Robust perfect adaptation in bacterial chemotaxis through integral

Robust perfect adaptation in bacterial chemotaxis through integral
Robust perfect adaptation in bacterial chemotaxis
through integral feedback control
Tau-Mu Yi*†, Yun Huang†‡, Melvin I. Simon*§, and John Doyle‡
*Division of Biology 147-75 and ‡Department of Control and Dynamical Systems 107-81, California Institute of Technology, Pasadena, CA 91125
Contributed by Melvin I. Simon, February 7, 2000
A
hallmark of many biological sensing devices is the ability to
adapt to a persistent input stimulus, thereby increasing the
range of sensitivity (1). Under most conditions, the signaling
apparatus mediating bacterial chemotaxis exhibits perfect adaptation to chemoattractants: the output is reset exactly to the
prestimulus value so that the steady-state behavior of the system
is independent of the concentration of a homogeneous distribution of the attractant (2–4). Bacteria traverse gradients of
chemoeffectors by engaging in a biased random walk consisting
of alternating periods of smooth runs and random tumbles (3).
Detecting elevated levels of chemoattractant decreases the
probability of a tumble, thus propelling the bacteria in the
favorable direction. This modulation of the length of runs is
mediated by a signal transduction pathway consisting of transmembrane receptors (methyl-accepting proteins) and the products of 6 Che genes: cheA, cheB, cheR, cheW, cheY, and cheZ (5).
The receptor forms a complex with the adaptor protein CheW
and the histidine kinase CheA. CheA phosphorylates the response regulator CheY, and this phosphorylated form, CheY-P,
stimulates tumbling by interacting with the flagellar motor.
When chemoattractant binds receptor, CheA activity is suppressed, the levels of CheY-P decrease, and the bacterium is less
likely to tumble. Adaptation results from the methylation of
receptor by CheR, which increases CheA activity, promoting
CheY phosphorylation. The methylation state of the receptor is
balanced by the demethylation enzyme CheB. CheZ acts to
dephosphorylate CheY-P (6). The dynamics of receptor methylation are considered slow (minutes) relative to CheY phosphorylation (milliseconds) (7, 8). This signal transduction system
has been the subject of extensive and fruitful mathematical
modeling (9–13).
Robustness, the insensitivity of system properties to parameter variation and other uncertainties in components and the
environment, is essential for the operation of both man-made
and biological systems in the real world. Robustness in engi-
neering systems has a large theoretical literature that began with
electrical network design (14). Quantitative application of engineering robustness methods in molecular biology began with
studies of biosynthetic pathways (15), although robustness of
biological responses as a selective property in evolution was
emphasized qualitatively even earlier (16).
In an elegant study, Barkai and Leibler investigated the
robustness of perfect adaptation in bacterial chemotaxis (17).
They constructed a two-state model (active or inactive) of the
receptor complex (receptor ⫹ CheA ⫹ CheW); the system
output, modulated by ligand binding and methylation, was the
concentration of active receptor complexes. In this model,
perfect adaptation was the intrinsic property of the connectivity
of the signaling network and did not require specific values for
the kinetic rate constants or concentrations of the constituent
enzymes. Alon et al. elegantly provided experimental evidence
for the robustness of perfect adaptation to parameter changes
when they demonstrated exact adaptation even when the levels
of the chemotactic proteins were varied dramatically (18). In this
work, we have reexamined these findings from the perspective of
robust control theory, which has allowed us to analyze in a more
rigorous fashion how a biochemical network can produce a
desired level of output in an uncertain environment (e.g., varying
levels of stimulant) with uncertain components (e.g., varying
concentrations of protein).
Results
It has been argued that there are two approaches to constructing
a system that exhibits perfect adaptation: (i) fine tuning the
parameters and (ii) designing a specific structure that creates this
property inherently (17). Bifurcation analysis can help distinguish between these two possibilities by systematically testing the
dependence of the steady-state solutions of the system to
parameter changes. In this manner, one can examine the robustness of the model without running numerous simulations. A
good example of the first approach is provided by Spiro et al.,
who formulated an ingenious method for identifying specific
values for the parameters that result in perfect adaptation (12).
