Design of versatile biochemical switches that respond
to amplitude, duration, and spatial cues
Azi Lipshtat1, Gomathi Jayaraman, John Cijiang He, and Ravi Iyengar
Department of Pharmacology and Systems Therapeutics, Mount Sinai School of Medicine, New York, NY 10029
Edited by Robert J. Lefkowitz, Duke University Medical Center, Durham, NC, and approved November 11, 2009 (received for review August 3, 2009)
Cells often mount ultrasensitive (switch-like) responses to stimuli.
computationally studied the switching behavior of GTPases, and
found that this first-order kinetic system can show ultrasensitivity.
can yield switches that respond to signal amplitude or duration.
The three-component GTPase system is analogous to the physical
fermiongas.This analogy allowsfor an analytical understanding of
the functional capabilities of first-order ultrasensitive systems.
Experiments show amplitude- and time-dependent Rap GTPase
switching in response to Cannabinoid-1 receptor signal. This first-
order switch arises from relative reaction rates and the concen-
trations ratios of the activator and deactivator of Rap. First-order
ultrasensitivity is applicable to many systems where threshold for
transitionbetweenstatesis dependent on the duration,amplitude,
or location of a distal signal. We conclude that the emergence of
ultrasensitivity from coupled first-order reactions provides a
versatile mechanism for the design of biochemical switches.
between two functional states upon crossing a threshold. Often, a
regulator triggers state change. Near the threshold point, a small
duration, causes switching of the responding component. Such
underlying a steep response curve is the "zero-order ultra-
sensitivity" first proposed by Goldbeter and Koshland (2), who
showed that under zero-order conditions—i.e., when one or more
of the enzymes in a coupled system are saturated—the transition
between the active and inactive conformations exhibits high sen-
sitivity to the concentration ratio of the enzymes. Other mecha-
nisms that yield ultrasensitivity include cooperativity, multistep
regulation, and stoichiometric inhibitors (3–5). Positive feedback
loops play an important role in producing switching behavior (4)
and are often considered necessary for bistability (6, 7). Mecha-
nisms that dependon loops requirecomplexnetwork organization
such as topological motifs in addition to the enzymatic activity to
produce switches. However, switching behavior is observed in the
absence of loops, and the design principles for such switches are
poorly understood. We have used analytical and numerical
methods as well as experiments to describe first-order ultra-
that responds to both duration and concentration of stimulus.
fficient regulation of intracellular processes benefits from "all
or none" response (1), where a cellular component switches
Ultrasensitivity in GTPases
Small GTPases can function as molecular switches in varied
cellular processes including signaling networks (8). Their con-
version from GDP (inactive) to GTP (active) conformations
promotes interaction with downstream effectors to propagate
information flow. Rapid responses of GTPases to incoming
regulation can turn downstream pathways on and off (9–11).
Thus, GTPases play an essential role in controlling many cellular
responses (10–13). Examples include cellular proliferation by
Ras (14), neurite growth by Rap1 (15), and nucleocytoplasmic
transport by Ran (16). Because of their central role in numerous
pathways, small GTPases (GTPases) have been studied exten-
sively, both experimentally and computationally (1, 13, 17, 18).
For many GTPases, the intrinsic cycle between the GDP-
bound state and GTP-bound state is very slow. Cycling rates are
greatly enhanced by guanine nucleotide exchange factors (GEFs)
and GTPase activating proteins (GAPs) (19). Signaling pathways
that use heterotrimeric G proteins or small GTPases show both
graded and switch-like responses. What mechanisms underlie the
switching behavior? Zero-order ultrasensitivity can be obtained
by low enzyme (GEF or GAP) to substrate (GTPase) ratio.
However, experimental observations and estimations show that
this is not always the case (8, 20). Although GEF and GAP
concentrations are lower than the GTPases levels, the difference
is not sufficient. When multiple GEFs or GAPs are simulta-
neously active, the effective concentrations of the regulators can
be similar to that of the GTPase, resulting in a first-order system.
How do first-order reactions yield ultrasensitive response, and
why don't we always observe this response?
