Biphasic responses in cell signalling: A unified approach

Biphasic responses are encountered at all levels in biological systems. At the cellular level, biphasic dose-responses are widely encountered in cell signalling and post-translational modification systems and represent safeguards against over-activation or overexpression of species. In this paper we provide a unified theoretical synthesis of biphasic responses in cell signalling systems, by assessing signalling systems ranging from basic biochemical building blocks to canonical network structures to well-characterized exemplars on one hand, and examining different types of doses on the other. By using analytical and computational approaches applied to a range of systems across levels (described by broadly employed models) we reveal (i) design principles enabling the presence of biphasic responses, including in almost all instances, an explicit characterization of the parameter space (ii) structural factors which preclude the possibility of biphasic responses (iii) different combinations of the presence or absence of enzyme-biphasic and substrate-biphasic responses, representing safeguards against overactivation and overexpression respectively (iv) the possibility of broadly robust biphasic responses (v) the complete alteration of signalling behaviour in a network due to biphasic interactions between species (biphasic regulation) (vi) the propensity of different co-existing biphasic responses in the Erk signalling network. These results both individually and in totality have a number of important consequences for systems and synthetic biology.


Introduction
The enzymatic modification of substrates is the basic building block of cell signalling and cellular biochemical networks [1,2,3]. The basic intuition from the analysis of these systems is that increasing both the concentration of the enzyme performing the modification, and total concentration of the substrate increases the steady state concentration of the modified substrate.
However observing both pathways and networks which are comprised of these building blocks indicates that there may be significant divergence from this basic behaviour. One of the most widespread ways in which this may manifest is through a biphasic dose response [4,5,6,7,8]. Far from being an accidental occurrence, biphasic dose responses have many advantages, including a built-in attenuation mechanism, a cap on the maximal level of output as well as the presence of a zone of "optimal" activation. This is borne out by the fact that these very same implications have important consequences in different contexts.
Biphasic responses (both steady state dose responses and dynamic responses) have been recorded all across the landscape of cellular signaling including in key signaling pathways such as Ras, PI3K/Akt, MAPK/Erk, p53 [9,10,11,12,13,14,15,16,17,18] and gene regulation [19,20,21,22], arising from a number of factors including specific network motifs [23] and protein structural factors [24]. Biphasic responses also leads to biphasic regulation in networks. As examples of biphasic regulation in networks, protein modification of myosin light chain (phosphorylation) is regulated biphasically by p21-activated kinase [25]; Lysophosphatidylcholine, can induce biphasic regulation of NF-kB activity through a PKC dependent pathway in human vascular endothelial cells [26]. Sometimes, the biphasic dose-response behavior is only present under certain conditions or in the presence of a different protein. For example, corticosteroids have been shown to biphasically regulate surfactant protein A mRNAs in human lung cells, in the presence of bibutyryl cAMP [27]. Tyrosine phosphorylation of STAT3 protein, in the presence of SOCS3, is biphasic in response to continuous IL-6 signaling, and monophasic without SOCS3 1 [28]. A recent study focusing on PKA and Erk regulation has identified how they both vary biphasically to cAMP levels (with opposite phases) [29]. Further more, multiphasic responses have been observed experimentally in other contexts in plant biology [30]. Biphasic responses have been documented and discussed in a wide-array of contexts under the umbrella of hormesis [31].
Biphasic responses in signaling are often desired, and can serve as protection against over-activation, overexpression and mutant takeover [32,6]. In other instances, this behavior is undesired and can result in inadequate and/or unexpected signalling [33,31]. All in all, given the diverse contexts in which it is encountered, the different ingredients which give rise to it, and the diverse perspectives from which it is studied and is of interest (natural/engineered biology, desired/undesired behaviour, key behavioural signature), a broad-based systems study of this behaviour, and the ingredients either enabling or precluding it is vital.
A typical way in which biphasic responses in cell signalling networks are studied is to investigate network motifs which generate such behaviour [23,34,35,36]. In this paper, by contrast, we investigate the sources of biphasic dose responses, by starting with the basic building blocks of substrate modification, which represent the most basic elements of signalling and post-translational modification networks. By investigating a series of basic building blocks, we reveal the potential for basic modifications to intrinsically give rise to biphasic dose responses, either in response to enzyme concentration or substrate concentration variation. In so doing we obtain a unified synthesis of biphasic responses in response to both enzyme and substrate variation (which could represent safeguards against protein overactivation and overexpression respectively). We then focus on the network level, building on these results to investigate how these sources of biphasic responses intrinsic to chemical modification affect network behaviour. The study is rounded out by an analysis of a concrete exemplar system (Erk signalling), leading to a re-evaluation of existing hypotheses and assumptions complemented by a unified synthesis of biphasic responses here.
