Page 1
Theory
Defining Network Topologies that
Can Achieve Biochemical Adaptation
Wenzhe Ma,1,2,3Ala Trusina,2,3Hana ElSamad,2,4Wendell A. Lim,2,5,* and Chao Tang1,2,3,4,*
1Center for Theoretical Biology, Peking University, Beijing 100871, China
2California Institute for Quantitative Biosciences
3Department of Bioengineering and Therapeutic Sciences
4Department of Biochemistry and Biophysics
5Howard Hughes Medical Institute and Department of Cellular and Molecular Pharmacology
University of California, San Francisco, CA 94158, USA
*Correspondence: lim@cmp.ucsf.edu (W.A.L.), chao.tang@ucsf.edu (C.T.)
DOI 10.1016/j.cell.2009.06.013
SUMMARY
Many signaling systems show adaptation—the ability
to reset themselves after responding to a stimulus.
We computationally searched all possible threenode
enzymenetworktopologiestoidentifythosethatcould
perform adaptation. Only two major core topologies
emerge as robust solutions: a negative feedback loop
with a buffering node and an incoherent feedforward
loopwithaproportionernode.Minimalcircuitscontain
ing these topologies are, within proper regions of
parameter space, sufficient to achieve adaptation.
Morecomplexcircuitsthatrobustlyperformadaptation
allcontainatleastoneofthesetopologiesattheircore.
This analysis yields a design table highlighting a finite
setofadaptivecircuits.Despitethediversityofpossible
biochemical networks, it may be common to find that
onlyafinitesetofcoretopologiescanexecuteapartic
ular function. These design rules provide a framework
for functionally classifying complex natural networks
and a manual for engineering networks.
For a video summary of this article, see the
PaperFlick file with the Supplemental Data available
online.
INTRODUCTION
The field of systems biology is largely focused on mapping and
dissecting cellular networks with the goal of understanding
how complex biological behaviors arise. Extracting general
design principles—the rules that underlie what networks can
achieve particular biological functions—remains a challenging
task, given the complexity of cellular networks and the small
fraction of existing networks that have been well characterized.
Nonetheless, growing evidence suggests the existence of
design principles that unify the organization of diverse circuits
across all organisms. For example, it has been shown that there
are recurrent network motifs linked to particular functions, such
astemporal expressionprograms(ShenOrretal.,2002),reliable
cell decisions (Brandman et al., 2005), and robust and tunable
biological oscillations (Tsai et al., 2008).
These findings suggest an intriguing hypothesis: despite the
apparent complexity of cellular networks, there might only be
a limited number of network topologies that are capable of
robustly executing any particular biological function. Some
topologies may be more favorable because of fewer parameter
constraints. Other topologies may be incompatible with a partic
ular function. Although the precise implementation could differ
dramatically in different biological systems, depending on
biochemical details and evolutionary history, the same core set
of network topologies might underlie functionally related cellular
behaviors (Milo et al., 2002; Wagner, 2005; Ma et al., 2006; Hor
nung and Barkai, 2008). If this hypothesis is correct, then one
may be able to construct a unified functiontopology mapping
that captures the essential barebones topologies underpinning
the function. Such core topologies may otherwise be obscured
by the details of any specific pathway and organism. Such
a map would help organize our everexpanding database of
biological networks by functionally classifying key motifs in
anetwork.Suchamapmightalsosuggestwaystotherapeutically
modulateasystem.Acircuitfunctiontopologymapwouldalsobe
invaluable for synthetic biology, providing a manual for how to
robustlyengineerbiologicalcircuitsthatcarryoutatargetfunction.
To investigate this hypothesis, we have computationally
explored the full range of simple enzyme circuit architectures
thatarecapableofexecutingonecriticalandubiquitousbiological
behavior—adaptation. We ask if there are finite solutions for
achieving adaptation. Adaptation refers to the system’s ability to
respond to a change in input stimulus then return to its prestimu
lated output level, even when the change in input persists.
Adaptation is commonly used in sensory and other signaling
networks to expand the input range that a circuit is able to sense,
to more accurately detect changes in the input, and to maintain
homeostasis in the presence of perturbations. A mathematical
description of adaptation is diagrammed in Figure 1A, in which
two characteristic quantities are defined: the circuit’s sensitivity
to input change and the precision of adaptation. If the system’s
responsereturnsexactlytotheprestimuluslevel(infiniteprecision),
it is called the perfect adaptation. Examples of perfect or near
perfect adaptation range from the chemotaxis of bacteria (Berg
760 Cell 138, 760–773, August 21, 2009 ª2009 Elsevier Inc.
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andBrown,1972;MacnabandKoshland,1972;Kirschetal.,1993;
Barkai and Leibler, 1997; Yi et al., 2000; Mello and Tu, 2003; Rao
et al., 2004; Kollmann et al., 2005; Endres and Wingreen, 2006),
amoeba (Parent and Devreotes, 1999; Yang and Iglesias, 2006),
and neutrophils (Levchenko and Iglesias, 2002), osmoresponse
in yeast (Mettetal et al., 2008), to the sensor cells in higher organ
isms (Reisert and Matthews, 2001; Matthews and Reisert, 2003),
and calcium homeostasis in mammals (ElSamad et al., 2002).
Here, insteadoffocusing ononespecific signaling system that
shows adaptation, we ask a more general question: What are all
network topologies that are capable of robust adaptation? To
answer this question, we enumerate all possible threenode
Parameter
sampling
(10,000 sets)
I1
I2
O1
A
BC
A
B
C
Input (I)
Output
D
B
A
Input
C
Output
B
A
Input
C
Output
B
A
Input
C
Output
Opeak
O2
logKI
logkI
P = {kIA, KIA, k'BA, K'BA...}
II
Large response
No adaptation
I
No response
III
Adaptation
(Functional)
Input
Output
A
B
C
log10(sensitivity)
III
II
I
high
low
high
log10(precision)
low
ODE
simulation
16038 networks
B
1B
A
FB
Active
Form
Inactive
Form
Input
Output
time
Output
Input
time
01 2
1
0
1
2
1

FB
FC
dA
dt
dB
dt
= kIAI
(1 A)
(1 A)+ KIA
(1 B)
(1 B)+ KAB
(1 C)
(1 C)+KAC
kBAB
A
A+ KBA
= kABAkFBBFB
B
B+ KFBB
C
C +KFCC
dC
dt
= kACAkFCCFC
Sensitivity =
(Opeak
(I2
O1)/O1
I1)/I1
Precision =(O2
O1)/O1
I1)/I1
(I2
1
Figure 1. Searching Topology Space for Adaptation Circuits
(A) Inputoutput curve defining adaptation.
(B) Possible directed links among three nodes.
(C) Illustrative examples of threenode circuit topologies.
(D) Illustration of the analysis procedure for a given topology.
