Exploring cyclic networks of multisite modification reveals origins of information processing characteristics

Multisite phosphorylation (and generally multisite modification) is a basic way of encoding substrate function and circuits/networks of post-translational modifications (PTM) are ubiquitous in cell signalling. The information processing characteristics of PTM systems are a focal point of broad interest. The ordering of modifications is a key aspect of multisite modification, and a broad synthesis of the impact of ordering of modifications is still missing. We focus on a basic class of multisite modification circuits: the cyclic mechanism, which corresponds to the same ordering of phosphorylation and dephosphorylation, and examine multiple variants involving common/separate kinases and common/separate phosphatases. This is of interest both because it is encountered in concrete cellular contexts, and because it serves as a bridge between ordered (sequential) mechanisms (representing one type of ordering) and random mechanisms (which have no ordering). We show that bistability and biphasic dose response curves of the maximally modified phosphoform are ruled out for basic structural reasons independent of parameters, while oscillations can result with even just one shared enzyme. We then examine the effect of relaxing some basic assumptions about the ordering of modification. We show computationally and analytically how bistability, biphasic responses and oscillations can be generated by minimal augmentations to the cyclic mechanism even when these augmentations involved reactions operating in the unsaturated limit. All in all, using this approach we demonstrate (1) how the cyclic mechanism (with single augmentations) represents a modification circuit using minimal ingredients (in terms of shared enzymes and sequestration of enzymes) to generate bistability and oscillations, when compared to other mechanisms, (2) new design principles for rationally designing PTM systems for a variety of behaviour, (3) a basis and a necessary step for understanding the origins and robustness of behaviour observed in basic multisite modification systems.


Models
We present the kinetic models employed in our study. In general the kinetic models are developed by describing enzymatic modification of the substrate in standard way: binding reversibly to the substrate to form a complex which irreversibly dissociates to give the product and release the enzyme. Whenever an enzyme effects multiple modifications, it is assumed to act distributively. The basic cyclic models (C1, C2, C3) correspond to cyclic networks described in the main text (Fig. 1). The equations for a 2-site cyclic distributive mechanism, model C1 (Fig. 7), which corresponds to a cylic network with a common kinase and a common phosphatase effecting the modifications, are: This is simply the mathematical description of the network in Fig. 1(a). In an analogous way, the equations for model C2 (see Fig. 1 for a schematic) which corresponds to a cyclic 2 site modification network with different kinases and a common phosphatase effecting the (de) modifications are: 2 The only difference (from model C1) is that there are different kinases involved in the phosphorylation leg of the cycle: K 1 phosphorylates A while K 2 phosphorylates A 01 . The common phosphatase dephosphorylates A 11 and A 10 Model C3 differs from C2 in that there are different phosphatases involved in the dephosphorylation leg. Phosphatase P 2 dephosphorylates A 11 while phosphatase P 1 dephosphorylates A 10 . Other than this, the model is identical to C2. The equations for model C3 are (see Fig. 1 for a schematic): Models of cyclic networks with additional reactions. We now present the equations for models which build on the basic cyclic models above with an additional reaction, either kinase-mediated or phosphatase-mediated (see Fig. 1(c)).
Model C21, builds on the basic model C2 (different kinase common phosphatase) with an additional reaction mediated by kinase K 2 , involving the conversion of A to A 10 (note that the phosphorylation activity of K 2 is associated with the first index of the subscript In an exactly analogous way the equations for models C31 and C32 are obtained. The augmentations for C31 and C32 are the same as C21 and C22 respectively, and the only difference is that these augmentations are built into model C3 (different kinase and different phosphatase) rather than model C2.

5
The equations for model C31 are 6 The equations for model C32 are In a similar way, the equations for models C11 and C22 are obtained (not shown). They represent the same augmentations as C21 and C22 respectively, overlaid on model C1 (see Fig. 1).
Comments on other augmentations to cyclic networks. We have listed 6 models involving either an additional kinase reaction augmentation (3 cases) or an additional phosphatase reaction augmentation (3 cases). Clearly, for a given basal cyclic network, there are 4 possibilities for a single augmentation. This gives rise to an additional 6 networks (see Fig. S3). For the most part, we do not study these additional cases in detail except to note that: (i) The networks studied in the main text considered representative cases involving an augmentation by a kinase mediated reaction or a phosphatase-mediated reaction: taken together this already reveals the essential landscape of behaviour and possibilities, from the focal point of our study. (ii) The additional models can be analyzed in analogous terms to the cases studied. (iii) From the perspective of bistability and oscillations, 4 of these additional networks (the common kinase common phosphatase and the separate kinase separate kinase cases) exactly map onto networks we have studied in the main text. (iv) Essential results and insights from the remaining two are discussed below in the context of results. (v) With regard to biphasic dose responses the models we have studied in the main text already reveal the landscape and types of responses which may be seen.

