Fig 1.
(a-c) Input-output curves for hypothetical systems. Input is thought to be a parameter of the system that is varied, and the output is the concentration of a species at steady state. Dashed lines correspond to unstable steady states, and solid lines to stable steady states. (a) displays a simple hysteresis switch; (b-c) show input-output curves for systems where bistability arises in two disjoint intervals of input. (d-e) For a system with two parameters, the system has more than one positive steady state in the orange regions, and one in the purple regions. In panel (d) the multistationarity region is connected, while in (e) it has two connected components.
Fig 2.
Reaction networks arising in cell signaling.
The subindex ‘p’ indicates a phosphorylated site. When writing ‘p’ and ‘pp’ it is assumed the substrate has two phosphorylation sites, and phosphorylation/dephosphorylation is ordered. When writing ‘0p’ for example, it means the substrate also has two sites numbered 1 and 2, and the second one is phosphorylated. All networks are known to be multistationary. For all networks but (c) and (h), the multistationarity region is path connected. For network (c), the multistationarity region has two path-connected components, while for network (h) our approach is inconclusive.
Fig 3.
Illustration of the relevant objects for the running example given in (1).
Fig 4.
The flux cone of the running example in .
The extreme vectors E1 = (1, 0, 1), E2 = (1, 1, 0) are shown in blue.
Fig 5.
(a) A strict separating hyperplane (in purple) of the support of . Red dots correspond to positive exponents, the blue square corresponds to the only negative exponent. (b) The preimage of the negative real half-line under f.
Table 1.
Summary of the algorithm on selected systems.
Fig 6.
Input-output curves (bifurcation diagrams) for the reduced reciprocal allosteric regulation system in Fig 2(c) (that is, with κ2 = κ5 = 0 for simplicity).
The following parameters are fixed: κ1 = 1, κ3 = 1, κ7 = 1, κ8 = 1, κ9 = 1, κ10 = 0.1, Ltot = 72, Ktot = 62, Stot = 426. In (a)-(d), the bifurcation parameter t, which can be negative, describes a path for . Subfigures (a)-(c) show the bifurcation diagrams for the concentration of Sp, K and L at steady state. In the intervals with three steady states, the one in the middle is unstable. Subfigure (d) shows the path in the three-dimensional space (K2, Ptot, κ6). The blue regions indicate the region of the path that belongs to the multistationarity region, and the displayed plane κ6 = 1 separates the two regions. Subfigures (e)-(f) show the multistationarity regions when we fix κ6 = 9.5 and κ6 = 0.195 respectively. These are two slices of the two path-connected components, obtained by keeping only two free parameters.