Keywords MAPK; phosphorylation; sequestration; signal transduction; zero-order ultrasensitivity Correspondence N.. Finally, we analyse the effect of sequestration on the dynamics of a co
Trang 1Nils Blu¨thgen1,* Frank J Bruggeman2, Stefan Legewie1, Hanspeter Herzel1, Hans V Westerhoff2,3 and Boris N Kholodenko4
1 Institute for Theoretical Biology, Humboldt University Berlin, Germany
2 Department of Molecular Cell Physiology, Institute of Molecular Cell Biology, Faculty of Earth and Life Sciences, Vrije Universiteit, Amsterdam, the Netherlands
3 Manchester Centre for Integrative Systems Biology, Manchester Interdisciplinary Biocentre, School of Chemistry, University of
Manchester, UK
4 Department of Pathology, Anatomy and Cell Biology, Thomas Jefferson University, Philadelphia, USA
In most biological organisms intracellular signal
pro-cessing is carried out by networks composed of
enzymes that activate and inactivate each other by
co-valent modification Signals received at the cell
mem-brane ripple through signalling networks via covalent
modification events to reach various locations in the
cell and ultimately cause cellular responses The
bio-chemical building blocks of these networks are
fre-quently enzyme pairs, such as a kinase and a
phosphatase, that form covalent modification cycles in
which the target enzyme is covalently modified at
sin-gle or multiple sites in a reversible manner
In some experiments, the stimulus–response curves display strong sigmoidal dependencies in vivo, for example, in the activation of the mitogen-activated protein kinase (MAPK) cascade [2] and Sic1 [3], and
in vitro, for example, in the phosphorylation of isocitrate dehydrogenase [4], muscle glycolysis [5] and
in postsynaptic calcium signalling [6] Sigmoidal stimulus–response curves imply that the responses are highly sensitive to changes in signals around the threshold level Thus it is more sensitive than a typical Michaelis–Menten-like response, a property that has been termed ultrasensitivity [1]
Keywords
MAPK; phosphorylation; sequestration;
signal transduction; zero-order ultrasensitivity
Correspondence
N Blu¨thgen, Institute for Theoretical
Biology, Humboldt University Berlin,
Invalidenstr 43, 10115 Berlin, Germany
Fax: +49 30 838 56943
Tel: +49 30 838 56971
E-mail: nils@itb.biologie.hu-berlin.de
*Present address
Molecular Neurobiology, Institute of Biology,
Free University of Berlin, Germany.
Note
Nils Blu¨thgen and Frank J Bruggerman
contributed equally to this study.
(Received 21 November 2005, accepted
15 December 2005)
doi:10.1111/j.1742-4658.2006.05105.x
The building blocks of most signal transduction pathways are pairs of enzymes, such as kinases and phosphatases, that control the activity of pro-tein targets by covalent modification It has previously been shown [Gold-beter A & Koshland DE (1981) Proc Natl Acad Sci USA 78, 6840–6844] that these systems can be highly sensitive to changes in stimuli if their cata-lysing enzymes are saturated with their target protein substrates This mechanism, termed zero-order ultrasensitivity, may set thresholds that filter out subthreshold stimuli Experimental data on protein abundance suggest that the enzymes and their target proteins are present in comparable con-centrations Under these conditions a large fraction of the target protein may be sequestrated by the enzymes This causes a reduction in tivity so that the proposed mechanism is unlikely to account for ultrasensi-tivity under the conditions present in most in vivo signalling cascades Furthermore, we show that sequestration changes the dynamics of a cova-lent modification cycle and may account for signal termination and a sign-sensitive delay Finally, we analyse the effect of sequestration on the dynamics of a complex signal transduction cascade: the mitogen-activated protein kinase (MAPK) cascade with negative feedback We show that sequestration limits ultrasensitivity in this cascade and may thereby abolish the potential for oscillations induced by negative feedback
Abbreviations
JAK, janus kinase; MAPK, mitogen-activated protein kinase; MAPKK, mitogen-activated protein kinase kinase; MCA, metabolic control analysis.
