A model of excitation and adaptation in bacterial chemotaxis

Transcription

1 Proc. Natl. Acad. Sci. USA Vol. 94, pp , July 1997 Biochemistry A model of excitation and adaptation in bacterial chemotaxis PETER A. SPIRO*, JOHN S. PARKINSON, AND HANS G. OTHMER* Departments of *Mathematics and Biology, University of Utah, Salt Lake City, UT Communicated by Charles S. Peskin, New York University, Hartsdale, NY, April 24, 1997 (received for review August 26, 1996) ABSTRACT Bacterial chemotaxis is widely studied because of its accessibility and because it incorporates processes that are important in a number of sensory systems: signal transduction, excitation, adaptation, and a change in behavior, all in response to stimuli. Quantitative data on the change in behavior are available for this system, and the major biochemical steps in the signal transduction processing pathway have been identified. We have incorporated recent biochemical data into a mathematical model that can reproduce many of the major features of the intracellular response, including the change in the level of chemotactic proteins to step and ramp stimuli such as those used in experimental protocols. The interaction of the chemotactic proteins with the motor is not modeled, but we can estimate the degree of cooperativity needed to produce the observed gain under the assumption that the chemotactic proteins interact directly with the motor proteins. Chemotaxis (or more accurately, chemokinesis), the process by which a cell alters its speed or frequency of turning in response to an extracellular chemical signal, has been most thoroughly studied in the peritrichous bacterium Escherichia coli. E. coli exhibits sophisticated responses to many beneficial or harmful chemicals. In isotropic environments the cell swims about in a random walk produced by alternating episodes of counterclockwise (CCW) and clockwise (CW) flagellar rotation. CCW rotation pushes the cell forward in a fairly straight run, CW rotation triggers a random tumble that reorients the cell. The durations of both runs and tumbles are exponentially distributed, with means of 1.0 s and 0.1 s, respectively (1). In a chemoeffector gradient, the cell carries out chemotactic migration by extending runs that happen to carry it in favorable directions. Using specific chemoreceptors to monitor its chemical environment, E. coli perceives spatial gradients as temporal changes in attractant or repellent concentration. The cell in effect compares its environment during the past second with the previous 3 4 s and responds accordingly. Attractant increases and repellent decreases transiently raise the probability of CCW rotation, or bias, and then a sensory adaptation process returns the bias to baseline, enabling the cell to detect and respond to further concentration changes. The response to a small step change in chemoeffector concentration in a spatially uniform environment occurs over a 2- to 4-s time span (2). Saturating changes in chemoeffector concentration can increase the response time to several minutes (3). Many bacterial chemoreceptors belong to a family of transmembrane methyl-accepting chemotaxis proteins (MCPs) (reviewed in ref. 4). Among the best-studied MCPs is Tar, the E. coli receptor for the attractant aspartate. Tar has a periplasmic binding domain and a cytoplasmic signaling domain that communicates with the flagellar motors via a phosphorelay sequence involving the CheA, CheY, and CheZ proteins (see Fig. 1). CheA, a histidine kinase, first autophosphorylates and then transfers its phosphoryl group to CheY. The publication costs of this article were defrayed in part by page charge payment. This article must therefore be hereby marked advertisement in accordance with 18 U.S.C solely to indicate this fact by The National Academy of Sciences $ PNAS is available online at http: Phospho CheY interacts with switch proteins in the flagellar motors to augment CW rotation, CCW being the default state in the absence of Phospho CheY. CheZ assists in dissipating CW signals by enhancing the dephosphorylation of CheY. Tar controls the flux of phosphate through this circuit by forming a stable ternary complex (Tar CheW CheA) that modulates CheA autophosphorylation in response to changes in ligand occupancy or methylation state. CheA phosphorylates more slowly when the receptor is occupied than when it is not. Changes in MCP methylation state are responsible for sensory adaptation. Tar has four residues that are reversibly methylated by a methyltransferase, CheR, and demethylated by a methylesterase, CheB. CheR activity is unregulated, whereas CheB, like CheY, is activated by phosphorylation via CheA. Thus, receptor methylation level is regulated by feedback signals from the signaling complex, which can probably shift between two conformational states having different rates of CheA autophosphorylation. Attractant binding and demethylation shift the equilibrium toward a low CheA activity state; attractant release and methylation shift the equilibrium toward a high CheA activity state. A receptor complex that is bound to attractant but not highly methylated can be thought of as a sequestered state (5) which is unable to autophosphorylate at a significant rate. E. coli can sense and adapt to ligand concentrations that range over five orders of magnitude (5). In addition, the machinery for detection and transduction is exquisitely sensitive to chemical stimuli. The cell can respond to exponential increases ( ramps ) in attractant levels that correspond to rates of change in fractional occupancy of only 0.1% per second. Consequently, the cell can respond even if only a small fraction of its receptors have changed occupancy state during a typical sampling period. These experimental observations raise several questions. First, how does the cell achieve the extreme sensitivity that is observed? Second, why are there multiple methylation states of the receptor? In this paper we describe a mathematical model based on known kinetic properties of Tar and the phosphorelay signaling components that accounts for this exquisite sensitivity. In the model, excitation results from the reduction in the autophosphorylation rate of CheA when Tar is bound to a ligand, and adaptation arises from methylation of the receptor. The disparity in the time scales of these processes produces a derivative- or temporal-sensing mechanism with respect to the ligand concentration. The model makes essential use of the multiple methylation states to achieve adaptation, and can be used to derive quantitative relations between certain key rates and the behavioral responses to experimental protocols, and to predict the cooperativity needed to achieve the observed gain. A Qualitative Description of the Response to Different Stimuli We assume that Tar is the only receptor type, that the Tar CheA CheW complex does not dissociate, and that Tar, CheA, and CheW are found only in this complex. We also assume that methylation of the multiple sites occurs in a specified order (6, 7). Abbreviations: CCW, counterclockwise; CW, clockwise; MCP, methyl-accepting chemotaxis protein. To whom reprint requests should be addressed. 7263

2 7264 Biochemistry: Spiro et al. Proc. Natl. Acad. Sci. USA 94 (1997) FIG. 1. Signaling components and pathways for E. coli chemotaxis. Chemoreceptors (MCPs) span the cytoplasmic membrane (hatched lines), with a ligand-binding domain deployed on the periplasmic side and a signaling domain on the cytoplasmic side. The cytoplasmic Che signaling proteins are identified by single letters e.g., A CheA. MCPs form stable ternary complexes with the CheA and CheW proteins to generate signals that control the direction of rotation of the flagellar motors. The signaling currency is in the form of phosphoryl groups ( P), made available to the CheY and CheB effector proteins through autophosphorylation of CheA. CheY p initiates flagellar responses by interacting with the motor to enhance the probability of clockwise rotation. CheB p is part of a sensory adaptation circuit that terminates motor responses. MCP complexes have two alternative signaling states. In the attractant-bound form, the receptor inhibits CheA autokinase activity; in the unliganded form, the receptor stimulates CheA activity. The overall flux of phosphoryl groups to CheB and CheY reflects the proportion of signaling complexes in the inhibited and stimulated states. Changes in attractant concentration shift this distribution, triggering a flagellar response. The ensuing changes in CheB phosphorylation state alter its methylesterase activity, producing a net change in MCP methylation state that cancels the stimulus signal (see ref. 4 for a review). FIG. 2. The