In Fig. 1a, we plotted the system activity of the Spiro model, the
concentration of CheY-P, vs. the total concentration of the
aspartate receptor Tar for three levels of the chemoattractant
aspartate: 0, 1 ␮M, and 1 mM. The intersection of the three
curves represents the value of Tar at which perfect adaptation
exists. Indeed, in the Spiro model, the concentration of Tar was
chosen to be 8 ␮M. This diagram exhibited no bifurcations,
qualitative changes in the steady-state solutions, and hence
represents a simple form of bifurcation analysis, equilibrium
analysis, in which the equilibria are tracked (19).
When we performed equilibrium analysis on the Barkai–
Leibler model, a different picture emerged. All three curves
completely overlapped, indicating that perfect adaptation held
†T.-M.Y.
§To
and Y.H. contributed equally to this work.
whom reprint requests should be addressed. E-mail: simonm@starbase1.caltech.edu.
The publication costs of this article were defrayed in part by page charge payment. This
article must therefore be hereby marked “advertisement” in accordance with 18 U.S.C.
§1734 solely to indicate this fact.
PNAS 兩 April 25, 2000 兩 vol. 97 兩 no. 9 兩 4649 – 4653
BIOPHYSICS
Integral feedback control is a basic engineering strategy for ensuring that the output of a system robustly tracks its desired value
independent of noise or variations in system parameters. In biological systems, it is common for the response to an extracellular
stimulus to return to its prestimulus value even in the continued
presence of the signal—a process termed adaptation or desensitization. Barkai, Alon, Surette, and Leibler have provided both
theoretical and experimental evidence that the precision of adaptation in bacterial chemotaxis is robust to dramatic changes in the
levels and kinetic rate constants of the constituent proteins in this
signaling network [Alon, U., Surette, M. G., Barkai, N. & Leibler, S.
(1998) Nature (London) 397, 168 –171]. Here we propose that the
robustness of perfect adaptation is the result of this system
possessing the property of integral feedback control. Using techniques from control and dynamical systems theory, we demonstrate that integral control is structurally inherent in the Barkai–
Leibler model and identify and characterize the key assumptions of
the model. Most importantly, we argue that integral control in
some form is necessary for a robust implementation of perfect
adaptation. More generally, integral control may underlie the
robustness of many homeostatic mechanisms.
Fig. 2. A block diagram of integral feedback control. The variable u is the
input for a process with gain k. The difference between the actual output y1
and the steady-state output y0 represents the normalized output or error, y.
Integral control arises through the feedback loop in which the time integral
of y, x, is fed back into the system. As a result, we have ẋ ⫽ y and y ⫽ 0 at
steady-state for all u. In the Barkai–Leibler model of the bacterial chemotaxis
signaling system, the chemoattractant is the input, receptor activity is the
output, and ⫺x approximates the methylation level of the receptors.
Fig. 1.
Robust and nonrobust perfect adaptation. The dependence of
steady-state system activity on total receptor concentration was calculated by
using equilibrium analysis for three concentrations of chemoattractant: (i) L ⫽
0 (solid), (ii) L ⫽ 1 ␮M (dashed), and (iii) L ⫽ 1 mM (dashed– dot). The filled circle
indicates the value of total receptor used in the model. (a) Spiro model. The
system activity is measured in terms of the concentration of CheY-P. The
intersection of the three lines, which coincides with the filled circle, represents
the value of total receptor at which perfect adaptation exists. (b) Barkai–
Leibler model. The system activity is measured in terms of the concentration
of active receptor complexes. The three lines completely overlap. These plots
were performed by using the program XPPAUT (34, 35).
for a range of values of total receptor concentration (Fig. 1b).
