Forward and Backward Bisimulations for Chemical Reaction Networks

We present two quantitative behavioral equivalences over species of a chemical reaction network (CRN) with semantics based on ordinary differential equations. Forward CRN bisimulation identifies a partition where each equivalence class represents the exact sum of the concentrations of the species belonging to that class. Backward CRN bisimulation relates species that have the identical solutions at all time points when starting from the same initial conditions. Both notions can be checked using only CRN syntactical information, i.e., by inspection of the set of reactions. We provide a unified algorithm that computes the coarsest refinement up to our bisimulations in polynomial time. Further, we give algorithms to compute quotient CRNs induced by a bisimulation. As an application, we find significant reductions in a number of models of biological processes from the literature. In two cases we allow the analysis of benchmark models which would be otherwise intractable due to their memory requirements.Comment: Extended version of the CONCUR 2015 pape

Translating BNGL models into Kappa our experience

Extended abstractSo as to test the Kappa development tools on more examples, we translated the models provided with the BNGL distribution, into Kappa. In this talk, we report about our experience. The translation was quite straight-forward except for few interesting issues that we detail here. Firstly the use of static analysis has exposed some glitches in the modelling of some pathways in the models of the BNGL distribution. We explain how static analysis has helped us to detect, locate, and correct these flaws. Secondly, expanding BNGL rules using equivalent sites into rules with uniquely identified sites is not so easy when one wants to preserve faithfully the kinetics of interactions. We recall the semantics of BNGL for equivalent sites, and explain how to perform such translation

Formal lumping of polynomial differential equations through approximate equivalences

It is well known that exact notions of model abstraction and reduction for dynamical systems may not be robust enough in practice because they are highly sensitive to the specific choice of parameters. In this paper we consider this problem for nonlinear ordinary differential equations (ODEs) with polynomial derivatives. We introduce a model reduction technique based on approximate differential equivalence, i.e., a partition of the set of ODE variables that performs an aggregation when the variables are governed by nearby derivatives. We develop algorithms to (i) compute the largest approximate differential equivalence; (ii) construct an approximately reduced model from the original one via an appropriate perturbation of the coefficients of the polynomials; and (iii) provide a formal certificate on the quality of the approximation as an error bound, computed as an over-approximation of the reachable set of the reduced model. Finally, we apply approximate differential equivalences to case studies on electric circuits, biological models, and polymerization reaction networks

Symbolic Computation of Differential Equivalences

Ordinary differential equations (ODEs) are widespread in manynatural sciences including chemistry, ecology, and systems biology,and in disciplines such as control theory and electrical engineering. Building on the celebrated molecules-as-processes paradigm, they have become increasingly popular in computer science, with high-level languages and formal methods such as Petri nets, process algebra, and rule-based systems that are interpreted as ODEs. We consider the problem of comparing and minimizing ODEs automatically. Influenced by traditional approaches in the theory of programming, we propose differential equivalence relations. We study them for a basic intermediate language, for which we have decidability results, that can be targeted by a class of high-level specifications. An ODE implicitly represents an uncountable state space, hence reasoning techniques cannot be borrowed from established domains such as probabilistic programs with finite-state Markov chain semantics. We provide novel symbolic procedures to check an equivalence and compute the largest one via partition refinement algorithms that use satisfiability modulo theories. We illustrate the generality of our framework by showing that differential equivalences include (i) well-known notions for the minimization of continuous-time Markov chains (lumpability),(ii) bisimulations for chemical reaction networks recently proposedby Cardelli et al., and (iii) behavioral relations for process algebra with ODE semantics. With a prototype implementation we are able to detect equivalences in biochemical models from the literature thatcannot be reduced using competing automatic techniques

KaDE: A Tool to Compile Kappa Rules into (Reduced) ODE Models

Tools paper trackInternational audienceKappa is a formal language that can be used to model sys- tems of biochemical interactions among proteins. It offers several se- mantics to describe the behaviour of Kappa models at different levels of abstraction. Each Kappa model is a set of context-free rewrite rules. One way to understand the semantics of a Kappa model is to read its rules as an implicit description of a (potentially infinite) reaction net- work. KaDE is interpreting this definition to compile Kappa models into reaction networks (or equivalently into sets of ordinary differential equations). KaDE uses a static analysis that identifies pairs of sites that are indistinguishable from the rules point of view, to infer backward and forward bisimulations, hence reducing the size of the underlying reaction networks without having to generate them explicitly. In this paper, we describe the main current functionalities of KaDE and we give some benchmarks on case studies

Formal reduction for rule-based models

International audienceMolecular biological models usually suffer from a large combinatorial explosion. Indeed, proteins form complexes and modify each others, which leads to the formation of a huge number of distinct chemical species (i.e.~non-isomorphic connected components of proteins). Thus we cannot generate explicitly the quantitative semantics of these models, and even less compute their properties. In this paper we propose a formal framework to automatically reduce the combinatorial complexity of the differential semantics of rule-based models. Our reduction is based on two abstractions, which are combined thanks to a generic product. The first abstraction tracks the flow of information between the different regions of chemical species, so as to detect and abstract away some useless correlations between the state of sites. The second abstraction detects pairs of sites having the same capabilities of interaction, and abstracts away any distinction between them. The initial semantics and the reduce one are formally related by Abstract Interpretation