## Abstract

Multi-site phosphorylation systems are repeatedly encountered in cellular biology and multi-site modification is a basic building block of post-translational modification. In this paper, we demonstrate how distributive multi-site modification mechanisms by a single kinase/phosphatase pair can lead to biphasic/partial biphasic dose–response characteristics for the maximally phosphorylated substrate at steady state. We use simulations and analysis to uncover a hidden competing effect which is responsible for this and analyse how it may be accentuated. We build on this to analyse different variants of multi-site phosphorylation mechanisms showing that some mechanisms are intrinsically not capable of displaying this behaviour. This provides both a consolidated understanding of how and under what conditions biphasic responses are obtained in multi-site phosphorylation and a basis for discriminating between different mechanisms based on this. We also demonstrate how this behaviour may be combined with other behaviour such as threshold and bistable responses, demonstrating the capacity of multi-site phosphorylation systems to act as complex molecular signal processors.

## 1. Introduction

Cells process information, respond to their environments and regulate various internal processes by means of complex protein and genetic networks. A basic and ubiquitously occurring element in such networks is a covalent modification cycle, wherein unmodified and modified forms of a substrate are reversibly interconverted by a pair of enzymes. The covalent modification cycle is one of the basic building blocks of post-translational modification and signal transduction. Multi-site phosphorylation systems represent a particular extension of the basic covalent modification cycle wherein the same pair of enzymes (kinases and phosphatases) may provide multiple modifications for the basic substrate. Thus, this may be seen as a concatenation of covalent modification cycles with the same pair of enzymes. Such multi-site modifications are widespread in cellular biology with multi-site modifications playing important regulatory roles in many contexts and in many cell types [1,2]. They are believed to play important roles in inflammation pathways and are implicated in important disorders [3–5]. The multi-site modification status of a protein has been referred to as a dynamic ‘molecular barcode’, controlling signal transduction and downstream effects.

In this paper, we focus only on the case where modifications are effected by only one enzyme pair, whereas the ‘molecular barcode’ usually refers to modification by multiple enzyme pairs. While the same enzyme pairs may be involved in multiple modifications, there are qualitatively different molecular mechanisms by which such modifications may be effected. Thus, multi-site phosphorylation systems may be classified as distributive, wherein each modification requires a new enzyme binding event, and processive where a single binding can lead to a sequence of modifications [6]. Mixed mechanisms that combine features of these two scenarios are also observed [7]. Even within distributive mechanisms, multiple variants such as ordered [8,9] and random [10–12] mechanisms exist. These differences arise from what kinds and sequences of modifications are permitted: an ordered mechanism allows only a specific order in the modifications, whereas a completely random mechanism allows any sequence of modifications.

Given their importance, multi-site phosphorylation systems have been the focus of a number of experimental and modelling studies. The experimental studies usually focus on the effect of multi-site modifications in specific contexts such as in cell cycle proteins and inflammation processes [3,13,14]. Because proteins that are modified at specific locations can have different functions, there is naturally a special interest in the details, mechanisms and consequences of these multi-site mechanisms. There is also a considerable body of work on pathways such as the mitogen-activated protein kinase (MAPK) pathway in its various contexts [15–17]. Recent experimental efforts aim to go beyond qualitative descriptions and quantify different modifications of substrates using tools such as mass spectrometry [18,19].

The wide-ranging applications and complexity of multi-site phosphorylation have attracted a range of modelling and theoretical studies. Modelling ranges from studying different aspects of multi-site phosphorylation in individual contexts such as the MAPK and p53 pathways, to examining multi-site phosphorylation in cell cycle control and multi-site modification as a controller for protein degradation [20–22]. Other theoretical studies examine the consequences of different mechanisms, and the signal transduction capabilities of such systems [23]. The ability of such systems to produce threshold and ‘ultrasensitive’ behaviour was the focus of a number of studies [24]. The ability of distributive systems to generate bistability was demonstrated in [25]. Subsequently, it was shown that multi-phosphorylation systems were capable of showing unlimited multi-stability, and that as the number of sites increases, the number of realizable stable steady states also increases [26]. The effects of other aspects such as sequestration were also examined [27,28].

In this paper, we examine the intrinsic capability of multi-site phosphorylation to exhibit biphasic (bell-shaped) and partial biphasic (henceforth referred to as biphasic) dose–response characteristics. To this end, we examine multiple variants of multi-site phosphorylation mechanisms, to examine the mechanisms which give rise to this behaviour. We use numerical simulations, bifurcation analysis and analytical work to understand the origins of this behaviour and the factors which accentuate it. We present the models which we use in §2 and discuss the results in the following section. We then briefly present analytical results in §4 (this section can be skipped by readers not interested in the details). We conclude the paper with a synthesis and discussion of the relevance of the findings. An appendix discusses analysis of different variants of mechanisms in more detail.

## 2. Models

Multi-site phosphorylation mechanisms are present in different contexts in signal transduction, sometimes combined with other factors. In this paper, we perform an *in silico* analysis of the intrinsic behaviour of multi-site phosphorylation mechanisms, and use generic models for this purpose. There are a few variants of multi-site phosphorylation mechanisms that are relevant here. In this paper, we primarily use models of distributive multi-site phosphorylation, and discuss processive mechanisms to serve as a contrast. In examining distributive mechanisms, we note that multiple variants of distributive mechanisms also exist: completely ordered, completely random and other intermediate cases. We examine both ordered as well as random mechanisms. By performing *in silico* analysis on fairly generic models, it is possible to understand intrinsic complexities of multi-site phosphorylation in a controlled manner, disentangling this from other complicating factors.

