Physical limits to biochemical signaling

Transcription

1 Physical liits to biocheical signaling Willia Bialek* and Sia Setayeshgar Joseph Henry Laboratories of Physics and Lewis Sigler Institute for Integrative Genoics, Princeton University, Princeton, NJ Counicated by Curtis G. Callan, Jr., Princeton University, Princeton, NJ, May 25, 2005 (received for review February 2, 2005) Many crucial biological processes operate with surprisingly sall nubers of olecules, and there is renewed interest in analyzing the ipact of noise associated with these sall nubers. Twentyfive years ago, Berg and Purcell showed that bacterial cheotaxis, where a single-celled organis ust respond to sall changes in concentration of cheicals outside the cell, is liited directly by olecule counting noise and that aspects of the bacteria s behavioral and coputational strategies ust be chosen to iniize the effects of this noise. Here, we revisit and generalize their arguents to estiate the physical liits to signaling processes within the cell and argue that recent experients are consistent with perforance approaching these liits. A striking fact about biological systes is that single olecular events can have acroscopic consequences. The ost faous exaple is, of course, the storage of genetic inforation in a single olecule of DNA, so that changes in the structure of this single olecule (utations) can have effects on anial behavior and body plan fro generation to generation (). But the dynaics of individual olecular interactions can influence behavior on uch shorter tie scales. Thus, we (and other anials) can see when a single olecule of rhodopsin absorbs a photon (2), and soe anials can sell a single olecule of airborne odorant (3). Even if a single olecular event does not generate a specific behavior, the reliability of behavior still can be liited by inevitable fluctuations associated with counting rando olecular events. Thus, the visual syste has a regie where perception is liited by photon shot noise (4, 5), and the reliability with which bacteria can swi up a cheical gradient appears to be liited by noise in the easureent of the gradient itself (6). It is an open question whether biocheical signaling systes within cells operate close to the corresponding counting noise liits. The analysis of bacterial cheotaxis by Berg and Purcell (6) provided a clear intuitive picture of the noise in easuring cheical concentrations. Their arguent was that if we have a sensor with linear diensions a, we expect to count an average of N ca 3 olecules when the ean concentration is c. Each such easureent, however, is associated with a noise N N. A volue with linear diension a can be cleared by diffusion in a tie D a 2 D, so if we are willing to integrate over a tie we can ake N eas D independent easureents, reducing the noise in our estiate of N by a factor of N eas. The result is that our fractional accuracy in easuring N, and hence in easuring the concentration c itself, is given by c c N N N N eas Dac. [] A crucial clai of Berg and Purcell (6) is that this result applies when the sensor is a single receptor olecule, so that a is of olecular diensions, as well as when the sensor is the whole cell, so that a. The discussion by Berg and Purcell (6) ade use of several special assuptions that we suspect are not required, which leads to soe clear questions: For interactions of a substrate with a single receptor, does Eq. provide a general liit to sensitivity, independent of olecular and biocheical details? Can we understand explicitly how correlations aong nearby receptors result in a liit like Eq. but with a reflecting the size of the receptor cluster? Do the spatial correlations aong nearby receptors have an analog in the tie doain, so that there is a iniu averaging tie required for noise reduction to be effective? Finally, if we can establish Eq. or its generalizations as a real liit on sensitivity for any signaling process (Fig. ), we would like to know if cells actually operate near this liit. We address these questions within the general fraework of statistical echanics through analysis of intrinsic fluctuations of the receptor ligand syste. In ost cases that we know about, biocheical signaling olecules are thought to interact with their receptors through soe kinetic processes that lead to equilibriu between bound and unbound states. In this case, fluctuations in occupancy of a binding site are a for of theral noise. Rather than tracing through the consequences of different icroscopic hypotheses about the nature of the interaction between signaling olecules and their targets, this connection to theral noise