Biphasic responses are encountered at all levels in biological systems. In this paper we address the question: what types of biphasic responses are encountered in cell signalling, what are the factors which contribute to this widely observed behaviour and what are its consequences? Our unified analysis reveals that the propensity for different types of biphasic responses (with respect to signal, and with respect to protein amounts) is woven into the fabric of cellular biochemical networks, and has a number of non-trivial implications for overall network behaviour.

Results
From a dose-response perspective, enzyme mediated protein (substrate) modification has a natural response (concentration of the modified substrate) and two natural doses: 1. The total enzyme amount, and 2. The total substrate amount. In this paper, using the exemplar of phosphorylation for such protein modifications, we study the capacity of networks and systems involving enzyme mediated protein modification to show biphasic dose responses in the concentration of the (maximally) modified substrate form with each of these doses. The biphasic dose responses in the (maximally) modified substrate with increasing total amounts of substrate and enzyme are henceforth called substrate biphasic and enzyme biphasic responses respectively.
The organization of the results reflects the multi-pronged nature of our analysis ( fig. 1). We first focus on the biochemical modification level and analyze the capacity of simple modules of enzymatic substrate modification (such as the covalent modification cycle and basic extensions thereof) to exhibit biphasic dose responses. Following this, we focus on the network level and analyze the consequences of such dose responses for network behavior by analyzing simple network motifs with biphasic interactions. Finally, we dissect different aspects of biphasic responses in a specific signalling system: ErK regulation, which is a well characterized cellular pathway shown to present biphasic dose response behavior [37].
Our results involve a combination of direct demonstration of the presence of biphasic responses as well as an unambiguous ruling out of biphasic responses for different systems and different doses. The presence of a biphasic response is demonstrated computationally by performing a one-parameter bifurcation analysis (varying the relevant dose: total amounts of substrates or enzymes). In each of the cases studied, we use analytical work (see models and methods) to ascertain the presence or absence of biphasic behavior in the model. Analytical work is used to reveal the conditions (intrinsic kinetic parameters and total amounts of substrates and enzymes) under which a given system can exhibit biphasic dose responses. In other cases, it is used to show how the behavior is impossible in the system for any choice of kinetic parameters and total species amounts. A discussion of the different classes of parameters (intrinsic kinetic parameters, total species amounts) is provided in the Models and Methods.
All in all, our approach presents a unified analysis of biphasic responses in signalling systems, by assessing both enzyme and substrate biphasic responses, examining signalling networks from three complementary perspectives (biochemical building blocks, network topologies, concrete exemplars) and in all cases uses analytical and computational work to obtain both structural and parameter dependent insights.
The modification biochemistry level. Our analysis begins with the modification biochemistry level. We explore a suite of basic systems containing a broad spectrum of basic modification systems involving enzymes and substrates (single modification, multiple modifications either of the same substrate or different substrates in a cascade, distinct/common enzymes and distinct/common substrates) to draw a number of basic insights. In all these cases, we employ broadly-used models explicitly describing the elementary steps of enzyme-substrate modification (binding, unbinding, catalytic conversion) and make no a priori assumptions on kinetic regimes.
The covalent modification cycle with additional complex formation is capable of enzyme biphasic responses. 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). 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 fig. 2B). Some aspects of the capacity of this system to exhibit enzyme biphasic dose responses have been studied earlier (albeit with different modelling assumptions by Varusai et al., [35]) and also as part of analyzing multispecific interactions [38]. 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).
Multisite substrate modification and commonality of enzyme action promotes biphasic dose responses. 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 fig. S5). 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 fig. 2C) [5]. 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 (k 1 k 3 − k 2 k 4 > 0) is satisfied by the kinetic catalytic constants (k 1 , k 2 associated with phosphorylation, 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 et al., [39] 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 behaviours is guaranteed to be observed across intrinsic kinetic parameter space.
Additional specificity of enzymes in DSP reduces capability of biphasic responses. 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 fig. S2), 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 phophatase (model M4) acting on each modification site is capable of only exhibiting enzyme biphasic responses, and only with the variation of total amount of the kinase effecting the second modification (see fig. 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 fig. 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).
Taken together, this shows how complexity at the level of the biochemistry of a node in a signalling network (arising from 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).
Enzymatic modification cascades and pathways. 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 [40]. This generates an enzymatic modification cascade structure (such as the MAPK system [41,42,43,44]). 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 fig. S5).
