Computational Biology and Chemistry 30 (2006) 438–444
The multi-step phosphorelay mechanism of unorthodox two-component
systems in E. coli realizes ultrasensitivity to stimuli while
maintaining robustness to noises
Jeong-Rae Kim a , Kwang-Hyun Cho a,b,∗
a
b
Bio-MAX Institute, Seoul National University, Gwanak-gu, Seoul 151-818, Republic of Korea
College of Medicine, Seoul National University, Jongno-gu, Seoul 110-799, Republic of Korea
Received 18 September 2006; accepted 27 September 2006
Abstract
E. coli has two-component systems composed of histidine kinase proteins and response regulator proteins. For a given extracellular stimulus, a
histidine kinase senses the stimulus, autophosphorylates and then passes the phosphates to the cognate response regulators. The histidine kinase in
an orthodox two-component system has only one histidine domain where the autophosphorylation occurs, but a histidine kinase in some unusual
two-component systems (unorthodox two-component systems) has two histidine domains and one aspartate domain. So, the unorthodox twocomponent systems have more complex phosphorelay mechanisms than orthodox two-component systems. In general, the two-component systems
are required to promptly respond to external stimuli for survival of E. coli. In this respect, the complex multi-step phosphorelay mechanism seems
to be disadvantageous, but there are several unorthodox two-component systems in E. coli. In this paper, we investigate the reason why such
unorthodox two-component systems are present in E. coli. For this purpose, we have developed simplified mathematical models of both orthodox
and unorthodox two-component systems and analyzed their dynamical characteristics through extensive computer simulations. We have finally
revealed that the unorthodox two-component systems realize ultrasensitive responses to external stimuli and also more robust responses to noises
than the orthodox two-component systems.
© 2006 Elsevier Ltd. All rights reserved.
Keywords: Two-component systems; Unorthodox two-component systems; Phosphorelay; Ultrasensitivity; Robustness
1. Introduction
Bacteria must promptly respond to external stimuli for their
survival under various circumstances. In this regard, the twocomponent systems (TCSs) of bacteria are very efficient signal
transduction systems as they can facilitate the rapid response
through their simple structure. A TCS consists of a histidine
kinase (HK) and a response regulator (RR). Each HK senses
a specific stimulus, autophosphorylates, and then it passes the
phosphoryl group to the cognate RR. The phosphorylated RR
regulates the transcription of some genes to produce proteins
that can cope with the given stimulation (Hoch, 2000; Stock et
al., 2000)
Abbreviations: TCS, two-component system; HK, histidine kinase; HKP,
phosphorylated histidine kinase; RR, reponse regulator; RRP, phosphorylated
response regulator
∗ Corresponding author. Tel.: +82 2 887 2650; fax: +82 2 887 2692.
E-mail address: [email protected] (K.-H. Cho).
1476-9271/$ – see front matter © 2006 Elsevier Ltd. All rights reserved.
doi:10.1016/j.compbiolchem.2006.09.004
Most TCSs are composed of two components with only two
phosphate-binding domains—HKs with one histidine domain
and RRs with one aspartate domain, and we call them orthodox TCSs. However, some two-component systems such as
ArcB/ArcA (Georgellis et al., 1998; Kwon et al., 2000),
TorR/TorS (Jourlin et al., 1997; Ansaldi et al., 2001; Bordi et
al., 2003, 2004), BarA/UvrY (Sahu et al., 2003; Tomenius et
al., 2005) and EvgA/EvgS (Uhl and Miller, 1996; Perraud et al.,
1998, 2000; Bock and Gross, 2002) consist of two components
with several phosphate-binding domains—HKs with histidine
(H1)–aspartate (D1)–histidine (H2) domains and RRs with one
aspartate domain (D2), and we call them unorthodox TCSs.
Naturally, the phosphorelay mechanisms of unorthodox TCSs
are more complex than those of orthodox TCSs. In unorthodox TCSs, the autophosphorylation occurs at H1 for a particular
stimulus and the phosphorylated H1 passes the phosphoryl group
to D2 through consecutive phosphorylation of D1 and H2. The
dephosphorylation (or reverse phosphorelays) also occurs in a
reverse direction at the same time.
