BioSystems xxx (2005) xxx–xxx

Kinetic models of phosphorylation cycles: A systematicapproach using the rapid-equilibrium approximation

for protein–protein interactions

Carlos Salazar, Thomas Hofer∗

Theoretische Biophysik, Institut fur Biologie, Humboldt-Universitat zu Berlin, Invalidenstr. 42, D-10115 Berlin, Germany

Received 29 March 2005; received in revised form 21 April 2005; accepted 10 May 2005

Abstract

Activation–inactivation cycles of signalling proteins and transcription factors catalysed by kinases and phosphatases are a corecomponent of cellular signal transduction. We present a systematic kinetic analysis of a phosphorylation cycle that starts from thedescription of elementary protein–protein interaction and catalytic steps. A rapid-equilibrium approximation for protein interactionsis used to reduce the set of parameters. The resulting description consists of a kinetic equation for the phosphorylation of the targetand a set of conservation conditions for kinase and phosphatase. Generally no explicit rate laws exist for the two enzymes; linearor Michaelis–Menten rate equations can be obtained in special cases. Key parameters that determine the stimulus–response curveand the response time of the cycle are the concentrations of kinase and phosphatase relative to the target protein and the affinitiesof the two enzymes for the different phosphorylation states of the target. Characterizing the response curve by the global responsec a previouslyu hibition ofe elative targeta horylationa©

K

1

kc1

f

canhergletoin-at-

otfor

rgyites

0

2

oefficient, we obtain a phase diagram that shows the existence of three kinds of behaviours: graded, ultrasensitive, andndescribed biphasic response. Two kinds of competition effect turn out to shape the behaviour: (1) the degree of product inach enzyme, and (2) the competition between kinase and phosphatase to bind the target protein, as determined by their rffinities. The approach outlined here may be useful also for analysing more complex systems, including multiple phospnd kinase cascades.2005 Elsevier Ireland Ltd. All rights reserved.

eywords: Transition time; Stimulus–response relation; Zero-order ultrasensitivity; Michaelis–Menten kinetics; NFAT

. Introduction

Cycles of reversible phosphorylation catalysed byinases and opposing phosphatases form the centralomponent of intracellular signalling pathways (Krebs,981; Karin and Hunter, 1995; Gerhart, 1999). Addition

∗ Corresponding author. Tel.: +49 30 2093 8592;ax: +49 30 2093 8813.

E-mail address: thomas.hoefer@biologie.hu-berlin.de (T. Hofer).

or removal of phosphate groups in a target proteincontrol its enzymatic activity, the interaction with otproteins or DNA, or its stability. For example, a sinphosphorylation of protein kinases displaces an auhibitory loop, allowing substrate binding and/or activing catalysis (Adams, 2003). Many proteins contain nonly one but several phosphorylation sites, allowinggreat versatility of regulation (Cohen, 2000; Holmbeet al., 2002; Pawson, 2004). Each site may individuallaffect a specific function of the protein, or many scan regulate one function in a concerted manner (Nash

303-2647/$ – see front matter © 2005 Elsevier Ireland Ltd. All rights reserved.doi:10.1016/j.biosystems.2005.05.015

BIO-2489; No. of Pages 1

2 C. Salazar, T. Hofer / BioSystems xxx (2005) xxx–xxx

et al., 2001). An intriguing example of concerted reg-ulation is provided by the transcription factors of theNFAT family, in which more than 10 serine residues con-spire to control nuclear localization (Hogan et al., 2003).This appears to be mediated by an allosteric mechanism,where the phosphorylation state determines the equilib-rium between active and inactive global conformationsof the protein (Okamura et al., 2000; Salazar and Hofer,2003).

In the absence of external stimulation, cycles of pro-tein phosphorylation exist in a steady state in whichboth phosphorylation and dephosphorylation reactionstake place. External stimuli that become transduced intoactivity changes of kinases or phosphatases will shift thephosphorylation state of the target protein. The kineticsof this process has been the subject of a number of theo-retical and experimental studies. A phenomenon that hasattracted much attention is the possibility that phospho-rylation cycles can have very steep stimulus–responsecurves, even when only a single phosphorylated residueis involved (Stadtman and Chock, 1977; Goldbeter andKoshland, 1981; Small and Fell, 1990). This so-calledzero-order ultrasensitivity has indeed been observedin vitro (Shacter et al., 1984a; Cimino and Herva-gault, 1987). Subsequently, other mechanisms for cre-ating response curves with steep thresholds have beenproposed, including phosphorylation cascades, multi-site phosphorylation and bistability (Huang and Ferrell,1996; Ferrell and Bhatt, 1997; Ferrell and Machleder,1998; Kholodenko, 2000; Bhalla et al., 2002; Salazar

-li inarriedsh innt forects

al.,dieslifiedh aretion,allye all

yettionprob-rse.ng ton

partners may also be bound together in a multi-proteincomplex, possibly organized by scaffold or adaptor pro-teins. Enzymes do not only bind their substrates butcan also remain tightly associated with their products,such as shown for phosphorylated NFAT1 and one ofits kinases (Okamura et al., 2004). Second, multiplephosphorylation (dephosphorylation) may be achievedby sequential, random, or mixed mechanisms. Together,the different possibilities for protein–protein interactionsand phosphorylation order can create a large numberof molecular microstates, which has been referred to ascombinatorial complexity (e.g.,Hlavacek et al., 2003).A third difficulty arises from the fact that the chiefmeans of simplification in traditional enzyme kinet-ics is of very limited applicability for signal trans-duction. In many reactions in metabolism, enzymesoccur at much lower concentrations than their substrates,which allows the powerful use of quasi-steady stateapproximations, underlying, for example, the derivationof the Michaelis–Menten rate law (Cornish-Bowden,2004). For signal transduction, relevant concentrationmeasurements are still rare (for first examples seeGhaemmaghami et al., 2003; Lee et al., 2003). However,the target proteins of phosphorylation cycles are fre-quently enzymes (such as the mitogen-activated proteinkinase, MAPK, being the substrate of the MAPK kinasesand phosphatases) or transcription factors (which oftenhave low concentrations). Moreover, many proteinkinases and phosphatases have several substrates, whichcan create complex networks of phosphorylation cycles.

