## Section: New Results

### Static analysis of signaling pathways

#### Formal and exact reduction for differential models of signaling pathways in rule-based languages

Participant : Ferdinanda Camporesi.

The behavior of a cell is driven by its capability to receive, propagate and communicate signals. Proteins can bind together on some binding sites. Post- translational modifications can reveal or hide some sites, so new interactions can be allowed or existing ones can be inhibited.

Due to the huge number of different bio-molecular complexes, we can no longer derive or integrate ODE models. A compact way to describe these systems is supplied by rule-based languages. However combinatorial complexity raises again when one attempt to describe formally the behavior of the models. This motivates the use of abstractions.

In this PhD thesis, we propose two methods to reduce the size of the models, that exploit respectively the presence of symmetries between sites and the lack of correlation between different parts of the system. The symmetries relates pairs of sites having the same capability of interactions. We show that this relation induces a bisimulation which can be used to reduce the size of the original model. The information flow analysis detects, for each site, which parts of the system influence its behavior. This allows us to cut the molecular species in smaller pieces and to write a new system. Moreover we show how this analysis can be tuned with respect to a context.

Both approaches can be combined. The analytical solution of the reduced model is the exact projection of the original one. The computation of the reduced model is performed at the level of rules, without the need of executing the original model.

#### Translating BNGL models into Kappa our experience

Participant : Kim Quyen Ly [correspondant] .

So as to test the Kappa development tools on more examples, we translated the models provided with the BNGL distribution, into Kappa. In [20], 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.

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

Participants : Ferdinanda Camporesi, Jerome Feret [correspondant] .

Site-graph rewriting languages as Kappa or BNGL supply a convenient way to describe models of signaling pathways. Unlike classical reaction networks, they emphasise on the biochemical structure of proteins. In [10], 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.

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

Participants : Ferdinanda Camporesi, Jerome Feret [correspondant] , Kim Quyen Ly.

In [11], we introduce the tool KaDe, that may be used to compile models written in Kappa in ODE. Kappa is a formal language that can be used to model systems of biochemical interactions among proteins. It offers several semantics 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 network. 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 [11], we describe the main current functionalities of KaDE and we give some benchmarks on case studies. A complete tutorial and more complete benchmarks may be found at the following url: http://www.di.ens.fr/~feret/CMSB2017-tool-paper/.