Versatile regulation of multisite protein phosphorylation by the order of phosphate processing and protein–protein interactions Carlos Salazar1 and Thomas Hofer1,2 ¨ Theoretical Biophysics, Institute for Biology, Humboldt University Berlin, Germany German Cancer Research Center, Heidelberg, Germany Keywords multisite phosphorylation; order of phosphate processing; stimulus–response relationship; transition time; ultrasensitivity Correspondence T Hofer, Theoretical Biophysics, Institute ă for Biology, Humboldt University Berlin, Invalidenstr 42, 10115 Berlin, Germany Fax: +49 30 2093 8813 Tel: +49 30 2093 8592 E-mail: thomas.hoefer@rz.hu-berlin.de Website: http://www.biologie.hu-berlin.de/ theorybp/ (Received 30 October 2006, revised 13 December 2006, accepted 18 December 2006) doi:10.1111/j.1742-4658.2007.05653.x Multisite protein phosphorylation is a common regulatory mechanism in cell signaling, and dramatically increases the possibilities for protein– protein interactions, conformational regulation, and phosphorylation pathways However, there is at present no comprehensive picture of how these factors shape the response of a protein’s phosphorylation state to changes in kinase and phosphatase activities Here we provide a mathematical theory for the regulation of multisite protein phosphorylation based on the mechanistic description of elementary binding and catalytic steps Explicit solutions for the steady-state response curves and characteristic (de)phosphorylation times have been obtained in special cases The order of phosphate processing and the characteristics of protein–protein interactions turn out to be of overriding importance for both sensitivity and speed of response Random phosphate processing gives rise to shallow response curves, favoring intermediate phosphorylation states of the target, and rapid kinetics Sequential processing is characterized by steeper response curves and slower kinetics We show systematically how qualitative differences in target phosphorylation ) including graded, switch-like and bistable responses ) are determined by the relative concentrations of enzyme and target as well as the enzyme–target affinities In addition to collective effects of several phosphorylation sites, our analysis predicts that distinct phosphorylation patterns can be finely tuned by a single kinase Taken together, this study suggests a versatile regulation of protein activation by the combined effect of structural, kinetic and thermodynamic aspects of multisite phosphorylation Reversible phosphorylation is arguably the most important mechanism for regulating protein activity [1] Also, other covalent modifications, such as methylation, acetylation, ubiquitination, sumoylation and citrullination, are increasingly being characterized [2] Studies in recent years have shown that multiple regulatory modifications of proteins are the rule rather than the exception [3,4] Proteins phosphorylated at several sites include, for example, membrane receptors, such as epidermal growth factor receptor and T cell receptor complex, protein kinases of the Src and mitogen-activated protein kinase (MAPK) families, and transcription factors, such as NFATs, b-catenin and Pho4 [5–10] The theoretical analysis of protein modification cycles dates back to the work of Stadtman & Chock [11,12] and Goldbeter & Koshland [13], who, among other findings, showed that very steep thresholds for Abbreviations MAPK, mitogen-activated protein kinase 1046 FEBS Journal 274 (2007) 1046–1061 ª 2007 The Authors Journal compilation ª 2007 FEBS C Salazar and T Hofer ă the phosphorylation of a single amino acid residue in a protein can arise under specific conditions Subsequent modeling studies have also focused on the problem of switch-like responses, which have been analyzed as a steady-state property [5,14–20] These studies have demonstrated that multiple phosphorylation as well as positive feedback can provide additional mechanisms for threshold generation Evidence of switch-like responses of protein phosphorylation has indeed been found in some experimental systems [21–23] Up to now, however, the dynamics of multiple phosphorylation have not been analyzed theoretically The signal transduction networks that are composed, in large part, of interacting kinases and