Section: New Results

Analysis of Biological Pathways

We have introduced a framework to design and analyze biological networks. We focus on protein-protein interaction networks described as graph rewriting systems. Such networks can be used to model some signaling pathways that control the cell cycle. The task is made difficult due to the combinatorial blow up in the number of reachable species (i.e. , non-isomorphic connected components of proteins).

Automatic Reduction of Differential Semantics

Keywords : biology, protein-protein interaction networks, differential semantics, verification.

Participants : Vincent Danos [ University of Edinburgh ] , Jérôme Feret, Walter Fontana [ Harvard Medical School ] , Russel Harmer [ Paris VII ] , Jean Krivine [ Paris VII ] .

We have developed an abstract interpretation-based framework that enables the computation of scalable differential semantics for protein-protein interaction networks. This framework uses indistinguishably techniques in order to detect and prove that some potential correlations between the states of some distinct parts in protein species have no impact on the dynamic of the networks. These information drive the computation of an abstract differential system over a set of self-consistent abstract observables. Results are sound since trajectories in the abstract system are projections of the trajectories in the concrete system. This framework gives new insights in order to describe evolution between systems: indeed several networks can be compared according to the relative amount of control between protein-protein interactions.

This framework has been published in [17] and presented in [51] , [50] , and [53] .

Automatic Reduction of Stochastic Semantics

Keywords : biology, protein-protein interaction networks, stochastic semantics, verification.

Participants : Jérôme Feret, Heinz Koeppl [ École Polytechnique Fédérale de Lausanne ] , Tatjana Petrov [ École Polytechnique Fédérale de Lausanne ] .

We have proposed an abstract interpretation-based framework for reducing the state-space of stochastic semantics for protein-protein interaction networks. Our approach consists in quotienting the state-space of networks. Yet interestingly, we do not apply the widely-used strong lumpability criterion which imposes that two equivalent states behave similarly with respect to the quotient, but a weak version of it. More precisely, our framework detects and proves some invariants about the dynamics of the system: indeed the quotient of the state-space is such that the probability of being in a given state knowing that this state is in a given equivalence class, is an invariant of the semantics.

Then we have introduced an individual-based stochastic semantics (where each agent is identified by a unique identifier) for the programs of a rule-based language and we use our abstraction framework to derive a sound population-based semantics and a sound fragments-based semantics, which give the distribution of the traces respectively for the number of instances of molecular species and for the number of instances of partially defined molecular species. These partially defined species are chosen automatically thanks to a dependency analysis which is also described in the paper.

Interestingly, we have showed that the criteria that we were using in [17] in order to abstract the differential semantics were not sound regarding to the stochastic semantics. Indeed, we had to strengthen these criteria in order to respect the stochastic semantics. As a consequence the reduction factor is far less impressive in the case of the stochastic semantics. This reflects the fact that the stochastic semantics is a much more expressive and interesting object than the differential semantics and also that it is much more difficult to abstract.

This work has been presented in [55] and [56] .

Rule Refinements

Keywords : biology, concurrency, protein-protein interaction networks, refinements.

Participants : Vincent Danos [ University of Edinburgh ] , Jérôme Feret, Russel Harmer [ Paris VII ] , Jean Krivine [ Harvard Medical School ] , Elaine Murphy [ University of Edinburgh ] .

We have proposed a formal framework to refine rule-based protein-protein interaction networks while preserving their stochastic and their differential semantics. Refinements is a key process in rule-based modeling. Refining an interaction allows tuning the kinetics of an interaction according to some constraints in the context of the interacting proteins.

In [84] , we had proposed a framework to make homogeneous refinements. In such a homogeneous refinement, the accuracy of the refinement is the same for each protein of a given type. In [34] , we have extended this framework in order to make heterogeneous refinements, where each agent in a given pattern can be refined independently.

Investigation of a Biological Repair Scheme

Keywords : biology, protein-protein interaction networks, refinements.

Participants : Vincent Danos [ University of Edinburgh ] , Jérôme Feret, Walter Fontana.

In [23] , we investigated 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 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.

Meta-Language

Keywords : biology, concurrency, protein-protein interaction networks, refinements.

Participants : Vincent Danos [ University of Edinburgh ] , Jérôme Feret, Walter Fontana [ Harvard Medical School ] , Russel Harmer [ Paris VII ] , Jean Krivine [ Harvard Medical School ] .

Rule-based modeling has already proved to be successful for taming the combinatorial complexity, typical of cellular signaling networks, caused by the combination of physical protein-protein interactions and modifications that generate astronomical numbers of distinct molecular species. However, traditional rule-based approaches, based on an unstructured space of agents and rules, remain susceptible to other combinatorial explosions caused by mutated and/or splice variant agents, that share most but not all of their rules with their wild-type counterparts; and by drugs, which must be clearly distinguished from physiological ligands.

In [16] , we define a syntactic extension of Kappa, an established rule-based modeling platform, that enables the expression of a structured space of agents and rules that allows us to express mutated agents, splice variants, families of related proteins and ligand/drug interventions uniformly. This also enables a mode of model construction where, starting from the current consensus model, we attempt to reproduce in numero the mutational—and more generally the ligand/drug perturbational—analyses that were used in the process of inferring those pathways in the first place.