on-andntra-

f afirstaph.we

sso-pho-theichnottantsts ofrva-

tein.tar-neraltheseasiceen

and Hofer, 2003; Markevich et al., 2004). A theoretical analysis of the processing of transient stimuphosphorylation cycles and cascades has been cby Heinrich et al. (2002). Among other results, it wademonstrated that the inactivating enzymes (whicmost cases are phosphatases) are very importadetermining the duration and speed of signalling. Effof scaffold proteins have also been considered (Brayand Lay, 1997; Levchenko et al., 2000; Heinrich et2002). The mathematical description in these stuhas been remarkably heterogeneous. Greatly simpoverall kinetics for kinases and phosphatases, whicof first-order in the respective substrate concentraMichaelis–Menten rate laws, or models which partiinclude elementary binding and reaction steps havbeen used.

The formulation of mechanistically appropriatetractable and insightful models of phosphorylacycles and cascades indeed meets with challenginglems. First, the protein–protein interactions are diveKinases and phosphatases can compete for binditheir target (Tanoue et al., 2000), but the three interactio

For all these reasons, it is very unlikely that strict ccentration hierarchies exist in which both kinaseopposing phosphatase occur always in lower concetion than their respective target.

In this work we analyse a general kinetic model ophosphorylation cycle, addressing, in particular, theand the third problem raised in the preceding paragrTo treat protein–protein interactions systematically,make use of the fact that very often the protein aciations occur on faster time scales than the phosrylation/dephosphorylation reactions. This allowsapplication of a rapid-equilibrium approximation, whimplies that only the protein–protein affinities butthe individual association and dissociation rate consneed to be considered. The resulting model consisa canonical kinetic equation augmented by consetion relations for the enzymes and the target proApplying this approach to a singly phosphorylatedget protein, key parameters are identified, and a gecharacterization of the behaviour in dependence onparameters is given. We find a novel type of biphresponse relation resulting from the competition betw

C. Salazar, T. Hofer / BioSystems xxx (2005) xxx–xxx 3

kinase and phosphatase for the target as well as recoverpreviously studied models as limiting cases.

2. Linear model: response curve, transition time,and optimal phosphatase activity

The simplest model of a phosphorylation cycle isobtained when kinase and phosphatase reactions areassumed to follow linear rate laws with the first-orderrate constants,α andβ, respectively (Scheme 1).

Denoting the concentrations of unphosphorylated andphosphorylated protein byX0 andX1, respectively, wehave the kinetic equation

dX1

dt= α(XT − X1) − βX1, (1)

where we have used the conservation conditionX0 + X1= XT = const. When phosphorylation and dephosphory-lation rates balance, the phosphorylated protein fractionx1 = X1/XT is given by

x1 = α/β

1 + α/β, (2)

which is a hyperbolic function of the ratio of kinase andphosphatase activities (Fig. 1a). When the activity ofkinase or phosphatase, or both enzymes, is regulated, thisrepresents the stimulus–response curve of the system.

Eq. (2) shows that kinase and phosphatase activitiesm andp tane-o d ins on-t e thatt n therd rgetp

Scheme 1.

which attains a maximum when the phosphatase rateconstant has the value (Fig. 1b)

β∗ = √α0α1. (4)

For β* the largest phosphorylation response isachieved on activation of the kinase. When the phos-phatase is more active, less substrate becomes phos-phorylated, while for a less active phosphatase moresubstrate is already phosphorylated under resting con-ditions. This result shows that there is an optimal phos-phatase activity for the response of the cycle.

Whenα andβ are fixed to constant values, the steadystate is approached exponentially. It is convenient todefine the transition time from an initial state of the sys-tem to the final steady state as usual as the inverse of theexponential decay rate:

τ = 1

α + β. (5)

The transition time depends equally on the kinase andphosphatase rate constants (Fig. 1c). A more formal def-inition of τ for monotonic and exponential approachesto the steady state is (Llorens et al., 1999):

τ =∫∞

0 (x1 − x1(t)) dt

x1 − x1(0). (6)

In the case of the kinetic equation(1) it yields Eq.(5).However, the definition will prove very useful for morecomplex systems.

lin-loyed

hatase

ust be tuned to each other. Typically, both kinasehosphatase of a phosphorylation cycle are simulusly active. For example, many kinases involveignal transduction have a resting activity that is cinuously counteracted by phosphatases. Supposhe rate constant of such a kinase changes betweeesting valueα0 and a maximally activated valueα1. Theifference in the concentration of phosphorylated tarotein between the two states is

�X1

XT

= (α1 − α0)β

(α0 + β)(α1 + β). (3)

Fig. 1. Response curve, transition time, and optimal phosp

3. Molecular model of a phosphorylation cycle

3.1. Model equations

So far, we have not given a justification for usingear rate equations. Indeed, some authors have emp

activity in a linear model. (a) ¯x1 vs.α/β; (b) �x1 vs.β; (c) τ vs.α +β.

4 C. Salazar, T. Hofer / BioSystems xxx (2005) xxx–xxx

Fig. 2. Molecular model of a phosphorylation cycle. The free form ofthen times phosphorylated substrate (n = 0, 1) is represented byXn,0,and the kinase–substrate and the phosphatase–substrate complexes aredenoted byXn,K andXn,P, respectively. The rate constants for phos-phorylation ofX0,K and dephosphorylation ofX1,P are denoted byaandb, respectively.Ln andQn are the dissociation constants for thecomplexesXn,K andXn,P, respectively.

