Fig 1.
The three levels (upper labels) in a Stochastic Stoichiometric Population Process realized as a rule-based system, with chemical reactions as an example.
Rules correspond to reaction mechanisms, in which a context K comprising the active atoms can support two bond configurations that we generically term patterns: a reactant pattern L is converted to a product pattern R by the reaction. The reactant pattern is embedded as a sub-graph in one or more literal molecules G by a map m, and the conversion maps l and r on patterns are used to generate embeddings d and m′, and conversions l′ and r′ to new literal molecules H, so that the remainders of the literal molecules outside the reacting bonds are “carried along” by the mechanism in a structure-preserving way. The second level formed by the action of rules on molecules is, in our treatment, the generator of a stochastic process in the form of a chemical reaction network (CRN) connecting literal molecule types by literal reaction types. The third level is a state space in which collections of the molecules evolve stochastically under the generator. Middle labels give the mathematical structures that express each level. For rules, they are morphisms from category theory; the commutative diagram is known as a double pushout. For reaction networks, the representation is called a multi-hypergraph (because the inputs and outputs may have multiple copies of the same molecule type, and so are termed “multisets”). For state spaces in a population process, the states are points in a lattice where the coordinates count molecule copy-numbers. Each level is connected to the next by a (generally) one-to-many generative relation: mechanisms generate CRNs (both molecules and reactions) through network expansion, in which the same rule may be instantiated in many different reactions. Sets of transitions from the CRN as a generator are embedded in the state space as paths of population states; the same reaction sequence may have indefinitely or infinitely many images through states with different numbers of molecules. The bottom labels give result-types at each level. For rules, they consist of the algebra of dependencies for activation of a rule on patterns created or eliminated by other rules. For reaction networks, they may be integer-flow solutions to a conversion problem. For state spaces, they are stochastically evolving population states, or distributions over states and their transitions that may evolve deterministically under a master equation. Left and middle panels are reproduced from [11], Fig 6.6 and Fig 12.3 respectively, and other terms used here are explained at length in that dissertation.
Table 1.
The five rules that generate biological sugar-phosphate conversions, from [34].
The rule name and a brief annotation for its action are shown.
Fig 2.
The diagram layout to be used.
Filled circles are species in the hypergraph notation following Feinberg [18]. Aldose (red) and ketose (blue) monophosphates, and bisphosphates (green), are arranged in the series shown, and the carbon number of the sugar is indicated in the outer ring. All edges generated for these compounds by network expansion from the rules in Table 1 are shown in doubly-bipartite graph form. Open circles are complexes, heavy lines are reactions, and light lines indicate stoichiometry.
Fig 3.
Left panel: the sugar re-arrangement part of the Calvin-Benson cycle as an integer-flow solution v with net currents from the environment to the network. Right panel: one version of the canonical Pentose-Phosphate Pathway as an integer flow solution with the equal and opposite Jext.
Fig 4.
763 solutions fi to the throughput problem, showing the number of reactions in the supporting graph fi (red) and the topological contribution to dissipation (blue and symbols) from Eq (16) evaluated for fi.
19 integer flows (light magenta dots) produce a graph fi that appears again as the supporting graph of a second solution (dark magenta dots).
Fig 5.
The 234 TKL trefoil shown in two representations.
Right-hand side: as a projection from the network from Fig 2. Left-hand side: with the 3-fold symmetry that conserves carbon number and aldose-plus-ketose number exhibited, as well as the Möbius-band topology that makes this a non-degenerate transport cycle.
Table 2.
Rule symbols paired with the conversions the rules make on 1-complexes or 2-complexes of sugar-monophosphates.
Legend: AlKe (aldose-ketose); KeAl (ketose-aldose); TAL (transaldolase); TKL (transketolase); AL (aldolase); PHL (phosphohydrolase). In the fifth line, TAL ∘ TKL indicates function composition: the output of a TKL reaction is input to the TAL reaction. In the sixth line, AlKe ⊕ KeAl indicates parallel application, AlKe on input An and KeAl on input Kn+1.
