In order to understand the functioning of organisms on the molecular level, we need to know which genes are expressed, when and where in the organism, and to which extent. The regulation of gene expression is achieved through genetic regulatory systems structured by networks of interactions between DNA, RNA, proteins, and small molecules. As most genetic regulatory networks of interest involve many components connected through interlocking positive and negative feedback loops, an intuitive understanding of their dynamics is hard to obtain. As a consequence, formal methods and computer tools for the modeling and simulation of genetic regulatory networks will be indispensable. This paper reviews formalisms that have been employed in mathematical biology and bioinformatics to describe genetic regulatory systems, in particular directed graphs, Bayesian networks, Boolean networks and their generalizations, ordinary and partial differential equations, qualitative differential equations, stochastic equations, and rule-based formalisms. In addition, the paper discusses how these formalisms have been used in the simulation of the behavior of actual regulatory systems. Key words: genetic regulatory networks, mathematical modeling, simulation, computational biology.

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Abstract. This paper presents an abstract machine for a variant of the stochastic pi-calculus, in order to correctly model the stochastic simulation of biological processes. The abstract machine is proved sound and complete with respect to the calculus, and then used as the basis for implementing a stochastic simulator. The correctness of the machine helps ensure that the simulator is correctly implemented, giving greater confidence in the simulation results. A graphical representation for the pi-calculus is also presented, as a potential front-end to the simulator. 1

In a biological cell, cellular functions and the genetic regulatory apparatus are implemented and controlled by a network of chemical reactions in which regulatory proteins can control genes that produce other regulators, which in turn control other genes. Further, the feedback pathways appear to incorporate switches that result in changes in the dynamic behavior of the cell. This paper describes a hybrid systems approach to modeling the intra-cellular network using continuous di#erential equations to model the feedback mechanisms and mode-switching to describe the changes in the underlying dynamics. We use two case studies to illustrate a modular approach to modeling such networks and describe the architectural and behavioral hierarchy in the underlying models. We describe these models using Charon [2], a language that allows formal description of hybrid systems. We provide preliminary simulation results that demonstrate how our approach can help biologists in their analysis of noisy genetic circuits. Finally we describe our agenda for future work that includes the development of models and simulation for stochastic hybrid systems.

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. We propose a compositional approach to the dynamics of gene regulatory networks based on the stochastic π-calculus, and develop a representation of gene network elements which can be used to build complex circuits in a transparent and efficient way. To demonstrate the power of the approach we apply it to several artificial networks, such as the repressilator and combinatorial gene circuits first studied in Combinatorial Synthesis of Genetic Networks [1]. For two examples of the latter systems, we point out how the topology of the circuits and the interplay of the stochastic gate interactions influence the circuit behavior. Our approach may be useful for the testing of biological mechanisms proposed to explain the experimentally observed circuit dynamics. 1

Abstract. Modelling is becoming a necessity in studying biological signalling pathways, because the combinatorial complexity of such systems rapidly overwhelms intuitive and qualitative forms of reasoning. Yet, this same combinatorial explosion makes the traditional modelling paradigm based on systems of differential equations impractical. In contrast, agentbased or concurrent languages, such as κ [1–3] or the closely related BioNetGen language [4–10], describe biological interactions in terms of rules, thereby avoiding the combinatorial explosion besetting differential equations. Rules are expressed in an intuitive graphical form that transparently represents biological knowledge. In this way, rules become a natural unit of model building, modification, and discussion. We illustrate this with a sizeable example obtained from refactoring two models of EGF receptor signalling that are based on differential equations [11, 12]. An exciting aspect of the agent-based approach is that it naturally lends itself to the identification and analysis of the causal structures that deeply shape the dynamical, and perhaps even evolutionary, characteristics of complex distributed biological systems. In particular, one can adapt the notions of causality and conflict, familiar from concurrency theory, to κ, our representation language of choice. Using the EGF receptor model as an example, we show how causality enables the formalization of the colloquial concept of pathway and, perhaps more surprisingly, how conflict can be used to dissect the signalling dynamics to obtain a qualitative handle on the range of system behaviours. By taming the combinatorial explosion, and exposing the causal structures and key kinetic junctures in a model, agent- and rule-based representations hold promise for making modelling more powerful, more perspicuous, and of appeal to a wider audience. 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.

Abstract. Living cells are extremely well-organized autonomous systems, consisting of discrete interacting components. Key to understanding and modeling their behavior is modeling their system organization. Four distinct chemical toolkits (classes of macromolecules) have been characterized, each combinatorial in nature. Each toolkit consists of a small number of simple components that are assembled (polymerized) into complex structures that interact in rich ways. Each toolkit abstracts away from chemistry; it embodies an abstract machine with its own instruction set and its own peculiar interaction model. These interaction models are highly effective, but are not ones commonly used in computing: proteins stick together, genes have fixed output, membranes carry activity on their surfaces. Biologists have invented a number of notations attempting to describe these abstract machines and the processes they implement. Moving up from molecular biology, systems biology aims to understand how these interaction models work, separately and together. 1