GEFs and GAPs are controlled by receptor-regulated intra-
cellular events (9, 21). Such regulation is critical for normal
physiology. Abnormal regulation of GEFs or GAPs has been
implicated in cancer (22), viral and bacterial pathogenesis (23),
vascularization defects during development (24), and mental
retardation (25). Often, regulation of either a GAP or a GEF is
sufficient for GTPase activation (9, 21, 26). We explored the
relationship between different levels of GEF and GAP activity by
numerically simulating receptor-regulated Rap activation, using
an ordinary differential equations model. The signaling network
(Fig. S1) includes our prior experimental data (15) and the
regulation of Rap by cAMP (27). Details of the simulations are
described in SI Text, and the models are available at the Virtual
Cell site. In Fig. 1, we show the formation of GTP-bound Rap in
response to signals from activated α2-adrenergic (α2R) and
β-adrenergic (βAR) receptors. The α2R signal leads to degra-
dation of Rap GAP* whereas the βAR signal activates the GEF.
We observe an abrupt transition from a low activity state to high
activity as the α2R signal crosses a threshold. Similar behavior is
observed when signal duration is lengthened while the signal
amplitude is fixed (Fig. 1C). Thus, level of active Rap is very
sensitive to the signal amplitude and duration, with distinct
subthreshold and above-threshold responses. As opposed to the
high sensitivity to α2R activation, the response to βAR stim-
ulation is slightly slower than a regular Michaelian curve (Fig.
1D). The amplitude of the βAR stimulation affect the Rap
activation level, but the sensitivity to the exact stimulus charac-
teristics is low. However, the duration of the signal produces a
Author contributions: A.L conducted all the theoretical analysis; A.L., J.C.H., and R.I. de-
signed research; G.J. and J.C.H. performed experiments; and A.L. and R.I. wrote the paper.
The authors declare no conflict of interest.
This article is a PNAS Direct Submission.
1To whom correspondence should be addressed at: Department of Pharmacology and
Systems Therapeutics, Mount Sinai School of Medicine, One Gustave L. Levy Place, Box
1215, New York, NY 10029. E-mail: firstname.lastname@example.org.
This article contains supporting information online at www.pnas.org/cgi/content/full/
| January 19, 2010
| vol. 107
| no. 3
switching response (Fig. 1E). These simulations indicate that
both GEF* and GAP* concentrations integrated over time
regulate GTPase activation.
These findings raise several questions: (i) What is the mech-
anism that enables the switching in response to α2R signals
amplitude or duration? (ii) When do GTPases display graded
responses? (iii) Is this type of switching a general mechanism in
many biochemical systems? (iv) Is there a design advantage of
such a system with seemingly redundant regulation of both the
GEF and GAP, because mutations in either component often
lead to a disease state?
To answer these questions we have studied the system ana-
lytically. Without restricting the concentrations, we assumed that
the reactions of the GTPase cycle are in the mass action regime.
Thus, the activation and deactivation terms of the GTPase are
proportional to the concentrations of active GEF and GAP
(GEF* and GAP*), respectively. These terms should be also
linear with respect to the (instantaneous) concentrations of the
inactive and active forms of the GTPase. Under saturation
conditions (zero-order reactions), the (de)activation rates may
be independent of substrate concentration, and so produce
ultrasensitivity (2). We limit our analysis to cases of first-order
reactions. Because many cellular components are found in a
similar range of concentrations, the mass action assumption is
valid for many intracellular systems (28). For these systems, the
GTPase cycle is governed by the following equations:
dt½SG?? ¼ kon½GEF??½SG?−koff½GAP??½SG?? ¼ −d
½SG?? þ ½SG? ¼ ½SGT?;
where [SG*] and [SG] are the concentrations of the active and
inactive forms of the GTPase, and [SGT] is its total concentra-
tion. Because the enzymatic activation and deactivation rates are
much faster than the changes in the GAP* and GEF* concen-
trations, one can assume a steady state solution with respect to
the GTPase activation level. Under these conditions, the steady
state activation level can be written as
1 þ kGAP=kGEF;
where kGEF= kon[GEF*] and kGAP= koff[GAP*] are the effec-
tive reaction rates. The activity of GEFs and GAPs may be pos-
itively or negatively regulated. We consider the case of negative
regulation of the GAP* by a signal X (receptor signal), which
targets the GAP* for irreversible deactivation (by degradation or
any other first-order reaction such as sequestration) (15, 21)
(Fig. 2A). The RasGAP neurofibromin undergoes degradation
upon treatment with various growth factors (29), and p120 Ras-
GAP is degraded by caspase (30). The signal triggers the degra-
dation of GAP* either directly or through a reaction cascade.