allows us to use the fluctuation dissipation theore (7 9), which relates noise levels to acroscopic kinetics in the sae way that Einstein connected the statistics of Brownian otion to the acroscopic frictional forces on the Brownian particle. Our ain result is a derivation of the accuracy liit to which biocheical receptors are able to easure concentrations of signaling olecules. Although analysis of fluctuations in cheically reacting systes has been an active area in theoretical cheistry (0 4), ost existing approaches are based on the Fokker Planck or Langevin equations and have focused on the detailed connection between kinetic paraeters and observable fluctuation spectra. In contrast, our goal is to establish, if possible, general liits on the sensitivity or accuracy of signaling systes that are independent of the often unknown kinetic details. We begin with a siple exaple of binding to a single receptor to present the fluctuation dissipation theore for cheical kinetic systes and to show that we can recover conventional results. By considering the coupled binding and ligand diffusion processes, we derive the accuracy liit to easuring the concentration of a diffusing ligand, where we find a contribution fro the cheical kinetics of the easureent process as well as a lower bound that depends only on olecule counting noise. Within the sae fraework, we extend this result fro a single receptor to ultiple, noninteracting receptors, addressing the ore coplex case of cooperative interactions aong receptors elsewhere (W.B. and S.S., unpublished results). Our results can be suarized by saying that the intuitive estiates of Berg and Purcell in fact correspond to a noise floor that is independent of kinetic details; real systes can be noisier but not ore precise than this liit. We copare our results with two recent quantitative experients on intracellular signaling in Escherichia coli, regulation of gene expression by transcription factors and control of the flagellar otor by the response regulator CheYP, and find *To who correspondence should be addressed. E-ail: wbialek@princeton.edu. Present address: Departent of Physics, Indiana University, Blooington, IN by The National Acadey of Sciences of the USA PNAS July 9, 2005 vol. 02 no. 29

2 Fig.. Measuring the concentration of a signaling olecule by a biological sensor, which in turn controls downstrea events, is a generic task. Here, several exaples are depicted scheatically for E. coli. Binding of attractant repellent olecules to a surface receptor coplex odulates the rate of autophosphorylation of the associated kinase. This change in kinase activity results in a corresponding concentration change of the internal signaling olecule, CheYP, that controls the direction of flagellar otor rotation. Also shown is transcription initiation, where the prooter region can be regarded as a sensor for transcription factors (TF). These proteins, whose concentrations vary depending on the cell cycle and external cues, deterine whether or not RNA polyerase (RNAP) turns on a gene. that the perforance of the cell is near the liit set by diffusive counting noise. Theory Binding to a Single Receptor. Consider a binding site for signaling olecules, and let the fractional occupancy of the site be n.ifwe do not worry about the discreteness of this one site, or about the fluctuations in concentration c of the signaling olecule, we can write a kinetic equation dnt k dt c nt k nt. [2] This equation describes the kinetics whereby the syste coes to equilibriu. The free energy F associated with binding, which is given by the difference in the free energies of the unbound and bound states of the receptor, is related to the rate constant through detailed balance, k c k exp F. [3] k B T If we iagine that theral fluctuations can lead to sall changes k and k in the rate constants, we can linearize Eq. 2 to obtain dn k dt c k n c nk nk. [4] But fro Eq. 3 we have k k F k k k B T. [5] Applying this constraint to Eq. 4, we find that the individual rate constant fluctuations cancel, and all that reains is the fluctuation in the therodynaic binding energy F; the resulting equation can be written in the for k B T dn k BTk c k n F. [6] k c n dt k c n It is useful to note the analogy between this cheical kinetic proble and the Langevin equation (5) for the position, X(t), of an overdaped Brownian particle bound by a Hookean spring. The spring generates a restoring force proportional to position, X, and as the particle oves through the fluid it experiences a viscous drag with drag coefficient, so that the (Newtonian) equation of otion becoes dx X ft, [7] dt where