J.-R. Kim, K.-H. Cho / Computational Biology and Chemistry 30 (2006) 438–444
The fast signal transduction is the primary advantage of TCSs
resulting from their simple two-step phosphotransfers. In this
point of view, the multi-step phosphorelays in unorthodox TCSs
seem to be disadvantageous for survival of bacteria. The question
is then why such multi-step phosphorelays in unorthodox TCSs
are still present, even rather ubiquitously, instead of being exterminated throughout the long-term evolution? The unorthodox
TCSs may have some special dynamical characteristics that are
not present in the orthodox TCSs. To answer the above question, we develop a mathematical model of unorthodox TCSs,
conduct extensive computer simulations, and analyze the hidden dynamics of phosphorelay processes in unorthodox TCSs.
From the simulation results over a broad range of parameters, it
turns out that the unorthodox TCSs are ultrasensitive to external
stimuli and also less sensitive (i.e., robust) to fluctuations of the
stimuli compared to orthodox TCSs.
2. Materials and methods
In this section, we develop mathematical models of the
orthodox and unorthodox two-component systems. Our aim is
to unravel the dynamical characteristics of unorthodox twocomponent systems compared to those of orthodox TCSs.
Hence, we do not attempt to construct mathematical models that
realize every single detail of real systems. Instead, we focus on
constructing simplified mathematical models that can capture
the essential differences of the two phosphorelay mechanisms.
2.1. Mathematical modeling of orthodox two-component
systems
The signal transduction process of an orthodox TCS can be
divided into two steps of autophosphorylation and phosphorelay.
The autophosphorylation occurs for an HK-specific stimulus S
by using ATP. The reaction can be described as follows:
k1
HK + ATPHKP + ADP
k2
where HKP denotes the phosphorylated HK. For simplification,
we assume ATP and ADP are constant as their concentrations
are much higher than that of HK. Then we have
k1
HKHKP
k2
and we can represent the reaction rate of autophosphorylation
by using the law of mass action as follows:
Fig. 1. The phosphorelay mechanism in unorthodox TCSs where the solid
arrows denote the forward phosphorelay and the dotted arrows indicate the
reverse phosphorelay.
further considering the auto-dephosphorylation of RRP:
d
RRP = k3 HKP RR − k4 HK RRP − k5 RRP.
dt
In many cases, the phosphorylation processes are modeled by
using Michaelis–Menten equations (Kholodenko, 2000; Tyson
et al., 2003). However, for simplicity, we employ in this paper the
law of mass action without considering the intermediate complex
formation. Any further details on the mathematical model can
be found at Supplementary Mathematical Models in Appendix.
2.2. Mathematical modeling of unorthodox two-component
systems
The unorthodox TCSs in E. coli (e.g., ArcB/ArcA,
TorR/TorS, BarA/UvrY and EvgA/EvgS) have a common phosphorelay mechanism as shown in Fig. 1. For a certain stimulus,
the homodimers of HKs autophosphorylate (H1) and relay the
phosphoryl groups to H2 through Dl, but the detailed phosphorelay mechanism between two HKs in a homodimer are largely
unknown. Although H1 may directly phosphorylate D2, we
ignore such a reaction as it is relatively very slow (Georgellis et
al., 1997). From the results of Tomenius et al. (2005), we assume
the phosphorelay mechanism in a homodimer of HKs is all the
same as shown in Fig. 2.
For mathematical modeling of the phosphorelay mechanism
in Fig. 2, we need 32 (=23 × 23 ÷ 2) state variables in total to
describe all the phospho-states of a homodimer of HKs. For
simplification, we group some domains through which phosphates are relayed. For instance, we consider a domain group
composed of H1, Dl and H2 where H1 and H2 locate at one
HK of the homodimer while Dl locates at the other HK of
the homodimer. In this way, we can describe the phosphostates of domain groups by eight state variables as summa-
d
HKP = k1 S HK − k2 HKP.
dt
The phosphotransfer reaction from HK to RR can be described
as follows:
k3
HKP + RRHK + RRP.
k4
Thus, we can represent the reaction rate of phosphorelay as follows by using the law of mass action (Kremling et al., 2004) and
439
Fig. 2. A phosphorelay model in a dimer of HKs of unorthodox TCSs.
