Relating continuous and discrete PEPA models of signalling pathways

Abstract

ODEs CTMC Kurtz a b s t r a c t PEPA and its semantics have recently been extended to model biological systems. In order to cope with massive quantities of processes (as is usually the case when considering biological reactions) the model is interpreted in terms of a small set of coupled ordinary differential equations (ODEs) instead of a large state space continuous time Markov chain (CTMC). So far the relationship between these two semantics of PEPA had not been established. This is the goal of the present paper. After introducing a new extension of PEPA, denoted PEPA + Π , that allows models to capture both mass action law and bounded capacity law cooperations, the relationship between these two semantics is demonstrated. The result relies on Kurtz's Theorem that expresses that a set of ODEs can be, in some sense, considered as the limit of pure jump Markov processes.

Scalable Differential Analysis of Process Algebra Models

“…This accomplishes an orthogonal goal with respect to ours, as the semantics associates a sequential component with an entire population instead of a single entity. Bounded capacity and mass action kinetics semantics for fluid models are merged in [19], in which a slight extension to the cooperation operation is provided to model signalling pathways which exhibit both kinds of reaction laws. An important contribution of this work is that the Markov process generated from the pathway-centric style of [20] is shown to converge in the limit to the underlying ODE by using Kurtz's results [6].…”

Analysis of the Fluid Approximation of Stochastic Process Algebra Models

“…The formulae (5) and (6) are activity centric while (7) is local derivative centric. Our approach to derive ODEs has extended previous results presented in the literature [21,23,25], by relaxing restrictions such as allowing shared activities may have different local rates, each action name may appear in different local derivatives within the definition of a sequential component, and may occur multiple times with that derivative definition, etc.…”

The Benefits of Sometimes Not Being Discrete

“…For example in [31], Geisweiller et al, working with a generalised form of PEPA models which allow two forms of synchronisation -both the usual PEPA synchronisation based on the bounded capacity, and the biological notion of mass action -show that the syntactically derived ODE models are indeed those which are obtained by the application of Kurtz's Theoreom, guaranteeing convergence in the limit. In [65], Tribastone et al show how it is possible to fully formalise the derivation of the ODEs for PEPA models, via a structured operational semantics.…”