An Algebraic Approach for Inferring and Using Symmetries in Rule-based Models

International audienceSymmetries arise naturally in rule-based models, and under various forms. Besides automorphisms between site graphs, which are usually built within the semantics, symmetries can take the form of pairs of sites having the same capabilities of interactions, of some protein variants behaving exactly the same way, or of some linear, planar, or 3D molecular complexes which could be seen modulo permutations of their axis and/or mirror-image symmetries. In this paper, we propose a unifying handling of symmetries in Kappa. We follow an algebraic approach, that is based on the single pushout semantics of Kappa. We model classes of symmetries as finite groups of transformations between site graphs, which are compatible with the notion of embedding (that is to say that it is always possible to restrict a symmetry that is applied with the image of an embedding to the domain of this embedding) and we provide some assumptions that ensure that symmetries are compatible with pushouts. Then, we characterise when a set of rules is symmetric with respect to a group of symmetries and, in such a case, we give sufficient conditions so that this group of symmetries induces a forward bisimulation and/or a backward bisimulation over the population semantics

Using alternated sums to express the occurrence number of extended patterns in site-graphs

To appearInternational audienceSite-graph rewriting languages as Kappa or BNGL supply a convenient way to describe models of signalling pathways. Unlike classical reaction networks, they emphasise on the biochemical structure of proteins. We use patterns to formalise properties about bio-molecular species. Intentionally, a pattern is a part of a species, but extensionally it denotes the multi-set of the species containing this pattern (with the multiplicity). Thus reasoning on patterns allows to handle symbolically arbitrarily big (if not infinite) multi-sets of species. This is a key point to design fast simulation algorithms or model reduction schemes. In this paper, we introduce the notion of extended patterns. Each extended pattern is made of a classical pattern and of a set of potential bonds between pairs of sites. Extended patterns have positive (when at least one of the potential bonds is realised) and negative (when none is realised) instances. They are important to express the consumption and the production of patterns by the rules that may break cycles in bio-molecular species by side-effects. We show that the number of positive (resp. negative) instances of extended patterns may be expressed as alternated sums of the number of occurrences of classical patterns

A generic framework to coarse-grain stochastic reaction networks by Abstract Interpretation

International audienceIn the last decades, logical or discrete models have emerged as a successful paradigm for capturing and predicting the behaviors of systems of molecular interactions. Intuitively, they consist in sampling the abundance of each kind of biochemical entity within finite sets of intervals and deriving transitions accordingly. On one hand, formallyproven sound derivation from more precise descriptions (such as from reaction networks) may include many fictitious behaviors. On the other hand, direct modeling usually favors dominant interactions with no guarantee on the behaviors that are neglected. In this paper, we formalize a sound coarse-graining approach for stochastic reaction networks. Its originality relies on two main ingredients. Firstly, we abstract values by intervals that overlap in order to introduce a minimal effort for the system to go back to the previous interval, hence limiting fictitious oscillations in the coarse-grained models. Secondly, we compute for pairs of transitions (in the coarse-grained model) bounds on the probabilities on which one will occur first. We illustrate our ideas on two case studies and demonstrate how techniques from Abstract Interpretation can be used to design more precise discretization methods, while providing a framework to further investigate the underlying structure of logical and discrete models

EPTCS

The induction of a signaling pathway is characterized by transient complex formation and mutual posttranslational modification of proteins. To faithfully capture this combinatorial process in a math- ematical model is an important challenge in systems biology. Exploiting the limited context on which most binding and modification events are conditioned, attempts have been made to reduce the com- binatorial complexity by quotienting the reachable set of molecular species, into species aggregates while preserving the deterministic semantics of the thermodynamic limit. Recently we proposed a quotienting that also preserves the stochastic semantics and that is complete in the sense that the semantics of individual species can be recovered from the aggregate semantics. In this paper we prove that this quotienting yields a sufficient condition for weak lumpability and that it gives rise to a backward Markov bisimulation between the original and aggregated transition system. We illustrate the framework on a case study of the EGF/insulin receptor crosstalk

Local Traces: An Over-Approximation of the Behavior of the Proteins in Rule-Based Models