Trang 2Sigmoid responses can be used to generate binary-like
decisions [7] and to filter out noise or delay responses
[8] Moreover, ultrasensitive signal transduction
cas-cades can display oscillations in combination with a
neg-ative feedback loop [9] and bistability (hysteresis) in
combination with positive feedback [10,11]
Surpris-ingly, ultrasensitivity coupled with negative feedback
also yields highly linear responses and signal fidelity in
the presence of high load [12]
Several mechanisms account for ultrasensitive
stimu-lus–response curves, including cooperativity, multisite
phosphorylation, feed-forward loops and enzymes
operating under saturation The latter mechanism
has been termed zero-order ultrasensitivity because a
necessary condition is that the modifying and
de-modi-fying enzyme of a covalent modification cycle display
zero-order kinetics This mechanism was explored for
the steady-states of cycles composed of enzymes with
irreversible product-insensitive kinetics in pioneering
work by Goldbeter & Koshland [1] Zero-order
sensi-tivity is appealing because of its simplicity: all it needs
is one modification site on a protein that acts as a
sub-strate (e.g a phosphorylation site) and, for example, a
kinase and a phosphatase in which at least one of the
enzymes has a KMvalue for their substrate that is low
compared with the total concentration of the protein
substrate This mechanism might provide cells with
simple ultrasensitive signalling units that can be
inter-connected to form networks that can display a great
variety of responses [13]
However, cells also use more complicated
mecha-nisms that activate proteins by multiple modification
events to bring about ultrasensitivity Examples of such
protein targets are Sic1 which has at least six
phos-phorylation sites [3] Nuclear factor of activated T-cells
(NFAT) has even more phosphorylation sites [14], and
the MAPK cascades containing MAPK kinase
(MAPKK) and MAPK both become fully activated by
double phosphorylation It remains a puzzle, why other,
more complicated means like multisite phosphorylation
need to be applied to get high sensitivity when there is a
simple mechanism like zero-order ultrasensitivity
Goldbeter & Koshland discussed briefly that product
sensitivity and a large amount of enzyme–substrate
complex compared with the total concentration of the
interconvertible enzyme may reduce the sensitivity of
the cycle They did not analyse any of the general
con-sequences of sequestration, however, and the severe
consequences of sequestration for ultrasensitivity
there-fore remain unclear The effect of product sensitivity
has been quantified in more detail by Ortega et al
[15], who showed that ultrasensitivity disappears if the
enzymes are product sensitive Data about protein
abundance in signal transduction cascades are now in hand, showing that members of the cascades are pre-sent in concentrations of the same order of magnitude [16] (see Table 1 for examples) Therefore, we decided
to investigate the effect of high enzyme concentration
on the sensitivity of signal transduction cascades in more detail Without loss of generality we assume that the modification is phosphorylation and the enzymes are kinases and phosphatases First, we investigate the amount of sequestered substrate in a simple covalent modification cycle (Fig 1) We then show that seques-tration reduces zero-order ultrasensitivity dramatically Subsequently, we illustrate the consequences of seques-tration on zero-order ultrasensitivity by numerical simulations and confirm the predictions
We show that sequestration also has dramatic effects
on signalling dynamics Sequestration can account for the transient transduction of a permanent signal Multisite phosphorylation and kinase sequestration can work as a sign-sensitive delay element [17], in which the rise in the signal is delayed but the dropping signal is transduced immediately
Finally, we analyse the effect of sequestration on a complex signal transduction cascade, the MAPK cas-cade Computational studies by Kholodenko [9] have shown that oscillation can arise in this system from a combination of ultrasensitivity and negative feedback
We show that sequestration abolishes those oscillations
by reducing zero-order ultrasensitivity
Results
Sequestration in covalent modification cycles Unlike metabolic systems, the modification cycles in-volved in signal transduction cascades often exhibit comparable amounts of protein substrates and enzymes [16] For example, the individual concentrations of
Table 1 Concentrations of members of the MAPK cascade (MAPKKK, MAPKK, MAPK) in different organisms and cell types as found in the literature In many of these, the concentrations are of the same order of magnitude RU, relative units.
