Abstract Interpretation of Cellular Signalling Networks
Cellular signalling pathways, where proteins can form complexes and undergo a large array of post translational modifications are highly combinatorial systems sending and receiving extra-cellular signals and triggering appropriate responses. Process-centr
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Plectix Biosystems Harvard Medical School 3 ´ Ecole Normale Sup´erieure CNRS, Universit´e Paris Diderot 5 ´ Ecole Polytechnique 2
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Abstract. Cellular signalling pathways, where proteins can form complexes and undergo a large array of post translational modifications are highly combinatorial systems sending and receiving extra-cellular signals and triggering appropriate responses. Process-centric languages seem apt to their representation and simulation [1,2,3]. Rule-centric languages such as κ [4,5,6,7,8] and BNG [9,10] bring in additional ease of expression. We propose in this paper a method to enumerate a superset of the reachable complexes that a κ rule set can generate. This is done via the construction of a finite abstract interpretation. We find a simple criterion for this superset to be the exact set of reachable complexes, namely that the superset is closed under swap, an operation whereby pairs of edges of the same type can permute their ends. We also show that a simple syntactic restriction on rules is sufficient to ensure the generation of a swap-closed set of complexes. We conclude by showing that a substantial rule set (presented in Ref. [4]) modelling the EGF receptor pathway verifies that syntactic condition (up to suitable transformations), and therefore despite its apparent complexity has a rather simple set of reachables.
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Introduction
Biological signalling pathways are large, natural, quantitative concurrent systems in charge of sending and receiving extra-cellular signals and triggering appropriate responses in the cell —eg differentiation, migration, or growth. They involve multiple proteins, from membrane bound receptors to adapters and relays to transcription factors. As thorough a description as possible of these pathways is key to their understanding and control. Such a task is difficult for a variety of reasons, one being of a purely representational nature. Those networks are highly combinatorial, meaning that their agents can assemble and be modified in a huge number of ways —about 1019 unique complexes for the EGF receptor pathway model we consider later. Usual representations based on reactions
´ Abstraction Project (INRIA, CNRS, and Ecole Normale Sup´erieure).
F. Logozzo et al. (Eds.): VMCAI 2008, LNCS 4905, pp. 83–97, 2008. c Springer-Verlag Berlin Heidelberg 2008
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between structureless entities must inevitably sample down this combinatorial complexity, and obtain models which stand in no clear relation to biological facts and are hard to keep abreast of new developments. Regev et al. have proposed using π-calculus [11] to avoid the combinatorial explosion besetting differential equations [1,2]. Other process-based languages have been proposed [3,12,13,14]. The rule-based languages κ [4,5,6,7] and BNG [9,10] bring additional ease in the building and modification of models [8]. The object of this paper is to explain and illustrate on a sizable example a method to explore the set of complexes that can be generated by a κ rule set, aka the system’s reachable comple
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