ligand-binding, phosphorylation, and methylation reactions of the Tar CheA CheW complex, denoted by T. LT indicates a ligand-bound complex. Vertical transitions involve ligand binding and release, horizontal transitions involve methylation and demethylation, and front-to-rear transitions involve phosphorylation and the reverse involve dephosphorylation. The details of the phosphotransfer steps are depicted in Fig. 3. Numerical subscripts indicate the number of methylated sites on Tar, and a subscript p indicates that CheA is phosphorylated. The rates for the labeled reaction pairs are given in Table 3. Our primary objective is to model the response to attractant, which probably involves only increases in the average methylation level above the unstimulated level of about methyl esters per transducer (8). For this reason and for simplicity, we consider only the three highest methylation states of Tar. In the model we postulate that only the phosphorylated form of CheB (CheB p ) has demethylation activity. We assume that the phosphorylation state of CheA does not affect the ligand-binding reactions of the receptor with which it forms a complex, and does not affect the activities of CheR or CheB p. We further assume that the formation of a complex between Tar and CheR does not affect the ligand-binding properties of Tar, and that it does not affect the autophosphorylation rate of the attached CheA until the transition to the more highly methylated state (and dissociation of the complex) occurs. We assume that CheZ activity is unmodulated, although it has been suggested (9) that an as-yet-unidentified mechanism for modulating CheZ activity could account for the observed gain. Later we consider the case of modulated CheZ activity. Finally, we assume that the rates of the phosphotransfer reactions between CheA and CheY or CheB are not affected by the occupancy or methylation state of the receptor. In Fig. 2 we illustrate the receptor states and the network of transitions between them that are used in the model. Details of the phosphotransfer reactions are shown in Fig. 3, together with the dephosphorylation reactions for CheB and CheY. The details of the reactions in these figures are given in Table 1. Before we introduce the equations we present a qualitative description of how the system works. First consider the response of the network to a step increase in attractant. The ligand-binding reactions are the fastest, so the first component of the response is a shift in the distribution of states toward the ligand-bound states (i.e., from states at the top to those at the bottom in Fig. 2). This increases the fraction of receptors in the sequestered states (LT 2, LT 2p, and possibly LT 3, LT 3p ), and because these states have the same phosphotransfer rates but much lower autophosphorylation rates than the corresponding ligand-free states, the distribution subsequently shifts toward states containing unphosphorylated CheA. This in turn leads to a reduction in CheY p, a decreased rate of tumbling, and an increase in the run length. This constitutes the excitation response of the system. Next methylation and demethylation, which are the slowest reactions, begin to exert an effect. CheR methylates ligand-bound receptors more rapidly than unbound receptors, and the decrease in CheA p that results from excitation causes a decrease in the level of CheB p, thereby reducing the rate of demethylation. As a result, the distribution shifts toward higher methylation states. However, autophosphorylation of CheA is faster, the higher the methylation state of the Tar CheA CheW complex, and therefore in the last phase of the response there is a shift toward the states containing CheA p via transitions along the right face of the network. Consequently, the net effect of an increase in attractant is to shift the distribution of receptor states toward those which are ligand-bound and more highly methylated, but the total level of receptor complex containing CheA p (the sum of the states at the rear face of the network) returns to baseline. As a result, the total phosphotransfer rate from CheA p species to CheY returns to the prestimulus level, which means that CheY p returns to its prestimulus level. Under the assumption that only CheY p and CheY interact with the motor complex, this in turn implies that the bias returns to its prestimulus level. Thus the cell can respond to an increase in ligand by a transient decrease in tumbling, but it adapts to a constant background level of ligand and retains sensitivity to further changes in the ligand concentration. Of course this qualitative description must be supplemented by numerical results which demonstrate that the model can also produce quantitatively correct results using experimentally based rate coefficients. This is done in the following section. FIG. 3. Detail of the phosphotransfer reactions corresponding to the labels 8 13 in Fig. 2.

3 Biochemistry: Spiro et al. Proc. Natl. Acad. Sci. USA 94 (1997) 7265 Table 1. Details of reactions depicted in Figs. 2 and 3 Description Methylation (first order kinetics) Methylation (Michaelis Menten kinetics) Reaction labels Kinetics and rate labels 1 4 T 2 R 3 k T 2 R º k 1b In ref. 10 we examine the response of a simplified model that incorporates just two methylation states to step changes and to a slow exponential ramp. The results of this analysis impose different conditions on the methylation and demethylation rates, because a significant ramp response results only when these rates are considerably slower than those which reproduce observed adaptation times to small steps. To capture both of these responses a model must contain at least three methylation states, and the transitions between the lowest and intermediate methylation states must be considerably faster than the transitions between the intermediate and highest states. In such a scheme the adaptation time to small steps is controlled by the transitions between the doubly and triply methylated states in Fig. 2, and the ramp response is controlled by the slower transitions between the triply and fully methylated states. The three-methylation-state model described in the following section also captures the correct time scale for the adaptation response to saturating levels of attractant. A Three-State Model Captures the Ramp, Step, and Saturation Responses A minimal model must have three methylation states, and we assume faster methylation and demethylation rates for transitions between the two lowest methylation states, and slower transition rates between the two highest methylation states. We use Michaelis Menten methylation kinetics, since CheR activity is thought to be saturated under normal conditions (11). This precludes an analytical derivation of the relationships between kinetic parameters which guarantee perfect adaptation, by which we mean that the bias returns precisely to baseline in the face of any constant attractant level below saturation, (10). Instead we have tuned the rate constants of the full system by trial and error so that it adapts well over a large range of ligand concentrations. The mathematical description of the model is based on mass action kinetics for all the steps except methylation. Because the equations that govern the evolution of the amounts in the various states of the receptor complex are similar and easy to derive, we only display one of them, and we indicate the process contributing to the rate of change beneath the corresponding rate expression. dt 2 dt Lk «5T 2 k 5 LT 2 Ligand binding release k «8 T 2 Phosphorylation B «pk 1T 3 Demethylation T k 2 yt 2p Y 0 Y p k b T 2p B 0 B p V max ««K R T 2 Phosphotransfer methylation [1] We write the equations for CheY p and CheB p as T 3 R dy p k dt y P Y 0 Y p k y ZY p, [2] k 1a T 2 R k 1c 3 T 3 R Demethylation 1 4 T 3 B p 3 k 1 T 2 B p Ligand binding 5 7 T 2 L º k 5 Autophosphorylation 8 13 T 2 3 k 8 k 5 LT 2 Phosphotransfer 8 13 T 2p B 3 k b T 2 B p Dephosphorylation T 2p T 2p Y 3 k y T 2 Y p B k b p 3 B Y p Z k y 3 Y Z db p k dt b P B 0 B p k b B p, [3] where P T 2 p LT 2 p T 3p LT 3p T 4p LT 4p is the total amount of phosphorylated receptor complex. In addition, the total amounts of T, Y, B, and R are conserved. These four conditions can be used to eliminate four variables, which was done for Y and B in Eqs. 2 and 3, and the resulting system can be integrated numerically, using the parameter values given in the tables. As we remarked earlier, we assume that the motor interacts only with CheY p and possibly CheY, and in view of the small number of motor complexes in a cell, we further assume that the motor does not affect the budgets of these species. It then follows from the conservation of Y that