In order to circumvent these problems, we present a general framework for the quantitative analysis of sensitivity, the relative amplification approach, which is based on the response coe
Trang 1Stefan Legewie, Nils Blu¨thgen and Hanspeter Herzel
Institute for Theoretical Biology, Humboldt University Berlin, Germany
In cellular signal transduction, a stimulus (e.g an
extracellular hormone) brings about intracellular
responses such as transcription These responses may
depend on the extracellular hormone concentration in
a gradual or an ultrasensitive (i.e all-or-none) manner
In gradual systems, a large relative increase in the
sti-mulus is required to accomplish large relative changes
in the response, while a small relative alteration in the
stimulus is sufficient in ultrasensitive systems
Ultra-sensitive responses are common in cellular information
transfer [1–5] as this allows cells to reject background
noise, while amplifying strong inputs [6,7] In addition,
ultrasensitivity embedded in a negative-feedback loop
may result in oscillations [8], while bistability can be
observed in combination with positive feedback [9,10]
Surprisingly, ultrasensitive signalling cascades equipped
with negative feedback may also exhibit an extended
linear response [11] Finally, spatial gradients known
to be important in development can be converted to
sharp boundaries if they elicit ultrasensitive responses
[5] Previous theoretical work has demonstrated that
ultrasensitivity in the fundamental unit of signal
trans-duction, the phosphorylation–dephosphorylation cycle,
can arise if the catalyzing enzymes operate near
satura-tion [12] and⁄ or if an external stimulus acts on both the phosphorylating kinase and the dephosphorylating phosphatase in opposite directions [13,14] In addition, multisite phosphorylation [1], stoichiometric inhibition [15], regulated protein translocation [16] and cascade amplification effects [17] have been shown to contri-bute to ultrasensitive behaviour in more complex sys-tems
Biochemical responses are usually analyzed by fitting the Hill equation, and the estimated Hill coefficient is taken as a measure of sensitivity However, this approach is not appropriate if the response under con-sideration significantly deviates from the best-fit Hill equation In addition, Hill coefficients greater than unity do not necessarily imply ultrasensitive behaviour
if basal activation is significant In order to circumvent these problems, we present a general framework for the quantitative analysis of sensitivity, the relative amplification approach, which is based on the response coefficient defined in metabolic control analysis [18] The relative amplification approach allows quantifica-tion of sensitivity, at both local and global levels In addition, our approach also applies for monotonically decreasing, bell-shaped or nonsaturated responses
Keywords
basal activation; Hill coefficient; metabolic
control analysis; response coefficient;
ultrasensitivity
Correspondence
S Legewie, Institute for Theoretical Biology,
Humboldt University Berlin, Invalidenstrasse
43, Berlin, D-10115, Germany
Fax: +49 30 20938801
Tel: +49 30 20938496
E-mail: s.legewie@biologie.hu-berlin.de
(Received 23 March 2005, revised 7 June
2005, accepted 14 June 2005)
doi:10.1111/j.1742-4658.2005.04818.x
Ultrasensitive responses are common in cellular information transfer because they allow cells to decode extracellular stimuli in an all-or-none manner Biochemical responses are usually analyzed by fitting the Hill equation, and the estimated Hill coefficient is taken as a measure of sensi-tivity However, this approach is not appropriate if the response under con-sideration significantly deviates from the best-fit Hill equation In addition, Hill coefficients greater than unity do not necessarily imply ultrasensitive behaviour if basal activation is significant In order to circumvent these problems we propose a general method for the quantitative analysis of sensitivity, the relative amplification plot, which is based on the response coefficient defined in metabolic control analysis To quantify sensitivity globally (i.e over the whole stimulus range) we introduce the integral-based relative amplification coefficient Our relative amplification approach can easily be extended to monotonically decreasing, bell-shaped or non-saturated responses
Trang 2The response coefficient
In ultrasensitive responses, a small relative increase in
the stimulus results in a large relative change in the
response This phrase is reflected in the definition of
the response coefficient used in metabolic control
ana-lysis [19,20]:
RXS ¼ S
X
dX
dS ¼
d ln X
where S is the stimulus and X is the response The
response coefficient equals the relative alteration in the
response divided by the relative change in the stimulus
Response coefficients greater than unity refer to
rela-tive amplification This means that a relarela-tive change in
the stimulus is increased by the factor RX
S; in other words the relative alteration in the response is RX
S-times greater Thus, highly ultrasensitive responses are