Similar plots for the other chemotactic proteins in the model
(CheB and CheR) and for the kinetic rate constants (kl, k⫺l, ar,
ar⬘, dr, kr, ab, db, and kb) and activation probabilities (␣m) exhibited
the same robustness of perfect adaptation to significant changes
in parameter values, thus confirming the simulation results of
Barkai and Leibler (17) (data not shown).
Developing a control system that robustly tracks a specific
steady-state output value is a problem commonly faced by
engineers (Appendix). The standard solution is integral feedback
control, in which the time integral of the system error, the
difference between the actual output and the desired steadystate output, is fed back into the system (Fig. 2). This type of
control structure ensures that the steady-state error approaches
zero despite fluctuations in the input or in the system parameters. The only required condition is that the closed loop system
is stable. Integral feedback loops are ubiquitous in complex
4650 兩 www.pnas.org
engineered systems, and so an obvious question is whether
integral control is present in the Barkai–Leibler model.
The Barkai–Leibler model for a receptor complex with M
methylation sites is represented by a set of differential equations
describing the dynamics of the various species in the biochemical
network [see the supplemental data (www.pnas.org)]. Rearranging these equations, we can derive an equation characteristic of
integral control with the activity of the system asymptotically
tracking a fixed steady-state value:
A st ⫽
␥ R bndK b
.
B tot ⫺ ␥ R bnd
[1]
Rbnd is the concentration of CheR bound to receptor complex,
Kb is the Michaelis constant for CheB, ␥ ⫽ kr兾kb is the ratio of
the turnover numbers for CheR and CheB, and Btot is the total
concentration of CheB. The expression for Ast depends only on
the concentrations and kinetic rate constants of CheR and CheB.
More importantly, as long as the concentration of bound CheR
is independent of ligand, there is no dependence on the level of
the chemoattractant, and adaptation will be perfect. Indeed,
CheR is thought to work at saturation (20), so that Rbnd ⬇
R
B
R
兾(V max
⫺ V max
), as described
Rtot. Then Ast reduces to KbV max
B
R
and V max
are the
by Barkai and Leibler (17), where V max
maximal catalytic velocities of CheB and CheR, respectively.
The recognition that integral control is responsible for the
robustness of perfect adaptation in the Barkai–Leibler model
allows one to evaluate the importance of the various assumptions
of the model. A number of assumptions were both explicitly
stated and implicitly built into the model. However, only the
following four are necessary to derive the integral control
equation (supplemental data, www.pnas.org): (i) CheB demethylates only active receptors. (ii) The kinetic rate constants of
CheR and CheB are relatively independent of the methylation
state and ligand occupancy of the inactive or active (in the case
of CheB) receptor complex. More specifically, the turnover
numbers of CheR and CheB satisfy the following two conditions:
u
o
u
o
⫽ krm
⫽ krm and kb(m⫹1)
⫽ kb(m⫹1)
⫽ kb(m⫹1) for m ⫽
(a) krm
Yi et al.
Yi et al.
equilibrium analysis to test the sensitivity of perfect adaptation
to deviations in the first assumption. We defined a⬘b to be the
association rate of CheB with inactive receptor (ab is the
association rate of CheB with active receptor). We then calculated the steady-state receptor activity as a function of ligand
concentration for different values of a⬘b. When a⬘b ⫽ 0, perfect
adaptation holds. Setting a⬘b ⫽ ab resulted in an adaptation
precision P of only 0.22, where P is defined to be the ratio of
steady-state receptor complex activity stimulated by saturating
amounts of chemoattractant vs. unstimulated activity (P ⫽ 1.0
for perfect adaptation). The requirement that CheB demethylates only active receptors is not absolute, because setting a⬘b ⫽
ab兾100 produced an adaptation precision of 0.93.
Another crucial assumption is that the rate constants found in
the expression for the steady-state activity Ast, the turnover
numbers for CheR and CheB and the Km for CheB, do not
depend on the ligand occupancy of the receptor. From Eq. 1, it
is clear that adaptation precision is approximated by the ratio of
the unoccupied and occupied rate constants: P ⬇ krokubKob兾krukobKub.
