Design of Versatile Biochemical Switches That Respond to Amplitude, Duration, and Spatial Cues

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. transport by Ran (16). Because of their central role in numerous The design principles underlying many switches are not known. We pathways, small GTPases (GTPases) have been studied exten- computationally studied the switching behavior of GTPases, and sively, both experimentally and computationally (1, 13, 17, 18). found that this first-order kinetic system can show ultrasensitivity. For many GTPases, the intrinsic cycle between the GDP- Analytical solutions indicate that ultrasensitive first-order reactions bound state and GTP-bound state is very slow. Cycling rates are can yield switches that respond to signal amplitude or duration. greatly enhanced by guanine nucleotide exchange factors (GEFs) The three-component GTPase system is analogous to the physical and GTPase activating proteins (GAPs) (19). Signaling pathways fermion gas. This analogy allows for an analytical understanding of that use heterotrimeric G proteins or small GTPases show both the functional capabilities of first-order ultrasensitive systems. graded and switch-like responses. What mechanisms underlie the Experiments show amplitude- and time-dependent Rap GTPase switching behavior? Zero-order ultrasensitivity can be obtained switching in response to Cannabinoid-1 receptor signal. This first- by low enzyme (GEF or GAP) to substrate (GTPase) ratio. order switch arises from relative reaction rates and the concen- However, experimental observations and estimations show that trations ratios of the activator and deactivator of Rap. First-order this is not always the case (8, 20). Although GEF and GAP ultrasensitivity is applicable to many systems where threshold for concentrations are lower than the GTPases levels, the difference transition between states is dependent on the duration, amplitude, is not sufficient. When multiple GEFs or GAPs are simulta- or location of a distal signal. We conclude that the emergence of neously active, the effective concentrations of the regulators can ultrasensitivity from coupled first-order reactions provides a be similar to that of the GTPase, resulting in a first-order system. fi versatile mechanism for the design of biochemical switches. How do rst-order reactions yield ultrasensitive response, and SYSTEMS BIOLOGY why don't we always observe this response? GTPase | signaling | ultrasensitivity GEFs and GAPs are controlled by receptor-regulated intracellular events (9, 21). Such regulation is critical for normal fficient regulation of intracellular processes benefits from "all physiology. Abnormal regulation of GEFs or GAPs has been Eor none" response (1), where a cellular component switches implicated in cancer (22), viral and bacterial pathogenesis (23), between two functional states upon crossing a threshold. Often, a vascularization defects during development (24), and mental regulator triggers state change. Near the threshold point, a small retardation (25). Often, regulation of either a GAP or a GEF is fi change in one parameter, such as regulator concentration or signal suf cient for GTPase activation (9, 21, 26). We explored the duration, causes switching of the responding component. Such relationship between different levels of GEF and GAP activity by responses are called ultrasensitive (2). A widely known mechanism numerically simulating receptor-regulated Rap activation, using underlying a steep response curve is the "zero-order ultra- an ordinary differential equations model. The signaling network sensitivity" first proposed by Goldbeter and Koshland (2), who (Fig. S1) includes our prior experimental data (15) and the — regulation of Rap by cAMP (27). Details of the simulations are showed that under zero-order conditions i.e., when one or more SI Text of the enzymes in a coupled system are saturated—the transition described in , and the models are available at the Virtual Cell site. In Fig. 1, we show the formation of GTP-bound Rap in between the active and inactive conformations exhibits high sen- α α sitivity to the concentration ratio of the enzymes. Other mecha- response to signals from activated 2-adrenergic ( 2R) and β-adrenergic (βAR) receptors. The α2R signal leads to degra- nisms that yield ultrasensitivity include cooperativity, multistep β regulation, and stoichiometric inhibitors (3–5). Positive feedback dation of Rap GAP* whereas the AR signal activates the GEF. We observe an abrupt transition from a low activity state to high loops play an important role in producing switching behavior (4) activity as the α2R signal crosses a threshold. Similar behavior is and are often considered necessary for bistability (6, 7). Mecha- observed when signal duration is lengthened while the signal nisms that depend on loops require complex network organization amplitude is fixed (Fig. 1C). Thus, level of active Rap is very such as topological motifs in addition to the enzymatic activity to sensitive to the signal amplitude and duration, with distinct produce switches. However, switching behavior is observed in the subthreshold and above-threshold responses. As opposed to the absence of loops, and the design principles for such switches are high sensitivity to α2R activation, the response to βAR stim- poorly understood. We have used analytical and numerical ulation is slightly slower than a regular Michaelian curve (Fig. methods as well as experiments to describe first-order ultra- 1D). The amplitude of the βAR stimulation affect the Rap sensitivity as the basis for a versatile design of a biochemical switch activation level, but the sensitivity to the exact stimulus charac- that responds to both duration and concentration of stimulus. teristics is low. However, the duration of the signal produces a Ultrasensitivity in GTPases Small GTPases can function as molecular switches in varied Author contributions: A.L conducted all the theoretical analysis; A.L., J.C.H., and R.I. de- cellular processes including signaling networks (8). Their con- signed research; G.J. and J.C.H. performed experiments; and A.L. and R.I. wrote the paper. version from GDP (inactive) to GTP (active) conformations The authors declare no conflict of interest. promotes interaction with downstream effectors to propagate This article is a PNAS Direct Submission. information flow. Rapid responses of GTPases to incoming – 1To whom correspondence should be addressed at: Department of Pharmacology and regulation can turn downstream pathways on and off (9 11). Systems Therapeutics, Mount Sinai School of Medicine, One Gustave L. Levy Place, Box Thus, GTPases play an essential role in controlling many cellular 1215, New York, NY 10029. E-mail: [email protected]. – responses (10 13). Examples include cellular proliferation by This article contains supporting information online at www.pnas.org/cgi/content/full/ Ras (14), neurite growth by Rap1 (15), and nucleocytoplasmic 0908647107/DCSupplemental.

