Compositionality is a key feature of process algebras which is often cited as one of their advantages as a modelling technique. It is certainly true that in biochemical systems, as in many other systems, model construction is made easier in a formalism which allows the problem to be tackled compositionally. In this paper we consider the extent to which the compositional structure which is inherent in process algebra models of biochemical systems can be exploited during model solution. In essence this means using the compositional structure to guide decomposed solution and analysis. Unfortunately the dynamic behaviour of biochemical systems exhibits strong interdependencies between the components of the model making decomposed solution a difficult task. Nevertheless we believe that if such decomposition based on process algebras could be established it would demonstrate substantial benefits for systems biology modelling. In this paper we present our preliminary investigations based on a case study of the pheromone pathway in yeast, modelling in the stochastic process algebra Bio-PEPA.
Biochemical systems are generally large and complex and, therefore, studying a biochemical system monolithically can be difficult, both in terms of the definition of the model and in terms of its analysis. Over the last decade process algebras have been proposed as suitable formalisms for constructing models of biochemical systems [21,23,20,9], with their compositional structure being claimed as one of their main advantages.
In many cases the process algebra model is being used as an intermediate language which gives rise to a mathematical representation of the dynamics of the system, on which analysis of the behaviour can be conducted. In many cases this mathematical representation is a stochastic simulation based on an implicit continuous time Markov chain (CTMC), but in others it may be an explicit CTMC, a set of ordinary differential equations (ODEs) or some combination of these. However, regardless of the mathematical formalism chosen, the size and complexity of the biochemical processes can give rise to problems of tractability in the underlying mathematical model. This problem is particularly acute in the case of explicit CTMC models such as those used in probabilistic model-checking. Thus it is attractive to consider the extent to which decomposed model analysis can be used. This approach considers the model to be comprised of a number of modules which may be analysed in isolation, their results being combined to give results for the complete systems. Furthermore when the model has been described in a compositional formalism we would hope that the compositional structure may be exploited in the identification of suitable modules. In this work we refer to process algebras because they are generally equipped with a formal compositional definition, which we can rely on for our modular analysis. It is worth noting, however, that the same approach could potentially be applied to any other language equipped with a notion of modules.
Deriving global dynamic behaviour from the study of isolated components is difficult but the rich potential benefits mean that it has been studied in a wide variety of contexts. For example, notions of modularity and model decomposition have been widely applied in the context of software engineering, network theory and, more recently, systems biology (e.g. [25,24,10,4,26,14]). Moreover, model decomposition techniques, such as product form approaches and time scale decomposition, have been defined and applied in the context of CTMC-based analyses and process algebras [15,16].
However, the full strength of compositionality of process algebra has yet to be used in the context of quantitative analysis of biochemical models, although there have been some attempts to exploit the compositional structure in qualitative analysis and via congruence relations (for example, see [2,19]). In this paper we present a case study of decomposed model analysis. The system we consider is the yeast pheromone system, previously studied by Kofahl and Klipp [17]. In that paper the authors present an ODE model and an informal decomposition into modules. Here we give a formal model of the system in the stochastic process algebra Bio-PEPA [8,9] and demonstrate how the compositional structure of the process algebra supports a rigorous definition of modules. Moreover the flexibility of the Bio-PEPA framework to generate a number of different underlying mathematical models allows different analysis techniques to be used in tandem to parameterise and analyse the submodels corresponding to the modules. We choose as our main focus the use of probabilistic model-checking to verify properties of the pathway, partly because this style of analysis is a feature of Bio-PEPA but also because this requires an explicit state space CTMC, so the problem of state space explosion is particularly acute.
As suggested above, we adopt a high-level language, such as Bio-PEPA, instead of considering directly the CTMC model, in order to take advantage of the multiple analysis techniques it supports, since these can be employed in a complementary fashion during model analysis. Specifically, we first use stochastic simulation in order to validate our model against the results in the literature and derive some information necessary for the definition of the associated explicit CTMC submodels. Subsequently, we study further properties of the submodels using model-checking and investigate which properties of the submodels can safely be interpreted in the context of the complete model. It is worth noting that the decomposition of the Bio-PEPA model into modules does not require us to build the full CTMC model, which indeed can be very large and difficult to represent and study explicitly.
The rest of the paper is organised as follows. First, we present a discussion of model decomposition and an overview of our approach (Section 2). In the following section (Section 3) we give a brief introduction to the Bio-PEPA language. Our appr
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