Chinese hamster ovary cells
1300 n M 2800 n M [7]
Trang 3the three kinases of the well-characterized MAPK
cas-cade are similar in a variety of cell types and
organ-isms (Table 1) Each of these kinases modifies its
target protein and is itself a target for the upstream
kinase Because the concentration of kinases and their
target proteins are comparable, the kinase can
seques-ter a significant amount of target by binding to it,
provided that the kinase shows high affinity towards
the substrate This sequestered fraction of the target is
no longer accessible to other kinases and phosphatases
Available data about phosphatase concentrations
sug-gest that they are also likely to be of the same order of
magnitude as or even exceed their substrate
concentra-tions [18,19]
The concentration of the kinase–substrate complex
[TK] in the steady-state can be calculated using the
Michaelis–Menten formula:
½TK ¼ ½T½KT
½T þ KM
ð1Þ
where [T] and [KT] are the free target concentration
and total kinase concentration, respectively, and KMis
the Michaelis–Menten constant of the kinase The con-centration of the complex [TK] approaches the total concentration of the kinase as [T] > KM The phos-phatase–substrate concentration can be calculated accordingly To illustrate this effect for a covalent modification cycle, we investigate a special case, i.e when both kinase and phosphatase have the same kin-etic constants and the same concentrations Conse-quently, the two substrates are in equal steady-state concentrations ([T] ¼ [T*]) and the two complex con-centrations are equal ([TK]¼ [T*P]) Therefore, the total target concentration can be expressed as: TT¼ 2[T] + 2[TK] After substitution of the resulting expres-sion for [TK] into the Michaelis–Menten formula, we obtain:
2 ½KT½T
KMþ ½T¼ TT 2½T ð2Þ From this, the amount of free substrate in the cycle, i.e [T] + [T*]¼ 2[T] can be calculated from the total centrations of kinase and target Importantly, the con-centrations of the free substrate forms [T] and [T*] decrease below KM if [KT] > [TT]) 2KM (see Supple-mentary material for mathematical details) If the cata-lytic activity of the phosphatase exceeds the activity of the kinase, the free substrates can be higher In this case, [T] and [T*] will still fall below KM if the kinase and phosphatase concentrations together exceed twice the target concentration, i.e [KT] + [PT] > 2[TT]) 4KM Thus, in a signalling cycle, sequestration reduces the free target concentrations such that the concentration
of the free target is below the KM value of the enzymes, provided that the enzymes are available in a concentration as high as their total protein substrate concentration
The effect of sequestration on zero-order ultrasensitivity
The sensitivity of simple modification cycles was explored in pioneering work by Goldbeter & Koshland [1] using methods from nonlinear dynamics Later, it was formulated in terms of metabolic control analysis (MCA) by Small & Fell [20] Small & Fell expressed the response of the active fraction to a change of the kinase concentration as a function of the concentra-tions of the two forms ([T] and [T*]) and the elastici-ties of the enzymes by the following simple relation:
RTKT ¼ ½T
et2
T ½T þ et1
As discussed in the Methods, this response coefficient expresses the fractional change of the active form
T
T
T
T
1a
2b
1b
2a
K
K
P
P
*
*
Fig 1 Schematic representation of the simplest covalent
modifica-tion cycle The target protein T can be covalently modified The
unmodified protein T binds to the kinase K in the first reaction (1a) to
form the complex TK The second reaction (1b) is the catalytic
modifi-cation step yielding K and the covalently modified target protein T*.
In the third reaction (2a), the phosphatase P binds T* to form the
complex T*P In the fourth reaction (2b) the cycle is closed by the
recycling of T via catalytic demodification and the release of P
Reac-tions 1b and 2b are assumed to be irreversible for simplicity.
Trang 4[T*] upon a fractional change of the kinase
concen-tration If the enzymes are unsaturated, their
elastici-ties are em2
T 1 and em1
T 1, and the response coefficient is RT
KT< 1, corresponding to a sublinear
response In this case, no ultrasensitivity is observed
In contrast, saturation of the enzymes leads to
elasti-cities closer to 0, hence RT
K T can exceed 1 and give rise to an ultrasensitive response In the derivation of
Eqn (3), Small & Fell [20] assumed that the
concen-tration of the substrate bound to the enzyme is
negli-gible But as discussed above, this assumption does
not hold where the concentrations of enzymes and
substrate are similar, as observed in signal
transduc-tion cascades if the enzymes are saturated Therefore,
the assumptions made to derive Eqn (3) may not
necessarily hold
If the effect of sequestration is taken into account
the response coefficient modifies to:
et2
T ½T þ et1
T½T þ et2
T et1
T½TK þ ½TP ð4Þ
A detailed mathematical derivation of Eqn (4) can be
found in the Supplementary material Comparison of
Eqn (4) with Eqn (3) reveals the effect of sequestration
on zero-order ultrasensitivity as an additional term in
the denominator which increases with the extent of
sequestration, i.e ([TK] + [T*P]) Therefore, at
con-stant elasticities, sensitivity should decrease with
sequestration Another effect is hidden in the
equa-tions: an increase in sequestration also increases the
elasticities et2
T and et1
T, because the available substrate decreases This eventually causes an additional