Michaelis–Menten laws instead (e.g.Goldbeter andKoshland, 1981). However, the Michaelis–Menten equa-tion is justified mainly when the enzyme concentrationis small (seeSegel, 1988, for an extensive discussion),which is not generally expected in signal transduction. Amore realistic and detailed description accounts for theelementary steps of enzyme binding and catalysis sep-arately.Fig. 2 shows the simplest scheme of this kind.This description requires the explicit consideration ofthe protein–protein interactions. To be specific, we haveassumed that either kinase or phosphatase can bind to thetarget protein, but not both enzymes together (Tanoue etal., 2000).

Its kinetic description of this scheme is more involvedthan the simple model (1). Let us denote the concen-tration of unbound target protein byXn,0 and the con-centrations of kinase–target and the phosphatase–targecomplexes byXn,K andXn,P, respectively. The subscriptn accounts for the phosphorylation state. Consideringa single phosphorylation site, we haven = 0, 1. It isstraightforward to write down the balance equations forthe six concentrations of free target protein and the com-plexes (unphopshorylated and phosphorylated), as wellas the three conservation conditions for target, kinase andphosphatase. Together, they constitute the elementary-step model (Appendix A).

This model can be simplified considerably when thebinding and the dissociation of the enzymes, whichoccurs typically on a sub-second scale, is faster thanthe phosphorylation and dephosphorylation reactions. A

tion

reactions is seconds to minutes, so that this assumptionwill apply widely (e.g.,Shaw et al., 1995; Lee et al.,2003; Okamura et al., 2004). Then the fundamental vari-ables are the total concentrations of unphosphorylatedand phosphorylated target protein:

Yn = Xn,0 + Xn,K + Xn,P, for n = 0, 1, (7)

together with the free concentrations of kinase and phos-phatase K and P, respectively. The lumping of the free andenzyme-bound target fractions into a single variable inEq.(7) is possible because free target and the complexesare in equilibrium. As detailed inAppendix A, the con-centrations of the various enzyme–substrate complexesare given by

Xn,K = Yn

K/Ln

1 + K/Ln + P/Qn

,

Xn,P = Yn

P/Qn

1 + K/Ln + P/Qn

, for n = 0, 1. (8)

L0 and L1 are the dissociation constants for thekinase–substrate complex for the unphosphorylated andphosphorylated forms of the substrate, respectively.Likewise,Q0 andQ1 are the dissociation constants forthe phosphatase–substrate complex. Using these rela-tions, the dynamics of the phosphorylated target proteinis described by the kinetic equation

dY1

dt= a

K/L0

1 + K/L0 + P/Q0Y0︸ ︷︷ ︸

os-.on-

al

characteristic time scale for many (de)phosphoryla

t

phosphorylation rate

− bP/Q1

1 + K/L1 + P/Q1Y1︸ ︷︷ ︸

dephosphorylation rate

, (9)

wherea andb are the catalytic rate constants for phphorylation and dephosphorylation respectively. Eq(9)is subject to the conservation conditions for the total ccentrations of substrate

Y0 + Y1 = YT = const., (10)

and the two enzymes

K

(1 +

1∑n=0

Yn/Ln

1 + K/Ln + P/Qn

)= KT = const.,

(11)

P

(1 +

1∑n=0

Yn/Qn

1 + K/Ln + P/Qn

)= PT = const.

(12)Eqs. (9)–(12) form a nonlinear algebro-differenti

equation system forYn(t), K(t) andP(t).

C. Salazar, T. Hofer / BioSystems xxx (2005) xxx–xxx 5

3.2. Special case I: linear rate laws

Before turning to the general analysis of the system(9)–(12), it is useful to consider special cases, wherethe relative concentrations of enzymes and target playa central role. In particular, for very large concentra-tions of kinase and phosphatase, the target-bound frac-tions are negligible compared with the free enzymes.InsertingK ≈ KT andP ≈ PT into Eq. (9), one obtainsa linear kinetic equation forY1(t). Effective first-orderrate constants for kinase and phosphatase are defined asfollows:

α = a︸︷︷︸catalytic rate of kinase

KT /L0

1 + KT /L0 + PT /Q0︸ ︷︷ ︸fraction of kinase-bound target protein

and β = b︸︷︷︸catalytic rate of phosphatase

PT /Q1

1 + KT /L1 + PT /Q1︸ ︷︷ ︸fraction of phosphatase-bound target protein

.

(13)

Thus for large concentrations of both enzymes, thekinetic description indeed simplifies to Eq.(1).

Notably, a very similar simplification is obtained –independent of the values of the enzyme concentrations– when the dissociation constants for the enzyme–targetcomplexes do not depend on the phosphorylation stateof the substrate:L0 = L1 = L andQ0 = Q1 = Q. It is eas-ily shown that the conservation relations(11) and(12)then become decoupled from the kinetic equation(9).They can then be transformed into a third-order alge-braic equation for eitherK or P which no longer dependson Yn and solved once. Inserting the resulting val-u sr i-l asi

thee teindc ho-r tieso eira ingt gaino on-ck iment cana ,2

eeno . Fore ily,w are

fully dephosphorylated by the phosphatase calcineurinin less than one minute, while their phosphoryla-tion can take 10–20 min (Shaw et al., 1995; Okamuraet al., 2004). In T cells, stable complexes betweenNFAT1 and one of its kinases, CK1, have been found,while a complex between calcineurin and NFAT1 couldnot be isolated under the same conditions (Okamuraet al., 2004). However, the higher rate constant ofcalcineurin may compensate for the weaker bind-ing, so that effective dephosphorylation can still beachieved.