www.pnas.org/cgi/doi/10.1073/pnas.0908647107 PNAS Early Edition | 1of6 Downloaded by guest on September 23, 2021 μ 300 [Clonidine] ( M) ½SG A 0.1 ¼ 1 ; [2] 200 0.032 0.01 SGT 1 þ kGAP=kGEF 0.0032 100 0.001 0.00032

) 0.0001 k k * k k * 2 0 where GEF = on[GEF ] and GAP = off[GAP ] are the effec-

m 0 2000 4000 6000 8000 10000 time (sec.) tive reaction rates. The activity of GEFs and GAPs may be pos- itively or negatively regulated. We consider the case of negative 200 B C 200 regulation of the GAP* by a signal X (receptor signal), which n =2.9 n =1.7 100 H H 100 targets the GAP* for irreversible deactivation (by degradation or 0 0 −3 −2 −1 0 1 2 any other first-order reaction such as sequestration) (15, 21) ⋅ GTP (molecules/ μ 10 10 10 10 10 10 [Clonidine] (μM) Clonidine Signal (sec.) (Fig. 2A). The RasGAP neurofibromin undergoes degradation Rap 200 D E 300 upon treatment with various growth factors (29), and p120 Ras- 200 n =0.7 n =2.2 GAP is degraded by caspase (30). The signal triggers the degra- 100 H H 100 dation of GAP* either directly or through a reaction cascade. 0 0 −4 −3 −2 −1 0 1 2 1 2 3 10 10 10 10 10 10 10 10 10 10 The analysis is valid as long as the effective rate of GAP* μ [Isoproterenol] ( M) Isoproterenol Signal (sec.) decrease is proportional to the signal amplitude and depends Fig. 1. Numerical simulations of Rap regulation. A detailed simulation of on the GAP* concentration [GAP*] (see calculation in SI Text). the Rap1 pathways was performed by using Virtual Cell (see Fig. S1 for the Although we assume that the GAP deactivation rate is propor- pathways). α2AR were stimulated for fixed duration and with various tional to [GAP*], in SI Text we show that this assumption is not amplitudes, evenly distributed on a logarithmic scale. Then, Rap was acti- necessary, and that ultrasensitivity can be achieved from any vated by a βAR stimulus. (A) The activation level is clustered into two groups α nonzero positive dependence on [GAP*]. Here, we present of low and high activation. (B and C) 2AR-stimulated steady state Rap the standard case of mass action law. In this regime, the deacti- activity is ultrasensitive with respect to concentration and duration. (D) βAR β vation rate (which is the time derivative of [GAP*]) is propor- stimulation of Rap is subsensitive with respect to signal amplitude. (E) AR X activation of Rap is ultrasensitive with respect to signal duration. tional to [GAP*]. As a result, applying a stimulus 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 duration, GAP* has a new steady state concentration, namely switching response (Fig. 1E). These simulations indicate that [GAP*]t=[GAP*]0exp(−k[X]τ) for times t > τ (Fig. 2BI). (In SI both GEF* and GAP* concentrations integrated over time Text we extend this derivation cases where the GAP* decay is regulate GTPase activation. slower than exponential, such as in reversible deactivation.) fi i These ndings raise several questions: ( ) What is the mech- Because the hydrolysis rate of the GTPase SG* is linearly anism that enables the switching in response to α2R signals dependent on the GAP* concentration, this rate also exponen- amplitude or duration? (ii) When do GTPases display graded tially decreases with the signal amplitude and time (SI Text, responses? (iii) Is this type of switching a