^{*}

Conceived and designed the experiments: PRtW. Performed the experiments: SBvA. Analyzed the data: SBvA. Wrote the paper: SBvA PRtW.

The authors have declared that no competing interests exist.

Push–pull networks, in which two antagonistic enzymes control the
activity of a messenger protein, are ubiquitous in signal transduction pathways.
A classical example is the chemotaxis system of the bacterium

In both prokaryotes and eukaryotes, extra- and intracellular signals are often
processed by biochemical networks in which two enzymes together control the
activity of a messenger protein via opposite modification reactions. A
well-known example is the chemotaxis network of

The protein network that controls chemotaxis of _{p}. In wild-type cells, CheA is
localized exclusively at the receptor cluster, and also CheZ is predominantly
localized at the receptor cluster _{p} by CheZ

The canonical model of the intracellular chemotaxis network of

In this network, the phosphorylated form of the messenger, CheY_{p} (_{p} also exhibits autophosphorylation and
autodephosphorylation, but these reactions are much slower than phosphorylation by
CheA and dephosphorylation by CheZ, respectively. The input to the signal
transduction pathway is

In wild-type _{p} using FRET imaging. While the input of the network was thus the
concentration of ligand, the measured output was proportional to the total,
integrated concentration of CheY_{p} bound to CheZ,

Vaknin and Berg found that the colocalization of the antagonistic enzymes has a
marked effect on the dose-response curve

In the next section, we show that the experiments of Vaknin and Berg

In the subsequent sections, we present two refined models of the intracellular
chemotaxis network of

In the next section, we therefore present an alternative model. The key ingredients
of this model are: 1) in wild-type cells, a small, yet significant, fraction of CheZ
is bound to the receptor cluster, while the remainder freely diffuses in the
cytoplasm _{p} first binds
CheZ at the receptor cluster; only when CheZ at the receptor cluster is saturated,
does CheY_{p} bind CheZ in the cytoplasm; since CheZ at the cluster has a
higher catalytic activity than CheZ in the cytoplasm, the response of
CheY_{p} is sigmoidal. Finally, we also incorporate cooperative binding of
CheY_{p} to CheZ

Vaknin and Berg performed experiments on four bacterial strains: wild-type cells,
_{p}CheZ—a
CheY_{p} molecule bound to a CheZ dimer—as a function of
the ligand concentration L) is complicated by the fact that they are determined
by both the response of the receptor cluster,

The first module is the receptor cluster. Its activity,

The second module of the chemotaxis network, the intracellular signal
transduction pathway, is described by the set of reactions in Equations
1–3. The input of this network is _{p}, _{p} bound to CheZ,

If the receptor cluster and the intracellular chemotaxis pathway indeed behave as
two independent modules connected in series, then the response function

A. The four response curves of Figure 5a in

In the following sections we will also consider models that have less extreme
functional forms for

The model that can describe the response of _{p} in the non-stimulated
state should be within the working range of the motor, i.e. between 1 and 5
µM

We now address the question whether the canonical model for the chemotaxis
pathway of

To elucidate the effect of CheZ localization, we have computed the input-output
relations for a network in which CheA and CheZ are colocalized at the receptor
cluster (corresponding to wild-type cells) and for a network in which CheA is
localized at the receptor cluster, while CheZ is distributed in the cytoplasm
(corresponding to CheZ mutant cells); for both networks, the chemical reactions
are given by Equations 1–3. The steady-state input-output relations of
these networks were obtained numerically by discretizing the system on a 1D grid
and propagating the chemical rate equations, which are given in the

As pointed out in the previous section, the input of the intracellular network is
not directly the ligand concentration [L], but rather

The principal results of our calculations are shown in _{p}CheZ (a CheY_{p}
molecule bound to a CheZ dimer) and the concentration of CheY_{p} as a
function of

The red curves correspond to wild-type cells in which CheA and CheZ are
colocalized at the receptor cluster, while the green curves correspond
to the mutant cells in which CheA is localized at the pole, while CheZ
freely diffuses in the cytoplasm. The bullets correspond to the
non-stimulated state of the system. The diffusion constant of the
diffusing components is