char-acterized by large response coefficients [17]
Limitations of the Hill approach
Sensitivity in biochemical stimulus–response
relation-ships is usually analyzed by fitting the Hill equation
[21]:
X¼ Xbasalþ Xð max XbasalÞ S
nH
SnH
50þ Sn H: ð2Þ Here, Xbasal and Xmax are the basal and the maximal
responses, respectively, while S50refers to the stimulus
required to reach half-maximal activation Depending
on the Hill coefficient, nH, the system is referred to as
ultrasensitive (nH> 1), hyperbolic (nH¼ 1) or
subsen-sitive (nH< 1) By using this approach, the term
ultrasensitivity is used synonymously with the phrase
‘more sensitive than the Michaelis–Menten equation’
(Eqn 2 with nH¼ 1) In addition, sensitivity is
ana-lyzed globally (i.e over the full range of stimulus
levels)
However, fitting the Hill equation is inappropriate
if the shape of the stimulus–response under
conside-ration significantly deviates from that of the Hill
equation As an example consider the scheme in
Fig 1A, which depicts a common motif in signal
transduction, the positive feedback (Appendix I) The
corresponding response (Fig 1B) appears to be
highly ultrasensitive upon weak stimulation, but
sub-sensitive for stronger stimuli The Hill equation is
usually assumed to fit biochemical responses well in
the range between 10 and 90% activation However,
the best-fit Hill equation (Fig 1B - - -) using this
range appears to be hyperbolic and thus does not
reflect the behaviour of the positive feedback model Accordingly, the Hill coefficient of the best-fit Hill equation (nH¼ 1.19) suggests that the positive feed-back scheme exhibits only very weak ultrasensitivity Similar conclusions also hold for an alternative defini-tion of the Hill coefficient proposed by Goldbeter & Koshland [12] for the analysis of responses, whose shape differs from the Hill equation:
Fig 1 Limitations of the Hill approach: deviation from the Hill equa-tion (A) Schematic representation of the positive feedback model (Appendix I) (B) The response of the positive feedback model (solid line) significantly deviates from the best-fit Hill equation (dashed line) with n H ¼ 1.19, so that fitting to the Hill equation is inappropri-ate for the quantification of ultrasensitivity For comparison, the Michaelis–Menten equation is also shown (dotted line) Parameters assumed in the positive feedback model (see Eqn A2 in Appendix I): k K ¼ 1, k A ¼ 1.3, m ¼ 3, X 50 ¼ 0.5, X tot ¼ 1 and k P ¼ 1 The Hill equation was fitted by using the least-squares method with the sti-mulus values Si¼ 1.0233 IÆ 10 )2and I 2 [100, 101, ., 300], which cover 10–90% of the maximal response (i.e 0.1 < X < 0.9) in the positive feedback model.
Trang 3nH¼ log 81ð Þ log S90
S10
Here, S10and S90equal the stimulus levels required to
achieve 10% and 90% activation, respectively For the
response of the positive feedback model depicted in
Fig 1B, one obtains nH¼ 1.01
Thus, both the Hill coefficients obtained by fitting
or by using Eqn (3) suggest that the positive
feed-back model is not ultrasensitive In addition, none of
the two approaches allows quantitative local analysis
for which stimuli ultrasensitivity is especially
pro-nounced Based on the preceding discussion, one
may conclude that the Hill approach is inappropriate
for the quantitative analysis of sensitivity if a
response consists of two parts that differ in their
steepness relative to the Michaelis–Menten equation
and thus cannot be described simultaneously by a
single Hill coefficient As further outlined in the
Dis-cussion, such ‘discontinous’ behaviour has indeed
been shown experimentally for a variety of
biochemi-cal responses
Experimentally measured biochemical responses
often exhibit basal activation [4] Figure 2 shows the
Hill equation with () or without (- - -) basal
activa-tion The Hill coefficient (nH¼ 4), the maximal
activa-tion level (Xmax¼ 1) and the half-maximal stimulus
(S50¼ 1) were assumed to be equal in both plots,
which allows direct comparison of their sensitivities
Importantly, a twofold increase in the stimulus level,
from S¼ 0.5 to S ¼ 1, results in an approximately
fivefold increased response without basal activation, but in only a 1.5-fold increased response with basal activation Thus, the biologically relevant sensitivity (i.e the response coefficient as defined in Eqn 1), strongly decreases with increasing basal activation As similar conclusions hold over the whole stimulus range, one can conclude that Hill coefficients significantly greater than unity do not necessarily imply ultrasensi-tive responses if basal activation is significant
Relative amplification approach
Relative amplification plot Owing to the conclusions made in the section above, it seems more reasonable to analyze the sensitivity of biochemical responses by means of the response coeffi-cient defined in Eqn (1) rather than by fitting the Hill equation
The response coefficient of the Hill equation devoid
