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

Versatile regulation of multisite protein phosphorylation
by the order of phosphate processing and proteinprotein
interactions
Carlos Salazar1 and Thomas Hofer1,2
ă
1 Theoretical Biophysics, Institute for Biology, Humboldt University Berlin, Germany
2 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:
Website: />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.

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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

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Kinetic models of multisite phosphorylation

C. Salazar and T. Hofer
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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 3 (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. 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-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 ẳ Xn;0 ỵ Xn;K ỵ Xn;P

1ị

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. 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 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 1 for the sequential mechanism
and N! for the random mechanism.

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dY0
¼ a1 Y0 ỵ b1 Y1
dt
dYn
ẳ an Yn1 anỵ1 ỵ bn ịYn
dt
ỵ bnỵ1 Ynỵ1 ; for 1 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
ă

Kinetic models of multisite phosphorylation

A

KT ẳ K ỵ

N
X

Xn;K

nẳ0

N
X

Yn =Ln
ẳK 1ỵ
1 ỵ K=Ln ỵ P=Qn
nẳ0

!

4aị
PT ẳ P ỵ

N
X

Xn;P

nẳ0

N
X

Yn =Qn
ẳP 1ỵ
1 ỵ K=Ln ỵ P=Qn
nẳ0

!

4bị
B

Fig. 2. 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 + 1 and bn, respectively. Ln and Qn are the dissociation constants for the complexes Xn,K and Xn,P, respectively.

K=Ln1
;
an
|{z}
1 ỵ K=Ln1 þ P=QnÀ1
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
catalytic
fraction of
rate of kinase
kinaseÀbound
target protein
P=Qn
bn ¼
bn
|{z}
1 þ K=Ln þ P=Qn
catalytic rate |{z}
fraction of
of phosphatase
phosphatasebound
target protein
an ẳ

3ị

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 do
not change. At steady state, the concentrations of the
various phosphorylation states are given explicitly by
8
n¼0
< YT =D

N

P  i aj 
Y n ¼ YT n ai
Pj¼1 b
: D Pi¼1 b 1 n N; D ẳ 1 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

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Kinetic models of multisite phosphorylation

C. Salazar and T. Hofer
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weakly. Then, Eqns (2)–(4) can be simplified 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 ) 1 and bn % bnPT ⁄ Qn ) 1. 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 ẳ Y n =YT :
y
n ẳ
y

rn r 1ị
rNỵ1 À 1

ð6Þ

The crucial parameter combination of rate constants,
enzyme concentrations and afnities is
rẳ

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. 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 (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 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 nd for the fraction
of n-times phosphorylated targets
 
rn
N
n ẳ
9ị
y
n 1 ỵ rịN
where


N
n



is the binomial coefcient, and r was dened 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 ⁄ 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 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 1 and 2 (Fig. 3E). For random and mixed

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


C. Salazar and T. Hofer
ă

Kinetic models of multisite phosphorylation

A

C

B

D

E

Fig. 3. 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 4. (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 1 [in (A–E)]; N ¼ 4 ([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

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C. Salazar and T. Hofer

ă

0

N yN ðtÞÞdt
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 ẳ

1 Nr 1ịrNỵ1 ỵ 1ị 2rrN 1ị
bPT =Q
r 1ị2 rNỵ1 1ị

1
HN
HN
ẳ
sN ẳ
bPT =Q 1 ỵ r a ỵ b

15
4

10
2
N=1

5

1

B

where

1

10

100

5
Random

6

b1
13ị

0.1

Stimulus strength, KT /PT

12ị

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 > 2 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 1. 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
0.01

ð11Þ

where r was defined in Eqn (7) (for details, see supplementary Doc. S3). For a single-site target, we obtain
1
1

1
ẳ
s1 ẳ
bPT =Q 1 ỵ r a ỵ b

Time constant,

sN ẳ

25
6

Time constant,

1
R

A

b

weak (see previous section). The transition time for
changes in concentration of the fully phosphorylated
target is appropriately defined as

4

4

3

2
2
N=1
1
0.01

0.1

1

10

100

Stimulus strength, KT /PT
Fig. 4. 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, 6.

