Figure 3. Role of biphasic interactions in network and motif response.

The results presented in this plot show the consequence of biphasic signal regulation and biphasic response within interactions in enabling network response. In each plot the inset image represents the basal behavior in the absence of biphasic signal regulation or biphasic interaction, for the same kinetic parameters of the system. (A–B). Role of biphasic signal regulation in perturbing behavior of enzymatic systems, (A) Covalent modification cycle and (B) double site modification (DSP) (common enzymes). (A) shows how biphasic signal regulation can enable homeostatic (prolonged flat region) and biphasic response from a covalent modification cycle (note that the response is an increasing function of dose for small signal values). (B) shows how biphasic signal regulation can enable not just biphasic behavior (top right panel) but multiphasic behavior (bottom right panel), and result in a dose response curve with multiple multistable regions (indicated by pairs of limit points) along with a biphasic response (bottom left panel). (C) contrasts this with the response from incoherent feedforward regulation of the DSP (a driver of biphasic responses), which can similarly present multiphasic responses (note that the curve decreases before increasing and decreasing again) (plot to the right shows behavior without incoherent feedforward regulation, where such behavior is absent). (D–F) Effect of biphasic interaction on common network regulatory motifs. Biphasic interaction can alter the fundamental feature of feedforward motifs in multiple ways. (D) In incoherent feedforward networks, it can destroy the expected biphasic response (left panel) and also enable the creation of multiphasic responses (right panel). Similarly, in feedback networks, it can destroy multi-stability in a positive feedback motif (E). On the other hand, it can erode homeostatic responses and enable multistability in negative feedback motifs (the later shown in F). [Colored straight arrows in network schematics represent biphasic interactions. LP: saddle-node bifurcation, solid lines, and dotted lines denote stable and unstable steady state, respectively].

Figure 3—figure supplement 1. Effect of biphasic response in interactions within feedback network motifs (open systems): The figure shows the introduction of novel behavior (bistability in NFB, see (B)), and the removal of expected behavior (bistability in PFB, see (A)) even when the biphasic responses in interactions are present within open system feedback network motifs.

This complements our analysis in the paper where similar results were shown with motifs constructed with closed system (see Figure 3—figure supplement 1). The inset plots show basal behavior without the biphasic interaction. (B) shows expanded region in the aside main plot. [Brown arrows in the network schematics represent biphasic interactions. LP: saddle node bifurcation. Solid lines and dotted lines denote steady and unstable steady states, respectively.].

Figure 3—figure supplement 2. Effect of biphasic response in interactions within integral feedback control motif: perturbation of expected homeostatic response.

The simple integral feedback control motif is capable of exact homeostasis in the substrate form M, as signal (S) and substrate forms P and A change (see plot A, M not shown). The lower regions of signal activation are unstable for the system and oscillate. There exists a threshold of signal above which the single steady state is stable and there is steady exact homeostasis in M. In plots B and C, we show how the expected response from the motif can be affected by (1) biphasic signal regulation (B), (2) biphasic response in an interaction within the motif (C). (B) shows how upstream biphasic signal regulation can diminish the range over which stable steady state homeostatic response is seen (in the concentration of M - not shown), by making the system unstable through a hopf bifurcation leading to oscillations, for both low signal and high signal ranges. Depending on the nature and strength of the biphasic signal regulation (manipulated here by changing parameter values associated with the signal regulation), the stable steady state regime in signal can diminish and for a sufficiently strong biphasic signal regulation, the system can be completely unstable, completely undermining the exact homeostasis expected in M from the motif. (C) shows how biphasic responses in an interaction within the motif (indicated in the adjacent schematic with a brown arrow) can further arrest the expected biphasic response by ‘capping’ the signal range over which the motif can function (due to the saturating effect from the biphasic response - see text). Furthermore, the steady state of the system can lose stability even for lower signal ranges (on the existing stable branch), due to oscillations. Thus, biphasic interactions within even a robust homeostatic motif such as integral feedback control can fundamentally subvert expected motif response [Brown arrows in the network schematics represent biphasic interactions. LP: saddle node bifurcation. H: Hopf point bifurcation. Solid lines and dotted lines denote steady and unstable steady states, respectively. Regions shaded in blue represent oscillations.].