KT and a enter the effective rate constant of phos-phorylation in a different way (Eq.(13)). Thereforethe stimulus–response relations with respect to the twoparameters will be different. Changinga will lead toa hyperbolic relation as depicted inFig. 1a. VaryingKT may result in different kinds of stimulus–responsecurves, to which we turn now.

3.3. Competition for target protein by kinase andphosphatase

es, the

the-the. Thethe

ationto

n ofse

ingtase

is byichd to,

hos-sed

ti-r.

es K = const. andP = const. into Eq.(9) again giveise to a linear kinetic equation, with a very simar definition of effective first-order rate constantn Eq.(13).

Eq. (13) shows how both the catalytic rates ofnzymes and their association with the target proetermine the effective rate constantsα andβ. This indi-ates two different strategies for controlling the phospylation state of the target. First, the catalytic activif the enzymes,a andb, can be regulated. Second, thffinities for the target protein can be changed. Assum

hat an inactive enzyme cannot bind to the target, af affinity would amount to an increase in the total centration of the enzyme (eitherKT andPT). For proteininases, both strategies have been described experally, where an activating tyrosine phosphorylationllow substrate binding or accelerate catalysis (Adams003).

In particular, dephosphorylation reactions have bbserved to proceed faster than phosphorylationsxample, transcription factors of the NFAT famhich carry more than ten phosphorylation sites,

-

When the concentrations of the modifying enzymare rather large compared with their target proteinenzymes may compete for binding to the target. Eq.(13)shows how this effect is accounted for in the mamatical formalism: phosphorylation is inhibited byphosphatase and dephosphorylation by the kinaseextent of such competition can strongly influencesteady state of the system. Let us consider a situwhere an inactive form of the kinase cannot bindthe target protein and activation results in the gaitarget affinity. Thus the activating signal will increathe total concentration of kinase,KT, which is availablefor phosphorylation and, potentially, also for inhibitdephosphorylation by competing with the phosphafor the target.

These two effects can be separated in the analysfirst focusing on a system without competition, whis realized when kinase and phosphatase do not binrespectively, the phosphorylated form and the unpphorylated form of the target. This property is expresby L1 → ∞ andQ0 → ∞, in which case the competion terms in the denominators in Eq.(13) disappea

6 C. Salazar, T. Hofer / BioSystems xxx (2005) xxx–xxx

Fig. 3. Stimulus–response relation for high enzyme concentrations. (a) ¯y1 vs. logKT/PT for different values ofε. (b) y1 vs. logKT for differentYT/PT ratios. Parameters:a = 1, b = 4 (a)L0 = 0.1,ε = 0, 0.01, 1 (b)L0 = Q1 = 0.1,L1 = Q0 = 10,YT/PT = 0.01, 0.1, 10, 100.

The steady-state phosphorylated fraction is then a hyper-bolic function of KT, which approachesa/(a + b) forvery largeKT; an example is shown inFig. 3a (dashedline). The phosphorylated target fraction remains wellbelow unity even for very large ratios of kinase tophosphatase,KT/PT. The target protein cycles contin-uously between phosphorylated and unphosphorylatedstates because the unphosphorylated form interacts onlywith the kinase and the phosphorylated form only withthe phosphatase.

This situation changes when the kinase can alsobind the phosphorylated target and thereby prevent itsdephosphorylation. The relative affinities for the kinaseand phosphatase are measured by the ratioε = Q1/L1.When ε is small, the phosphorylated target will bindpredominantly to the phosphatase. However, even thena very large kinase concentration can lead to com-plete phosphorylation, as seen inFig. 3a (solid line).The response curve has a ‘biphasic’ shape with a dou-ble threshold. The second increase to unity seen forlarge KT is caused by the kinase sequestering moreand more of the phosphorylated target so that it is notavailable for dephosphorylation. Asε becomes larger,full phosphorylation of the target will be achieved bythis mechanism at smaller values ofKT (Fig. 3a, dottedline).

To study the impact of the target concentration onthe behaviour of the phosphorylation more system-atically, we have solved numerically the steady-stateequations for the general system(9)–(12) at varying

tasensehet. Forcter-rget,

3.4. Special case II: Michaelis–Menten kineticsobtained for small enzyme concentrations

Comparatively low concentrations of both kinase andphosphatase constitutes a second special case in whichthe governing Eqs.(9)–(12)can be simplified. When thefree enzyme concentrations are much smaller than theirrespective dissociation constants

K << Ln and P << Qn, for n = 0, 1,

the majority of the enzyme molecules will be bound tothe substrate at any given time. We can then neglect thetermsK/Ln andP/Qn in the denominators in Eqs.(9),(11)and(12), obtaining in place of Eq.(9)

dY1

dt= VK

(YT − Y1)/L0

1 + (YT − Y1)/L0 + Y1/L1︸ ︷︷ ︸phosphorylation rate

− VPY1/Q1

1 + (YT − Y1)/Q0 + Y1/Q1︸ ︷︷ ︸dephosphorylation rate

. (14)

Phosphorylation rate and dephosphorylation rateobey Michaelis–Menten rate laws, where the maxi-mal velocities are defined asVK = aKT and VP = bPT.Although both reactions are irreversible, their rates arealso affected by the concentrations of the respective reac-

und totion

rget,ess

concentrations of total target protein to phosphaYT/PT. Fig. 3b depict the resulting stimulus–respocurves with respect toKT, which depend strongly on trelative concentrations of the enzymes and the targelow concentrations of target, we observe the charaistic double-threshold curves, while for abundant taa single, very steep threshold is obtained.

tion products,Y1 for the kinase andY0 = YT − Y1 for thephosphatase, because the products can remain bothe enzymes. This is a typical case of product inhibiknown to occur in many enzymatic reactions (Cornish-Bowden, 2004).