general mechanism in section 1). With decreasing levels of GAP*, the ratio of SG*/ many biochemical systems? (iv) Is there a design advantage of SGT increases (Fig. 2BII). Plugging the effect of the signal into such a system with seemingly redundant regulation of both the the activation level of the GTPase results in a signal-dependent GEF and GAP, because mutations in either component often switch of the GTPase from a GDP-bound to GTP-bound state BIII lead to a disease state? (Fig. 2 ). Regardless of the GTPase concentration, the frac- To answer these questions we have studied the system ana- tional activation level as function of the signal duration (for a fi lytically. Without restricting the concentrations, we assumed that xed amplitude) is given by the reactions of the GTPase cycle are in the mass action regime. ½SG 1 Thus, the activation and deactivation terms of the GTPase are ¼ ; [3A] SGT 1 þ A expð − b½τÞ proportional to the concentrations of active GEF and GAP 0 (GEF* and GAP*), respectively. These terms should be also where A is the initial k /k ratio (k and k ,as linear with respect to the (instantaneous) concentrations of the 0 GAP GEF GAP GEF defined above, incorporate both concentrations of the active inactive and active forms of the GTPase. Under saturation GEF* or GAP* and respective kinetic rates) and b is the product conditions (zero-order reactions), the (de)activation rates may of the signal amplitude [X] and the effective GAP* deactivation be independent of substrate concentration, and so produce (or degradation) rate. For a broad range of parameters, this ultrasensitivity (2). We limit our analysis to cases of first-order function is ultrasensitive with respect to τ. This way, using mass reactions. Because many cellular components are found in a action reactions only, a regulated GTPase can act as a time- similar range of concentrations, the mass action assumption is dependent switch. Similarly, the dependence of activation on valid for many intracellular systems (28). For these systems, the the signal amplitude [X] (with fixed duration) is given by GTPase cycle is governed by the following equations: ½SG ¼ 1 ; [3B] SGT 1 þ A0expð − b½XÞ d d ½SG ¼kon½GEF ½SG − koff ½GAP ½SG ¼ − ½SG dt dt b τ where in this case, is the product of the signal duration and ½SG þ½SG¼½SGT ; [1] the effective GAP* deactivation rate. In both cases, the parameter b is a measure of the signal impact. (For complete derivation where [SG*] and [SG] are the concentrations of the active and please see SI, section 1.) The activation level of GTPase depends X τ inactive forms of the GTPase, and [SGT] is its total concentra- on the extracellular stimulus characteristics ([ ]or ) through a tion. Because the enzymatic activation and deactivation rates are logistic function, also known as the Fermi–Dirac distribution. much faster than the changes in the GAP* and GEF* concen- The dependence of the activation curve (Eqs. 3A and 3B)on trations, one can assume a steady state solution with respect to A0 and b resembles the role of Fermi energy and the inverse −1 the GTPase activation level. Under these conditions, the steady temperature β =(kBT) in the Fermi–Dirac distribution. High state activation level can be written as temperatures slow the transition (as a function of energy) and as

