Adaptive simulation of hybrid stochastic and deterministic models for biochemical systems

Abstract

In the past years it has become evident that stochastic effects in regulatory networks play an important role, leading to an increasing in stochastic modelling attempts. In contrast, metabolic networks involving large numbers of molecules are most often modelled deterministically. Going towards the integration of different model systems, gen-regulatory networks become part of a larger model system including signalling pathways and metabolic networks. Thus, the question arises of how to efficiently and accurately simulation such coupled or hybrid systems. We present an algorithmic approach for the simulation of hybrid stochastic and deterministic reaction models that allows for adaptive step-size integration of the deterministic equations while at the same time accurately tracing the stochastic reaction events. We present a mathematical derivation of the hybrid system on the stochastic process level, and present numerical examples that outline the power of hybrid simulations.Résumé. Au cours des dernières années, il est devenu clair que les effets aléatoires jouaient un rôle important dans les réseaux de régulation, et les modèles employés aujourd'hui pour décrire ces réseaux sont de nature stochastique. En revanche, les réseaux métaboliques, qui mettent en jeu un grand nombre de molécules, sont le plus souvent décrits par des modèles déterministes. Dans la modélisation de systèmes complexes, réseaux régulateurs de gènes, chemins de signaux et réseaux métaboliques sont intégrés dans un même modèle. Se pose alors la question de simuler efficacement et avec précision de tels modèles couplés (on parle aussi de modèles hybrides). Nous présentons ici une approche pour la simulation de modèles de réactions hybrides stochastiques/déterministes permettantà la fois d'avoir recoursà des pas de temps adaptatifs dans l'intégration deséquations déterministes et de simuler précisément les réactions décrites par des processus stochastiques. Des simulations numériques illustrent la puissance de ces simulations hybrides.

A Dynamical Low-Rank Approach to the Chemical Master Equation

“…Since its computational effort scales with the number of reactions occurring, the stochastic simulation algorithm can be to expensive for highly reactive biological systems. As a remedy, approximate techniques have been suggested based on tau-leaping or partitioning of the system [17,36,7,5,18,37,34,20,1,38,6,10]. Tau-leaping methods assume that many reaction events will occur without significantly changing the reaction propensities, allowing to locally approximate the inhomogenous Poisson process by a simpler distribution [17,36,7,5].…”

“…In [18,37,34] the fast reactions are solved analytically using some quasi-steady state assumption, resulting in a reduced stochastic system of slow reactions to be solved. Rather than explicitly computing the quasi steady states, a different line of approaches solves the fast reaction system numerically, either using multi-scale techniques, or approximating the fast dynamics by some Langevin or deterministic reaction rate equation [20,1,38,6,10]. In order to analyze the stochastic reaction system, all above Monte Carlo approaches require a statistically large number of realizations.…”

Eradication-resolution dynamics with stochastic flare-ups

“…The usual deterministic treatment is then seen as a only a special case of the general situation which is a partitioning into a low-numbers component is to be treated stochastically and the high-numbers component can treated deterministically (Kiehl et al (2004);Wilkinson (2006)). Simulation of such hybrid systems is perfectly straightforward (Alfonsi et al (2005)); since the hazard rates generally co-depend on the deterministic component, it is necessary to integrate this time-varying hazard in order to realise the time till the next transition in the stochastic component (this time interval being a random variable).…”

(Hybrid) automata and (stochastic) programs * The hybrid automata lattice of a stochastic program

“…• The set of discrete modes, as in the PDMP, is Q ={q on ,q off }, while P is the only variable • There are three continuous transitions: degradation in the two modes and production in the active mode of the gene: (prod,q on , (1) Product of TDSHA: we define now a notion of product for TDSHA. Due to the asynchronous nature of sCCP, such a product will be itself asynchronous, i.e.…”

“…The idea is simple: if we jump out of the allowed region, we immediately apply a sequence of discrete transitions. Formally, let η ∈ TS(q,x), where TS(q,x) ={η ∈ TS | e 1 [η]=q∧guard[η](x)}, and suppose r(η,x) ∈ D. Let e 2 [η]=q η and r(η,x) = x η , and consider TD(q η ,x η ) =∅ (as x η ∈ D q η ). We can now simply modify the tentative reset kernel R((q,x),A) of Equation (9) by replacing δ q η ,x η (A) with R((q η ,x η ),A).…”