Cell 138, 760–773, August 21, 2009 ª2009 Elsevier Inc. 761
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network topologies (restricting ourselves to enzymatic nodes)
and study their adaptation properties over a range of kinetic
parameters (Figure 1B). We use three nodes as a minimal frame
work:onenodethatreceivesinput,asecondnodethattransmits
output, and a third node that can play diverse regulatory roles.
There are a total of 16,038 possible threenode topologies that
contain at least one direct or indirect causal link from the input
node to the output node. For each topology, we sampled
a wide rangeof parameter space (10,000 sets of networkparam
eters) and characterized the resulting behavior in terms of the
circuit’s sensitivity to input change and its ability to adapt. In
all we have analyzed a total of 16,038*10,000 z1.6 3 108
different circuits. This search resulted in an exhaustive circuit
function map, which we have used to extract core topological
motifs essential for adaptation. Overall, our analysis suggests
that despite the importance of adaptation in diverse biological
systems, there are only a finite set of solutions for robustly
achieving adaptation. These findings may provide a powerful
framework in which to organize our understanding of complex
biological networks.
RESULTS
Searching for Circuits Capable of Adaptation
Adaptation is defined by the ability of circuits to respond to input
change but to return to the prestimulus output level, even when
the input change persists. Therefore, in this study we monitor
two functional quantities for each network: the circuit’s sensi
tivity to input stimulus and its adaptation precision (Figure 1A).
Sensitivity is defined as the height of output response relative
to the initial steadystate value. Adaptation precision represents
the difference between the pre and poststimulus steady states,
defined here as the inverse of the relative error. We have limited
ourselves to exploring circuits consisting of three interacting
nodes (Figures 1B and 1C): one node that receives inputs (A),
one node that transmits output (C), and a third node (B) that
can play diverse regulatory roles. Although most biological
circuits are likely to have more than three nodes, many of these
cases can probably be reduced to these simpler frameworks,
giventhatmultiplemoleculesoftenfunctioninconcertasasingle
virtual node. By constraining our search to threenode networks,
we are in essence performing a coarsegrained network search.
This sacrifice in resolution, however, allows us to perform a
complete search of the topological space.
For this analysis, we limited ourselves to enzymatic regulatory
networksandmodelednetworklinkagesusingMichaelisMenten
rate equations. As described in Experimental Procedures, each
node in our model network has a fixed total concentration that
can be interconverted between two forms (active and inactive)
by other active enzymes in the network or by basally available
enzymes.Forexample,apositivelinkfromnodeAtonodeBindi
cates that the active state of enzyme A is able to convert enzyme
B from its inactive to active state (see Figure 1D). If there is no
negative link to node B from the other nodes in the network, we
assume that a basal (nonregulated) enzyme would inactivate B.
We used ordinary differential equations to model these interac
tions, characterized by the MichaelisMenten constants (KM’s)
and catalytic rate constants (kcat’s) of the enzymes. Implicit in
our analysis are assumptions that the enzyme nodes operate
under MichaelisMenten kinetics and that they are noncoopera
tive (Hill coefficient = 1). In the Supplemental Experimental
Procedures available online, section 10, we show that these
assumptions do not significantly alter our results—similar results
emerge when using mass action rate equations instead of
MichaelisMenten equations, or when using nodes of higher co
operativity.
Our analysis mainly focused on the characterization of the
circuit’s sensitivity and adaptation precision, which can be map
ped on the twodimensional sensitivity versus precision plot
(Figure 1D). We define a particular circuit architecture/parameter
configuration to be ‘‘functional’’ for adaptation if its behavior falls
within the upperright rectangle in this plot (the green region in
Figure 1D)—these are circuits that show a strong initial response
(sensitivity>1)combinedwithstrongadaptation(precision>10).
Inmostofoursimulationswegaveanonzeroinitial input(I1=0.5)
and then changed it by 20% (I2= 0.6). The functional region
corresponds to an initial output change of more than 20% and
a final output level that is not more than 2% different from the
initial output. Nonfunctional circuits fall into other quadrants of
this plot, including circuits that show very little response
(upperleft quadrant) and circuits that show a strong response
but low adaptation (lowerright quadrant). For any particular
circuit architecture, we focused on how many parameter sets
can perform adaptation—a circuit is considered to be more
robust if a larger number of parameter sets yield the behavior
defined above.
To identify the network requirements for adaptation, we took
two different but complementary approaches. In the first
approach, we searched for the simplest networks that are
capable of achieving adaptation, limiting ourselves to networks
containing three or fewer links. We find that all circuits of this
type that can achieve adaptation fall into two architectural
classes: negative feedback loop with a buffering node (NFBLB)
and incoherent feedforward loop with a proportioner node
(IFFLP). In the second approach, we searched all possible
16,038 threenode networks (with up to nine links) for architec
tures that can achieve adaptation over a wide range of parame
ters. These two approaches converge in their conclusions: the
more complex robust architectures that emerge are highly
enriched for the minimal NFBLB and IFFLP motifs. In fact all
adaptation circuits contain at least one of these two motifs.
The convergent results indicate that these two architectural
motifs present two classes of solutions that are necessary for
adaptation.
Identifying Minimal Adaptation Networks
We started by examining the simplest networks capable of
achieving adaptation (defined as sensitivity > 1 and precision
> 10) for any of their parameter sets. For networks composed
ofonlytwonodes(aninputreceivingnodeAandoutputtransmit
ting node C, with no third regulatory node), there are 4 possible
links and 81 possible networks, none of which is capable of
achieving adaptation for the parameter space that we scanned
(Figure S1).
Next, we examined minimal threenode topologies with
only three or fewer links between nodes (maximally complex
762 Cell 138, 760–773, August 21, 2009 ª2009 Elsevier Inc.
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threenode topologies contain nine links). None of the twolink,
threenode networks were capable of adaptation (Figure S2)—
the minimal number of links for this to be functional is three.
The simplest topologies capable of adaptation, under at least
some parameter sets, are eleven threenode, threelink net
works. These network architectures are listed in Figure 2 along
B
A
Input
C
Output
B
A
Input
C
Output
Output directly feeds back to
input (no buffer node)
Lacks negative
feedback loop
B
A
Input
C
Output
B
A
Input
C
Output
AC CB BA
+
+



+

+



+
+


B
A
Input
C
Output
Minimal adaptation networks (all)
Case I: Negative feedback loops
A
Related nonadaptation networks (examples)
Defects:
A B BA AC
+
+


+ or 
+ or 
B
C CB A
+



+
+
+ or 
+
+
B
A
Input
C
Output
107
106
105
104
103
102
101
1
Probability
log10(Sensitivity)
log10(Precision)
B
B
A
Input
C
Output
B
A
Input
C
Output
B
A
Input
C
Output
Lacks incoherent
feedforward loop
(only coherent)
A to C and B to C
have the same sign
Case II: Incoherent feedforward loops
Related nonadaptation networks (example)
Defects:
B
A
Input
C
Output
Minimal adaptation networks (all)
Figure 2. Minimal Networks (%3 Links) Capable of Adaptation
(A) Adaptive networks composed of negative feedback loops. Three examples of adaptation networks are shown in the upper panel. Each is one member
(shaded) of a group of similar adaptation networks, whose signs of regulations are listed underneath. For comparison, three examples of nonadaptive networks
are shown in the low panel, with their ‘‘defects’’ for adaptation function listed underneath.