phosphatases typically mediate transient cellular responses to external stimuli [24] Therefore, elucidation of the kinetic properties of phosphorylation cycles and cascades will be crucial for understanding their cellular function Multisite phosphorylation can be achieved in a variety of ways One or several kinases and phosphatases can process their target sites in a strictly ordered sequence [25–27] Repetitive motifs have been identified that impose sequential phosphorylation by certain kinases Conversely, the sequence of (de)phosphorylation can be random [28–30] Studies on rhodopsin indicate that the sequence of multiple phosphorylation can be critical for protein function The timing of rhodopsin deactivation critically depends on the number of phosphorylatable residues, and, paradoxically, proceeds faster with six residues in the wild-type protein than with three residues in a mutant [31] Regarding the underlying mechanism, rhodopsin phosphorylation and dephosphorylation apparently proceed in a nonsequential order [32] The kinetics of multiple phosphorylation have also been invoked for controlling the timing and specificity of cell-cycle progression and circadian rhythms [22,33–35] The theoretical analysis of multisite phosphorylation is complicated by several issues [36] The various possibilities for protein–protein interactions and phosphorylation sequence can create a very large number of complexes and phosphorylation states In many cases, it has been found that phosphorylation at one site enhances or suppresses the binding affinity of the kinase or its catalytic activity at another site, so that the phosphorylation kinetics of one residue can depend on the phosphorylation state of other residues in the protein [8] It is not clear how these factors modulate the response in the protein’s phosphorylation state Furthermore, traditional enzyme kinetics, which rest on the smallness of the enzyme concentration compared Kinetic models of multisite phosphorylation to those of the reactants, cannot be applied in a straightforward manner to protein phosphorylation in cell signaling, because there are often no large concentration differences between kinases and their targets In place of enzyme kinetics, the mathematical description of elementary reaction and binding steps is feasible but introduces a large number of variables and parameters, many of which are difficult to measure experimentally In this article, we develop a concise kinetic description of multisite phosphorylation that attempts to address these challenges Our approach starts from the description of the elementary steps of enzyme–target binding and catalysis and then uses the rapidequilibrium approximation for protein–protein interactions for a systematic simplification of the model [20] This allows us to obtain, in special cases, explicit solutions for the steady-state response curves and phosphorylation times, and to identify key parameters that determine system behavior and should be given priority in experimental measurements By scanning the space of these parameters, we arrive at experimentally testable predictions concerning both the steady-state response and the kinetics of multisite phosphorylation We demonstrate here that the order in which the individual residues are addressed by kinase and phosphatase is of overriding importance for both sensitivity and speed of response Sequential phosphate processing gives rise to steeper response curves and slower kinetics than random processing Moreover, we illustrate systematically how qualitative differences in target phosphorylation (graded, switchlike and bistable responses) are determined by quantitative parameters of protein–protein interactions such as enzyme concentrations and enzyme–target affinities Finally, we analyze how specific kinetic designs of phosphorylation cycles can potentiate differential control of the phosphorylation sites by the same kinase This study provides a link between the structural, kinetic and thermodynamic aspects of complex multisite phosphorylation on the one hand, and the specific and versatile regulation of protein activation required in signaling pathways on the other Results Mathematical model We consider a target protein with several phosphorylation sites, and are interested in how the abundance