Exact hybrid particle/population simulation of rule-based models of biochemical systems

Network-based approaches to modeling and simulating biochemical reaction systems, such as ordinary differential equations, require enumerating all possible species and reactions that can potentially exist in a system. The applicability of these approaches is limited, however, by the problem of combinatorial complexity, an explosion in the number of species and reactions due to protein-protein interactions prevalent in biochemical systems. Rule-based modeling is an approach that avoids this problem by representing molecules as structured objects and encoding their interactions as pattern-based rules. Rule-based models can, on the one hand, be used to generate fully enumerated reaction networks or, alternatively, simulated directly using particle-based kinetic Monte Carlo methods. The latter "network-free" approach produces exact stochastic trajectories with a computational cost that is independent of network size. However, memory and run time costs increase linearly (at best) with the number of particles, limiting the size of system that can be feasibly simulated. Here, we present a hybrid particle/population simulation method that combines the best attributes of both the network-based and network-free approaches. The method takes as input a rule-based model augmented with a user-specified subset of species to treat as population variables rather than as particles. The model is then transformed by a process of partial network expansion into a form that can be simulated using a population-adapted network-free simulator. We have implemented the transformation method within the open-source rule-based modeling platform BioNetGen and the resulting hybrid model can be simulated using the particle-based simulator NFsim. Benchmark tests show that significant memory savings can be achieved using the new approach and a monetary cost analysis provides a practical measure of its utility.