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

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

by the order of phosphate processing and protein–protein interactions

Carlos Salazar1and Thomas Ho¨fer1,2

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, acetymethy-lation, 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 phosphthe-orylation based on the mechanistic description of elementary binding and catalytic steps Explicit solutions for the steady-state response curves and characteristic (de)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.

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 reacdescrip-tion 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

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

n¼ 0, N, where N is 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 Xn,0, Xn,K and Xn,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:

Yn¼ Xn;0þ Xn;Kþ Xn;P ð1Þ i.e the sum of free and enzyme-bound forms As shown in supplementary Doc S1, the total concentra-tions Ynare governed by the differential equations

dY0

dt ¼ a1Y0þ b1Y1 ð2aÞ

dYn

dt ¼ anYn1 ðanþ1þ bnÞYn

þ bnþ1Ynþ1; for 1 n  N  1

ð2bÞ

dYN

dt ¼ aNYN1 bNYN ð2cÞ

where an and bn are effective rate constants of phos-phorylation and dephosphos-phorylation

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 2 N 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.

an ¼ an

|{z}

catalytic

rate of kinase

K=Ln1

1þ K=Ln1þ P=Qn1

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

fraction of kinasebound target protein

;

|{z}

catalytic rate

of phosphatase

P=Qn

1þ K=Lnþ P=Qn

|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

fraction of phosphatasebound target protein

ð3Þ

These account for both enzyme–target binding and the

catalysis K and P denote the free concentrations of

kinase and phosphatase, respectively, and Ln and Qn

are the respective dissociation constants for the

kinase–target and phosphatase–target interactions an

and bn are the catalytic rate constants for addition or

removal of the nth phosphoryl group, respectively

Because the physical properties of the target protein

will generally change with the number of

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

KT¼ K þXN

n¼0

Xn;K ¼ K 1 þXN

n¼0

Yn=Ln

1þ K=Lnþ P=Qn

!

ð4aÞ

PT¼ P þXN

n¼0

Xn;P ¼ P 1 þXN

n¼0

Yn=Qn

1þ K=Lnþ P=Qn

!

ð4bÞ 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 nonlin-ear through the nonlinnonlin-ear 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 pdep-hosphorylation 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=D n¼ 0

YT

DPn i¼1ai

bi 1 n  N; D¼ 1 þPN

i¼1

Pi j¼1

aj

bj

8

<

:

ð5Þ where

YT ¼XN n¼0

Yn

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

B

A

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 (n ¼ 0, 1, , N) is represented

by Xn,0 The kinase–substrate and phosphatase–substrate

com-plexes are denoted by Xn,K and Xn,P, respectively The rate

con-stants for phosphorylation of X n,K and dephosphorylation of X n,P are

denoted by an + 1 and bn, respectively Lnand Qnare the

dissoci-ation constants for the complexes Xn,Kand Xn,P, respectively.

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 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

(Ln¼ L and Qn¼ Q for all n) and the same also holds

for the catalytic rate constants (an¼ a and bn¼ b)

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

n-times phosphorylated target, yn¼ Yn=YT:



yn¼r

nðr  1Þ

The crucial parameter combination of rate constants,

enzyme concentrations and affinities is

r¼a

b¼aKT=L

bearing in mind the assumption of weak enzyme

bind-ing r is 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 (Ln ¼ L, Qn ¼ Q, an¼ a and

bn¼ 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 þ 1Þa; bn! nb ð8Þ

These relations indicate that an n-times phosphorylated

substrate can be further phosphorylated on N) n

dif-ferent residues and dephosphorylated on n residues In

this way, the random scheme is mapped to a linear

chain of reactions, in which the effective

phosphoryla-tion rate decreases with increasing phosphorylaphosphoryla-tion 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



yn¼ N n

where

N n

 

is the binomial coefficient, and r was defined in Eqn (7)