decrease in the sensitivity RT
KT
To elucidate this further, we examined the special case
when both kinase and phosphatase have the same
kin-etic constants In this case, we expect, on the basis of
symmetry, that the highest response coefficient occurs
when there are equal amounts of phosphorylated
and unphosphorylated target We can then express
all concentrations and elasticities in terms of [T], the
Michaelis–Menten constant, KM, of kinase and
phos-phatase In this case Eqn (4) reads:
RTKT ¼ 1þ
½T
K M
2 1þ KM ½K T
ðK M þ½TÞ2
RT
KT increases with [T] and decreases with [KT] This
shows that the response coefficient gets smaller as the
amount of free substrate [T] decreases due to
seques-tration As discussed previously, similar
concentra-tions of the enzymes and target imply that the free
target falls below the KMvalue The response is then
sublinear, i.e RT
KT < 1, because 2K M
½TþK M 1 Also, if
KMis very small, most of the substrate is sequestered, leading to essentially zero concentrations of T and T*
Goldbeter & Koshland [1] discussed the possibility that ultrasensitivity might be preserved if the phospha-tase–target complex T*P is assumed to be active How-ever, as calculated in the Supplementary material, the combined response of T and T*P, RTKTþTP is always
< RT
K T Thus, the attenuation of sensitivity by seques-tration cannot be restored by an active phosphatase– target complex
The consequences of sequestration for ultrasensitivity: numerical investigations
To further investigate the consequences of sequestra-tion on ultrasensitivity, the steady-state of the cycle depicted in Fig 1 was calculated numerically The
KM value was chosen to be much smaller than the total concentration (KM¼ 0.02[TT]) for both the kin-ase and the phosphatkin-ase The phosphatkin-ase concentra-tion [PT] was increased from 0 to 2[TT], to vary the amount of sequestration Figure 2B shows that this increase is accompanied by an increase in the seques-tered fraction ([TK] + [T*P])⁄ [TT] The response of the cycle [T*] to the input [KT] decreases if the total levels of the phosphatase approach half of the total target concentration [TT] (Fig 2A) Taken together these two plots illustrate our argument: when the kin-ase and phosphatkin-ase concentrations become compar-able with the total concentration of the target protein, the sequestered fraction increases, which cau-ses the sensitivity to decrease In Fig 2C the activated fraction of the target T* is plotted, illustrating that the fraction of activated target decreases dramatically
as the phosphatase concentration exceeds [TT]⁄ 2 These results are in good agreement with the esti-mates made above This suggests that in vivo, where
in many cases the concentrations of the kinase, the phosphatase and the target protein are comparable, the sensitivity of covalent-modification cycles is likely
to be achieved by mechanisms other than zero-order ultrasensitivity Simulations for different catalytic activities of kinases and phosphatases are shown in Fig 2D–I If the phosphatase is catalytically 10-fold more active than the kinase, the region of enhanced sensitivity is broadened slightly (Fig 2D–F) In con-trast, if the kinase is more active than the phos-phatase, the region if ultrasensitivity is drastically reduced (Fig 2G–I) It seems that in case of mamma-lian MAPK cascades high concentrations of the
Trang 5phos-phatases yield high sequestration which do not allow
for zero-order ultrasensitivity
The consequences of sequestration for signalling
dynamics
Receptor desensitization is a relatively slow process
and downstream signal transduction cascades are often
in a quasi-steady-state with the receptor activity
How-ever, some downstream parameters adapt very quickly
(e.g insulin receptor substrate phosphorylation after insulin and Erk after epidermal growth factor), sug-gesting that downstream pathways are capable of adaptation Figure 3A shows the dynamics of the co-valent modification cycle for a fast kinase with low affinity and a slow phosphatase with high affinity If a permanent stimulus is given, the target displays only transient activation Thus a covalent modification cycle
is capable of terminating prolonged signals The fast kinase phosphorylates the available target, but the
1
0
2
[K ]T T [P ]
0.1 0.2
[K ]T T [P ]
[K ]T T [P ]
Legend A,D,G 0
20
[P ]T
0
Response Coefficient
Sequestered Target
Activated Fraction
1
0 2
0.1 0.2
10
0
20 0
1
0 2
0.1 0.2
10
0
20 0
A
D
G
B
E
H
C
F
I
> 3
< 3
< 2
< 1.5
< 1.25
< 1
< 0.75
< 0.5
> 0.9
< 0.9
< 0.5
< 0.15
< 0.1
< 0.05
< 0.02
< 0.01
> 0.9
< 0.9
< 0.8
< 0.7
< 0.6
< 0.5
< 0.4
< 0.3
< 0.2
< 0.1
10
Legend B,E,H Legend C,F,I
Fig 2 Steady-state signalling characteristics of a covalent-modification cycle for equal catalytic activity of kinase and phosphatase (A–C), for 10-fold higher catalytic activity of the kinase (D–F), for 10-fold reduced catalytic activity of the kinase (G–I) (A) Contour plot of the response coefficient R T
KT as function of the total concentration of the phosphatase and the kinase (normalized to the phosphatase concentration) (B) Sequestered fraction of the target protein (C) Fraction of the activated target protein Parameter values: TT¼ 1, k 1a,f ¼ 10, k 1a,r ¼ 0.1,
k 1b,r ¼ 0, k 2a,f ¼ 10, k 2b,f ¼ 0.1 and k 2b,r ¼ 0 varied to simulate different catalytic activity of the kinase: (A–C) k 1b,f ¼ 0.1, (D–F) k 1b,f ¼ 1, (G–I) k1b,f¼ 0.01 K T and PTrefer to the total kinase and phosphatase concentration, respectively.