of the various phosphorylation states of the target is FEBS Journal 274 (2007) 1046–1061 ª 2007 The Authors Journal compilation ª 2007 FEBS 1047 Kinetic models of multisite phosphorylation C Salazar and T Hofer ă regulated by its kinase(s) and phosphatase(s) Experimental studies have shown that there are different mechanisms for the processing of the individual phosphorylation sites (Fig 1) Several kinases phosphorylate repetitive motifs of serine ⁄ threonine residues in a fixed order, e.g S ⁄ T-X-X-S ⁄ T for casein kinase I [25–27] When dephosphorylation proceeds in the reverse order, we will refer to this case as a strictly sequential mechanism (Fig 1, upper panel) Sequential action of phosphatases has indeed been described [8,30] Alternatively, the sequence of (de)phosphorylation can be random (Fig 1, second panel) [28,29] Mixed mechanisms can also occur, such as the random dual phosphorylation of MAPK extracellular-signal-regulated kinase (ERK) by mitogen-activated or extracellular signal-regulated protein kinase (MEK) and its sequential dephosphorylation by mitogen-activated protein kinase phosphatase (MKP3) (Fig 1, third panel) [5,30] A cyclic mechanism for the phosphorylation and dephosphorylation of rhodopsin has been proposed (Fig 1, lowest panel) [32] These alternative mechanisms of reversible phosphorylation differ in the number and kind of partially phosphorylated states and pathways of phosphorylation and dephosphorylation It will be an aim of this study to elucidate the consequences of processing order for the regulatory properties of the target protein We now derive a general model describing the dynamics of multisite reversible phosphorylation Initially, we focus on the sequential mechanism, in which case the phosphorylation states can be enumerated by the number of consecutively phosphorylated residues n ¼ 0, N, where N is the number of phosphorylatable residues In each phosphorylation state, the target can occur in free form or bound to kinase or phosphatase; the respective concentrations of the target will be denoted by Xn,0, Xn,K and Xn,P, respectively They are determined by the rates of the reversible enzyme–target associations ⁄ dissociations and the irreversible phosphorylation ⁄ dephosphorylation reactions as depicted in Fig Frequently, the protein–protein interactions take place more rapidly than the addition and cleavage of phosphoryl groups [20,37] In this case, the rapid-equilibrium approximation is justified [38], and the system dynamics can be formulated in terms of the total concentrations attained by the various phosphorylation states: Yn ¼ Xn;0 þ Xn;K þ Xn;P ð1Þ i.e the sum of free and enzyme-bound forms As shown in supplementary Doc S1, the total concentrations Yn are governed by the differential equations Fig Order of phosphate processing Sequential phosphorylation and dephosphorylation (first panel), random phosphorylation and dephosphorylation (second panel), mixed scheme with random phosphorylation and sequential dephosphorylation (third panel), and cyclic mechanism (fourth panel) The mechanisms are illustrated schematically for three phosphorylation sites In the sequential mechanism, there are N + different phosphorylation states (where N is the total number of phosphorylation sites); random mechanisms can create 2N different phosphorylation states It is of note that the number of different possible sequences to achieve full phosphorylation of the target is for the sequential mechanism and N! for the random mechanism 1048 dY0 ẳ a1 Y0 ỵ b1 Y1 dt dYn ẳ an Yn1 anỵ1 ỵ bn ịYn dt ỵ bnỵ1 Ynỵ1 ; for n dYN ẳ aN YN1 À bN YN dt ð2aÞ ð2bÞ N À1 ð2cÞ where an and bn are effective rate constants of phosphorylation and dephosphorylation FEBS Journal 274 (2007) 1046–1061 ª 2007 The Authors Journal compilation ª 2007 FEBS C Salazar and T Hofer ¨ Kinetic models of multisite phosphorylation A KT ¼ K þ N X Xn;K n¼0 N X Yn =Ln ¼K 1ỵ ỵ K=Ln ỵ P=Qn nẳ0 ! 