In the limiting case of a target with a single phos-phorylation site (N¼ 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 sin-gle 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 unphosphoryl-ated to nearly fully phosphorylunphosphoryl-ated 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 n¼ 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 nH¼

ln 81⁄ ln R, where the global response coefficient R is the ratio of the concentration of active kinase K0.9at which there is 90% fully phosphorylated target to the kinase concentration K0.1at which 10% of the target is fully phosphorylated, R ¼ K0.9⁄ K0.1 [13] For the sequential mechanism, the effective Hill coefficient ran-ges between 1 and 2 (Fig 3E) For random and mixed

sequential-random mechanisms, nH is generally

smal-ler 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 implica-ted in this [22,33] How long does it take for a multi-site 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

C

E

D

A

B

Fig 3 Steady-state behavior and the order of phosphate processing (A) Steady-state behavior of the fully phosphorylated fraction yNas a function of the kinase ⁄ phosphatase concentration ratio K T ⁄ P T (stimulus strength) for different numbers of phosphorylation sites N in the case of a sequential mechanism (B) Phosphorylation fractions y N as a function of K T ⁄ P T for a sequential mechanism and N ¼ 4 (C) Steady-state behavior of the fully phosphorylated fraction yNas a function of KT⁄ P T for the sequential (solid black line), cyclic (solid gray line) and random (dashed black line) mechanisms (D) Steady-state behavior of the sum of the partially phosphorylated fractions P N1

n¼1 y N as a function

of K T ⁄ P T for the sequential (solid black line), cyclic (solid gray line) and random (dashed black line) mechanisms (E) Comparison of the effect-ive 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 dephosphory-lation or vice versa) are situated between the curves corresponding to the sequential and random schemes Parameters: a n ¼ b n ¼ 1, L n ¼

Q n ¼ 1 [in (A–E)]; N ¼ 4 ([in (B–D)].

weak (see previous section) The transition time for

changes in concentration of the fully phosphorylated

target is appropriately defined as

sN ¼

R 1 0 ðyN yNðtÞÞdt

where yN (0) and yN are the steady states before and

after the transition [20,38,40]

For the sequential mechanism, we obtain

sN¼ 1

bPT=Q

Nðr  1Þ rð Nþ1þ 1Þ  2r rð N 1Þ

ðr  1Þ2ðrNþ1 1Þ ð11Þ where r was defined in Eqn (7) (for details, see

supple-mentary Doc S3) For a single-site target, we obtain

s1¼ 1

bPT=Q

1

1þ r¼

1

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

activ-ity For a multisite target, this is no longer the case

Let us consider the switching-on of an initially inactive

kinase (r¼ 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, r¼ 1 At this point, sNbecomes proportional

to 2N + N2: the phosphorylation time increases

quad-ratically with the number of phosphorylation sites

The transition time sNfor the random mechanism is

obtained as

sN ¼ 1

bPT=Q

HN

1þ r¼

HN

where

HN ¼ RNi¼11=i

is the Nth harmonic number (for details, see

supple-mentary Doc S3) Hence, the transition time of a

ran-dom multisite phosphorylation has the same

depend-ence on the effective kinase and phosphatase activities,

a and b, respectively, as the transition time for

single-site 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, HNgrows

approxi-mately as fast as ln N, so that the phosphorylation time

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 line-arly with the number of sites

Plasticity of regulation

In the previous sections, we have analyzed the model

in a special case (weak enzyme binding and phos-phorylation-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 concen-trations 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 sys-tem parameters were varied syssys-tematically

A

Stimulus strength, K P T/ T

0 5 10 15 20 25

N=1

2 4 6

B

1 2 3 4 5

2 4 6

Stimulus strength, K P T/ T

Random Sequential

Fig 4 Transition time and the order of phosphate processing The transition time s is plotted as a function of K T ⁄ P T for different values of N in a sequential mechanism (A) and in a random mechanism (B) of phosphate processing Parameters: an¼ b n ¼ 1,

L n ¼ Q n ¼ 1, N ¼ 1, 2, 4, 6.