2of6 | www.pnas.org/cgi/doi/10.1073/pnas.0908647107 Lipshtat et al. Downloaded by guest on September 23, 2021 A Table 1. Analogy between GTPase activation and Fermion gas Signal SG* Fermion gas GTPase (small G) Particles GTP (or GDP) molecules GEF GAP Quantum states Small GTPase molecules SG A state can be either empty or GTPase can be either GTP or occupied by a single particle GDP bound Transition rate depends on Transition rate depends on − 100 100 Boltzmann factor exp( E/kBT) GAP/GEF ratio [proportional B (II) to exp(-k[X]τ) or to exp(-kx)] (I)

[GAP] Switching point depends on Switching point depends on initial 50 50 Fermi energy, EF value of GAP/GEF (Eq. 3) [GAP]

0 0 −2 −1 0 1 2 −4 −3 −2 −1 0 10 10 10 10 10 10 10 10 10 10 signal [X] [SG*]/[SG ] total 1 (III) GTPases, but it does occur for other proteins with nucleotide ] 0.8

total triphosphatase activity involved in polymerization processes 0.6

0.4 (such as actin, an ATPase, and tubulin, a GTPase), where

[SG*]/[SG 0.2 monomers can bind to and detach from polymers. There is no theoretical limit on the number of monomers that are associated 0 −2 −1 0 1 2 10 10 10 10 10 signal [X] with a single polymer. Thus, without any further calculation, one may expect that the length distribution of polymers (analogous to Fig. 2. Ultrasensitive response in GTPase activation. (A) Schematic diagram occupation distribution of bosons) would follow Bose–Einstein of the small GTPase cycle and its regulation. (B) Illustration of the mathe- distribution. Detailed calculations show that this is indeed the matical reasoning ultrasensitivity. First-order GAP* deactivation yields SI Text exponential dependence of the GAP on the signal (BI). The dependence of case ( , section 4). Thus, by comparing the abstract struc- the GTPase activation on the GAP* (BII) is hyperbolic. However, dependence tures of these two systems, one can predict the behavior of the of the activation level on the upstream signal (BIII) is ultrasensitive. The biological system under various conditions, based on the straight lines show how different signals (one order of magnitude apart knowledge we already have about physical systems and their each other) are clustered into two groups of high and low activation level. properties. Furthermore, this analogy shows that the switching mechanism presented here is based on the general architecture SYSTEMS BIOLOGY of the system, rather than on any particular properties of the temperature decreases, the curve becomes steeper. At T =0 fi – GTPases. Thus, rst-order ultrasensitivity can be applicable to the Fermi Dirac function becomes a step function. Is this sim- many different cellular systems. ilarity between the GTPase cycle and Fermi–Dirac system a coincidence or can the analogy provide some insight into the Ultrasensitivity in Space principles underlying the design of the three-component GTPase The first-order ultrasensitivity mechanism is mathematically system? Mapping of biological questions onto known physical based on the exponential dependence of GAP* concentration on systems has proven useful in several other cases (31). the stimulus X or duration τ. The comparison with Fermi–Dirac Fermions are particles that obey the Pauli Exclusion Principle. distribution implies that any two-state system with transition No two fermions can occupy the same quantum state simulta- rates that are exponentially dependent on an input variable can neously. Thus, a quantum state can be either empty or occupied – be ultrasensitive with respect to the value of that variable. This by a single particle. Fermi Dirac distribution determines the observation opens the way for a broad range of applications, probability of a given quantum state to be occupied by a fermion, including spatial localization. as a function of energy level and temperature. In the GTPase Spatial gradients may provide exponential dependence and system, the GTPase molecules are associated with molecules form intracellular regions of high activity (microdomain). If (GTP or GDP), where each copy of GTPase can be found in one there is a point source at one side of the cell, and one component of two possible states. Although the GTPase cycle also includes spreads out by diffusion, then the concentration of the compo- intermediate transition states (32), the two-state model is a good nent decreases exponentially with the distance x from the source. approximation that is widely used in biochemistry (21). Activa- The same analysis that has been used for the exponentially fi tion level of a GTPase is the probability of nding a GTPase decreasing GAP* with respect to the signal applies here as well, molecule in the GTP-bound state and hence is analogous to the with distance x replacing the signal amplitude X. A sharp change occupation probability of the quantum states. The occupation in the activation level of the regulated GTPase can be predicted probability exhibits an ultrasensitive curve with respect to energy. to occur at one particular spatial location. This activation could −E k T Energy is introduced by the Boltzmann factor exp( / B ), then initiate further local stimulation of downstream effectors. which is the relative probability to find a particle in a given state. Fig. 2B, which depicts the construction of ultrasensitive response Because the probability of finding a GTPase in a GTP-bound through GAP regulation, can be also used to illustrate spatial state is exponentially dependent on the signal characteristics ([X] switching. Instead of regulating the GAP* concentration as a or τ), the signal in the biological system is analogous to energy in function of the signal, Fig. 2BI can be viewed as a spatial dis- the physical system. As such, the GTPase activation follows tribution of GAP*. If the gradient is exponential, and the Fermi–Dirac distribution as function of τ or [X] (Table 1). This dependence of SG* on GAP* is as shown in Fig. 2BII, then the analogy between the two unrelated systems opens the way for overall spatial distribution of the active form SG* is the same as adaptation and adoption of known results from one system to the shown in Fig. 2BIII, where the x axis denotes the spatial coor- other. For example, in statistical physics there are particles that dinate rather than the signal amplitude. are not subject to the exclusion principle. These are bosons, and Location-dependent ultrasensitivity implies formation of many of them can occupy the same quantum state. Because of multiple biochemical compartments without physical boundaries. this difference between bosons and fermions, bosons follow The differences between adjacent compartments can be sig- Bose–Einstein statistics rather than Fermi–Dirac statistics. The nificant, whereas within each compartment there is no spatial biological analogy of bosons is a system where many molecules variation. How can a single continuous gradient form a multi- can bind to a single complex. This does not happen for the small compartmental pattern? The first-order ultrasensitivity provides