(B)Adaptivenetworkscomposedofincoherentfeedforwardloops.Theonlytwominimaladaptationnetworksinthiscaseareshownintheupperpanel.Examples
of nonadaptive networks are shown in the lower panel.
Cell 138, 760–773, August 21, 2009 ª2009 Elsevier Inc. 763
Page 5
with examples of the distribution of sensitivity/precision behav
iors for the 10,000 parameter sets that were searched (see also
Figure S3). An architecture is considered capable of adaptation
if this distribution extends into the upperright quadrant (high
sensitivity,highprecision).Thecommonfeaturesofthenetworks
capable of adaptation are either a single negative feedback loop
or a single incoherent feedforward loop. Here, we define a
negative feedback loop as a topology whose links, starting
from any node in the loop, lead back to the original node with
thecumulativesignofregulatorylinkswithintheloopbeingnega
tive. We define an incoherent feedforward loop as a topology in
which two different links starting from the inputreceiving node
both end at the outputtransmitting node, with the cumulative
sign of the two pathways having different signs (one positive
and one negative). The first row of Figure 2 shows several exam
ples of threelink, threenode networks capable of adaptation;
the second row shows related counter parts that cannot achieve
adaptation. Overall incoherent feedforward loops appear to
perform adaptation more robustly than negative feedback
loops—theyarecapableofhighersensitivityandhigherprecision
asindicatedbythelargerdistributionofsampledparametersthat
lie in the upperright corner of the sensitivity/precision plot.
While it not surprising that positive feedback loops cannot
achieve adaptation (Figure 2A), it is interesting to note that nega
tive feedback loop topologies differ widely in their ability to
perform adaptation (Figure 2A, lower panel). Notably, there is
only one class of simple negative feedback loops that can
robustly achieve adaptation. In this class of circuits, the output
node must not directly feedback to the input node. Rather, the
feedback must go through an intermediate node (B), which
serves as a buffer. The importance of this buffering node will
be discussed in detail later.
Amongfeedforwardloops(Figure2B),coherentfeedforwardis
clearlyverypooratadaptation(Figure2B,lowerpanel).Thethree
incoherent feedforward loops in Figure 2B also differ drastically
in their performance. Of these, only the circuit topology in which
the output node C is subject to direct inputs of opposing signs
(one positive and one negative) appears to be highly preferred.
As will be seen later, the reason this architecture is preferred is
because the only way for an incoherent feedforward loop to
achieverobustadaptationisfornodeBtoserveasaproportioner
for node A—i.e., node B is activated in proportion to the activa
tion of node A and to exert opposing regulation on node C.
Key Parameters in Minimal Adaptation Networks
Two major classes of minimal adaptive networks emerge from
the above analysis: one type of negative feedback circuits and
one type of incoherent feedforward circuits. Why are these two
classes of minimal architectures capable of adaptation? Here
we examine their underlying mechanisms, as well as the param
eter conditions that must be met for adaptation.
Negative Feedback Loop with a Buffer Node
The NFBLB class of topologies has multiple realizations in three
node networks (Figure 2A), all featuring a dedicated regulation
node ‘‘B’’ that functions as a ‘‘buffer.’’ We show how the motif
works by analyzing a specific example (Figure 3A), which has a
negative feedback loop between regulation node B and output
transmitting node C.
The mechanism by which this NFBLB topology adapts and
achieves a high sensitivity can be unraveled by the analysis of
the kinetic equations
dA
dt=IkIA
ð1 ? AÞ
ð1 ? AÞ+KIA? FAk0
ð1 ? BÞ
ð1 ? BÞ+KCB? FBk0
ð1 ? CÞ
ð1 ? CÞ+KAC? Bk0
FAA
A
A+K0
FAA
dB
dt=CkCB
FBB
B
B+K0
FBB
dC
dt=AkAC
BC
C
C+K0
BC
(1)
where FAand FBrepresent the concentrations of basal enzymes
that carry out the reverse reactions on nodes A and B, respec
tively (they oppose the active network links that activate A and
B). In this circuit, node A simply functions as a passive relay of
the input to node C; the circuit would work in the same way
if the input were directly acting on node C (just replacing A
with I in the third equation of Equation 1). Analyzing the param
eter sets that enabled this topology to adapt indicates that the
two constants KCBand K0
activation of B by C and inhibition of B by the basal enzyme)
tend to be small, suggesting that the two enzymes acting on
nodeB must approach saturation to achieve adaptation. Indeed,
it can be shown that in the case of saturation this topology can
achieve perfect adaptation. Under saturation conditions, i.e.,
(1B) > > KCBand B > > K0
approximated by the following:
FBB(MichaelisMenten constants for
FBB, the rate equation for B can be
dB
dt=CkCB? FBk0
FBB:
(2)
The steadystate solution is
C?=FBk0
FBB=kCB;
(3)
which is independent of the input level I. The output C of the
circuit can still transiently respond to changes in the input (see
the first and the third equations in Equation 1) but eventually
settles to the same steady state determined by Equation 3.
Note that Equation 2 can be rewritten as
dB
dt=kCBðC ? C?Þ;
B=B?ðI0Þ+kCB
Rt
0
ðC ? C?Þdt:
(4)
Thus, the buffer node B integrates the difference between the
output activity C and its inputindependent steadystate value.
Therefore, this NFBLB motif, node C —j node B / node C,
implements integral control—a common mechanism for perfect
adaptation in engineering (Barkai and Leibler, 1997; Yi et al.,
2000). All minimal NFBLB topologies use the same integral
control mechanism for perfect adaptation.
The parameter conditions required for more accurate adapta
tion and higher sensitivity can also be visualized in the phase
planes of nodes B and C (Figure 3A). The nullclines for nodes
B and C (dB/dt = 0 and dC/dt = 0, respectively) are shown for
two different input values. For this topology, only the C nullcline
764 Cell 138, 760–773, August 21, 2009 ª2009 Elsevier Inc.
Page 6
(redcurve)dependsexplicitlyontheinputthroughA(Equation1).