of basal activation (Eqn 2 with Xbasal¼ 0) is a func-tion of the stimulus S, the Hill coefficient, nH, and the half-maximal stimulus, S50 (data not shown) Thus, a plot of the response coefficient vs the stimulus not only depends on the sensitivity of the response (i.e the Hill coefficient, nH), but also on the half-maximal stimulus, S50, and is therefore inappropriate for the quantitative analysis of sensitivity To circumvent this problem, we propose to plot the response coefficient, defined in Eqn (1), against the activated fraction, f, which is given by:
f ¼ X Xbasal
Xmax Xbasal
Expressing the response coefficient of the Hill equation devoid of basal activation as a function of the activa-ted fraction f ¼ Sn H
Sn H þ SnH 50
yields [22]:
RXS ¼ nH 1 fð Þ: ð5Þ This is a linear relationship, which solely depends on the Hill coefficient, nH, and thus allows the quantifi-cation of sensitivity Also, more in general, a plot of the response coefficient against the activated fraction,
f, which will be referred to as ‘relative amplification plot’ in the following, solely depends on the sensiti-vity of the response considered: the activated fraction defined in Eqn (4) refers to a per cent response, so that the relative amplification plot does not depend
on the threshold (i.e the half-maximal stimulus), as,
at the threshold, f¼ 0.5 holds for all responses Like-wise, the relative amplification plot is independent of
Fig 2 Limitations of the Hill approach: basal activation Stimulus–
response of the Hill equation with (; X basal ¼ 0.5) and without (- - -;
Xbasal¼ 0) basal activation As n H ¼ 4, S 50 ¼ 1 and X max ¼ 1 in both
plots, the sensitivities can be compared directly, which reveals that
basal activation decreases the sensitivity (see the main text).
Trang 4the maximal response, as the maximal activated
frac-tion equals unity for all responses In other words, all
responses are treated as if they had the same
half-maximal stimulus and the same half-maximal activation
level
A relative amplification plot of the
Michaelis–Men-ten equation (Eqn 5 with nH¼ 1) is shown in Fig 3A
(black dashed line) If other relative amplification plots
reside below or above the linear plot of the Michaelis–
Menten equation, the corresponding systems can be
considered to be sub- or ultrasensitive Here, we use
the term ultrasensitivity synonymously with ‘more
sen-sitive than the Michaelis–Menten equation’ to allow
direct comparison with the Hill approach described
above (Discussion) As an example, consider the
rela-tive amplification plot of the posirela-tive feedback scheme
( in Fig 3A), whose stimulus–response is depicted in
Fig 1B The positive feedback scheme is ultrasensitive
in the range of weak activation levels (f < 0.45), but
subsensitive upon strong stimulation (f > 0.45) This
demonstrates that the relative amplification plot can be
used to quantify the sensitivity locally (i.e for a given
response), even if the shape of the response under
con-sideration differs from that of the Hill equation
Relative amplification coefficient
Often it is more reasonable to analyze sensitivity
globally (i.e over the whole stimulus range) As
mentioned above, the Hill coefficient obtained by fit-ting Eqn (2) or by using Eqn (3) is generally used to measure sensitivity globally Based on Eqn (5) we can define a more general metric of global sensitivity, which circumvents the problems associated with fitting the Hill equation Consider the relative amplification plots of the Hill equation devoid of basal activation (Eqn 2 with Xbasal¼ 0) shown in Fig 3A: the area below the ultrasensitive Hill function (nH¼ 2; grey line), divided by the area of the hyperbolic Michaelis– Menten equation (nH¼ 1; - - -), equals two (i.e it equals the Hill coefficient of the ultrasensitive Hill function) Thus, we can define the ‘relative amplifica-tion coefficient’ as a measure of global sensitivity:
nR¼
RfH
fL
RX Sðf Þdf
Rf H
f L
RXR
S R ðf Þdf
ð6Þ
Here, fLand fHspecify the range of activated fractions over which the sensitivity of the response X under consideration is compared to that of the reference response XR The relative amplification coefficient nR equals the mean response coefficient of the response X divided by the mean response coefficient of the refer-ence response XR
In principle, the reference response XR can be any monotonically increasing or decreasing function (see
Fig 3 Relative amplification approach (A) Relative amplification plot of the positive feedback model shown in Fig 1B () The response coefficient (Eqn 1) is plotted as a function of the activated fraction (Eqn 4) A comparison with the reference Michaelis–Menten equation (- - -) reveals that the positive feedback model is ultrasensitive for f < 0.45 and subsensitive for f > 0.45 The corresponding relative amplifica-tion coefficients n R (Eqn 6) are indicated on the top The grey line corresponds to a Hill equation devoid of basal activation with n H ¼ 2 See Fig 1B for parameters chosen in the feedback model (B) Relative amplification plot of the Michaelis–Menten equation for varying basal activation levels Xbasal The maximal activation level Xmaxwas kept constant and assumed to be unity The corresponding relative amplification coefficients n R (Eqn 6) calculated over the whole stimulus range (f L ¼ 0 and f H ¼ 1) are indicated in the legend.