perfect adaptation of the bias only requires perfect adaptation of CheY p. The responses obtained from this three-methylation-state model are shown in Fig. 4A C and the concentrations and rates used are listed and compared with measured values in Tables 2 and 3. The system exhibits both a significant response to a slow ramp and an appropriate adaptation time for a small step. It also exhibits the correct adaptation time to a saturating step in ligand concentration. Analysis of the Gain In the preceding figures we have chosen the step size so that the maximum deviations of the CheY p concentration from baseline are equal in Fig. 4A and B, and thus the maximum change in bias in response to the corresponding ramp (0.015 s 1 ) and step (11% change in receptor occupancy) will be the same. Experimentally it is found that the maximum change in bias in response to this ramp is about 0.3 (figure 4A in ref. 16), and so our model, coupled with a scheme for the interaction of CheY p with the motor, would produce a maximum gain of about in response to the step used in Fig. 4B. This is consistent with the finding (17) that the maximum gain is about 6, though not with the much higher gain of 55 reported (18). The model thus demonstrates that ramp experiment results are consistent with the lower estimates regarding the gain for small steps. However the source of the gain remains undetermined, because even though the deviations of CheY p concentration from baseline are significant ( 9%), they are small compared with the reported change (0.3) in bias for this ramp stimulus (16). Thus one or more mechanisms to amplify the internal CheY p signal must be involved, and the major possibilities are cooperative effects in the interaction of CheY species with the motor and modulation of CheZ activity. To explore these possibilities, we define the gain as g db d ln p, [4] where b is the bias, or probability that a flagellum will be rotating CCW, and p k y P is the pseudo-first order rate constant for phosphorylation of CheY. We have chosen this definition of gain for mathematical convenience, but it can be shown (10) that for small steps this definition of g is consistent with the definition of gain given in ref. 17, namely, the change in bias per percent change in receptor occupancy. When the phosphorylation reactions are at pseudoequilibrium, one can show that y p p z, [5] where y Y p Y 0 [0,1] is the dimensionless amount of CheY in phosphorylated form, and p and z k -y Z are the production and loss coefficients, respectively, of y. We note that

4 7266 Biochemistry: Spiro et al. Proc. Natl. Acad. Sci. USA 94 (1997) FIG. 4. Response of the system depicted in Fig. 2 to various chemoattractant (aspartate) stimulus protocols: a ramp of rate s 1 (Left), a step at t 10 s from zero concentration to a concentration (0.11 M) that is 11% of the K d for ligand binding (Center), and a step at t 100 s from zero concentration to a concentration (1 mm) that is 1,000 times the K d for ligand binding (Right). Time series in the top row of figures are for Che Y p (solid line), CheB p (short dashes), and aspartate (long dashes). The concentrations and rates used are listed and compared to measured values in Tables 2 and 3. The maximum change in [Che Y p ] from baseline for both the ramp and small step responses is 9%. Time series in the bottom row are the corresponding bias responses, using a Hill coefficient of 11 for binding of both Che Y p and the competitive inhibitor Che Y to the motor. (Compare the left trace to figure 4A in ref. 12; the center trace to figure 4 in ref. 2, figure 2 in ref. 14, and figure 8 in ref. 13; and the right trace to figure 8 in ref.3. ln y ln p z 1. [6] p z Thus without modulation of CheZ activity, the fractional change in CheY p levels must be less than the change in receptor occupancy, so that no signal amplification is possible upstream of the interaction of CheY and CheY p with the motor. To examine the effect on the gain of cooperative interactions of CheY species at the motor and modulation of CheZ activity, we expand Eq. 4, using Eq. 5, to obtain g db d ln z y 1 y 1 v 1 w. [7] dy d ln p Here v y(1 y)db dy is the gain due only to cooperative interaction of CheY with the motor, and w dlnz dlnp is, for small steps at least, the fractional change in CheZ activity per fractional change in receptor occupancy (10). We have assumed here that CheZ does not interact directly with the motor (19), and acts only by dephosphorylating CheY. This expression makes it clear that the two possible sources of cooperativity act multiplicatively. To determine the gain achievable, consider first the factor v. We assume that CheY species interact with the motor by binding to the flagellar switch, and we allow for the possibility that both CheY p and unphosphorylated CheY may bind (19). Interaction with the motor may involve cooperativity, either Table 2. Conserved quantities used in the model (see ref. 3) Species T R B Y Z Concentration, M via cooperative binding, or possibly via some cooperative interaction among subunits of the switch which bind individually to CheY molecules. Here we consider the first possibility (the second is treated in ref. 10), and we assume that a motor unit binds either Che Y or Che Y p according to jy p M º M Y p j, [8] ky M º M Y k. [9] Here j and k are the number of molecules of CheY p and CheY, respectively, which bind to the motor, M represents motor unbound to either CheY species, M(Y p ) j represents motor in complex with Y p, and M(Y) k represents motor in complex with Y. We assume that the binding reactions Eqs. 8 and 9 equilibrate rapidly, and study only the steady-state quantities m M 1 M 0 1 A j y j A k 1 y k, [10] m j M Y p j M 0 m k M Y k M 0 A j y j 1 A j y j A k 1 y k, [11] A k 1 y k 1 A j y j A k 1 y k. [12] Here M 0 is the total concentration of motor, and A j and A k are the ratios of the association and dissociation constants for CheY p and unphosphorylated CheY, respectively. We assume that the unbound motor M exists in CCW mode with probability one (19), and that the M j and M k states have probabilities f j and f k, respectively, of being in CCW mode. Then the bias is given by

5 Biochemistry: Spiro et al. Proc. Natl. Acad. Sci. USA 94 (1997) 7267 Table 3. Rates used in the model, and corresponding values from the literature Reaction Rate constant Value Literature value Ref. T 2 R 3 T 3 R k 1c 0.17 s s 1 12 T 3 R 3 T 4 R k 2c 0.1k 1c 0.02k 1c * 13 LT 2 R 3 LT 3 R k 3c 30k 1c 15k 1c 30k 1c 13 LT 3 R 3 LT 4 R k 4c 30k 2c 15k 2c 30k 2c 13 T n R º T n R k 1b k 1a,..., k 4b k 4a 1.7 M 1.7 M 12 T 3 B p 3 T 2 B p k M 1 s M 1 s 1 11 T 4 B p 3 T 3 B p k M 1 s 1 3.0k 1 13 LT 3 B p 3 LT 2 B p k 3 k 1 k 1 13 LT 4 B p 3 LT 3 B p k 4 k 2 k 2 13 L T 3 LT k 5, k 6, k M 1 s M 1 s 1 3 LT 3 L T k 5, k 6, k 7 70 s 1 70 s 1 3 T 2 3 T 2p k 8 15 s 1 17 s 1 14 T 3 3 T 3p k 9 3k 8 T 4 3 T 4p k k 8 LT 2 3 LT 2p k 11 0 LT 3 3 LT 3p k k 8 LT 4 3 LT 4p k k 10 k B T np 3 B p T n k b M 1 s M 1 s 1 11 Y T np 3 Y p T n k y M 1 s M 1 s 1 14 B p 3 B k b 0.35 s s 1 14 Y p Z 3 Y Z k y M 1 s M 1 s 1 14 *Methylation rates of different methylation sites vary by a factor of up to 50. Ligand binding increases methylation rates of different methylation sites by a factor of Demethylation rates of different methylation sites vary by a factor of up to 3. Ligand binding has little effect on demethylation rate. Estimated from figure 10 in ref. 3. Estimated from figure 3 in ref. 11. b m f j m j f k m k 1 f j A j y j f k A k 1 y k 1 A j y j A k 1 y k.