f(t) is a fluctuating force. The dissipative and fluctuating parts of the force on the Brownian particle are related through the fluctuation dissipation theore ftft 2k B T, [8] where k B is the Boltzann constant and T is the teperature; angle brackets denote enseble averages. Intuitively, this relation is a consequence of the fact that the fluctuating and the dissipative forces both arise because of collisions of the Brownian particle with the olecules of the fluid. More generally, the linear response, X(t), of a syste fro equilibriu due to the therodynaically conjugate force, F(t), defines the generalized susceptibility, (t), Xt 0 tft tdt, [9] where we have taken X(t) 0 (Fig. 2). The generalized susceptibility depends on the properties of the syste and copletely characterizes its response to sall external perturbations. Fourier transforing 0 te it dt, [0] the response to an external force near equilibriu becoes X () ()F (). In its general for, the fluctuation dissipation theore relates the iaginary part of the generalized susceptibility, (), which deterines how uch energy is dissipated by a syste as heat due to an external force, to the power spectru of the spontaneous fluctuations of the corresponding coordinate, X, for the closed syste in theral equilibriu S X 2k BT I, [] where I [...] refers to the iaginary part. In the present cheical syste, the coordinates are the concentrations of the BIOPHYSICS PHYSICS Bialek and Setayeshgar PNAS July 9, 2005 vol. 02 no

3 equations, as in Eq. 2, plus the fact that binding is an equilibriu process. Note that in using the fluctuation dissipation theore to arrive at this result, no assuptions are required about the underlying statistics of transitions between the bound and unbound states of the receptor. The Markovian nature of these transitions is reflected in the acroscopic cheical kinetic equations. Fig. 2. For the ass-spring syste iersed in a viscous fluid, easuring the linear response of the position, X(t), to a known, sall external force, F(t), deterines the generalized susceptibility. Fro this susceptibility, the fluctuation dissipation theore can be used to obtain the power spectru of fluctuations in the closed syste at equilibriu, as in Eq.. These fluctuations provide a lower bound on the accuracy of any easureent of the position. If easureents are carried out on N identical ass-spring systes, the expected error is reduced by a factor of N. However, as N increases, this iproveent ceases to hold, as neighboring ass-spring systes becoe physically close enough to experience correlated fluctuations fro collisions with the particle bath, as shown. These correlations have been easured for two optically trapped colloidal particles (6). interacting species, or equivalently the fractional occupancy of receptors, the phenoenological equations of otion are the cheical kinetic equations, and the therodynaically conjugate forces are the free-energy differences aong the species (7, 8) (Table ). Fourier transforing, fro Eq. 6 we find the generalized susceptibility, (), describing the response of the coordinate n to its conjugate force F, ñ F k B T k c n i k c k. [2] The fluctuation dissipation theore relates this response function to the power spectru of fluctuations in the occupancy, n, S n 2k c n 2 k c k 2, [3] where the total variance is n 2 2 c c 2, [4] d n 2 2 S n k B T ñ F 0 [5] k c n n n, [6] k c k and the correlation tie is given by c (k c k ). These are the usual results for switching in a Markovian way between two states; here it follows fro the acroscopic kinetic Coupled Binding and Diffusion. The sae ethods can be used in the ore general case where the concentration itself has dynaics due to diffusion. Now we write dnt k dt cx 0, t nt k nt, [7] where the receptor is located at x 0, and cx, t t D 2 cx, t x x 0 dnt, [8] dt where the last ter expresses the injection of one olecule at the point x 0 as it unbinds fro the receptor. Linearizing the equations as before, and solving Eq. 8 by transforing to spatial Fourier variables, we find the linear response function ñ F k c n k B T where c is the ean concentration, and i k c k, [9] k n d3 k 2 3 i Dk 2. [20] We note that by obtaining the spatial Fourier transfor of Eq. 8 over infinite volue, we are assuing the nuber of ligand olecules to be infinite at constant concentration. Hence, we are considering only the regie where the nuber of ligand olecules exceeds the nuber of receptors. The self-energy () is ultraviolet divergent, which can be traced to the delta function in Eq. 8; we have assued that the receptor is infinitely sall. A ore realistic treatent would give the