Báo cáo khoa học: Điều hòa linh hoạt quá trình phosphoryl hóa protein đa vị trí thông qua thứ tự xử lý phosphate và tương tác protein

Versatile regulation of multisite protein phosphorylation

by the order of phosphate processing and protein–protein

interactions

Carlos Salazar

and Thomas Ho

¨fer

1,2

1 Theoretical Biophysics, Institute for Biology, Humboldt University Berlin, Germany

2 German Cancer Research Center, Heidelberg, Germany

Reversible phosphorylation is arguably the most

important mechanism for regulating protein activity

[1]. Also, other covalent modiﬁcations, such as methy-

lation, acetylation, ubiquitination, sumoylation and cit-

rullination, are increasingly being characterized [2].

Studies in recent years have shown that multiple regu-

latory 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 mito-

gen-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

Keywords

multisite phosphorylation; order of

phosphate processing; stimulus–response

relationship; transition time; ultrasensitivity

Correspondence

T. Ho

¨fer, Theoretical Biophysics, Institute

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 path-

ways. 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 the-

ory 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)phos-

phorylation times have been obtained in special cases. The order of phos-

phate 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 differ-

ences 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.

Abbreviations

MAPK, mitogen-activated protein kinase.

1046 FEBS Journal 274 (2007) 1046–1061 ª2007 The Authors Journal compilation ª2007 FEBS

the phosphorylation of a single amino acid residue in

a protein can arise under speciﬁc conditions. Subse-

quent 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 phosphoryla-

tion as well as positive feedback can provide addi-

tional 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 exter-

nal stimuli [24]. Therefore, elucidation of the kinetic

properties of phosphorylation cycles and cascades

will be crucial for understanding their cellular func-

tion. Multisite phosphorylation can be achieved in a

variety of ways. One or several kinases and phospha-

tases 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)phos-

phorylation can be random [28–30]. Studies on rho-

dopsin 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 possi-

bilities for protein–protein interactions and phosphory-

lation 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 kin-

ase 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 pro-

tein [8]. It is not clear how these factors modulate the

response in the protein’s phosphorylation state. Fur-

thermore, traditional enzyme kinetics, which rest on

the smallness of the enzyme concentration compared

to those of the reactants, cannot be applied in a

straightforward manner to protein phosphorylation in

cell signaling, because there are often no large concen-

tration differences between kinases and their targets.

In place of enzyme kinetics, the mathematical descrip-

tion of elementary reaction and binding steps is feas-

ible 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 rapid-

equilibrium approximation for protein–protein interac-

tions 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 experi-

mentally 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 phos-

phate processing gives rise to steeper response curves

and slower kinetics than random processing. More-

over, we illustrate systematically how qualitative

differences in target phosphorylation (graded, switch-

like and bistable responses) are determined by quan-

titative 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 dif-

ferential 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 phosphoryla-

tion sites, and are interested in how the abundance of

the various phosphorylation states of the target is

C. Salazar and T. Ho

¨fer Kinetic models of multisite phosphorylation

FEBS Journal 274 (2007) 1046–1061 ª2007 The Authors Journal compilation ª2007 FEBS 1047

regulated by its kinase(s) and phosphatase(s). Experi-

mental studies have shown that there are different

mechanisms for the processing of the individual phos-

phorylation sites (Fig. 1). Several kinases phosphory-

late 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 mecha-

nisms can also occur, such as the random dual phos-

phorylation of MAPK extracellular-signal-regulated

kinase (ERK) by mitogen-activated or extracellular sig-

nal-regulated protein kinase (MEK) and its sequential

dephosphorylation by mitogen-activated protein kinase

phosphatase 3 (MKP3) (Fig. 1, third panel) [5,30]. A

cyclic mechanism for the phosphorylation and dep-

hosphorylation 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 pro-

cessing order for the regulatory properties of the target

protein.