2 Bistability in cyclic mechanisms and variants
The text discusses multiple instances of the presence or absence of bistability (and multistationarity in general) in different cyclic models and their variants. Here we use analytical approaches to demonstrate that bistability can be ruled out in some of these cases, irrespective of parameters. Specifically we show that (i) All the basic cyclic models (C1,C2,C3) possess only a single biologically feasible steady state. (ii) Specific augmentations of the cyclic models, such as model C11 also possess only one steady state. This complements computational results which directly demonstrates the presence of bistability in other models.
Our analysis of the models employs a number of basic points repeatedly, and we summarize them here. We make a comment on notation in the analysis. Here and below we use [A 00 ] to denote the concentration of the unphosphorylated form of substrate, which is referred to as [A] above. Additionally we refer to the catalytic constants of modification of A 00 , A 01 , A 11 , A 10 as k c1 , k c2 , k c3 , k c4 respectively.
Naturally this specific notation has no consequence for the analysis and the results which emerge therefrom.
The approach to the analysis is as follows. Thus we have shown the absence of multistationarity in the model C1. We now show how the same approach can be used for the models C2 and C3.
Model C2. The only difference in this case is the presence of two kinases. (i) The conservation condition for kinases implies that (iii) Now using the expressions for the free enzymes, we have an , where all constants are positive. Inverting this expression, we have an equation of the form , which is similar to the type of equation obtained above. Depending on the relative magnitudes of γ 1 /a 1 , γ 0 /a 2 , γ 1 /b 1 , all variables can be eliminated in favour of the variable associated with the largest of these constants. Suppose the largest of these constants is γ 1 /b 1 (the other cases can be studied in exactly analogous terms as mentioned above).
Taking the inverse of this equation, we have an equation of the form We then examine the largest of the constants γ/a 1 , γ 0 /a 2 , γ 1 /b 1 , γ 2 /b 2 . Suppose it is γ 1 /b 1 : then all other substrate variables can be written in terms of the substrate variable associated with γ 1 /b 1 , namely A 11 (the exact same approach can be used for other choices of the largest constant). Then Now use the conservation condition: . So incorporating all the variable dependencies on A 11 gives an equation of the form η where α 0 < 1. Now looking at the equation Thus we see that the relationships of the different phosphoforms to each other are of an essentially similar nature as the basic cyclic model C1. (ii) This means that P = P tot /(1 + γ[A 11 ]), respectively by a change of labels (for substrates and enzymes). Consequently for attractor based behaviour (as opposed to individual dose response characteristics), the behaviours of the models also map. We thus infer that model C13 does not exhibit bistability while C14 can (and can do so even with the augmented reaction in the unsaturated limit). (b) The separate kinase separate phosphatase case: here, for a similar reason, the models C33 and C34 can be mapped on to C31 and C32 respectively, to show in each case that a single augmentation in the unsaturated limit can give rise to bistability. (c) The separate kinase common phosphatase case: here the additional models C23 and C24 cannot exactly be mapped on to C21 and C22.
In this regard, we make the following two points: (i) C23 can indeed exhibit bistability and even do so when the additional reaction is in the unsaturated limit (see Fig. S4). (ii) Model C24 on the other hand involves an augmentation which does not create a new complex. Thus similar to the case of model C11 analyzed above, the augmented model has the same mathematical structure as the basic cyclic network and bistability is precluded.
The previous section focussed on bistability. Here we turn to biphasic responses of the fully phosphorylated form as the total concentration of a kinase is changed. Thus in contrast to the previous section we focus on a dose response curve (associated with a specific phosphoform) rather than an emergent non-linear dynamical behaviour characteristic of the overall system.
We have indicated in the main text that biphasic responses can be obtained in a number of models while they are not found in other models (in some cases, in response to specific kinase variation). We substantiate these latter conclusions analytically.
The results are organized as follows: (1)

Basic Cyclic models.
We now analyze the basic cyclic models C1 , C2 and C3, using the methods of analysis employed in the context of bistability (and do not repeat those steps).
As before we assume a biphasic response for the dose response curve in response to K As before we assume a biphasic response for the dose response curve in response to K

