We present the results of the analysis of each of these systems, in sequence. In each case (a) we indicate whether or not a biphasic response (of either type) is possible (b) we specify whether there are any restrictions in intrinsic kinetic parameters for this to happen (this is based on analytical work) (c) in systems where biphasic responses are possible, we present the basic mechanistic requirements and intuition emerging from the analysis. (a) and (b) are also depicted in Figure 2: here when biphasic responses are possible, a sample biphasic response is shown. We present a summary of additional parametric analysis for the systems which can exhibit biphasic responses, at the end of this subsection.

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The simplest enzymatic protein modification system, the covalent modification cycle (model M0) is incapable of exhibiting biphasic responses with either increasing total enzyme amounts or total substrate amounts (see Supplementary file 1, section 1). This is true irrespective of model parameters. However, an additional complex formation of the modified substrate with the kinase enzyme (model M1), allows for the network to exhibit enzyme biphasic responses (see Figure 2B). Some aspects of the capacity of this system to exhibit enzyme biphasic dose responses have been studied earlier (albeit with different modeling assumptions by Varusai et al., 2015) and also as part of analyzing multispecific interactions (Seaton and Krishnan, 2012). Interestingly, we find that enzyme biphasic responses may be obtained for any choice of intrinsic kinetic parameters. However, this system is incapable of exhibiting substrate biphasic responses (see Supplementary file 1, section 1).

Examining the two cases above illuminates why an enzyme-biphasic response emerges in the later case. The extra protein interaction creates a competing effect: while an increase in kinase enzyme concentration, generally favors increased concentrations of the modified substrate, it also has a secondary effect of binding to the modified substrate and sequestering it, thus reducing the (free) modified substrate concentration. This creates the in-built competing effect resulting in a biphasic response.

Proteins rarely undergo just one (de)modification and often are (de)modified at different sites (effected by either the same enzyme or by different enzymes). To explore the consequence of such multiplicities in modifications, we consider the simplest extension of a basic covalent modification cycle, with a substrate undergoing two modifications in an ordered fashion, where the order of phosphorylation is opposite that of dephosphorylation and the (de)modifications effected by the same or different enzymes (see models M2-M5 in Figure 1—figure supplement 1). In the instance where a common enzyme effects multiple modifications, the mechanism is assumed to be distributive.

We observe that the double site modification (DSP) network with common kinase and common phosphatase effecting the modifications (Model M2) is capable of exhibiting both substrate and enzyme biphasic responses in the maximally modified substrate, A_pp (see Figure 2C; Suwanmajo and Krishnan, 2013). Our analytical work reveals the parametric dependence for this behavior to be observed. Substrate biphasic responses are guaranteed to be observed for some total amount of kinase and phosphatase for any set of intrinsic kinetic parameters of the system. On the other hand, enzyme biphasic responses can only be present when the following condition (${\displaystyle {\displaystyle k_1{k}_3-k_2{k}_4>0}}$) is satisfied by the kinetic catalytic constants (${\displaystyle k_1,k_2}$ associated with phosphorylation, ${\displaystyle k_3,k_4}$ associated with dephosphorylation) of the system. This condition is a necessary and sufficient condition, and when it is violated the system is incapable of exhibiting enzyme biphasic responses (see Supplementary file 1, section 2).

It is interesting to note that this kinetic condition involving the catalytic constants is also observed elsewhere in studies relating to the presence of multistability in the DSP with common enzymes. Conradi and Mincheva, 2014 identified that if the condition is violated, then the network is guaranteed to exhibit multistability at some total amounts of substrate and enzyme. These results together suggest that irrespective of the kinetic regime of the system, the DSP with common enzymes is capable of exhibiting either multistability or biphasic dose response with enzyme necessarily (for some total amounts of substrates and enzymes). This reveals the plurality of information processing behaviors capable with substrate modification systems with only two modification sites, and how catalytic constants (and more broadly intrinsic kinetic constants) can allow us to guarantee and generate these behaviors in such systems. It is also a striking example of how one among a suite of non-trivial behaviors is guaranteed to be observed across intrinsic kinetic parameter space.