receptor a finite size, which is equivalent to cutting off the k integrals at soe (large) a, with a the linear diension of the receptor. If we iagine echaniss that read out the receptor occupancy and average over a tie long copared with the correlation tie c of the noise, then the relevant quantity is the low frequency liit of the noise spectru. Hence, we are interested in Da 2 0 k n 2Da, [2] and Table. Linear response in echanical and cheical systes Physical quantity Mass-spring syste Cheical syste Coordinate Displaceent Receptor occupancy, n n n Conjugate force f Free energy change F k B T (k k k k ) Spring constant k B T[n ( n )] Daping constant k B T(k n ) The response of a cheical syste near equilibriu is directly analogous to that of the failiar ass-spring syste in a viscous fluid, in the liit that the inertial ter, MẌ, can be neglected (valid for M 2 4) Bialek and Setayeshgar

4 ñ F k c n i 0 k k B T c k. [22] Applying the fluctuation dissipation theore once again, we find S n 2k c n k c k 2. [23] The total variance in occupancy is unchanged because it is an equilibriu property of the syste. Coupling to concentration fluctuations does serve to renoralize the correlation tie of the noise, c 3 c [ (0)]. The new c can be written as n n n c k 2Dac, [24] so the second ter is a lower bound on c, independent of the kinetic paraeters k, n n c 2Dac. [25] Again, the relevant quantity is the low-frequency liit of the noise spectru, 2n n S n 0 k c k n n2. [26] Dac If we average for a tie, then the root-ean-square error in our estiate of n will be n rs S n 0, [27] and we see that this noise level has a iniu value independent of the kinetic paraeters k, n rs n n Dac. [28] To relate these results back to the discussion by Berg and Purcell (6), we note that an overall change in concentration is equivalent to a change in F by an aount equal to the change in cheical potential, so that cc Fk B T. This equivalence eans that there is an effective spectral density of noise in easuring c S eff c c 2 S k B T F, [29] where the noise force spectru S F () is given by the fluctuation dissipation theore as S F ñ F 2 S n 2k BT In the present case, we find that I F ñ. [30] 2c 2 S eff c k c n c Da. [3] As before, the accuracy of a easureent that integrates for a tie is set by c rs S c eff 0, [32] and we find again a lower bound that is deterined only by the physics of diffusion, c rs c Dac. [33] Note that this result is (up to a factor of ) exactly the Berg Purcell forula in Eq.. Binding to Multiple, Noninteracting Receptors. To coplete the derivation of Berg and Purcell s original results (6), we consider a collection of receptor sites at positions x, where, 2,..., : dn t k dt cx, t n t k n t, [34] cx, t t D 2 cx, t x x dn t. [35] dt i We can solve Eq. 35 by going to a spatial Fourier representation as before, and we find cx, i 2 2 D ñ i 2 2 ñ x x 0 k sinkx x i Dk 2 dk, [36] where is the cut-off wave nuber; as before, the cut-off arises to regulate the delta function in Eq. 35 and is related to the size of the individual receptor. In the liit x x (D) 2, for with,,...,, we have cx, i 2 2 D ñ i 4D ñ x x. [37] Cobining this equation with the Fourier transfor of Eq. 34 and suing to find the total occupancy Ñ() n () of the receptor cluster, we obtain Ñ iñ k c k ik n 2 2 D ik n 4D k c n F ñ x x. [38] In cluster geoetries such that the innerost su is independent of x, we can write the su as k B T ñ Ñ, [39] x x BIOPHYSICS PHYSICS Bialek and Setayeshgar PNAS July 9, 2005 vol. 02 no

5 Suary. For both a single receptor and an array of receptors, we find that the sensitivity of signaling is liited, and this liit can be described as an effective noise in concentration, c rs, deterined by c rs c 2 Fk j, c, Dc. [43] Fig. 3. Scheatic representation of a cluster of receptors of size b, distributed uniforly on a ring of size a. For a b, the relative accuracy in easureent of the substrate concentration iproves as until b a, at which point the bindingunbinding events of nearby receptors are no longer independent. where x x, [40] 2 and this siplification allows us to solve Eq. 38 directly to find the response of Ñ to the force F. Then, as before, we use the fluctuation dissipation theore to find the spectru of F and convert that to an equivalent concentration error as in Eq. 3. The result is c 2 rs 2 c k c n. Dc 2 [4] We note that whereas the first ter is positive and depends on the details of the cheical kinetics of ligand-receptor binding, the second ter defines a lower bound on the easureent accuracy of the ligand concentration by the receptor cluster that depends only on the physics of diffusion. As an