Cells often mount ultrasensitive (switch-like) responses to stimuli. The design principles underlying many switches are not known. We computationally studied the switching behavior of GTPases, and found that this first-order kinetic system can show ultrasensitivity. Analytical solutions indicate that ultrasensitive first-order reactions can yield switches that respond to signal amplitude or duration. The three-component GTPase system is analogous to the physical fermion gas. This analogy allows for 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 concentrations ratios of the activator and deactivator of Rap. First-order ultrasensitivity is applicable to many systems where threshold for transition between states is 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.
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Efficient regulation of intracellular processes benefits from "all or none" response (1), where a cellular component switches between two functional states upon crossing a threshold. Often, a regulator triggers state change. Near the threshold point, a small change in one parameter, such as regulator concentration or signal duration, causes switching of the responding component. Such responses are called ultrasensitive (2). A widely known mechanism underlying a steep response curve is the "zero-order ultrasensitivity" 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 sensitivity to the concentration ratio of the enzymes. Other mechanisms 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). Mechanisms that depend on loops require complex network 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 ultrasensitivity as the basis for a versatile design of a biochemical switch that responds to both duration and concentration of stimulus.
Ultrasensitivity in GTPases
Small GTPases can function as molecular switches in varied cellular processes including signaling networks (8). Their conversion 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 extensively, 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 simultaneously 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 intracellular 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 degradation 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 stimulation 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 characteristics is low. However, the duration of the signal produces a switching response (Fig. 1E). These simulations indicate that both GEF* and GAP* concentrations integrated over time regulate GTPase activation.
Fig. 1.
Numerical simulations of Rap regulation. A detailed simulation of 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 activated 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.
These findings raise several questions: (i) What is the mechanism 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 analytically. 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:
where [SG*] and [SG] are the concentrations of the active and inactive forms of the GTPase, and [SGT] is its total concentration. Because the enzymatic activation and deactivation rates are much faster than the changes in the GAP* and GEF* concentrations, 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
where kGEF = kon[GEF*] and kGAP = koff[GAP*] are the effective reaction rates. The activity of GEFs and GAPs may be positively 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 RasGAP is degraded by caspase (30). The signal triggers the degradation 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 proportional 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 deactivation rate (which is the time derivative of [GAP*]) is proportional to [GAP*]. As a result, applying a stimulus X for a duration causes decay in active [GAP*] that is exponential with respect to both time and X. By the end of the signal duration, 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 exponentially decreases with the signal amplitude and time (SI Text, section 1). With decreasing levels of GAP*, the ratio of SG*/SGT increases (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 fractional activation level as function of the signal duration (for a fixed amplitude) is given by
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 . 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
where in this case, b is the product of the signal duration τ and the effective GAP* deactivation rate. In both cases, the parameter 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 A0 and b resembles the role of Fermi energy and the inverse temperature β = (kBT)−1 in the Fermi–Dirac distribution. High temperatures slow the transition (as a function of energy) and as the temperature decreases, the curve becomes steeper. At T = 0 the Fermi–Dirac function becomes a step function. Is this similarity 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).
Fig. 2.
Ultrasensitive response in GTPase activation. (A) Schematic diagram of the small GTPase cycle and its regulation. (B) Illustration of the mathematical 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.
Fermions are particles that obey the Pauli Exclusion Principle. No two fermions can occupy the same quantum state simultaneously. 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). Activation 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 ), 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 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 structures 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.
Table 1.
Analogy between GTPase activation and Fermion gas
Fermion gas
GTPase (small G)
Particles
GTP (or GDP) molecules
Quantum states
Small GTPase molecules
A state can be either empty or occupied by a single particle
GTPase can be either GTP or GDP bound
Transition rate depends on Boltzmann factor exp(−E/kBT)
Transition rate depends on GAP/GEF ratio [proportional to exp(-k[X]τ) or to exp(-kx)]
Switching point depends on Fermi energy, EF
Switching point depends on initial value of GAP/GEF (Eq. 3)
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 component 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 distribution 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 coordinate rather than the signal amplitude.
Location-dependent ultrasensitivity implies formation of multiple biochemical compartments without physical boundaries. The differences between adjacent compartments can be significant, whereas within each compartment there is no spatial variation. How can a single continuous gradient form a multicompartmental pattern? The first-order ultrasensitivity provides 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 deactivates 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 distinct 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 compartment.
Fig. 3.
Spatial properties of coupled switches. (A) A general scheme where 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 activation level of each GTPase has a different dependence on the local GAP concentration. (BIII) This difference yields different switching points, resulting in distinct compartments.
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 A0 yield a smooth and shallow transition, whereas high values lead to an abrupt and steep change. For values of A0 that 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 A0 of 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 (analogous 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 ultrasensitive to both signal duration and signal strength, making such a system a versatile switch.
Fig. 4.