Fig 6.
An AlKe null cycle shown two ways.
Right-hand side: as a projection from the network from Fig 2. Left-hand side: showing how the TKL ∘ TAL composite reaction balances the direct sum (⊕) of two otherwise-independently feasible AlKe and KeAl reactions.
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.
Fig 8.
Top panel shows the three reactions in f14 that are not part of the prelude creating F6P from GAP + DHAP and Ru5P from GLP and DHAP.
Middle panel expands the set of species and reactions to correspond to the supporting graph in Fig 7. A TKL trefoil with aldose backbone lengths 2, 3, 4 covers the faces on the left-hand side of the cube, shown with green circulation arrows. A second TKL trefoil with aldose backbone lengths 2, 3, 5 covers the faces on the right-hand side of the cube, shown with red circulation arrows. The two trefoils are non-stoichiometrically coupled through any potential drop that arises on e96. Arrows are shown in the directions that make trefoils null. The sense of edges in the MØD listing, relative to the drawn arrows, is indicated with ± signs. All currents and potentials will be similarly signed relative to drawn arrow directions. Bottom panel shows the AlKe null cycle of Fig 6 overlaid on the 235 TKL trefoil, with which it shares edge 97. The pair of dark-gold circulations shows the stoichiometrically coupled reaction directions.
Fig 9.
Lattice diagrams for all edges (plus the composite PHL ∘ AL) in null flows.
Left panel: AlKe edges (cyan). Center panel: TAL edges (blues) and PHL ∘ AL (orange). Right panel, TKL edges (greens). To aid visibility for edges that overlap, TAL and TKL edges are grouped into “tiers” anchored at fixed points (circles), with darkness distinguishing between tiers. TAL plus PHL ∘ AL and TKL both have 10 distinct edges.
Table 3.
Three “reactions” that interconvert 1-complexes and 2-complexes through the input or output of species in the net conversion schema (26), and to simplify later plots, the species DHAP which is always produced from GAP by AlKe at flux 2.
Fig 10.
Lattice diagrams for a sequence of complexes that carry one autocatalytic An backbone through the sequence of its conversions in one full turn of the CBB cycle.
To simplify the diagrams, the inputs are taken to be 3A3 + 2K3 (corresponding to 3 GAP + 2 DHAP). First panel, starting with A2, is f14 from Fig 8 of S1 File C 2. Second panel, starting with A3 is f13 from the left panel of Fig 3. Third panel, starting with A4 and representative of An for any n ≥ 4, is f194. The links in each path are to be read in the order that makes a closed cycle. The starting aldose is a 1-complex An, on the lower axis (black dot). Additions of DHAP are red up arrows, forming the 2-complex input to PHL ∘ AL edges (orange), producing 1-complexes (ketoses) on the vertical axis. Additions of GAP are black right arrows, forming the 2-complex inputs to TKL edges (green). Extractions of Ru5P are blue down arrows, converting 2-complexes back to single aldoses on the horizontal axis, from which new complexes are created by DHAP addition. The sub-flow Ru5P_out ∘ TKL ∘ GAP_in ∘ PHL ∘ AL ∘ DHAP_in, shown in the fourth panel, forms one turn of an “algorithmic” loop incrementing the value n for the starting aldose 1-complex An. AlKe conversions (cyan), followed by Ru5P_out ∘ TKL ∘ GAP_in in each of the first three panels, reset two iterations of this algorithmic loop to its starting complex. Also shown in the fourth panel is a 3-cycle of A3 → K3 which, added to the previous graphs, would permit a starting configuration of 5A3 + 0K3, at the cost of slightly greater complexity in plots.
Fig 11.
