Báo cáo khoa học: Quantitative analysis of ultrasensitive responses pot

Quantitative analysis of ultrasensitive responses
Stefan 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:
(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.
FEBS Journal 272 (2005) 4071–4079 ª 2005 FEBS 4071
The 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]:
R
X

S
¼
S
X
Á
dX
dS
¼
d ln X
d ln S
; ð1Þ
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 relative change in
the stimulus is increased by the factor R
X
S
; in other
words the relative alteration in the response is R
X
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]:

X ¼ X
basal
þ X
max
À X
basal
ðÞÁ
S
n
H
S
n
H
50
þ S
n
H
: ð2Þ
Here, X
basal
and X
max
are the basal and the maximal
responses, respectively, while S
50
refers to the stimulus
required to reach half-maximal activation. Depending
on the Hill coefficient, n
H
, the system is referred to as

ultrasensitive (n
H
> 1), hyperbolic (n
H
¼ 1) or subsen-
sitive (n
H
< 1). By using this approach, the term
ultrasensitivity is used synonymously with the phrase
‘more sensitive than the Michaelis–Menten equation’
(Eqn 2 with n
H
¼ 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 ¾) 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 (n
H
¼ 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 1andk
P
¼ 1. The Hill
equation was fitted by using the least-squares method with the sti-

mulus values S
i
¼ 1.0233
I
Æ 10
)2
and 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.
Quantification of ultrasensitivity S. Legewie et al.
4072 FEBS Journal 272 (2005) 4071–4079 ª 2005 FEBS
n
H
¼
log 81ðÞ
log S
90

S
10
ÀÁ
: ð3Þ
Here, S
10
and S
90
equal 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 n

H
¼ 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 (¾) or without (- - -) basal activa-
tion. The Hill coefficient (n
H
¼ 4), the maximal activa-
tion level (X
max
¼ 1) and the half-maximal stimulus
(S
50
¼ 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 S ¼ 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 X
basal
¼ 0) is a func-
tion of the stimulus S, the Hill coefficient, n
H
, and the
half-maximal stimulus, S
50
(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, n
H
), but also on the half-maximal
stimulus, S
50
, 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 À X
basal
X
max
À X
basal
: ð4Þ
Expressing the response coefficient of the Hill equation
devoid of basal activation as a function of the activa-
ted fraction f ¼ S
n
H

S
n
H
þ S
n
H

50
ðÞ(Eqn 2 and Eqn 4)
yields [22]:
R
X
S
¼ n
H
Á 1 À fðÞ: ð5Þ
This is a linear relationship, which solely depends on
the Hill coefficient, n
H
, 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, f ¼ 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 (¾; X
basal
¼ 0.5) and without (- - -;
X
basal
¼ 0) basal activation. As n

H
¼ 4, S
50
¼ 1andX
max
¼ 1 in both
plots, the sensitivities can be compared directly, which reveals that
basal activation decreases the sensitivity (see the main text).
S. Legewie et al. Quantification of ultrasensitivity
FEBS Journal 272 (2005) 4071–4079 ª 2005 FEBS 4073
the 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 maximal activation
level.
A relative amplification plot of the Michaelis–Men-
ten equation (Eqn 5 with n
H
¼ 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 positive feedback scheme
(¾ 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 X
basal
¼ 0) shown in Fig. 3A: the area
below the ultrasensitive Hill function (n
H
¼ 2; grey
line), divided by the area of the hyperbolic Michaelis–
Menten equation (n
H
¼ 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:
n

R
¼
R
f
H
f
L
R
X
Sðf Þ
df










R
f
H
f
L
R
X
R
S

R
ðf Þ
df










ð6Þ
Here, f
L
and f
H
specify the range of activated fractions
over which the sensitivity of the response X under
consideration is compared to that of the reference
response X
R
. The relative amplification coefficient n
R
equals the mean response coefficient of the response X
divided by the mean response coefficient of the refer-
ence response X
R
.