Trang 6phosphorylated target is subsequently sequestered by
the low-activity high-affinity phosphatase At
steady-states most of the target substrate is sequestered by the
phosphatase Thus substrate sequestration by a
phos-phatase might be a means to achieve signal
termin-ation and desensitiztermin-ation downstream of receptors
without involving a negative feedback loop
For many signals, their duration determines the
bio-logical response [21] We have pointed out that
seques-tration might cause short, transient signals However,
interpretation of the signal by the signal transduction
network requires circuits that respond only to
pro-longed activation As pointed out by Deshaies &
Ferrell [22], such signal duration decoding requires a
threshold time Also, deactivation must be fast in com-parison with activation as removal of the signal has
to be translated into an immediate response Such properties have been described for coherent feed-for-ward loops, which display sign-sensitive delay [17] Figure 3B shows that competition for the enzyme by two phosphorylation sites may also account for such a sign-sensitive delay and dramatically improves dur-ation decoding The solid line shows the dynamics of double-phosphorylation in which both phosphoryla-tion sites compete for the kinase, the dotted line shows the dynamics of the corresponding system in case there
is no competition (details in the Supplementary mater-ial) If the stimulus increases it must be of a certain length to be transduced if the sites compete for the kinase However, if the stimulus falls, the change is transduced immediately Thus, sequestration and multisite phosphorylation might be a mechanism for sign-sensitive delays, similar to coherent feed-forward loops in transcriptional networks [17]
Changes in the steady-state stimulus–response curve might also have a large impact on the dynamics because the onset of oscillations in a signal transduc-tion cascade harbouring a negative feedback is deter-mined by the sensitivity of the stimulus–response curve
in the steady-state We investigated the effects of sequestration in a complex signal transduction cascade with negative feedback as described below
The effect of sequestration in MAPK signal transduction cascade
The MAPK cascade consists of three kinases that activate their downstream kinases by phosphorylation (Fig 4) It has the potential to be ultrasensitive because of the combination of multisite phosphoryla-tion, zero-order kinetics [23] and cascade amplification effects [24] According to Kholodenko [9] a negative feedback that is wrapped around this ultrasensitive cascade can bring about sustained oscillations over a wide range of stimuli if sequestration is neglected (Fig 6A)
As the kinases are present at similar concentrations,
we investigated whether sequestration affects ultrasen-sitivity and oscillatory behaviour We modelled the cascade such that sequestration was taken into account (similar to Huang & Ferrell [23], with parameters adopted to reflect the catalytic and Michaelis–Menten constants from Kholodenko [9], see Supplementary material for details) We chose concentrations of the phosphatases for MAPK and MAPKK that were as high as that of their substrate (300 nm) First, we ana-lysed the cascade without feedback Figure 5 compares
time 0
25
50
A
x0.1
B
time 0
20
40
60
80
100
Fig 3 (A) The dynamics of free phosphorylated target protein in
case of more active kinase than phosphatases k 1a,f ¼ 0.005,
k1a,r¼ 0.4, k 1b,f ¼ 0.1, k 2a,f ¼ 0.0005, k 2a,r ¼ 0.004, k 2b,f ¼ 0.001
T T ¼ 100 K T ¼ 300 P T ¼ (300) At zero time-point, the system is at
steady-state for zero stimulus (initial conditions: [T](0) ¼ T T ,
[T*](0) ¼ [TK](0) ¼ [T*P](0) ¼ 0) (B) The dynamics of
double-phos-phorylation in case the kinase shows higher affinity towards the
un-phosphorylated target Solid line: the case of kinase sequestration,
dotted line: no kinase sequestration Grey lines indicate the
stimu-lus (i.e kinase concentration), scaled by 0.1 in (A).