4aị PT ẳ P ỵ N X Xn;P nẳ0 N X Yn =Qn ẳP 1ỵ ỵ K=Ln ỵ P=Qn nẳ0 ! ð4bÞ B Fig Model for multiple phosphorylation cycles (A) Schematic representation of a phosphorylation–dephosphorylation cycle (B) Mathematical model for a sequential mechanism of multiple phosphorylation based on the schema of Fig 1A The free form of the n-times phosphorylated substrate (n ¼ 0, 1, , N) is represented by Xn,0 The kinase–substrate and phosphatase–substrate complexes are denoted by Xn,K and Xn,P, respectively The rate constants for phosphorylation of Xn,K and dephosphorylation of Xn,P are denoted by an + and bn, respectively Ln and Qn are the dissociation constants for the complexes Xn,K and Xn,P, respectively K=LnÀ1 ; an |{z} ỵ K=Ln1 ỵ P=Qn1 |{z} catalytic fraction of rate of kinase kinaseÀbound target protein P=Qn bn ẳ bn |{z} ỵ K=Ln ỵ P=Qn catalytic rate |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} fraction of of phosphatase phosphataseÀbound target protein an ¼ ð3Þ These account for both enzyme–target binding and the catalysis K and P denote the free concentrations of kinase and phosphatase, respectively, and Ln and Qn are the respective dissociation constants for the kinase–target and phosphatase–target interactions an and bn are the catalytic rate constants for addition or removal of the nth phosphoryl group, respectively Because the physical properties of the target protein will generally change with the number of phosphorylated residues, the kinetic parameters can depend on the target’s phosphorylation state The concentrations of free and target-bound kinase and phosphatase obey the conservation relations Equations (2)–(4) define the dynamics of sequential multisite phosphorylation ⁄ dephosphorylation Although the differential Eqns (2) are linear in the concentration variables Yn, the full system is rendered strongly nonlinear through the nonlinear dependence of the effective rate constants (Eqn 3) on the enzyme concentrations and the conservation relations (Eqn 4) This has the remarkable consequence that, in general, no enzymekinetic rate laws can be derived for the kinase and phosphatase Moreover, Eqn (3) shows that the phosphorylation can be directly inhibited by the phosphatase (and dephosphorylation by the kinase) due to competition of the two enzymes for the target Indeed, there is experimental evidence for kinases and phosphatases competing for binding to their targets [39] Assuming the rapid-equilibrium approximation, the dynamics of target phosphorylation are determined by the balance between the phosphorylation and dephosphorylation rates of the several phosphorylation forms of the target protein After a sufficiently long time span, these rates balance, and the system will reach a steady state at which the concentrations not change At steady state, the concentrations of the various phosphorylation states are given explicitly by n¼0 < YT =D N  P  i aj  Y n ¼ YT n Pj¼1 b : D Piẳ1 b n N; D ẳ ỵ i iẳ1 j 5ị where YT ẳ N X Yn nẳ0 is the total concentration of target protein Eqn (5) is subject to the conservation conditions (Eqn 4), so that the solution must generally be computed numerically Analytic solutions for the steady state ) comparison of sequential and random mechanisms We begin the analysis with the special case that the enzymes bind to the target protein comparatively FEBS Journal 274 (2007) 1046–1061 ª 2007 The Authors Journal compilation ª 2007 FEBS 1049 Kinetic models of multisite phosphorylation C Salazar and T Hofer ă weakly Then, Eqns (2)(4) can be simplied considerably, and informative explicit results can be derived with respect to the steady-state response of the system (discussed here) and its kinetics (see next subsection) Weak binding corresponds to high values of the dissociation constants Ln and Qn, implying that the free enzyme concentrations are approximately equal to the total concentrations: K % KT and P % PT (see Eqn 4) The effective rate constants then simplify to an % anKT ⁄ Ln ) and bn % bnPT ⁄ Qn ) This can be further simplified when the dissociation constants are independent of the target’s phosphorylation state (Ln ¼ L and Qn ¼ Q for all n) and the same also holds for the catalytic rate constants (an ¼ a and bn ¼ b) Then we have, for the steady-state fraction of the  n-times phosphorylated target, n ¼ Y n =YT : y n ¼ y rn ðr 1ị rNỵ1 6ị The crucial parameter combination of rate constants, enzyme concentrations and affinities is r¼ a aKT =L ẳ b bPT =Q 7ị bearing in mind the assumption of weak enzyme binding r is a measure of the stimulus strength The analysis of nonsequential phosphorylation mechanisms is generally more complicated, due to the large number of