Persistence is the property, for differential equations in R n, that solutions starting in the positive orthant do not approach the boundary of the orthant. For chemical reactions and population models, this translates into the non-extinction property: provided that every species is present at the start of the reaction, no species will tend to be eliminated in the course of the reaction. This paper provides checkable conditions for persistence of chemical species in reaction networks, using concepts and tools from Petri net theory, and verifies these conditions on various systems which arise in the modeling of cell signaling pathways.

Abstract interpretation is a theory of abstraction that has been introduced for the analysis of programs. In particular, it has proved useful for organizing the multiple semantics of a given programming language in a hierarchy corresponding to different detail levels, and for defining type systems for programming languages and program analyzers in software engineering. In this paper, we investigate the application of these concepts to systems biology formalisms. More specifically, we consider the Systems Biology Markup Language SBML, and the Biochemical Abstract Machine BIOCHAM with its differential, stochastic, discrete and boolean semantics. We first show how all of these different semantics, except the differential one, can be formally related by simple Galois connections. Then we define three type systems: one for checking or inferring the functions of proteins in a reaction model, one for checking or inferring the activation and inhibition effects of proteins in a reaction model, and another one for checking or inferring the topology of compartments or locations. We show that the framework of abstract interpretation elegantly applies to the formalization of these further abstractions, and to the implementation of linear or quadratic time type checking as well as type inference algorithms. Furthermore, we show a theorem of independence of the graph of activation and inhibition effects from the kinetic expressions in the reaction model, under general conditions. Through some examples, we show that the analysis of biochemical models by type inference provides accurate and useful information. Interestingly, such a mathematical formalization of the abstractions commonly used in systems biology already provides some guidelines for the extensions of biochemical reaction rule languages.

Petri nets are a discrete event simulation approach developed for system representation, in particular for their concurrency and synchronization properties. Various extensions to the original theory of Petri nets have been used for modeling molecular biology systems and metabolic networks. These extensions are stochastic, colored, hybrid and functional. This paper carries out an initial review of the various modeling approaches based on Petri net found in the literature, and of the biological systems that have been successfully modeled with these approaches. Moreover, the modeling goals and possibilities of qualitative analysis and system simulation of each approach are discussed.

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Abstract. The Pathalyzer is a program for analyzing large-scale signal transduction networks. Reactions and their substrates and products are represented as transitions and places in a safe Petri net. The user can interactively specify goal states, such as activation of a particular protein in a particular cell site, and the system will automatically find and display a pathway that results in the goal state – if possible. The user can also require that the pathway be generated without using certian proteins. The system can also find all individual places and all pairs of places which, if knocked out, would prevent the goals from being achieved. The tool is intended to be used by biologists with no significant understanding of Petri nets or any of the other concepts used in the implementation. 1

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Summary. A positive dynamical system is said to be persistent if every solution that starts in the interior of the positive orthant does not approach the boundary of this orthant. For chemical reaction networks and other models in biology, persistence represents a non-extinction property: if every species is present at the start of the reaction, then no species will tend to be eliminated in the course of the reaction. This paper provides checkable necessary as well as sufficient conditions for persistence of chemical species in reaction networks, and the applicability of these conditions is illustrated on some examples of relatively high dimension which arise in molecular biology. More specific results are also provided for reactions endowed with mass-action kinetics. Overall, the results exploit concepts and tools from Petri net theory as well as ergodic and recurrence theory. 1

Abstract. Biologists use diagrams to represent interactions between molecular species, and on the computer, diagrammatic notations are also employed in interactive maps. These diagrams are fundamentally of two types: reaction graphs and activation/inhibition graphs. In this tutorial, we study these graphs with formal methods originating from programming theory. We consider systems of biochemical reactions with kinetic expressions, as written in the Systems Biology Markup Language (SBML), and interpreted in the Biochemical Abstract Machine (Biocham) at different levels of abstraction, by either an asynchronous boolean transition system, a continuous time Markov chain, or a system of Ordinary Differential Equations over molecular concentrations. We show that under general conditions satisfied in practice, the activation/inhibition graph is independent of the precise kinetic expressions, and is computable in linear time in the number of reactions. Then we consider the formalization of the biological properties of systems, as observed in experiments, in temporal logics. We show that these logics are expressive enough to capture semi-qualitative semi-quantitative properties of the boolean and differential semantics of reaction models, and that model-checking techniques can be used to validate a model w.r.t. its temporal specification, complete it, and search for kinetic parameter values. We illustrate this modelling method with examples on the MAPK signalling cascade, and on Kohn’s map of the mammalian cell cycle. 1

This paper derives new results for certain classes of chemical reaction networks, linking structural to dynamical properties. In particular, it investigates their monotonicity and convergence under the assumption that the rates of the reactions are monotone functions of the concentrations of their reactants. This is satisfied for, yet not restricted to, the most common choices of the reaction kinetics such as mass action, Michaelis-Menten and Hill kinetics. The key idea is to finding an alternative representation under which the resulting system is monotone. As a simple example, the paper shows that a phosphorylation/dephosphorylation process, which is involved in many signaling cascades, has a global stability property. We also provide a global stability result for a more complicated example that describes a regulatory pathway of a prevalent signal transduction module, the MAPK cascade.

Abstract. The complexity of biological regulatory networks calls for the development of proper mathematical methods to model their structures and to give insight in their dynamical behaviours. One qualitative approach consists in modelling regulatory networks in terms of logical equations (using either Boolean or multi-level discretisation). Petri Nets (PNs) offer a complementary framework to analyse large systems. In this paper, we propose to articulate the logical approach with PNs. We first revisit the definition of a rigourous and systematic mapping of multi-level logical regulatory models into specific standard PNs, called Multi-level Regulatory Petri Nets (MRPNs). In particular, we consider the case of multiple arcs representing different regulatory effects from the same source. We further propose a mapping of multi-level logical regulatory models into Coloured PNs, called Coloured Regulatory Petri Nets (CRPNs). These CRPNs provide an intuitive graphical representation of regulatory networks, relatively easy to grasp. Finally, we present the PN translation and the analysis of a multi-level logical model of the core regulatory network controlling the differentiation of T-helper lymphocytes into Th1 and Th2 types. 1