We present simulations and analysis for ordered as well as random mechanisms and consider double-site modifications for the most part. We briefly consider what happens if more than two different modifications may occur. Our studies are based on ordinary differential equation (ODE)-based models of multi-site modification. These models are based on a deterministic and lumped description of the multi-site modification process. We do this, because the ODE models can be analysed effectively and transparently, and serve as a platform for examining additional issues such as stochastic effects. As the focus of this paper is on a basic steady-state characteristic of multi-site mechanisms, our ODE description suffices.

Our ODE models include standard kinetic descriptions of enzymes reversibly binding to substrates to form complexes, and the catalytic conversion of the complex to the modified substrate. In our basic model, we have an ordered mechanism of double-site phosphorylation by a kinase–phosphatase pair. Thus, the same kinase is involved in both the first-site and the second-site modification, and the phosphatase acts to undo each of the conversions. All the elementary steps are described by mass-action kinetics. The model is thus a mathematical depiction of the scenario depicted in figure 1*a* with the assumption above. In the case of random mechanisms, the kinase may modify the unmodified substrate to two different singly modified substrates (with potentially different rates), and can convert each of these to the doubly modified substrate. The phosphatase can likewise act to reverse each of these modifications. These scenarios generalize in a straightforward way to an arbitrary number of modification sites. Both models represent generic descriptions of multi-site modifications and have been used repeatedly in the literature.

We discuss the model in a little more detail. The equations for a two-site-ordered distributive mechanism (figure 8) are 2.1

In this equation [*A*],[*A*_{p}], and [*A*_{pp}] are respectively the concentrations of unphosphorylated, the singly phosphorylated and the doubly phosphorylated substrate, [*AK*] and [*A*_{P}*K*] describe the concentrations of relevant substrate–kinase complexes and [*A*_{p}*P*] and [*A*_{pp}*P*] describe the concentrations of the relevant substrate–phosphatase complexes. Such a model may be regarded as one describing a mechanism where the order of dephosphorylation is opposite to the order of phosphorylation. The equations for the free enzymes (kinase/phosphatase) are also shown. In the above model, [*K*] and [*P*] refers to the active kinase and phosphatase concentrations, respectively. In general, both kinases and phosphatases are subject to different control mechanisms and either of them may be activated (or made available) to trigger and control signal transduction. In this model, for the sake of simplicity, we avoid an explicit description of this activation and thus the enzymes in our model represent active modification-capable enzymes. It is worth emphasizing that the model is a fairly generic model for describing ordered distributive multi-site phosphorylation. We remark that the behaviour we focus on here is readily observed in more complex models that describe explicit activation of the kinase by the upstream signal.

In this model, the total (active) kinase and phosphatase concentrations are parameters (they are conserved in the dynamical evolution of the system) and likewise the total substrate concentration is also a constant. In an *n*-site-ordered modification system (a generalization of the above model to include *n* modifications), there are thus *n* independent substrate concentrations, along with *n* kinase–substrate complexes and *n* phosphatase–substrate complexes. The random mechanism allows for all possible ordering of modification, and thus accounts for all possible partial phosphoforms and their dynamics explicitly, allowing for different modifications to take place at different rates. This naturally leads to a substantially more complicated kinetic model. In the random mechanism, there are 2* ^{n}* − 1 independent substrate concentrations (for all possible phosphoforms), and an equal number of kinase–substrate and phosphatase complexes. The equations for random two-site mechanisms and ordered

*n*-site mechanisms are shown in the appendix. One may also identify mechanisms which fall between the descriptions outlined above. One such mechanism is a ‘cyclic’ mechanism shown for two-site modification in figure 1. This is essentially an ordered mechanism where the order of phosphorylation and dephosphorylation is not the exact opposite of one another: thus, it is important to explicitly account for the different partially phosphorylated species. The kinetic model of this cyclic mechanism and its analysis is presented in the appendix.

The model of multi-site modification is specified by the choice of mechanism and the parameters. The parameters include total concentrations of substrate and (active) kinase and phosphatase. Because both kinases and phosphatases may be subject to different control mechanisms and be seen in modifying different substrates, it is possible to encounter different relative amounts of these quantities. For purposes of analysis, we start with a scenario where the phosphatase concentration is less that the total substrate (we will examine the effects of varying total phosphatase concentration subsequently). Situations involving relatively limited amounts of phosphatase are seen, for instance, in the case of phosphatase Cdc14 in metaphase yeast cells and (depending on the level of stimulus) in the agonist-induced dephosphorylation of protein kinase isotypes [29,30]. The sharing of the same phosphatase between many signalling pathways, as well as other control mechanisms regulating phosphatase availability/activity [31] could also result in a scenario akin to this. The total (active) kinase concentration is a parameter which we will directly vary in our analysis. With regard to the kinetic parameters, we generally choose kinetic parameters where the enzyme–substrate binding and unbinding rates are equal, and normally choose the catalytic rate to be less than the binding and unbinding rates. This is similar to parameters used in the literature. Our results focus on how biphasic dose–response characteristics may arise by varying certain parameters, and for the most part, we keep the other parameters fixed and in the background. Our results do not depend strongly on the particular choice of auxiliary kinetic parameters.

Having chosen a representative choice of parameters (similar to others chosen in the literature), we examine how the variation of certain fixed parameters may give rise to biphasic behaviour. We use bifurcation analysis, simulations and analytical approaches to understand this effect. Our simulations were performed using both ODE15s in MATLAB and COPASI for comparison. The equations are directly programmed in the former, while generated by the software automatically in the latter. This is complemented by bifurcation analysis in MATCONT. Finally, we use analytical work to examine the effects observed, and how it changes when other parameters are varied and when different enzymatic mechanisms are used. This multi-pronged approach allows us to see how the insights from different approaches add up to a consistent picture.