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
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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
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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 1. 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 4 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, a1existence a region of bistability that occupies the same
area in parameter space as the region of zero-order
ultrasensitivity in Fig. 5A. From these and further
related calculations, we conclude that the following are
necessary conditions for bistability: (a) low enzyme
concentrations; (b) a higher kinase affinity of the
unphosphorylated target than of the fully phosphorylated target, and analogously, a higher phosphatase

affinity of the fully phosphorylated target than of the
unphosphorylated target (in supplementary Doc. S4,
these conditions are discussed analytically).
To investigate the effects of the catalytic rate constants, we allowed the first phosphorylation step to
have a different rate from the following three steps
(a1 „ a2,a3,a4) and, likewise, the first dephosphorylation step to be different from the following steps
(b4 „ b3,b2,b1). As seen in supplementary Fig. S1A,
both phosphorylation and dephosphorylation must
exhibit kinetic cooperativity in the sense that partial
(de)phosphorylation of the target accelerates the
remaining catalytic steps. The higher the enzyme saturation, the less stringent the requirement for kinetic
cooperativity will be. We also examined the effect of
the number of phosphorylation sites. Under otherwise
unchanged conditions, an increase in the number of
phosphorylation sites favors bistability (supplementary
Fig. S1B). Compared to a sequential mechanism, random phosphorylation reduces the region of the bistable
response in parameter space (supplementary Fig. S1C).
This is because the effective catalytic rates are biased

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C. Salazar and T. Hofer
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A

B

Fig. 5. Plasticity of the stimulus sensitivity depends on protein–protein interactions. Contour plots are given of the effective Hill coefficient
nH as a function of the enzyme saturation (measured by YT ⁄ PT) and cooperativity of enzyme binding (measured by L0 ⁄ LN ¼ QN ⁄ Q0). To
reduce the dimensionality for displaying the results of parameter variations, we took L0 ⁄ LN ¼ QN ⁄ Q0, so that the degrees of cooperativity in
the binding of kinase and phosphatase to the target are the same. The dark areas denote regions of high stimulus sensitivity (high Hill coefficients). The stimulus–response curves for particular parameter values are depicted (using a log scale on the x-axis) in the small boxes (solid
lines) and compared with the hyperbolic Michaelis–Menten kinetics (dashed line). (A) Non-cooperative kinetics: the catalytic rate constants of
the enzymes have been kept independent of n. (B) Cooperative kinetics: the black area denotes a region of bistable response, which occupies the same region in parameter space as the region of zero-order ultrasensitivity in the noncooperative system shown in Fig. 5A. Parameters: an ¼ bn ¼ 1 [in (A)]; a1 ¼ bN ¼ 0.01, an ¼ 1 (n „ 1), bn ¼ 1 (n „ N) [in (B)]; N ¼ 4, L0 ¼ L1 ¼ . . . ¼ LN ) 1 ¼ 0.1, QN ¼ QN ) 1 ¼ . . . ¼
Q1 ¼ 0.1 [in (A–B)].

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Kinetic models of multisite phosphorylation

towards the partially phosphorylated forms, exhibiting
a smaller degree of cooperative kinetics than the corresponding sequential scheme.
Kinetic behavior
We expected not only the final steady state but also
the transition time s into the steady state to be
affected by the kinetic parameters of the system and
enzyme concentrations. To study this question, s was
computed numerically for a switch of active kinase
from zero to the same concentration as the active

phosphatase (KT ¼ PT). The relevant target state was
taken to be the fully phosphorylated protein, so we
used Eqn (10) to compute s. As shown in Fig. 6, the
transition time strongly depends on the kinetic design
of the phosphorylation cycle. The most influential
parameter is the substrate saturation of the enzymes:
saturated cycles are usually slower than unsaturated
ones. In addition, cooperativity of enzyme binding
results in an increase in the transition time. A
comparison of Figs 5A and 6 shows that the association of steeper steady-state thresholds with larger
transition times, as seen in the previous sections for
the special case of weak binding of the enzymes,
holds more generally.
Individual regulation of the phosphorylation sites
So far, we have focused our analysis on the behavior
of the fully phosphorylated fraction. In many cases,
the partially phosphorylated forms also contribute to
the activation of the target protein. Multiple phosphorylation sites can act additively, e.g. by producing
gradual changes in the DNA-binding affinity of transcription factors [41,42]. Alternatively, individual sites
may control different functions of the target protein,
such as nuclear transport, DNA binding, and transcriptional activity [43,44], which can be achieved when
each residue is phosphorylated by a distinct kinase.
Remarkably, we found that individual control of multiple sites by the same kinase is also feasible. However,
this requires a specific kinetic design of target phosphorylation.
Partially phosphorylated states generally attain
higher concentrations in the case of random and cyclic
phosphate processing than in sequential processing
(Fig. 3D). To illustrate the effect of the kinetic design
on the differential regulation of phosphorylation sites,
we consider a target protein with two phosphorylatable

sites, A and B, that are randomly modified. When all
reactions occur at identical rates, the partially phosphorylated forms will be equally abundant (Fig. 7A).