Lipshtat et al. PNAS Early Edition | 3of6 Downloaded by guest on September 23, 2021 a simple mechanism, based on the observation that not only does change. For values of A0 that are about 10 or higher, the signal each GTPase potentially have several GAPs and GEFs, but there dependence is of the “all or none” nature, or ultrasensitive. are also GAPs and GEFs that regulate several GTPases (33, 34). Changing the value of b (for example by changing the duration of Such a mix-and-match configuration is a powerful design feature. signal) does not affect the steepness of the curve but shifts the Consider an exponential spatial gradient of a GAP* that deac- switching point. Two parameters govern the switch dynamics: the A tivates two independent GTPases (Fig. 3 ). The change in initial ratio A0 of GEF* and GAP* activities, and the signal GAP* activity is shown as a function of distance x (Fig. 3BI). impact b (Eq. 3). Switching behavior is characterized by a close- Because GTP hydrolysis rates of different GTPases are different to-zero derivative (of activation with respect to signal) away from even with the same GAP (35), and each GTPase has its own the switching point and positive derivative in close proximity to GEF, the switching points of the GTPases (which are functions that point (SI Text and Fig. S2). Thus, we define the switching k k of the GAP/ GEF ratio) are distinct. The multiple threshold point as the signal amplitude at which the derivative d[SG*]/d[X] points yield different spatial distribution of SG* for the two gets its maximum. Simple analysis shows that the critical signal GTPases. Between any two adjacent threshold points, one type for switching is [X]switch = (1/b)lnA0 (see derivation in SI Text). of GTPase (e.g., SG1) is above its threshold, and thus activated, ½SG =½SGT ¼1 This is also the signal strength at which 2 (anal- and the other is not. This results in the formation of three dis- ogous to Fermi energy), which is another reason to view this tinct compartments with varying levels of activated SG1 or value as the switching point. For low values of A , there is high SG2 (Fig. 3BIII). This way, the single continuous gradient of 0 SG activity even before stimulation, and thus there is no maxima GAP* can drive the formation a discrete set of compartments point for the derivative and no switching dynamic is observed defined by the different activity state of GTPases within the (Fig. S2). However, for larger values, the dynamics are ultra- compartment. sensitive to both signal duration and signal strength, making such Signal Duration-Dependent Ultrasensitivity a system a versatile switch. X b The analysis above shows that GTPases respond rapidly when Like the activation level, the switching point [ ]switch = (1/ ) lnA0 depends on the same two parameters A0 and b. Whereas b the total signal impact crosses a threshold, either due to long fl A duration or high amplitude. This interchangeability of the re ects the regulation of one enzyme (e.g., GAP), the ratio 0 is amplitude and duration of the signal is a major design advantage a function of the initial concentrations of both GAP* and GEF*, that allows the system to function as an information integrator. and thus also represents signals regulating GEF. One signal can A In Fig. 4, we show that time-dependent switching occurs for a set the ratio 0 and shift the switching point. Then, the other broad range of parameters. In time-dependent switch, the x axis signal switches the system between the two states at the desired is the signal duration and the parameter b includes kinetic rate point. Thus the switching point is tuneable. This is an advanta- and the fixed amplitude of the signal, whereas in the amplitude- geous design for two reasons: Separate regulation of GAP* and dependent switch the x axis represents the signal strength and b is GEF* provides a higher level of flexibility. This design also the product of the kinetic rate and the fixed signal duration. So enables switching over a broad range of GAP* and GEF* con- the curves for signal amplitude-dependent switches are also the centrations. Because the control parameter is the ratio of GEF* same as those of the signal duration-dependent switches with the to GAP*, rather than a concentration of either component, difference in the interpretation of the parameters. In all cases, switching is not limited to zero-order regime and can be achieved the GTPase switches from low activity in the case of a brief (or at higher concentrations as well (Fig. S3). Broad applicability of weak) stimulus, to high activation in the case of a sustained (or this mechanism is illustrated in the CAM-kinase-II system, where strong) signal. The steepness of this transition is determined by a shift of the switching point due to changes in phosphatase the value of A0: low values of A0 yield a smooth and shallow activity has observed (36). Such a shift is easily explained by the transition, whereas high values lead to an abrupt and steep 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 A Signal GEF is defined by GEF SG2* SG 1* 1 2 GAP A =1 A =10 A =100 SG2 SG 1 0 0 0 1 1 0.8 b=1 0.6 100 1 0 B 0.4 0 5 10 50 0.2 0 0 5 10 [GAP]