exaple, for receptors of radius b uniforly distributed around a ring of radius a b (Fig. 3), this lower bound is c rs c Dc b g 2 0, [42] 2a where for, we have used () g 0 a; g 0 is a geoetric factor of order unity for typical cluster geoetries and receptor distributions (W.B. and S.S., unpublished results). With an increasing nuber of receptors,, the accuracy in easuring the substrate concentration ultiately is liited by the linear size of the cluster. Extension of this result to cooperatively interacting receptors is treated separately elsewhere (W.B. and S.S., unpublished results); rearkably, we find that this lower bound in easureent accuracy persists, independent of the details of cooperative interactions aong the cluster subunits. In this suary, F({k j }, c, ) depends on the details of the kinetic interactions between the signaling olecules and its receptor through the kinetic paraeters, {k j }, the substrate concentration, c, and the nuber of receptors,. The iportant point is that this ter is positive, so that even if we do not know the details of the ligand-receptor cheical kinetics, we know that the noise level can never be saller than that set by the second ter. In the second ter is an effective size for the receptor or the receptor array; then, to within a factor, this ter is exactly that written down by Berg and Purcell (Eq. ) for a perfect concentration easuring device. Thus, although the original Berg and Purcell (6) arguents ade use of very specific assuptions, we see that in the general case the details of the cheical signaling only add to the noise level. Coparison with Experient Here, we consider two experientally well-characterized exaples to deonstrate how the theoretical liits on receptor occupancy noise and the resulting precision of concentration easureents copare with the perforance of real cellular signaling systes. Regulation of Gene Expression in Bacteria. Expression of genes is controlled in part by the occupancy of specific sites in the prootor regions adjacent to the sequences of DNA that code for protein (9). Reversing the usual picture of changes in transcription factor concentration as driving changes in gene expression, we can view gene expression as a sensor for the concentration of the transcription factor proteins that bind to the prooter site. In a bacteriu like E. coli, transcription factors are present in N TF 00 copies in a cell of volue of 3 (20); presuably, the concentration of free transcription factor olecules is saller than N TF V. Diffusion constants for sall proteins in the E. coli cytoplas are D 3 2 s (2); prooter sites have a linear diension a 3 n, and putting these factors together, we find the crucial cobination of paraeters Dac 3s. If the transcription factor is a repressor then gene expression levels are deterined by n, whereas if it is an activator then expression is related to n. Because n rs n ( n ) (Eq. 28), fractional fluctuations in either A n or A n are deterined by A A A Dac. [44] The iniu fluctuations in expression level thus are given by A A 0. A 00 2 N TF 30 s 2. [45] Recent experients (22) indicate that E. coli achieves 0% precision in control of gene expression at sall values of A. For this perforance to be consistent with the physical liits, the transcription achinery ust therefore integrate the prooter site occupancy for ties of the order of in, even assuing that the translation fro occupancy to expression level itself is noiseless. This integration can be provided by the lifetie of the RNA transcripts theselves, which is 3 in in prokaryotes (23). Recent theoretical work, otivated by ref. 22, has ad Bialek and Setayeshgar

6 dressed the difference between intrinsic sources of noise in the regulation of gene expression in bacteria and extrinsic sources of noise that arise fro population-level variations aong cells (24 26). In this context, our results deonstrate the existence of a previously unappreciated intrinsic noise floor due to diffusion of transcription factors. In principle, this noise floor could be reduced by averaging over longer ties, but very recent work on engineered regulatory eleents in E. coli (27) shows that the correlation tie of intrinsic noise is 0 in, suggesting that the tie averaging done by this syste is not uch ore than the iniu required to achieve the observed precision. Control of the Flagellar Motor by CheY. The output of bacterial cheotaxis is control of the flagellar rotary otor (28). The phosphorylated for of the signaling protein CheY (CheYP) binds to the otor and odulates the probability of clockwise vs. counterclockwise rotation (29). Recent easureents (30) show