Trang 5also the Discussion) In order to obtain values for
the relative amplification coefficient that are
compar-able to the Hill coefficient discussed above, we will
use the Michaelis–Menten equation as the reference
response, so that ultrasensitivity (i.e nR> 1) again
refers to ‘more sensitive than the Michaelis–Menten
equation’
As an example consider the response of the positive
feedback model: the relative amplification coefficients
calculated for fL¼ 0 and fH¼ 0.45, as well as for
fL¼ 0.45 and fH¼ 1 reflect ultrasensitive behaviour in
the range of weak activation (nR¼ 2.54) and
subsensi-tivity in the range of strong activation (nR¼ 0.72), as
indicated in Fig 3A Furthermore, the relative
amplifi-cation coefficient calculated over the full range of
acti-vated fractions (i.e fL¼ 0 and fH¼ 1) classifies the
response of the positive feedback scheme as
ultrasensi-tive (nR¼ 1.93)
Basal activation
As fitting the Hill equation is inappropriate for the
quantification of sensitivity if basal activation is
signifi-cant (see above), we will now analyze the Hill equation
with basal activation by using the relative amplification
approach Expressing the response coefficient of
the Hill equation with basal activation (Eqn 2 with
Xbasal „ 0) as a function of the activated fraction
f ¼ SnH=ðSnHþ SnH
50Þ (Eqn 2 and Eqn 4) yields:
RXS ¼ nHð1 fÞ f
Xmax
Xbasal 1
1þ f X max
Xbasal 1
Importantly, the response coefficient solely depends on
the ratio of maximal and basal activation, so that the
impact of basal activation on the sensitivity can easily
be analyzed As an example, the relative amplification
plots of the Michaelis–Menten equation with Xmax¼ 1
are shown in Fig 3B for varying basal activation
lev-els, Xbasal Sensitivity strongly decreases with increasing
basal activation levels and this effect is especially
pro-nounced for weak responses This is a result of the fact
that upon weak stimulation (i.e for ffi 0) the Hill
equation with basal activation (Eqn 2 with Xbasal „ 0)
is approximately given by X Xbasal, while
X Xmax S
S50
nH
for the Hill equation devoid of basal activation (Eqn 2 with Xbasal¼ 0) Even if
Xmax⁄ Xbasal¼ 10, the relative amplification coefficient
calculated over the full stimulus range (i.e for fL¼ 0
and fH¼ 1) is reduced by one-third (see the legend to
Fig 3B) when compared to the Michaelis–Menten
equation without basal activation (Xbasal¼ 0) Thus,
the impact of basal activation on sensitivity is likely to
be physiologically relevant, as signalling intermediates are known to exhibit basal activation levels of 5–10% and, in some cases, even > 20% [23–27] Similarly, even saturating concentrations of extracellular hor-mones often induce less than 10-fold transcriptional induction or repression of target genes [28], so that
Xmax⁄ Xbasal < 10 Hence, we can conclude that Hill coefficients obtained by fitting the Hill equation to responses with basal activation [4] overestimate sensi-tivity in biochemical systems However, depending on the accuracy of the fit, the Hill equation obtained may
be reanalyzed in a relative amplification plot to esti-mate biologically relevant sensitivity
Discussion
Owing to the problems associated with fitting the Hill equation to biochemical responses (Figs 1B and 2), we have presented a general framework for the quantita-tive analysis of sensitivity, the relaquantita-tive amplification approach, which is based on the response coefficient (Eqn 1) defined in metabolic control analysis We pro-pose to analyze the response coefficient as a function
of the activated fraction f (‘relative amplification plot’),
as this allows quantitative comparison of sensitivities, regardless of model structure and⁄ or parameters In addition, expressing analytically derived response coef-ficients as a function of an activated fraction deter-mines which parameters determine sensitivity, while those affecting the activated fraction (i.e the response) and the sensitivity to the same extent cancel out Thus, the relative amplification approach provides more detailed insight into mathematical models of biochemi-cal systems To quantify sensitivity globally (i.e over the whole stimulus range), we introduced the integral-based relative amplification coefficient (Eqn 6), which