The analysis is valid as long as the effective rate of GAP*
decrease is proportional to the signal amplitude and depends
on the GAP* concentration [GAP*] (see calculation in SI Text).
Although we assume that the GAP deactivation rate is propor-
tional to [GAP*], in SI Text we show that this assumption is not
necessary, and that ultrasensitivity can be achieved from any
nonzero positive dependence on [GAP*]. Here, we present
the standard case of mass action law. In this regime, the deacti-
vation rate (which is the time derivative of [GAP*]) is propor-
tional to [GAP*]. As a result, applying a stimulus X for a
duration τ causes decay in active [GAP*] that is exponential
with respect to both time and X. By the end of the signal dura-
tion, GAP* has a new steady state concentration, namely
[GAP*]t=[GAP*]0exp(−k[X]τ) for times t > τ (Fig. 2BI). (In SI
Text we extend this derivation cases where the GAP* decay is
slower than exponential, such as in reversible deactivation.)
Because the hydrolysis rate of the GTPase SG* is linearly
dependent on the GAP* concentration, this rate also exponen-
tially decreases with the signal amplitude and time (SI Text,
section 1). With decreasing levels of GAP*, the ratio of SG*/
SGTincreases (Fig. 2BII). Plugging the effect of the signal into
the activation level of the GTPase results in a signal-dependent
switch of the GTPase from a GDP-bound to GTP-bound state
(Fig. 2BIII). Regardless of the GTPase concentration, the frac-
tional activation level as function of the signal duration (for a
fixed amplitude) is given by
1 þ A0expð−b½τ?Þ;
where A0 is the initial kGAP/kGEF ratio (kGAP and kGEF, as
defined above, incorporate both concentrations of the active
GEF* or GAP* and respective kinetic rates) and b is the product
of the signal amplitude [X] and the effective GAP* deactivation
(or degradation) rate. For a broad range of parameters, this
function is ultrasensitive with respect to τ. This way, using mass
action reactions only, a regulated GTPase can act as a time-
dependent switch. Similarly, the dependence of activation on
the signal amplitude [X] (with fixed duration) is given by
1 þ A0expð−b½X?Þ;
where in this case, b is the product of the signal duration τ and
the effective GAP* deactivation rate. In both cases, the param-
eter b is a measure of the signal impact. (For complete derivation
please see SI, section 1.) The activation level of GTPase depends
on the extracellular stimulus characteristics ([X] or τ) through a
logistic function, also known as the Fermi–Dirac distribution.
The dependence of the activation curve (Eqs. 3A and 3B) on
A0and b resembles the role of Fermi energy and the inverse
temperature β = (kBT)−1in the Fermi–Dirac distribution. High
temperatures slow the transition (as a function of energy) and as
Clonidine Signal (sec.)
Isoproterenol Signal (sec.)
the Rap1 pathways was performed by using Virtual Cell (see Fig. S1 for the
pathways). α2AR were stimulated for fixed duration and with various
amplitudes, evenly distributed on a logarithmic scale. Then, Rap was acti-
vated by a βAR stimulus. (A) The activation level is clustered into two groups
of low and high activation. (B and C) α2AR-stimulated steady state Rap
activity is ultrasensitive with respect to concentration and duration. (D) βAR
stimulation of Rap is subsensitive with respect to signal amplitude. (E) βAR
activation of Rap is ultrasensitive with respect to signal duration.
