A Structural Theory of Admissible Transitions in Biological Reaction Networks

Biological reaction networks often exhibit complex transient behavior that cannot be explained solely by the analysis of steady states or long-term persistence. Existing structural approaches identify persistent system properties and have considered transitions between organizations, but do not provide a general criterion for admissible transitions between arbitrary species configurations. We introduce admissible transitions, a new structural concept that describes feasible changes between species subsets using only reaction network structure and feasible reaction fluxes, independently of kinetic parameters. We prove that solutions of reaction-based ordinary differential equation systems induce canonical sequences of admissible transitions with a fundamental asymmetry: closure-building transitions are uniquely determined by network structure, whereas downward transitions are generally non-unique and depend on the realized trajectory. This establishes a structural layer linking network topology to transient system evolution. The framework is illustrated using a classical HIV immune-response model. By extending structural reaction network analysis from persistent states to transient dynamics, the proposed theory provides a general, kinetics-independent framework for analyzing reachability, organization formation, and transient behavior in biological systems.