Thus, a 10-fold increase in kr by chemoattractant would lead to
a 10-fold change in P. On the other hand, P is less sensitive to
the dependence of the above rate constants on the receptor
methylation state. For example, decreasing kr2 and kr3 to 1兾100
of the value of kr0 resulted in an adaptation precision of 0.90.
Similar changes in kbm and Kbm also had a modest effect on P.
Integral control is sufficient to explain robust perfect adaptation, but is it necessary? Perhaps other types of control
structures produce equally robust mechanisms for maintaining
exact adaptation. Instead of feeding back the integral of the
system error, one might try to feed back a linear proportion of
the system error (proportional control). In the Appendix, we
show that any equally robust solution to achieving zero steadystate error and thus perfect adaptation must be equivalent to
integral control.
The necessity of integral feedback control is important to
biologists, because they must reverse engineer systems ‘‘designed’’ by evolution. When a system exhibits robust asymptotic
tracking, it must have integral feedback as a structural property
of the system. When combined with biological realizability, this
may greatly constrain, on the basis of external behavior, the
possible internal mechanisms that can be used to achieve the
observed behavior. Thus, one goal for the future is to catalog the
types of basic biochemical networks that can implement integral
control and other more sophisticated regulatory mechanisms.
Barkai and Leibler provided one example of a simple enzyme
system, an ‘‘adaptive module,’’ in which the activity of an enzyme
E is influenced by modification. If the forward and reverse
modification reactions depend on the system activity, then
steady-state activity is independent of the ligand concentration (17).
We are currently investigating several other types of simple
networks that use integral feedback to regulate the activities and
concentrations of molecular components of the system. In Fig.
3, we describe a hypothetical biochemical network in which the
precursor molecules Xi are converted into the biologically important intermediate Y by the enzymes Ei1, and Y is converted
into the product molecules Zi by the enzymes Ei2. The steadystate concentration of Y, Y0, depends only on the enzymatic
activities of E3 and E4, which lie on a shunt pathway, because of
an integral feedback loop (A negatively regulates E11). The
assumptions are that E4 is operating at saturation and that the
feedback system is stable. Note that the kinetic parameters of the
‘‘synthesis’’ enzymes E11 to En1 and the ‘‘degradation’’ enzymes E12
to Em
2 do not influence Y0. Thus, variations in the flux from X to
Y to Z caused by intracellular and extracellular perturbations will
not affect the steady-state level of Y, ensuring that homeostasis
for Y is maintained.
PNAS 兩 April 25, 2000 兩 vol. 97 兩 no. 9 兩 4651
BIOPHYSICS
0,1, . . . , (M ⫺ 1); and (b) kr0兾kb1 ⫽ kr1兾kb2 ⫽ 䡠 䡠 䡠 ⫽ kr(M⫺1)兾kbM ⫽
␥. That is, the turnover numbers do not depend on whether
ligand is bound to receptor (the superscript u denotes unoccupied, and o denotes occupied), and the ratio of the forward and
back reaction catalytic rate constants for each methylation state
(the subscript m denotes methylation number) is constant.
Finally, the Michaelis constant for CheB must also be independent of ligand occupancy and the methylation state of the active
o
receptor complex: Kub1 ⫽ Kob1 ⫽ Kub2 ⫽ 䡠 䡠 䡠 ⫽ KbM
⫽ Kb. (iii) The
activity of E0, the unmethylated receptor, is negligible relative to
the methylated receptor forms. (iv) The concentration of bound
CheR, Rbnd, does not depend on the ligand level. Relaxing any
of these four assumptions results in a deviation from exact
adaptation.
In simpler terms, we have defined a variable z to approximate
the total methylation level of the receptors (supplemental data,
www.pnas.org). Thus, the rate of change in z equals the methylation rate r minus the demethylation rate bA, which is proportional to the receptor complex activity A (assumptions 1 and 3):
ż ⫽ r ⫺ bA. At steady-state, ż ⫽ 0, and A approaches the fixed
value r兾b. If r and b are independent of the ligand level
(assumptions ii and iv), then perfect adaptation holds. The
expression r ⫺ bA represents the normalized output or error
of the system, and its integral, z, is fed back into the system
because receptor complex activity is a function of the methylation level (Fig. 2).