Fig. 6. Plasticity of the transition time depends on protein–protein
interactions. Contour plots are given of the transition time as a
function of the enzyme saturation (measured by YT ⁄ PT) and product
inhibition (measured by L0 ⁄ LN ¼ QN ⁄ Q0). The dark areas denote
regions with slow phosphorylation kinetics (high transition times).
Parameters: an ¼ bn ¼ 1, N ¼ 4, L0 ¼ L1 ¼ . . . ¼ LN ) 1 ¼ 0.1,
QN ¼ QN ) 1 ¼ . . . ¼ Q1 ¼ 0.1.

However, when the rates differ, interesting phenomena
can arise. If site A is phosphorylated more rapidly
than site B, and site B is dephosphorylated more rapidly than site A, the two partially phosphorylated
forms of the target will be separated in the response
curve, occurring at different kinase activities (Fig. 7B).
The intermediate form with residue A phosphorylated
will be preferred at lower kinase activities (dashed–dotted curve), whereas the intermediate form with residue B phosphorylated will dominate at higher kinase
activities (dashed curve). At very high kinase activity,
both residues become phosphorylated. However, this
apparently trivial statement is not generally true. If we
additionally consider that phosphorylation proceeds
sequentially and dephosphorylation in a random way,
the target protein will not be fully phosphorylated even
at extremely high activities of the kinase (Fig. 7C).
The kinetic designs in Fig. 7B,C resemble the cyclic
mechanism shown in Fig. 1 (fourth panel); similar
mechanisms may govern the individual control of a
larger number of sites (N > 2) by a single kinase.

Discussion
In this article, we have attempted a systematic analysis
of how multisite phosphorylation is regulated by the

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A

C. Salazar and T. Hofer
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B

C

Fig. 7. Individual control of phosphorylation sites by a single kinase. The steady-state behavior of the phosphorylation fractions yN as a function of KT ⁄ PT is shown for a target with two phosphorylatable residues, A and B. The following kinetic designs are considered. (A) Random:
random processing, assuming that all rate constants are identical. (B) Cyclic symmetric: random processing, assuming higher rate constants
for early phosphorylation of residue A and for early dephosphorylation of residue B than for the other processes. (C) Cyclic asymmetric: kinetic design as in Fig. 7B, but assuming sequential phosphorylation beginning with residue A. Dotted and solid lines represent the behaviors
of the fully dephosphorylated and fully phosphorylated forms, respectively. Dashed and dashed–dotted lines correspond to the partially phosphorylated forms, PO and OP.

phosphorylation order and by the parameters of protein–protein interactions and the enzyme-catalyzed
steps. Several conclusions have emerged from the analysis that are relevant for both mathematical modeling
and the experimental investigation of signaling networks.
Mathematical modeling
In contrast to the situation for the majority of biochemical reactions in metabolism, analytic rate laws

for kinase and phosphatase reactions in signal transduction cannot generally be derived. Whereas in many
metabolic reactions, enzymes occur at much lower concentrations than their substrates, justifying the use of
the Michaelis–Menten rate law [45], such strict concentration hierarchies are not frequently met in signaling
networks. Quasi-steady-state arguments as used in the
derivation of Michaelis–Menten kinetics can be extended to other special cases, such as those of very high
enzyme concentrations [46,47]. Generally, however, a
mechanistic description of phosphorylation cycles in
terms of elementary binding and catalytic steps
appears to be more appropriate for signal transduction. The kinetic equations are nonlinear but can be
simplified by considering protein associations occurring
on faster time scales than the enzyme-catalyzed steps
[20,37]. This allows the application of a rapid-equilibrium approximation, which implies that only the
1056

protein–protein affinities and not the individual association and dissociation rate constants need to be considered. Thus, the number of parameters can be
reduced considerably, which facilitates the analytic and
numerical treatment of the kinetic equations. Explicit
solutions for the steady-state response curves and
transition times of this system are obtained in the special cases of very large or very small enzyme concentrations compared to those of their substrates.
Protein–protein interactions
Multisite phosphorylation per se is not associated with
switch-like responses [18,48]. We showed here that the
characteristics of protein–protein interactions, such as
the relative concentrations of enzymes and target protein, together with the enzyme–target affinities, are key
determinants of such behavior. There are two opposing
situations in which phosphorylation cycles can exhibit
switch-like response: (a) very low enzyme concentrations and negative cooperativity of enzyme binding;
and (b) high enzyme concentrations and positive cooperativity of enzyme binding. The first case has been
described as zero-order ultrasensitivity [13], and does
not depend on the target having multiple phosphorylation sites. However, when multisite phosphorylation

is additionally combined with ‘cooperativity’ in the
catalytic rate constants, the system can be bistable
and thus realizes a perfect switch. For the second