## 3. Results

Here, we discuss the computational analysis of the model. Complementary analytical work is presented in §4. For the remainder of this paper, enzyme concentrations refer to active enzyme concentrations. We focus on studying the steady-state response of the distributive multi-site phosphorylation system as a function of the total kinase concentration, for the basal set of parameters chosen. We start by analysing ordered multi-site phosphorylation systems, and examine random mechanisms subsequently. For purposes of illustration, we discuss double-site modifications, and briefly examine other aspects associated with the number of sites subsequently.

Figure 2 presents simulations showing how the maximally phosphorylated substrate concentration at steady state varies with total kinase concentration. Figure 2*a* shows that as the total kinase concentration increases, the concentration of the double phosphorylated substrate increases monotonically and asymptotes to its maximum level. The single phosphorylated substrate concentration exhibits a biphasic response: the initial increase at low kinase levels reaching a maximum, before a decrease at higher kinase concentrations. This is simply understood from the fact that the dominant effect at higher kinase concentrations is the conversion of the single modified substrate to the double modified substrate.

We then examine the same system for a case where the catalytic rate of the second phosphorylation step is reduced. In this case, we find (figure 2*b*) that the concentration of the double phosphorylated form increases and appears to plateau off, but actually reduces for higher kinase concentrations. In other words, the maximally phosphorylated substrate exhibits a (partial) biphasic response to the input. This result is interesting as it demonstrates that a simple distributive multi-site phosphorylation mechanism is intrinsically capable of generating such a biphasic mechanism.

Further insights into the origins of this behaviour can be obtained by looking at the variation of the four enzyme–substrate complexes as a function of total kinase concentration. This is shown in figure 3 which demonstrates that while both enzyme–substrate complexes of the first cycle are biphasic as expected, the complexes of the second cycle exhibit a monotonic variation, which saturates at high kinase concentration.

When we examine the trends exhibited by the enzyme–substrate complexes, we see that those of the second are in qualitative contrast to the double phosphorylated substrate. In fact, it immediately raises the question: what causes the concentration of the double phosphorylated substrate to decrease? A closer examination of the results reveals that there are, in fact, two competing effects. An increase in the kinase concentration means more conversion of substrate into the kinase–substrate complex of the second cycle. However, this occurs at the expense of the single phosphorylated substrate. This conversion or ‘sequestration’ of the single modified substrate into the kinase complex now means that there is a reduction in the amount of the single modified substrate for the phosphatase to bind to. In other words, an increase of kinase leads not only to an increase of the *A*_{p}*K* complex which saturates, but also frees up phosphatase which might otherwise also bind to *A*_{p}. This increased availability of phosphatase now means that the extra phosphatase can bind to *A*_{pp} and degrade it more efficiently. This explains the origins of the biphasic behaviour. A close inspection of the graph reveals that the complex *A*_{p}*P* actually decreases quite substantially (in the region where the *A*_{pp} curve turns around), consistent with this picture. It is also worth pointing out that while a (relatively) low catalytic constant in the second cycle causes this biphasic effect, a low binding constant (of kinase to *A*_{p}) does not. This again is consistent with the explanation for the reason of the biphasic behaviour, as a low binding constant does not provide an effective ‘sequestration’ of substrate in the *A*_{p}*K* complex.

Having investigated the origins of this biphasic behaviour, we then study the effects of different parameters on this response. Figure 4*a*,*b* shows the effects of varying the binding and catalytic constants of the first cycle. We find that decreasing each of these can lead to the abolishment of this biphasic response. The reason for this is that when these constants are low (and comparable with the catalytic constant of the second cycle), there is no longer a clear cut sequestration effect of substrate in the *A*_{p}*K* complex as before. On the other hand, increasing the binding constant of the dephosphorylation of the first cycle can strongly accentuate the biphasic behaviour (figure 4*c*) as it only more strongly accentuates the reduced availability of phosphatase. We also find that reducing the dephosphorylation catalytic constant for the second cycle tends to diminish and abolish the biphasic behaviour (reducing the impact of the greater available phosphatase, by having significant phosphatase sequestered in the *A*_{pp}*P* complex). Again, a reduction of the binding constant of the kinase to *A*_{p} tends to reduce or abolish the biphasic behaviour, as it reduces the relative effect of sequestration of substrate (and in particular *A*_{p}) in the complex *A*_{p}*K*.

It is worth pointing out that while we have observed biphasic behaviour by reducing the catalytic constant of the second phosphorylation, it can, in fact, be obtained when catalytic constants of both phosphorylation reactions are equal, but with a relatively high dephosphorylation rate constant of *A*_{pp} (results not shown).

We now investigate the effect of the total phosphatase concentrations. As seen in figure 5*a* an increased availability of phosphatase actually reduces or abolishes this biphasic effect, which is again consistent with the ‘release’ of phosphatase being a role in generating the behaviour mentioned above. It is worth pointing out, however, that if the available phosphatase is too low, then a biphasic behaviour is not seen (figure 5*b*). In fact, our explanation of the biphasic behaviour shows two competing effects, one being the strong enough effect of the release of phosphatase. If this is too weak, then the biphasic behaviour is not seen.