An additional source of complexity in enzyme-mediated substrate modification arises from the specificity of enzyme action. For instance, the same enzyme could effect modifications on different proteins. Conversely, in multisite modification of the same protein, there could be distinct enzymes effecting each (de)modification. Considering such scenarios with different enzymes effecting each (de)modification in double-site modification (see models M3-M5 in Figure 2—figure supplement 2), we notice that the capacity of the DSP system to exhibit biphasic dose response progressively disappears, with increasing specificity in enzyme action. The DSP with separate kinases and a common phosphatase (model M4) acting on each modification site is capable of only exhibiting enzyme biphasic responses, and only with the variation of the total amount of the kinase effecting the second modification (see Figure 2E). Associated kinetic constraints are discussed in Supplementary file 1, section 3. On the other hand, it is incapable of exhibiting substrate biphasic dose response irrespective of kinetic parameters. Similarly, the DSP with a common kinases and different phosphatases effecting each modification (model M3) is capable of exhibiting substrate biphasic responses (see Figure 2D), while it is incapable of exhibiting enzyme biphasic responses with changing total kinase amount. Analytical work further reveals that substrate biphasic responses in the model is guaranteed to be observed for any choice of underlying kinetic parameter choice, for some total amounts of kinase and phosphatase. The DSP with separate kinases and phosphatases effecting (de)modifications (model M5) is incapable of exhibiting either substrate or enzyme biphasic (see Supplementary file 1, section 3 for the relevant proofs).

An examination of all these cases reveals the factors responsible for generating the biphasic response (also see Suwanmajo and Krishnan, 2013). First, the possibility of an enzyme biphasic response can be explained as follows: in the common-kinase common-phosphatase case, while increasing the kinase concentration favors an increase in the fully modified substrate, it can also have an auxiliary effect: sequestering the partially modified substrate in a kinase complex. This has the effect of making more phosphatase available to dephosphorylate the fully modified substrate and this provides the in-built competing effect. The crucial factor here is the fact that the phosphatase dephosphorylates both fully modified and partially modified substrate. This is further borne out by the fact that even the separate kinase common phosphatase case can result in an enzyme biphasic response when the concentration of the second kinase is varied (for the same reason).

With regard to substrate biphasic responses, we find that the common kinase separate phosphatase case shows this. Here, increasing the total substrate concentration has two effects: the natural effect of increasing all substrate concentrations including that of the fully modified substrate, and another effect: that of reducing the free enzyme concentration. This, when combined with co-operativity, the effect of having two substrates reduce the free kinase concentration (the unmodified and partially modified form) provides a sufficiently strong nonlinear effect, allowing for a competing effect resulting in the biphasic response (confirmed by analytical work). Naturally, this effect is present in the common kinase common phosphatase case as well.

Taken together, this shows how complexity at the level of the biochemistry of a node in a signaling network (arising from the number of modifications, and enzyme sharing) can enable biphasic dose responses in the (maximally modified) substrate form. Sequestration and the commonality of enzymes here are key features that promote biphasic dose response (both enzyme and substrate).

Modifications provide additional functionality to proteins, and in many instances, a protein undergoing a modification can gain functionality that allows it to effect modifications on another protein (Millar et al., 2019). This generates an enzymatic modification cascade structure (such as the MAPK system [Kocieniewski et al., 2012; Kõivomägi et al., 2011; O’Shaughnessy et al., 2011; Markevich et al., 2004]). Here too, common/separate enzymes may effect the demodifications of each of the substrates.

To explore the possibility of biphasic responses in such enzymatic modification cascades, we studied the two-tier enzymatic cascade with the demodifications being effected by either common or separate enzymes (M6 and M6a, respectively, see Figure 1—figure supplement 1).

This enzymatic cascade with separate phosphatases is incapable of exhibiting either substrate biphasic responses (in response to the variation in total substrate amounts in either tier) or enzyme biphasic responses in response to change in total amounts of kinase. On the other hand, while the analogous system with common phosphatases effecting demodifications on each substrate is still incapable of enzyme biphasic responses or substrate biphasic responses with respect to the first tier substrate (see Supplementary file 1, section 4), it is capable of exhibiting substrate biphasic responses in the second tier substrate (see Figure 2G). Associated kinetic constraints are discussed in Supplementary file 1.

This latter case can be understood as follows: increasing the total substrate concentration in the second tier has the effect of increasing the sequestering of the modified species in the first tier (in a complex with the unmodified species of the second tier). This has the consequence of making more phosphatase available, which can then dephosphorylate the modified species in the second tier. This competing effect (reminiscent of that responsible for an enzyme biphasic in multisite substrate modification above) allows for the manifestation of the substrate biphasic response.

Other forms of coupling are possible with enzymatic modifications. For example, a single enzyme can effect modifications on multiple substrates allowing for a sophisticated degree of cross-talk and coupling between pathways (Allende and Allende, 1995; Ghomlaghi et al., 2021; Shockley et al., 2019; Seaton and Krishnan, 2011). We studied the propensity for biphasic responses in the system with enzyme sharing between two covalent modification cycles. The system with a common kinase and common phosphatase for the cycles (model M7) can exhibit substrate biphasic responses in one (but not both) of the substrate modification cycles (individually, see Figure 2I). The system is, however, incapable of exhibiting enzyme biphasic dose responses. Analytical work further reveals that the coupled covalent modification cycles are necessarily guaranteed to present substrate biphasic responses in one of the modification cycles (but not both) for some total amount of kinase, phosphatase, and (other) substrate amount. The identity of the modification cycle which is capable of substrate biphasic responses is determined by an explicit analytical criterion which depends on intrinsic kinetic parameters.