The network given by Equations 1–3 is very similar to a canonical
push-pull network, in which two enzymes covalently modify a substrate in an
antagonistic manner _{p} is thus proportional to _{p} is constant, i.e. independent of
[Y]. This is not because the phosphorylation reaction is in
the zero-order regime; this reaction is, in fact, in the linear regime _{p} at the cell pole is constant because a) in
steady state _{p} does not depend upon the spatial distribution
of CheZ, which is indeed what

However, while the model of Equations 1–3 predicts that in wild-type
cells the response of [Y_{p}Z] to the addition of
attractant does not depend on the location of CheZ, the experiments by Vaknin
and Berg clearly demonstrate that it does _{p}Z] to the ligand concentration
[L] depends upon the response of
[Y_{p}] to the activity of the receptor cluster, _{0}([L]), is the same for
both wild type and CheZ mutant cells, the discrepancy between the predictions of
the canonical model and the experimental observations of Vaknin and Berg must
lie in the dependence of [Y_{p}Z] on

Irrespective of the model parameters, it is always true that the rate of
phosphorylation equals the rate of dephosphorylation if the system is in steady
state. For the canonical model, i.e. Equations 1–3, this means that
for both the spatially uniform network in which CheA and CheZ are colocalized,
and the spatially non-uniform network in which CheZ is distributed in the
cytoplasm, the following relation holds in steady state:_{p}
bound to CheZ, [Y_{p}Z]. For the regime of interest, _{p}Z] on

The linear relation between [YpZ] and _{0}([L]). Vaknin and
Berg report the _{p}Z] to _{0}([L]).
While plotting the renormalized FRET signal may mask potentially useful
information, this observation does allow us to draw an important conclusion:
_{0}([L])

The experiments of Wang and Matsumura illustrate the importance of this
conclusion _{s}, which is
localized at the receptor cluster _{p}
association rate _{p}Z] on _{p}Z] on _{0}([L]): merely
changing the slope of [Y_{p}Z] as a function _{0}([L])
is the same for both cells and

The critical ingredient in the above analysis is that
[Y_{p}Z] varies linearly with _{p}Z] on _{p}Z] as a function of _{p}Z] tend to increase, and _{p}Z] as a
function of

Changes in the rate constants (

Do the CheZ mutant cells exhibit a tenfold lower phosphatase activity (_{p} would
be at its maximal value, and the clockwise bias would be close to unity.
However, the experiments of Sanatinia

Recent experiments strongly suggest that the intracellular chemotaxis network of
_{p} concentration. It is clearly
important to understand how the response curve _{p} concentration. In this section, we present a simple
model for the cooperative dependence of the phosphatase activity on
CheY_{p} concentration, which can be solved analytically. Furthermore,
we show that incorporation of cooperativity into the phosphatase reactions can
lead to a model of type I (see

In _{p}
concentration. The experiments of Eisenbach and coworkers _{p} binding _{p}
concentration.

The model for the cooperative dephosphorylation of CheY_{p} by CheZ is
based upon the following assumptions: 1) a single CheZ dimer can bind up to two
CheY_{p} molecules; 2) CheZ can dephosphorylate CheY_{p} in
both CheY_{p}-bound states, thus dephosphorylation can occur when only a
single CheY_{p} molecule is bound or when two CheY_{p} molecules
are bound. This model can be described by two coupled Michaelis-Menten
reactions, those of Eq. 3 in combination with_{p} molecule facilitates the
binding of the second CheY_{p} molecule. However, in their model binding
of the second CheY_{p} molecule does not enhance the catalytic activity
of CheZ

In the model presented in this section, we assume that in wild-type cells all
CheZ proteins are localized at the receptor cluster, while in the CheZ mutant
cells all CheZ proteins freely diffusive in the cytoplasm. For both cells, the
chemical reactions are given by Eqs. 1–3 and Eq. 5. However, while the
rate constants of the phosphorylation reactions in Eqs. 1 and 2 are identical
for both cells, they differ for the dephosphorylation reactions of Eqs. 3 and 5.
In particular, in order to obtain a good fit to the FRET data _{p}-CheZ association rates

In this model, with _{p} in
mutant cells when ^{2} s^{−1}; all
enzyme-substrate dissociation rates are zero.