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 fig. 2G). Associated kinetic constraints are discussed in supplementary file 1).
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 [45,46,47,48]. 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 fig. 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 phosphatasemodel 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).
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 fig. S1).
The network level. We now shift our perspective from the level of modification systems to the network level. From our studies above, we find that biphasic dose responses are readily seen in basic protein modification systems in cell signalling, requiring only minimal ingredients (enzyme sharing, sequestration and multiple modification) which are widespread and readily encountered in cellular systems. Cellular signal processing is largely dependent on networks involving protein modifications and thus the consequence of biphasic response at a node within a network can fundamentally alter network response. In addition to this, the widespread occurrence of biphasic responses (hormesis) at many levels is well-documented. These observations lead us to examine the consequences of such biphasic responses at the network level, by exploring the implications of biphasic interaction/regulation patterns. All in all, biphasic responses at the network level could be generated by (a) network topologies which facilitate this (b) complex biochemistry at a single node (as seen above) or (c) complex biochemistry (or implicit network structures) which may give rise to biphasic responses in a species not explicitly present as a node. It is this last case which we examine first, before turning to the other two cases.Here in effect, the problem reduces to the incorporation of the biphasic response in a network label/arrow (i.e. interaction pattern). For all relevant models, and how biphasic interaction/regulation is described, please see Models and Methods.
Biphasic interactions within signal regulation. 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 fig. 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 response 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.
Impact on network motifs. To explore this interplay further, we consider standard signaling motifs (feedback and feedforward) with biphasic interactions between the nodes (See models N3-N5 in fig. S5) 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 [35,23]. 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 multiphasic response, and 3. The two distinct biphasic responses can cancel out resulting in a simple monotonic dose response (see fig. 3D).
Feedback motifs. The positive feedback network motif, characteristic of reinforcement, is known for introducing multistability [49,50]. Biphasic interaction between the nodes in such a network motif is capable of destroying such multistability arising from the feedback structure (see fig. 3E). Similarly the negative feedback network motif, characteristic of homeostatic response [51], can have this behavior impacted with the introduction of combinations of behaviors including homeostasis, multistability and monotonic responses (see fig. 3F). Particularly notable is the fact that multistability (usually associated with positive feedback) emerges from a negative feedback network in this case.
The subversion of expected behavior and introduction of new behavior is seen even when the motifs involve open systems (see fig. S3). Analytical work reveals how with 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).
Impact on homeostatic regulation. 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 fig. S4) (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 (eg. inferring networks from data), its information processing capability and the role of its constituent elements.
Complex signal regulation and interaction of biphasic responses from distinct sources. 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 behaviour (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 signal regulation of DSP (with common kinase and phophatase). 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 fig. 3B,C).
We incorporate these aspects in simple models N1 and N2 (see fig. S5); In the former instance (model N1, fig. 3B) the biphasic response is inbuilt into the signal regulation activating kinase, where as in the latter (model N2, fig. 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 ( fig. 3B), when the DSP presents biphasic responses independently. When the biphasic behaviour at the network level arises out of the incoherent feedforward regulation ( fig. 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 signalling interaction, or arising from network structure) can lead to a plethora of unexpected and non-intuitive signaling behavior.
Erk Regulation. 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 [6].
Our analysis reveals that the Erk pathway is capable of exhibiting both substrate and enzyme biphasic responses in the maximally modified substrate form (see fig. 4A,B). The system is in fact capable of biphasic dose response with both doses simultaneously, for the same underlying kinetic parameters (see fig. 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 significant robustness in this behaviour, as intrinsic kinetic parameters are varied, something 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 fig. 4. The consequences of these results for both systems and synthetic biology are discussed in the conclusions. We note here that the the Erk system represents examples of biphasic interaction and regulation in cell signalling, both for its regular downstream pathways, as well as pathways it interacts with as a consequence of cross-talk.
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.

Discussion
Biphasic dose responses are ubiquitous at all levels of biology, and in cell signalling and biochemical networks in particular. In this context, the overall attenuation with dose suggests beneficial safeguarding mechanisms which limit and set bounds on output and are associated with "optimal" ranges of activation. In other instances the departure from monotonic dose-responses may be undesirable (particularly in pharmacology and drug treatment) [52,53,54]. Given the widespread occurrence of such responses, we aimed to study this with a unified approach (both enzyme and substrate biphasic responses), at different levels (biochemical modification, network, specific exemplar systems) using analytical and computational approaches to unambiguously establish the presence or absence of these responses. This provides a unified synthesis of different types of biphasic responses in cell signalling and post-translational modification systems, their diverse origins and consequences. Our focus was on how such responses could be obtained from the most basic aspects of intrinsic kinetics of substrate modifications and network organization, complementing studies of other sources of these responses (scaffolding, protein structure modification, product inhibition [4,24]).