Basic models with an extra reversible step.
We now examine models with an augmented step. We consider two types of augmentations: one with an extra kinase mediated reaction (mediated by K 2 , in the case of multiple kinases for specificity): this is labelled by a second subscript 1 in the model (models C11, C21, C31), and the other, an extra phosphatase mediated reaction (mediated by P 2 in the case of multiple phosphatases, for specificity): this is labelled by a second subscript 2 in the model (models C12, C22, C32). In all the cases the first subscript refers to the basic cyclic model being perturbed (common kinase common phosphatase model labelled 1, different kinase common phosphatase model labelled 2, different kinase different phosphatase model labelled 3). We first begin with models C31 and C32 (see Fig. 1) both of which can exhbit a biphasic respose to a change in total concentration of K 2 . In the text we asserted that a biphasic response in response to a change in total concentration of kinase K 1 is not possible. We establish that below. At the outset, we note that the presence of an extra reaction step complicates the analysis, relative to the basic cyclic networks.
Model C31. In this model, there is an extra reaction involving enzyme K 2 converting A 00 to A 10 .
Suppose there is a biphasic response as K Model C11. We have already analyzed model C11 in the context of bistability. We have shown that the extra reaction introduces an extra complex of a preexisting type. Thus the behaviour of this model is very similar to that of model C1, and by eliminating variables, we see the same types of dependencies between variables, with the only alterations being in the constants. For an exactly analogous reason, biphasic responses can be ruled out, exactly as they were in model C1. We will consequently not repeat what is essentially an identically parallel analysis.
Model C21. Model C21 exhibits a robust biphasic response with regard to the variation of the total concentration of K 2 . This is seen for all values of parameters. The reason for this can be seen as follows.
When K 2 is absent [A 11 ] = 0 simply because there is no reaction producing it. As increases. Interestingly as K 2,tot → ∞, [A 11 ] → 0. This can be seen intuitively from the network C21 itself. For very large K 2,tot , most of the reaction (involving the conversion of A 00 ) involves the conversion of A 00 to A 10 , with negligible flux to the A 01 reaction: in fact in relative terms the flux to this pathway approaches 0. Consequently, [A 11 ] approaches zero as it is dependent on this reaction, not being produced by any other pathway. This then implies that there necessarily has to be a biphasic response (or a multiphasic response), and monotonic dose response curves are precluded.
Model C31. Model C31 shares a similar structure as model C21 (with respect to enzyme K 2 ).
Consequently, a biphasic response to the variation of K 2 total concentration is guaranteed. Here too, we see that when K 2,tot = 0, The reason for this is exactly the same as above. For large values of K 2,tot , the conversion of A 00 is predominantly to A 10 with negligible flux in the upper branch of reactions, proceeding through A 01 (Fig. 1). Thus, there necessarily has to be a biphasic (or multiphasic response) and monotonic dose-response curves are precluded.
3.2.1 The role of enzyme sequestration in generating biphasic responses.
Our analysis above has primarily focussed on ruling out biphasic responses in a number of augmentations of the basic cyclic models. Simulations, on the other hand do show biphasic responses for all the cases summarized in Table 1. This includes models C21, C22, C31, C32 in response to total concentration of K 2 , Model C21 in response to total concentration of K 1 , and model C12 in response to total concentration of K. Each of these models has one extra reaction which enables this biphasic response (noting that the basic cyclic models do not exhibit the biphasic response). However, the question which remains is whether a biphasic response persists even if there is no enzyme sequestration in this extra reaction, i.e. if this extra reaction operates in an unsaturated regime.
We justify, below the answer to this: (i) With respect to variation of concentration of K 2 , models C21 and C31 exhibit a biphasic response even without enzyme sequestration, while models C22 and C32 do not. (ii) Model C12 does not exhibit a biphasic response to K, if the additional reaction is not associated with sequestration.
We make a comment with regard to our analysis below. The extra reaction acting in an unsaturated regime implies that there is negligible sequestration of enzyme and negligible complex. In our analysis the negligible sequestration of the enzyme emerges as a key factor in the analysis. We note that while the relevant complex concentration is low, the associated reaction rate in general is not. In the analysis, we retain the complex as in the analysis above everywhere, except in the enzyme conservation equation, and draw all our conclusions from this.
A slightly different way of doing this is to eliminate the relevant complex in both enzyme and substrate conservation equations, and using the associated rate of the reaction (where it is used) as proportional to the product of the relevant free enzyme and substrate. This analysis is essentially similar to our analysis, since all our analysis requires about the complex (apart from the negligible sequestration of enzyme therein) is the fact that the concentration of the complex is proportional to that of the free enzyme and substrate, anyway and (ii) the presence of the complex in the substrate conservation does not alter the argument in any essential way. In its absence there is one less term to deal with, but the essential argument deployed works in an identical way.
Models C21 and C31. As discussed above, both these models result in robust biphasic responses, and as K 2,tot → ∞, [A 11 ] → 0. This does not in any way depend on K 2 being sequestered in the additional reaction, and purely depends on how K 2 appears in the reaction network. Consequently biphasic responses can be obtained without sequestration of enzyme in the extra reaction.
Model C22. From the conservation of enzymes, we have: P = P tot /(1 + γ 1 [A 11 ] + γ 2 [A 10 ]), terms, and the RHS has zero derivative. This results in a contradiction. Thus a biphasic response of the type assumed can be ruled out.
Model C32. From the conservation of enzymes, we have: Biphasic response in model C21. We have already showed that the biphasic response in model C21 to variation of K 2 does not need sequestration of enzyme in the additional step. We now demonstrate that this is also true for the enzyme K 1 . The reasoning here is different, and we note that the biphasic response to K 1 is not as widespread as the biphasic response to K 2 . Our approach here is to analyze a tractable case to demonstrate the origins of this biphasic behaviour in model C21. We make the following assumptions: (i) K 2 acts in the unsaturated limit, as does K 1 . (ii) The dephosphorylation of A 11 also occurs in the unsaturated limit. By employing the conservation condition for substrate we have the equation Firstly we note here that if there were no phosphatase sequestration at all i.e. α 7 = 0 it is easy to see that even here as [K 2 ] → ∞, [A 11 ] → 0. This further substantiates the fact that a biphasic response in [K 2 ] can be obtained without any sequestration effects in the additional reaction, and we see this to be the case in an even simpler setting where there is no sequestration of any kind.
We make one assumption on parameters to explicitly reveal the origins for biphasic response, more transparently. We assume that term the denominator of the expression for P, we can neglect 1 in relation to This can be simplified as If we neglect the last group of terms, then no biphasic response to [K 1 ] can be obtained. However this last grouping of terms demonstrates the ingredients for a tradeoff. Examining the term