There are several pieces of experimental data that address the
validity of these assumptions for the response of the bacterial
chemotaxis signaling pathway to the chemoattractant aspartate.
First, it has not been possible to measure directly whether CheB
demethylates only active receptors. However, immediately after
the addition of chemoattractant, the rate of demethylation
declines dramatically and then recovers, consistent with the
hypothesis that CheB works less effectively on receptor that has
been transiently inactivated by introducing aspartate or serine
(21, 22). This effect cannot be attributed to the phosphorylation
of the N-terminal regulatory domain of CheB by the receptor
complex, because it is observed in strains containing a truncated
version of CheB lacking this domain (22, 23). Conversely, the
removal of attractant elicits a sudden spike in receptor demethylation by CheB. The duration of these spikes and troughs
roughly corresponds to the adaptation time, as would be
expected.
Second, Terwilliger and Koshland (24) have measured the
methylation and demethylation rates at each of the four sites on
the aspartate receptor. The kinetics of demethylation did not
vary substantially from site to site [2- to 4-fold differences,
although the data of Stock and Koshland (25) suggest potentially
greater differences] and were not affected by the presence of
ligand. On the other hand, the rates of methylation varied
approximately 50-fold from the most strongly methylated site to
the weakest site. The data on the effect of ligand on the
methylation rate are somewhat contradictory. Terwilliger and
Koshland observed a 10- to 20-fold increase after aspartate
addition in vivo (24), whereas in vitro results indicated a much
smaller effect (26). The data do not address the question whether
aspartate is modifying the Km or the kcat of CheR.
Third, Borkovich and Simon prepared completely demethylated aspartate receptor by expressing Tar in a cheR⫺ cheB⫹
strain (27). This receptor displayed little activity in vitro relative
to the methylated receptor forms. Fourth, the fact that the
concentration of receptor complexes capable of binding CheR in
both the presence and absence of chemoattractant is much
greater than the Michaelis constant of CheR suggests that CheR
operates at full saturation (Rbnd ⬇ Rtot) (20).
Barkai and Leibler emphasized that the assumption that CheB
demethylates only active receptors is critical to preserving the
robustness of perfect adaptation in their model. We used
Fig. 3. Robust regulation of the steady-state level of a pathway component.
In this hypothetical biochemical pathway, the level of the component Y is
maintained at a constant steady-state level by an integral feedback loop
mediated by the regulator molecule A and the enzymes that produce and
remove A, E3 and E4. Both the upstream molecules (Xi) and enzymes (E1i ) and
the downstream molecules (Zi) and enzymes (E2i ) do not influence the steadystate concentration of Y, Y0, assuming that the system is stable. The equation
for Y0 was derived by using Michaelis–Menten kinetics and the assumption
3
4
that E4 is operating at saturation. V max
and V max
are the maximal velocities,
3
4
and Km
and Km
are the Michaelis constants for E3 and E4.
Discussion
The ‘‘adaptive module’’ described by Barkai and Leibler is an
elegant example of integral feedback even though they did not
identify it as such. Indeed, their Eq. 1 (17) is an explicit integral
control equation. In this work, we have placed their findings in
a more general theoretical framework. Most importantly, we
show that integral control is not only sufficient but also necessary
for robust perfect adaptation. Thus, if their specific model is
later found to be contradicted by experimental data, another
mechanism implementing integral feedback is likely to be
present.
We have also identified the four specific assumptions in the
Barkai–Leibler model required to achieve integral control, and
we have argued for most of the assumptions either that there is
experimental evidence supporting the validity of the assumption
or that violation of the assumption has a modest effect on the
precision of adaptation. The exceptions are the indirect but
suggestive data consistent with the hypothesis that CheB acts
preferentially on active receptor complexes and the uncertain
data on whether the turnover number of CheR is independent
of the ligand occupancy of the receptor complex. Clearly, further
experiments are needed to address these open questions regarding the Barkai–Leibler model.