The results for this model are shown in _{p}
molecule bound, _{p} molecule is bound to
CheZ. Another important point to note is that the FRET response of CheZ mutant
cells is strongly concave over the relevant range of _{p} by CheZ: for small receptor activities _{p}] is low, CheZ is mostly
singly occupied by CheY_{p}, and since the catalytic activity of
CheY_{p}CheZ, _{p},
and since _{p}CheZ, a
given increase in receptor activity

While the model discussed in the previous section can describe the FRET response
as measured by Vaknin and Berg

The key ingredients of our model are:

_{s} and CheA_{L}, which
can form the following dimers: CheA_{L}CheA_{L},
CheA_{L}CheA_{s}, and
CheA_{L}CheA_{s}. The first two,
CheA_{L}CheA_{L} and
CheA_{L}CheA_{s}, have catalytic activity and can
transfer phosphoryl groups to CheY _{s}CheA_{s}, does not have
catalytic activity, but can bind CheZ. Earlier experiments suggest
that CheZ binds selectively to CheA_{s}
_{L}
_{s}CheA_{s} homodimers is about 360, while
the number of CheZ dimers is about 1600 _{s}CheA_{s}, and that each of the
CheA_{s}CheA_{s} homodimers strongly binds one
CheZ dimer, we arrive at the estimate that about 20% of
the CheZ dimers is bound to the cluster, consistent with the
estimate based on the FRET data of Vaknin and Berg

_{p} has a much higher
affinity for CheZ bound to CheA than for CheZ freely diffusing
in the cytoplasm._{p}Z] in the cytoplasm roughly equals
that of [Y_{p}Z] at the receptor cluster;
yet, as mentioned above, Figure 2b of Ref. _{p} than CheZ in the
cytoplasm, as can also be seen directly from Figure 2d of Ref. _{p} association rate in a manner
analogous to the gain of function mutations in CheZ studied by
Silversmith _{p} to CheZ. A more speculative
hypothesis is that CheA increases the CheZ- CheY_{p}
association rate because of the close physical proximity between
CheA, where CheY is phosphorylated, and cluster-bound CheZ: a CheY
molecule that has just been phosphorylated by a CheA_{p}
dimer at the cluster, can very rapidly bind cluster-bound CheZ; in
fact, if CheY_{p} would be directly transferred from
CheA_{p} to CheZ, the association rate could even exceed
the diffusion-limited rate.

_{p} association rate, or to a
higher catalytic activity. We assume that binding of CheZ to CheA
not only increases the CheZ-CheY_{p} association rate, as
discussed above, but also the catalytic activity of CheZ.

_{p} as CheZ in
wild-type cells that is not bound to CheA at the
cluster._{p} is thus assumed to be unaffected.
This assumption is not critical for obtaining a good fit of our
model to the data of Vaknin and Berg

For reasons of clarity, we first disregard the cooperativity in the
phosphatase activity of CheZ. The CheZ mutant cells are thus described by
the reactions of Eqs. 1–3, while the wild-type cells are described
by the reactions of Eqs. 1–2, Eq. 3 for the reactions involving
diffusive CheZ and the following reactions involving localized CheZ:

The model presented here is similar to that of Lipkow _{p}; consequently, while in our model
the bound fraction of CheZ is fairly constant in time, in the model of
Lipkow _{p}; 2) in the model of Lipkow _{p}CheZ pair to a CheA homodimer, can nucleate the
formation of oligomers of CheY_{p}CheZ pairs at the cluster.
However, as mentioned above, recent

_{p}CheZ pairs and CheY_{p} is affected by varying the
critical parameters in this model: the fraction of CheZ bound to the cluster
(_{p} associates with CheZ at the
cluster (

The black line corresponds to the prediction of our model for CheZ
mutant cells, while the red line corresponds to the model prediction
for wild-type cells, in which _{p}Z]—to a good
approximation, at this point _{p} in the
first regime is essentially zero. This is because the phosphatase
activity of CheZ at the receptor cluster is much higher than that of
CheZ in the cytoplasm. The baseline parameters are: ^{2}
s^{−1}; all enzyme-substrate dissociation
rates were set to zero.