Having understood the reasons for the biphasic behaviour in the ordered double phosphorylation mechanism, we examine whether such behaviour may occur in slightly more complex multi-site modification mechanisms. Figure 6*a* shows the results of analysis of a random double-modification mechanism. Here, the catalytic constant of only one of the end cycles is reduced. We find that the biphasic behaviour is not only present, but in fact enhanced. Again, the effects of varying other kinetic parameters (in the relevant leg of the cycle) in this more complex system are broadly the same as the simpler ordered system. Because this random mechanism (with lowered catalytic constant of one of the second steps) can be thought of as a more complex version of the ordered mechanism, it is worth asking what the role of the extra pathway is. We find that in the regime of kinase concentration where the biphasic behaviour occurs, very little of the intermediate modified substrate in the second cycle exists. Overall, we find from numerical analysis that the biphasic characteristic is more pronounced, because of the presence of an extra pathway for the fully phosphorylated substrate to be unmodified. It is worth pointing out that if the catalytic constants of conversion of both singly modified forms (*A*_{01}, *A*_{10}) are reduced, then we see a behaviour very similar to an ordered mechanism.

We also examined another more complex version of the basic model that we studied: an ordered triple phosphorylation system. As seen in figure 6*b*, we find that this triple phosphorylated system can, indeed, also show the biphasic behaviour. The reason for this is very similar to that of the double phosphorylated system studied above. Furthermore, it is possible under some conditions (for example if the catalytic constants of the forward reaction of the last two cycles were reduced) that the maximally phosphorylated form exhibits a limited biphasic response, which reduces in a certain range before increasing again (figure 6*c*). Clearly, the presence of more sites can lead to more complex behaviour building on the basic mechanism which we analysed in the double phosphorylation system.

Having studied the origins of this partial biphasic behaviour, we asked whether this could be combined with other qualitatively different features of signal transduction in multi-site phosphorylation. Figure 7 shows how the biphasic behaviour can be combined with a threshold behaviour characteristic of such systems, even in double-site phosphorylation systems. This demonstrates how the multi-site phosphorylation system can exhibit a complex dose–response curve comprising both thresholds and biphasic behaviour. Figure 7 also shows how both ordered triple and random double phosphorylation can exhibit bistable behaviour that can be combined with such biphasic behaviour. This clearly demonstrates how tuning parameters in the final cycle giving rise to/accentuating biphasic behaviour can be combined with other intrinsic complex features of signal transduction.

## 4. Model analysis

Section 3 presented an *in silico* analysis demonstrating the presence of hidden competing effects which give rise to biphasic dose–response characteristics. Here, we complement our numerical results with some analytical results that build on and consolidate the insights obtained. We focus on the ordered distributive model for two-site modifications.

It is worth examining a simpler case first. This simpler case is that of the same model where all the catalytic constants are high. In this case, there is negligible hold-up of enzyme in complexes, and each of the modification reactions can be described via mass-action kinetics. The analysis of this situation can be easily performed explicitly, noting that essentially all the enzyme is free (i.e. the free enzyme concentration is essentially the total enzyme concentration) and also that in the ordered model, the steady state involves equilibrium for each modification/demodification cycle. It is easy to see that in this limit leading to a hyperbolic response for the maximally phosphorylated form

We thus see that no biphasic response (as total kinase concentration is varied) occurs in this case.

We will demonstrate the factors responsible for the biphasic response analytically. In order to do this, we note that even the ordered double phosphorylation model is quite complicated to analyse explicitly. Therefore, we focus on some limiting cases which retain the main effects, and simplify others. Thus, we assume that the first phosphorylation reaction occurs via mass-action kinetics (high catalytic constant, and hence negligible kinase hold-up in the complex) and make the same assumption for the dephosphorylation reaction for the second step. We thus focus on a slightly simpler situation, which can be analysed more transparently.

Analysing the steady state of the system results in the conservation conditions
and
along with the steady-state equations
4.1After some algebra, this results in the steady state
4.2where the various coefficients in the equation for [*A*_{pp}] can be obtained explicitly in terms of kinetic constants (see appendix). Note that if the second and fourth terms in the denominator were absent, then this would reduce to a form similar to the case considered above.

Equation (4.2) reveals the presence of the competing effects. First, we see that the free enzyme concentrations are not constant (as the parameter *K*_{tot} is changed) and depend on concentration of the partial phosphorylated species. Further by looking at the denominator of the expression for [*A*_{pp}] we can see the competition between the second and third terms. As the total kinase concentration increases, the concentration of the partial phosphorylated species decreases eventually increasing the free phosphatase concentration. The free kinase concentration increases both because the total kinase concentration increases and because the partial phosphorylated species concentration decreases. We thus see that the second and third terms in the denominator are in competition, and that it is possible in certain parameter regimes that the increase of the free phosphatase is the dominant contribution, thus decreasing the overall concentration of *A*_{pp} (the last term in the denominator is small relative to the others in this regime). This is in agreement with the intuitive explanation provided earlier, providing a more explicit demonstration. Further, we see from the analytical expression that this arises only when there is significant sequestration of substrate in the *A*_{p}*K* complex. Finally, a relatively small catalytic constant for the second phosphorylation (*k*_{6}) has the effect of increasing the relative effect of the free phosphatase as the total kinase concentration becomes high. This is because *α*_{1} is inversely proportional to *k*_{6} (see appendix). While *α*_{2} also increases, the effect is offset by a high free kinase concentration.

Taken together, our analysis shows how such a biphasic effect arises when one accounts for significant substrate sequestration in the complex. It is worth emphasizing that while such an effect gives rise to biphasic behaviour in multi-site modification, such behaviour will not be seen in single modification cycles. This is shown analytically in the appendix. Thus, we see that the biphasic behaviour relies on a combination of multiple modifications and certain reactions where the kinetics is far from mass action. Further insight is obtained by examining a slightly different model: the cyclic model, which describes ordered phosphorylation, but where the order of dephosphorylation is the same as the order of phosphorylation (resulting in different partially phosphorylated forms). Here, the ‘sequestration’ of the singly phosphorylated form in the kinase complex does not result in any biphasic response. This is demonstrated analytically in the appendix, where we show that biphasic responses of the maximally phosphorylated species as a function of total kinase concentration cannot exist. In this case, the phosphatase does not bind to the singly phosphorylated form (*A*_{01}) in the forward modification, and thus the competing effects described earlier do not come into play. This provides further consolidated understanding of the role of the mechanism we have described earlier. Finally, we also demonstrate analytically in the appendix, that different processive models of multi-site phosphorylation will not exhibit such biphasic behaviour.