[13] One can show that the optimal values of f j and f k are 0 and 1, respectively (10), and using these v is given by v y 1 4 j 1 y ky a k 1 a k. [14] where a k A k (1 y) k and y is the baseline CheY p concentration. Because this expression is monotonically increasing in a k, v(y ) is maximized with respect to a k when a k is as large as possible, indicating that both CheY species should have strong binding affinities for the flagellar switch. In the limit as a k 3, Eq. 14 becomes v y 1 j 1 y ky. [15] 4 Finally, maximizing Eq. 15 with respect to y (subject to y [0, 1]) shows that the maximum possible gain at the flagellar switch is v y 1 n, n max j, k. [16] 4 Thus, a gain of 2.7 requires a Hill coefficient of at least 11 if the only cooperativity is in the interaction of CheY species with the switch. In Fig. 4 D F we show the bias resulting from the CheY p response in Fig. 4 A C using a Hill coefficient of 11 for both CheY species (j k 11). The full expression for the gain now becomes g 1 n 1 w, [17] 4 and so the fractional change in CheZ activity relative to the fractional change in receptor occupancy is w 4 g 1. [18] n Thus, without cooperative effects at the switch, we must have w 10, and for moderate cooperativity at the switch, for example n 6 (20), we must have w 0.8. A gain of 6 (17) requires w 23 and w 3 in these respective cases. Finally, we note that an additional cooperative step probably occurs in the interactions between flagella, since the bias of an individual flagellum in the absence of stimulation [ 0.64 (2)] is less than that of a swimming cell [ 0.9, assuming mean run and tumble durations of 1.1 s and 0.14 s, respectively (21)]. We can estimate what the threshold number of flagella might be for this cooperative interaction if we adopt the voting hypothesis (21, 22), whereby the biases of the individual flagella are identical and independent, and the probability that the flagella will form a bundle is one when the number of flagella turning CCW equals or exceeds a threshold and zero otherwise. Then the bias B of the cell is given by N B N j b j 1 b N j, [19] j (not B N j b j (1 b) N j, as claimed by Weis and Koshland (22)], where N is the total number of flagella and b is the bias of an individual flagellum. For b 0.64 and either n 6or n 8 (23), we find that B 0.9 when N 2, and thus a simple majority rules. Discussion In our model of aspartate signal transduction via Tar, excitation is the result of the reduced autophosphorylation rate of the ligand-bound state of the receptor, which reduces the level of CheA p and CheY p, thereby reducing the tumbling rate. Adaptation results from the enhanced rate of methylation of bound states and the fact that methylation increases the rate of autophosphorylation, which returns CheA p and CheY p to their prestimulus

6 7268 Biochemistry: Spiro et al. Proc. Natl. Acad. Sci. USA 94 (1997) levels. The results we present demonstrate that the model can reproduce the experimentally observed responses to both step increases and slow ramps using experimentally determined values for most of the parameters. The disparity between the time scale of excitation, which is fast, and that of adaptation, which is slow, implies that the transduction system can function as a derivative sensor with respect to the ligand concentration: the DC component of a signal is ultimately ignored if it is not too large. This provides a bacterium with a temporal sensing mechanism without the need for any type of memory beyond that embodied in the disparity in time scales between excitation and adaptation. In ref. 10 we show that, as is seen experimentally (16), the magnitude of the response to a slow ramp is an increasing function of ramp rate, and so we may understand the ramp response as the result of a difference between the rate at which receptors enter the sequestered state via increases in receptor occupancy, and the rate at which they exit via transitions between methylation states. Steeper ramps result in larger differences between the entrance and exit rates, and thus in larger responses. The threshold ramp response (16) occurs when these rates are equal. In response to a ramp, the deviation from baseline of the level of sequestered receptor is an approximately linear function of ramp rate. If the downstream steps in the transduction pathway operate in a linear range, then the bias response will in turn be approximately linear in the ramp rate, consistent with the finding of Block et al. (16). Three methylation states were necessary to accurately reproduce the responses to both step and ramp stimuli, because these responses place competing restrictions on the effective methylation and demethylation rates. For step stimuli given at a baseline attractant concentration of zero (17, 18), adaptation is dominated by the fast transitions between the two lowest methylation states, which provide a sufficiently fast adaptation response. At the higher attractant concentrations of ramp experiments (16, 18), receptors are on average more highly methylated, and so the slower transitions between the two highest methylation states are more prominent, providing a significant ramp response. The sensitivity, or gain, of the signal transduction system is found experimentally to be quite high, but the source of this sensitivity is unknown. We have shown that the sensitivity observed in response to ramp stimuli (16, 18) is consistent with the moderate estimate of 6 for the maximum gain of the system (17) obtained from step experiments, but that cooperativity equivalent to a Hill coefficient on the order of 11 is necessary to produce the desired gain. There are several potential sources of cooperativity. Binding of CheY p to the flagellar switch is thought to be cooperative (20), and we have shown that competitive inhibition by unphosphorylated CheY can enhance the sensitivity. Because the motor contains 26 or 27 subunits of switch protein FliF (19), it is conceivable that high cooperativity occurs either in binding to the switch or else in interactions among switch subunits bound individually to CheY p. CheZ phosphatase activity may also be modulated in a manner that exhibits cooperativity. The system gain could be significant if CheZ activity were positively correlated with the level of sequestered CheA, because small fractional changes in receptor occupancy can correspond to large fractional changes in sequestered CheA. As an example, the cells possess a short form of CheA (CheA s ) which may bind several molecules of CheZ (24). If it does so when the associated transducer is unbound to attractant or highly methylated, and releases these molecules into the cytoplasm upon attractant binding, the effective concentration of CheZ will be raised following the latter event. Alternatively, the CheA s CheZ complex may amplify CheZ phosphatase activity [Wang (1996) cited in ref. 25], which could produce high gain if the complex were to form upon attractant binding. Polymerization of CheZ in the presence of CheY p is another potential mechanism for signal amplification, though it is more likely that this reaction instead enhances the adaptation response (25). A further source of gain might be found in cooperative interactions among receptors in close proximity to one another, a possibility raised by the existence of a nose spot of elevated receptor density (26). We have shown that the effects of cooperativity at different locations in the signal transduction pathway are likely to be multiplicative, and thus the total system gain may be the result of moderate cooperativity occurring at two or more of the above-mentioned locations. For example, a Hill coefficient of 6 at the flagellar switch (20) and a moderate degree of CheZ modulation are sufficient to produce the desired gain. Although the model analyzed herein is specific to signal transduction in bacterial chemotaxis, the structure of the network in Fig. 2 is very similar to those of other signal transduction processes, such as those modeled in refs This suggests that the type of analysis done here will have applicability to other systems. A more complete discussion of aspects of adaptation not treated here, including an evaluation of general models of adaptation, is given in ref. 10. This work was supported in part by National Institutes of Health Grant GM29123 to H.G.O. and National Institutes of Health Grant GM19559 to J.S.P. 1. Berg, H. C. (1990) Cold Spring Harbor Symp. Quant. Biol. 55, Block, S. M., Segall, J. E. & Berg, H. C. (1982) Cell 31, Stock, J. B. (1994) in Regulation of Cellular Signal Transduction Pathways by Desensitization and Amplification, eds. Sibley, D. R. & Houslay, M. D. (Wiley, New York), pp Stock, J. B. & Surette, M. G. (1996) Escherichia coli and Salmonella: Cellular and Molecular Biology (Am. Soc. Microbiol., Washington, DC). 5. Bourret, R. B., Borkovich, K. A. & Simon, M. I. (1991) Annu. Rev. Biochem. 60, Engström, P. & Hazelbauer, G. L. (1980) Cell 20, Springer, M. S., Zanolari, B. & Pierzchala, P. A. (1982) J. Biol. Chem. 257, Boyd, A. & Simon, M. I. (1980) J. 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(1983) J. Bacteriol. 155, Weis, R. M. & Koshland, D. E. (1990) J. Bacteriol. 172, Stewart, R. C. & Dahlquist, F. W. (1987) Chem. Rev. 87, Amsler, C. D. & Matsumura, P. (1995) in Two-Component Signal Transduction, eds. Hoch, J. A. & Silhavy, T. J. (Am. Soc. Microbiol., Washington, DC), pp Blat, Y. & Eisenbach, M. (1996) J. Biol. Chem. 271, Parkinson, J. S. & Blair, D. F. (1993) Science 259, Katz, B. & Thesleff, S. (1957) J. Physiol. (London) 138, Knox, B. E., Devreotes, P. N., Goldbeter, A. & Segel, L. A. (1986) Proc. Natl. Acad. Sci. USA 83, De Young, G. & Keizer, J. (1992) Proc. Natl. Acad. Sci. USA 89,

