Rigid Geometric Constraints for Kappa Models

International audienceRule-based modeling languages such as Kappa [11,7] and BNGL [3,2] allow for a concise description of combinatorially complex biochemical processes as well as efficient simulations of the resulting models [8,7]. A key aspect of the rule-based modeling approach is to partially expose the structure of the chemical species involved. However, the above-mentioned languages do not provide means to directly express the three-dimensional geometry of chemical species. As a consequence models typically capture only the network-topological structure of the species involved. For certain biochemical processes, such as the assembly of molecular complexes, in which steric constraints play a key role, it would seem natural to also model the geometric structure of species. We propose an extension to the Kappa modeling language allowing the annotation of the structure of chemical species with three-dimensional geometric information. This naturally introduces rigidity constraints on the species and reduces the state space of the resulting model by excluding species that are not geometrically sound. We show that models extended in this way can still be simulated efficiently, albeit at the cost of a greater number of null-events occurring during the simulation. The geometric constraints introduced by the extension are inherently non-local in that they may entangle the position and orientation of sub-structures at arbitrary distances in large species such as polymers. We give a formal definition of the notion of locality based on the intuition that local modifications should only affect sub-structures within a finite radius around the point where the modification occurred. We show that there are indeed geometrically enhanced Kappa models that are non-local, and conversely, that every local model can be simulated accurately using a finite classical Kappa model at the expense of a possible combinatorial explosion of its rule set. We also give some sufficient conditions for the locality of a model but show that locality is undecidable in general