We now derive a general model describing the

dynamics of multisite reversible phosphorylation. Ini-

tially, we focus on the sequential mechanism, in which

case the phosphorylation states can be enumerated by

the number of consecutively phosphorylated residues

n¼0, ... N, where Nis the number of phosphorylata-

ble residues. In each phosphorylation state, the target

can occur in free form or bound to kinase or phospha-

tase; the respective concentrations of the target will be

denoted by X

n,0

n,K

and X

n,P

, respectively. They are

determined by the rates of the reversible enzyme–target

associations ⁄dissociations and the irreversible phos-

phorylation ⁄dephosphorylation reactions as depicted

in Fig. 2.

Frequently, the protein–protein interactions take

place more rapidly than the addition and cleavage of

phosphoryl groups [20,37]. In this case, the rapid-equi-

librium approximation is justiﬁed [38], and the system

dynamics can be formulated in terms of the total con-

centrations attained by the various phosphorylation

states:

Yn¼Xn;0þXn;KþXn;Pð1Þ

i.e. the sum of free and enzyme-bound forms. As

shown in supplementary Doc. S1, the total concentra-

tions Y

are governed by the differential equations

dY0

dt ¼a1Y0þb1Y1ð2aÞ

dYn

dt ¼anYn1ðanþ1þbnÞYn

þbnþ1Ynþ1;for 1 nN1

ð2bÞ

dYN

dt ¼aNYN1bNYNð2cÞ

where a

and b

are effective rate constants of phos-

phorylation and dephosphorylation

Fig. 1. 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+ 1 different phosphorylation states

(where Nis the total number of phosphorylation sites); random

mechanisms can create 2

different phosphorylation states. It is of

note that the number of different possible sequences to achieve

full phosphorylation of the target is 1 for the sequential mechanism

and N! for the random mechanism.

Kinetic models of multisite phosphorylation C. Salazar and T. Ho

¨fer

1048 FEBS Journal 274 (2007) 1046–1061 ª2007 The Authors Journal compilation ª2007 FEBS

an¼an

|{z}

catalytic

rate of kinase

K=Ln1

1þK=Ln1þP=Qn1

|ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ}

fraction of

kinasebound

target protein

;

bn¼bn

|{z}

catalytic rate

of phosphatase

P=Qn

1þK=LnþP=Qn

|ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ}

fraction of

phosphatasebound

target protein

ð3Þ

These account for both enzyme–target binding and the

catalysis. Kand Pdenote the free concentrations of

kinase and phosphatase, respectively, and L

and Q

are the respective dissociation constants for the

kinase–target and phosphatase–target interactions. a

and b

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 phosphoryl-

ated 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

KT¼KþX

n¼0

Xn;K¼K1þX

n¼0

Yn=Ln

1þK=LnþP=Qn

ð4aÞ

PT¼PþX

n¼0

Xn;P¼P1þX

n¼0

Yn=Qn

1þK=LnþP=Qn

ð4bÞ

Equations (2)–(4) deﬁne the dynamics of sequential

multisite phosphorylation ⁄dephosphorylation. Although

the differential Eqns (2) are linear in the concentration

variables Y

, the full system is rendered strongly nonlin-

ear 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 enzyme-

kinetic rate laws can be derived for the kinase and

phosphatase. Moreover, Eqn (3) shows that the phos-

phorylation can be directly inhibited by the phosphatase

(and dephosphorylation by the kinase) due to competi-

tion 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 dep-

hosphorylation 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 do

not change. At steady state, the concentrations of the

various phosphorylation states are given explicitly by



Yn¼

YT=Dn¼0

DPn

i¼1

bi1nN;D¼1þP

i¼1

j¼1



:ð5Þ

where

YT¼X

n¼0

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

Fig. 2. Model for multiple phosphorylation cycles. (A) Schematic

representation of a phosphorylation–dephosphorylation cycle. (B)

Mathematical model for a sequential mechanism of multiple phos-

phorylation based on the schema of Fig. 1A. The free form of the

n-times phosphorylated substrate (n¼0, 1, . . ., N) is represented

by X

n,0

. The kinase–substrate and phosphatase–substrate com-

plexes are denoted by X

n,K

and X

n,P

, respectively. The rate con-

stants for phosphorylation of X

n,K

and dephosphorylation of X

n,P

are

denoted by a

n+1

and b

, respectively. L

and Q

are the dissoci-

ation constants for the complexes X

n,K

and X

n,P

, respectively.