Left-hand panel: an equilibrium concentration profile from both non-stereochemical and stereochemical formation free energies, with hydrolyzing potential set to ensure
(dark green circles), and a geometric decay per carbon determined by [GAP] to make concentrations within a group roughly independent of chain length. Right-hand panel: the values
from Eq (15) at the thermodynamic landscape in the upper panel, showing the qualitative similarity to
retained but also the sensitivity to stereochemical corrections.
Table 4.
Flow properties of the Pythagorean theorem illustrated in Fig 12.
Edges in the graph of Fig 2 are listed in the first column, and fluxes v at through current are given in columns for each flow solution.
in the final row is multiplied by [GAP]2 in the thermochemical background where the force-flux relation is solved. The trefoil current v° (the only null flow in the union graph) equals the flux in edge e23. At the minimum-dissipation solution
. The relation (50) is fulfilled with the dissipation in the final row
.
Fig 12.
The Pythagorean theorem (50) in the linear-response regime where it is the ordinary Euclidean theorem.
Two irreducible flows f14 and f193 differ by a TKL trefoil current v° with magnitude 1. In the hierarchy ,
is the full graph from Fig 2, and
is the union of the supporting graphs for f14 and f193.
may be the supporting graph for either f14 or f193. Vertical solid arrow is square root of the integral
in Eq (50) from zero to
in
. Hypotenuse solid arrows are square roots of integrals
in Eq (50) in either graph
. Horizontal dashed arrows are square roots of integrals ∫dη′ in either direction of v° in Eq (50). Details of the final flow parameters and the functional dependence of
on v° are given in Table 4.
Fig 13.
The chemical potentials in the solution on the union of supports for f193 and f184, across all edges of the AlKe null cycle.
Current through the null cycle is shown in black; potential drops in green. The potential drop across the AlKe edges is ∼ [GAP] ∼ 10−3 smaller because they are first-order in organics, whereas the TAL and TKL edges are second-order. Therefore almost-all chemical work delivered by the current v23 = 1 (dark red circle) to the boundary of e97, which is saved from dissipation in e97 by the flow around the AlKe null cycle, is transduced stoichiometrically to the boundary of e15, where it is dissipated.
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.
Table 5.
Currents v and potential drops Δμ on the edges in TKL trefoils for the masks of f14, f535, and f535 ∪ f193. f535 adds one (234) TKL trefoil to f14 by adding edge e98, and f193 adds a second (235) TKL trefoil by adding edges e23 and e97.
The final columns, f535 ∪ f193, provide the labeling for the transduction in Fig 14 in either potential. (eQNS) labels the thermochemical background for group contribution from non-stereochemical SMILES, and (eQS) labels the background with stereochemistry, approximating KEGG data values. The relation v22 − v98 = 1 = v6 in all cases is the topologically constrained AlKe velocity E4P ⇀ Eu4P. Chemical-potential cycles must cancel around trefoils; thus Δμ23 + Δμ97 − Δμ96 = 0 and Δμ22 + Δμ98 − Δμ96 = 0.
Fig 15.
The stationarity condition (62), written in the dual coordinates of Eq (65).
Ellipse is a surface of constant excess dissipation from Eq (66) below. This is the minimum of
on the surface 1Tα = 1 of fixed flux Jext.
and
are the eigenvectors of g∘, normalized here to the 1Tα = 1 surface. αi is the coordinate vector of some other integer flow solution fi, restricted to support on a subgraph of the graph defining g°.
Fig 16.
The integer flow solution (blue) for the canonical Calvin cycle (irreducible f13) from the left panel of Fig 3, expanded in a basis of the eigenvectors of g°.
The 28 reactions are listed in the order of Fig 11 in S1 File E 1. The velocity of each reaction is plotted as the ordinate. Red shows the contribution from eigenvectors 1 and 2; green shows the contribution from eigenvectors 1, 2, 3, and 10, respectively the two-largest, and four-largest coefficients in absolute magnitude in the eigenbasis expansion.