(I) ] 0 1 0 −1 0 1 2

] 10 10 distance x (A.U.)10 10 total 0.8 1 total 0.6 b=0.1 0.5 SG2 SG1 0.4 (II) 0.2 0

[SG*]/[SG −3 −2 −1 0 1 2 ] 10 10 10 [GAP] 10 10 10 [SG*] / [SG 0 1 1 total SG1* 0.8 0.5 SG1* low high SG1* high SG2* low SG2* high 0.6 SG2* b=0.01 (III) low 0.4 0

[SG*]/[SG −1 0 1 2 10 10 distance x (A.U.)10 10 0.2 0 −2 −1 0 1 2 −2 −1 0 1 2 −2 −1 0 1 2 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 Fig. 3. Spatial properties of coupled switches. (A) A general scheme where Signal Duration (min) two GTPases are regulated by a common GAP. (B) Simulation of spatial domain formation. (BI) Due to diffusion mechanism, the spatial distribution Fig. 4. Simulations of time-dependent switch under various conditions. of GAP is exponential. (BII) Because each GTPase has its own GEF, the acti- Steady-state activation level of the GTPase are plotted as a function of signal

vation level of each GTPase has a different dependence on the local GAP duration for A0(kGAP/kGEF) = 1, 10, or 100 (Left, Middle, and Right, respec- concentration. (BIII) This difference yields different switching points, result- tively), and b=0.01, 0.1, and 1 (Bottom, Middle, and Top). All curves are ing in distinct compartments. semilog plots, and two examples in linear scale are in Insets for [τ] < 10.

4of6 | www.pnas.org/cgi/doi/10.1073/pnas.0908647107 Lipshtat et al. Downloaded by guest on September 23, 2021 3.5 4 logð81Þ n ¼ ; [4] 3 A B 3.5 H ðS =S Þ log 90 10 3 2.5 2.5 2 where Sp is the stimulus (i.e., duration or amplitude of signal) 2 1.5

