We introduce a formalism to represent and analyze protein-protein and protein-DNA interaction networks. We illustrate the expressivity of this language, by proposing a formal counterpart of Kohn's compilation on the mammalian cell cycle control. This e#ectively turns an otherwise static knowledge into a discrete transition system incorporating a qualitative description of the dynamics. We then propose to use the Computation Tree Logic CTL as a query language for querying the possible behaviours of the system. We provide examples of biologically relevant queries expressed in CTL about the mammalian cell cycle control and show the e#ectiveness of symbolic model checking tools to evaluate CTL queries in this context.

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. One obtains in this paper a process algebra RCCS, in the style of CCS, where processes can backtrack. Backtrack, just as plain forward computation, is seen as a synchronization and incurs no additional cost on the communication structure. It is shown that, given a past, a computation step can be taken back if and only if it leads to a causally equivalent past. 1

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. A refinement of Cardelli’s brane calculus [1] is introduced where membrane actions are directed. This modification brings the language closer to biological membranes and also obtains a symmetric set of membrane interactions. An associated structural congruence, termed the projective equivalence, is defined and shown to be preserved under all possible system evolutions. Comparable notions of projective equivalence can be developed in other hierarchical process calculi and might be of interest in other applications. 1

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 [1–3]. Rule-centric languages such as κ [4–8] 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. 1

A general and abstract framework to defining congruence formats for various process equivalences coalgebraic approach to process equivalence, based on a notion of test suite. The resulting technique is illustrated on the example of completed trace equivalence. Rather than providing formal proofs, the paper is guiding the reader through the process of deriving a congruence format in the test suite approach.

Abstract. Rule-based modelling is particularly effective for handling the highly combinatorial aspects of cellular signalling. The dynamics is described in terms of interactions between partial complexes, and the ability to write rules with such partial complexes-i.e., not to have to specify all the traits of the entitities partaking in a reaction but just those that matter- is the key to obtaining compact descriptions of what otherwise could be nearly infinite dimensional dynamical systems. This also makes these descriptions easier to read, write and modify. In the course of modelling a particular signalling system it will often happen that more traits matter in a given interaction than previously thought, and one will need to strengthen the conditions under which that interaction may happen. This is a process that we call rule refinement and which we set out in this paper to study. Specifically we present a method to refine rule sets in a way that preserves the implied stochastic semantics.

RCCS is a variant of Milner's CCS where processes are allowed a controlled form of backtracking. It turns out that the RCCS reinterpretation of a CCS process is equivalent, in the sense of weak bisimilarity, to its causal transition system in CCS. This can be used to develop an efficient method for designing distributed algorithms, which we illustrate here by deriving a distributed algorithm for assembling trees. This requires solving a highly distributed consensus, and a comparison with a traditional CCS-based solution shows that the code we obtain is shorter, easier to understand, and easier to prove correct by hand, or even to verify.

Linear implication can represent state transitions, but real transition systems operate under temporal, stochastic or probabilistic constraints that are not directly representable in ordinary linear logic. We propose a general modal extension of intuitionistic linear logic where logical truth is indexed by constraints and hybrid connectives combine constraint reasoning with logical reasoning. The logic has a focused cut-free sequent calculus that can be used to internalize the rules of particular constrained transition systems; we illustrate this with an adequate encoding of the synchronous stochastic pi-calculus. We also present some preliminary experiments of direct encoding of biological systems in the logic. 1