The same system with separate enzymes acting on the two modification cycles (either kinase or phosphatase-model M7a & M7b, respectively) is incapable of exhibiting substrate biphasic responses, showing how enzyme sharing and simultaneous coupling through both enzymes is required for realizing substrate biphasic responses(see Supplementary file 1, section 4 for relevant proofs).

The fact that covalent modification cycles involving a shared kinase and a shared phosphatase can result in a substrate biphasic response (for one of the substrates) can be understood as follows. Increasing one of the substrates can under certain situations result in a combination of effects: one the reduction of free kinase (as seen previously) and the other an increase in free phosphatase (since the reduction in free kinase implies a lower level of modification of the other substrate, and this implies less phosphatase used there). Both these effects have been seen in different guises previously, but here their combined co-operative effect provides the nonlinearity to create the competing effect.

The analysis above for all systems has demonstrated either the presence or absence of biphasic responses, and analyzed to what extent the intrinsic kinetic parameters provide a fundamental restriction to the possibility of biphasic responses occurring. We now build on that to provide further insights. (1) We first note that when a biphasic response occurs, the role of intrinsic kinetic parameters are either (a) they provide no essential restriction by themselves or (b) there is a region of parameter space, explicitly delineated, which is the region where a biphasic response is possible. (2) In the later case, in all instances, there is a simple inequality involving relative magnitudes of the ratio of phosphorylation to dephosphorylation rate constants in different modifications which determines whether biphasic responses are possible or not. Equivalently, the inequality can be written in terms of relative magnitudes of products of specific catalytic constants. This emerges from detailed analysis and reflects the fact that when such a condition is satisfied, the possibility of an in-built competing effect sufficiently strong to be capable of giving rise to a biphasic response exists. (3) The type of this inequality is such that multiplying all catalytic constants by a specific constant does not affect the inequality. Thus, this inequality does not set a restriction on the absolute levels of these rate constants. Furthermore, it is clear from the nature of the inequality, that experimentally feasible ranges of kinetic parameters (catalytic constants) can satisfy such an inequality and this inequality represents a broad range of parameter space. (4) We now examine the effect of other parameters (total amounts of enzymes or substrates which are not part of the dose). By proceeding with further analytical and computational work, we can, in some cases, show further restrictions of enzyme or substrate amounts to realize a biphasic response. For instance, we show (see Supplementary maple document 2 for the points below) that higher phosphatase amounts favors enzyme biphasic responses in the DSP common kinase/phosphatase system, and also substrate biphasic in the two-tier cascade studied above. (5) Similarly we show that in both the DSP common kinase/common phosphatase (enzyme biphasic) and in the coupled covalent modification cycles (substrate biphasic), there is a minimum level of concentration of the modified substrate to allow for a biphasic response (this incidentally also places a lower bound on the total substrate concentration for this to happen). In the coupled covalent modification cycle we can also determine which cycle will exhibit the biphasic response. (6) Further, parametric exploration into the effects of total amounts and intrinsic kinetic parameters was performed semi-analytically, starting with the analytical equations and specifying some parameters. This then provides equations for the remaining parameters and total amounts for a biphasic response to be obtained, and this was explored for all models. As part of this analysis, we also completely fixed intrinsic kinetic parameters in ranges obtained experimentally and studied computationally in Witzel and Blüthgen, 2018 and then used semi-analytical approaches to determine the sets of two total amounts of species (enzymes or substrates) where biphasic responses are obtained (one of the set of three total amounts is the dose, this leaves, two other total amounts in the most basic case where there is a single kinase and a single phosphatase: in the case of multiple kinase and/or multiple phosphatases, there are more total amounts, and these total amounts could be explored together, or by having some of them fixed at particular values). (7) We performed semi-analytical and computational explorations for parameters sets for different modifications in the range of those experimentally determined in the MEK system, and explored in Witzel and Blüthgen, 2018: this is summarized in Figure 2—figure supplement 3 and Figure 2—figure supplement 4. We find that biphasic responses can be obtained in ranges of intrinsic kinetic parameters, in a very similar range as that obtained there. In each model, we were able to obtain biphasic responses in reasonable physiological ranges of total amounts (and for total amounts which could be employed in synthetic settings). We also explored a second parameter set where the unbinding rate constants were set to zero, and again biphasic responses were obtained in reasonable ranges (results not shown). (8) Building on these semi-analytical and computational analyses and using continuity arguments, we can see that biphasic response can be readily obtained in neighboring regions in parameter space (both intrinsic kinetic parameters and total amounts). All in all, this complements our analysis of the intrinsic kinetic rate constants in restricting the possibility of biphasic responses (presented earlier), providing additional insights and parametric regions for biphasic responses and how they may be explored further.