Our analysis focused on a suite of commonly occurring substrate modification systems, using broadly employed models, and revealed explicitly which systems were capable of generating biphasic responses. By focussing on substrate modifictaion systems which are building blocks for cell signalling and post-translational modification, we are able to establish clear boundaries and transitions between the non-occurrence, occurrence and widespread occurrence of different biphasic responses arising from the intrinsic kinetics, and the structural features responsible for each of these possibilities.For these systems some exhibited biphasic responses only to the enzyme, while others did so only to the variation of the substrate amount. Other systems exhibited biphasic responses to both substrate and enzyme, this representing a robust multi-pronged in-built safeguarding mechanism. For some systems, we were able to demonstrate that biphasic responses are precluded, irrespective of parameters. The implication of such a result is that if biphasic responses were observed experimentally, it would have to be due to some other factor, either at the network level (eg. network regulation) or at the modification level (eg. scaffolding).

Drivers of biphasic responses in biochemical systems.
It is interesting to note that enzyme sharing (a simple consequence of the organization of cellular networks with different roles for both enzymes and substrates) promotes biphasic responses, enabling them when they didn't exist otherwise. Multisite modification, especially with commonality of enzymes, promotes this as well. Interestingly, a commonality of kinase promotes a substrate biphasic response, while a commonality of phosphatase promotes an enzyme biphasic response in multisite substrate modification systems. A commonality of both kinases and phosphatases thus enables both substrate and enzyme biphasic responses.The effect of these ingredients also depends on the underlying modification systems: for instance a commonality of phosphatase promotes a substrate biphasic response in enzymatic cascades (in contrast to its creation of enzyme biphasic responses in multisite modification systems). Similarly for random (i.e. non-ordered) modification systems, even distinct enzymes performing each modification can generate both biphasic responses. In many instances the biphasic behaviour observed is surprisingly robust, and for any given kinetic parameters, a suitable choice of enzyme and substrate amounts (experimentally accessible factors) guarantees that it will occur.
Biphasic responses amidst other information processing characteristics. Interestingly, the biochemical building blocks we have examined giving biphasic behaviour can intrinsically also generate other complex behaviour such as bistability (see Table in Fig. 4). In almost all cases (except the ordered double-site modification mediated by a common kinase/phosphatase pair), we have found the simultaneous co-existence of both behaviour possible in a dose response. In the context of ordered double-site substrate modification (mediated by a common kinase/phosphatase pair), we uncover an unusual instance of a system where one of a suite of non-trivial systems behaviours (in this case biphasic responses and bistability) is guaranteed across the intrinsic kinetic parameter space, arising from the largely complementary parametric requirements for each of these behaviours.
Biphasic regulation and interaction in networks. We then investigated the impact of such biphasic responses in networks by associating certain interactions (which may contain multiple species or subnetworks implicitly) with biphasic responses. We find that biphasic interaction patterns can completely abolish expected behaviour in networks, and generate unexpected new behaviour which may not be characteristic of the network (eg. bistability in negative feedback networks). It can also significantly impact expected functioning of networks in different ways (exemplified by its impact on homeostatic control networks). We also explored the interplay of two sources of biphasic behaviour: the network regulation and substrate modification and show that this could lead to reinforcement, cancellation, or even multiphasic or other (eg multistable) responses. Consequently the presence of biphasic responses has an even greater import on the behaviour of signalling networks than may be expected from basic considerations.
Methodology of analysis and testable predictions. Our analysis makes a number of predictions regarding the absence, presence and robust occurrence of biphasic responses in a variety of biochemical modification systems, as well as consequences of biphasic regulation in networks. This results in a number of experimentally testable predictions for example, the realization of enzyme or substrate biphasic responses in different systems (including Erk signalling) which can be obtained by suitably varying enzyme and/or substrate amounts, or the generation of unexpected network behaviour as a consequence of biphasic regulation. These could be tested directly in the relevant system (eg. Erk or other basic modification system) using experiments both in cells and in testtube settings. The consequences of biphasic regulation in networks could be tested using networks containing biphasic regulation, which could be engineered (and tuned) synthetically. Our methods of analysis, combining both analytical and computational analysis could also be deployed for deconstructing and analyzing the possible occurrence and robustness of different biphasic responses in other systems comprising network regulation and substrate modification systems (eg. p53, Ras, MAPK pathways).