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J.-R. Kim, K.-H. Cho / Computational Biology and Chemistry 30 (2006) 438–444
Table 1
State variables representing the phospho-states of domain groups where O
denotes the phosphorylated state and X indicates the unphosphorylated state
State variables
HK1
HK2
HK3
HK4
HK5
HK6
HK7
HK8
without considering the formation of complexes such that we
can compare the dynamics of two systems (see v1 , v2 , v3 and v4
in Table 2). For modeling of the phosphorelay process, we consider the flow of phosphates depicted in Fig. 2. A phosphate in the
domain of one HK can be transferred to the other unphosphorylated domain. For instance, the state HK2 can be changed to the
Domains
H1
D1
H2
X
O
X
X
O
O
X
O
X
X
O
X
O
X
O
O
X
X
X
O
X
O
O
O
k5
state HK3 (R5 : HK2−→HK3). As there is no other molecule
involved in this relay, the reaction R5 can be described as follows:
v5 = k5 HK2.
The phosphorelay reaction from HKs to RRs can occur when
H2 domains are phosphorylated (HK4, HK6, HK7 and HK8).
For example, we have a reaction of HK4 and RR as follows:
rized in Table 1. We need two more state variables (RR and
RRP) to describe the states of RRs where RR represents the
unphosphorylated RR and RRP denotes the phosphorylated
RR.
We can divide the signal transduction process of unorthodox
TCSs into two parts—autophosphorylation and phosphorelay.
The autophosphorylation occurs for an HK-specific stimulus
when H1 domain is not phosphorylated. So, there are four states
(HK1, HK3, HK4 and HK7) that can be autophosphorylated.
Like the orthodox TCSs, we employ the law of mass action
R9 :
k9
HK4 + RR−→HK1 + RRP.
Note that we employ the same law of mass action as in the
orthodox TCSs to describe all these reactions (see Table 2). In
Table 2, all the reactions and the corresponding reaction rates
are summarized with the classifications of autophosphorylation,
forward phosphorelay and reverse phosphorelay. Based on the
reaction rates in Table 2, we can construct a system of ordinary
differential equations describing the signal transduction process
Table 2
Reactions and reaction rates in unorthodox TCSs where S denotes a stimulus and Pi indicates the inorganic phosphate
Reaction
Autophosphorylation
Reaction rate
k1
R1 : HK1−→HK2
v1 = k1 S HK1 − kd1 HK2
R2 : HK3−→HK5
v2 = k2 S HK3 − kd2 HK5
R3 : HK4−→HK6
v3 = k3 S HK4 − kd3 HK6
R4 : HK7−→HK8
v4 = k4 S HK7 − kd4 HK8
k2
k3
k4
Forward phosophorelay
k5
R5 : HK2−→HK3
v5 = k5 HK2
R6 : HK3−→HK4
v5 = k6 HK3
R7 : HK5−→HK6
v7 = k7 HK5
R8 : HK6−→HK7
v8 = k8 HK6
R9 : HK4 + RR−→HK1 + RRP
v9 = k9 HK4 RR
R10 : HK6 + RR−→HK2 + RRP
v10 = k10 HK6 RR
k6
k7
k8
k9
k10
k11
R11 : HK7 + RR−→HK3 + RRP
v11 = k11 HK7 RR
R12 : HK8 + RR−→HK5 + RRP
v12 = k12 HK8 RR
k12
Reverse phosphorelay
k13
R13 : HK6−→HK5
v13 = k13 HK6
R14 : HK4−→HK3
v14 = k14 HK4
k14
k15
R15 : HK8−→HK6 + Pi
v15 = k15 HK8
R16 : HK5−→HK2 + Pi
v16 = k16 HK5
R17 : HK3−→HK1 + Pi
v17 = k17 HK3
k16
k17
k18
R18 : HK7−→HK4 + Pi
v18 = k18 HK7
R19 : HK1 + RRP−→HK4 + RR
v19 = k19 HK1 RRP
k19
k20
R20 : HK2 + RRP−→HK6 + RR
v20 = k20 HK2 RRP
R21 : HK3 + RRP−→HK7 + RR
v21 = k21 HK3 RRP
R22 : HK5 + RRP−→HK8 + RR
v22 = k22 HK5 RRP
R23 : RRP−→RR
v23 = kd5 RRP
k21
k22
kd5
J.-R. Kim, K.-H. Cho / Computational Biology and Chemistry 30 (2006) 438–444
441
the unorthodox TCS models, more than 99.9% response curves
were sigmoidal. This result implies that the response curves of
the unorthodox TCS shows much slower responses than those
of the orthodox TCS at the beginning of stimulation.