Numerical simulations of Rap regulation. A detailed simulation of
| www.pnas.org/cgi/doi/10.1073/pnas.0908647107Lipshtat et al.
the temperature decreases, the curve becomes steeper. At T = 0
the Fermi–Dirac function becomes a step function. Is this sim-
ilarity between the GTPase cycle and Fermi–Dirac system a
coincidence or can the analogy provide some insight into the
principles underlying the design of the three-component GTPase
system? Mapping of biological questions onto known physical
systems has proven useful in several other cases (31).
Fermions are particles that obey the Pauli Exclusion Principle.
No two fermions can occupy the same quantum state simulta-
neously. Thus, a quantum state can be either empty or occupied
by a single particle. Fermi–Dirac distribution determines the
probability of a given quantum state to be occupied by a fermion,
as a function of energy level and temperature. In the GTPase
system, the GTPase molecules are associated with molecules
(GTP or GDP), where each copy of GTPase can be found in one
of two possible states. Although the GTPase cycle also includes
intermediate transition states (32), the two-state model is a good
approximation that is widely used in biochemistry (21). Activa-
tion level of a GTPase is the probability of finding a GTPase
molecule in the GTP-bound state and hence is analogous to the
occupation probability of the quantum states. The occupation
probability exhibits an ultrasensitive curve with respect to energy.
Energy is introduced by the Boltzmann factor exp(−E/kBT),
which is the relative probability to find a particle in a given state.
Because the probability of finding a GTPase in a GTP-bound
state is exponentially dependent on the signal characteristics ([X]
or τ), the signal in the biological system is analogous to energy in
the physical system. As such, the GTPase activation follows
Fermi–Dirac distribution as function of τ or [X] (Table 1). This
analogy between the two unrelated systems opens the way for
adaptation and adoption of known results from one system to the
other. For example, in statistical physics there are particles that
are not subject to the exclusion principle. These are bosons, and
many of them can occupy the same quantum state. Because of
this difference between bosons and fermions, bosons follow
Bose–Einstein statistics rather than Fermi–Dirac statistics. The
biological analogy of bosons is a system where many molecules
can bind to a single complex. This does not happen for the small
GTPases, but it does occur for other proteins with nucleotide
triphosphatase activity involved in polymerization processes
(such as actin, an ATPase, and tubulin, a GTPase), where
monomers can bind to and detach from polymers. There is no
theoretical limit on the number of monomers that are associated
with a single polymer. Thus, without any further calculation, one
may expect that the length distribution of polymers (analogous to
occupation distribution of bosons) would follow Bose–Einstein
distribution. Detailed calculations show that this is indeed the
case (SI Text, section 4). Thus, by comparing the abstract struc-
tures of these two systems, one can predict the behavior of the
biological system under various conditions, based on the
knowledge we already have about physical systems and their
properties. Furthermore, this analogy shows that the switching
mechanism presented here is based on the general architecture
of the system, rather than on any particular properties of
GTPases. Thus, first-order ultrasensitivity can be applicable to
many different cellular systems.
Ultrasensitivity in Space
The first-order ultrasensitivity mechanism is mathematically
based on the exponential dependence of GAP* concentration on
the stimulus X or duration τ. The comparison with Fermi–Dirac
distribution implies that any two-state system with transition
rates that are exponentially dependent on an input variable can
be ultrasensitive with respect to the value of that variable. This
observation opens the way for a broad range of applications,
including spatial localization.
Spatial gradients may provide exponential dependence and
form intracellular regions of high activity (microdomain). If
there is a point source at one side of the cell, and one component
spreads out by diffusion, then the concentration of the compo-
nent decreases exponentially with the distance x from the source.
The same analysis that has been used for the exponentially
decreasing GAP* with respect to the signal applies here as well,
with distance x replacing the signal amplitude X. A sharp change
in the activation level of the regulated GTPase can be predicted
to occur at one particular spatial location. This activation could
then initiate further local stimulation of downstream effectors.