We now discuss the relevance of our results for both systems and synthetic biology Systems Biology.
Basic cellular biochemical building blocks feature in-built safeguards against protein overactivation and overexpression. The first basic observation is that many ubiquitously occurring structures present either enzyme biphasic or substrate biphasic behaviour, and in some cases both behaviour. 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 [55]. 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 signalling networks, beyond network regulatory structures. As an illustration, we have investigated a single tier of the Erk signalling network and demonstrated how this exhibits robust substrate biphasic responses (without restriction on intrinsic kinetic parameters, in contradiction to the picture suggested in [6]) and can also exhibit enzyme biphasic responses in broad parameter ranges. This illustrates how a single level of a specific signalling 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.
Inferring drivers of biphasic response in networks. The fact that there are multiple intrinsic sources of safeguarding and biphasic responses in signalling networks (i.e. even intrinsic to a node in these network), 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 eg. observed biphasic responses) and suggests that the inference process must be able to account for different sources of such behaviour. 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.
Consequences and implications for network behaviour. 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 lead to biphasic interactions between network nodes and this can significantly alter network behaviour, even resulting in behaviour which might be expected from a very different network structure. Biphasic responses can be used in networks to access specific downstream behaviour in specific ranges of inputs. As such it can combine with complex dynamical behaviour (eg. bistable switching, oscillations) in different ways giving rise to behaviour which may appear unintuitive, if the presence of biphasic behaviour is not recognized.
The presence of safeguarding mechanisms/biphasic behaviour at different locations in a network have important consequences for the overall robustness of the network as well as the maintenance of behaviour within bounds. Our results revealing how biphasic responses at different levels (eg 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 behaviour) determines the overall safeguarding behaviour in the network and keeps relevant species behaviour within suitable bounds is a topic of future investigation.
Biphasic responses: matching capacity with desired functionality. 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 have 4 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, Fig. 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, 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 fig. 4). This results in 16 cases, 4 of which correspond to both desired outcomes being met, 4 of them corresponding to both desired outcomes not met, and 8 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 over-activation, including the extent to which intrinsic capabilities for such responses have been used. This needs to be evaluated on a case by case basis.

Synthetic Biology.
Structural and parameteric analysis of a variety of systems transparently reveals engineering design principles.The results have further specific consequences for synthetic biology. Biphasic responses have been generated from synthetic gene regulatory incoherent feedforward circuits [56] and synthetic microbial communities [57]. There have been multiple recent efforts to engineer biochemical circuits to obtain complex information-processing behaviour [58,59,60] and these include engineering substrate modification as well [61,43]. 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 behaviour, and in some cases the widespread occurrence of such behaviour, 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 over expression of specific substrate forms. Interestingly, in addition to our analysis of building block circuits, our analysis of a key layer of Erk signalling reveals such robust safeguarding mechanisms, making this an appealing candidate for incorporating into synthetic circuits.
A comparative analysis of different systems reveals benefits and tradeoffs for design. 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 behaviour, 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 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. Engineering network behaviour.The analysis here can also be used to shape overall network/circuit behaviour. 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 behaviour and create different safeguarding mechanisms against over expression in networks. Interestingly some preliminary analysis [36] 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.

Models and Methods
Systems considered. The systems considered and investigated in this paper span a range from basic biochemical building blocks to basic network motifs. With regard to basic biochemical building blocks, we examine a suite of basic biochemical systems involving enzyme mediated protein modification, all of which build on basic covalent modification cycles. Thus we consider covalent modification cycles (both basic cycles and those with"dead-end" complexes), double site modification systems (ordered and distributive) where different (de)modifications are effected by either the same or different enzymes. We also consider two-step enzymatic modification cascades where the phosphatases in the two steps may be either the same or different. We also examine coupled modifictaion cycles involving shared enzymes. Taken together, these system (models M0-M7) represent a suite of basic biochemical building blocks (incorporating different kinds of modification "logic") and their extensions, which are repeatedly encountered in biochemical pathways. It is worth pointing out that by considering these various building blocks (themselves, amongst the simplest extensions of covalent modification cycles), it is possible to obtain transparent insights (arising from the most basic considerations of the kinetics) into key structural features enabling different kinds of biphasic responses (see Fig. 2).
