Fig 1.
Unipartite versus bipartite representations.
(A) Unipartite graph of a network comprised of a single 3-cycle; edges are labeled with the corresponding transition matrix elements of unspecified sign (left). The bipartite representation (influence topology) of this network is also displayed (right); here, species-to-reaction node edges correspond to the Jacobian elements (∂v/∂x) and reaction-to-species node edges correspond to the stoichiometries (s). (B) Unipartite versus bipartite depiction of a branching influence of one species on two immediately downstream species. See §2 for further details.
Fig 2.
Species connectivity and 1-cycles.
(A) All possible signed directed bipartite connections between two species. (B) All possible 1-cycle networks.
Fig 3.
(A) For a network composed of two overlapping cycles, the possible cycle compaction terms (q0 and q1) are listed. (B) Upon slight modification of A, a network composed of four unique overlapping cycles is obtained, with now three possible compaction terms (q0, q1, and q2). The collection of edges that contribute to the non-overlapping and overlapping parts of the influence topology cycles for each network are shown as Venn diagrams at the bottom of each panel. See §3 for further details.
Fig 4.
Non-redundancy of stoichiometric reduction and cycle compaction.
The displayed network (top) provides a concrete example of the non-redundancy of stoichiometric reduction and cycle compaction. At the top, all Jacobian parameters (r0 = 1, r1, r2, r3) and stoichiometric parameters (s1, s2, s3) of the network are labeled. Stoichiometric reduction of reaction v1 allows the setting of either s1 (bottom left) or s2 (bottom right) to unity. The choice of s1 = 1 followed by cycle compaction leads to three final parameters (3D stability phase space), whereas the better choice of s2 = 1 leads to only two final parameters (2D stability phase space).
Fig 5.
(A) All possible 2-cycle network influence topologies. (B) Rotation network influence topology.
Fig 6.
(A) Influence topology. Cycle compaction allows definition of q0 = σ1 σ2 r3. Temporal scaling of all Jacobian edges to |q0| leaves only ρ1 = r1/|q0| and ρ2 = r2/|q0|. (B) Stability phase space. Axes correspond to the two parameters, ρ1 and ρ2, that remain after parameter reduction. For plotting ρ1 and ρ2, the variable transformation has been used to allow visualization of the entire range of the ρi from 0 to ∞ (this arctan transform also conveniently permits visualization of the range −∞ < ρi < 0, which would correspond to a different sign for this Jacobian element and therefore a different influence topology). Flows in the plot map the zones over which Δ1 (black), Δ2 (red), and Δ3 (blue) are negative. Only Δ2 (red) and Δ3 (blue) can go negative (in this case, simultaneously). The green background indicates that ρ1 and ρ2 can independently assume any positive definite values based on their definitions in terms of the parameters used to define the original Jenkin-Maxwell equations (Equations 54–56).
Fig 7.
(A) Influence topology. Cycle compaction allows definition of q0 = σ1 r3. Temporal scaling of all Jacobian edges to |q0| leaves only ρ1 = r1/|q0| and ρ2 = r2/|q0|. (B) Stability phase space. Flows in the plot map the zones over which Δ1 (black) and Δ2 (red) are negative. The unstable zones for Δ1 (black) and Δ2 (red) completely overlap; in this region, two unstable eigenvalues obtain according to the Routh array. The green line indicates the possible set of solutions obtainable for the original van der Pol equations (Equations 72 and 73). See Fig. 6 for further details.
Fig 8.
(A) Influence topology. Cycle compaction allows definition of q0 = σ1 r4. Temporal scaling of all Jacobian edges to |q0| leaves only ρ1 = r1/|q0|, ρ2 = r2/|q0|, and ρ3 = r3/|q0|. The stability phase space is shown for (B) ρ3 = 1 (σ1 = 1), (C) ρ3 = 1/2 (σ1 = 2), and (D) ρ3 = 2 (σ1 = 1/2). Flows in the plot map the zones over which Δ1 (black) and Δ2 (red) are negative. See Fig. 6 for further details.
Fig 9.
(A) Influence topology. Temporal scaling of all Jacobian edges to |r0| gives ρ1 = r1/|r0| and ρ2 = r2/|r0|; σ1 and σ2 must be specified as well. The stability phase space is shown for σ1 = 1 and the following values for σ2: (B) σ2 = 1/2, (C) σ2 = 1, and (D) σ2 = 2. Flows in the plot map the zones over which Δ1 (black) and Δ2 (red) are negative. See Fig. 6 for further details.
Fig 10.
(A) Influence topology. Temporal scaling of all Jacobian edges to |r0| gives the parameters ρ1 = r1/|r0|, ρ2 = r2/|r0|, and ρ3 = r3/|r0|. The positive stoichiometric terms σ1 and σ2 must also be specified independently. The stability phase space is shown for σ1 = σ2 = 1 and (B) ρ3 = 1/10; (C) ρ3 = 1/4; and (D) ρ3 = 1. Flows in the plot map the zones over which Δ1 (black) and Δ2 (red) are negative. See Fig. 6 for further details.
Fig 11.
(A) Influence topology. Cycle compaction leads to definition of q0 = r4r5r6, which is further removed after temporal scaling, leaving only ρ1 = r1/|q0|1/3, ρ2 = r2/|q0|1/3, and ρ3 = r3/|q0|1/3. (B) Stability phase space for ρ3 = 1. Flows in the plot map the zones over which Δ1 (black), Δ2 (red), and Δ3 (blue) are negative. For ρ3 < 1, the domain of instability will be increased towards the upper right (and oppositely for larger ρ3). No green zone is indicated here due to the dependence of the exact steady state solution(s) on the choice of Hill coefficients in the definition of the original network in Equations 139–141. See Fig. 6 for further details.
Fig 12.
(A) Network consisting of two distinct levels of overlapping cycles. (B) Species trajectories (χ1, cyan; χ2, magenta; χ3, black) of an explicit algebraic version (Equations 158–160) of a network having the influence topology shown in A (see §6 for details). (C) Similar two-level network as in A, but with the levels swapped. (D) Species trajectories (χ1, cyan; χ2, magenta; χ3, black) of an explicit algebraic version (Equations 161–163) of a network having the influence topology shown in C (see §6 for details). (E) Example of a more complicated multilevel network.