Interestingly, in all models capable of biphasic responses studied above, almost all systems which are capable of exhibiting multistability are also capable of exhibiting it alongside biphasic response in the dose-response curve (see Figure 2—figure supplement 1).

We probe the impact of a biphasic interaction pattern in a network. To do this, we first probe the effect of biphasic regulation by signal of a basic covalent modification cycle (a representative node of a network). We then explore the impact of biphasic regulation internal to networks, by examining ubiquitous network motifs (feedforward and feedback motifs). Finally, we explore the interplay of biphasic regulation, interaction, and biphasic behavior intrinsic to a node in a network, the later arising from substrate modification biochemistry.

To explore this, we begin by analyzing how a simple signal can regulate a covalent modification cycle via a biphasic interaction pattern: (model N0, see Figure 3A). This simple interplay of a signal regulation and biphasic transduction can convert a saturated response into a biphasic response with a range of prolonged homeostatic responses in the concentration of the maximally modified substrate form, showing the potential for complicated signaling behaviors to emerge out of simple network considerations and biphasic interactions among nodes.

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To explore this interplay further, we consider standard signaling motifs (feedback and feedforward) with biphasic interactions between the nodes (See models N3-N5 in Figure 1—figure supplement 1) and study how the biphasic interaction can alter expected network behavior. Networks motifs, such as feedforward and feedback networks, are recurring network structures in cellular signaling that are known to be responsible for key information processing characteristics such as multistability, biphasic responses, homeostasis, and oscillations.

The incoherent feedforward network motif (model N3), represents a simple network where a signal activates both a kinase and a phosphatase (and more generally activating opposing effects) and is known to present biphasic responses (Varusai et al., 2015; Kim et al., 2008). This nominal network behavior can be affected by the presence of a biphasic interaction between the nodes in three ways; (1) The behavior can be reinforced or strengthened leading to a more prominent overall biphasic response, (2) The two distinct biphasic responses (from the network structure and from the interaction between the nodes) can combine to give a multiphasic response, and (3) The two distinct biphasic responses can cancel out resulting in a simple monotonic dose response (see Figure 3D).

The positive feedback network motif, characteristic of reinforcement, is known for introducing multistability (Tyson et al., 2003; Ferrell et al., 2009). Biphasic interaction between the nodes in such a network motif is capable of destroying such multistability arising from the feedback structure (see Figure 3E). Similarly the negative feedback network motif, characteristic of homeostatic response (Nijhout et al., 2019), can have this behavior impacted with the introduction of combinations of behaviors including homeostasis, multistability and monotonic responses (see Figure 3F). Particularly notable is the fact that multistability (usually associated with positive feedback) emerges from a negative feedback network in this case.

The basic intuition in both instances is that the biphasic interaction pattern causes deviation from expected behavior, simply because in a certain range, the nature of regulation changes from positive to negative. This negative regulation can combine with an overall negative feedback regulation pattern to give behavior reminiscent of a positive feedback (e.g. multistability).

The subversion of expected behavior and introduction of new behavior is seen even when the motifs involve open systems (see Figure 3—figure supplement 2). Analytical work reveals how negative feedback (involving an open system) can, in the presence of a biphasic interaction, present with multistationarity (which the motif is inherently incapable of in its absence; see Supplementary file 1, section 5).

We focus on a widely-studied motif incorporating homeostasis via integral control regulation which arises from a zeroth order reaction inherent in the system. Here, stable homeostasis is observed once the signal is above a certain threshold (the homeostatic state is unstable for lower signal values). We explore different aspects of biphasic regulation in interactions. If the signal regulation of the motif incorporates a biphasic interaction, this has the effect of creating both upper and lower limits of signal beyond which homeostasis does not occur. The presence of biphasic regulation within interactions intrinsic to the motif has two consequences (i) it limits the range of signal over which feasible homeostatic steady states occur (see Figure 3—figure supplement 2) (ii) it results in two steady states, one of which is shown to be necessarily unstable.

The above results indicate how hidden biphasic responses between nodes in networks can not only fundamentally alter expected network responses, but can also lead to incorrect inferences regarding the structure of the network (e.g. inferring networks from data), its information processing capability and the role of its constituent elements.

Cellular signaling often involves both complex network regulation and additional complexity at the level of a node, with the node itself being capable of complex behavior (such as the DSP system). This implies the possibility (in addition to the cases considered above) of biphasic responses both in interactions as well as within a node in a network.

To explore this interplay further, we consider the signal regulation of DSP (with common kinase and phosphatase). This allows us to explore the interplay of biphasic response arising from two fundamentally different sources, (1) the network (for instance incoherent feedforward signal regulation) or biphasic response within an interaction between nodes, and (2) the intrinsic biochemistry of the node (DSP, is itself capable of exhibiting biphasic response - see Figure 3B, C).