Taken together, the analysis pins down both the competing effects and shows how making changes in the mechanism completely removes the competing effects and prevents the possibility of biphasic dose–response characteristics.

## 5. Conclusion

Multi-site modification mechanisms are found in a variety of contexts in cell biology and their role is gradually being understood and appreciated. Multi-site phosphorylation represents a specific complex extension of a basic covalent modification cycle. The roles, consequences and effects of the extra steps are of interest both in the specific cellular biological contexts and from broader signal transduction perspectives. Systematically, understanding the effects of multi-site modification in different contexts is challenging because of their intrinsic complexity, context-specific factors as well as the coupling and interaction with other pathways and cellular entities. It is clear that disentangling such complexity requires a systematic elucidation of these different elements and their interplay, and that understanding the intrinsic complexity and coupling to other cellular factors is relevant in multiple-specific contexts.

This paper focused on elucidating a specific aspect of signal transduction which is intrinsic to multi-site phosphorylation. This elucidation is done *in silico* by investigating a generic model of (distributive) multi-site phosphorylation, and its variants at steady state. The model is similar to others in the literature, and simply involves the description of different modifications of the substrate by the enzyme in a standard way. The assumptions in the model are minimal, and on par with those typically made for modelling signal transduction (steady state, and a closed system for enzymes and substrates). It could happen that in some specific contexts, even these assumptions are not strictly met. Nevertheless, the description above can be regarded as fairly generic descriptions of distributive multi-site phosphorylation, without the consideration of other factors.

Our analysis focused on one aspect of the intrinsic complexity of distributive multi-site phosphorylation. We showed how distributive multi-site phosphorylation is intrinsically capable of generating (partial) biphasic dose–response characteristics in the maximally phosphorylated substrate, even under monostable conditions. Using a combination of numerical simulations, bifurcation analysis and analytical work, we were able to uncover a hidden competing effect which is responsible for this. Such behaviour was obtained by considering parameter regimes where the catalytic rate of phosphorylation in the last cycle was small, compared with the others, though this could be obtained also in other parameter regimes. This allowed for the substrate to become converted and ‘sequestered’ in the enzyme–substrate complex of the last cycle, as the total amount of active kinase was increased. This led to the generation of increased phosphatase by removing the intermediate phosphoforms to which it could bind, which could act on the maximally dephosphorylated substrate, providing the competing effect. We remark that while our model does not explicitly describe the activation of enzymes, the same behaviour is readily observed in an expanded model that explicitly describes activation of kinase enzyme by upstream signals (results not shown).

Such biphasic behaviour was readily obtained in both ordered and random mechanisms, and in fact, random mechanisms showed a greater accentuated biphasic behaviour. Increasing the number of modifications (for instance considering an ordered triple phosphorylation system) could still readily lead to such biphasic behaviour. We were also able to demonstrate that this biphasic behaviour could be readily combined with other studied behaviours of multi-site phosphorylation mechanisms such as thresholds and bistability. This demonstrates that a multi-site modification system, in addition to possessing a larger ‘state space’ is also intrinsically capable of more complex signal processing.

As the behaviour observed depends on ‘sequestration’ of the substrate in the enzyme–substrate complex of the last cycle, we see (and analysis confirms) that such behaviour would not be obtained by modelling each modification step using mass-action kinetics (this neglects the complex). Some simple analysis (see appendix) immediately reveals that a single covalent modification cycle by itself is not capable of demonstrating biphasic behaviour either. We thus see that the behaviour observed relies on both multiple modifications as well as sufficient hold-up in complexes.

We were able to further consolidate our insights by examining a number of variants of enzymatic mechanisms. We demonstrated analytically how the biphasic behaviour is destroyed in ordered mechanisms, when the order of phosphorylation and dephosphorylation is altered, showing how a structural change in the network, which bypasses the mechanism described above, is incapable of intrinsically exhibiting a biphasic dose–response characteristic. By systematically analysing a series of processive multi-site phosphorylation mechanisms, we were able to demonstrate that such mechanisms were not capable of exhibiting such behaviour. Taken together, this analysis indicates that if a biphasic (even partial) response is observed in a specific multi-site phosphorylation system, then either certain mechanisms of multi-site phosphorylation can be immediately ruled out, or the effect has to be extrinsic to the multi-site phosphorylation. Thus, our analysis may prove useful in certain contexts to exclude certain mechanisms based on the observation of this simple to observe behaviour. Such analysis could prove to be of broader relevance in systems (bio)chemistry, as it focuses on one aspect of the intrinsic complexity of multi-site modification of substrates by an enzyme pair.