The B nullcline (black curve) does not depend on A. The steady
state of the system is given by the intersection of the B and C
nullclines. Thus, the change in steady state for any input change
is only determined by the movement of the inputdependent
C nullcline (e.g., dashed red curve in Figure 3A). The adaptation
precision is therefore directly related to the flatness of the B null
cline near the intersection of the two nullclines. The smaller the
0 0.5
B
0
0.2
0.4
0.6
0.8
1
C
0 50
t
100
C
0 0.5
B
0
0.2
0.4
0.6
0.8
1
C
0 10
t
20
C
0 0.5
B
0
0.2
0.4
0.6
0.8
1
C
010
t
20
C
00.5
B
0
0.2
0.4
0.6
0.8
1
C
010
t
K'F B
20
C
00.5
B
0
0.2
0.4
0.6
0.8
1
C
010
t
20
C
00.5
B
0
0.2
0.4
0.6
0.8
1
C
05
t
C
BA
KCB
decreases
dB/dt=0
dC/dt=0
IFFLP
dB/dt=0
dC/dt=0
dB/dt=0
dC/dt=0
k'BC
increases
K'F B
decreases
B
B
A
Input
C
Output
dB/dt=0
dC/dt=0
kAB, KAB
Proportioner
node
k'F B, K'F B
B
B
A
Input
C
Output
I1
I2
dB/dt=0
dC/dt=0
O1
O2
Opeak
KCB
Buffering
node
K'F B
B
kAC
increases
k'F B
decreases
B
kAB
decreases
dB/dt=0
dC/dt=0
KAB
decreases
NFBLB
kAC
k'BC
Unconstrained
Linear
Saturated
Parameter ranges for Km
B
Increases
B
Figure 3. Phase Diagram and Nullcline Analysis of Representative Networks from the Two Classes of Minimal Adaptive Topologies
The two networks are shown on the top with the key regulations colored to indicate the parameter constraints for achieving perfect adaptation.
(A) Phase planes of the variables B and C for a NFBLB topology. The B nullclines are drawn in black lines and C nullclines in red (solid red for the initial input I1and
dashed red for the changed input I2). The steady states with input I1and I2are the intersections of the nullclines and are highlighted by black and gray dots,
respectively. When the input is changed from I1to I2, the trajectory (blue lines) of the system variables follows the vector field (dB/dt, dC/dt) (with input I2), which
is denoted by the green arrows. The trajectory’s projection on the C axis is the system’s output and is shown separately right next to the phase plane. (Refer to
Figure 1A for the functional meaning of O1, O2, and Opeak.) Two sets of key parameters (KM’s on B) are used to illustrate their effect on adaptation precision:
K0
effect on sensitivity: kAC= 10 and k0BC= 10 for the top and the middle panels and kAC= 0.1 and k0BC= 0.1 for the lower panel.
(B) Phase planes for an IFFLP topology. K0
k0
F0B= 0.1 and KCB= 0.1 for the top panel and K0
F0B= 0.01 and KCB= 0.01 for the middle and lower panels. Two sets of rate constants are used to illustrate their
F0B= 1 and KAB= 0.1 for the top panel. K0
F0B= 2000 for the lower panel.
F0B= 100 and KAB= 0.001 for the middle and the lower panels. kAB= 0.5 and
F0B=10 for the top and the middle panels. kAB= 100 and k0
Cell 138, 760–773, August 21, 2009 ª2009 Elsevier Inc. 765
Page 7
dependence ofConB,thesmallertheadaptationerror.Oneway
to achieve a small dependence of C on B, or equivalently a sharp
dependence of B on C, in an enzymatic cycle is through the
zeroth order ultrasensitivity (Goldbeter and Koshland, 1984),
which requires the two enzymes regulating the node B to work
at saturation. This is precisely the condition leading to Equation
2. All NFBLB minimal topologies have similar nullcline structures
and their adaptation is related to the zeroth order ultrasensitivity
in a similar fashion.
The ability of the network to mount an appropriate transient
response to the input change before achieving steadystate
adaptation depends on the vector fields (dB/dt, dC/dt) in the
phase plane (green arrows, Figure 3). A large response, corre
sponding to sensitive detection of input changes, is achieved
by a large excursion of the trajectory along the C axis. This in
turn requires a large initial jdC/dtj and a small initial jdB/dtj
near the prestimulus steady state. For this class of topologies,
this can be achieved if the response time of node C to the input
change is faster than the adaptation time. The response time of
node C is set by the first term in the dC/dt equation—faster
response would require a larger kAC. The timescale for adapta
tion is set by the equation for node B and the second term of
the equation for node C—slower adaptation time would require
a smaller k0BC/K0BCand/or a slower timescale for node B. This
illustrates an important uncoupling of adaptation precision and
sensitivity: once the MichaelisMenten constants are tuned to
achieve operation in the saturated regimes, the timescales of
the system can be independently tuned to modulate the sensi
tivity of the system to input changes.
Incoherent Feedforward Loop with a Proportioner Node
The other minimal topological class sufficient for adaptation is
the incoherent feedforward loop with aproportional node(IFFLP)
(Figure 2B). In an incoherent FFL, the output node C is subject to
two regulations both originating from the input butwith opposing
cumulative signs in the two pathways. There are two possible
classes of incoherent FFL architectures, but only one is able to
robustly perform adaptation (Figure 2B, upper panel): the func
tional architectures all have a ‘‘proportioner’’ (node B) that regu
lates the output (node C) with the opposite sign as the input to C.
We denote this class IFFLP.
The IFFLP topology achieves adaptation by using a different
mechanism from that of the NFBLB class. Rather than moni
toring the output and feeding back to adjust its level, the feedfor
ward circuit ‘‘anticipates’’ the output from a direct reading of the
input.nodeBmonitorstheinputandexertsanopposingforceon
node C to cancel the output’s dependence on the input. For the
IFFLP topology shown in Figure 3B, the kinetic equations are as
follows:
dA
dt=IkIA
ð1 ? AÞ
ð1 ? AÞ+KIA? FAk0
ð1 ? BÞ
ð1 ? BÞ+KAB? FBk0
ð1 ? CÞ
ð1 ? CÞ+KAC? Bk0
FAA
A
A+K0
FAA
BdB
dt=AkAB
FBB
B+K0
FBB
dC
dt=AkAC
BC
C
C+K0
BC
:
(5)
The adaptation mechanism is mathematically captured in the
equation for node C: if the steadystate concentration of the
negative regulator B is proportional to that of the positive regu
lator A, the equation determining the steadystate value of C,
dC/dt = 0, would be independent of A and hence of the input I.
In this case, the equation for node B generates the condition
under which the steadystate value B* would be proportional
to A*: the first term in dB/dt equation should depend on A only
and the second term on B only. The condition can be satisfied
if the first term is in the saturated region ((1 ? B) [ KAB) and
the second in the linear region (B ? K0
B?=A?$kABK0
FBB), leading to
FBB=ðFBk0
FBBÞ:
(6)
This relationship, established by the equation for node B,
shows that the steadystate concentration of active B is propor
tional to the steadystate concentration of active A. Thus B will
negatively regulate C in proportion to the degree of pathway
input. This effect of B acting as a proportioner node of A can
be graphically gleaned from the plot of the B and C nullclines
(Figure 3B). In this case, maintaining a constant C* requires the
B nullcline to move the same distance as the C nullcline in
response to an input change. Here again, the sensitivity of the
circuit (the magnitude of the transient response) depends on
the ratio of the speeds of the two signal transduction branches:
A / C and A / B —j C, which can be independently tuned from
the adaptation precision.
