Syntactic Markovian Bisimulation for Chemical Reaction Networks

Abstract

In chemical reaction networks (CRNs) with stochastic semantics based on continuous-time Markov chains (CTMCs), the typically large populations of species cause combinatorially large state spaces. This makes the analysis very difficult in practice and represents the major bottleneck for the applicability of minimization techniques based, for instance, on lumpability. In this paper we present syntactic Markovian bisimulation (SMB), a notion of bisimulation developed in the Larsen-Skou style of probabilistic bisimulation, defined over the structure of a CRN rather than over its underlying CTMC. SMB identifies a lumpable partition of the CTMC state space a priori, in the sense that it is an equivalence relation over species implying that two CTMC states are lumpable when they are invariant with respect to the total population of species within the same equivalence class. We develop an efficient partition-refinement algorithm which computes the largest SMB of a CRN in polynomial time in the number of species and reactions. We also provide an algorithm for obtaining a quotient network from an SMB that induces the lumped CTMC directly, thus avoiding the generation of the state space of the original CRN altogether. In practice, we show that SMB allows significant reductions in a number of models from the literature. Finally, we study SMB with respect to the deterministic semantics of CRNs based on ordinary differential equations (ODEs), where each equation gives the time-course evolution of the concentration of a species. SMB implies forward CRN bisimulation, a recently developed behavioral notion of equivalence for the ODE semantics, in an analogous sense: it yields a smaller ODE system that keeps track of the sums of the solutions for equivalent species.

Peregrine: A Tool for the Analysis of Population Protocols

“…As mentioned in the introduction, population protocols are isomorphic to chemical reaction networks (CRNs), a popular model in natural computing. Cardelli et al have recently developed model checking techniques and analysis algorithms for stochastic CRNs [7][8][9]. The problems studied therein are incomparable to the parameterized questions addressed by Peregrine.…”

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

“…KaDE and BioNetGen generate exactly the same reduced networks. Lastly, we apply the fast version of Erode of the bisimulation inference algorithm [29] on the original networks and the complete version on the reduced ones [28]. But we found not further reduction this way.…”

“…In particular, it offers some primitives to discover the best forward bisimulation (resp. the best uniform backward bisimulation) induced by an equivalence relation over the bio-molecular species of a reaction network [28,29]. Erode can capture more forward bisimulations than KaDE since equivalent sites can induce only a particular kind of equivalence relations over species.…”

Lumping the Approximate Master Equation for Multistate Processes on Complex Networks

“…Lumping is a popular model reduction technique for Markov-chains and systems of ODEs [21,6,28,7,8]. It has also been applied to the underlying model of epidemic contact processes [27,19] and has recently been shown to be extremely effective for the DBMF equation as well as for the PA approach [20].…”