Hybrid approximation of stochastic concurrent constraint programs

Modeling in computational systems biology is dominated by two formalisms, the first one being (mainly ordinary) differential equations and the second one being continuous-time stochastic processes [1, 2]. Both methodologies have their roots reaching back to physical and chemical arguments that, at least for modeling biochemical reactions, give strong foundations to the approach. Recently, stochastic modeling has received a lot of attention, thanks also to the use of process algebras (in their stochastic variants) as modeling languages that allow to construct models in a compositional and reusable way [3, 4]. It is well known that the choice of “mathematics” to use is not just a matter of personal taste, but rather it is connected with the specific features of the system under analysis. Differential models are appealing, as they count on a (mature) bulk of mathematical instruments for their analysis, numerical solvers above all. However, biological processes often have inherently discrete components (they involve molecules interacting) and hence a continuous approximation may lead to incorrect results, especially when small-sized populations are taken into account. Stochastic models can be more precise, as they are intrinsically discrete and also able to capture the effects of noise. However, their exact simulation with the (celebrated) Gillespie algorithm [5] is computationally more expensive and, in general, the analysis of stochastic models is more difficult. Relating stochastic process algebras (SPA) and differential equation models is a difficult and interesting problem. Specifically, we are interested in finding methods that automatically translate SPA models into ODEs. This translation must somehow preserve both the structure and the behavior of the models: if we start from a SPA program, the associated set of ODEs should be behaviorally equivalent to it, in the sense of showing the same (qualitative) dynamics. Some techniques have been proposed in the literature to cope with this problem. The first one that appeared has been developed by Hillston [6] for PEPA [7]. Subsequently, this technique has been extended to stochastic π-calculus [8, 9] and to stochastic concurrent constraint programming (sCCP [10]) [11, 12]. Unfortunately, all these methods, though working well for some systems, fail to preserve behavior in general. A paradigmatic example is the Repressilator [13], a synthetic genetic circuit involving three genes repressing cyclically each other, which exhibits oscillatory behavior. A simple stochastic model for Repressilator can be written using the formalism of gene gates [14], where genes are considered as logical gates pro-