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 PTwas 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

concen-trations of the enzymes relative to the target protein,

and the respective dissociation constants Figure 5A

shows the results for a protein with N¼ 4

phosphory-lation 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

phosphoryla-tion state to changes in kinase activity This

ultrasensi-tivity depends also on the dissociation constants for

the kinase–target and phosphatase–target interactions

To be specific, the dissociation constants of the

var-ious 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

kin-ase readily leaves the fully phosphorylated target

Con-versely, if LN<< L0, the kinase will remain

preferentially associated with the phosphorylated

tar-get, 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

ultrasensi-tivity First, ultrasensitivity is obtained when the

enzymes are saturated by the target protein and both

enzymes dissociate readily from their respective

end-products (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

multi-site phosphorylation Second, ultrasensitivity occurs

also with the diametrically opposed parameter con-stellation 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 pro-tein In addition, the catalytic rate constants of (de)phosphorylation could be different for each partic-ular 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 hys-teresis The first theoretical evidence of this phenom-enon in multisite phosphorylation has been recently presented for the doubly phosphorylated MAPK [17] Figure 5B shows how concentration and affinities affect the shape of the response curve as in Fig 5A, but assuming now that the first phosphorylation and dephosphorylation steps are slower than the other steps, a1<an„ 1and bN<bn„ N This brings into existence 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 phosphoryl-ated 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 con-stants, we allowed the first phosphorylation step to have a different rate from the following three steps (a1„ a2,a3,a4) and, likewise, the first dephosphoryla-tion 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 sat-uration, 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, ran-dom phosphorylation reduces the region of the bistable response in parameter space (supplementary Fig S1C) This is because the effective catalytic rates are biased

B

Fig 5 Plasticity of the stimulus sensitivity depends on protein–protein interactions Contour plots are given of the effective Hill coefficient

n H as a function of the enzyme saturation (measured by Y T ⁄ P T ) and cooperativity of enzyme binding (measured by L 0 ⁄ L N ¼ Q N ⁄ Q 0 ) To reduce the dimensionality for displaying the results of parameter variations, we took L 0 ⁄ L N ¼ Q N ⁄ Q 0 , 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 coeffi-cients) 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 occu-pies the same region in parameter space as the region of zero-order ultrasensitivity in the noncooperative system shown in Fig 5A Parame-ters: a n ¼ b n ¼ 1 [in (A)]; a 1 ¼ b N ¼ 0.01, a n ¼ 1 (n „ 1), b n ¼ 1 (n „ N) [in (B)]; N ¼ 4, L 0 ¼ L 1 ¼ ¼ L N ) 1 ¼ 0.1, Q N ¼ Q N ) 1 ¼ ¼

Q1¼ 0.1 [in (A–B)].

towards the partially phosphorylated forms, exhibiting

a smaller degree of cooperative kinetics than the

cor-responding 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 (KT¼ 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

associ-ation 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

phos-phorylation sites can act additively, e.g by producing

gradual changes in the DNA-binding affinity of

tran-scription factors [41,42] Alternatively, individual sites

may control different functions of the target protein,

such as nuclear transport, DNA binding, and

tran-scriptional activity [43,44], which can be achieved when

each residue is phosphorylated by a distinct kinase

Remarkably, we found that individual control of

mul-tiple sites by the same kinase is also feasible However,

this requires a specific kinetic design of target

phos-phorylation

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

phos-phorylated forms will be equally abundant (Fig 7A)

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 rap-idly 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–dot-ted curve), whereas the intermediate form with resi-due 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

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⁄ P T ) and product inhibition (measured by L0⁄ L N ¼ Q N ⁄ Q 0 ) The dark areas denote regions with slow phosphorylation kinetics (high transition times) Parameters: an¼ b n ¼ 1, N ¼ 4, L0¼ L 1 ¼ ¼ L N ) 1 ¼ 0.1,

QN¼ Q N ) 1 ¼ ¼ Q 1 ¼ 0.1.