Computational Translation Framework Identifies Biochemical Reaction Networks with Special Topologies and Their Long-Term Dynamics

Abstract

Long-term behaviors of biochemical systems are described by steady states in deter-ministic models and stationary distributions in stochastic models. Obtaining their analytic solutions can be done for limited cases, such as linear or finite-state systems, as it generally requires solving many coupled equations. Interestingly, analytic solutions can be easily obtained when underlying networks have special topologies, called weak reversibility (WR) and zero deficiency (ZD), and the kinetic law follows a generalized form of mass-action kinetics. However, such desired topological conditions do not hold for the majority of cases. Thus, translating networks to have WR and ZD while preserving the original dynamics was proposed. Yet, this approach is limited because manu-ally obtaining the desired network translation among the large number of candidates is challenging. Here, we prove necessary conditions for having WR and ZD after translation, and based on these conditions, we develop a user-friendly computational package, TOWARDZ, that automatically and efficiently identifies translated networks with WR and ZD. This allows us to quantitatively examine how likely it is to obtain WR and ZD after translation depending on the number of species and reac-tions. Importantly, we also describe how our package can be used to analytically derive steady states of deterministic models and stationary distributions of stochastic models. TOWARDZ provides an effective tool to analyze biochemical systems.