A Scalable Computational Framework for Establishing Long-Term Behavior of Stochastic Reaction Networks

Reaction networks are systems in which the populations of a finite number of species evolve through predefined interactions. Such networks are found as modeling tools in many biological disciplines such as biochemistry, ecology, epidemiology, immunology, systems biology and synthetic biology. It is now well-established that, for small population sizes, stochastic models for biochemical reaction networks are necessary to capture randomness in the interactions. The tools for analyzing such models, however, still lag far behind their deterministic counterparts. In this paper, we bridge this gap by developing a constructive framework for examining the long-term behavior and stability properties of the reaction dynamics in a stochastic setting. In particular, we address the problems of determining ergodicity of the reaction dynamics, which is analogous to having a globally attracting fixed point for deterministic dynamics. We also examine when the statistical moments of the underlying process remain bounded with time and when they converge to their steady state values. The framework we develop relies on a blend of ideas from probability theory, linear algebra and optimization theory. We demonstrate that the stability properties of a wide class of biological networks can be assessed from our sufficient theoretical conditions that can be recast as efficient and scalable linear programs, well-known for their tractability. It is notably shown that the computational complexity is often linear in the number of species. We illustrate the validity, the efficiency and the wide applicability of our results on several reaction networks arising in biochemistry, systems biology, epidemiology and ecology. The biological implications of the results as well as an example of a non-ergodic biological network are also discussed.Author Summary: In many biological disciplines, computational modeling of interaction networks is the key for understanding biological phenomena. Such networks are traditionally studied using deterministic models. However, it has been recently recognized that when the populations are small in size, the inherent random effects become significant and to incorporate them, a stochastic modeling paradigm is necessary. Hence, stochastic models of reaction networks have been broadly adopted and extensively used. Such models, for instance, form a cornerstone for studying heterogeneity in clonal cell populations. In biological applications, one is often interested in knowing the long-term behavior and stability properties of reaction networks even with incomplete knowledge of the model parameters. However for stochastic models, no analytical tools are known for this purpose, forcing many researchers to use a simulation-based approach, which is highly unsatisfactory. To address this issue, we develop a theoretical and computational framework for determining the long-term behavior and stability properties for stochastic reaction networks. Our approach is based on a mixture of ideas from probability theory, linear algebra and optimization theory. We illustrate the broad applicability of our results by considering examples from various biological areas. The biological implications of our results are discussed as well.