Parameter values in Figures
We present here some details regarding the parameters employed in the computational results presented in the main text. The parameters are all dimensionless parameters. The paper analyses the information processing capabilities of cyclic multi-site phosphorylation systems to make focussed conclusions regarding the possibility of different types of behavior. In all the numerical simulation and bifurcation analysis, unless otherwise mentioned, the initial conditions of substrate, also equal to total concentration of substrate (A tot ), and phosphatase, (equal to total phosphatase, P tot ), are 40 and 1, respectively. While the initial condition for the substrate corresponds to the substrate being present in the unmodified form, the initial condition for the kinase and phosphatase is for all the enzyme which is in the active form. In any case the concentrations of the various complexes are zero initially. Furthermore, kinetic parameters such as binding, unbinding and catalytic constants are equal to 1, unless otherwise mentioned below. and k 12 = 2 (ii) k 4 = 0.1, k 6 = 4, k 7 = 50, k 12 = 2 and K tot = 1.3 (c) Model C2: k4 = 0.1, k6 = 4 and k7 = k12 = 50 (i) K 2,tot = 4 (ii) K 2,tot = 4.5 (iii) K 2,tot = 5.5  Figure 6 (a) Model C 31 : k 1 = 100, k 4 = 2.5, k 6 = 100, k 10 = 0.1, k 12 = k 13 = 100, P 1,tot = P 2,tot = 1.5 and K 2,tot = 0.5 (b) Modified model C 2 (i) Model C 22 : k 1 = 100, k 4 = 0.1, k 6 = 4, k 7 = 100, k 10 = 2.5, k 12 = 180, k 13 = 0.5, k 15 = 5 and K 2,tot = 1.5 (ii) Model C 21 : k 4 = 0.1, k 6 = 4, k 7 = 6, k 10 = 2.5, k 12 = 200, k 13 = k 14 = k 15 = 0.1 and K 1,tot = 1.5 Supplementary Figures. All parameter values unless stated otherwise are 1. Total concentration for substrates and enzymes unless otherwise stated are 5.