Fig. 2B, which depicts the construction of ultrasensitive response
through GAP regulation, can be also used to illustrate spatial
switching. Instead of regulating the GAP* concentration as a
function of the signal, Fig. 2BI can be viewed as a spatial dis-
tribution of GAP*. If the gradient is exponential, and the
dependence of SG* on GAP* is as shown in Fig. 2BII, then the
overall spatial distribution of the active form SG* is the same as
shown in Fig. 2BIII, where the x axis denotes the spatial coor-
dinate rather than the signal amplitude.
Location-dependent ultrasensitivity implies formation of
multiple biochemical compartments without physical boundaries.
The differences between adjacent compartments can be sig-
nificant, whereas within each compartment there is no spatial
variation. How can a single continuous gradient form a multi-
compartmental pattern? The first-order ultrasensitivity provides
of the small GTPase cycle and its regulation. (B) Illustration of the mathe-
matical reasoning ultrasensitivity. First-order GAP* deactivation yields
exponential dependence of the GAP on the signal (BI). The dependence of
the GTPase activation on the GAP* (BII) is hyperbolic. However, dependence
of the activation level on the upstream signal (BIII) is ultrasensitive. The
straight lines show how different signals (one order of magnitude apart
each other) are clustered into two groups of high and low activation level.
Ultrasensitive response in GTPase activation. (A) Schematic diagram
Table 1.Analogy between GTPase activation and Fermion gas
Fermion gasGTPase (small G)
A state can be either empty or
occupied by a single particle
Transition rate depends on
Boltzmann factor exp(−E/kBT)
GTP (or GDP) molecules
Small GTPase molecules
GTPase can be either GTP or
Transition rate depends on
GAP/GEF ratio [proportional
to exp(-k[X]τ) or to exp(-kx)]
Switching point depends on initial
value of GAP/GEF (Eq. 3)
Switching point depends on
Fermi energy, EF
Lipshtat et al.PNAS
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a simple mechanism, based on the observation that not only does
each GTPase potentially have several GAPs and GEFs, but there
are also GAPs and GEFs that regulate several GTPases (33, 34).
Such a mix-and-match configuration is a powerful design feature.
Consider an exponential spatial gradient of a GAP* that deac-
tivates two independent GTPases (Fig. 3A). The change in
GAP* activity is shown as a function of distance x (Fig. 3BI).
Because GTP hydrolysis rates of different GTPases are different
even with the same GAP (35), and each GTPase has its own
GEF, the switching points of the GTPases (which are functions
of the kGAP/kGEF ratio) are distinct. The multiple threshold
points yield different spatial distribution of SG* for the two
GTPases. Between any two adjacent threshold points, one type
of GTPase (e.g., SG1) is above its threshold, and thus activated,
and the other is not. This results in the formation of three dis-
tinct compartments with varying levels of activated SG1 or
SG2 (Fig. 3BIII). This way, the single continuous gradient of
GAP* can drive the formation a discrete set of compartments
defined by the different activity state of GTPases within the
Signal Duration-Dependent Ultrasensitivity
The analysis above shows that GTPases respond rapidly when
the total signal impact crosses a threshold, either due to long
duration or high amplitude. This interchangeability of the
amplitude and duration of the signal is a major design advantage
that allows the system to function as an information integrator.
In Fig. 4, we show that time-dependent switching occurs for a
broad range of parameters. In time-dependent switch, the x axis
is the signal duration and the parameter b includes kinetic rate
and the fixed amplitude of the signal, whereas in the amplitude-
dependent switch the x axis represents the signal strength and b is
the product of the kinetic rate and the fixed signal duration. So
the curves for signal amplitude-dependent switches are also the
same as those of the signal duration-dependent switches with the
difference in the interpretation of the parameters. In all cases,
the GTPase switches from low activity in the case of a brief (or
weak) stimulus, to high activation in the case of a sustained (or
strong) signal. The steepness of this transition is determined by
the value of A0: low values of A0yield a smooth and shallow
transition, whereas high values lead to an abrupt and steep
change. For values of A0that are about 10 or higher, the signal
dependence is of the “all or none” nature, or ultrasensitive.