7 letters to nature Robustness in simple biochemical networks N. Barkai & S. Leibler Departments of Physics and Molecular Biology, Princeton University, Princeton, New Jersey 08544, USA... Cells use complex networks of interacting molecular components to transfer and process information. These computational devices of living cells 1 are responsible for many important cellular processes, including cell-cycle regulation and signal transduction. Here we address the issue of the sensitivity of the networks to variations in their biochemical parameters. We propose a mechanism for robust adaptation in simple signal transduction networks. We show that this mechanism applies in particular to bacterial chemotaxis 2 7. This is demonstrated within a quantitative model which explains, in a unified way, many aspects of chemotaxis, including proper responses to chemical gradients The adaptation property 10,13 16 is a consequence of the network s connectivity and does not require the fine-tuning of parameters. We argue that the key properties of biochemical networks should be robust in order to ensure their proper functioning. Cellular biochemical networks are highly interconnected: a perturbation in reaction rates or molecular concentrations may affect numerous cellular processes. The complexity of biochemical networks raises the question of the stability of their functioning. One possibility is that to achieve an appropriate function, the reaction rate constants and the enzymatic concentrations of a network need to be chosen in a very precise manner, and any deviation from the fine-tuned values will ruin the network s performance. Another possibility is that the key properties of biochemical networks are robust; that is, they are relatively insensitive to the precise values of biochemical parameters. Here we explore the issue of robustness of one of the simplest and best-known signal transduction networks: a biochemical network responsible for bacterial chemotaxis. Bacteria such as Escherichia coli are able to sense (temporal) gradients of chemical ligands in their vicinity 2. The movement of a swimming bacterium is composed of a series of smooth runs, interrupted by events of tumbling, in which a new direction for the next run is chosen randomly. By modifying the tumbling frequency, a bacterium is able to direct its motion either towards attractants or away from repellents. A well established feature of chemoxis is its property of adaptation 10,13 16 : the steady-state tumbling frequency in a homogeneous ligand environment is insensitive to the value of ligand concentration. This property allows bacteria to maintain their sensitivity to chemical gradients over a wide range of attractant or repellent concentrations. The different proteins that are involved in chemotactic response have been characterized in great detail, and much is known about the interactions between them (Fig. 1a). In particular, the receptors that sense chemotactic ligands are reversibly methylated. Biochemical data indicate that methylation is responsible for the adaptation property: changes in methylation of the receptor can compensate for the effect of ligand on tumbling frequency. Theoretical models proposed in the past assumed that the biochemical parameters are fine-tuned to preserve the same steady-state behaviour at different ligand concentrations 17,18. We present an alternative picture in which adaptation is a robust property of the chemotaxis network and does not rely on the fine-tuning of parameters. We have analysed a simple two-state model of the chemotaxis network closely related to the one proposed previously 2,19. The twostate model assumes that the receptor complex has two functional states: active and inactive. The active receptor complex shows a kinase activity: it phosphorylates the response regulator molecules, Figure 1 a, The chemotaxis network. Chemotactic ligands bind to specialized receptors (MCP) which form stable complexes (E), with the proteins CheA and CheW. CheA is a kinase that phosphorylates the response regulator, CheY, whose phosphorylated form (CheY p ) binds to the flagellar motor and generates tumbling. Binding of the ligand to the receptor modifies the tumbling frequency by changing the kinase activity of CheA. The receptor can also be reversibly methylated. Methylation enhances the kinases activity and mediates adaptation to changes in ligand concentration. Two proteins are involved in the adaptation process: CheR methylates the receptor, CheB demethylates it. A feedback mechanism is achieved through the CheA-mediated phosphorylation of CheB, which enhances its demethylation activity. b, Mechanism for robust adaptation. E is transformed to a modified form, E m, by the enzyme R; enzyme B catalyses the reverse modification reaction. E m is active with a probability of a m (l), which depends on the input level l. Robust adaptation is achieved when R works at saturation and B acts only on the active form of E m. Note that the rate of reverse modification is determined by the system s output and does not depend directly on the concentration of E m (vertical bar at the end of the arrow). Figure 2 Chemotactic response and adaptation. The system activity, A, of a model system (the reference system described in Methods) which was subject to a series of step-like changes in the attractant concentration, is plotted as a function of time. Attractant was repeatedly added to the system and removed after 20 min, with successive concentration steps of l of 1, 3, 5 and 7 M. Note the asymmetry to addition compared with removal of ligand, both in the response magnitude and the adaptation time. The chemotactic drift velocity of this system is presented in the inset. Inset: the different curves correspond to gradients V l ¼ 0, 0.01, and 0.05 M/ m. An average change in receptor occupancy of less than 1% per second is sufficient to induce a mean drift velocity of the order of microns per second. Nature Macmillan Publishers Ltd 1997 NATURE VOL JUNE

8 letters to nature which then bind to the motors and induce tumbling. The receptor complexes can be either in the active or in the inactive state, although with probabilities that depend on both their methylation level and ligand occupancy. The average complex activity can be considered as the output of the network, whereas its input is the concentration of the ligand. A quantitative description of the model consists of a set of coupled differential equations describing interactions between protein components (Box 1). The two-state model correctly reproduces the main features of bacterial chemotaxis. When a typical model system is subject to a step-like change in attractant concentration, l (Fig. 2), it is able to respond and to adapt to the imposed change. The adaptation is nearly perfect for all ligand concentrations. The addition (removal) of attractant causes a transient decrease (increase) in system activity, and thus of tumbling frequency. We observe a strong asymmetry in the response to the addition compared with the removal of ligand. This asymmetry has been observed experimentally 14. The chemotactic response of the system has been measured by the average drift velocity in the presence of a linear gradient of attractant (Fig. 2, inset). The system is very sensitive: an average change in the receptor occupancy of 1% per second is enough to induce a drift velocity of 1 micron per second. Figure 3a illustrates the most striking result of the model: we have found that the system shows almost perfect adaptation for a wide range of values of the network s biochemical parameters. Typically, one can change simultaneously each of the rate constants severalfold and still obtain, on average, only a few per cent deviation from perfect adaptation. For instance, over 80 per cent of model systems, obtained from a perfectly adaptive one by randomly changing all of its biochemical parameters by a factor of two, still show 15% Box 1 Two-state model of the bacterial chemotactic network The main component of the two-state model 2,19 is the receptor complex, MCP þ CheA þ CheW (Fig. 1a), considered here as a single entity, E. The complex is assumed to have two functional states active and inactive. A receptor complex in the active state shows a kinase activity of CheA; by phosphorylating the response regulators, CheY, it sends a tumbling signal to the motors. The output of the network is thus the average numberof receptors in the active state, the system activity A. It is assumed that this quantity determines the tumbling frequency of the bacteria. The transformation between A and the tumbling frequency depends on the kinetics of CheY phosphorylation and dephosphorylation, as well as on the interaction of CheY with the motors, which are not considered explicitly in the present model. The receptor complexes are assumed to exist in different forms. Consider a complex methylated on m sites (m ¼ 1; M). Such