phosphorylation states However, the fully random scheme depicted in Fig (second panel) can be analyzed in a similar manner when we again assume that the kinetic parameters not depend on the target’s phosphorylation state (Ln ¼ L, Qn ¼ Q, an ¼ a and bn ¼ b for all n) As shown in supplementary Doc S2, the system dynamics can be deduced by lumping all n-times phosphorylated target molecules into a single class regardless of the position of the phosphorylated residues The corresponding concentration variables will again be denoted by Yn, as indicated in Fig 1A (second panel) The Yn values are determined by a system of algebro-differential equations of the form of Eqns (2)–(4) when the following replacements are made in Eqn (2): an ! ðN À n ỵ 1ịa; bn ! nb 8ị These relations indicate that an n-times phosphorylated substrate can be further phosphorylated on N ) n different residues and dephosphorylated on n residues In this way, the random scheme is mapped to a linear chain of reactions, in which the effective phosphorylation rate decreases with increasing phosphorylation of the target (because fewer unphosphorylated sites 1050 remain) while the effective dephosphorylation rate increases (because more sites become available to the phosphatase) At steady state, we find for the fraction of n-times phosphorylated targets   rn N n ẳ 9ị y n ỵ rÞN where  N n  is the binomial coefficient, and r was defined in Eqn (7) In the limiting case of a target with a single phosphorylation site (N ¼ 1), its phosphorylated fraction is a hyperbolic function of r [Eqn (6) and Eqn (9) then coincide] For sequential multisite phosphorylation (N > 1), the concentration of the fully phosphorylated protein becomes a sigmoid function of r (Fig 3A) Thus, multiple phosphorylation can give rise to more threshold-like responses to changes in catalytic activity or concentration of kinase or phosphatase than a single phosphorylation site This is particularly seen for low kinase ⁄ phosphatase activity ratios, where the phosphorylation sets in more sharply when N is large However, the overall range of kinase-to-phosphatase activities over which a switch from the unphosphorylated to nearly fully phosphorylated target is achieved varies only moderately with N This limited overall steepness of the response curve for complete phosphorylation is linked with the fact that over a sizeable range of kinase ⁄ phosphatase activity ratios, much of the target protein exists in partially phosphorylated states (Fig 3B) Only at such extreme ratios does the target becomes fully phosphorylated or unphosphorylated For the random mechanism, the response curve for the fully phosphorylated form is less steep than for sequential processing (Fig 3C) Correspondingly, partially phosphorylated forms are overall more abundant in the steady state (Fig 3D); in Eqn (9), this is reflected by the binomial coefficient, which reaches its maximum for n ¼ N ⁄ Further analysis showed that the cyclic mechanism depicted in the lower panel of Fig 1A has an even less steep response curve We quantified the overall steepness of the response curve by means of the effective Hill coefficient nH ¼ ln 81 ⁄ ln R, where the global response coefficient R is the ratio of the concentration of active kinase K0.9 at which there is 90% fully phosphorylated target to the kinase concentration K0.1 at which 10% of the target is fully phosphorylated, R ¼ K0.9 ⁄ K0.1 [13] For the sequential mechanism, the effective Hill coefficient ranges between and (Fig 3E) For random and mixed FEBS Journal 274 (2007) 1046–1061 ª 2007 The Authors Journal compilation ê 2007 FEBS C Salazar and T Hofer ă Kinetic models of multisite phosphorylation A C B D E Fig Steady-state behavior and the order of phosphate processing (A) Steady-state behavior of the fully phosphorylated fraction yN as a function of the kinase ⁄ phosphatase concentration ratio KT ⁄ PT (stimulus strength) for different numbers of phosphorylation sites N in the case of a sequential mechanism (B) Phosphorylation fractions yN as a function of KT ⁄ PT for a sequential mechanism and N ¼ (C) Steadystate behavior of the fully phosphorylated fraction yN as a function of KT ⁄ PT for the sequential (solid black line), cyclic (solid gray line) and P random (dashed black line) mechanisms (D) Steady-state