The steady-state fraction of the phosphorylated tay1 = Y1/YT , can be written in terms of the dimensionl

C. Salazar, T. Hofer / BioSystems xxx (2005) xxx–xxx 7

parametersr = VK/VP, ln = Ln/YT, andqn = Qn/YT:

y1 = − l1

2

(2q1 + q0q1 − q0)r + q0(1 + l0)

l1(q0 − q1)r + (l0 − l1)q0

+√

l1q0[4l0q1(1 + l1 + q0)r + l1q0(l20 − 2l0((1 − q1)r − 1) + ((1 + q1)r − 1))]

2[l1(q0 − q1)r + (l0 − l1)q0](15)

In this special case, there is no difference in responsecurves regarding to whether the catalytic activity or thetarget affinity of the kinase (or the phosphatase) is reg-ulated, asVK (VP) is a linear function ofa andKT (bandPT). However, the response curve depends stronglyon the values of the dissociation constantsL0, L1, Q0andQ1 in two respects. First, the achievable degree ofsaturation of the enzymes is determined by the disso-ciation constants relative to the total concentration ofthe target protein. In particular, whenl0 � 1 andq1 � 1,kinase and phosphatase can be saturated with their sub-strates, the unphosphorylated and phosphorylated tar-get proteins, respectively. Second, the ratiosL0/L1 andQ0/Q1 determine the extent of product inhibition of theenzymes.

As previously shown byGoldbeter and Koshland(1981), the special case of no product inhibition(corresponding to the limitL1 → ∞ and Q0 → ∞)and substrate saturation of both enzymes (l0 � 1 andq1 � 1) yields strongly sigmoid stimulus–responsecurves (Fig. 4a, solid line). This phenomenon can betraced to the fact that the kinase and phosphatase kinet-i ero-o rderu nots r sig-mid and

no saturation (dashed line), and saturation and productinhibition (dotted line).

The kinetic design also strongly affects the transitiontime τ to the steady state (Fig. 4b). These results havebeen computed numerically on the basis of Eq.(6). Inparticular, the saturation of the enzymes with the tar-get protein leads to significantly slower kinetics than areobserved in the absence of saturation. Moreover, the tran-sition time corresponding to the sigmoid response curveexhibits a maximum at the position of the threshold inthe response curve.

3.5. Classification of stimulus–response curves inparameter space

Two types of parameters have emerged from the anal-ysis to control the stimulus–response curves of phospho-rylation cycles: the extent of saturation of kinase andphosphatase with the target protein and the degree ofproduct inhibition. To compare different response curveswe have computed the global response coefficientR(Goldbeter and Koshland, 1981). It is defined by theratio of total kinase concentration required to obtain 90%of phosphorylated target to the kinase concentration atwhich 10% of the target are phosphorylated:

R = KT (y1 = 0.9)

KT (y1 = 0.1)(16)

F enzym rea es), no nd producti nd dasa

cs operate predominantly in the fully saturated, “zrder” range. It has therefore been termed zero-oltrasensitivity. If either one of the two conditions isatisfied, the stimulus–response curve is no longeoid but, like the hyperbolic curve inFig. 1a, attains

ts greatest slope for vanishing kinase activity.Fig. 4aepicts two examples: absence of product inhibition

ig. 4. Stimulus–response relation and transition time for smallnalyzed: enzyme saturation and no product inhibition (solid lin

nhibition (dotted line). Parameters:L0 = Q1 = 1,L1 = Q0 = 106 (solid and dotted lines).

e concentrations. (a) ¯y1 vs.VK/VP. (b) τ vs.VK/VP. The following cases asaturation and no product inhibition (dashed line), saturation a

hed lines),L1 = Q0 = 1 (dotted line),YT = 1 (dashed line),YT = 50 (solid

8 C. Salazar, T. Hofer / BioSystems xxx (2005) xxx–xxx

Fig. 5. Effect of enzyme saturation and product inhibition on the stimulus sensitivity for a single phosphorylation cycle. Contour plot of the effectiveHill coefficient as function of the enzyme saturation (measured byYT/PT) and product inhibition (measured byL0/L1 = Q1/Q0). The dark areasdenote regions of high stimulus sensitivity (high Hill coefficients). The stimulus–response curves for particular parameter values are depicted inthesmall boxes (solid lines) and compared with the hyperbolic Michaelis–Menten kinetics (dashed line). Parameters:an = bn = 1, N = 1.

and thus indicates by how much the kinase activity mustbe increased from a basal level to strongly phosphorylatethe target.R is related to the effective Hill coefficient ofthe stimulus–response curve by

nH = ln 81

ln R. (17)

A value ofnH = 1 implies that the stimulus–responsecurve has the same response coefficient as a hyperbolic,Michaelis–Menten-like function, while values ofnH > 1are indicative of sigmoid responses.

We have varied the kinase concentrationKT forparameter sets differing with respect to enzyme satura-tion and product inhibition, and for each set the effectiveHill coefficient of the response curve was computed.Enzyme saturation was measured by the ratioYT/PT

(the KT where effective phosphorylation is achievedwill match PT to some extent). For comprehensibilityof the graphical representation, we have taken the samedegree of product inhibition for kinase and phosphatase,

as expressed by the ratios of the substrate dissociationconstantsL0/L1 = Q1/Q0. Plotting the effective Hill coef-ficient as a function of these two parameters yields awell-structured ‘phase diagram’, as shown inFig. 5. Tworegions of high sensitivity correspond to the parameterranges of zero-order ultrasensitivity, on the one hand,and strong product inhibition as well as low enzyme sat-uration, on the other. Very low sensitivity withnH < 1(sometimes termed subsensitivity) is obtained for lowenzyme saturation and absence of product inhibition.Here continuous cycling of substrate between phospho-rylated and unphosphorylated forms is particularly pro-nounced. Also note that the dashed line gives the locusof a hyperbolic response curve which is independent ofthe degree of enzyme saturation and can be expressed inanalytical form (see Section3.1).