In principle, the reference response X
R
can be any
monotonically increasing or decreasing function (see
AB
Fig. 3. Relative amplification approach. (A) Relative amplification plot of the positive feedback model shown in Fig. 1B (¾). 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 X
basal
. The maximal activation level X
max
was 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 0andf
H
¼ 1) are indicated in the legend.
Quantification of ultrasensitivity S. Legewie et al.
4074 FEBS Journal 272 (2005) 4071–4079 ª 2005 FEBS
also 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. n
R
> 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 f
L
¼ 0 and f
H
¼ 0.45, as well as for
f
L
¼ 0.45 and f
H
¼ 1 reflect ultrasensitive behaviour in
the range of weak activation (n
R
¼ 2.54) and subsensi-
tivity in the range of strong activation (n
R
¼ 0.72), as
indicated in Fig. 3A. Furthermore, the relative amplifi-
cation coefficient calculated over the full range of acti-
vated fractions (i.e. f
L
¼ 0 and f

H
¼ 1) classifies the
response of the positive feedback scheme as ultrasensi-
tive (n
R
¼ 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
X
basal
„ 0) as a function of the activated fraction
f ¼ S
n
H
=ðS
n
H
þ S
n
H
50
Þ (Eqn 2 and Eqn 4) yields:
R
X
S

¼ n
H
Á
1 À fðÞÁf Á
X
max
X
basal
À 1

1 þ f Á
X
max
X
basal
À 1

ð7Þ
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 X
max
¼ 1
are shown in Fig. 3B for varying basal activation lev-
els, X
basal
. 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 f fi 0) the Hill
equation with basal activation (Eqn 2 with X
basal
„ 0)
is approximately given by X % X
basal
, while
X % X
max
Á S

S
50
ÀÁ
n
H
for the Hill equation devoid of
basal activation (Eqn 2 with X
basal
¼ 0). Even if
X
max
⁄ X
basal
¼ 10, the relative amplification coefficient
calculated over the full stimulus range (i.e. for f
L
¼ 0
and f

H
¼ 1) is reduced by one-third (see the legend to
Fig. 3B) when compared to the Michaelis–Menten
equation without basal activation (X
basal
¼ 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
X
max
⁄ X
basal
< 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 relative 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).
S. Legewie et al. Quantification of ultrasensitivity
FEBS Journal 272 (2005) 4071–4079 ª 2005 FEBS 4075
In 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 (R
X
S
> 1), where a relative change in the stimu-
lus elicits an R
X
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. R
X
R
S
R
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-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) gives 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 Ca
2+
oscillations [47], or the number of
repetitive Ca

2+
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
Quantification of ultrasensitivity S. Legewie et al.
4076 FEBS Journal 272 (2005) 4071–4079 ª 2005 FEBS
pronounced 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 X
0
, 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 ultrasensitive 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-PLA
2
-cascade [10].
By using the mass-conservation relationships X
tot
¼

X
0
+ X and Y
tot
¼ Y
0
+ Y, the differential equations
can be written as:
dX
dt
¼ k
K
Á S þ k
K1
Á YðÞÁX
tot
À XðÞÀk
P
Á X
dY
dt
¼ k
K2
Á
X
m
X
m
þ K
m

Á Y
tot
À YðÞÀk
P1
Á Y
ðA1Þ
Here, we have assumed linear kinetics in all (de)phos-
phorylation reactions despite the phosphorylation of
Y
0
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 k
K
Á S þ k
A
Á
X
m
X
m
50
þ X
m


Á X
tot
À XðÞÀk
P
Á X
ðA2Þ
Here, k
A
and X
50
are lumped constants, which can be
deduced from Eqn (A1).
S. Legewie et al. Quantification of ultrasensitivity
FEBS Journal 272 (2005) 4071–4079 ª 2005 FEBS 4079