that the probability p of clockwise rotation depends very steeply on the concentration c of CheYP, c h p c h h, [46] c 2 with h 0 and c 2 3 M. In the phenoenological description of the otor as a siple rando telegraph process, switching between clockwise and counterclockwise rotation is governed by Poisson statistics. For c c 2, the switching frequency is easured experientally to be f.5 s.ifwe view the otor as a sensor for the internal essenger CheYP, then the observed behavior of the otor deterines an equivalent noise level of c rs p 2p p c 0 2, [47] where 0 is the correlation tie of the otor state; for the siple telegraph odel it can be shown that 0 2p( p)f. Using Eq. 46 we find c rs 2 c h f. [48] Thus, for c c 2, a single otor provides a readout of CheYP concentration accurate to 2% within 2 s. The otor C ring has a diaeter 2a 45 n, with 34 individual subunits to which the CheYP olecules bind (3). Fro Eq. 42 we find c rs 2s 2, [49] c 27. Schrödinger, E. (944) What is Life? (Cabridge Univ. Press, Cabridge, U.K.). 2. Rieke, F. & Baylor, D. A. (998) Rev. Mod. Phys. 70, Boeckh, J., Kaissling, K. E. & Schneider, D. (965) Cold Spring Harbor Syp. Quant. Biol. 30, Barlow, H. B. (98) Proc. R. Soc. London B 22, Rieke, F., Warland, D., de Ruyter van Stevenick, R. & Bialek, W. (997) Spikes: Exploring the Neural Code (MIT Press, Cabridge, MA). 6. Berg, H. C. & Purcell, E. M. (977) Biophys. J. 20, Callen, H. B. & Greene, R. F. (952) Phys. Rev. 86, Kubo, R. (966) Rep. Prog. Phys. 29, Landau, L. D. & Lifshitz, E. M. (980) Statistical Physics: Part I (Pergaon Press, Oxford). 0. Chen, Y. D. (975) J. Theor. Biol. 55, Chen, Y. D. (977) J. Che. Phys. 65, Chen, Y. D. (977) J. Che. Phys. 66, Chen, Y. D. (978) J. Che. Phys. 68, McQuarrie, D. A. & Keizer, J. E. (98) Theoretical Cheistry: Advances and Perspectives 6A, Langevin, P. (908) C. R. Acad. Sci. (Paris) 46, 530; trans. Leons, D. S. & Gythiel, A. (997) A. J. Phys. 65, Meiners, J. C. & Quake, S. R. (999) Phys. Rev. Lett. 82, Katchalsky, A. & Curran, P. F. (965) Nonequilibriu Therodynaics in Biophysics (Harvard Univ. Press, Cabridge, MA). 8. Bialek, W. (987) Ann. Rev. Biophys. Biophys. Che. 6, where we have taken the size of the individual receptor binding site to be b n, and D 3 2 s as above. Hence, for the collection of receptors coprising the otor, the physical liit to easureents of the CheYP concentration corresponds to 4% precision within 2 s. Taken at face value, our estiates of the actual sensitivity and liiting sensitivity of the otor agree within a factor of three. Recent work shows that at constant CheYP concentration the power spectru of otor bias deviates substantially fro the Lorentzian prediction of the Poisson or telegraph odel (32); in particular, there is peaking in the spectru so that the low frequency liiting noise ay be lower than estiated fro the ean switching frequency. It also ight be the case that even the very steep dependence in Eq. 46 is broadened by sall errors in the concentration easureents (see note 7 in ref. 30), so that we underestiate the actual sensitivity of the otor. Thus, it is possible that there is an even closer agreeent between the cell s perforance and the physical liit, which could be tested in experients pointed ore specifically at this issue. Concluding Rearks We have derived fro statistical echanics the physical liits to the precision of concentration easureent for biological sensors that rely on the binding of a diffusing ligand to a receptor. Our approach copleents and extends the classic work by Berg and Purcell (6), establishing that their intuitive result indeed does set a lower bound on the actual noise level. For a single receptor the accuracy in easureent of concentration is liited by the noise associated with the arrival of discrete substrate olecules at the receptor. Our approach extends in a straightforward way to ultiple receptors without relying on additional considerations; for this case, our result deonstrates ore transparently the role of ultiple receptors in iproving the easureent accuracy, as well as that of correlations in insuring that this iproveent saturates at a level set by the receptor cluster size. Relevant internal or external signaling olecules are often present in low copy nubers, and their concentration in turn regulates downstrea biocheical networks crucial to the cell s functions. For two experientally well-studied exaples, we show that the cell s perforance is close to the physical liits. We thank T. Gregor, R. R. de Ruyter van Steveninck, D. W. Tank, and E. F. Wieschaus for any helpful discussions. 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