Changing the value of b (for example by changing the duration of
signal) does not affect the steepness of the curve but shifts the
switching point. Two parameters govern the switch dynamics: the
initial ratio A0of GEF* and GAP* activities, and the signal
impact b (Eq. 3). Switching behavior is characterized by a close-
to-zero derivative (of activation with respect to signal) away from
the switching point and positive derivative in close proximity to
that point (SI Text and Fig. S2). Thus, we define the switching
point as the signal amplitude at which the derivative d[SG*]/d[X]
gets its maximum. Simple analysis shows that the critical signal
for switching is [X]switch= (1/b)lnA0(see derivation in SI Text).
This is also the signal strength at which ½SG??=½SGT? ¼1
ogous to Fermi energy), which is another reason to view this
value as the switching point. For low values of A0, there is high
SG activity even before stimulation, and thus there is no maxima
point for the derivative and no switching dynamic is observed
(Fig. S2). However, for larger values, the dynamics are ultra-
sensitive to both signal duration and signal strength, making such
a system a versatile switch.
Like the activation level, the switching point [X]switch= (1/b)
lnA0depends on the same two parameters A0and b. Whereas b
reflects the regulation of one enzyme (e.g., GAP), the ratio A0is
a function of the initial concentrations of both GAP* and GEF*,
and thus also represents signals regulating GEF. One signal can
set the ratio A0and shift the switching point. Then, the other
signal switches the system between the two states at the desired
point. Thus the switching point is tuneable. This is an advanta-
geous design for two reasons: Separate regulation of GAP* and
GEF* provides a higher level of flexibility. This design also
enables switching over a broad range of GAP* and GEF* con-
centrations. Because the control parameter is the ratio of GEF*
to GAP*, rather than a concentration of either component,
switching is not limited to zero-order regime and can be achieved
at higher concentrations as well (Fig. S3). Broad applicability of
this mechanism is illustrated in the CAM-kinase-II system, where
a shift of the switching point due to changes in phosphatase
activity has observed (36). Such a shift is easily explained by the
The standard measure of ultrasensitivity is the Hill coefficient
nH(37, 38). Values larger than 1 indicate ultrasensitivity (6).
Following Goldbeter and Koshland, the effective Hill coefficient
is defined by
distance x (A.U.)
distance x (A.U.)
two GTPases are regulated by a common GAP. (B) Simulation of spatial
domain formation. (BI) Due to diffusion mechanism, the spatial distribution
of GAP is exponential. (BII) Because each GTPase has its own GEF, the acti-
vation level of each GTPase has a different dependence on the local GAP
concentration. (BIII) This difference yields different switching points, result-
ing in distinct compartments.
Spatial properties of coupled switches. (A) A general scheme where
[SG*] / [SGtotal]
Signal Duration (min)
Steady-state activation level of the GTPase are plotted as a function of signal
duration for A0(kGAP/kGEF) = 1, 10, or 100 (Left, Middle, and Right, respec-
tively), and b = 0.01, 0.1, and 1 (Bottom, Middle, and Top). All curves are
semilog plots, and two examples in linear scale are in Insets for [τ] < 10.
Simulations of time-dependent switch under various conditions.
| www.pnas.org/cgi/doi/10.1073/pnas.0908647107 Lipshtat et al.
where Spis the stimulus (i.e., duration or amplitude of signal)
required for p percent activation (2). In our case, the Hill coef-
ficient is determined exclusively by the value of A0(see deriva-
tion in SI Text) and is given by
nHis defined for A0> 9 because for lower values activation level
is above 10% even without any stimulus (Fig. 4, left column). For
A0> 9.5, we obtain ultrasensitivity, namely nH> 1. For A0= 100,
the Hill coefficient is >4 (Fig. S4). When the deactivation is not
in the first-order regime, the values of A0that are required for
ultrasensitivity may be different. This analysis shows that
whereas the switching point is a function of both parameters
and can be regulated independently, the steepness depends on
the initial ratio A0only and not on b. Changing the GAP* deac-
tivation rate will affect the switching point but not the steepness.
