Probabilistic model checking of complex biological pathways

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

Modular Verification of Biological Systems

Systems of interest in systems biology (such as metabolic pathways, signalling pathways and gene regulatory networks) often consist of a huge number of components interacting in different ways, thus exhibiting very complex behaviours. In biology, such behaviours are usually explored by means of simulation techniques applied to models defined on the basis of system observation and of hypotheses on its functioning. Model checking has also been recently applied to the analysis of biological systems. This analysis technique typically relies on a state space representation whose size, unfortunately, makes the analysis often intractable for realistic models. A method for trying to avoid the state space explosion problem is to consider a decomposition of the system, and to apply a modular verification technique. In particular, properties to be verified often concern only a small portion of the modelled system rather than the system as a whole. Hence, for each property it would be useful to be able to isolate a minimal fragment of the model that is necessary to verify such a property. In this thesis we introduce a modular verification technique in which the system of interest is described by means of an automata-based formalism, called sync-programs, that supports modular construction. Our modular verification technique is based on results of Grumberg et al.~and on their application to the theory of concurrent systems proposed by Attie and Emerson. In particular, we adapt Attie and Emerson's approach to deal with biological systems by allowing automata to synchronise by performing transitions simultaneously. Modular verification allows qualitative aspects of systems to be analysed with the guarantee that properties proved to hold in a suitable model fragment also hold in the whole model. The correctness of the verification technique is proved. The class of properties preserved is ACTL$^{-}$, the universal fragment of temporal logic CTL. The preservation holds only for positive answers and negative answers are not necessarily preserved. In order to verify properties we use the NuSMV model checker, which is a well-established and efficient instrument. We provide a formal translation of sync-programs to simpler automata, which can be given as input to NuSMV. We prove the correspondence of the verification problems. We show the application of our verification technique in some biological case studies. We compare the time required to verify the property on the whole model with the time needed to verify the same property by only considering those modules which are involved in the behaviour of the system related to the property. In order to handle modelling and verification of more realistic biological scenarios, we propose also a dynamic version of our formalism. It allows entities to be created dynamically, in particular by other already running entities, as it often happens in biological systems. Moreover, multiple copies of the same entities can be present at the same time in a system. We show a correspondence of our model with Petri Nets. This has a consequence that tools developed for Petri Nets could be used also for dynamic sync-programs. Modular verification allows properties expressed as DACTL- formulae (dynamic version of ACTL-) to be verified on a portion of the model. The results of analysis of the case study of the MAP kinase cascade activated by surface and internalised EGF receptors, which consists of 143 species and 80 reactions, suggest applicability and scalability of the approach. The results raise the prospect of rendering tractable problems that are currently intractable in the verification of biological systems. In addition, we expect that the techniques developed in the thesis could be applied with profit not only to models of biological systems, but more generally to models of concurrent systems

Efficient Parallel Statistical Model Checking of Biochemical Networks

We consider the problem of verifying stochastic models of biochemical networks against behavioral properties expressed in temporal logic terms. Exact probabilistic verification approaches such as, for example, CSL/PCTL model checking, are undermined by a huge computational demand which rule them out for most real case studies. Less demanding approaches, such as statistical model checking, estimate the likelihood that a property is satisfied by sampling executions out of the stochastic model. We propose a methodology for efficiently estimating the likelihood that a LTL property P holds of a stochastic model of a biochemical network. As with other statistical verification techniques, the methodology we propose uses a stochastic simulation algorithm for generating execution samples, however there are three key aspects that improve the efficiency: first, the sample generation is driven by on-the-fly verification of P which results in optimal overall simulation time. Second, the confidence interval estimation for the probability of P to hold is based on an efficient variant of the Wilson method which ensures a faster convergence. Third, the whole methodology is designed according to a parallel fashion and a prototype software tool has been implemented that performs the sampling/verification process in parallel over an HPC architecture

Investigating modularity in the analysis of process algebra models of biochemical systems

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

Petri nets for systems and synthetic biology

We give a description of a Petri net-based framework for modelling and analysing biochemical pathways, which uni¯es the qualita- tive, stochastic and continuous paradigms. Each perspective adds its con- tribution to the understanding of the system, thus the three approaches do not compete, but complement each other. We illustrate our approach by applying it to an extended model of the three stage cascade, which forms the core of the ERK signal transduction pathway. Consequently our focus is on transient behaviour analysis. We demonstrate how quali- tative descriptions are abstractions over stochastic or continuous descrip- tions, and show that the stochastic and continuous models approximate each other. Although our framework is based on Petri nets, it can be applied more widely to other formalisms which are used to model and analyse biochemical networks