For a reaction network N with species set S , a log-parametrized (LP) set is a non-empty set of the form E(P, x * ) = {x ∈ R S > | log x − log x * ∈ P ⊥ } where P (called the LP set's flux subspace) is a subspace of R S , x * (called the LP set's reference point) is a given element of R S > , and P ⊥ (called the LP set's parameter subspace) is the orthogonal complement of P . A network N with kinetics K is a positive equilibria LP (PLP) system if its set of positive equilibria is an LP set, i.e., E+(N , K) = E(PE, x * ) where PE is the flux

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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. Methods have been developed for transforming biochemical reaction networks into weakly reversible and deficiency zero networks, which allows the analytic steady states to be derived easily. Identifying this transformation, however, can be challenging for large and complex networks. In this paper, we address this difficulty by breaking the network into smaller independent subnetworks and then deriving the analytic steady states of each subnetwork. We show that the analytic steady states of the original network can then be derived by stitching these solutions together. To facilitate this process, we develop a user-friendly and publicly-available package, COMPILES. With our approach, we can easily test the presence of bistability of a CRISPRi toggle switch model, which was previously investigated via tremendous amount of numerical simulations. Furthermore, our approach can be used to identify absolute concentration robustness (ACR), the property of a system to maintain the concentration of particular species at 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.

The long-term behaviors of biochemical systems are often 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 generalized 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.

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