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C. Salazar and T. Hofer
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parameter constellation, threshold behavior can also
be found with a singly phosphorylated target [20]. This
behavior becomes more accentuated as the number of
phosphorylation sites increases. Notably, outside these
two special regions, the effective Hill coefficient of the
response curve for complete phosphorylation generally
remains below 2. Thus, multiple phosphorylation can,
with certain specific parameter choices, exhibit switchlike behavior but, for other parameter values, can
equally well give rise to graded responses.
We have been able to define the general conditions for bistability and hysteresis arising solely from
multisite phosphorylation, which has recently been
described for dual MAPK phosphorylation [17].
Bistability has also been found in other systems undergoing multisite phosphorylation (e.g. Cdk–cyclin complex), but as a consequence of direct positive feedback
loops in the system [49]. In our case, this bistability
results from a finely tuned interplay of protein–protein
interaction parameters and kinetic properties of both
kinase and phosphatase [48]. First, the unphosphorylated and phosphorylated targets must compete efficiently for binding to the enzymes. Moreover, the total
enzyme concentration must be low enough for competition to become noticeable. Second, the unphosphorylated target must be phosphorylated slowly and,
likewise, the fully phosphorylated target must be
dephosphorylated slowly. Multistability has been previously associated with feedback loops in gene expression, where it can serve as a mechanism of molecular

memory. Bistability of phosphorylation cycles may
represent an alternative for memory generation or
serve for sustained signal propagation [21].
Order of phosphate processing
Structural factors, such as the order in which the phosphorylation sites are processed, are additional degrees
of freedom that affect the steady-state and kinetic
response of the fully phosphorylated substrate. Both
cyclic and random mechanisms exhibit shallower dose–
response curves, as compared to the sharp dose–
response curves obtained with the sequential mechanism. At the same time, however, the cyclic mechanism is
more responsive to small concentration of kinase than
either sequential or random mechanisms. For both cyclic and random mechanisms, the intermediate phosphorylation states generally attain higher concentrations
than for the sequential mechanism. In these cases,
a much higher ratio of kinase activity to phosphatase
concentration is needed to achieve complete phosphorylation of the target protein. Because the number of intermediate states grows exponentially with random

Kinetic models of multisite phosphorylation

processing order and only linearly with sequential order,
the differences between random and sequential phosphate processing become more pronounced when the
number of phosphorylation sites is large.
Concerning the response kinetics, there is a profound qualitative difference between the random and
sequential mechanisms. The response kinetics of random phosphorylation and dephosphorylation are significantly faster than for a strictly sequential order of
both processes. This property results from the fact that
there are many more phosphorylation routes available
for the random mechanism, where, in particular, phosphorylation can start from any site. By contrast, in the
sequential case, an intermediate site becomes a substrate for the kinase only after all the preceding sites
have been phosphorylated. This kinetic effect may
explain experimental observations on rhodopsin deactivation by phosphorylation. Somewhat paradoxically,
deactivation was observed to be three-fold slower when

only three phosphorylation sites were present in a
mutant instead of the six sites of the wild-type protein
[31]. However, only three of the six sites were required
for reproducible deactivation. The findings of this
article are consistent with an unordered mechanism
of phosphorylation in which the total number of
phosphorylated sites rather than their identity determines rhodopsin activity [50]. This may explain why
a higher number of available sites results in faster
deactivation.
Processivity of multisite phosphorylation
The sensitivity of multisite phosphorylation is also
modulated by the degree of processivity, which describes the number of phosphorylation sites that are
processed during a single enzyme–substrate binding
event. Very processive substrates can be fully phosphorylated in a single binding event [9,29,51], whereas
more distributive substrates interact weakly with their
modifying enzymes and have to associate multiple
times to achieve complete phosphorylation [5,22,30,52].
Because the last scenario offers more possibilities for
protein–protein interactions, and many substrates are
distributively modified, we centered our analysis on
this mechanism. Distributive kinetics ensure iterated
competition between the enzymes, which produces a
threshold response and ⁄ or a switch. In the absence of
cooperative kinetics or zero-order ultrasensitivity, a
distributive mechanism gives rise to an activation
threshold but, at the same time, produces a poor
switch [18].
One immediate question is whether the degree of
processivity and the order of phosphate processing are