Building on this, we also consider simple signaling network motifs (feedforward and feedback networks). Here the goal is to explore how biphasic regulation which may be present within the network (due to regulatory steps implicit in the network) may impact network behaviour. Thus, we examine these motifs with biphasic interaction between nodes of the network motif (models N0, N3-5). Finally we also examine cases which bring elements of network regulation along with the biochemical protein modification systems studied above, together. To do this we focus on the case of double site protein modification system, and examine both biphasic regulation of the kinase, as well as incoherent feedforward regulation (via incoherent feedforward networks). These are examined in models N1 and N2.
Complementing the above systems, we study a biochemical exemplar system, the Erk regulatory pathway (model adopted from [6,62]). A summary schematic of the various models used is presented in fig. S4.
Mathematical modelling. We first describe the modelling of the basic biochemical building block systems considered above. The systems considered above are modelled as a system of ordinary differential equations using simple mass action kinetic description for the elementary reactions. Each (de)modification step involves three elementary reactions, the reversible binding (and unbinding) of the free active enzyme with the substrate to form an enzyme substrate complex, followed by the irreversible catalytic reaction to release the free active enzyme and the (de)modified substrate (see fig. 1). An abundance of ATP is implicitly assumed. Using such a description for every (de)modification step, the mathematical model for the biochemical systems are constructed. Since the total amount of substrate and enzymes are also conserved in the system, the ODEs are also associated with the respective conservation equations. We note that the models of the biochemical systems generated, represent widely employed models of such systems. We do not incorporate other effects such as product inhibition or scaffolding.
We present below how the covalent modification cycle (where a substrate gets modified and demodified by a kinase and a phosphatase respectively; model M0) can be described mathematically using such a description. The mathematical models for the other systems considered are constructed in a similar manner and simply involve a combination of such descriptions for individual modification/de-modification steps In using simple mass action kinetics to represent the kinetics of the individual elementary steps (which are part of an overall modification step), we make no a priori assumption regarding the kinetic regime of the enzyme action, and thus the results are broadly representative of the overall behavior of the system. The computational models created in MATLAB are cross-validated with model descriptions generated in COPASI [63] (which only requires definition of the reaction schematic for model generation).

Modelling biphasic interactions between nodes in a network. The biphasic interactions between nodes A → 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 k 1 [A]e −k2[A] . This is in contrast to a basic (monophasic regulation) for instance by a linear function (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 (k 1 and 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.
Parameters. The models constructed have a number of parameters, involving both kinetic constants and total amounts of substrate and enzymes. It is useful to draw a distinction between these two classes of parameters for multiple reasons. 1. The total species amounts represent easily accessible levers that can be tuned, while the kinetics are intrinsic to the system and generally less accessible. 2. In the context of biphasic dose responses, the total amounts represent doses to the system and thus are natural experimental factors that change, and consequently are basic parameters with respect to which which bifurcation analysis is carried out. 3. In many systems (eg. double site phosphorylation) the possibility of different behaviour (eg. bistability) is actually determined by constraints on intrinsic kinetic parameters, and once these constraints are satisfied, it is possible to vary substrate and enzyme amounts suitable to ensure that the desired behaviour is realized. Thus there is a hierarchy in the set of parameters to enable different information processing behaviour.
Doses. Enzyme mediated protein modifications have two natural doses (with respect to the maximally modified substrate form), i.e. 1, Total substrate amount, and 2. total enzyme amount.
Approach. The approach to assessing biphasic response in the networks is as follows.
We begin by analytically characterizing the presence or absence of the biphasic behavior for the relevant substrate concentration with the dose, and in the process obtain binary answers to the feasibility of observing the behavior in parameter space for a given model. In the instance where the behavior is shown to be present, we show evidence of it computationally with a 1-parameter bifurcation analysis with the relevant dose. We further supplement this with analytical work to establish sufficient and necessary conditions (involving parameters, such as total amounts and kinetic constants) that guarantee the presence of biphasic dose response behavior. When the behaviour is not possible we analytically establish that this is the case irrespective of parameters.
Bifurcation analysis. 1-parameter bifurcation analysis was carried out computationally using the MatCont package in Matlab [64]. 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 approach to assessing biphasic response. 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 d[V ar] 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 ( d[V ar] 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 [65], 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 focusses 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.
Code availability. The custom code used in this manuscript is available at: https://github.com/VaidhiswaranR/ Work-in-progress--Biphasic2022

Summary of Analytical Approach.
In this subsection we provide a birds-eye view of the analytical approaches used in the various models. At the outset we reiterate that our goal is to explore the presence or absence of both substrate and enzyme biphasic responses. In all the models, the focus is on a key substrate variable, as discussed in the Results. In all cases, the models involve solving for the steady states of the ODEs with appropriate conservation conditions. Typically each distinct type of enzyme and each set of substrates is associated with their conservation condition.
