Abstract. Probabilistic model checking is a formal verification technique that has been successfully applied to the analysis of systems from a broad range of domains, including security and communication protocols, distributed algorithms and power management. In this paper we illustrate its applicability to a complex biological system: the FGF (Fibroblast Growth Factor) signalling pathway. We give a detailed description of how this case study can be modelled in the probabilistic model checker PRISM, discussing some of the issues that arise in doing so, and show how we can thus examine a rich selection of quantitative properties of this model. We present experimental results for the case study under several different scenarios and provide a detailed analysis, illustrating how this approach can be used to yield a better understanding of the dynamics of the pathway. 1

Abstract. Differential equations are a classical approach for biochemical system modelling and have frequently been used to describe reactions of interest in biochemical pathways. Process algebras have also been applied in a small number of cases to describe such systems. In this paper we establish a connection between these approaches. This has the benefit of allowing process algebra models to be validated against trusted ODEs or, conversely, allowing ODEs derived from process algebra models to be evaluated and compared using bisimulation or other methods. In addition the process algebra models may now be efficiently solved using numerical differential equations procedures such as adaptive fifth-order Runge-Kutta. 1

Abstract. We describe a quantitative modelling and analysis approach for signal transduction networks. We illustrate the approach with an example, the RKIP inhibited ERK pathway [CSK + 03]. Our models are high level descriptions of continuous time Markov chains: proteins are modelled by synchronous processes and reactions by transitions. Concentrations are modelled by discrete, abstract quantities. The main advantage of our approach is that using a (continuous time) stochastic logic and the PRISM model checker, we can perform quantitative analysis such as what is the probability that if a concentration reaches a certain level, it will remain at that level thereafter? or how does varying a given reaction rate affect that probability? We also perform standard simulations and compare our results with a traditional ordinary differential equation model. An interesting result is that for the example pathway, only a small number of discrete data values is required to render the simulations practically indistinguishable.

We tackle the problem of relating models of systems (mainly biological systems) based on stochastic process algebras (SPA) with models based on differential equations. We define a syntactic procedure that translates programs written in stochastic Concurrent Constraint Programming (sCCP) into a set of Ordinary Differential Equations (ODE), and also the inverse procedure translating ODE’s into sCCP programs. For the class of biochemical reactions, we show that the translation is correct w.r.t. the intended rate semantics of the models. Finally, we show that the translation does not generally preserve the dynamical behavior, giving a list of open research problems in this direction.

In this work we present Bio-PEPA, a process algebra for the modelling and the analysis of biochemical networks. It is a modification of PEPA, originally defined for the performance analysis of computer systems, in order to handle some features of biological models, such as stoichiometry and the use of general kinetic laws. The domain of application is the one of biochemical networks. Bio-PEPA may be seen as an intermediate, formal, compositional representation of biological systems, on which different kinds of analysis can be carried out. Bio-PEPA is enriched with some notions of equivalence. Specifically, the isomorphism and strong bisimulation for PEPA have been considered. Finally, we show the translation of three biological models into the new language and we report some analysis results.

Abstract. Given the combinatorial nature of cellular signalling pathways, where biological agents can bind and modify each other in a large number of ways, concurrent or agent-based languages seem particularly suitable for their representation and simulation [1–4]. Graphical modelling languages such as κ [5–8], or the closely related BNG language [9– 14], seem to afford particular ease of expression. It is unclear however how such models can be implemented. 6 Even a simple model of the EGF receptor signalling network can generate more than 10 23 non-isomorphic species [5], and therefore no approach to simulation based on enumerating species (beforehand, or even on-the-fly) can handle such models without sampling down the number of potential generated species. We present in this paper a radically different method which does not attempt to count species. The proposed algorothm uses a representation of the system together with a super-approximation of its ‘event horizon ’ (all events that may happen next), and a specific correction scheme to obtain exact timings. Being completely local and not based on any kind of enumeration, this algorithm has a per event time cost which is independent of (i) the size of the set of generable species (which can even be infinite), and (ii) independent of the size of the system (ie, the number of agent instances). We show how to refine this algorithm, using concepts derived from the classical notion of causality, so that in addition to the above one also has that the even cost is depending (iii) only logarithmically on the size of the model (ie, the number of rules). Such complexity properties reflect in our implementation which, on a current computer, generates about 10 6 events per minute in the case of the simple EGF receptor model mentioned above, using a system with 10 5 agents. 1

Abstract. Starting from a biochemical signalling pathway model expressed in a process algebra enriched with quantitative information we automatically derive both continuous-space and discrete-state representations suitable for numerical evaluation. We compare results obtained using implicit numerical differentiation formulae to those obtained using approximate stochastic simulation thereby exposing a flaw in the use of the differentiation procedure producing misleading results. 1

This paper presents efficient mechanisms for the direct implementation of formal models of highly concurrent dynamic systems. The formalisms captured are CSP (for concurrency) and B (for state transformation). The technology is driving the development of occam-π, a multiprocessing language based on a careful combination of ideas from Hoare’s CSP (giving compositional semantics, refinement and safety/liveness analysis) and Milner’s π-calculus (giving dynamic network construction and mobility). We have been experimenting with systems developing as layered networks of self-organising neighbourhood-aware communicating processes, with no need for advanced planning or centralised control. The work reported is part of our TUNA (‘Theory Underpinning Nanotech Assemblers’) project, a partnership with colleagues from the Universities of York, Surrey and Kent, which is investigating formal approaches to the capture of safe emergent behaviour in highly complex systems. A particular study modelling artificial blood platelets is described. A novel contribution reported here is a fast resolution of (CSP external) choice between multiway process synchronisations from which any participant may withdraw its offer at any time. The software technology scales to millions of processes per processor and distributes over common multiprocessor clusters. 1.

Abstract. The Performance Evaluation Process Algebra (PEPA) language is a stochastic process algebra, generating Continuous Time Markov Chains (CTMC) to allow quantitative analysis. Protocols such as BitTorrent are highly parallel in nature, and represent one area where CTMC analysis is limited by the well-known state space problem. The number of unique states each client can exist in, and the number of clients required to accurately model a typical BitTorrent network preclude the use of CTMCs. Recent work has shown that PEPA models also allow the derivation of an activity matrix, from which ODE and stochastic simulation models, as alternative forms of analysis, are possible. Using this technique, a BitTorrent network is created, analysed, and the results compared against previous BitTorrent models.

The MAPK/ERK pathway regulates fundamental cellular function such as cell proliferation, survival, differentiation and motility, raising the question how these diverse functions are specified and coordinated. They are encoded through the activation kinetics of the pathway, a multitude of feedback loops, scaffold proteins, subcellular compartmentalisation, and crosstalk with other pathways. These regulatory motifs alone or in combination can generate a multitude of complex behaviour. Systems biology tries to decode this complexity through mathematical modelling and prediction in order to gain a deeper insight into the inner works of signalling networks. Key words: MAPK/ERK, signalling networks, systems biology, computational modelling 2 The MAPK pathway Originally, the name Mitogen Activated Protein Kinase, MAPK, referred to a kinase that was rapidly stimulated by insulin and other growth factors. When cloned this kinase was dubbed Extracellular signal Regulated Kinase, ERK, and the name MAPK evolved into the family