Simulations of time-dependent switch under various conditions. 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, respectively), 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.
Like the activation level, the switching point [X]switch = (1/b)lnA0 depends on the same two parameters A0 and b. Whereas b reflects the regulation of one enzyme (e.g., GAP), the ratio A0 is a function of the initial concentrations of both GAP* and GEF*, and thus also represents signals regulating GEF. One signal can set the ratio A0 and 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 advantageous 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* concentrations. 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 first-order ultrasensitivity.
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
where Sp is the stimulus (i.e., duration or amplitude of signal) required for p percent activation (2). In our case, the Hill coefficient is determined exclusively by the value of A0 (see derivation in SI Text) and is given by
nH is 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 A0 that 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 A0 only and not on b. Changing the GAP* deactivation 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 deactivation rate is proportional to the strength of the signal, this mechanism 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 certain conditions (such as A0> 9.5 in the GAP deactivation example) 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 Go that targets RapGAP (GAP*) for degradation. The βAR, activated by isoproterenol, 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 clonidine 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 A0 only, it is expected that the same steepness will be observed for both dose-dependent and duration-dependent switches. For a GEF* concentration of 2 × 10−4 μM, we obtained a Hill coefficient of 3.5 for the amplitude-dependent activation and 3.6 for the duration-dependent activation. Increasing the GEF* concentration to 3 × 10−4 μM 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 Go is known to initiate degradation of RapGAP, which can increase Rap1 activity (15). Neuro 2A cells were treated for a fixed time with varying concentrations of HU-210, or for varying times with fixed concentration, 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.
Fig. 5.
Experimentally observed ultrasensitivity. In Neuro 2A cells, Rap1 was 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.
Because the control parameter A0 depends 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 contrast, 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 activation, short signals cannot yield significant GTPase activity. In Fig. 1E, we showed that active Rap at the end of the isoproterenol signal is ultrasensitive to the signal duration. Signal amplitude-dependent ultrasensitivity requires a nonlinear activation 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 case, the regulation of kGEF = kon[GEF*] is done by de facto 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, of course, depends on the kinetic parameters (Table S2). These simulations show that the first-order ultrasensitivity mechanism can apply to both the activating and deactivating arms of a three-component G protein system.
Fig. 6.
Ultrasensitivity by activation of GEF. Activation of Rho-family GTPase by GDI phosphorylation was simulated. Activation exhibits ultrasensitive response with respect to level of protein kinase signal.
Discussion
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 design.
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 simulations 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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Anthony Hasseldine designed preliminary simulations 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.
To 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: [email protected].
Author contributions: A.L conducted all the theoretical analysis; A.L., J.C.H., and R.I. designed research; G.J. and J.C.H. performed experiments; and A.L. and R.I. wrote the paper.
Competing Interests
The authors declare no conflict of interest.
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A. Lipshtat, G. Jayaraman, J.C. He, R. Iyengar, Design of versatile biochemical switches that respond to amplitude, duration, and spatial cues, Proc. Natl. Acad. Sci. U.S.A.
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where [SG*] and [SG] are the concentrations of the active and inactive forms of the GTPase, and [SGT] is its total concentration. Because the enzymatic activation and deactivation rates are much faster than the changes in the GAP* and GEF* concentrations, 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
where kGEF = kon[GEF*] and kGAP = koff[GAP*] are the effective reaction rates. The activity of GEFs and GAPs may be positively 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 RasGAP is degraded by caspase (30). The signal triggers the degradation 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 proportional 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 deactivation rate (which is the time derivative of [GAP*]) is proportional to [GAP*]. As a result, applying a stimulus X for a duration 
causes decay in active [GAP*] that is exponential with respect to both time and X. By the end of the signal duration, 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 exponentially decreases with the signal amplitude and time (SI Text, section 1). With decreasing levels of GAP*, the ratio of SG*/SGT increases (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 fractional activation level as function of the signal duration (for a fixed amplitude) is given by
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 
. 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
where in this case, b is the product of the signal duration τ and the effective GAP* deactivation rate. In both cases, the parameter 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 A0 and b resembles the role of Fermi energy and the inverse temperature β = (kBT)−1 in the Fermi–Dirac distribution. High temperatures slow the transition (as a function of energy) and as the temperature decreases, the curve becomes steeper. At T = 0 the Fermi–Dirac function becomes a step function. Is this similarity 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).
), 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 
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 structures 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.
(analogous 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 ultrasensitive to both signal duration and signal strength, making such a system a versatile switch.
where Sp is the stimulus (i.e., duration or amplitude of signal) required for p percent activation (2). In our case, the Hill coefficient is determined exclusively by the value of A0 (see derivation in SI Text) and is given by
nH is 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 A0 that 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 A0 only and not on b. Changing the GAP* deactivation 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 deactivation rate is proportional to the strength of the signal, this mechanism can function.