a complex can either be occupied or unoccupied by the ligand. We denote the concentration of these The presence of a m in the equation is due the fact that CheB demethylates only the active receptors; we have also included two different association rate constants of CheR to the active (a r ) and the inactive (a r ) receptors (see below). d jk is the Kronecker s delta (d jk ¼ 1, when j ¼ k, and is zero otherwise). Similar equations can be written to describe kinetics of {E u mb}, {E u mr}, E 0 m, {E 0 mb} and {E 0 mr}, with additional parameters a 0 m defining the probabilities of E 0 m to be in the active state. For fixed a m and a 0 m the biochemical parameters of this system include nine different rate constants (k l ; k l ; a r ; a r ; d r ; k r ; a b ; d b ; k b ) and three enzyme concentrations (total concentrations of CheR, CheB and receptor complexes). The present model is by no means the only two-state model of the chemotactic network that exhibits robust adaptation and proper chemotactic response. Rather, it is one of the simplest variants that is consistent with the experimental data on the response and adaptation of wild-type E. coli. The main assumptions underlying this model are as follows. complexes by E 0 m and E u m, respectively. Each form of the receptor complex can X The input to the system is the ligand concentration; rapid binding (and be in the active state with a probability depending on both its methylation level and its ligand occupancy. We assume that an occupied receptor complex has the probability 0 m of being in the active state; for an unoccupied receptor, this probability is m. If l is the ligand concentration, B(R) the concentration of CheB (CheR), and {E u mb} the concentration of the E u mcheb complex and so on, the unbinding) of the ligand to the receptor induces an immediate change in the activity of the complex. For simplicity, the binding affinity is assumed to be independent of the receptor s activity and its degree of methylation. This assumption can be relaxed without affecting the main conclusions of our model. model reactions can then be illustrated schematically (see figure). X The methylation and demethylation reaction occur on slower timescales. A central assumption is that CheB can only demethylate active receptors. In addition, the demethylation rate constants do not depend explicitly either on ligand occupancy or on the methylation of the receptor, so that all active receptors are demethylated at the same rate. In the variant of the model discussed here, the phosphorylation of CheB is not considered explicitly; in molecular terms, we assume that the phosphorylated form of CheB, CheB p, does not move freely in the cell. Rather,. a CheB p molecule can only demethylate the same receptor that has phosphorylated it. We note, however, that this assumption can also be readily relaxed (N.B. et al., manuscript in preparation). Robust adaptation is maintained as long as both CheB and CheB p demethylate only the active receptors. X The methylating enzyme, CheR, acts both on active and inactive receptors. Here we assume that the association rate constant for this reaction depends only on the activity of the receptor (a r for inactive, a r for The differential equations describing our model can be written in a standard way from the figure. For instance, the kinetic equation for E u m is active), whereas the dissociation rate constant d r and the catalytic rate constant k r are the same for all forms of the receptor. This assumption can de u m ¼ k dt l le u m þ k l E 0 again be relaxed in various ways. In particular, the accumulated biochemical mþ data indicate that CheR works at saturation and operates at its maximal 8 E u ÿ 12 1 d m;0 ab a m E u 8 mb þ d b 9 mb þ kr E u m 1R 93 þ velocity. In this case, the conclusions of our model are not altered, even if d r, a r and a r depend on the ligand occupancy and on the methylation level 21. h i ÿ 1 1 d mm ar a m E u ÿ 1 8 mr þ a r 1 a m E u mr þ d b E u 9 8 mr þ kb E u 9 mþ1b ÿ 1 m ¼ 1; M Nature Macmillan Publishers Ltd NATURE VOL JUNE 1997

9 letters to nature deviation from perfect adaptation (Fig. 3a, lower panel). When varied separately, most of the rate constants may be changed by several orders of magnitude without inducing a significant deviation from perfect adaptation. In our model we have assumed Michaelis Menten kinetics for simplicity. However, we have found that cooperative effects in the enzymatic reactions can be added without destroying the robustness of adaptation. Similarly, robust adaptation is obtained for systems with different numbers of methylation sites. Multiple methylation sites are thus not required for robust adaptation, but possibly are for allowing strong initial responses for a wide range of attractant and repellent stimuli (N.B. et al., manuscript in preparation). The adaptation itself, as measured by its precision (Fig. 3a), is thus a robust property of the chemotactic network. This does not mean, however, that all the properties are equally insensitive to variations in the network parameters. For instance, Fig. 3b shows that the adaptation time, t, which characterizes the dynamics of relaxation to the steady-state activity, displays substantial variations in the altered systems. Robustness is thus a characteristic of specific network properties and not of the network as a whole: whereas some properties are robust, others can show sensitivity to changes in the network parameters. Plots similar to the ones depicted in Fig. 3 can be obtained in quantitative experiments. A large collection of chemotactic mutants can be analysed for variations in the biochemical rate constants of the chemotactic network components. Alternatively, the rate constants of the enzymes could be systematically modified or their expression varied. At the same time, their various physiological characteristics can be measured, such as steady-state tumbling frequency, precision of adaptation, adaption time, and so on. In this way, the predictions of the model can be quantitatively checked. What features of the chemotactic network make the adaptation property so robust? We propose here a general and simple mechanism for robust adaptation. Let us introduce this mechanism for one of the simplest networks (Fig. 1b), which can be viewed either as an adaptation module, or, as a simplifying reduction of a more complex adaptive network, such as the one presented for bacterial chemotaxis. Consider an enzyme, E, which is sensitive to an external signal l, such as a ligand. Each enzyme molecule is at equilibrium between two functional states: an active state, in which it catalyses a reaction, and an inactive state, in which it does not. The signal level l affects the equilibrium between two functional states of the enzyme: we suppose that a change in l causes a rapid response of the system by shifting this equilibrium. Thus, l is the input of this signal transduction system and the concentration of active enzymes (that is, the system activity, A) can be considered as its output. The enzyme E can be reversibly modified, for example by addition of methyl or phosphate groups. The modification of E affects the probabilities of the active and inactive states, and hence can compensate for the effect of the ligand. In general, then, Aðl Þ ¼ aðl ÞE þ a m ðl ÞE m, where E m and E are the concentrations of the modified and unmodified enzyme, respectively, and a m (l) and a(l) are the probabilities that the modified and unmodified enzyme is active. After an initial rapid response of the system to a change in the input level, l, slower changes in the system activity proceed according to the kinetics of enzyme modification. The system is adaptive when its steady-state activity, A st, is independent of l. A mechanism for adaptation can be readily obtained by assuming a fine-tuned dependence of the biochemical parameters on the signal level, l. This kind of mechanism has been proposed for an equivalent receptor system 17,18. A mechanism for robust adaptation, on the other hand, can be obtained when the rates of the modification and the reverse-modification reactions depend solely on the system activity, A, and not explicitly on the concentrations E m and E. This system can be viewed as a feed-back system, in which the output A determines the rates of modification Figure 3 Robustness of adaptation. a, The precision of adaptation, P, and b, adaptation time, t, to a step-like addition of saturating amount of attractant are plotted as a function of the total parameter variation k, for an ensemble of model systems (see Methods). The time evolution of the system activity A is depicted in the inset for the reference system (solid curve) and for an altered model system, obtained by randomly increasing or decreasing by a factor of two all biochemical parameters of the reference system (dashed curve). Each point in the top graphs in a and b corresponds to a different altered system, out of the total number of 6,157. The reference system is denoted by a black diamond; the particular altered system from the inset is denoted by an open square. Bottom graphs: a, the probability that P is larger than 0.95; b, the probability that t deviates from the adaptation time of the reference system ( 10 min) by less than 5% (solid curve) and by a factor 5 (dashed curve). c, Individuality in the chemotaxis model. The inverse steady-state activity A 1 is plotted as a function of the adaptation time, t. Each point represents an altered system, obtained from the reference system (arrow) by varying the concentration of CheR (between 100 to 300 molecules per cell). Nature Macmillan Publishers Ltd 1997 NATURE VOL JUNE