is equivalent to the mean response coefficient of the response under consideration divided by the mean response coefficient of a reference response
The relative amplification approach requires that the maximal activation level of a saturated biochemical response can be measured experimentally, which may
be difficult in some cases However, complete satura-tion was undoubtedly observed in a variety of studies [4,27,29–31] and can generally be achieved if signal transduction is studied in vitro [1,2,32] or by using pep-tide hormone stimulation in culture [23,26,28,33] As fitting the Hill equation to data devoid of saturation serves only as a guess to what the global behaviour might be, the sensitivity of the response can only be quantified locally (e.g by plotting the response coeffi-cient as a function of the stimulus)
Trang 6In the present article we have used the term
‘ultra-sensitivity’ synonymously with the phrase ‘more
sensi-tive than the Michaelis–Menten equation’ in order to
directly compare the relative amplification approach
with the established methodology, which is based on
the Hill equation (Eqn 2) However, it may be more
reasonable to define ultrasensitivity as relative
amplifi-cation (RX
S > 1), where a relative change in the
stimu-lus elicits an RX
S-times greater relative alteration in the
response, so that ultrasensitivity has direct biochemical
meaning Then, the relative amplification coefficient
(Eqn 6) as a measure of global ultrasensitivity should
be calculated by formally setting the reference response
coefficient to unity (i.e RXR
SR ¼ 1)
Application of the relative amplification approach to
the Hill equation with basal activation (Eqn 2) reveals
that sensitivity significantly decreases with increasing
basal activation (Fig 3B) Even if the basal activation
level is only 10% of the maximal response, as
com-monly observed in biochemical responses [4,23–28], the
relative amplification coefficient (i.e the mean response
coefficient) is reduced by one-third when compared to
a system devoid of basal activation Thus, Hill
coeffi-cients obtained by fitting the Hill equation to
responses with basal activation [4] overestimate
sensi-tivity in biochemical systems However, depending on
the accuracy of the fit, the Hill equation obtained may
be reanalyzed in a relative amplification plot to
esti-mate biologically relevant sensitivity In addition,
fit-ting the Hill equation to data with basal activation
may be reasonable to quantify the degree of apparent
cooperativity (i.e to obtain a hint of the biochemical
mechanisms involved)
By using the positive feedback model (Fig 1A) as
an example, we have shown that the relative
amplifica-tion approach allows quantitative analysis of local and
global sensitivities, even if the shape of the response
under consideration deviates from that of the Hill
equation (Fig 3A) The Hill approach is inappropriate
for the analysis of the feedback model, as the response
is more sensitive than the Michaelis–Menten equation
for weak stimuli, while being less sensitive for strong
stimuli (Fig 3A) Similar ‘discontinous’ behaviour was
also reported for multisite phosphorylation [1,34] and
stoichiometric inhibition [15] Yet, other responses,
such as insulin-induced PKBa (protein kinase B alpha)
activation in hepatocytes [33], phorbol ester-induced
p54JNK (c-Jun N-terminal kinase) activation [23], and
anisomycin-induced JNK activation in 293 cells [26],
are shallow for weak stimuli, but switch-like as the
activation level is further increased By using the Hill
approach, such ‘discontinuities’ are averaged out
(Fig 1B) However, quantitative insights into the local
behaviour of biochemical responses are needed because common upstream activators often induce multiple downstream pathways, each of which exhibits a dis-tinct threshold activator concentration [5,23,35] Even
if the best-fit Hill equation deviates significantly from
a given ‘discontinous’ response only in the range of 0–30% and 70–100%, important biological informa-tion may be lost, as it was, for example, shown that 10% receptor occupation already drives some phero-mone responses in yeast [15]