required for p percent activation (2). In our case, the Hill coef- GTP (fold change) 1.5 ± fi A 1 n =8.6±0.8 n =5 0.2 cient is determined exclusively by the value of 0 (see deriva- H H 1 Rap − SI Text 0.5 0.5 tion in ) and is given by 0 20 40 60 0 100 200 300 Signal Duration (min.) HU210 (nM) logð81Þ nH ¼ : [5] Fig. 5. Experimentally observed ultrasensitivity. In Neuro 2A cells, Rap1 was log½logð9A0Þ=logðA0=9Þ 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 nH is defined for A0 > 9 because for lower values activation level is above 10% even without any stimulus (Fig. 4, left column). For were calculated as described in SI Text. A0 > 9.5, we obtain ultrasensitivity, namely nH > 1. For A0 = 100, the Hill coefficient is >4(Fig. S4). When the deactivation is not degradation of RapGAP, which can increase Rap1 activity (15). fi A in the rst-order regime, the values of 0 that are required for Neuro 2A cells were treated for a fixed time with varying con- ultrasensitivity may be different. This analysis shows that centrations of HU-210, or for varying times with fixed concen- whereas the switching point is a function of both parameters tration, and Rap1 activity was measured. Both dose-dependent and can be regulated independently, the steepness depends on and duration-dependent ultrasensitivity were observed (Fig. 5). A b the initial ratio 0 only and not on . Changing the GAP* deac- The numerical simulations and the experiments demonstrate the tivation rate will affect the switching point but not the steepness. interchangeability of dose and duration predicted by our analysis Furthermore, the analysis is based only on linear dependence on and the versatile nature of this switch. fi deactivation steps without any assumptions regarding speci c Because the control parameter A0 depends on the GAP*/ biochemical mechanisms. The signal need not be chemical. GEF* ratio, the effect of GEF activation is mathematically The same result is obtained when X represents physical force, analogous to GAP deactivation. However, in practice there is an UV radiation, or temperature change. As long as the deactiva- important difference: deactivation rate is typically proportional fi tion rate is proportional to the strength of the signal, this mech- to the active form concentration ( rst-order reaction). In con- SYSTEMS BIOLOGY anism can function. trast, a typical activation process is negatively dependent on the The analysis presented here is independent of concentrations concentration of the active form—the more active the form, the of the substrate (e.g., GTPase) and the regulatory enzymes (e.g., slower the reaction. This dependence yields a subzero-order GAP* or GEF*); rather, the steepness determining parameter is reaction. This is the reason why in our simulations of the Rap the ratio A0. In cases where the upstream pathways regulating system subsensitivity is observed in response to isoproterenol the two direct regulators are coupled, perturbation in one stimulation (Fig. 1D). However, GEF activation can yield a time- pathway can be compensated for by a reciprocal perturbation in response dynamic switching. Under low GAP* conditions, the other. This is a robust design for a switch because the GTPase activation by GEF* stimulation is very rapid. Thus, well important variable (the ratio) remains unaffected even under before the system approaches steady state it is sensitive to signal conditions where the actual concentrations of the components duration. Whereas long enough signals evoke almost full acti- change. Note that the system organization enables ultrasensitivity vation, short signals cannot yield significant GTPase activity. In but does not guarantee it. The kinetic values need to satisfy cer- Fig. 1E, we showed that active Rap at the end of the iso- tain conditions (such as A0 > 9.5 in the GAP deactivation exam- proterenol signal is ultrasensitive to the signal duration. Signal ple) to obtain ultrasensitivity, otherwise the system of reactions amplitude-dependent ultrasensitivity requires a nonlinear acti- will show a graded response. vation of GEF. The source of nonlinearity can be at any place along the pathway and not necessarily directly connected to the Numerical Simulations and Experimental Tests GEF. The sources of nonlinearity for the various regulatory We numerically simulated the activation of Rap by an agonist processes are summarized in Table S1. A common mechanism (clonidine) of the α2AR. The signal activates the Go that targets for such nonlinearity in activating pathways is the removal of an RapGAP (GAP*) for degradation. The βAR, activated by iso- inhibitor. The GDP dissociation inhibitors (GDIs) are known to proterenol, stimulates the production of cAMP to determine the bind Rho family GTPases and block GTPase activation (39). levels of GEF* that activate Rap (SI Text and Fig S1). Numerical Dissociation of a GDI from Rac is regulated by phosphorylation simulations show how, despite the multistep regulation of clo- by PAK kinase (39, 40), allowing receptor signals to inactivate nidine to GAP, an ultrasensitive activation of Rap is seen in the GDI and enabling GEF* to activate the Rac GTPase. In this response to receptor ligand concentration (signal amplitude) and the duration of receptor activation (SI Text, Section 5, and Fig. 90 S5). As predicted, the steepness changes with the initial GAP* to 80 GEF* ratio. Because the Hill coefficient is a function of A0 only, 70 it is expected that the same steepness will be observed for both 60 dose-dependent and duration-dependent switches. For a GEF* 50 −4 n =1.65 concentration of 2 × 10 μM, we obtained a Hill coefficient of 40 H 3.5 for the amplitude-dependent activation and 3.6 for the 30 duration-dependent activation. Increasing the GEF* concen- 20 − × 4 μ fi 10