International audienceThanks to rule-based modelling languages, we can assemble large sets of mechanistic protein-protein interactions within integrated models. Our goal would be to understand how the behavior of these systems emerges from these low-level interactions. Yet, this is a quite long term challenge and it is desirable to offer intermediary levels of abstraction, so as to get a better understanding of the models and to increase our confidence within our mechanistic assumptions. To this extend, static analysis can be used to derive various abstractions of the semantics, each of them offering new perspectives on the models. We propose an abstract interpretation of the behavior of each protein, in isolation. Given a model written in Kappa, this abstraction computes for each kind of proteins a transition system that describes which conformations this protein may take and how a protein may pass from one conformation to another one. Then, we use simplicial complexes to abstract away the interleaving order of the transformations between conformations that commute. As a result, we get a compact summary of the potential behavior of each protein of the model

Local traces: an over-approximation of the behaviour of the proteins in rule-based models

International audienceThanks to rule-based modelling languages, we can assemble large sets of mechanistic protein-protein interactions within integrated models. Our goal would be to understand how the behaviour of these systems emerges from these low-level interactions. Yet this is a quite long term challenge and it is desirable to offer intermediary levels of abstraction, so as to get a better understanding of the models and to increase our confidence within our mechanistic assumptions. In this paper, we propose an abstract interpretation of the behaviour of each protein, in isolation. Given a model written in Kappa, this abstraction computes for each kind of protein a transition system that describes which conformations this protein can take and how a protein can pass from one conformation to another one. Then, we use simplicial complexes to abstract away the interleaving order of the transformations between conformations that commute. As a result, we get a compact summary of the potential behaviour of each protein of the model

Reachability analysis via orthogonal sets of patterns

International audienceRule-based modelling languages, such as Kappa, allow for the description of very detailed mechanistic models. Yet, as the rules become more and more numerous, there is a need for formal methods to enhance the level of confidence in the models that are described with these languages.We develop abstract interpretation tools to capture invariants about the biochemical structure of the bio-molecular species that may occur in a given model. In previous works, we have focused on the relationships between the states of the sites that belong to the same instance of a protein. This comes down to detect for a specific set of patterns, which ones may be reachable during the execution of the model. In this paper, we generalise this approach to a broader family of abstract domains that we call orthogonal sets of patterns. More precisely, an orthogonal set of patterns is obtained by refining recursively the information about some patterns containing a given protein, so as to partition the set of occurrences of this protein in any mixture. We show that orthogonal sets of patterns offer a convenient choice to design scalable and accurate static analyses. As an example, we use them to infer properties in models with transport of molecules (more precisely, we show that each pair of proteins that are connected, always belong to the same compartment), and models involving double bindings (we show that whenever a protein of type A is bound twice to proteins of type B, then the protein A is necessarily bound twice to the same instance of the protein B)

Combinatorial Complexity and Compositional Drift in Protein Interaction Networks

The assembly of molecular machines and transient signaling complexes does not typically occur under circumstances in which the appropriate proteins are isolated from all others present in the cell. Rather, assembly must proceed in the context of large-scale protein-protein interaction (PPI) networks that are characterized both by conflict and combinatorial complexity. Conflict refers to the fact that protein interfaces can often bind many different partners in a mutually exclusive way, while combinatorial complexity refers to the explosion in the number of distinct complexes that can be formed by a network of binding possibilities. Using computational models, we explore the consequences of these characteristics for the global dynamics of a PPI network based on highly curated yeast two-hybrid data. The limited molecular context represented in this data-type translates formally into an assumption of independent binding sites for each protein. The challenge of avoiding the explicit enumeration of the astronomically many possibilities for complex formation is met by a rule-based approach to kinetic modeling. Despite imposing global biophysical constraints, we find that initially identical simulations rapidly diverge in the space of molecular possibilities, eventually sampling disjoint sets of large complexes. We refer to this phenomenon as “compositional drift”. Since interaction data in PPI networks lack detailed information about geometric and biological constraints, our study does not represent a quantitative description of cellular dynamics. Rather, our work brings to light a fundamental problem (the control of compositional drift) that must be solved by mechanisms of assembly in the context of large networks. In cases where drift is not (or cannot be) completely controlled by the cell, this phenomenon could constitute a novel source of phenotypic heterogeneity in cell populations