Furthermore, the analysis is based only on linear dependence on
deactivation steps without any assumptions regarding specific
biochemical mechanisms. The signal need not be chemical.
The same result is obtained when X represents physical force,
UV radiation, or temperature change. As long as the deactiva-
tion rate is proportional to the strength of the signal, this mech-
anism can function.
The analysis presented here is independent of concentrations
of the substrate (e.g., GTPase) and the regulatory enzymes (e.g.,
GAP* or GEF*); rather, the steepness determining parameter is
the ratio A0. In cases where the upstream pathways regulating
the two direct regulators are coupled, perturbation in one
pathway can be compensated for by a reciprocal perturbation in
the other. This is a robust design for a switch because the
important variable (the ratio) remains unaffected even under
conditions where the actual concentrations of the components
change. Note that the system organization enables ultrasensitivity
but does not guarantee it. The kinetic values need to satisfy cer-
tain conditions (such as A0> 9.5 in the GAP deactivation exam-
ple) to obtain ultrasensitivity, otherwise the system of reactions
will show a graded response.
Numerical Simulations and Experimental Tests
We numerically simulated the activation of Rap by an agonist
(clonidine) of the α2AR. The signal activates the Gothat targets
RapGAP (GAP*) for degradation. The βAR, activated by iso-
proterenol, stimulates the production of cAMP to determine the
levels of GEF* that activate Rap (SI Text and Fig S1). Numerical
simulations show how, despite the multistep regulation of clo-
nidine to GAP, an ultrasensitive activation of Rap is seen in
response to receptor ligand concentration (signal amplitude) and
the duration of receptor activation (SI Text, Section 5, and Fig.
S5). As predicted, the steepness changes with the initial GAP* to
GEF* ratio. Because the Hill coefficient is a function of A0only,
it is expected that the same steepness will be observed for both
dose-dependent and duration-dependent switches. For a GEF*
concentration of 2 × 10−4μM, we obtained a Hill coefficient of
3.5 for the amplitude-dependent activation and 3.6 for the
duration-dependent activation. Increasing the GEF* concen-
tration to 3 × 10−4μM decreased the Hill coefficient to 2.6 or 2.7
for amplitude- and duration-dependent switches, respectively.
These analytical and computational studies predict that
receptor-regulated Rap should behave as a time-dependent and
agonist concentration-dependent switch. HU-210 acting through
the CB1 receptor that also couples to Gois known to initiate
degradation of RapGAP, which can increase Rap1 activity (15).
Neuro 2A cells were treated for a fixed time with varying con-
centrations of HU-210, or for varying times with fixed concen-
tration, and Rap1 activity was measured. Both dose-dependent
and duration-dependent ultrasensitivity were observed (Fig. 5).
The numerical simulations and the experiments demonstrate the
interchangeability of dose and duration predicted by our analysis
and the versatile nature of this switch.
Because the control parameter A0depends on the GAP*/
GEF* ratio, the effect of GEF activation is mathematically
analogous to GAP deactivation. However, in practice there is an
important difference: deactivation rate is typically proportional
to the active form concentration (first-order reaction). In con-
trast, a typical activation process is negatively dependent on the
concentration of the active form—the more active the form, the
slower the reaction. This dependence yields a subzero-order
reaction. This is the reason why in our simulations of the Rap
system subsensitivity is observed in response to isoproterenol
stimulation (Fig. 1D). However, GEF activation can yield a time-
response dynamic switching. Under low GAP* conditions,
GTPase activation by GEF* stimulation is very rapid. Thus, well
before the system approaches steady state it is sensitive to signal
duration. Whereas long enough signals evoke almost full acti-
vation, short signals cannot yield significant GTPase activity. In
Fig. 1E, we showed that active Rap at the end of the iso-
proterenol signal is ultrasensitive to the signal duration. Signal
amplitude-dependent ultrasensitivity requires a nonlinear acti-
vation of GEF. The source of nonlinearity can be at any place
along the pathway and not necessarily directly connected to the
GEF. The sources of nonlinearity for the various regulatory
processes are summarized in Table S1. A common mechanism
for such nonlinearity in activating pathways is the removal of an
inhibitor. The GDP dissociation inhibitors (GDIs) are known to
bind Rho family GTPases and block GTPase activation (39).
