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Rules, hypergraphs, and probabilities: The three-level analysis of chemical reaction systems and other stochastic stoichiometric population processes
Fig 14
Two linked TKL-trefoils that appear in the union of the supporting graphs for two flows (f193 and f535).
The Möbius boundary of each trefoil is formed by tracing a continuous path of solid (stoichiometric) links as they pass through species and complexes. A set of chemical-potential drops (green lettering) and currents (black lettering) for a 
-minimizing solution are shown for each reaction. Top panel shows two basis elements for null flows on the complete supporting graph: the 234 trefoil (green shades) on the left and the 235 trefoil (red shades) on the right. The period-2 backbone cycles shown in Fig 7 of S1 File B 2 appear as simple circulations in their respective faces of the cube. Bottom panel shows “environment” reactions K6 + A3 ⇀ A4 + K5 and A5 + K4 ⇀ K7 + A2 removed as explicit sources of dissipation. A red cycle with current j2 = 0.3384 fully accounts for the current supplied to the complexes bounding the “output” conversion A5 + K4 ⇀ K7 + A2. A green cycle with current j1 = 0.1447 flows through the subgraph, alongside a supply of the boundary complexes K6 + A3 ⇀ A4 + K5 at rate 0.8553 across the potential 0.4755 from the complement to this subgraph in the complete graph (making the total current flow between these two complexes the topologically-constrained value of 1). Other sources (TAL edge and preludes) also couple to the boundary complexes, and serve to maintain their chemical potentials.
doi: https://doi.org/10.1371/journal.pcsy.0000022.g014