Our analysis was performed on a model of multi-site phosphorylation, which represents multi-site phosphorylation in a well-controlled environment. Multi-site modification is repeatedly seen in cellular contexts, and both kinase and phosphatase activity/concentration may be regulated by multiple factors. It is being increasingly being realized that phosphatase activity is regulated in a number of ways in cells [31]. This includes regulatory control mechanisms and localization. Furthermore, there are different contexts where the active phosphatase concentration may be relatively low. One example is that of Cdc14 in metaphase budding yeast [29] (in fact, a gradual increase in phosphatase concentration occurs at different stages of the cell cycle). A similar example can be seen in PKC signal transduction [30]. The alkaline phosphatase (involved in multi-site phosphorylation) concentration is considerably reduced in certain disorders [32]. In general, a combination of factors could reduce the available phosphatase. The effect of coupling of pathways through shared components is a recurring theme in signalling, and analysis suggests that phosphatase sharing may in fact be responsible for coupling and crosstalk in different cellular contexts [33]. In our case, the hidden competing effect which we examined arises from the fact that the same phosphatase is shared between different phosphoforms. A simple extension of our analysis shows that it is possible to obtain a biphasic response in concentration of the unphosphorylated substrate as (active) phosphatase concentration is increased, keeping the (active) kinase concentration fixed, for exactly analogous reasons. In fact, for fixed kinetic parameters, it is possible to obtain biphasic responses of maximally phosphorylated substrate (as active kinase concentration is increased) and unmodified substrate (as active phosphatase concentration is increased). This reveals that such competing effects can be at play in both directions. In some contexts, the entity of interest may be the dephosphorylated substrate, and our analysis directly applies here too.

The above-mentioned analysis demonstrates how the sharing of phosphatase (and kinase) enzyme between different phosphoforms can give rise to the behaviour observed. It is worth extending the analysis above to reveal another consequence of such coupling in our model. If the total enzyme concentrations are kept fixed, and the total substrate concentration is increased, then in some cases, the concentration of the maximally phosphorylated substrate can also exhibit a biphasic response. In this case, increasing the substrate concentration has two competing effects: while it tends not only to increase the maximally phosphorylated substrate concentration, it also has the effect of sequestering the kinase in complexes. This reduces the free kinase concentration and has a competing effect. This is discussed in the appendix. Thus, kinase sequestration rather than phosphatase release plays a role in the biphasic response.

From the above-mentioned discussion, it is very plausible that the mechanism described may be at work in specific contexts. In order to clearly and unambiguously demonstrate this experimentally careful experiments are required, which are able to disentangle any complexities from extraneous factors. Phosphatase sharing is believed to be an important source of crosstalk and coupling [33–35], which is also true for kinases. The effect of coupling of processes through shared components is a pervasive theme in current systems biology (see [36] and references therein). Experiments have demonstrated that such coupling through shared components can have significant effects *in vivo,* and the addition of other substrates could have significant effects in MAPK signalling [37,38]. Further, recent experiments in the MAPK system [39] varying the total amount of ERK (substrate) demonstrate a biphasic response of the maximally phosphorylated form of ERK, and the experiments are shown to be consistent with a mechanism involving kinase sequestration by the substrate. This is an experimental example in multi-site phosphorylation suggesting sequestration of kinase as a mechanism for a biphasic response [39] and is a concrete example of how sharing of a common enzyme between different phosphoforms can result in a biphasic response. Our analysis provides a framework for understanding how and under what conditions this may happen. In addition, experiments exploring tau (multi-site) phosphorylation reveal a biphasic response of tau phosphorylation on forskolin [40]. The experiments indicate that at high levels of forskolin, the phosphatase PP2-A expression and activity is increased, giving rise to this effect. While additional regulatory mechanisms which give rise to this may indeed exist, our results indicate how phosphatase release may itself be able to give rise to biphasic effects.

While biphasic behaviour has been observed in many contexts in cell signalling, they have in fact been observed elsewhere in signalling pathways also involving multi-site phosphorylation systems [28]. The experimental study in [28] clearly shows a biphasic-type response in the doubly modified phosphoform. While there could be multiple reasons for this behaviour, our results indicate possible ways in which the multi-site phosphorylation mechanism itself could contribute to this.

A biphasic response represents an auto-shutdown/reduction in response at high input levels, and is observed in many biological contexts. Such a response is referred to as hormesis in some contexts. It arises in many contexts in signal transduction itself and can sometimes be related to or be partially responsible for dose-dependent downstream decision-making. Indeed, the biphasic response results in an ‘optimal’ range of stimulus for downstream effects and may be desirable for different reasons. Biphasic responses can arise due to scaffolds, or incoherent feed-forward pathways [41,42], and are often associated with these factors. It is interesting to find this capability built into the basic multi-site modification mechanism, and not requiring additional mechanisms such as scaffolds or other regulation. We note that this biphasic response is due to a hidden competing effect in multi-site phosphorylation. It is worth pointing out, incidentally, that in some cases different modifications can have opposing effects on downstream pathways, which could be a different source of biphasic signal transduction characteristic in signalling, involving multi-site phosphorylation. Our results indicate that the biphasic behaviour may be accentuated by other factors such as additional negative feedback or scaffolds. It is also possible that such biphasic behaviour is built-in at many levels in signal transduction and that such behaviour built into the multi-site phosphorylation mechanism can increase the robustness of biphasic responses in pathways. Finally, it is worth emphasizing that multi-site modification mechanisms are part of many signal transduction pathways, which are complex and process signals in a highly nonlinear way. Our uncovering of the hidden competing effect allows for a better understanding of the signal transduction characteristics, capabilities and constraints in pathways (in multiple contexts) involving multi-site modification or interacting with them.

Our results suggest how multi-site phosphorylation, in addition to acting as sophisticated biological control mechanisms, could act as complex signal processors. By combining effects involving one part of the cycle (giving rise to thresholds) and those at the other end of the cycle, it is possible for such mechanisms to exhibit complex signal processing characteristics. This has implications for both the signal processing capability of multi-site phosphorylation and their engineering using synthetic means.

## Appendix A

### A.1. Model equations

Here, we present the models we use in our analysis. We briefly present and discuss the *n*-site-ordered distributive model and the two-site random mechanism.