Dissociation of a GDI from Rac is regulated by phosphorylation
by PAK kinase (39, 40), allowing receptor signals to inactivate
the GDI and enabling GEF* to activate the Rac GTPase. In this
Signal Duration (min.)
Rap−GTP (fold change)
activated for different times by the cannabinoid receptor-1 agonist HU-210
(A) or different amounts of HU-210 (B). The shaded lines are the results of
individual experiments, and the black line is their average. Hill coefficients
were calculated as described in SI Text.
Experimentally observed ultrasensitivity. In Neuro 2A cells, Rap1 was
Rho−family GTPase activation (%)
by GDI phosphorylation was simulated. Activation exhibits ultrasensitive
response with respect to level of protein kinase signal.
Ultrasensitivity by activation of GEF. Activation of Rho-family GTPase
Lipshtat et al. PNAS
| January 19, 2010
| vol. 107
| no. 3
case, the regulation of kGEF= kon[GEF*] is done by de facto Download full-text
controlling kon rather than the GEF* concentration. “All-or-
none” response of Rho-family GTPases has been observed (41).
To examine whether phosphorylation of GDIs can produce
ultrasensitivity, we implemented a published model of the Rho
GTPase, including its intermediate complexes (18). We added to
this model active and inactive GDIs, and reversibly deactivated the
GDIs by phosphorylation. Under first-order reaction assumption,
activation of the Rho-family GTPase level as a function of the
protein kinase concentration (hence, receptor signal amplitude)
exhibits ultrasensitivity (Fig. 6). Introducing cooperativity would
increase the sensitivity even further.This,ofcourse,depends onthe
order ultrasensitivity mechanism can apply to both the activating
and deactivating arms of a three-component G protein system.
We have elucidated the mathematical basis for the binary
response at the systems level, within a graded system of first-order
reactions. Our theoretical framework shows that, with the
“appropriate” kinetic parameters, various systems can display
ultrasensitive responses, whereas the same system of reactions
exhibits graded response with other set of kinetic parameters (see
comprehensive analysis in SI Text). For GTPases, our analysis
shows why ultrasensitive behaviors are experimentally observed in
varied systems (Table S3 and Fig. S6). This general theory for the
design of a flexible switch explains not only the sharp response, but
also the design advantage of the double regulation and its use in
tuning the switching point. Such tuning is likely to be biologically
important. Tuneable switches make cellular responses robust
because they are able to maintain their switching behavior under
dynamic conditions. This design advantage applies not only to a
single switch, but also to coupled switching processes that can
create functional spatial compartments in the absence of physical
barriers. The ultrasensitivity mechanism described here does not
depend on any particular biochemical mechanism or on substrate
concentration. It applies equally to cases of high or low enzyme-to-
substrate ratio, as long as there is no saturation. This general
applicability across a range of concentrations for the cellular
response system and the interconvertibility between chemical or
physical signal and time (i.e., signal amplitude and signal duration)
make the first-order switching mechanism a versatile and flexible
Materials and Methods
Neuro-2A cells (ATCC) were cultured and treated and Rap activity measured
as described in SI Text. Numerical simulation were done by using Virtual Cell
(42). More details can be found in SI Text.
ACKNOWLEDGMENTS. Anthony Hasseldine designed preliminary simula-
tions of the Rap system. We thank Drs. Eric Sobie and Bob Blitzer for a
critical reading of the manuscript, and Dr. Walter Kolch for valuable
discussions that led to the analyses of coupled GTPase systems. This work
was supported by National Institutes of Health Grants GM54508 and
P50GM071558, the Systems Biology Center grant. Virtual Cell is supported
by National Institutes of Health Grant P41RR013186 from the National
Center for Research Resources.
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| www.pnas.org/cgi/doi/10.1073/pnas.0908647107 Lipshtat et al.