We incorporate these aspects in simple models N1 and N2 (see Figure 1—figure supplement 1); In the former instance (model N1, Figure 3B) the biphasic response is inbuilt into the signal regulation activating kinase, where as in the latter (model N2, Figure 3C), the biphasic arises from the incoherent feedforward network structure (signal activating both enzymes).

When the biphasic interaction from the network arises directly from the regulating signal, the combination of biphasic responses from the network and the node can result in (i) introduction of biphasic responses (when the DSP does not present it inherently) (ii) strengthening the biphasic response and also generating distinct zones of multistability and homeostasis (Figure 3B), when the DSP presents biphasic responses independently. When the biphasic behavior at the network level arises out of the incoherent feedforward regulation (Figure 3C), similarly the behavior could be strengthened or destroyed. In both instances, multiphasic response is also possible.

Thus beyond simple signal regulation, minimal complexity at the level of the biochemistry of the node and biphasic responses (either within the signaling interaction, or arising from network structure) can lead to a plethora of unexpected and non-intuitive signaling behavior.

The ErK pathway is an example of a pathway of central biological interest, which is well studied, contains the biochemical building blocks studied above and is known to exhibit biphasic responses. This makes it an excellent exemplar case worth investigating in detail. We explored the propensity of the established Erk pathway to present substrate biphasic responses (with total amounts of Erk) and enzyme biphasic responses (with total amounts of Mek) in the concentration of the maximally modified Erk (pYpTErk). Studies in the past have focussed on the capacity of the Erk regulatory pathway to show substrate biphasic responses computationally (Witzel and Blüthgen, 2018).

Our analysis reveals that the Erk pathway is capable of exhibiting both substrate and enzyme biphasic responses in the maximally modified substrate form (see Figure 4A and B). The system is in fact capable of biphasic dose response with both doses simultaneously, for the same underlying kinetic parameters (see Figure 4C). Analytical work further reveals that substrate biphasic responses are guaranteed to exist for some total amounts of Mek and phosphatase, irrespective of intrinsic kinetic parameter values. This is indicative of a degree of robustness in this behavior, as intrinsic kinetic parameters are varied, which is not apparent in a purely computational study. Analytical work revealed an explicit condition (dependent on total amount of phosphatase and kinetic parameters) that guarantees the existence of enzyme biphasic response. This condition is, however, only sufficient and not necessary for observing this (see Supplementary file 1, section 6) These results are summarised and consolidated in Figure 4. The consequences of these results for both systems and synthetic biology are discussed in the conclusions. We note here that the Erk system represents examples of biphasic interaction and regulation in cell signaling, both for its regular downstream pathways, as well as pathways it interacts with as a consequence of cross-talk.

Figure 4

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It is worth pointing out that the Erk network has the structure of a random substrate double site modification system, with partial irreversibility. Having established the result above, along with analogous results for ordered modification systems (one leg of the Erk network), we can assert that essential insights are relevant even if one of the irreversible steps is made reversible.

All in all, we can conclude that the commonality of kinase in different modification steps and the commonality of phosphatase in demodification steps provides the (different) in-built competing effects for enzyme biphasic and substrate biphasic responses to be realized.

The first basic observation is that many ubiquitously occurring biochemical building block structures present either enzyme biphasic or substrate biphasic behaviour, and in some cases both behaviour, and this arises from the presence of key basic ingredients. An example is enzyme sharing which is common in cellular networks, especially since kinases significantly outnumber phosphatases, leading to phosphatases deployed in different kinase mediated reactions (Ghaemmaghami et al., 2003). Interestingly while certain building blocks do not admit biphasic responses (unless some additional feature such as substrate inhibition occurs), others do, and in certain cases, they can even be obtained just by tuning substrate and enzyme total amounts (completely independent of intrinsic kinetic parameters). Since these building blocks are ubiquitous in cells, this already indicates a number of in-built sources of safeguarding against overexpression of active entities in signaling networks, beyond network regulatory structures.

As an illustration, we have investigated a single tier of the Erk signaling network and demonstrated how this exhibits robust substrate biphasic responses (without restriction on intrinsic kinetic parameters, in contradiction to the picture suggested in Witzel and Blüthgen, 2018) and can also exhibit enzyme biphasic responses in broad parameter ranges. This illustrates how a single level of a specific signaling pathway may incorporate multiple levels of safeguarding and that the structure of the biochemical reactions involved provides cells (and evolution) with easily tuneable factors to realize this safeguarding. This also provides an example of biphasic regulation/interaction in networks, by considering the various downstream targets of Erk (as well as elements impacted by Erk cross-talk). Another consequence of the above results is the possibility of biphasic dose-responses occurring, which may be not (particularly) desirable, but are present due to the characteristics of the basic building blocks. This could, in some contexts, illuminate why such responses may occur.