These calculations show that different types ofresponse curves can be obtained for specific choices ofthe kinetic parameters and enzyme concentrations. How-ever, because of the scarcity of experimental information

C. Salazar, T. Hofer / BioSystems xxx (2005) xxx–xxx 9

Fig. 6. The hyperbolic curve is the most frequent stimulus–response behaviour. Histogram of the effective Hill coefficient obtained when allparameters are assigned at random (1000 choices), and (a) the kinase concentrationKT is varied (activation of target binding), or (b) the catalyticrate of phosphorylationa is varied (catalytic activation). The logarithm of each parameter was assumed to be equally distributed over several ordersof magnitude. Parameters:N = 1, 10−2 < L0, L1, Q0, Q1, YT < 102 �M (in (a) and (b)); 10−2 < a/b < 102 (in (a)); 10−2 < KT/PT < 102 (in (b)).

on these parameters, we cannot at present compare ourtheoretical results directly with data. We therefore askedwhich kind of response curve results when all parametersare assigned at random. To obtain realistically large vari-ations of these parameters, we assumed their logarithmsto be equally distributed over several orders of magni-tude. Interestingly, for both ways of kinase activation –increase in target affinity and in catalytic activity – themost common effective Hill coefficients are around unity(Fig. 6). Comparatively few random parameter combina-tions (≈5%) yield Hill coefficients larger than two. Thusthe typical stimulus–response curve of a single phos-phorylation cycle tends to be of (near) hyperbolic shape,while ultrasensitivity requires very specific parametercombinations.

4. Discussion

In this work we have developed a general kineticdescription of reversible protein phosphorylation. Start-ing from a description of the elementary protein–proteininteraction and reaction steps, we have shown how theapplication of the rapid-equilibrium approximation forthe former yields a simplified model, consisting of asingle kinetic equation augmented by conservation rela-tions. The model shows that there exist no explicitexpressions for general rate laws of kinase and phos-phatase. Thus, unlike in metabolic modelling where rateequations are widely used, the kinetic description of sig-nalling pathways will generally have to start from a con-s ucedt

r ofu tionr nts)h e-

ters that govern the qualitative behaviour of phospho-rylation cycles. Particular attention has been paid tothe stimulus–response curve that characterizes how thephosphorylated fraction of the target protein changes asa function of the kinase (or phosphatase) activity. Theshape of these relations has been found to be determinedby the relative concentrations of kinase, phosphatase andtarget protein as well as the respective dissociation con-stants. We have obtained a systematic characterization bymeans of a ‘phase diagram’ which, interestingly, does notonly contain known behaviours such as zero-order ultra-sensitivity but also shows a previously undescribed typeof biphasic, double-threshold, stimulus–response curve(c.f. Fig. 3).

These results demonstrate that the quantitativeaspects of the protein–protein interactions – concen-trations and affinities – are determinants of the kineticproperties of signal transduction. The relation betweenconcentrations and affinities can be phrased in termsof competition effects. When kinase and phosphatasecan become saturated with the target protein (that is,their concentration are sufficiently small), the phospho-rylation and dephosphorylation rates obey Michaelis–Menten kinetics with product inhibition. For example,the phosphorylation is inhibited when the kinase remainsbound to some extent to the phosphorylated target.The kinetic behaviour – mainly, whether ultrasensitiveresponses are obtained or not – depends on the extentof product inhibition, or, in other words, on how effec-tively the unphosphorylated and phosphorylated forms

andl caseough

n thehos-ch a

ideration of elementary steps and can then be redo simpler form.

Although moderate, the decrease in the numbenknown parameters (eight binding and dissociaate constants reduced to four equilibrium constaas been instrumental in identifying critical param

of the target compete for the binding of the kinasephosphatase. In some sense, the opposite speciaoccurs when the enzyme concentrations are large enso that they are not saturated by the target. Thekinetic behaviour is shaped by how kinase and pphatase compete for binding to the target. In su

10 C. Salazar, T. Hofer / BioSystems xxx (2005) xxx–xxx

Fig. 7. Signalling gain can be affected by target expression. The ratiobetween phosphorylated and total target proteinY1/YT (∼gain) is plot-ted vs. the ratio between total target and phosphatase concentrationYT/PT (∼target expression) for differentL0/L1 = Q1/Q0 = 0, 1, 100(product inhibition). Parameters:N = 1, KT = 5.1,a = 0.2,b = 0.2.

regime, sequestration effects can play an important role,for example, when the phosphorylated substrate can beprotected from dephosphorylation by remaining boundto the kinase or other interaction partners. Such interac-tions have indeed been observed (e.g.,Okamura et al.,2004).

The importance of such affinity effects is illustratedby an example depicted inFig. 7. Over long time scales,an important mode of regulation of signal transductionwill be the change of expression of the protein compo-nents.Fig. 7shows the effect of changing the expressionof the target protein. The phosphorylated fraction (gain)is plotted for fixed kinase and phosphatase activities,which may be thought of as characterizing the ‘on-state’of the cycle. The gain can increase strongly with tar-get expression (Fig. 7, dashed line). However, this isdependent on the relative affinities of the phosphorylatedand unphosphorylated targets for the enzymes, expressedas the degree of product inhibition. Thus the gain mayremain constant (solid line) or even decrease (dottedline).

Protein–protein interactions do not only control theshape of the response curve but also the kinetics ofswitching phosphorylation state. We found that the tran-sition times can change by several orders of magnitudewhen protein concentrations and affinities vary whilethe catalytic rate constants of kinase and phosphataseremain fixed. As a general tendency, the responses arefast when kinase and phosphatase are abundant (i.e.non-saturated with target) and do not strongly bind to

latedpar-etated

and unphosphorylated states, which accounts for the lowtransition time to the new steady state when the enzymeactivities are changed. As this involves continuous ATPhydrolysis, rapid responses are achieved at the expenseof metabolic energy. Quantitative estimates indicate thatthe ATP usage in signal transduction networks may besmall compared to other ATP consuming processes, butthis issue has not yet been considered on the basis ofextensive experimental data (Shacter et al., 1984b).