behavior of the sum of the partially phosphorylated fractions NÀ1 yN as a function n¼1 of KT ⁄ PT for the sequential (solid black line), cyclic (solid gray line) and random (dashed black line) mechanisms (E) Comparison of the effective Hill coefficient for sequential (filled black boxes), cyclic (filled gray boxes) and random phosphorylation (open boxes) with variation of the number of phosphorylation sites N Hill coefficients corresponding to mixed schemes (random phosphorylation and sequential dephosphorylation or vice versa) are situated between the curves corresponding to the sequential and random schemes Parameters: an ¼ bn ¼ 1, Ln ¼ Qn ¼ [in (A–E)]; N ¼ ([in (B–D)] sequential-random mechanisms, nH is generally smaller Thus, multisite phosphorylation is not a sufficient condition to generate switch-like responses Phosphorylation kinetics ) sequential versus random mechanisms Given that physiologic stimuli are generally transient, the kinetics of signal transduction in relation to the stimulus timing can play a crucial role in cellular responses Moreover, the molecular steps of the cell cycle and the circadian oscillator need to be precisely timed, and multisite phosphorylation has been implicated in this [22,33] How long does it take for a multisite target to reach a new phosphorylation state after a change in kinase or phosphatase activities? Explicit solutions can be obtained for the fully phosphorylated target under the assumption that enzyme binding is FEBS Journal 274 (2007) 1046–1061 ª 2007 The Authors Journal compilation ª 2007 FEBS 1051 Kinetic models of multisite phosphorylation C Salazar and T Hofer ă N yN tịịdt y N À yN ð0Þ y ð10Þ where yN (0) and N are the steady states before and y after the transition [20,38,40] For the sequential mechanism, we obtain sN ¼ Nr 1ịrNỵ1 ỵ 1ị 2rrN 1ị bPT =Q r 1ị2 rNỵ1 1ị HN HN ẳ sN ẳ bPT =Q ỵ r a ỵ b 15 10 N=1 B where 10 100 Random b1 ð13Þ 0.1 Stimulus strength, KT /PT ð12Þ so that the transition always becomes faster when the effective rate constants of kinase (a) or phosphatase (b) are increased This fact holds true independently of whether phosphorylation or dephosphorylation of the target occurs as a result of the change in enzyme activity For a multisite target, this is no longer the case Let us consider the switching-on of an initially inactive kinase (r ¼ 0) In supplementary Doc S3, we show that for N > phosphorylation sites, the transition time exhibits a maximum for intermediate values of r (Fig 4A) The maximum occurs near the point where the effective rate constants for kinase and phosphatase balance, r ¼ At this point, sN becomes proportional to 2N + N2: the phosphorylation time increases quadratically with the number of phosphorylation sites The transition time sN for the random mechanism is obtained as Sequential 20 0.01 ð11Þ where r was defined in Eqn (7) (for details, see supplementary Doc S3) For a single-site target, we obtain 1 ¼ s1 ¼ bPT =Q þ r a þ b Time constant, sN ¼ 25 Time constant, R A b weak (see previous section) The transition time for changes in concentration of the fully phosphorylated target is appropriately defined as 4 2 N=1 0.01 0.1 10 100 Stimulus strength, KT /PT Fig Transition time and the order of phosphate processing The transition time s is plotted as a function of KT ⁄ PT for different values of N in a sequential mechanism (A) and in a random mechanism (B) of phosphate processing Parameters: an ¼ bn ¼ 1, Ln ¼ Qn ¼ 1, N ¼ 1, 2, 4, in the random mechanism increases only moderately with the number of phosphorylation sites This is in stark contrast to the sequential mechanism, where the phosphorylation time increases even stronger than linearly with the number of sites HN ¼ RN 1=i i¼1 is the Nth harmonic number (for details, see supplementary Doc S3) Hence, the transition time of a random multisite phosphorylation has the same dependence on the effective kinase and phosphatase activities, a and b, respectively, as the transition time for singlesite phosphorylation The number of phosphorylation sites only comes into play through the constant factor HN Phosphorylation of multisite targets is achieved much faster by a random mechanism than by a sequential one (Fig 4B) Moreover, HN grows approximately as fast as ln N, so that the phosphorylation time 1052 