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

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(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

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

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-

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.

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

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

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

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-

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

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

(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-

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)

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

protein kinase II-protein phosphatase 1 switch facilitates
specificity in postsynaptic calcium signaling. Proc Natl
Acad Sci USA 100, 10512–10517.
3 Hardie DG, Salt IP, Hawley SA & Davies SP (1999)
AMP-activated protein kinase: an ultrasensitive system
for monitoring cellular energy charge. Biochem J 338,
717–722.
4 Ralston DM & O’Halloran TV (1990) Ultrasensitivity
and heavy-metal selectivity of the allosterically modula-
ted MerR transcription complex. Proc Natl Acad Sci
USA 87, 3846–3850.
5 Burz DS, Rivera-Pomar R, Jackle H & Hanes SD
(1998) Cooperative DNA-binding by Bicoid provides a
mechanism for threshold-dependent gene activation in
the Drosophila embryo. EMBO J 17, 5998–6009.
6 Heinrich R, Neel BG & Rapoport TA (2002) Mathema-
tical models of protein kinase signal transduction. Mol
Cell 9, 957–970.
7 Thattai M & van Oudenaarden A (2002) Attenuation of
noise in ultrasensitive signaling cascades. Biophys J 82,
2943–2950.
8 Kholodenko BN (2000) Negative feedback and ultrasen-
sitivity can bring about oscillations in the mitogen-acti-
vated protein kinase cascades. Eur J Biochem 267,
1583–1588.
9 Ferrell JE Jr & Machleder EM (1998) The biochemical
basis of an all-or-none cell fate switch in Xenopus
oocytes. Science 280, 895–898.
10 Bhalla US, Ram PT & Iyengar R (2002) MAP kinase
phosphatase as a locus of flexibility in a mitogen-acti-

erythrocytes. Acta Biol Med Ger 31, 479–494.
20 Hofmeyr JH, Kacser H & van der Merwe KJ (1986)
Metabolic control analysis of moiety-conserved cycles.
Eur J Biochem 155, 631–641.
21 Hill AV (1910) The possible effects of the aggregation
of the molecules of haemoglobin on its oxygen dissocia-
tion curve. J Physiol (Lond) 40, 4–7.
22 Blu
¨
thgen N & Herzel H (2003) How robust are switches
in intracellular signaling cascades? J Theor Biol 225,
293–300.
23 Gong Q, Cheng AM, Akk AM, Alberola-Ila J, Gong
G, Pawson T & Chan AC (2001) Disruption of T cell
signaling networks and development by Grb2 haploid
insufficiency. Nat Immunol 2, 29–36.
24 Makevich NI, Moehren G, Kiyatkin A, Hoek JB &
Kholodenko BN (2004) Signal processing at the Ras cir-
cuit: what shapes the Ras activation patterns? IEE Syst
Biol 1, 104–113.
25 Warwick HK, Nahorski SR & Challiss RA (2005)
Group I metabotropic glutamate receptors, mGlu1a and
mGlu5a, couple to cyclic AMP response element bind-
ing protein (CREB) through a common Ca2
+
- and
protein kinase C-dependent pathway. J Neurochem 93,
232–245.
26 Bagowski CP, Besser J, Frey CR & Ferrell JE Jr (2003)
The JNK cascade as a biochemical switch in mamma-

32 Birnbaumer L, Swartz TL, Abramowitz J, Mintz PW &
Iyengar R (1980) Transient and steady state kinetics of
the interaction of guanyl nucleotides with the adenylyl
cyclase system from rat liver plasma membranes. Inter-
pretation in terms of a simple two-state model. J Biol
Chem 255, 3542–3551.
33 Walker KS, Deak M, Paterson A, Hudson K, Cohen P
& Alessi DR (1998) Activation of protein kinase B beta
and gamma isoforms by insulin in vivo and by 3-phos-
phoinositide-dependent protein kinase-1 in vitro: com-
parison with protein kinase B alpha. Biochem J 331,
299–308.
34 Waas WF, Lo HH & Dalby KN (2001) The kinetic
mechanism of the dual phosphorylation of the ATF2
transcription factor by p38 mitogen-activated protein
(MAP) kinase alpha. Implications for signal ⁄ response
profiles of MAP kinase pathways. J Biol Chem 276,
5676–5684.
35 Green JB & Smith JC (1990) Graded changes in dose of
a Xenopus activin A homologue elicit stepwise transi-
tions in embryonic cell fate. Nature 347, 391–394.
36 Salazar C & Ho
¨
fer T (2003) Allosteric regulation of the
transcription factor NFAT1 by multiple phosphoryla-
tion sites: a mathematical analysis. J Mol Biol 327,
31–45.
37 Goldbeter A & Koshland DE Jr (1984) Ultrasensitivity
in biochemical systems controlled by covalent modifica-
tion. Interplay between zero-order and multistep effects.

7453–7459.
44 Murphy LO, MacKeigan JP & Blenis J (2004) A network
of immediate early gene products propagates subtle
differences in mitogen-activated protein kinase signal
amplitude and duration. Mol Cell Biol 24 , 144–153.
45 Marshall CJ (1995) Specificity of receptor tyrosine
kinase signaling: transient versus sustained extracellu-
lar signal-regulated kinase activation. Cell 80, 179–
185.
46 Yang SN, Tang YG & Zucker RS (1999) Selective
induction of LTP and LTD by postsynaptic [Ca
2+
]i ele-
vation. J Neurophysiol 81, 781–787.
47 Dolmetsch RE, Xu K & Lewis RS (1998) Calcium oscil-
lations increase the efficiency and specificity of gene
expression. Nature 392, 933–936.
48 Ducibella T, Huneau D, Angelichio E, Xu Z, Schultz
RM, Kopf GS, Fissore R, Madoux S & Ozil JP (2002)
Egg-to-embryo transition is driven by differential
responses to Ca (2+) oscillation number. Dev Biol 250,
280–291.
49 Deshaies RJ & Ferrell JE Jr (2002) Multisite phosphor-
ylation and the countdown to S phase. Cell 107, 819–
822.
50 Fields RD, Eshete F, Stevens B & Itoh K (1997) Action
potential-dependent regulation of gene expression: tem-
poral specificity in Ca2+, cAMP-responsive element
binding proteins, and mitogen-activated protein kinase
signaling. J Neurosci 17, 7252–7566.

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 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