Currently, many of the quantitative questions raisedby our analysis are not yet addressed by the availableexperimental data. However, large-scale measurementsof cellular protein concentrations are appearing (e.g.,Ghaemmaghami et al., 2003), and affinities and kineticparameters have already been determined in some cases(e.g., Adams, 2003). At the same time, it is becom-ing clear that regulation of protein function commonlyinvolves multiple phosphorylation, where different phos-phorylation modules in a protein may serve distinctroles or may conspire in regulating a particular function(Okamura et al., 2004; Salazar and Hofer, 2005). Thelarge range of possible behaviours demonstrated of evena single phosphorylation cycle underscores the necessityfor quantitative analysis.

Acknowledgements

The work was supported by the Deutsche Forschungs-gemeinschaft through a project grant in the Collaborative

.H.

n-g

hos-

their respective reaction products (the phosphoryand unphosphorylated forms of the target); this isticularly clearly seen inFig. 4b. In this case, the targcycles comparatively rapidly between phosphoryl

,

Research Center Theoretical Biology (SFB 618) to T

Appendix A. Kinetic equations for aphosphorylation cycle

Denote bykn,K andk−n,K the binding and dissociatiorate constants of the kinase to then-times phosphorylated form of the substrate (n = 0, 1). The correspondinrate constants for phosphatase arekn,P and k−n,P. Thecatalytic rate constants for phosphorylation and depphorylation are denoted bya andb, respectively.Fig. 2is described by the kinetic equations

dXn,0

dt= −(kn,KK + kn,PP)Xn,0 + k−n,KXn,K

+ k−n,PXn,P, for n = 0, 1 (A.1a)

dX0,K

dt= k0,KKX0,0 − (k−0,K + a)X0,K (A.1b)

dX0,P

dt= k0,PPX0,0 + bX1,P − k−0,PX0,P (A.1c)

C. Salazar, T. Hofer / BioSystems xxx (2005) xxx–xxx 11

dX1,K

dt= k1,KKX1,0 + aX0,K − k−1,KX1,K (A.1d)

dX1,P

dt= k1,PPX1,0 − (k−1,P + b)X1,P, (A.1e)

together with the conservation relations

1∑n=0

(Xn,0 + Xn,K + Xn,P) = YT = const. (A.2a)

K +1∑

n=0

Xn,K = KT = const. (A.2b)

P +1∑

n=0

Xn,P = PT = const. (A.2c)

When the binding of the enzymes to the substrate ismuch faster than the phosphorylation and dephosphory-lation reactions, we have the equilibrium relations

Xn,K = K

Ln

Xn,0,

Xn,P = P

Qn

Xn,0, for n = 0, 1, (A.3)

whereLn = k−n,K/kn,K and Qn = k−n,P/kn,P are the dis-sociation constants for kinase and phosphatase, respec-tively.

Summing the free, kinase-bound, and phosphatase-bound substrate concentrations for the unphosphorylateda riates

Y

w

hek rate

X

X

w Eq.(

R

A is inbitor

Bhalla, U.S., Ram, P.T., Iyengar, R., 2002. MAP kinase phosphatase asa locus of flexibility in a mitogen-activated protein kinase signalingnetwork. Science 297, 1018–1023.

Bray, D., Lay, S., 1997. Computer-based analysis of the binding stepsin protein complex formation. Proc. Natl. Acad. Sci. U.S.A. 94,13493–13498.

Cimino, A., Hervagault, J.F., 1987. Experimental evidence for a zero-order ultrasensitivity in a simple substrate cycle. Biochem. Bio-phys. Res. Commun. 149, 615–620.

Cohen, P., 2000. The regulation of protein function by multisitephosphorylation—a 25 year update. Trends Biochem. Sci. 25,596–601.

Cornish-Bowden, A., 2004. Fundamentals of Enzyme Kinetics. Port-land Press, London.

Ferrell Jr., J.E., Bhatt, R.R., 1997. Mechanistic studies of the dual phos-phorylation of mitogen-activated protein kinase. J. Biol. Chem.272, 19008–19016.

Ferrell Jr., J.E., Machleder, E.M., 1998. The biochemical basis of anall-or-none cell fate switch inXenopus oocytes. Science 280, 895–898.

Gerhart, J., 1999. 1998 Warkany lecture: signaling pathways in devel-opment. Teratology 60, 226–239.

Ghaemmaghami, S., Huh, W.K., Bower, K., Howson, R.W., Belle, A.,Dephoure, N., O’Shea, E.K., Weissman, J.S., 2003. Global analysisof protein expression in yeast. Nature 425, 737–741.

Goldbeter, A., Koshland, D.E., 1981. An amplified sensitivity arisingfrom covalent modification in biological systems. Proc. Natl. Acad.Sci. U.S.A. 78, 6840–6844.

Heinrich, H., Neel, B.G., Rapoport, T.A., 2002. Mathematical modelsof protein kinase signal transduction. Mol. Cell 9, 957–970.

Hlavacek, W.S., Faeder, J.R., Blinov, M.L., Perelson, A.S., Goldstein,B., 2003. The complexity of complexes in signal transduction.Biotechnol. Bioeng. 84, 783–794.

Hogan, P.G., Chen, L., Nardone, J., Rao, A., 2003. Transcriptionalregulation by calcium, calcineurin, and NFAT. Genes Dev. 17,

ulti-crip-

en-A. 93,

os-the

canase

lyco-zyme

hetical

maynase. Sci.