We have simulated the unorthodox TCS model with randomly chosen distinct reaction parameters. However, some of
the parameters in real unorthodox TCSs can be similar or even
identical. For instance, HK1, HK3, HK4 and HK7 autophosphorylate at each H1 domain, and thereby their reaction parameters
k1 , k2 , k3 and k4 can be quite similar. What will happen in such
a case? To answer this question, let us endow the unorthodox
TCS model with the following constraints on parameters:
k1 = k2 = k3 = k4 ,
k9 = k10 = k11 = k12 ,
Fig. 3. Hyperbolic vs. sigmoidal temporal response curves.
in unorthodox TCSs (see Supplementary Mathematical Models
in Appendix for further details).
3. Results
3.1. Temporal response curves of unorthodox TCSs are
sigmoidal
TCSs are required to rapidly respond to their specific stimuli.
So, we suppose that the reaction curves of TCSs should rapidly
increase right after the stimuli and then they may reach some
maximal states if the stimuli are sustained. This type of response
exhibits a hyperbolic temporal response curve (Fig. 3). On the
other hand, if the reaction does not promptly occur but holds for a
while until the stimulus is recognized fully enough, the response
will exhibit a sigmoidal temporal response curve (Fig. 3).
We have investigated the response curve type of unorthodox TCSs by simulating their phosphorelay mechanisms based
on mathematical models with a broad range of parameter settings (see Section 2 for the mathematical models). Specifically,
we have generated 100,000 models by randomly choosing 27
parameters over a range of 0–10. For each model, we have
obtained a response curve by solving the system of ODEs. To
determine the type of each temporal response curve, we have
computed the derivative (i.e., the local slope) at t = 0.1, 0.2,
0.3. If the derivative monotonically decreases as time increases,
we determine that the curve is ‘hyperbolic’ and otherwise, it
is regarded as ‘sigmoidal’. The simulation results are summarized in Table 3. In the orthodox TCS models, 76,251 (76.3%)
out of 100,000 response curves were hyperbolic. However, in
k19 = k20 = k21 = k22 ,
k5 = k8 ,
k6 = k7 ,
k13 = k14 ,
k15 = k16 = k17 = k18 ,
kd1 = kd2 = kd3 = kd4 .
This reformulates the model with only nine independent parameters. We have produced again 100,000 response curves of the new
model through random selection of the independent parameters.
The fourth row in Table 3 shows the summary of this simulation result. In this case, all the response curves of the unorthodox
TCS model were sigmoidal. Note that the difference between the
two cases (the cases with or without the parameter constraints)
is very small (less than 1%) even though we have reduced the
number of independent parameters from 27 to 9. Hence, we can
conclude that the sigmoidal characteristics of the unorthodox
TCSs is not caused by the parameter variations but primarily
caused by the three-step phosphorelay mechanism.
Let us consider how much the reverse phosphorelay reaction parameters affect the response curve types. As the
unorthodox TCSs have many reverse phosphorelay reaction
steps, the reverse reaction parameters including dephosphorylation parameters (kd1 , kd2 , kd3 , kd4 , kd5 , kd13 , kd14 , . . . , kd22 )
can affect the response curve types. The forward phosphorelay reaction parameters in TCSs are larger than the reverse
phosphorelay reaction parameters in general. So, we have set
the forward phosphorelay parameters (k1 , k2 , . . ., k12 ) to the
range from 1 to 10 and the reverse phosphorelay parameters
(kd1 , kd2 , kd3 , kd4 , kd5 , k13 , k14 , . . . , k22 ) to the range from 0 to
1 in the unorthodox TCS model. It turns out that all the response
curves were sigmoidal. This implies that the sigmoidal temporal
response curves of the unorthodox TCSs do not so much depend
on the parameter ranges but depend mainly on the different phosphorelay mechanism.
