Taming the complexity of biochemical models through bisimulation and collapsing: theory and practice

We propose the use of hybrid automata and we discuss the use of the notions of bisimulation and collapsing for a \u201cqualitative\u201d analysis of the temporal evolution of biological systems. Many biological systems can be modeled using systems of ordinary differential algebraic equations (e.g., S-systems), thus allowing the study of their solutions and behavior automatically with suitable software tools (e.g., PLAS, Octave/Matlabtm). Usually, numerical solutions (traces or trajectories) for appropriate initial conditions are analyzed in order to infer significant properties of the biological systems under study. When several variables are involved and the traces span over a long interval of time, the analysis phase necessitates automation in a scalable and efficient manner. Earlier, we have advocated and experimented with the use of automata and temporal logics for this purpose (XS-systems and Simpathica) and here we continue our investigation more deeply. We propose the use of hybrid automata and we discuss the use of the notions of bisimulation and collapsing for a "qualitative" analysis of the temporal evolution of biological systems. As compared with our previous approach, hybrid automata allow maintenance of more information about the differential equations (S-system) than standard automata. The use of the notion of bisimulation in the definition of the projection operation (restrictions to a subset of "interesting" variables) makes it possible to work with reduced automata satisfying the same formulae as the initial ones. Finally, the notion of collapsing is introduced to move toward still simpler and equivalent automaton taming the complexity in terms of states whose number depends on the attained level of approximation. \ua9 2004 Elsevier B.V. All rights reserved