-mes. 77,

nd phosphorylated forms, we obtain the approplow variables

n = Xn,0 + Xn,K + Xn,P, for n = 0, 1, (A.4)

hich obey

dYn

dt= (−1)n+1(aX0,K − bX1,P). (A.5)

Using the equilibrium relations, we find for tinase- and phosphatase-bound forms of the subst

n,K = Yn

K/Ln

1 + K/Ln + P/Qn

and

n,P = Yn

P/Qn

1 + K/Ln + P/Qn

, (A.6)

hich, inserted into the above kinetic equation, yield9).

eferences

dams, J.A., 2003. Activation loop phosphorylation and catalysprotein kinases: is there functional evidence for the autoinhimodel? Biochemistry 42, 601–607.

2205–2232.Holmberg, C.I., Tran, S.E.F., Eriksson, J.E., Sistonen, L., 2002. M

site phosphorylation provides sophisticated regulation of transtion factors. Trends Biochem. Sci. 27, 619–627.

Huang, C.Y., Ferrell Jr., J.E., 1996. Ultrasensitivity in the mitogactivated protein kinase cascade. Proc. Natl. Acad. Sci. U.S.10078–10083.

Karin, M., Hunter, T., 1995. Transcriptional control by protein phphorylation: signal transmission from the cell surface tonucleus. Curr. Biol. 5, 747–757.

Kholodenko, B.N., 2000. Negative feedback and ultrasensitivitybring about oscillations in the mitogen-activated protein kincascades. Eur. J. Biochem. 267, 1583–1588.

Krebs, E.G., 1981. Phosphorylation and dephosphorylation of ggen phosphorylase: a prototype for reversible covalent enmodification. Curr. Top. Cell. Regul. 18, 401–419.

Lee, E., Salic, A., Kruger, R., Heinrich, R., Kirschner, M.W., 2003. Troles of APC and Axin derived from experimental and theoreanalysis of the Wnt pathway. PLoS Biol. 1, 116–132.

Levchenko, A., Bruck, J., Sternberg, P.W., 2000. Scaffold proteinsbiphasicallu affect the levels of mitogen-activated protein kisignaling and reduce its threshold properties. Proc. Natl. AcadU.S.A. 97, 5818–5823.

Llorens, M., Nuno, J.C., Rodrıguez, Y., Melendez-Hevia, E., Montero, F., 1999. Generalization of the theory of transition tiin metabolic pathways: a geometrical approach. Biophys. J23–36.

12 C. Salazar, T. Hofer / BioSystems xxx (2005) xxx–xxx

Markevich, N.I., Hoek, J.B., Kholodenko, B.N., 2004. Signalingswitches and bistability arising from multisite phosphorylation inprotein kinase cascades. J. Cell Biol. 164, 353–359.

Nash, P., Tang, X., Orlicky, S., Chen, Q., Gertler, F.B., Mendenhall,M.D., Sicheri, F., Pawson, T., Tyers, M., 2001. Multisite phospho-rylation of a CDK inhibitor sets a threshold for the onset of DNAreplication. Nature 414, 514–521.

Okamura, H., Aramburu, J., Garcia-Rodriguez, C., Viola, J.P.B.,Raghavan, A., Tahiliani, M., Zhang, X., Qin, J., Hogan, P.G., Rao,A., 2000. Concerted dephosphorylation of the transcription factorNFAT1 induces a conformational switch that regulates transcrip-tional activity. Mol. Cell 6, 539–550.

Okamura, H., Garcıa-Rodrıguez, C., Martinson, H., Qin, J., Virshup,D.M., Rao, A., 2004. A conserved docking motif for CKI bindingcontrols the nuclear localisation of NFAT1. Mol. Cell. Biol. 24,4184–4195.

Pawson, T., 2004. Specificity in signal transduction: fromphosphotyrosine-SH2 domain interactions to complex cellular sys-tems. Cell 116, 191–203.

Salazar, C., Hofer, T., 2003. Allosteric regulation of the transcriptionfactor NFAT1 by multiple phosphorylation sites: a mathematicalanalysis. J. Mol. Biol. 327, 31–45.

Salazar, C., Hofer, T., 2005. Activation of the transcription fac-tor NFAT1: concerted or modular regulation? FEBS Lett. 579,621–626.

Segel, L.A., 1988. On the validity of the quasi-steady assump-tion of enzyme kinetics. Bull. Math. Biol. 50, 579–593.

Shacter, E., Chock, P.B., Stadtman, E.R., 1984a. Regulation throughphosphorylation/dephosphorylation cascade systems. J. Biol.Chem. 259, 12252–12259.

Shacter, E., Chock, P.B., Stadtman, E.R., 1984b. Energy consumptionin a cyclic phosphorylation/dephosphorylation cascade. J. Biol.Chem. 259, 12260–12264.

Shaw, K.T.Y., Ho, A.M., Raghavan, A., Kim, J., Jain, J., Park, J.,Sharma, S., Rao, A., Hogan, P.G., 1995. Immunosuppressive drugsprevent a rapid dephosphorylation of transcription factors NFAT1in stimulated immune cells. Proc. Natl. Acad. Sci. U.S.A. 92,11205–11209.

Small, J.R., Fell, D.A., 1990. Covalent modification and metaboliccontrol analysis. Modification to the theorems and their applicationto metabolic systems containing covalently modifiable enzymes.Eur. J. Biochem. 191, 405–411.

Stadtman, E.R., Chock, P.B., 1977. Superiority of interconvert-ible enzyme cascades in metabolic regulation: analysis ofmonocyclic systems. Proc. Natl. Acad. Sci. U.S.A. 74, 2761–2766.

Tanoue, T., Adachi, M., Moriguchi, T., Nishida, E., 2000. A conserveddocking motif in MAP kinases common to substrates, activatorsand regulators. Nat. Cell. Biol. 2, 110–116.