dynamics of biological systems. A large number of realisations of the stochastic process are computed These realisations can be used to compute various temporal characteristics of the system, in particular oscillation amplitude and frequency distributions. Therewith it is possible to calculate both the amplitude distribution and the mean frequency of the oscillations This method has the disadvantage of requiring the discussed systems to be almost linear and the frequency of the oscillations not to be noteworthy disturbed by the stochastic effects. As a more realistic example, we discuss oscillations in the MAP kinase cascade with an incorporated negative feedback with limited amounts of entities of each molecular species For this system, the stationary probability distribution can be estimated by a linear approximation, or it can be computed numerically. We compare the amplitude distributions predicted by our method, based on these two approaches, to an "experimental" amplitude distribution obtained from a long-term stochastic simulation

Results and Discussion
Derivation of the Results
Numerical Approach
Conclusion
Leloup JC and Goldbeter A

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