We will outline the approach in the case where there is a single kinase enzyme and a single set of substrates (examples of this include the simple covalent modification cycle, the protein-protein interaction model, the double site modifcation system with common kinase and common phosphatase and the double site modification system with common kinase and separate phosphatases). This is because the nature of the analysis is most transparent here. We then discuss how other cases involving more kinases and/or sets of substrates are studied.
The analysis progresses by a sequential elimination of variables at steady state starting with the concentrations of the various complexes in terms of concentrations of the constituent enzyme and substrate, and then writing other substrate concentrations in terms of the concentration of the substrate of interest. Then after exploiting any phosphatase conservation conditions, the steady state of the system can be determined from the two remaining conservation conditions A con = 0 (substrate conservation condition) and K con = 0 (kinase conservation condition). Note that the way the conservation condition is written, the total amounts of substrate A tot and kinase K tot are present in these expressions, linearly with coefficient -1.
If we denote the substrate variable by A var and the kinase variable by K var , then the equations take the form A con (A var , K var , A tot ) = 0 and K con (A var , K var , K tot ) = 0. Now suppose an enzyme biphasic exists, then we require that dA var /dK tot = 0 at the biphasic peak. Differentiating both the conservation equations with respect to K tot results in two equations ∂Acon Our analysis is based on examining whether such a pair of equations can be satisfied, with the requirement that dA var /dK tot = 0. The analysis, carried out in the Maple files in detail, proceeds as follows. We establish that the terms multiplying dA var /dK tot in these two equations are necessarily finite (since these are rational expressions where the denominator is shown to be non-zero), and so when dA var /dK tot = 0, the associated terms can be eliminated. We are then left with two equations ∂Acon ∂Kvar ∂Kvar ∂Ktot = 0 and ∂Kcon ∂Kvar ∂Kvar ∂Ktot = 1. Now there are different possibilities. 1. ∂Acon ∂Kvar is shown to be necessarily non-zero (via a symbolic computation). This means that ∂Kvar ∂Ktot = 0. This however contradicts the second equation, since it can be shown that ∂Kcon ∂Kvar is finite. Thus the assumption of a biphasic response and biphasic peak is incorrect. This establishes that enzyme biphasic responses in this instance are ruled out. 2. ∂Acon ∂Kvar can be zero. In this case, the possibility of the above contradiction is averted. There are different possibilities here (note that in all such cases, the problem reduces to the possibility of a polynomial expression in two variables being zero). (a) Structurally the polynomial expression contains two sets of terms with different signs. In this case, we establish that one can find suitable values for A var and K var so that a solution exists. There are ways to directly establish that a suitable solution exists for some values of these two variables. In such a case, we then use the conservation conditions with these values of A var and K var to establish that one can find suitable total amounts of substrate and enzyme A tot and K tot to ensure this happens. The conclusion is that for the given intrinsic kinetic parameters, one can always find total amounts of species (Substrate, enzyme) to ensure that a biphasic response is possible (b) Structurally, the polynomial expression does not necessarily separate into two sets of terms with different signs. However it may do so, for certain ranges of kinetic parameters. In such a case, we establish that the equation can separate into two sets of terms with different signs for some regimes of intrinsic kinetic parameters. This then can be used to deduce the fact that a feasible solution of this equation exists in this regime of kinetic parameters. The upshot of this is that a biphasic response can be observed in certain ranges of intrinsic kinetic parameters, again by a suitable choice of total enzyme and substrate amounts.