It is important to appreciate that not all individual biochemical
networks may be sufficiently well constructed to produce perfect
integral control. Indeed, the Barkai–Leibler mechanism may not
completely explain the robust and exact perfect adaptation
observed experimentally because of violations of the assumptions. This model, however, does not consider other levels of
regulation such as the phosphorylation of the N-terminal regulatory domain of CheB by the receptor complex, stimulating the
rate of demethylation. Moreover, perfect adaptation is measured
experimentally in terms of bacterial motility or flagellar activity,
which are at least two steps downstream of the receptor complex
in the signaling pathway. Additional modes of regulation, perhaps involving further integral feedback control, may occur
through the dephosphorylation of CheY-P by CheZ or even at
the flagellar motor itself. Many engineering systems contain
cascades and hierarchies of integral control loops to further
improve robustness.
Homeostasis, the maintenance of constant physiological conditions, is essential for all life. A crucial aspect of homeostasis is
that the concentrations and activities of enzymes and small
molecules are held in a narrow physiologically important range.
4652 兩 www.pnas.org
Given that integral control is both necessary and sufficient for
robust tracking of a specific steady-state value and can also be
implemented by simple biochemical networks, we believe that
integral control may represent an important strategy for ensuring homeostasis for biological systems that often possess imperfect components in a noisy environment. For example, within the
cell, the levels of important second messenger molecules such as
calcium and of key metabolites such as ATP fluctuate dramatically in response to both internal and external events. Integral
control operating through the enzymes that create or remove
these molecules can provide a robust mechanism for restoring
the concentrations of these species to their optimal steady-state
level.
We expect to observe integral feedback control at all levels of
biology. In complex man-made systems such as modern jet
airplanes, integral control loops are found at every level from
transistors and circuits to instruments and actuators, and finally
to the entire vehicle itself (e.g., autopilot). A single oil refinery
has more than 10,000 integral feedback loops, and the electric
power grid uses integral feedback throughout to regulate frequency and voltage. Internet congestion control uses a variant of
integral feedback and is implemented on essentially every networked computer. Similarly, we believe integral control and
related strategies are important not only for cellular homeostasis
but also for homeostasis of the whole organism and even for
ecosystem balance. For example, Koeslag et al. have argued
that integral control is used to regulate hormone secretion in
humans (28).
Finally, integral control and equilibrium analysis represent
only the most elementary ideas from control and dynamical
systems theory that might be relevant to the understanding of
biological complexity. A promising aspect of this broader theory
is in providing further necessity results to help biologists greatly
narrow their search for specific mechanisms. We expect that
concepts such as robustness tradeoffs—robustness to specific
uncertainties is achieved at the expense of heightened sensitivities elsewhere (29, 30)—may prove particularly powerful.
Appendix: A Primer on Integral Control
We are interested in studying the conditions under which a
system has the property that the output is independent of the
input level in steady-state. This is called perfect adaptation in
biology, but we will refer to it as asymptotic tracking in this
section, following control theory terminology.
In Fig. 2, we display the block diagram for a simple example
of integral feedback control. The variable u represents the input;
y is the normalized system output or error, the difference
between the actual output y1 and the steady-state output y0. The
gain k is a positive real number representing some process that
takes u as an input and produces the output y1. The distinguishing
feature of integral control is that the time integral, x, of the
system error is fed back to the system. Intuitively, it makes sense
to use the integral of the error and not the error itself, because
the past errors ‘‘charge up’’ the controller to offset a constant
disturbance even as the error approaches 0.
As a result, we obtain the equation that characterizes integral
control:
ẋ ⫽ y.