Plasticity of regulation In the previous sections, we have analyzed the model in a special case (weak enzyme binding and phosphorylation-independent kinetic parameters), which has allowed us to elucidate the role of phosphorylation order in the steady-state response and the kinetics However, the kinetic parameters and enzyme concentrations may also play a decisive role in shaping the behavior of the system We have therefore conducted numerical simulations of Eqn (2)–(4) in which the system parameters were varied systematically FEBS Journal 274 (2007) 1046–1061 ª 2007 The Authors Journal compilation ª 2007 FEBS C Salazar and T Hofer ¨ To quantify the steepness of the response curve of the system, we used the effective Hill coefficient nH as introduced above, where steeper, more switch-like responses are associated with nH values considerably larger than The concentration of active kinase KT was considered as the changeable control parameter, whereas the phosphatase concentration PT was fixed Ultrasensitive responses We found that the shape of the response curve is strongly affected by the two groups of parameters that determine the protein–protein interactions: the concentrations of the enzymes relative to the target protein, and the respective dissociation constants Figure 5A shows the results for a protein with N ¼ phosphorylation sites The target ⁄ enzyme concentration ratio is expressed in terms of the phosphatase concentration (the kinase concentration range in which changes in target phosphorylation occurs is effectively determined by the phosphatase concentration) Two regions are visible in this ‘phase diagram’, where the effective Hill coefficient becomes much larger than unity (dark areas), indicating high sensitivity of the phosphorylation state to changes in kinase activity This ultrasensitivity depends also on the dissociation constants for the kinase–target and phosphatase–target interactions To be specific, the dissociation constants of the various phosphorylation states of the target for the kinase were all set equal to L0, except for the value LN for > the fully phosphorylated target If LN > L0, the kinase readily leaves the fully phosphorylated target Con< versely, if LN < L0, the kinase will remain preferentially associated with the phosphorylated target, which, in the language of enzyme kinetics, is referred to as product inhibition of the enzyme For the phosphatase, Q0 was similarly allowed to differ from the other equal dissociation constants, which < were all set to QN (for Q0 < QN, we then have product inhibition of the phosphatase) The two-dimensional diagram in Fig 5A depicts the special case in which the degree of product inhibition is the same for both kinase and phosphatase, where we found the most pronounced occurrences of ultrasensitivity First, ultrasensitivity is obtained when the enzymes are saturated by the target protein and both enzymes dissociate readily from their respective endproducts (upper left-hand corner of the diagram) For a target protein with a single phosphorylation site, these are precisely the conditions for the occurrence of so-called zero-order ultrasensitivity [13,20] Hence, zero-order ultrasensitivity can also be found for multisite phosphorylation Second, ultrasensitivity occurs Kinetic models of multisite phosphorylation also with the diametrically opposed parameter constellation of large enzyme concentrations and strong product inhibition (lower right-hand corner of the diagram) Bistable responses Thus far, we have considered the effects of enzyme concentrations and enzyme affinities for the target protein In addition, the catalytic rate constants of (de)phosphorylation could be different for each particular residue Specific combinations of these three kinds of parameter can give rise to bistability in the response of the system Bistability would impart very special properties, such as sharp response thresholds and hysteresis The first theoretical evidence of this phenomenon in multisite phosphorylation has been recently presented for the doubly phosphorylated MAPK [17] Figure 5B shows how concentration and affinities affect the shape of the response curve as in Fig 5A, but assuming now that the first phosphorylation and dephosphorylation steps are slower than the other steps, a1