Structural determinants of relaxation dynamics in chemical reaction networks

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 inputs (e.g., nutrients) to a specified set of targets (e.g., biomass precursors). 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 networks stoichiometric matrix, namely, (i) the rank gap, determining the existence of a steady-state; (ii) the left null-space, being related to conserved quantities in the dynamics; and (iii) the stoichiometric cone, dictating the range of achievable chemical concentrations. We further demonstrate that these three quantities also predict 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 unified method to predict the qualitative features of the relaxation dynamics presented here can provide a basis for understanding the design of metabolic reaction networks as well as industrially useful chemical production pathways. Author summaryThe relationship between network structure and chemical reaction dynamics is of central interest in chemical reaction network theory, as it underlies chemical manufacturing, cellular metabolism, and bioengineering. The links between structure and steady-state properties have been extensively investigated. However, how far the network structure determines the out-of-steady-state, transient dynamics of chemical reactions is unexplored. Here we construct a chemical reaction network model that is simple but generates a wide variety of network instances. By computationally exploring the networks structural- and dynamical features, we found that three network-structural features are sufficient to predict the qualitative characteristics of the relaxation dynamics after the chemical concentrations are perturbed from their steady-state. Depending on the values of those three features, the chemical reaction dynamics on the network exhibit exponential, plateau, and power-law relaxation. Also, we found that such features are determinants of the dynamics of biological metabolic reaction systems. Our findings provide a foundation for the structure-based prediction of chemical reaction dynamics.