[A1]
At steady-state, ẋ ⫽ y ⫽ 0 despite variations in the input u and
parameter k, thus ensuring robust asymptotic tracking. The
condition that k ⬎ 0 ensures stability, and the value of k affects
the speed of the response but not the tracking property. Note
that in the more general case described below we do not have to
identify explicitly the nature of the feedback. If we replace k with
a more complex process including nonlinear dynamics, ẋ ⫽ y ⫽
0 still holds when the feedback system reaches steady-state. Thus,
Yi et al.
ẋ ⫽ Ax ⫹ bu
[A2]
y ⫽ cx ⫹ du.
At steady state, ẋ ⫽ 0 and y ⫽ 共d ⫺ cA⫺1b兲u, and we can
ignore the dynamics and treat the problem purely algebraically.
Thus, for all constant u, y ⫽ 0 if and only if either
关c
d兴 ⫽ 0 or det
冋 册
A
c
b
⫽ 0.
d
The former is the trivial case when y(t) ⫽ 0 for all t, and the latter
is satisfied if and only if ᭚ k 僆 ᑬn, k ⫽ 0 such that k[A b] ⫽
[c d]. Thus, defining z ⫽ kx, we have ż ⫽ kẋ ⫽ k(Ax ⫹ bu) ⫽
cx ⫹ du ⫽ y.
If y ⫽ 0 for all parameter variations, then ż ⫽ y for all
parameter variations. The latter condition is equivalent to
integral control being a structural property of the system. Thus,
a necessary and sufficient condition for robust asymptotic tracking is that the system possesses integral feedback. This wellknown result in control theory is a special case of the internal
model principle, which states that the controller must contain a
model of the external signal to achieve robust tracking (32).
Further aspects of the full nonlinear case are beyond the scope
of this paper (33), but we can observe briefly that if a nonlinear
system has robust tracking, then so must its linearization about
the resulting equilibrium. Thus, the necessity results extend to
the nonlinear case in the sense that robust tracking implies that
the linearization must have integral control.
The state vector x contains the n variables (species) of the system;
y is the output, in this case activity; u is the input, ligand
concentration. As a linearization about an equilibrium, x, y, and
u are differences between the species, activity, and ligand level
and their corresponding values at the equilibrium. A (n ⫻ n
system matrix), b (n ⫻ 1 input matrix), c (1 ⫻ n output matrix),
and d are the system parameters. We assume that A has all its
eigenvalues in the open left half of the complex plane so that the
system is asymptotically stable.
We acknowledge valuable discussions with Drs. S. Lall, H. Berg, D.
Petrasek, U. Alon, N. Barkai, and S. Leibler. Special thanks to Drs. U.
Alon, H. Berg, J. Stock, and P. Iglesias for comments on the manuscript.
This work was supported by an Air Force Office of Scientific Research
(AFOSR)兾DDRE MURI AFS-5X-F496209610471 grant entitled ‘‘Uncertainty Management in Complex Systems’’ and Defense Advanced
Research Planning Agency兾AFOSR grant AFS-5-F4962098-L0487. T.M.Y. was supported by a fellowship from the Caltech Initiative in
Computational Molecular Biology funded by the Burroughs–Wellcome
Foundation.
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Yi et al.
PNAS 兩 April 25, 2000 兩 vol. 97 兩 no. 9 兩 4653
BIOPHYSICS
integral feedback gives a robust mechanism for asymptotic
tracking.
A heating system controlled by a thermostat is one well-known
example of integral feedback control. Because temperature,
which is proportional to the integral of heat (the output of the
heater), is compared to the desired temperature and fed back
into this closed-loop system, the difference between the room
temperature and the desired temperature approaches zero despite large external environmental disturbances or moderate
variations in the heater behavior. More typically, the integral
control action in most controllers is created by an explicit
implementation of an integrator in the controller itself. This use
of integral control is almost a century old (31).
We now demonstrate that robust asymptotic tracking, tracking
that holds for parameter variations as well as input variations,
holds if and only if integral control is a structural property of the
system. For simplicity, we approximate the nonlinear chemotaxis
signaling network by the following general linear model, assumed to be linearized around an equilibrium for a fixed ligand
level:
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