To further validate the aforementioned classification of different curve types, we employ another method called a relative amplification approach (Legewie et al., 2005). The relative
Table 3
Comparison of the response curve types of the two TCSs over randomly generated 100,000 response curves
Orthodox TCS
Unorthodox TCS
Unorthodox TCS (with parameter constraints)
Values in parenthesis are in percentage.
Hyperbolic
Sigmoidal
76,251 (76.3)
12 (0.01)
0 (0)
23,749 (23.7)
99,988 (99.99)
100,000 (100)
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J.-R. Kim, K.-H. Cho / Computational Biology and Chemistry 30 (2006) 438–444
Fig. 4. The relative amplification plots where the red solid line denotes a hyperbolic response curve, the blue dashed line indicates an orthodox TCS response
curve and the green dotted line shows an unorthodox TCS response curve. (For
interpretation of the references to color in this figure legend, the reader is referred
to the web version of this article.)
amplification plot of a stimulus–response curve describes the
response coefficients of an activated fraction (i.e., the steady
state divided by the maximal value of the response curve). The
relative amplification plot of a typical Hill function with the Hill
coefficient of 1 is shown in Fig. 4 (red solid line). If the relative amplification plot of a certain response curve lies above
this line, the response curve shows ultrasensitive behavior. The
relative amplification plots of orthodox (blue dashed line) and
unorthodox (green dotted line) TCSs are shown in Fig. 4. From
these relative amplification plots, it turns out that the temporal
response curves of the orthodox TCS are not exactly the typical Hill type curves (with the Hill coefficient of 1) but a little
close to ultrasensitive curves. However, the temporal response
curves of the unorthodox TCS are much more ultrasensitive than
those of the unorthodox TCS. For a more quantitative comparison, we have computed the relative amplification coefficient
of each response curve defined by the ratio of the area under
the relative amplification plot of a response curve and that of
the hyperbolic curve. As the relative amplification coefficient is
larger, the response curve is more ultrasensitive. We have computed the relative amplification coefficients of 100,000 response
curves generated from each TCS model. The averages of the relative amplification coefficients were about 1.4 for orthodox and
2.5 for unorthodox TCSs, respectively. These results imply that
temporal response curves of unorthodox TCSs incline to sigmoid
while those of orthodox TCSs are hyperbolic.
3.2. The unorthodox TCSs are ultrasensitive to external
stimuli
We found that the temporal response curves of unorthodox
TCSs are sigmoidal. This means that the unorthodox TCSs
have slow responses at the beginning of stimulation. We note
however that the steady state level of RRP is also important
as it determines how bacteria cope with the given stimulation.
Fig. 5. Stimulus–response curves with the Hill coefficient n = 0.5, 1, 2, respectively. The graphs are obtained from the Hill function R(S) = 50Sn /3n + Sn
(n = 0.5, 1, 2).
So, let us compare the stimulus–response curves of two TCS
models.
The steady state level of RRP as a function of stimulus S
in the orthodox TCS model is given as follows (by assuming
that the self-degradation rate constant k5 of RRP is relatively
small, without loss of generality) from the steady state equations
(dHKP/dt = 0, dRRP/dt = 0):
RRP(S) =
RR0 S
(k2 k4 /k1 k3 ) + S
(1)
where RR0 denotes the total concentration of RR. Note that the
steady state level of RRP does not depend on the total concentration of HK but on RR0, S and reaction constants. From Eq. (1),
we notice that the stimulus–response curves of orthodox TCSs
are hyperbolic (see Fig. 3; see also the curve in Fig. 5 with the
Hill coefficient n = 1). Note that the response curve is still almost
hyperbolic without the assumption of a small self-degradation
rate since the average relative amplification coefficient is 1.1
even in this case.
To examine the sensitivity of the unorthodox TCSs with
respect to external stimuli, we have randomly selected 1000
parameter sets for which the system of ODEs have steady
states (i.e., dRRP/dt ≈ 0) for S = 1 within a given time span
(0 ≤ t ≤ 1000). We have solved the systems of ODEs over a
stimulus range 0 ≤ S ≤ 1 and obtained 1000 stimulus–response
curves. To compare the dynamical characteristics of the unorthodox TCSs with respect to those of the orthodox TCSs, we
compute the Hill coefficients of the stimulus–response curves.