10 letters to nature reactions, which in turn determine the slow changes in A. With such activity-dependent kinetics, the value of the steady-state activity, A st, is independent of the ligand level, therefore the system is adaptive. Activity-dependent kinetics can be achieved in a variety of ways. As a simple example, consider a system for which only the modified enzyme can be active (a ¼ 0); the enzyme R, which catalyses the modification reaction E E m, works at saturation, and the enzyme B, which catalyses the reverse-modification reaction E m E, can only bind to active enzymes. In this case, the modification rate is constant at all times, whereas the reverse modification rate is a simple function of the activity de m dt A ¼ V R max V B max K b þ A where V B max and V B max are the maximal velocities of the modification and the reverse-modification reactions, respectively, and K b is the Michaelis constant for the reverse modification reaction; we have assumed V R max V B max. For simplicity, we have assumed that the enzymes follow Michaelis Menten (quasi-steady-state) kinetics. The functioning of the feedback can now be analysed: the system activity is continuously compared to a reference stead-state value V R max A st ¼ K b : V B max V R max For A A st, the amount of modification increases, leading to an increase in A; for A A st, the modification decreases, leading to a decrease in A. In this way, the system always returns to its steadystate value of activity, exhibiting adaptation. Moreover, with these activity-dependent kinetics, the adaptation properties is insensitive to the values of system parameters (such as enzyme concentrations), so adaptation is robust. Note, however, that the steady-state activity itself, which is not a robust property of the network, depends on the enzyme concentrations. Thus, the mechanism presented here still provides a way to control the system activity on long timescales, for example by changing the expression level of the modifying enzymes while preserving adaptation itself on shorter timescales. A quantitative analysis demonstrates that, on methylation timescales, the kinetics of the two-state model of chemotaxis can, for a wide range of parameters, be mathematically reduced to the simple activity-dependent kinetics shown in equation (1) (N.B. et al., manuscript in preparation). Robust adaptation thus follows naturally as consequence of the simple mechanism described above. The deviations from perfect adaptation (Fig. 3) are in fact connected to departures from the assumptions underlying this mechanism (such as V R max V B max). This simple mechanism suggests that the various detailed assumptions about the system s biochemistry can be easily altered, provided that the activity-dependent kinetics of receptor modifications is preserved. All variants of the model obtained in this way still exhibit robust adaptation (N.B. et al., manuscript in preparation). Two main observations argue in favour of a robust, rather than a fine-tuned, adaptation mechanism for chemotaxis. First, the adaptation property is observed in a large variety of chemotactic bacterial populations. It is easier to imagine how a robust mechanism allows bacteria to tolerate genetic polymorphism, which may change the network s biochemical parameters. In addition, in genetically identical bacteria some features of the chemotactic response, such as the values of adaptation time and of steadystate tumbling frequency, vary significantly from one bacterium to another, while the adaptation property itself is preserved 20. This individuality can be readily explained in the framework of the present model. The concentrations of some cellular proteins, for example the methylating enzyme CheR, are very low 2, and thus may ð1þ be subject to considerable stochastic variations. In consequence, both adaptation time and steady-state tumbling frequency, which are not robust properties of the network, should vary significantly. Moreover, the present model predicts that both these quantities should show a strong correlation in their variation (Fig. 3c), which has been observed experimentally 20. How general are the results presented here? In addition to explaining response and adaptation in chemotaxis, the present model accounts, in a unifying way, for other taxis behaviour of bacteria mediated by the same network. Indeed, as the network s dynamics is solely determined by the system activity, the system will respond and adapt to any environmental change that affects this activity. Mechanisms of robust adaptation similar to the one introduced above could apply to a wider class of signal transduction networks. Robustness may be a common feature of many key cellular properties and could be crucial for the reliable performance of many biochemical networks. Robust properties of a network will be preserved even if its components are modified through random mutations, or are produced in modified quantities. Systems whose key properties are robust could have an important advantage in having a larger parameter space in which to evolve and to adjust to environmental changes. The degree of robustness in many biochemical networks can be quantitatively investigated. This can be achieved by characterizing a behavioural, a physical or biochemical property while varying systematically the expression level and the rate constants of the network s components. The complexity of biological systems introduce several conceptual and practical difficulties, however. Among the most important is the difficulty of isolating smaller subsystems that could be analysed separately. For instance, in the present analysis, we have neglected the existence of different types of receptors and any crosstalk between them. We have also disregarded the interactions between the chemotaxis network and other components of the cell. In addition, the complexity and stochastic variability of biological networks may preclude their complete molecular description. Rate constants and concentrations of many enzymes can only be measured outside their natural cellular environment and many other network parameters remain unknown. Robustness may provide a way out of both these quandaries: robust properties do not depend on the exact values of the network s biochemical parameters and should be relatively insensitive to the influence of the other subsystems. It should then be possible to extract some of the principles underlying cell function without a full knowledge of the molecular detail Methods Numerical integration of the kinetics equations defining the two-state model (see Box 1) was used to investigate its properties. Computer programs in Cþþ language were executed on an SGI (R4000) workstation using a standard routine (lsode from LLNL). Typical CPU time for finding a numerical solution of a model system is of the order of 1 min. A particular model system was obtained by assigning values to the rate constants and the total enzyme concentrations. Most of our results were obtained for a reference system defined by the following biochemical parameters: the equilibrium binding constant of ligand to receptor is 1 M and the time constant for the reaction is 1 ms (k 1 ¼ 1 ms 1 M 1, k 1 ¼ 1 ms 1 ). CheR methylates both active and inactive receptors at the same rate, with a Michaelis constant of 1.25 M, and a time constant of 10 s (a r ¼ a r ¼ 80 s 1 M 1, d r ¼ 100 s 1, k r ¼ 0:1 1 ), CheB (CheB p ) demethylates only active receptors with a Michaelis constant of 1.25 M and a time constant of 10 s (a b ¼ 800 s 1 M 1, d b ¼ 1;000 s 1, k b ¼ 0:1 s 1 ). The number of enzyme molecules per cell are: 10,000 receptor complexes, 2,000 CheB and 200 CheR (cell volume of 1: l). The probabilities that a receptor with m ¼ 1; 4 methylated sites is in its active state are: 1 ¼ 0:1, 2 ¼ 0:5, 3 ¼ 0:75, 4 ¼ 1 if it is unoccupied, and 0 1 ¼ 0, 0 2 ¼ 0:1, 0 3 ¼ 0:5, 0 4 ¼ 1 if it is occupied. Nature Macmillan Publishers Ltd NATURE VOL JUNE 1997

