In this work we present Bio-PEPA, a process algebra for the modelling and the analysis of biochemical networks. It is a modification of PEPA, originally defined for the performance analysis of computer systems, in order to handle some features of biological models, such as stoichiometry and the use of general kinetic laws. The domain of application is the one of biochemical networks. Bio-PEPA may be seen as an intermediate, formal, compositional representation of biological systems, on which different kinds of analysis can be carried out. Bio-PEPA is enriched with some notions of equivalence. Specifically, the isomorphism and strong bisimulation for PEPA have been considered. Finally, we show the translation of three biological models into the new language and we report some analysis results.

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

Recently, a range of information-processing circuits have been implemented in DNA by using strand displacement as their main computational mechanism. Examples include digital logic circuits and catalytic signal amplification circuits that function as efficient molecular detectors. As new paradigms for DNA computation emerge, the development of corresponding languages and tools for these paradigms will help to facilitate the design of DNA circuits and their automatic compilation to nucleotide sequences. We present a programming language for designing and simulating DNA circuits in which strand displacement is the main computational mechanism. The language includes basic elements of sequence domains, toeholds and branch migration, and assumes that strands do not possess any secondary structure. The language is used to model and simulate a variety of circuits, including an entropy-driven catalytic gate, a simple gate motif for synthesizing large-scale circuits and a scheme for implementing an arbitrary system of chemical reactions. The language is a first step towards the design of modelling and simulation tools for DNA strand displacement, which complements the emergence of novel implementation strategies for DNA computing.

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

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.

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Abstract. Bennett’s proposed chemical Turing machine is one of the most important thought experiments in the study of the thermodynamics of computation. Yet the sophistication of molecular engineering required to physically construct Bennett’s hypothetical polymer substrate and enzyme has deterred experimental implementations. Here we propose a chemical implementation of stack machines — a Turing-universal model of computation similar to Turing machines — using strand displacement cascades as the underlying chemical primitive. More specifically, the mechanism described herein is the addition and removal of monomers from the end of a polymer, controlled by strand displacement logic. We capture the motivating feature of Bennett’s scheme — that physical reversibility corresponds to logically reversible computation, and arbitrarily little energy per computation step is required. Further, as a method of embedding logic control into chemical and biological systems, polymer-based chemical computation is significantly more efficient than geometry-free chemical reaction networks. 1

Abstract. We present an algorithm, called BioLab, for verifying temporal properties of rule-based models of cellular signalling networks. BioLab models are encoded in the BioNetGen language, and properties are expressed as formulae in probabilistic bounded linear temporal logic. Temporal logic is a formalism for representing and reasoning about propositions qualified in terms of time. Properties are then verified using sequential hypothesis testing on executions generated using stochastic simulation. BioLab is optimal, in the sense that it generates the minimum number of executions necessary to verify the given property. Bio-Lab also provides guarantees on the probability of it generating Type-I (i.e., false-positive) and Type-II (i.e., false-negative) errors. Moreover, these error bounds are pre-specified by the user. We demonstrate Bio-Lab by verifying stochastic effects and bistability in the dynamics of the T-cell receptor signaling network. 1

Abstract. This note details an interaction pattern for the allocation of a scarce biological resource where and when it is needed. It is entirely based on a mass action stochastic dynamics. Domain-domain binding plays a crucial role in the design of the pattern which we therefore present using a rule-based approach where binding is an explicit primitive. We also a use a series of refinements, starting from a very simple interaction set, which we feel gives an interesting and intuitive rationale for the working of the final repair scheme. 1

Abstract. In this chapter we introduce process algebras, a class of formal modelling techniques developed in theoretical computer science, and discuss their use within systems biology. These formalisms have a number of attractive features which make them ideal candidates to be intermediate, formal, compositional representations of biological systems. As we will show, when modelling is carried out at a suitable level of abstraction, the constructed model can be amenable to analysis using a variety of different approaches, encompassing both individualsbased stochastic simulation and population-based ordinary differential equations. We focus particularly on Bio-PEPA, a recently defined extension of the PEPA stochastic process algebra, which has features to capture both stoichiometry and general kinetic laws. We present the definition of the language, some equivalence relations and the mappings to underlying mathematical models for analysis. We demonstrate the use of Bio-PEPA on two biological examples.