Roles of network topology in the relaxation dynamics of simple chemical reaction network models

Understanding the relationship between the structure of chemical reaction networks and their reaction dynamics is essential for unveiling the design principles of living organisms. However, while some network-structural features are known to relate to the steady-state characteristics of chemical reaction networks, mathematical frameworks describing the links between out-of-steady-state dynamics and network structure are still underdeveloped. Here, we characterize the out-of-steady-state behavior of a class of artificial chemical reaction networks consisting of the ligation and splitting reactions of polymers. Within this class, we examine minimal networks that can convert a given set of sources (e.g., nutrients) to a specified set of targets (e.g., biomass precursors). By exploring the dynamics of the models with a simple setup, we find three distinct types of relaxation dynamics after perturbation from a steady-state: exponential-, power-law-, and plateau-dominated. We computationally show that we can predict this out-of-steady-state dynamical behavior from just three features computed from the network's stoichiometric matrix, namely, (1) the rank gap, determining the existence of a steady-state; (2) the left null-space, being related to conserved quantities in the dynamics; and (3) the stoichiometric cone, dictating the range of achievable chemical concentrations. We further demonstrate that these three quantities relates to the type of relaxation dynamics of combinations of our minimal networks, larger networks with many redundant pathways, and a real example of a metabolic network. The relationship between the topological features of reaction networks and the relaxation dynamics presented here are useful clues for understanding the design of metabolic reaction networks as well as industrially useful chemical production pathways.