A framework for deriving analytic long-term behavior of biochemical reaction networks

The long-term behaviors of biochemical systems are described by their steady states. Deriving these states directly for complex networks arising from real-world applications, however, is often challenging. Recent work has consequently focused on network-based approaches. Specifically, biochemical reaction networks are transformed into weakly reversible and deficiency zero networks, which allows the derivation of their analytic steady states. Identifying this transformation, however, can be challenging for large and complex networks. In this paper, we address this difficulty by breaking the complex network into smaller independent subnetworks and then transforming the subnetworks to derive the analytic steady states of each subnetwork. We show that stitching these solutions together leads to the the analytic steady states of the original network. To facilitate this process, we develop a user-friendly and publicly available package, COMPILES (COMPutIng anaLytic stEady States). With COMPILES, we can easily test the presence of bistability of a CRISPRi toggle switch model, which was previously investigated via tremendous number of numerical simulations and within a limited range of parameters. Furthermore, COMPILES can be used to identify absolute concentration robustness (ACR), the property of a system that maintains the concentration of particular species at a steady state regardless of any initial concentrations. Specifically, our approach completely identifies all the species with and without ACR in a complex insulin model. Our method provides an effective approach to analyzing and understanding complex biochemical systems. Author summary Steady states describe the long-term behaviors of biochemical systems, which are typically based on ordinary differential equations. To derive a steady state analytically, significant attention has been given in recent years to network-based approaches. While this approach allows a steady state to be derived as long as a network has a special structure, complex and large networks rarely have this structural property. We address this difficulty by breaking the network into smaller and more manageable independent subnetworks, and then use the network-based approach to derive the analytic steady state of each subnetwork. Stitching these solutions together allows us to derive the analytic steady state of the original network. To facilitate this process, we develop a user-friendly and publicly available package, COMPILES. COMPILES identifies critical biochemical properties such as the presence of bistability in a genetic toggle switch model and absolute concentration robustness in a complex insulin signaling pathway model.