11 letters to nature Response and adaptation. In a typical assay, a model system was subject to a step-like change in attractant concentration. A system in steady-state, characterized by the system activity A st, was perturbed by an addition or removal of attractant. As a result, the system activity changed abruptly and then relaxed, with the characteristic adaptation time, t, to a new steady-state value A st p. Here p measures the precision of adaptation; perfect adaptation corresponds to p ¼ 1 (see inset in Fig. 3a). Robustness of adaptation. The sensitivity of adaptation precision and adaptation time to variations in the biochemical constants defining a model system was investigated. An ensemble of altered systems was obtained from the reference system by random modifications of its reaction rate constants and enzymatic concentrations, k 0 n. Each alternation of the reference system was characterized by the total parameter variation, k, which is defined as: log ðkþ ¼ S L n¼1 j log ðk n =k 0 nþ j, where k n are the biochemical parameters of the altered system. The altered system was subject to a step-like addition of saturating concentrations of attractant (1 mm), and both the precision of adaptation, p, and the adaptation time, t, were measured. The assay was repeated for various reference model systems, with different values of biochemical parameters and of m, and different variants of the model. The robustness of adaptation (Fig. 3) is independent of these choices. Chemotactic drift velocity. The behaviour of a model system in the presence of a linear gradient of attractant, l, was simulated. The movement of the system was assumed to be composed of a series of smooth runs at a constant velocity of 20 m s 1, interrupted by tumbling events. The tumbling frequency was taken to be a sigmoidal function of the system activity (Hill coefficient, q ¼ 2. Different values of q lead to the same qualitative picture; the sensitivity increases with q). The trajectories were also subject to a rotation diffusion, with D ¼ 0:125 rad 2 s 1 (ref. 9). Attractant concentration was increasing along the x direction, (with l ¼ 1 M at x ¼ 0). The chemotactic drift velocity was estimated by measuring the average x position of a hundred identical simulated systems as a function of time. Received 31 December 1996; accepted 17 April Bray, D. Protein molecules as computational elements in living cells. Nature 376, (1995). 2. Stock, J. B. & Surette, M. in E. coli and S. typhimurium: Cellular and Molecular Biology (ed. Neidhardt, F. C.) (American Soceity of Microbiology, Washington DC, 1996). 3. Parkinson, J. S. Signal transduction schemes of bacteria. Cell 73, (1993). 4. Hazelbauer, G. L., Berg, H. C. & Matsumura, P. M. Bacterial motility and signal transduction. Cell 73, (1993). 5. Bourret, R. B., Borkovich, K. A. & Simon, M. I. Signal transduction pathways involving protein phosphorylation in prokaryotes. Annu. Rev. Biochem. 60, (1991). 6. Adler, J. Chemotaxis in bacteria. Annu. Rev. Biochem. 44, (1975). 7. Bray, D., Bourret, R. B. & Simon, M. I. Computer simulation of the phosphorylation cascade controlling bacterial chemotaxis. Mol. Biol. Cell 4, (1993). 8. Adler, J. Chemotaxis in bacteria. Science 153, (1996). 9. Berg, H. C. & Brown, D. A. Chemotaxis in E. coli analysed by three-dimensional tracking. Nature 239, (1972). 10. Macnab, R. M. & Koshland, D. E. The gradient-sensing mechanism in bacterial chemotaxis. Proc. Natl Acad. Sci. USA 69, (1972). 11. Block, S. M., Segall, J. E. & Berg, H. C. Impulse responses in bacterial chemotaxis. Cell 31, (1982). 12. Koshland, D. E. A response regulator model in a simple sensory system. Science 196, 1055 (1977). 13. Berg, H. C. & Tedesco, P. M. Transient response to chemotactic stimuli in E. coli. Proc. Natl Acad. Sci. USA 72, (1975). 14. Springer, M. S., Goy, M. F. & Adler, J. Protein methylation in behavioural control mechanism and in signal transduction. Nature 280, (1979). 15. Koshland, D. E., Goldbeter, A. & Stock, J. B. Amplification and adaptation in regulatory and sensory systems. Science 217, (1982). 16. Khan, S., Spudich, J. L., McCray, J. A. & Tentham, D. R. Chemotactic signal integration in bacteria. Proc. Natl Acad. Sci. USA 92, (1995). 17. Segel, L. A., Goldbeter, A., Devreotes, P. N. & Knox, B. E. A mechanism for exact sensory adaptation based on receptor modification. J. Theor. Biol. 120, (1986). 18. Hauri, D. C. & Ross, J. A model of excitation and adaptation in bacterial chemotaxis. Biophys. J. 68, (1995). 19. Asakura, S. & Honda, H. Two-statemodel for bacterialchemoreceptor proteins. J. Mol. Biol. 176, (1984). 20. Spudich, J. L. & Koshland, D. E. Non-genetic individuality: chance in the single cell. Nature 262, (1976). 21. Kleene, S. J., Hobson, A. C. & Adler, J. Attractants and repellents influence methylation and demethylation of methyl-accepting proteins in an extract of E. coli. Proc. Natl Acad. Sci. USA 76, (1979). A family of cytokine-inducible inhibitors of signalling Robyn Starr*, Tracy A. Willson*, Elizabeth M. Viney*, Leecia J. L. Murray*, John R. Rayner, Brendan J. Jenkins, Thomas J. Gonda, Warren S. Alexander*, Donald Metcalf*, Nicos A. Nicola* & Douglas J. Hilton* * The Walter and Eliza Hall Institute for Medical Research and The Cooperative Research Center for Cellular Growth Factors, Parkville, Victoria, Australia 3052 The Hanson Centre for Cancer Research, IMVS, Adelaide, Southern Australia, Australia Cytokines are secreted proteins that regulate important cellular responses such as proliferation and differentiation 1. Key events in cytokine signal transduction are well defined: cytokines induce receptor aggregation, leading to activation of members of the JAK family of cytoplasmic tyrosine kinases. In turn, members of the STAT family of transcription factors are phosphorylated, dimerize and increase the transcription of genes with STAT recognition sites in their promoters 1 4. Less is known of how cytokine signal transduction is switched off. We have cloned a complementary DNA encoding a protein SOCS-1, containing an SH2-domain, by its ability to inhibit the macrophage differentiation of M1 cells in response to interleukin-6. Expression of SOCS-1 inhibited both interleukin-6-induced receptor phosphorylation and STAT activation. We have also cloned two relatives of SOCS-1, named SOCS-2 and SOCS-3, which together with the previously described CIS (ref. 5) form a new family of proteins. Transcription of all four SOCS genes is increased rapidly in response to interleukin-6, in vitro and in vivo, suggesting they may act in a classic negative feedback loop to regulate cytokine signal transduction. To identify cdnas encoding proteins capable of suppressing cytokine signal transduction, we used an expression cloning approach. The strategy used the murine monocytic leukaemic M1 cell line that differentiates into mature macrophages and ceases proliferation in response to various cytokines, including interleukin-6 (IL-6), and in response to the steroid, dexamethasone 6,7. Parental M1 cells were infected with the RUFneo retrovirus, into which a library of cdnas from the factor-dependent haemopoietic cell line FDC-P1 had been inserted 8. Retrovirally infected M1 cells that were unresponsive to IL-6 were selected in semi-solid agar culture by their ability to generate compact colonies in the presence of IL-6 and geneticin. One stable IL-6-unresponsive clone, 4A2, was obtained after examining 10 4 infected cells (Fig. 1). A 1.4 kilobase pair (kbp) cdna insert, which we have named suppressor of cytokine signalling-1, or SOCS-1, was recovered by polymerase chain reaction (PCR) from the retrovirus that had integrated into genomic DNA of 4A2 cells. The SOCS-1 PCR product was used to Acknowledgements. We thank J. Stock, M. Surette, A. C. Maggs, U. Alon, L. Hartwell, M. Kirschner, A. Levine, A. Libchaber, A. Murray and T. Surrey for discussion; A. C. Maggs for help with numerical issues; and J. Stock, M. Surette and H. Berg for introducing us to bacterial chemotaxis and pointing out many useful references. This work has been partially supported by grants from the NIH and the NSF. N.B. is a Rothschild Fellow and a Dicke Fellow at Princeton University. Correspondence and requests for materials should be addressed to S.L. ( leibler@princeton.edu). Figure 1 Phenotype of IL-6 unresponsive M1 cell clone, 4A2. Colonies of parental M1 cells (left panel) and clone 4A2 (right panel) cultured in semi-solid agar for 7 days in saline or 100 ng ml 1 IL-6. Nature Macmillan Publishers Ltd 1997 NATURE VOL JUNE