The *n*-site distributive-ordered mechanism is a simple extension of the two-site mechanism to incorporate *n* modifications. The equations for the *n*-site modification can be obtained in an exactly analogous way, and can be succinctly written as follows
A1
A2

These equations describe the concentration dynamics of the unmodified, partially modified and completely modified substrates, the various kinase and phosphatase complexes. The equations for the free kinase and phosphatase are written in a manner similar to the double-site modification and are not shown explicitly here.

The random double-site modification mechanism model is obtained in a similar way to the ordered model, except for the fact that two different partially modified substrates are incorporated.

The equations for the model are

A3

### A.2. Double-site-ordered mechanism and single-site mechanism

Our analysis in the text shows how at steady state the concentration of the doubly modified substrate may be obtained as A4

This was obtained by imposing steady-state conditions for concentrations of all relevant components and imposing conservation of total substrate, kinase and phosphatase.

In equation (A 4), the various constants may be written explicitly in terms of kinetic constants as

A5

In the main text, we asserted that biphasic behaviour would not be possible in a single modification cycle. Using the same notation for the single cycle, as we do for the first cycle but without making any assumptions about rate constants, it is easy to see that at steady state

Thus, we have from a conservation of enzyme that

The conservation of total substrate results in
A6where *α* = *k*_{1}/(*k*_{2} + *k*_{3}), *β* = *k*_{10}/(*k*_{11} + *k*_{12}) (we will need only the functional form). Further, at steady state, we have
A7By differentiating the conservation equation for substrate with respect to *K*_{tot}, we see that if d*A*_{p}/d*K*_{tot} = 0, then it should follow that d*A*/d*K*_{tot} = 0. Suppose we assume the possibility of d*A*_{p}/d*K*_{tot} = 0. Then by differentiating the steady-state equation above, we arrive at a contradictory conclusion that *α* = 0. Thus, the single modification cycle is incapable of exhibiting a biphasic dose–response characteristic.

### A.3. The cyclic mechanism

Here, we analyse a variation of the ordered mechanism where the order of phosphorylation is not opposite to the order of dephosphorylation (figure 8), and again consider a two-site mechanism. This mechanism is almost identical to the ordered mechanism considered in the paper, except that the order of phosphorylation and dephosphorylation implies that there are different singly phosphorylated forms. This means, the model has to explicitly account for the two different singly phosphorylated forms *A*_{01} and *A*_{10}. The model for this mechanism can be written in a very similar manner to those above.
A8

Again, this model is supplemented by conservation conditions for total kinase, phosphatase and substrate. The kinetic constants in this model refer to similar conversion and binding/unbinding reactions as in the ordered model considered in the main text, with the only difference that there are two separate singly modified phosphoforms, with only *A*_{01} being further phosphorylated, and *A*_{10} being dephosphorylated.

We present a brief analysis of this model. In what follows, we focus on the functional relationships between different phosphoforms. First, we note that at steady state A9 A10 A11 A12By noting that, steady state implies A13

From above, we see that all steady-state complex concentrations are directly proportional to that of *A*_{11}*P* (proportionality constant only dependent on kinetic constants), which in turn is proportional to the product of the concentrations of *A*_{11} and *P*. Further, we see that in a similar way, [*A*_{11}] and [*A*_{10}] are proportional, and similarly, so are [*A*] and [*A*_{01}]. Finally, from the proportionality of complexes, we see that both [*A*] and [*A*_{01}] are proportional to [*A*_{11}][*P*]/[*K*] from above.

Now, the conservation of kinase results in
A14for a suitable constant *α* which is a function of kinetic parameters (obtained by rewriting all other kinase complexes in terms [*A*_{11}*P*]). Similarly, a conservation of phosphatase results in
A15for a suitable constant *β* obtained in terms of kinetic parameters. This is obtained by writing both phosphatase complexes in terms of [*A*_{11}][*P*] directly from above.

For the conservation of the substrate, we note that the substrate may be present in different modified forms as well as complexes. Noting from the comments above, how all of them can be related to [*A*_{11}] we end up with an equation of the form
A16for suitable constants *γ*_{1}, *γ*_{2}, *γ*_{3}. Now suppose d[*A*_{11}]/d*K*_{tot} = 0 at some value of *K*_{tot}. This means that the derivative of the denominator in the expression for [*A*_{11}] must necessarily be zero. We see by differentiating the expression of [*P*] that at this value of *K*_{tot} that d[*P*]/d*K*_{tot} = 0. Further, as d([*P*]/[*K*])/d*K*_{tot} = −([*P*]/[*K*]^{2})d*K*/*K*_{tot} + (1/[*K*])d[*P*]/d*K*_{tot}, we see that the only way the derivative of the denominator is zero is if d[*K*]/d*K*_{tot} = 0. However, by direct differentiation of the expression for [*K*], at this location, we see that d[*K*]/d*K*_{tot} at this location cannot be zero. Thus, we see that *A*_{11} cannot be a biphasic function of *K*_{tot}.

### A.4. The processive mechanism: ordered mechanism

We now briefly examine a two-site-ordered processive mechanism. The processive mechanism is realized by kinase binding to substrate, forming a complex which is catalytically converted to *A*_{p}*K*, which then becomes converted to *A*_{pp}, releasing the kinase. An analogous chain of events happens in the dephosphorylation. The key point here is that the enzyme is held in complexes and is released only in the final step.

The equations for a two-site processive mechanism are (figure 8) A17

These reflect the mechanism described above.

Now, at steady state, we have

Further, from the steady state, it is easy to see that [*A*_{pp}*P*] is proportional to [*A*_{p}*K*].