Analysis of All Possible ThreeNode Networks:
An NFBLB or IFFLP Architecture Is Necessary
for Adaptation
The above analyses focused on minimal (less than or equal to
three links) threenode networks and identified simple architec
tures that are sufficient for adaptation. But are these architec
tures also necessary for adaptation? In other words, are the
identified minimal architectures the foundation of all possible
adaptive circuits, or are there more complex higherorder solu
tions that do not contain these minimal topologies? To investi
gate this question, we expanded our study to encompass all
possible threenode networks, each with combinations of up to
nine intranetwork links. Again, for each network architecture,
we sampled 10,000 possible parameter sets. Figure 4A shows
a comprehensive map of the functional space, expressed as
the distribution of all topologies and all sampled parameter
sets on the sensitivity/precision plot. Only the regions above
the diagonal are occupied, since by definition sensitivity cannot
be lower than adaptation error (Experimental Procedures).
The vast majority of the circuits lie on the diagonal where sensi
tivity = 1/error. This very common functional behavior is simply
a monotonic change of the output in response to the input
change, a hallmark of a direct, nonadaptive signal transduction
response. The distribution plot quickly drops off away from the
diagonal as the number of circuits with increasing sensitivity
and/or adaptation precision drops. Overall, only 0.01% of all
1.6 3 108possible architecture/parameter sets fall within the
upperright corner of the plot in Figure 4A—i.e., those circuits
that can achieve both high sensitivity and high adaptation preci
sion. We are interested in topologies that are overrepresented in
these regions. By overrepresentation, we require that the
topology be mapped to this region more than 10 times when
766 Cell 138, 760–773, August 21, 2009 ª2009 Elsevier Inc.
Page 8
sampledwith10,000parametersets.Thereare395outof16,038
such topologies.
Analysis of these 395 robust topologies shows that they are
overrepresented with feedback and feedforward loops (Supple
mental Experimental Procedures, section 4). Strikingly, all 395
topologies contain at least one NFBLB or IFFLP motif (or both)
(Figure 4B). These results indicate that at least one of these
motifs is necessary for adaptation.
Motif Combinations that Improve Adaptation
Comparing the sensitivity/precision distribution plot of all
networks (Figure 4A) with that of the minimal networks (Figure 2),
it is clear that some of the more complex topologies occupy
a larger functional space than the minimal topologies. We
wanted to investigate what additional features can improve the
functional performance in these networks. To address this ques
tion, we separated the 395 adaptation networks into the two
categories, NFBLB and IFFLP. We then clustered the networks
within each category using a pairwise distance between
networks. The results, shown in Figures 4C and 4D, clearly indi
cate the presence of common structural features (subclusters) in
each category, some of which are shown on the righthand side
in the figure. One striking feature shared by some of the more
complexadaptationnetworksintheNFBLBcategoryisapositive
selfloop on the node B in the case where the other regulation on
B is negative. This type of topology, with a saturated positive
selfloop on B and linear negative regulation from other nodes,
implements a special type of integral control to achieve perfect
adaptation—heretheLog(B),ratherthanBitself,istheintegrator
(Supplemental Experimental Procedures, section 5). Another
commonfeatureofthemorecomplexnetworks,whichispresent
in both categories, is the presence of additional negative feed
back loops that go through node B. We found that this feature
also enhances the performance—the networks with more such
negative feedback loops have larger Q values (defined as the
number of sampled parameter sets that yield the target adapta
tion behavior) than the minimal networks (Supplemental Experi
mental Procedures, section 12).
Analytic Analysis: Two Classes of Adaptation
Mechanisms
Inordertoelucidateallpossibleadaptationmechanismsformore
complex networks, we analyzed analytically the structure of the
steadystate equations for threenode networks. The steady
state equations for any threenode network in our model can be
written as dA/dt = fA(A*, B*, C*, I) = 0, dB/dt = fB(A*, B*, C*) = 0
anddC/dt=fC(A*,B*,C*)=0,whereA*,B*,andC*arethesteady
statevaluesofthethreenodes,andfA,fB,andfCrepresenttheMi
chaelisMenten terms contributing to the production/decay rate
of A, B, and C, respectively. In response to a small change in
Networks with
IFFLP + NFBL
AA ABACBABB BCCACBCC
IFFLP + other motifs (229)
log10(Precision)
2.5 2 101
1
0
1
2
2.5
log10(Sensitivity)
1
101
102
103
104
105
106
107

A
C
B
All possible 3node networks (16038)
395
robust
adaption
networks
Common links in
subcluster of networks
Topological clustering for robust
adaptation networks
D
Positive regulationsNegative regulations No regulation
All possible 3node networks (16038)
NFBL
(11070)
0
Adaptation
networks (395)
IFFL
(2916)
166
6
318
2369
8312
223
Networks with
NFBLB + NFBL
Networks with NFBLB
+ selfloop on B
AA AB ACBAB B BCCAC BCC
NFBLB + other motifs (166)
Figure 4. Searching the Full CircuitSpace for All Robust Adaptation Networks
(A) The probability plot for all 16,038 networks with all the parameters sampled. Three hundred and ninetyfive networks are overrepresented in the functional
region shown by the orange rectangle.
(B) Venn diagram of networks with three characters: adaptive, containing negative FBL, and containing incoherent FFL.
(C) Clustering of the adaptation networks that belong to the NFBLB class. The network motifs associated with each of the subclusters are shown on the right.
(D) Clustering of adaptation networks that belong to the IFFLP class.
Cell 138, 760–773, August 21, 2009 ª2009 Elsevier Inc. 767
Page 9
Condition for perfect adaptation
and stability: N=0, J<0
fB
AB
Linearized equations
Adaptation error
fB
A,
fB
B,
fC
A,
fC
B
0
J =
fA
A
fB
A
fC
A
fA
B
fB
B
fC
B
fA
C
fB
C
fC
C
N =
fB
A
fC
A
fB
B
fC
B
N =
fB
A
fC
A
fB
B
fC
B
J < 0 implies at least one
negative feedback loop
The two pathways have different signs
Incoherent feedforward loop
Stable steady state fB
B< 0
fB
A
fC
B
fC
A
=
fB
B
< 0
N =
0
fC
A
0
fC
B
N =
fB
A
fC
A
0
0
J =
fA
B
fB
C
fC
A
fA
A
fB
C
fC
B
0
J =
fA
B
fB
C
fC
A
fA
B
fB
A
fC
C
0
This implies a feedforward loop
fB
B
=0 and
fB
A
= 0
fB
B
=0 and
fC
B
= 0
This implies NO feedforward loop
1
2
2
1
fB= A+ g(C)B
fA
A
fB
A
fC
A
fA
B
fB
B
fC
B
fA
C
fB
C
fC
C
A*
B*
C*
+
fA
I
0
0
I =0
I
II
I
II
Either = = 0
or = 0
I
II
I
II
= = 0
I
II
= 0
implies
fB= A+ C +
fB= B( A+ C + )
No selfloop on B
Positive selfloop on B
Robustly achieving N=0 requires
N  =
fC
fB
B
fC
A
C*/C*
I/I
=
I
C*
fA
I
N
J
fB
B
= 0
A
Theory
NFBLB class
IFFLP class
BC
Figure 5. General Analysis for Adaptive Circuits
(A) Relevant equations. Thesteadystate outputchange DC*withrespect totheinputchangeDIcanbederived from thelinearized steadystateequations. Azero
adaptation error around a stable steady state requires a zero minor jNj and a nonzero determinant jJj < 0. There are two terms I and II in jNj, and jNj = 0 implies
either both terms are zero or they are equal but nonzero. We are only interested in robust adaptation, i.e., the cases where the condition leading to jNj = 0 holds
within a range of parameters and input values.