Trang 7a model neglecting sequestration (Fig 5A–C) and one
including the effect of sequestration (Fig 5D,E)
Whereas the response of the first molecule
(MAPKKK) is relatively unchanged because its kinase
and phosphatase are present only at low
concentra-tions, the response of the second and third molecules
(MAPKK and MAPK) is changed dramatically There
are two main effects of sequestration visible in the
response of MAPKK and MAPK: the ultrasensitivity
of the stimulus response curves is reduced and the
amount of maximally activated kinases in this cascade
is decreased
If we add a negative feedback loop to this model,
similar to the model by Kholodenko [9], no oscillations
arise (Fig 6B) The effect of lower activation of
MAPK can be compensated for by a stronger feedback
(lower values of kloop, see Supplementary material)
However, lowering of kloop does not restore
oscilla-tions (Fig 6B) This leads us to conclude that the
reduction in ultrasensitivity due to sequestration is
responsible for the diminishing of oscillations
We observed in the analysis of simple, isolated
cova-lent modification cycles that an increase in the total
target concentration will limit the sequestered fraction
of the target and restore ultrasensitivity However, in
cascades such as the MAPK cascade the kinases are both enzymes for the modification of the downstream kinase and substrate for the upstream kinase Hence, the complex of, for example, MAPK and MAPKK reduces the free concentration of both MAPK and MAPKK Therefore, an increase of the MAPK con-centration in this cascade gives rise to more sequestra-tion of MAPKK by MAPK Consequently, it is not surprising that we found that an increase in MAPK of one order of magnitude cannot restore the oscillations
In addition, we investigated the effects of sequestra-tion by phosphatases We found that oscillasequestra-tions can
be restored if the phosphatase concentrations of MAPK- and MAPKK-phosphatase are lowered to one fifth of the kinase concentrations (while increasing their catalytic activity by factor five to keep the Vmax value constant) However, in contrast to the model that neglects sequestration, the stimulus needs to be rather low (Fig 6C) In this case, sequestration due to the phosphatase is reduced and the upstream kinases
of MAPK and MAPKK are only slightly activated and can sequester only limited fractions of MAPK and MAPKK
Discussion
The function of the signal transduction network is to sense changes in the environment of the organism in the form of signals of physicochemical origin, e.g concen-trations of molecules or mechanical stress, and to integ-rate these with the current cellular status to ‘compute’
an adaptive response [12] Such adaptive responses involve covalent modification of enzymes, changes in gene expression, and cell-fate decisions that occur on different time scales Many signal transduction networks have common building blocks: enzyme couples that acti-vate and inactive their protein targets via covalent modi-fication It is reasonable to expect that network responses can be highly sensitive to changes in the sig-nals Ultrasensitivity can be used generate thresholds, oscillations and linear responses [12] Therefore, it may not be surprising that ultrasensitivity has been docu-mented experimentally for some signalling systems [10] Theoretical studies by Goldbeter & Koshland [1] unveiled a potential mechanism responsible for ultra-sensitivity: the kinase and phosphatase have to be sat-urated with their target protein This case has been referred to as zero-order ultrasensitivity Since then, many groups have analysed zero-order ultrasensitivity [15,25,26,27] Although the effect of complex forma-tion in a substrate cycle has been addressed previously [28], the impact of sequestration on zero-order ultra-sensitivity has not
P P P
MAPKK
P
MAPKK
P
MAPKK
P
P
MAPKKKK
Fig 4 Sketch of the MAPK cascade A MAPKKKK stimulates the
phosphorylation of MAPKKK, which after phosphorylation
phos-phorylates MAPKK at two sites The double-phosphorylated
MAPKK phosphorylates MAPK also at two sites The
double-phos-phorylated MAPK in turn inhibits the activity of MAPKKKK.