The Hill coefficients can be obtained by the following formula
(Goldbeter and Koshland, 1981; Mutalik et al., 2004):
nHill =
log 81
log(I90 /I10 )
J.-R. Kim, K.-H. Cho / Computational Biology and Chemistry 30 (2006) 438–444
where I90 (I10 ) is the stimulus size whose steady state value
corresponds to 90% (10%) of the maximal steady state value.
Based on this formula, we have computed the Hill coefficients
of 1000 stimulus–response curves and found that the average
Hill coefficient is 2.1. This implies that the unorthodox TCSs
are more sensitive to external stimuli than the orthodox TCSs.
Note that if the Hill coefficient is larger (less) than one then the
response curve is called ultrasensitive (subsensitive) (Fig. 5).
We have also computed the relative amplification coefficients of
the 1000 stimulus response curves and found that the average
is 2.0. These relative amplification coefficients increase as the
reverse phosphorelay reactions decrease compared to the forward reactions. As we have assumed the same kinetics for the
two TCS models, we can conclude that the ultrasensitivity of
the unorthodox TCS is caused by the multi-step phosphorelay
mechanism.
3.3. The unorthodox TCSs are more robust to noises
Signaling pathways are required to possess robustness to
noises – we regard a signal with a very short duration as a noise –
for proper reactions coping with external stimuli. In this respect,
let us compare the robustness of two TCSs. We have randomly
selected 1000 parameter sets for each TCS model such that the
443
steady states are almost identical (≈39). Fig. 6A (the upper figure) shows the average of 1000 temporal response curves of each
TCS model for S = 1. For each of the 1000 parameter sets, we
have solved the corresponding ODEs under various noise conditions (the lower figures in Fig. 6B–H) and obtained the average
temporal response curves (the upper figures in Fig. 6B–H). If
a weak pulse noise is given, the orthodox TCSs respond more
rapidly while the unorthodox TCSs show slower responses with
a little bit higher phosphorylation (Fig. 6B). On the other hand,
if a strong pulse noise is given, we find that the orthodox TCSs
attain much higher phosphorylation (Fig. 6C). This means that
the unorthodox TCSs respond less sensitively to the variation
of short-time pulse noise compared to the orthodox TCSs. We
have further inquired the effect of a strong noise given when
the TCSs maintain their steady states. Fig. 6D shows that the
orthodox TCSs respond more sensitively than the unorthodox
TCSs when such a strong noise is given at steady states. We
find that the unorthodox TCSs seldom respond to such a noise.
This also holds for an under-shooting pulse noise (Fig. 6E) and
even at transient states (Fig. 6F and G). Note however that, if the
duration of such stimulation is long enough then the unorthodox
TCSs recognize the stimulation as a right signal (Fig. 6H). In
summary, the unorthodox TCSs ignore short-time stimulations
and thereby they behave more robust to external noises than the
orthodox TCSs.
Fig. 6. The average temporal response curves over randomly selected 1000 sample response curves for each TCS model where the black solid lines denote the response
curves of the orthodox TCS model and the red dotted lines indicate the response curves of the unorthodox TCS model in upper subfigures. (For interpretation of the
references to color in this figure legend, the reader is referred to the web version of this article.)
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J.-R. Kim, K.-H. Cho / Computational Biology and Chemistry 30 (2006) 438–444
4. Discussion
References
The TCS in bacteria is a simple but very effective signal transduction system enabling prompt reactions to external stresses
based on its simple mechanism. In this regard, the orthodox
TCSs should be most advantageous. We have noticed however
that there are unorthodox TCSs having much more complicated
phosphorelay mechanisms than the orthodox TCSs. So, we have
investigated the dynamical characteristics of such unorthodox
TCSs by employing an in silico approach. We have developed
simplified mathematical models that can capture the essential
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Acknowledgments
This work was supported by the 21C Frontier Microbial
Genomics and Application Center Program, Ministry of Science
& Technology (Grant MG05-0204-3-0), Republic of Korea.
Appendix A. Supplementary data
Supplementary data associated with this article can be
found, in the online version, at doi:10.1016/j.compbiolchem.
2006.09.004.