The fact that there are multiple intrinsic sources of safeguarding and biphasic responses in signaling networks (i.e. even intrinsic to a node in these networks), suggests that there may be a hierarchy of such mechanisms–ranging from network structures (such as incoherent feedforward networks) to additional ingredients in modification systems (such as scaffolding) to the intrinsic modification systems themselves. This is especially relevant in inferring network structures from data (for example, observed biphasic responses) and suggests that the inference process must be able to account for different sources of such behavior. Our analysis suggests that focussed experiments can discriminate between these different sources of biphasic responses and simple controlled test tube experiments can reveal the presence of intrinsic biphasic responses. Such experiments would be very valuable as part of the inference process.

Analyzing the consequences of biphasic responses at the network level yields further insights. We have already demonstrated that the presence of biphasic responses of the kind studied leads to biphasic interactions between network nodes and this can significantly alter network behavior, even resulting in behavior which might be expected from a very different network structure. Biphasic responses can be used in networks to access specific downstream behavior in specific ranges of inputs. As such it can combine with complex dynamical behaviour (e.g. bistable switching, oscillations) in different ways giving rise to behavior which may appear unintuitive, if the presence of biphasic behavior is not recognized.

The presence of safeguarding mechanisms/biphasic behavior at different locations in a network has important consequences for the overall robustness of the network as well as the maintenance of behavior within bounds. Our results revealed how biphasic responses at different levels (e.g. network and substrate modification) may either reinforce or cancel one another, suggests that the implications of different sources and locations of biphasic responses/safeguarding in a network, need to be carefully studied via a dedicated systems analysis. The analysis of how distributed sources of such safeguarding (both enzyme and substrate biphasic behavior) determines the overall safeguarding behavior in the network and keeps relevant species behavior within suitable bounds is a topic of future investigation.

Natural and engineered biology and the emerging discipline bridging the two, motivates the need to examine the presence or absence of biphasic responses relative to desired functionality. To start with, if we focus on a given kind of biphasic response (either enzyme biphasic or substrate biphasic) then we have four distinct possible cases depending on the presence or absence of this behavior and whether its presence is desired or not (Table in upper left panel, Figure 4). Two of these cases (biphasic response present, desired) (biphasic response absent, undesired) represent desirable outcomes, and the key insight in the former case is that this may be obtained in broad swathes of parameter space by just varying total amounts of enzyme and substrate. The other two cases (biphasic response absent, desired), (biphasic response present, undesired) represent a mismatch between desired functionality and capability. In the former case, (apart from possible tuning of enzyme or substrate amounts) alternative structures can be engineered or realized in evolution, for instance, the creation of incoherent feedforward networks (for realizing enzyme biphasic responses) or the incorporation of extra structures in the substrate modification systems (for realizing substrate biphasic responses). In the latter case, a tuning of total enzyme and substrate amount can be used to prevent biphasic responses in desired ranges of operation, and if this can be done consistent with information processing goals, a desired outcome can be obtained.

The same picture can now be expanded to consider the possibility of both enzyme and substrate biphasic responses in relation to whether those biphasic responses are desirable or not (see Figure 4). This results in 16 cases, four of which correspond to both desired outcomes being met, four of them corresponding to both desired outcomes not met, and eight cases where one desired outcome is met and one is not. In order to ‘engineer’ the system towards desired goals, different approaches can be taken depending on the specific case and mismatch. Clearly depending on the context, the managing of desired outcomes requires a combination of structural and (easy to manipulate) parameter manipulation and this may involve a careful optimization to keep it consistent with the information processing goals. Our analysis provides a basis from which to evaluate what steps may have been incorporated in evolution in different contexts to achieve safeguarding against over-expression or overactivation, including the extent to which intrinsic capabilities for such responses have been used. This needs to be evaluated on a case by case basis.

The results have further specific consequences for synthetic biology. Biphasic responses have been generated from synthetic gene regulatory incoherent feedforward circuits (Basu et al., 2004) and synthetic microbial communities (Zong et al., 2022). There have been multiple recent efforts to engineer biochemical circuits to obtain complex information-processing behavior (Maguire and Huck, 2019; Helwig et al., 2018; Holmqvist et al., 2012) and these include engineering substrate modification as well (Valk et al., 2014; O’Shaughnessy et al., 2011). Our demonstration of basic building blocks generating biphasic responses is of particular use in engineering biochemical networks. Of particular relevance is the presence and guarantee of obtaining such behavior, and in some cases the widespread occurrence of such behavior, which can be accessed by varying simple easy to vary factors (substrate and enzyme amounts). Our analysis in totality across systems provides structural requirements for the generation of such behaviour and where necessary also provides kinetic parameter ranges where this may be obtained. This combination of structural and parametric criteria provides engineering design principles for creating such behaviour from basic biochemical building blocks. Furthermore, our analysis can be used to engineer both substrate and enzyme biphasic responses, representing a robust safeguarding mechanism for preventing overexpression of specific substrate forms. Interestingly, in addition to our analysis of building block circuits, our analysis of a key layer of Erk signaling reveals such robust safeguarding mechanisms, making this an appealing candidate for incorporating into synthetic circuits.