Trang 8Experimental data (Table 1) indicate that the
concen-trations of enzymes and target proteins in signal
trans-duction cascades are similar When the affinity of
enzymes for their target is sufficiently high, it implies
that a high fraction of the target concentration is bound
to the enzymes, and thereby sequestered This, in
turn, decreases the amount of target accessible to the
enzymes, and reduces ultrasensitivity Moreover, the
amount of activated target decreases dramatically
Con-sequently, the concentrations of the complexes can no
longer be neglected in the analysis of ultrasensitivity,
as long as the concentrations of players in the signal
transduction cascades are comparable
We investigated the consequences of sequestration
on zero-order ultrasensitivity using the analytical
approach of MCA and numerical simulations In terms
of MCA, ultrasensitivity is equivalent to a response
coefficient higher than 1 [15] We derived an analytical
expression for the response coefficient (Eqn 4) that
accounts for the effect of sequestration Comparison
with a response coefficient that neglects sequestration
(Eqn 3) suggests that sequestration may significantly
reduce and even eliminate ultrasensitivity Eqn (5)
cor-roborates this for a simple example in which the
kin-etic parameters of both enzymes are equal It shows
that the response coefficient decreases below 0.5: hence, ultrasensitivity is absent The results of numer-ical simulations illustrated that if the total concentra-tions of both enzymes are increased simultaneously, ultrasensitivity decreases and ultimately vanishes when these concentrations exceed 70% of the total target concentration This correlated with high sequestration
of the target protein by the enzymes, which illustrates that sequestration reduces ultrasensitivity
Another problem of zero-order ultrasensitivity arises due to the sequestration of the enzyme by the sub-strate: The saturated enzyme may then not be available for other reactions This is of special importance if the enzyme itself is the substrate of a modification cycle like MAPKK, which is itself controlled by phosphory-lation and is the enzyme that phosphorylates MAPK Here sequestration reduces the zero-order ultrasensitiv-ity in both cycles: the cycle in which the enzyme drives the modification and that in which the enzyme is subject to modification In such signalling cascades sequestration can be significant even if the kinase concentrations increase along the cascades due to the sequestration of the enzymes The extent of ultrasensi-tivity that can be generated by signal transduction cas-cades is thereby limited by sequestration This effect
0
50
100
0
100
200
300
0
100
200
300
no sequestration
sequestration
stimulus (MAPKKKK in nM)
A
B
C
D
E
F
Fig 5 Stimulus–response curves for the three layers of the MAPK cascade in the model considered by Kholodenko [9] (A–C), which neglects sequestration and the cor-responding model that takes the effects of sequestration into account (D–F).
Trang 9might be responsible for the fact that sustained
oscillations have not yet been documented in the
MAPK cascade as opposed to the NF-jB cascade [29]
Because each enzyme usually targets more than one
reaction, as, for example, most phosphatases,
modifica-tion cycles compete for the enzymes After a pathway
is activated it recruits its phosphatases, which are no
longer accessible to others We show that sequestration
of the kinase in a double-phosphorylation cycle may
account for a sign-sensitive delay element, such that
the activation of a target enzyme upon a signal is
delayed, but it is in-activated immediately after
removal of the signal Such a delay element provides
cells with units that neglect short fluctuations in
sig-nals, but transduce long signals
In addition, sequestration might mediate cross-talk
between pathways if an enzyme is shared This has
been observed in the JAK⁄ STAT pathway, in which
the receptors share the janus kinase (JAK) and
mul-tiple receptors compete for it Upregulation of one
receptor downregulates the response of the other by sequestration of JAK [30]
Our results suggest that to generate ultrasensitivity, cells need to exploit mechanisms that do not require enzyme saturation Such mechanisms include multisite phosphorylation, which generates ultrasensitivity with-out the need for sequestration Moreover, not only ultrasensitivity, but also bistability and hysteresis arise from multisite covalent modification in signalling cas-cades [31] Ultrasensitivity and bistability induced by multisite phosphorylation may be a widespread mech-anism for the control of protein activity in signalling networks, whereas zero-order ultrasensitivity is unli-kely to be the major means of generating switch-like behaviour in such systems
One thing is clear, the covalent cycle is extremely versatile for eliciting different kinds of behaviour [12,32] This great versatility may partly explain why signalling pathways, in both prokaryotic and
eukaryot-ic systems, employ this motif in so many instances
MAPKKKK (nM) 0
100 200 300
MAPKKKK (nM) 0
25
50
A
C
B
MAPKKKK (nM) 0
30 60
sustained oscillations
no sustained oscillations
Fig 6 Bifurcation diagrams for the models
that neglect (A) and include (B,C) the effects
of sequestration Solid lines show stable
steady-states, dotted lines indicate unstable
steady-states The dashed lines mark the
amplitude of the oscillations observed in the
model that neglects sequestration The four
lines in (B) show situations for different
feedback parameters (from top to bottom:
kloop¼ 9, 1, 0.1, 0.01 n M ) (C)
Two-dimen-sional bifurcation diagram for the model that
includes the effect of sequestration
Con-centrations of the MAPK- and
MAPKK-phos-phatases (vertical axis) and the stimulus
(horizontal axis) are changed The dashed
area shows the region where sustained
oscillations occur Insets show qualitatively
the dynamics in the corresponding areas.