(B) NFBLB class of adaptive circuits (I = II = 0). In this category vfC/vA s 0, which means that there is always a link from node A to node C. (Otherwise,
there would be no direct or indirect path from A to C.) Then II = 0 implies that vfB/B is always zero. I = 0 implies that at least one of vfB/vA and v fC/vB is
zero. This condition implies that there is no feedforward loop in this category. In our enzymatic model vfB/vB = 0 can be robustly achieved either
by saturating the enzymes on the node B so that fBdoes not depend on B explicitly or by adding a positive selfloop on node B so that the dependence
of fB on B can be factored out. An example of the latter is when node B is regulated by itself positively and by C negatively, so that
768 Cell 138, 760–773, August 21, 2009 ª2009 Elsevier Inc.
Page 10
the input: I/I+DI, the steady state changes to A*+DA*, B*+DB*,
andC*+DC*,correspondingly.Theconditionsforperfectadapta
tion, DC* = 0, can then be obtained by analyzing the linearized
steadystate equations. We refer the reader to Supplemental
Experimental Procedures (section 6) for technical details and
only summarize the main results below (as schematically illus
trated in Figure 5).
These analyses again indicate that there are only two ways to
achieve robust perfect adaptation without finetuning of param
eters. The first requires one or more negative feedback loops but
occludes the simultaneous presence of feedforward loops in the
network (Figure 5B). In this category, the node B is required to
function as a feedback ‘‘buffer,’’ i.e., its rate change does not
depend directly on itself (vfB/vB = 0) at steady state. This implies
that fB either does not explicitly depend on B (fB = g(A,C))
or takes a form of fB= B 3 g(A,C) so that the steadystate condi
tion g(A*,C*) = 0 guarantees that vfB/vB = 0 at steady state. In
either case, within the MichaelisMenten formulation, the
steadystate condition for B, g(A*,C*) = 0, establishes a mathe
matical constraint aA* + gC* + d = 0 that is satisfied by A* and/
or C*, with a, g, and d constant. This equation plays a key role
in setting the steadystate value C* to be independent of the
input. All the minimal adaptation networks in the NFBLB class
discussed before are simple examples of this case. In particular,
the minimal network analyzed in Figure 3A is characterized by
fB= g(C) when both enzymes on B work in saturation. Hence,
the steadystate equation for the node B reduces to gC* + d = 0
(Equation 3). The case in which fB= B 3 g(A,C) corresponds to
adaptation networks in which node B has a positive selfloop.
The other way to achieve robust perfect adaptation requires
an incoherent feedforward loop, but in this case allowing
for other feedback loops in the network (Figure 5, panel C). In
this category, vfB/vB s 0 and the condition for robust perfect
adaptation implies a form of fBto be fB= aA + g(C)B. The
steadystate condition fB= 0 establishes a proportionality rela
tionship between B and A: B* = G(C*) A*, where G is a nonzero
function of C*. Thus, the node B here is required to function as
a ‘‘proportioner.’’ All minimal adaptation networks in the IFFLP
class are special cases of this category. For example, the
network in Figure 3B sets B* = constant 3 A* (Equation 6).
Therefore, the above analyses indicate that all robust adapta
tion networks should fall into one of these two categories, which
can be viewed as the generalization of the two classes of
the minimal topologies for adaptation. Indeed, we found that
all 395 robust adaptation networks can be classified based on
their membership of the broad NFBLB and IFFLP categories
(indicated by the two different colors in Figure 4B).
Design Table of Adaptation Circuits
Our results can be concisely summarized into a design table for
adaptation circuits, as exemplified in Figure 6. Overall, there
are two architectural classes for adaptation: NFBLB and IFFLP.
In each class, the minimal networks are sufficient for perfect
adaptation. These minimal networks also form the topological
core for the more complex adaptation networks that, with addi
tional characteristic motifs, can exhibit enhanced performance.
Figure 6 illustrates three examples in which such motifs can be
added to minimal networks to generate networks of increasing
complexity and increasing robustness (Q values).
Letusfirstfocusontheexampleshowninthemiddlecolumnof
Figure 6. On the top is a minimal network in the NFBLB class.
AddingoneC—jBlink(orequivalently,addingonemorenegative
feedback loop) to the minimal network results in a network with
two negative feedback loops that go through the control node
B that has a larger Q value. Note that no more negative feedback
loops that go through B can be added to the network without
creating an incoherent feedforward loop. Adding a link B/C
generates one more negative feedback loop that goes through
B but results in an IFFLP motif. This changes the network to the
IFFLP class—consequently, the adaptation mechanism and
the key regulations on B are changed (C—jB changed from satu
rated to linear).
In the example shown in the left column of Figure 6, we start
with one of the minimal networks in the NFBLB class that have
internode negative regulations on B. Adding a positive self
loop on B to this type of network significantly improves the
performance. One additional negative feedback loop further
increases the performance. When we arrive at the network
shown at the bottom of the left column, no negative feedback
loops that go through B can be added without resulting in an
incoherent feedforward loop.
In the last example (Figure 6, right column), a minimal IFFLP
network is layered with more and more negative feedback loops
to increase the Q value. A comprehensive design table with all
minimal networks and all their extensions that increase the
robustness, along with the analysis of their adaptation mecha
nisms, is provided in Supplemental Experimental Procedures,
section 12 and Figure S15. (The readers can simulate and visu
alize the behavior of these and other networks of their own
choice with an online applet at http://tang.ucsf.edu/applets/
Adaptation/Adaptation.html.)
DISCUSSION
Design Principles of Adaptation Circuits
Despite thegreatvariety ofpossible threenodeenzyme network
topologies, we found that there are only two core solutions
that achieve robust adaptation. The main functional feature of
the adaptation circuits is to maintain a steadystate output that
is independent of the input value. This task is accomplished by
a dedicated control node B that functions to establish different
mathematical relationships among the steadystate values of
the nodes that regulate it (see Supplemental Experimental
Procedures, section 7 for a comprehensive analysis) with the
fB=kBBB ð1 ? BÞ=ð1 ? B+KBBÞ ? k0
minantjJjcorrespondtodifferentfeedbackloopsascoloredinthefigure.Thus,thereshouldbeatleastone,butcanbetwo,negativefeedbackloopsinthiscategory.