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C. Salazar and T. Hofer
ă

related or not. Previous studies suggest that processivity requires local sequence recognition. Enzymes such
as glycogen synthase kinase 3 and casein kinase I can
phosphorylate multiple serines with a high degree of
processivity by sequentially creating new recognition
sites [25]. Although processive phosphorylation is characterized by a hyperbolic response, a sequential order
of processing generally requires priming kinases, and
several functional domains are also involved, which
may contribute to produce a threshold or a switch in
the response. On the other hand, random phosphorylation apparently proceeds in a distributive manner, e.g.
dual phosphorylation of ERK by MEK [5]. Our results
indicate that random processing may attenuate the
switch-like response of a distributive mechanism. Further research is needed to explore in detail how the
phosphorylation order and the degree of processivity
are related, and, if so, whether this relationship contributes to the high specificity and timing of signaling
processes.
Regulation of protein conformation by multisite
phosphorylation
Subsequent phosphorylation induces changes in protein
conformation that, ultimately, can affect the protein
function. Rearrangements in the protein structure could

contribute to determining whether the next phosphorylation should be a sequential or random one [53]. A kinase may phosphorylate multiple sites randomly when
the target protein exhibits considerable flexibility in its
structure so that all sites are equally accessible to the
kinase and, thus, kinetically indistinguishable. For
example, the adoption of a helical or unfolded structure
may favor a random order of phosphate processing. By
contrast, sequential phosphorylation probably occurs
when the protein domains are more rigidly held in place
so that the kinase can start at one locus and then phosphorylate in a single direction until the entire block of
residues is modified. In a sequential mechanism, phosphorylation of the first residue by a priming kinase may
permit further phosphorylations to take place [25].
One scenario for the control of protein function is
that all phosphorylation residues concertedly regulate
a particular function, such as by affecting the equilibrium between distinct global conformations of the protein (Fig. 8, first panel) [8,41,42]. Our previous model
on the activation of the transcription NFAT1 predicts
that such concerted regulation can efficiently be
achieved by a cooperative sequential phosphorylation
mechanism [16]. Alternatively, different functions of
the target protein may be controlled by several phosphorylation sites grouped in domains (second panel)
1058

[54] or by individual sites (third panel) [43,44]. Such
differential regulation of a substrate can be accomplished by several kinases and ⁄ or phosphatases acting
in the same or distinct subcellular compartments [53].
There is also experimental evidence of transcription
factors responding to different activity levels of a single kinase with different programs of gene expression
[43]. We have shown here that a nonsequential, cyclic
design of target phosphorylation allows robust control
of such multiple functions by a single kinase.
We conclude that knowledge concerning phosphorylation order and the properties of protein–protein

interactions is crucial for the elucidation of the steadystate and kinetic behavior of multisite phosphorylation. Although these parameters are currently not
known for most signaling pathways, large-scale measurements of cellular protein concentrations, binding
affinities and kinetic parameters are already appearing
[55–57]. Moreover, novel techniques for the detection
and quantification of temporal changes in all of a protein’s phosphorylation sites are emerging [58,59].
In particular, the use of specific phosphoantibodies,
the mimicking of nonphosphorylated and phosphorylated states by mutating a particular phosphorylatable
site, as well as quantitative MS measurements of the
A

Import, Export
DNA binding
P P P P P P

Concerted regulation

Import

Export
DNA binding

P P P

P P P

B

Modular regulation

Import

DNA
binding

Export

P

P

P

C

Individual regulation
Fig. 8. Regulatory mechanisms in multisite phosphorylation. (A)
Concerted regulation: all phosphates regulate several functions in a
concerted manner. (B) Modular regulation: phosphates belonging to
a specific domain regulate a particular function. (C) Individual regulation: each phosphate regulates a distinct function.

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C. Salazar and T. Hofer
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kinetics of different phosphorylation sites on the same
protein, may be useful in testing our theoretical conclusions. The integration of experimental information
with modeling approaches will certainly provide new
insights into the intramolecular and intermolecular

signaling mechanisms.

Acknowledgements
This work was supported by the German Research
Foundation (DFG) through a project grant in the Collaborative Research Center Theoretical Biology (SFB
618).

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Supplementary material
The following supplementary material is available
online:
Fig. S1. Regulation of bistable responses in multisite
phosphorylation by kinetic, thermodynamic and structural parameters (page S-1).

Kinetic models of multisite phosphorylation

Doc. S1. Kinetic equations and model reduction for a
sequential mechanism of multiple phosphorylation
(page S-3).
Doc. S2. Kinetic equations and model reduction for a
random mechanism of multiple phosphorylation (page
S-7).

Doc. S3. Transition times for multiple phosphorylation
cycles (page S-10).
Doc. S4. Bistability in a double phosphorylation cycle
(page S-14).
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FEBS Journal 274 (2007) 1046–1061 ª 2007 The Authors Journal compilation ª 2007 FEBS

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