tration to 3 10 M decreased the Hill coef cient to 2.6 or 2.7 Rho−family GTPase activation (%)

0 −3 −2 −1 0 1 2 for amplitude- and duration-dependent switches, respectively. 10 10 10 10 10 10 These analytical and computational studies predict that [Receptor Signal] receptor-regulated Rap should behave as a time-dependent and Fig. 6. Ultrasensitivity by activation of GEF. Activation of Rho-family GTPase agonist concentration-dependent switch. HU-210 acting through by GDI phosphorylation was simulated. Activation exhibits ultrasensitive the CB1 receptor that also couples to Go is known to initiate response with respect to level of protein kinase signal.

Lipshtat et al. PNAS Early Edition | 5of6 Downloaded by guest on September 23, 2021 case, the regulation of kGEF = kon[GEF*] is done by de facto tuning the switching point. Such tuning is likely to be biologically controlling kon rather than the GEF* concentration. “All-or- important. Tuneable switches make cellular responses robust none” response of Rho-family GTPases has been observed (41). because they are able to maintain their switching behavior under To examine whether phosphorylation of GDIs can produce dynamic conditions. This design advantage applies not only to a ultrasensitivity, we implemented a published model of the Rho single switch, but also to coupled switching processes that can GTPase, including its intermediate complexes (18). We added to create functional spatial compartments in the absence of physical this model active and inactive GDIs, and reversibly deactivated the barriers. The ultrasensitivity mechanism described here does not fi GDIs by phosphorylation. Under rst-order reaction assumption, depend on any particular biochemical mechanism or on substrate activation of the Rho-family GTPase level as a function of the concentration. It applies equally to cases of high or low enzyme-to- protein kinase concentration (hence, receptor signal amplitude) substrate ratio, as long as there is no saturation. This general exhibits ultrasensitivity (Fig. 6). Introducing cooperativity would applicability across a range of concentrations for the cellular increase the sensitivity even further. This, of course, depends on the response system and the interconvertibility between chemical or kinetic parameters (Table S2). These simulations show that the first- physical signal and time (i.e., signal amplitude and signal duration) order ultrasensitivity mechanism can apply to both the activating fi fl and deactivating arms of a three-component G protein system. make the rst-order switching mechanism a versatile and exible design. Discussion Materials and Methods We have elucidated the mathematical basis for the binary response at the systems level, within a graded system of first-order Neuro-2A cells (ATCC) were cultured and treated and Rap activity measured reactions. Our theoretical framework shows that, with the as described in SI Text. Numerical simulation were done by using Virtual Cell (42). More details can be found in SI Text. “appropriate” kinetic parameters, various systems can display ultrasensitive responses, whereas the same system of reactions ACKNOWLEDGMENTS. Anthony Hasseldine designed preliminary simula- exhibits graded response with other set of kinetic parameters (see tions of the Rap system. We thank Drs. Eric Sobie and Bob Blitzer for a comprehensive analysis in SI Text). For GTPases, our analysis critical reading of the manuscript, and Dr. Walter Kolch for valuable shows why ultrasensitive behaviors are experimentally observed in discussions that led to the analyses of coupled GTPase systems. This work varied systems (Table S3 and Fig. S6). This general theory for the was supported by National Institutes of Health Grants GM54508 and fl P50GM071558, the Systems Biology Center grant. Virtual Cell is supported design of a exible switch explains not only the sharp response, but by National Institutes of Health Grant P41RR013186 from the National also the design advantage of the double regulation and its use in Center for Research Resources.

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