Trang 10Unfortunately, the lack of any clear guidance from
experimental data means we are unable to determine
exactly the true functional role played by these motifs
Although many signalling networks have been mapped
in great detail we still have very little understanding of
their actual dynamical behaviour Until
experimental-ists embrace a systems approach we will remain in the
dark regarding this question
Methods
The model files used to perform numerical simulations are
available from the authors upon request
Metabolic control analysis
To analyse ultrasensitivity, we adopt some methods and
terms from MCA [33–35], for application to conserved
moi-eties, see Hofmeyr et al [27] MCA has been successfully
applied to intracellular signal transduction systems in the
past [36–38] MCA links ‘global’ control properties of a
network to ‘local’ properties (e.g mechanistic details of
enzyme-catalysed reactions) The local properties are called
elasticity coefficients and are defined by evj
x i ¼ x i
v j
@v j
@½x i Elasti-cities evaluate the relative change in a reaction rate as a
result of an infinitesimal relative change in one of its
sub-strate, product, or effector concentrations (e.g [xi]) The
elasticities of an enzyme ejfollowing irreversible Michaelis–
Menten kinetics with the Michaelis–Menten constant KM
are evj
e j ¼ 1 with respect to the enzyme concentration and
evj
S ¼ K M
½S þ K Mfor the substrate S
Global properties are called response coefficients and
describe the response of the entire system to small
perturba-tions in parameters, RS i
p j ¼ pj
½S i d½S i
dp j Here, RS i
p j accounts for a relative change in steady-state metabolite concentration
[Si] upon infinitesimal relative change in the value of the
parameter pj
Model of a simple interconversion cycle
In the first part of this paper we analyse a simple covalent
modification cycle that consists of two enzymes K and P
that phosphorylate and dephosphorylate a target protein T,
respectively (Fig 1) T can be in a modified and unmodified
form, denoted by T and T, respectively To investigate the
effect of sequestration, we model the reactions catalysed by
the two enzymes K and P We explicitly take the enzyme–
target complex into account In the case of reversibility and
product sensitivity, this system has been shown not to be
ultrasensitive, and therefore such effects are not considered
here [15] However, Ortega et al [15] did not consider
sequestration The total concentrations of the three
enzymes involved are denoted by [TT], [KT] and [PT] The
enzyme–substrate complexes are called TK and T*P We
describe the dynamics of this kinetic scheme depicted in Fig 1 by a system of three ordinary differential equations using mass-action kinetics
Models of the MAPK cascade
We shall also analyse the effect of sequestration in a more complicated system, the MAPK cascade We construct two models: One according to Kholodenko [9], which neglects sequestration, and a second one similar to Huang & Ferrell [23], which takes enzyme–substrate complexes into account In the second model, the parameters are adopted such that they reflect the catalytic constants and KMvalues of the model by Kholodenko [9] The details of the kinetic model can be found
in the appendix The numerical analysis of the equations was carried out using mathematica and xpp-auto [39]
Acknowledgements
We would also like to thank Herbert M Sauro for critically reading the manuscript and for assisting NB
in the development of some of the theory outlined here during NB’s stay at Sauro’s Laboratory in Los Ange-les NB acknowledges support from DFG (SFB 618)
FB was supported by the European Union through the Network of Excellence BioSim, Contract No LSHB-CT-2004-005137 Cooperation between NB and FB was supported by the DFG Graduate Program GK268
‘Dynamics and Evolution of Cellular and Macromole-cular Processes’ BNK acknowledges support from the National Institute of Health Grant GM59570 HMS acknowledges support from the National Science Foundation Grant CCF-0432190
References
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