The black line and black symbols corresponds to the CheZ mutant cells
(see also _{p} molecules
diffuse into the cytoplasm before they will bind CheZ molecules. For
parameter values, see the caption of

The black line and black symbols corresponds to the CheZ mutant cells
(see also

_{p} to cluster-bound CheZ and
freely diffusive CheZ, respectively. When _{p} and cluster-bound CheZ, as compared to that between
CheY_{p} and freely diffusive CheZ: _{p} and hence CheY_{p} bound to
CheZ will initially increase only slowly with _{p}. At this point _{p}Z] and

We can now understand the effect of varying the critical parameters in this
model. As the fraction of CheZ that is bound to the cluster increases (from
green to red to blue in _{p} needed to saturate
cluster-bound CheZ increases, leading to a shift of the inflection point in _{p} and cluster-bound CheZ decreases (from
red to blue to green), the response curve _{p} and
cluster-bound CheZ is lowered, it becomes more likely that a phosphorylated
CheY molecule diffuses into the cytoplasm, where it will be dephosphorylated
by freely diffusive CheZ with a lower catalytic activity.

The differential-affinity-and-activity model is able to explain the measured
difference between the response curves for the CheZ mutant cells and the
CheZ wild-type cells. However, while the response curves of Vaknin and Berg
_{p} concentrations between 1 and 5
µM for both strains in the non-stimulated state _{p} concentration is 8 µM in the non-stimulated
state, which is well outside this range.

This fit can, however, be improved by taking into account the effect of
cooperativity in the phosphatase reactions, which we have neglected thus far
in the differential-affinity-and-activity model. The reactions of diffusive
CheZ, both in the wild-type cells and in the CheZ mutants cells, are given
by Eqs. 3 and 5, while the reactions involving CheZ localized at the
receptor cluster in wild-type cells are given by Eq. 8 in combination with_{p} and the phosphatase activity of CheZ are enhanced when CheZ
is localized to CheA at the receptor cluster. This means that the
association rates

_{p}
concentration does not dramatically affect the dose-response curves, a
conclusion that was also reached by Sourjik and Berg _{p} molecules, as suggested by the _{p} equals 2
µM in the non-stimulated state, which is within the working range
of the motor. The concentration of CheY_{p} in the CheZ mutant cells
in their non-stimulated state is around 5 µM, which is lower than
that in the simplified differential-affinity-and-activity model, but still
at the high end of the working range of the motor.

The black line and symbols correspond to CheZ mutant cells, while the
red line and symbols correspond to cells containing wild-type CheZ
(see also ^{2} s^{−1}; all
enzyme-substrate dissociation rates are zero.

The experiments by Vaknin and Berg on the effect of CheZ localization on the
dose-response curves of

Here, we have presented two different models that can explain the FRET data of
Vaknin and Berg _{p} in a cooperative manner

We have therefore presented an alternative model that is consistent with most, if
not all, of the currently available data. In this model, _{p} and a higher catalytic activity than CheZ in the cytoplasm.
All these assumptions seem to be supported by experiment

In essence, the model that we propose consists of a push-pull network with one
activating enzyme, CheA, and two deactivating enzymes, CheZ bound to the cluster
and CheZ that freely diffuses in the cytoplasm. Our analysis shows that the
competition between these two deactivating enzymes for binding and deactivating
the substrate can yield an ultrasensitive response even when the push-pull
network does not operate in the zero-order regime. In fact, this mechanism of
differential-affinity-and-catalytic-activity is evocative of the
“branch point effect”, in which the interdependence of the
activities of two branch-point enzymes that compete for a common substrate can
yield an abrupt change in the flux through one of the enzymes

If the response function _{p}Z] and _{p}Z] to changes in [L], in
terms of the signal amplification properties of the receptor cluster _{p}Z] on the activity of the receptor
cluster, _{p}CheZ would then also have to be taken into account.