(C)IFFLPclass(I=IIs0).Inthiscategory,noneofthefactorsinjNjarezero.ThisimpliesthepresenceofthelinkscoloredinthefigureandhenceaFFL.Thecondition
forjNj =0canberobustlysatisfiediftheFFLexertstwoopposingbutproportionalregulationsonC.TheproportionalityrelationshipcanbeestablishedbyfBtaking
the form shown in the figure.
CBC B=ðB+k0
CBÞzkBBB ? k0
CBC B=k0
CB= BðkBB? k0
CB=k0
CBCÞ, in the limits (1B) [ KBBand B ? K0CB. The terms in the deter
Cell 138, 760–773, August 21, 2009 ª2009 Elsevier Inc. 769
Page 11
goal of setting a constant steadystate output C*. Importantly,
these desired relationships necessary for perfect adaptation
are not achieved by finetuning any of the circuit’s parameters
but rather by the key regulations on the control node B
approaching the appropriate limits (saturation or linear) (Barkai
and Leibler, 1997). This is the reason behind the functional
robustness of the adaptation circuits of either major class.
Furthermore, the requirements central to perfect adaptation
are relatively independent from other properties. In particular,
the circuit’s sensitivity to the input change can be separately
tuned by changing the relative rates of the control node to those
of the other nodes.
Several authors have computationally investigated the circuit
architecture for adaptation (Levchenko and Iglesias, 2002;
Yang and Iglesias, 2006; Behar et al., 2007). In particular, Fran
c ¸ois and Siggia simulated the evolution of adaptation circuits
using fitness functions that combine the two features of adapta
tion we considered here: sensitivity and precision (Franc ¸ois and
Siggia, 2008). Starting from random gene networks, they found
that certain topologies emerge from evolution independent of
the details of the fitness function used. Their model circuits
have a mixture of regulations (enzymatic, transcriptional, dimer
ization, and degradation), and they did not enumerate but
focused on only a few adaptation circuits. Nonetheless, it is
very interesting to note that the adaptation architectures that
emerged in their study seem to also fall into the two general
classes we found here. Further studies are needed to systemat
ically investigate the general organization principles for the
adaptation circuits made of other (than enzymatic) or mixed
regulation types.
One additional NFBL One additional selfloop on BOne additional NFBL
Two additional NFBL
One additional NFBL
Q=72
Q=133
Q=27
Q=49
See supplement for
a comprehensive list
Q=16
Minimal network
Q=16
Q=72
Q=27
Unconstrained
Linear
Saturated
Q=26
Q=8
Q=8
Q=5
Combinations that improve the performance (Examples)
NFBLB
Q=5
Minimal network Minimal network
IFFLP
Parameter ranges for Km
Robustness (Q) of adaptation networks
(C*=const)
(A*=const)
(B*=const A*)
(B*=const A*/C*)
(B*=const A*/C*)
(C*=const)
(C*=const)
( A*+ C*= 0)
Figure 6. Design Table of Adaptation Networks
Two examples are shown on the left for the NFBLB class of adaptation networks, which require a core NFBLB motif with the node B functioning as a buffer. One
example is shown on the right for the IFFLP class, which require a core IFFLP motif with the node B functioning as a proportioner. The table is constructed by
adding more and more beneficial motifs to the minimal adaptation networks. The Q value (Robustness) of each network is shown underneath, along with the
mathematical relation the node B establishes.
770 Cell 138, 760–773, August 21, 2009 ª2009 Elsevier Inc.
Page 12
Biological Examples of Adaptation
A wellstudied biological example of perfect adaptation is in the
chemotaxis of E. coli (Barkai and Leibler, 1997; Yi et al., 2000)
(Figure 7). Intriguingly, we found that one of the minimal topolo
gies (NFBLB) we identified is equivalent to the BarkaiLeibler
model of perfect adaptation (Barkai and Leibler, 1997). In E. coli
the binding of the chemoattractant/repellant to the chemore
ceptor R and its methylation level M modulate the activity of
the histidine kinase CheA, which forms a complex with the
chemoreceptor R. CheA phosphorylates the response regulator
CheY, which in turn regulates the motor activity of the flagella.
The methylation level M of the receptor/CheA complexes is
determined by the activities of the methylase CheR and the
demethylase CheB. According to the BarkaiLeibler model,
CheR works at saturation with a constant methylation rate for
all receptor/CheA complexes, independent of the methylation
level M, whereas CheB binds only to the active receptor/CheA
complexes, resulting in a demethylation rate that is dependent
only on the system’s output (CheA activity). Therefore, the
network structure or topology of the E. coli chemotaxis is
very similar to one of the topologies we found (Figure 7), with
the buffer node B corresponding to the methylation level of the
chemoreceptors.
In our theoretical study of adaptation circuits with Michaelis
Menten kinetics, the IFFLP class consistently performs better
than the NFBLB class. However, there have so far not been clear
cases where IFFLP is implemented in any biological systems to
achieve good adaptation. Does IFFLP topology have some
intrinsic differences concerning adaptation from NFBLB that
are not captured by our study? Is it harder to implement in real
biological systems? Or, do we simply have to search more bio
logicalsystems?Acluemightbefoundwhenweaddcooperativ
ity in the MichaelisMenten kinetics (replacing ES/(S+K) with
ESn/(Sn+Kn) in the equations; see Supplemental Experimental
Procedures, section 10.2). A higher Hill coefficient n > 1 would
helpachievethetwosaturationconditionsnecessaryforadapta
tionintheNFBLBclassbutwouldhamperthelinearityrequiredto
establish the proportionality relationship necessary in the IFFLP
class. This requirement for noncooperative nodes in the IFFLP
class may effectively reduce its robustness and might be one of
the reasons behind the apparent scarceness of the IFFLP archi
tecture in natural adaptation systems. The FFL motifs, both
coherent and incoherent, are abundant in the transcriptional
networks of E. coli (ShenOrr et al., 2002) and S. cerevisiae
(Milo et al., 2002). These transcriptional FFL circuits can perform
a variety of functions (Alon, 2007), including pulse generation—
a function rather similar to adaptation. It is difficult to draw
conclusions from these findings, however, since preliminary
analysis (W.M., unpublished results) suggests that the require
ments for transcriptionbased adaptation networks may differ
from those of enzymebased adaptation networks.
Guiding Principles for Mapping, Modulating,
and Designing Biological Circuits
The general approach outlined here—to generate a function
topologymapconstructedfromapurelyfunctionalperspective—
could be applied to many different functions beyond adaptation.