Theoretical studies [36,37] suggest that biochemical responses can exhibit multiple thresholds (‘staircase response’), that is, the system exhibits two ranges of high sensitivity that are separated by a plateau of very low sensitivity Indeed, such behaviour has been confirmed experimentally for allosteric enzymes [37], for Senseless-induced transcription [38], for mTOR-induced DNA synthesis [39], for insulin-mTOR-induced DNA synthesis [40] and for phosphatidic acid-induced 1-phosphatidylinositol 4,5-bisphosphate production [41] While the Hill approach obviously fails for these staircase responses, the relative amplification approach allows the quantitative analysis of sensitivity Import-antly, the relative amplification approach also applies for monotonically decreasing responses, which occur frequently if an inhibitor diminishes signal transduc-tion In addition, bell-shaped functions, where the response increases with increasing stimulus up to a maximum and subsequently decreases for supramaxi-mal stimuli [42,43], can also be analyzed In this case the monotonically increasing and decreasing parts need
to be quantified separately, as Eqn (6) may be unde-fined if the sign of the response coefficient changes Finally, appropriate activated fractions can also be defined for many nonsaturated responses, which are nonlinear for weak stimuli but linear upon strong sti-mulation (S Legewie, N Blu¨thgen, R Scha¨fer and
H Herzel, unpublished observations) Then, the relat-ive amplification coefficient defined in Eqn (6) grelat-ives a measure of sensitivity in the nonlinear range
We believe that future signal transduction research will focus on the processing of transient signals There
is ample evidence that signaling networks must be able
to discriminate transient signals of different amplitudes and⁄ or different durations [44–46] Likewise, the fre-quency of Ca2+ oscillations [47], or the number of repetitive Ca2+ spikes [48], are known to determine biological outcomes Obviously, signaling networks will be able to efficiently discriminate such transient inputs if they respond in an ultrasensitive manner (e.g with respect to signal duration) [49] Previous studies suggest that such discrimination curves often do not match the Hill equation [50,51] or that they exhibit
Trang 7pronounced basal activation [52] In addition,
mono-tonically decreasing [47] and bell-shaped [53]
relation-ships are frequently observed This suggests that the
relative amplification approach presented in this report
is especially suited for the analysis of such transient
phenomena in signal transduction
Acknowledgements
We thank Rene Hoffmann and Jana Wolf for useful
discussions S Legewie was supported by the German
Federal Ministry of Education and Research (BMBF)
and N Blu¨thgen was supported by the Deutsche
Fors-chungsgemeinschaft (DFG: SFB 618)
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Appendix I
Figure 1A schematically depicts a positive feedback in
signal transduction: The stimulus S catalyzes the
phos-phorylation of the inactive precursor X0, which yields
the active species X (¼ response) The phosphorylated
species X activates the intermediate Y in an
ultrasensi-tive manner (e.g via an ultrasensiultrasensi-tive phosphorylation
cascade) [1], and Y, in turn, catalyzes the formation of
X For example, X may be the Raf protein, which is
known to induce its own enzymatic activator protein
kinase C (PKC) via a Mek-Erk-PLA2-cascade [10]
By using the mass-conservation relationships Xtot¼
X0+ X and Ytot¼ Y0+ Y, the differential equations can be written as:
dX
dt¼ kð K S þ kK1 YÞ Xð tot XÞ kP X dY
dt¼ kK2
Xm
Xmþ Km Yð tot YÞ kP1 Y
ðA1Þ
Here, we have assumed linear kinetics in all (de)phos-phorylation reactions despite the phos(de)phos-phorylation of
Y0 by X, which is modeled by using a Hill term to reflect ultrasensitivity As we also assume that the (de)phosphorylation of Y proceeds much faster than that of X, we can approximate Y by using a quasi-equilibrium assumption (i.e dY⁄ dt ¼ 0), so that Eqn A1 reduces to:
dX
dt ¼ kK S þ kA
Xm
X m
50 þ Xm
Xð tot XÞ kP X
ðA2Þ Here, kA and X50are lumped constants, which can be deduced from Eqn (A1)