Figure 2. Presence and absence of substrate and enzyme biphasic dose responses in the commonly observed building blocks of cellular signaling systems.

An examination of a suite of substrate modification systems and different doses allows us to clearly ascertain the origin and necessary features of such systems to present enzyme and substrate biphasic dose responses. Where results denote a biphasic response is absent (indicated by a cross in a box), this is absent irrespective of kinetic parameter values and total amounts of substrate(s) and enzyme(s). This is established through analytical work (see Supplementary file 1). When a biphasic dose-response is present, it is shown in a bifurcation diagram where the relevant dose is the bifurcation parameter. The presence of specific dose-responses can be characterized in parameter space in the following way, either the behavior is (1) present for all intrinsic kinetic parameter values (accessible for some total amounts of substrate and enzyme, transparent boxes with a tick), or (2) present only for specific intrinsic kinetic parameter values (accessible for some total amounts of substrate and enzymes, blue-shaded boxes with a tick) - see text and analytical work for more details. (A) Covalent modification system. Absence of substrate and enzyme biphasic response. (B) Protein-Protein Interaction Model. Enzyme biphasic dose responses are seen in the protein-protein interaction model, but substrate biphasic responses are absent. In contrast to the covalent modification cycle which is incapable of biphasic responses, this result indicates how a single additional complex formation by the enzyme (and resulting sequestration) can generate enzyme biphasic dose responses. (C) Double site modification (DSP): common enzymes. Presence of enzyme and substrate biphasic response (enzyme biphasic responses only for certain ranges of intrinsic kinetic parameter values - see text). (D) DSP: common kinase and separate phosphatase. Presence and absence of substrate and enzyme biphasic response, respectively. (E) DSP: separate kinase and common phosphatase. Absence and presence of substrate and enzyme biphasic response, respectively (enzyme biphasic only with respect to the total amount of the second kinase, and only for certain ranges of intrinsic parameter values- see text). (F) DSP: separate kinase and separate phosphatase. Absence of both substrate and enzyme biphasic response. These results together show how commonality of both enzymes promotes biphasic responses with both doses, and in particular how commonality in phosphatase and kinase action enables enzyme and substrate biphasic responses, respectively. (G, H) Two-tier enzymatic cascades with common and separate phosphatases, respectively. Presence of substrate biphasic in the second-tier substrate of the two-tier cascaded enzymatic modification system with a common phosphatase. The first tier substrate is incapable of substrate biphasic in the same model. Both tier substrates are incapable of substrate biphasic responses when the phosphatases are distinct. Enzyme biphasic responses are absent in both systems. (I–K) Coupled covalent modification cycles with common and separate enzymes. Presence and absence respectively of substrate and enzyme biphasic responses in the system with common kinases and common phosphatases. The system with separate phosphatases and the system with separate kinases are incapable of substrate and enzyme biphasic responses. (G–K) indicates how commonality of enzymes can enable biphasic responses in covalent modification systems, which are either otherwise decoupled (I, J) or where they are part of a cascade (G, H). This highlights how such features emerge, even though the constituent modules (covalent modification cycles) are incapable of it.

Figure 2—figure supplement 1. Coexistence of multi-stability and biphasic dose response in various enzymatic models for the same underlying kinetic regime.

The above panels illustrate computational evidence of the possibility of coexistence of biphasic response with multi-stability, in a 1D bifurcation along the respective dose (for the same/similar underlying kinetics), for the different networks considered. The results are shown for all models capable of biphasic response (from within our study) and multistability. Note that a coupled covalent system (with common kinase) while capable of substrate biphasic is incapable of multistabilty. (A) Double site modification (DSP) with common kinase and phosphatase: Substrate biphasic response and multistability for the same underlying kinetics. (B) DSP with common kinase and common phosphatase: Enzyme biphasic response and multistability - for different (but close) parameter regimes. Inset plots show magnified regions from the main figure. (C) DSP with common kinase and separate phosphatase. Substrate biphasic response and multistability for the same underlying kinetics. (D) DSP with separate kinase and common phosphatase. Enzyme biphasic response and multistability for the same underlying kinetics. (E) Two-tier cascaded enzymatic network. Substrate biphasic response and multistability for the same underlying kinetics. This shows how the enzymatic modification mechanism (and the associated sequestration and non-linearity in interaction) within these networks allows for not just the capacity to present biphasic dose response behavior but also to be present in conjunction with other qualitative signaling behaviors. [LP: saddle-node bifurcation, solid, and dotted lines denote stable and unstable steady states, respectively].