The resulting functiontopology maps or design tables could
have broad usage. First, an increasing number of biological
network maps are being generated by various highthroughput
methods. Analyzing these complex networks with the guidance
of functiontopology maps may help identify the underlying
function of the networks or lead to testable functional hypoth
eses. Second, many biological systems that display a clear
function (e.g., adaptation) have an unclear mechanism or incom
plete network map. In these cases a functiontopology map can
CheY
CheR CheB
Input
B
A
Input
C
Output
Input
Output
Methylation
level
Receptor
complex
CheY
CheB
CheR
CheA
Receptor
NFBLB
Figure 7. The Network of Perfect Adaptation in E. coli Chemotaxis Belongs to the NFBLB Class of Adaptive Circuits
Left: the original networkin E. coli.Middle: the redrawnnetwork to highlight the role and the controlof the key node ‘‘Methylation Level.’’ Right: one of the minimal
adaptation networks in our study.
Cell 138, 760–773, August 21, 2009 ª2009 Elsevier Inc. 771
Page 13
provide important information about the possible network struc
ture and its key components, thushelping to design experiments
to fully elucidate the underlying network. Finally, there is growing
interest in learning how to modify cellular networks to generate
new behaviors or optimize existing ones. In medicine, an under
standing of how specific changes in architecture can shift a
system from one behavior to another could greatly aid in devel
oping more intelligent therapeutic strategies for treatment of
complex diseases like cancer. In the emerging field of synthetic
biology, this type of functiontopology design table could serve
as a manual providing different possible solutions to building
a biological circuit with a target set of behaviors.
EXPERIMENTAL PROCEDURES
Enumeration of ThreeNode Topologies
We considered all possible threenode network topologies (Figure 1B). There
are a total of nine directed links among the three nodes. Each link has three
possibilities: positive regulation, negative regulation, or no regulation. Thus
there are 39= 19,683 possible topologies. We let the input act on node A
and use as the output the active concentration of node C. There are 3,645
topologies that have no direct or indirect links from the input to the output.
We use all the remaining 16,038 topologies in our study.
Equations of the Circuit
We assume that each node (labeled as A, B, C) has a fixed concentration
(normalized to 1) but has two forms: active and inactive (here ‘‘A’’ represents
the concentration of active state, and ‘‘1A’’is the concentration of the inactive
state). The enzymatic regulation converts its target node between the two
forms. For example, a positive regulation of node B by node A as denoted by
alinkA/BwouldmeanthattheactiveAconvertsBfromitsinactivetoitsactive
form and would be modeled by the rate R(Binactive/Bactive) = kABA(1 ? B) /
[(1 ? B) + KAB], where A is the normalized concentration of the active form of
node A and 1 ? B the normalized concentrations of the inactive form of node
B. Likewise, A—jB implies that the active A catalyzes the reverse transition of
node B from its active to its inactive form, with a rate R(Bactive/Binactive) =
k0ABAB / (B+K0AB). When there are multiple regulations of the same sign on
a node, the effect is additive. For example, if node C is positively regulated
by node A and node B, R(Cinactive/Cactive) = kACA(1 ? C) / [(1 ? C) + KAC] +
kBCB(1 ? C) / [(1 ? C) + KBC]. We assume that the interconversion between
active and inactive forms of a node is reversible. Thus if a node i has only posi
tiveincominglinks,itisassumedthatthereisabackground(constitutive)deac
tivating enzyme Fiof a constant concentration (set to be 0.5) to catalyze the
reverse reaction. Similarly, a background activating enzyme Ei= 0.5 is added
for the nodes that have only negative incoming links. The rate equation for
a node (e.g., node B) takes the form:
dB
dt=
X
i
Xi$kXiB$
ð1 ? BÞ
ð1 ? BÞ+KXiB
?
X
i
Yi$k0
YiB$
B
B+k0
YiB
;
(7)
whereXi=A,B,C,EA,EB,orECaretheactivatingenzymes(positiveregulators)
of B and Yi= A, B, C, FA, FB, or FCare the deactivating enzymes (negative
regulators) of B. In the equation for node A, an input term is added to the right
handside of the equation: IkIA(1A)/((1A)+KIA). The number of parameters in
anetworkis np= 2nl+2,where nlis the number of links inthenetwork(including
links from the basal enzymes if present).
Functional Performance
For each network topology, 10,000 parameter sets were sampled uniformly in
logarithmic scale in the npdimensional parameter space, using the Latin
hypercube sampling method (Iman et al., 1980). The sampling ranges of the
parameters are k?0.110 and K?0.001100. A circuit refers to a network
topology with a particular choice of parameters. The typical output curve of
an adaptive circuit has two steadystate values O1and O2, corresponding to
the two input values I1and I2, respectively, and, in response to the input
change, hasatransientpulsewiththepeak valueOpeak(Figure 1A).Illbehaved
circuits are excluded from further analysis. They can be circuits with too small
steadystate values (<0.001) of the active or inactive enzymes, the ones
spending too much time to approach a steady state, or those with persistent
or too weakly underdamped oscillations (Supplemental Experimental Proce
dures, section 1). The remaining circuits were evaluated for their sensitivity
to input change and adaptation precision.
(1) Precision: the inverse of the adaptation error. The error E is defined as
the relative difference between the output steady states before and after the
input change.
P=E?1=
?jO2? O1j=O1
jI2? I1j=I1
??1
(8)
(2) Sensitivity: the largest transient relative change of the output divided by
the relative change of the input.
S=jOpeak? O1j=O1
jI2? I1j=I1
(9)
It is obvious that S R E since jOpeak? O1j R jO2? O1j. This implies that
log(S) + log(P) R 0.
The overall performance of a topology is measure by itsRobustness or the Q
value, defined here as the numberof times the topology is mappedto thehigh
sensitivity/highprecision region of the functional space (the green rectangle in
Figure 1D).
Clustering of Networks
There are nine possible links for a network. For every network, we assign
a value to each of the nine links: 1 for positive regulation, ?1 for negative regu
lation, and 0 for no regulation. Thus a network is represented by a sequence of
length 9. We define the distance between two networks as the Hamming
distance betweentheir sequences, that is,the number of regulations that differ
in the two networks. The distance matrix is then used for clustering, using the
MATLAB function clustergram.
SUPPLEMENTAL DATA
Supplemental Data include Supplemental Experimental Procedures, fifteen
figures, and a video summary and can be found with this article online at
http://www.cell.com/supplemental/S00928674(09)007120.
ACKNOWLEDGMENTS
We thank Caleb Bashor, Noah Helman, Morten Kloster, Ilya Nemenman, and
Eduardo Sontag for helpful discussions, Angi Chau, KaiYeung Lau, Thomas
Shimizu, and David Burkhardt for critical reading of the manuscript. W.M.
acknowledges the support from the Li Foundation. A.T. acknowledges the
support from the Sandler Family Supporting Foundation. This work was
supported in part by the National Science Foundation (DMR0804183) (C.T.),
Ministry of Science and Technology of China (C.T.), the National Natural
Science Foundation of China (C.T.), the Howard Hughes Medical Institute
(W.A.L.),thePackardFoundation(W.A.L.),theNIH(W.A.L.),and theNIHNano
medicine Development Centers (W.A.L.).
Received: December 9, 2008
Revised: March 29, 2009
Accepted: June 3, 2009
Published: August 20, 2009
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