Recently, Kim _{p} levels _{p}-CheZ and CheY_{p}-FLiM
interactions

While the differential-affinity-and-catalytic-activity model can describe the
dose-response curves as reported by Vaknin and Berg _{p}] in non-stimulated CheZ mutant cells is on
the border of the working range of the motor, while experiments on mutant cells
lacking CheA_{s}, which plays a role in localizing CheZ to the receptor
cluster _{p} in the adapted state? In particular, how well
must that be inside the working range of the motor? It is conceivable that cells
with [Y_{p}] at the high end of the motor's
working range can chemotax, albeit less efficiently. Another possibility is that
CheZ mutant cells can chemotax, because [Y_{p}] forms
spatial gradients inside CheZ mutant cells _{p}] at some motors will be outside the
motor's working range, [Y_{p}] at other
motors might be inside the working range of the motor.

But perhaps the most likely explanation is that phosphorylation of CheB by
CheA_{p} provides a negative feedback loop on the activity of the
receptor cluster that tends to keep the concentration of CheY_{p} within
a certain range. The concentration of CheY_{p} in the adapted state is
determined by the activity of the receptor cluster in the adapted state, which
is controlled by the activity of the methylation and demethylation enzymes CheR
and CheB, respectively. CheA_{p} cannot only phosphorylate CheY, but
also CheB. Moreover, phosphorylated CheB has a higher demethylation activity
than unphosphorylated CheB. Since CheY and CheB compete with one another for
phosphorylation by CheA_{p}, the concentration of phosphorylated CheB
increases as [Y_{p}] increases and
[Y] decreases _{p}]. In our model, the activity of the
receptor cluster is assumed to be the same for wild-type and CheZ mutant cells,
and it was chosen such that the concentration of CheY_{p} in adapted
wild-type cells is within the working range of the motor. Yet, it is conceivable
that because of the negative feedback loop, the activity of the receptor cluster
in the adapted state is lower in CheZ mutant cells than in CheZ wild-type cells.
This would lower the concentration of CheY_{p} in the CheZ mutant cells
and could bring it within the motor's range.

Vaknin and Berg measured not only the response to the addition to serine, but
also the response of [Y_{p}Z] to changes in aspartate
concentration _{p}Z] to changes in ligand
concentration [L] is determined by two independent modules
connected in series: _{0}([L]): the response of

The canonical model of the intracellular chemotaxis network of _{p} are uniform in space, and the concentrations
can be obtained by solving the following chemical rate equations:

When CheZ cannot bind the receptor cluster and thus diffuses in the cytoplasm,
concentration gradients of CheY and CheY_{p} will form. We will assume that
the cell is cylindrically symmetric, and we will integrate out the lateral
dimensions _{p} and CheA_{p}CheY are
localized at one end of the cell; the unit of their concentrations is the number of
molecules per area. The other components diffuse in the cell. Their concentrations,
which are in units of number of molecules per volume, depend upon the position _{p} and CheA_{p}CheY are localized; only in Equations 20 and 21
is the _{p}-CheY association rate depends on the concentration of CheY at
contact. Zero-flux boundary conditions are imposed at both cell ends. The
steady-state input-output relations of the network described by Equations
17–23 were obtained numerically by discretizing the system on a (1-D) grid
and propagating these equations in space and time until steady state was reached.

The reaction-diffusion equations for the other models described in the main text,
i.e. in section

Two independent modules connected in series.

(0.09 MB PDF)

Mapping between canonical push-pull network and chemotaxis network.

(0.22 MB PDF)

Cooperativity in the phosphatase reactions.

(0.29 MB PDF)

We thank Howard Berg, Dennis Bray, Victor Sourjik, Ady Vaknin, and Sorin Tănase-Nicola for useful discussions and Ady Vaknin and Tom Shimizu for a critical reading of the manuscript.