Our examination of different building block structures in comparison with one another provides further insights. For instance, the presence of a ‘dead-end’ complex (protein-protein interaction system) reveals a robust presence of enzyme biphasic responses, while double site substrate modification with either common kinase or different kinases but common phosphatases provides enzyme biphasic responses in a substantial region of kinetic parameter space (the region in fact being exactly the same). A comparison of the latter two cases suggests that the separate kinase common phosphatase system provides additional tuneability, along with access to other behavior, with no loss of capability in terms of biphasic responses. Interestingly, a number of substrate modification systems: double site modification with common kinase common phosphatase, common kinase separate phosphatase, as well as coupled modification cycles sharing a common kinase and phosphatase all yield robust substrate biphasic responses across intrinsic kinetic parameter space. Comparing the first two cases, the possibility of having different phosphatases may again provide additional tuneability. In the third case, one of the cycles is guaranteed to give substrate biphasic responses, the identity of which is determined by an explicit analytic expression involving kinetic constants. Taken together, this illustrates how our analytical work across different systems reveals tangible structural and organizational insights directly relevant to design.

The analysis here can also be used to shape overall network/circuit behavior. Modules exhibiting biphasic responses with tuneable characteristics can be designed and incorporated into chemical reaction networks (either naturally occurring or itself synthetic) to shape outcomes, regulate behavior, and create different safeguarding mechanisms against over-expression in networks. Interestingly some preliminary analysis (Ramesh et al., 2023) shows how basic modules studied here when combined with positive and negative feedback can in fact maintain, not just the biphasic response, but specific characteristics such as the peak amplitude, thus testifying to both the robustness of the insights and its value in engineering design.

All in all, our unified exploration of enzyme and substrate biphasic responses through the analysis of ubiquitous building blocks in networks opens up new vistas for exploration in natural biology, engineered biology, and the emerging area straddling the two.

The biphasic interactions between nodes ${\displaystyle A\to B}$ in a network was implemented by simply incorporating a biphasic function connecting the activation of B with the concentration of A. For this purpose, we use a Poisson function, given by the following mathematical expression ${\displaystyle k_1[A]e^{-k_2[A]}}$. This is in contrast to a basic (monophasic regulation) for instance by a linear function (${\displaystyle k_2[A]}$). We note that the particular function we use to describe the biphasic regulation is representative, and the essential results and insights do not depend on the particular choice of biphasic function.

When choosing parameters in the context of systems with biphasic response in the interaction (${\displaystyle k_1}$ and ${\displaystyle k_2}$), in order to maintain parity and a like-for-like comparison between the system with the biphasic regulation and without it, we ensure that the average strengths of the interaction terms remains the same when averaged over the range of total signal activation. Further details on how this is implemented is provided in the Supplementary file 1, section 5.

One-parameter bifurcation analysis was carried out computationally using the MatCont package in Matlab (Govaerts et al., 2017). In all cases showing the presence of biphasic response, the result is accompanied with analytical work establishing the presence of the behavior in broad regions of the parameter space (see below).

Analytical work exploring the presence or absence of biphasic dose response (for a given choice of kinetic parameters and total amounts) relies on the mathematical requirement for a steady state of the system to satisfy the following expression at some finite concentration of dose ${\displaystyle \frac{d[Var]}{d[Dose]}=0}$, should a biphasic response exist. Here, Var denotes the particular variable under consideration. Analytical work proceeds by exploring the possibility of the model satisfying this condition for some feasible steady state of the system. This is done by solving the system of ODEs of the model at steady state and ascertaining an expression for the above expression (${\displaystyle \frac{d[Var]}{d[Dose]}}$) as a function of the steady state concentration of fewer variables and the parameters. This reduction allows us to ascertain whether the behavior is possible/impossible across parameter space, and where possible ascertain conditions guaranteeing (and conversely precluding) the behavior. The entirety of this analysis is carried out in the computational platform Maple (Maplesoft, 2022), and is presented in Supplementary file 1. This file is also supplied as a PDF for easy accessibility. A concise summary of the way analytical work is undertaken is discussed further below (see Summary of analytical approach).

Note: Our analysis throughout focuses on the existence of a biphasic dose response in the steady state of the system of the dose. We, however, make no comment or perform further analysis on the stability of the steady state when such a biphasic response is predicted. This, however, can be easily verified with computations and we supplement each of our analytical results with computational proof of the existence of the behavior for a range of parameters.

