Fig. 1
From: Roles of network topology in the relaxation dynamics of simple chemical reaction network models
(A) The fully-connected network for \(M=2\) and \(L=6\). (B) Two examples of the minimum network which synthesizes a single target from two sources. The sources and target chemicals are indicated by the red arrows and the blue arrow, respectively. In (A and B), the bipartite representation of the chemical reaction network is adopted. For example, the reaction \(S_1+S_2\rightarrow P\) is depicted as follows (see inset of the panel B); first, the chemical nodes representing \(S_1\) and \(S_2\) (the gray disks) are wired to the reaction node (the pink diamond), and then, an edge connects the reaction node and the chemical node of the chemical P (the gray disk). Directions of the reactions are chosen to be consistent with the steady-state flux distribution. (C–F) Four typical relaxation dynamics emerged from the minimum networks: Exponential, Metastable-Plateau, Confined-Plateau, and Power-law relaxation. The vertical and horizontal axes are on a logarithmic scale. The four dynamics are classified into three category based on the networks types that exhibit the corresponding dynamics. (G) Example distributions of the migration length (see text for definition). The top panel is the distribution over all initial conditions for one network showing metastable-plateau type relaxation, and the bottom panel is from a network showing both the plateau and power-law type relaxations. (H) Classification of the minimum networks based on the average relaxation time (see text for definition) and the average migration length.