Figure 2—figure supplement 2. Biphasic responses in random double site modification network with separate enzymes effecting each modification.

While the ordered double site modification (DSP) with separate enzymes acting on each modification site is incapable of exhibiting any biphasic dose response, random ordered (de)modification exhibits both substrate (left) and enzyme (right) biphasic dose response. This illustrates how additional complexity (inherently) associated with enzyme-mediated substrate modification (random) can further expand the propensity for such networks containing them to generate biphasic dose response.

Figure 2—figure supplement 3. Exploration of the effect of species total amounts in allowing for biphasic responses in different modification systems.

There are two kinds of parameters in substrate modification systems: intrinsic rate constants and species total amounts. Analytical work has characterized the possibility or impossibility of obtaining biphasic responses and the dependence on intrinsic rate constants. Semi-analytical work (presented here and in Figure 2—figure supplement 4) provides further analysis into how species amounts can allow for biphasic responses. For each of the systems considered, we first fix intrinsic kinetic rate constants at or close to those seen experimentally in a reference system (see Witzel and Blüthgen, 2018). We then semi-analytically explore the steady state equations by systematically reducing them and requiring the presence of a biphasic response. In all cases, the presence of biphasic responses (or specifically the onset of biphasic responses, i.e. biphasic peak) is shown by a curve involving two concentration variables. Each point along this curve denotes a concentration variable pair, that will guarantee the onset of biphasic response at the given value. Using these concentrations, the total amounts of all relevant species can be back-calculated (for every point on this curve) easily to ascertain what total amounts are to be needed for the behavior (also see Figure 2—figure supplement 4). Sample calculations for a representative point on each curve is presented in the Supplementary file 2. It is shown by sampling a given point from each of these graphs, that the resulting total amounts of substrates and enzymes are also within the ranges of those accessible by experiments (see Supplementary file 2 for more detail). All concentrations shown are in µM (A) Coupled covalent modification cycles (B) Two-tier enzymatic cascade (C) Double site modification system with common kinase and separate phosphatase (D) Double site modification system with separate kinase and common phosphatase (E) Double site modification with common kinase and common phosphatase (note that ${\displaystyle ϵ=K/P}$). The values of total amounts of one of the enzymes (and in where multiple enzymes (kinases/phosphatases) are involved, two of the enzymes), are fixed at levels in physiological ranges, and the total amounts of other species are explored. This is to allow for the creation and easy visualization of contour plots.

Figure 2—figure supplement 4. Semi-analytical approach for determining total amounts of species for realizing biphasic responses.

This figure provides complementary information to Figure 2—figure supplement 4 into how species total amounts can allow for biphasic responses. For each of the systems considered, as discussed in Figure 2—figure supplement 4 we first fix intrinsic kinetic rate constants at or close to those seen experimentally in a reference system. We then semi-analytically explore the steady state equations by systematically reducing them and requiring the presence of a biphasic response. In all cases, the presence of biphasic responses (or specifically the onset of biphasic responses i.e. biphasic peak) is shown by a curve involving two concentration variables in different ways. On one hand, we show the curve relating two concentration variables (as shown in Figure 2—figure supplement 4). On the other hand, the curve has also been translated into curves in total amounts and that is shown in a separate panel. This demonstrates that biphasic responses can be readily obtained for biologically reasonable total amounts (all total amounts and concentrations are in µM). (A) Coupled covalent modification cycles (B) Two-tier enzymatic cascade (C) Double site modification system with common kinase and separate phosphatase (D) Double site modification system with separate kinase and common phosphatase (E) Double site modification with common kinase and common phosphatase (${\displaystyle ϵ=K/P}$). The values of total amounts of one of the enzymes (and in where multiple enzymes (kinases/phosphatases) are involved, two of the enzymes), are fixed at levels in physiological ranges, and the total amounts of other species are explored. This is to allow for the creation and easy visualization of contour plots.