Now, as [*K*] + [*AK*] + [*A*_{p}*K*] = *K*_{tot}, we have
A18for a suitable constant *α* which is a function of kinetic parameters (concentrations of all kinase complexes, proportional to [*A*_{pp}*P*]). We also have
A19for a suitable constant *β*. This is obtained from the conservation of phosphatase, writing concentrations of all phosphatase complexes in terms of [*A*_{pp}] [*P*]. Finally, from the relations above and conservation of total substrate, we have
A20for suitable constants *γ*_{1}, *γ*_{2}, *γ*_{3}. That such a functional form arises is seen by noting that concentrations of all complexes (kinase and phosphatase) are proportional to [*A*_{pp}][*P*], whereas (from above) the concentration of free *A* is proportional to [*A*_{pp}] [*P*]/[*K*].

Now, suppose d[*A*_{pp}]/d*K*_{tot} = 0. We see from above (very similar to the previous case) that d[*P*]/d*K*_{tot} = 0. However, differentiating the expression for [*A*_{pp}] we see it is necessary that d[*K*]/d*K*_{tot} = 0, which is impossible from the expression for [*K*].

Thus, we conclude that the two-site-ordered processive mechanism cannot exhibit biphasic behaviour. This analysis can be extended in a very straightforward way to ordered *n*-site processive mechanisms. The only difference is the presence of more intermediate kinase and phosphatase complexes. However, at steady state, the concentrations of all these intermediate forms are proportional to the product of the concentrations of the maximal phosphorylated form and the free phosphatase. We thus see that the argument carries through in exactly the same way also in that case.

#### A.4.1. Processive cyclic mechanism

We briefly discuss a cyclic processive two-site mechanism, which is processive, but where the order of phosphorylation and dephosphorylation are different. This results in different intermediate complexes (figure 8).

The analysis can be performed in a very similar way to the previous cases. We note that at steady state, for reasons exactly as above (i) the concentration of all kinase complexes is proportional to [*A*][*K*]. (ii) The concentration of all phosphatase complexes is proportional to [*A*_{11}][*P*]. This follows in a very simple way by considering the steady state of all components. Further, we find by balancing the net rates of the forward and backward reaction (or equivalently considering the steady state of *A*_{11} + *A*_{11}*P*) that [*A*][*K*] is proportional to [*A*_{11}][*P*]. Now using this information and considering conservation of kinase, phosphatase and substrate results in exactly the same kind of functional relationship as before:
A21
A22
A23for suitable constants *α*, *β*, *γ*_{1}, *γ*_{2}, *γ*_{3}. The conclusions follow exactly the same way as in previous sections.

#### A.4.2. Random processive mechanism

Random mechanisms are generally more complicated that ordered mechanisms. We show that a random double-site processive mechanism will not exhibit any partial biphasic behaviour either. A look at the random processive model reveals a more complicated structure (figure 8). However, by considering the steady state for all complexes, we arrive at the following conclusion: (i) the concentration of all kinase complexes is proportional to [*A*][*K*] and for exactly the same reasons (ii) the concentration of all phosphatase complexes is proportional to [*A*_{11}][*P*]. Now, at steady state (for example by considering the steady state of [*A*_{11}] + [(*A*_{11}*P*)_{1}] + [(*A*_{11}*P*)_{2}]), we have a balance between the net forward and backward reactions:
A24

Noting the facts mentioned above regarding the kinase and phosphatase complexes, we see that [*A*][*K*] is proportional to [*A*_{11}][*P*] just as before. Here again, from the conservation of kinase, phosphatase and substrate, accounting for all complexes, we have
A25
A26
A27for suitable constants *α*, *β*, *γ*_{1}, *γ*_{2}, *γ*_{3}. The conclusions follow exactly the same way as in previous sections. We thus see that the extra complexity of the random mechanism, actually does not alter the conclusions.

### A.5. Effect of increasing substrate concentration

In the text, we demonstrated how the sharing of enzymes between different phosphoforms could result in a biphasic response of either the maximally modified substrate or the unmodified substrate, as the active enzyme concentration is varied. Here, we extend our analysis of the model by briefly considering the effect of varying the substrate concentration keeping the total enzyme concentration fixed. We focus on the ordered two-site distributive mechanism for the sake of simplicity.

While increasing the substrate concentration can increase the concentration of the maximally phosphorylated substrate at steady state, it is also possible to observe a biphasic dose–response characteristic. This is demonstrated in figure 9*a* for a case where the kinetic characteristics of all steps (binding constant, unbinding constant, catalytic constant) are similar. In this case, the total kinase concentration is less than that of the phosphatase, and by increasing the substrate concentration, we find that it is possible for the kinase to be sequestered by the substrate in the *AK* complex, reducing the freely available kinase concentration. Here, we see that as the substrate concentration increases *A*_{pp} exhibits a biphasic response. In contrast to the case in the text, we find that the complex *A*_{p}*K* also exhibits a biphasic response (while *A*_{p} does not), clearly revealing the diminished kinase as a source of this biphasic behaviour.

Interestingly, a similar behaviour is seen even when the kinase concentration is clearly greater than that of the phosphatase, when the catalytic constant of the first phosphorylation step is lower than the others. In this case, this low catalytic constant can combine with the increase in substrate concentration to effectively sequester kinase and result in a biphasic response (figure 9*b*). The primary effect here is the reduction of the kinase. In more complex situations, the effect of sequestration of kinase may possibly be accentuated also by a release of phosphatase (from the sequestration of the substrate).

Overall, we find that the sharing of kinases and phosphatases between different phosphoforms can, through limitations of their relative amounts, result in appropriate competing effects and biphasic responses. Naturally, in the case considered in the text, kinase limitation was not a factor. Overall, either kinase sequestration or phosphatase release may provide nontrivial competing effects.

- Received August 12, 2013.
- Accepted September 9, 2013.

- © 2013 The Author(s) Published by the Royal Society. All rights reserved.