Rules, hypergraphs, and probabilities: The three-level analysis of chemical reaction systems and other stochastic stoichiometric population processes
Fig 7
A network formed from the union of the supporting graphs for three irreducible flows f14, f18 and and f193.
A subgraph with 8 edges hosts f14 uniquely. It is the first lattice diagram in Fig 9 below. The union of that graph with e98 (the lattice edge from (A2, K6) to (A4, K4,)) hosts f535 and f629 with one additional TKL trefoil of backbones 234. The further union with e23 (the lattice edge from (A3, K7) to (A5, K5)) and e97 (the lattice edge from (A2, K7) to (A5, K4)) adds a second TKL trefoil with backbones 235 and common edge e96 with the first trefoil. Two firings of the TIM reaction and single firings of each AL/PHL sequence are fixed by the topology for the conversion (26), so the prelude on this graph is independent of the background. The 5 TKL edges and the remaining C4 AlKe edge constitute the fugue. Like the preludes, the AlKe reaction is topologically constrained to fire one time. The only two degrees of freedom responsive to the kinetics are the circulations in the two trefoils, illustrated below in Fig 14.