Comments. 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. Firstly, 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 A var and 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 A var is the concentration of the substrate variable of interest. while the variable 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 increase. 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 analogue 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.  Figure 1: Biphasic dose responses in basic building blocks in cell signalling: from enzyme-mediated substrate modification systems to network motifs/modules Reversible enzyme mediated substrate modification is an integral part of cellular signalling and is among its basic building blocks. Top left shows the basic mechanism of substrate modification by an enzyme involving the following steps: the binding (and unbinding) of the substrate (S) with the enzyme (E) to form a substrate-enzyme complex (ES), which finally disassociates to give the modified substrate (S 1 ) and the free enzyme (E). The reverse modification is not depicted for conciseness. The output of the system is the concentration of the modified substrate S 1 .There are two natural doses associated with this -total substrate amount and total enzyme amount (S 0 and E 0 respectively). The potential dose-response behavior in the simplest instance can thus be either a) monotonic or b) exhibit biphasic dependence of the modified substrate with the dose. Both substrate and enzyme doses can potentially elicit biphasic responses. Top right shows the schematics of commonly encountered substrate modification building block systems in cell signalling. Each arrow denotes a modification and involves binding, unbinding and catalytic action by the enzyme. Arrows of the same colour indicate action by the same enzyme type (a kinase and a phosphatase for example). Bottom left These systems form key components of wider signalling networks in the cell. From a network perspective they can be incorporated in two ways. 1. They can either be explicitly described, for example as part of a node or 2. be implicitly present within an interaction between nodes (implicit in an arrow, which can be the sum of actions of species not explicitly defined). A fruitful way to study signalling networks is through the study of recurring signalling motifs (with characteristic input-output responses) such as feedback and feedforward structures. Bottom right lists canonical network motifs (positive and negative feedback, and coherent and incoherent feedforward, integral feedback control) that we study in our work. We complement our study of biphasic responses in biochemical building-block systems and network motifs by studying biphasic responses in a concrete exemplar system, the ERK signaling pathway. Detailed mathematical model descriptions of all the systems studied are discussed in the supplementary material, and a detailed reaction schematic of the models considered is shown in fig. S5.      Figure 4: Presence and consequence of biphasic dose responses in cellular signaling arising from enzyme mediated substrate modification: Top left hand panel: Schematic representation of the study. The basic mechanism of enzyme-mediated substrate modification (when present in different reversible modification cycles) allows for simple biochemical systems to present biphasic dose-response behavior. This is analyzed in a series of commonly encountered biochemical building blocks of cell signalling (M0-M8, see fig. S5 for detailed schematic). In each case, we assess whether the system exhibits biphasic behaviour with respect to variation of total amounts of enzyme and substrate as doses, and if is does so robustly. The table below, provides a classification system to determine the different possibilities of biphasic responses (enzyme/substrate) vis-a-vis the desirability/undesirability of biphasic responses. This provides a framework (in synthetic biology) for determining what changes need to be made to a system, to obtain desirable characteristics. We then extend the study to consider the consequence of such biphasic dose responses in network motif structures (either within nodes, or in interactions-coloured in red) and study how biphasic interactions can alter expected outcomes from these systems (N1-N5). Top right hand panel: Case study: Extracellular signal-regulated kinase (Erk) Regulation. The ERK regulatory network is capable of both exhibiting substrate and enzyme biphasic response in the double phosphorylated Erk (pYpTErk) in the steady state dose response with M ek T otal (Enzyme biphasic) and Erk T otal (Substrate biphasic). A-B shows an instance of enzyme and substrate biphasic response, respectively. C shows how the steady state concentration of pYpTErk changes with M ek T otal and Erk T otal (for a fixed kinetic parameter set). This indicates the presence of simultaneous substrate and enzyme biphasic response in the system (red and blue lines indicate the peak concentration of pYpTErk achieved for a given amount of total substrate and total enzyme amounts respectively). Bottom panel. Table: Detailed summary of the various results discussed in the manuscript, including the possibility and impossibility of specific dose-response behaviors in a model, and the different ways in which expected outcome of a network motif structure is undercut by biphasic responses within interactions. Note: These results discuss analytical results which can be found in supplementary file 1.  Figure S1: 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 coupled covalent system (with common kinase) while capable of substrate biphasic is incapable of multistabilty. A. 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 phoshpatase. 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.  Figure S2: Biphasic responses in random double site modification network with separate enzymes effecting each modification. While the ordered 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 propensity for such networks containing them to generate biphasic dose response.  Figure S4: 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 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 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). Further, the steady state of the system can loose stability even for lower signal ranges (on the existing stable branch), due to oscillations.  Figure S5: Schematic representation of the various models considered in the manuscript. Models M0-M1 represent the basic unit of (de)modification, the covalent modification cycle and its simple extension with the substrate forming an additional complex with the enzyme. Models M2-M5 represent different systems of ordered distributive double site modification of a substrate (by different or common enzymes acting on each modification site). Models M6-M7 represent other architectures of modification systems in chemical reaction networks, such as cascaded system (with different and common enzymes affecting (de)modification) and coupled covalent modification networks (coupled through enzyme sharing in (de)modification). Models N0-N5, represent different networks and network motifs with biphasic dose-response subsumed in an interaction. A Enzyme biphasic dose response in PPI model k 1 = 1; k 4 = 1; k b1 = 1; k b2 = 20; k b4 = 1; k ub1 = ; k ub2 = 1; k ub4 = 1; A T otal = 30; P T otal = 1;