C. Salazar and T. Ho

¨fer Kinetic models of multisite phosphorylation

FEBS Journal 274 (2007) 1046–1061 ª2007 The Authors Journal compilation ª2007 FEBS 1049

weakly. Then, Eqns (2)–(4) can be simpliﬁed consider-

ably, 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 dis-

sociation constants L

and Q

, implying that the free

enzyme concentrations are approximately equal to

the total concentrations: KK

and PP

(see

Eqn 4). The effective rate constants then simplify to

a

⁄L

n)1

and b

b

⁄Q

n)1

. This can be

further simpliﬁed when the dissociation constants are

independent of the target’s phosphorylation state

¼Land Q

¼Qfor all n) and the same also holds

for the catalytic rate constants (a

¼aand b

¼b).

Then we have, for the steady-state fraction of the

n-times phosphorylated target, 

yn¼

Yn=YT:



yn¼rnðr1Þ

rNþ11ð6Þ

The crucial parameter combination of rate constants,

enzyme concentrations and afﬁnities is

r¼a

b¼aKT=L

bPT=Qð7Þ

bearing in mind the assumption of weak enzyme bind-

ing. ris a measure of the stimulus strength.

The analysis of nonsequential phosphorylation mech-

anisms is generally more complicated, due to the large

number of phosphorylation states. However, the fully

random scheme depicted in Fig. 1 (second panel) can be

analyzed in a similar manner when we again assume that

the kinetic parameters do not depend on the target’s

phosphorylation state (L

¼L,Q

¼Q,a

¼aand

¼bfor 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 Y

, as indicated in Fig. 1A

(second panel). The Y

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nþ1Þa;bn!nbð8Þ

These relations indicate that an n-times phosphorylated

substrate can be further phosphorylated on N)ndif-

ferent residues and dephosphorylated on nresidues. In

this way, the random scheme is mapped to a linear

chain of reactions, in which the effective phosphoryla-

tion rate decreases with increasing phosphorylation of

the target (because fewer unphosphorylated sites

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



yn¼N

rn

ð1þrÞNð9Þ

where



is the binomial coefﬁcient, and rwas deﬁned in Eqn (7).

In the limiting case of a target with a single phos-

phorylation site (N¼1), 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 sin-

gle phosphorylation site. This is particularly seen for

low kinase ⁄phosphatase activity ratios, where the

phosphorylation sets in more sharply when Nis large.

However, the overall range of kinase-to-phosphatase

activities over which a switch from the unphosphoryl-

ated to nearly fully phosphorylated target is achieved

varies only moderately with N. This limited overall

steepness of the response curve for complete phos-

phorylation 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, par-

tially phosphorylated forms are overall more abundant

in the steady state (Fig. 3D); in Eqn (9), this is reﬂec-

ted by the binomial coefﬁcient, which reaches its maxi-

mum for n¼N⁄2. 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 n

ln 81 ⁄ln R, where the global response coefﬁcient Ris

the ratio of the concentration of active kinase K

0.9

which there is 90% fully phosphorylated target to the

kinase concentration K

0.1

at which 10% of the target is

fully phosphorylated, R¼K

0.9

⁄K

0.1

[13]. For the

sequential mechanism, the effective Hill coefﬁcient ran-

ges between 1 and 2 (Fig. 3E). For random and mixed

Kinetic models of multisite phosphorylation C. Salazar and T. Ho

¨fer

1050 FEBS Journal 274 (2007) 1046–1061 ª2007 The Authors Journal compilation ª2007 FEBS

Báo cáo khoa học: Versatile regulation of multisite protein phosphorylation by the order of phosphate processing and protein–protein interactions

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