In the case of the biochemical substrate modification systems, our analytical work establishes to what extent intrinsic rate constants prevent or enable biphasic responses. We find three possibilities (a) biphasic responses are completely precluded (b) biphasic responses are possible for any values of the intrinsic rate constants: in this case, we can guarantee that there exist suitable values of total amounts of substrate and/or enzyme for this to happen (c) there is a region in the parameter space (of intrinsic rate constants), explicitly delineated, where biphasic responses are possible. Here again, we can guarantee that there are suitable values of total amounts of species to make this possible. In order to further investigate ranges of total amounts of species where biphasic responses are possible, we can start with the necessary requirement of a biphasic response (as in our analytical approach). This can be used further semi-analytically to determine total amounts of species where a biphasic response is realized. In particular, we can input kinetic parameters to determine the location of the biphasic peak (Supplementary file 2). As part of this, we also show that biphasic responses can be obtained in ranges of kinetic parameters seen experimentally. Here, we choose intrinsic kinetic parameters similar to those found in experimentally well-characterized reference systems (see Witzel and Blüthgen, 2018) and fix some enzyme amounts at physiologically reasonable values. We then obtain curves in the space of total amounts (see Figure 2—figure supplement 3 and Figure 2—figure supplement 4) which correspond to the occurrence of a biphasic (peak) response. This can be confirmed by a computational bifurcation analysis.

The same approach is used in all relevant cases of analysis. In the discussion below, the derivatives refer to derivatives of the substrate or kinase variables to the relevant substrate or kinase total amount. Note that the above analysis results in a set of linear equations for those derivatives, one homogeneous and one inhomogeneous.

1. First, in the above case, the possibility/impossibility of a substrate biphasic response is established in an exactly analogous way.

2. In the cases where we show a biphasic is possible, in many instances we are able to find values of the variables ${\displaystyle A_{var}}$ and ${\displaystyle K_{var}}$ where the associated equation has a solution (averting a contradiction between the two equations). This then establishes total amounts of substrate and enzyme where a biphasic peak can be expected. Numerical bifurcation analysis then establishes the actual presence of the relevant biphasic response.

3. In the above equation, the variable ${\displaystyle A_{var}}$ is the concentration of the substrate variable of interest. while the variable ${\displaystyle K_{var}}$ could either be the free kinase concentration, or in some instances, the ratio of free kinase to free phosphatase concentrations.

4. As the number of either substrates or kinases (or both) increases, the number of equations increases. However, they have the same structure. The LHS of the equations involves a linear combination of different derivatives of the relevant variables with respect to substrate or kinase amounts. The RHS of all but one of the equations is zero, and the RHS of one equation is 1 (when the derivative with respect to the total amount associated with that conservation equation is the focal point of interest).

5. In this case the impossibility of biphasic responses has been demonstrated in one of two ways. In both cases, we start with the assumption of a biphasic response, implying the possibility of a zero derivative of the relevant substrate variable to either a relevant kinase or substrate total amount. (a) We progressively eliminate some of the derivatives in terms of others, until we obtain a set of equations, all but one, homogeneous. By assessing the coefficients of the derivatives in those equations, we show that a particular derivative (or set of derivatives) needs to be zero, which then makes the inhomogeneous equation impossible to satisfy (this is essentially an analog of the simpler case considered above) (b) In some cases, a consideration of two homogeneous equations themselves shows that they are impossible to satisfy simultaneously. In each case, this contradicts the assumption of a biphasic response

6. Likewise, the possibility of a biphasic response is established by showing that the simultaneous equations can have a solution. The essential insight is to show that the homogeneous equations can be satisfied in a way (by certain coefficients being zero), so that the inhomogeneous equation can also be satisfied. This then provides a condition to show that the biphasic response may either be observed (a) irrespective of intrinsic kinetic parameter values (relying only on choices of total amounts of species) or (b) requiring non-trivial restrictions on the intrinsic kinetic parameter values.

7. Note that all our analysis involves analyzing the first derivative requirement (to be zero). We do not explicitly check the second derivative condition (to show that there is a genuine change in sign of slope, as opposed to there being inflection point), simply because this is what can be expected generically (with the possible exception of ‘rare’ points in parameter space, which would correspond to sets of zero volume/measure). In this connection, we mention that when we make the statement that biphasic responses are observed everywhere in intrinsic kinetic parameter space, we mean that the responses are obtained essentially everywhere, with the possible exception of these special points. This does not affect our essential insight about robustness, Finally, our analysis does not explicitly assess stability. For all the cases analyzed, a predicted biphasic response has been shown computationally to be associated with a stable steady state over the range of parameters studied.

• Vaidhiswaran Ramesh