The nonequilibrium mechanism for ultrasensitivity in a biological switch: Sensing by Maxwell’s demons Yuhai Tu* T. J. Watson Research Center, IBM, P. O. Box 218, Yorktown Heights, NY 10598 Communicated by Charles H. Bennett, IBM Thomas J. Watson Research Center, Yorktown Heights, NY, May 20, 2008 (received for review December 19, 2007) The Escherichia coli flagellar motor senses the intracellular concen- focusing on understanding the mechanism for E. coli flagellar tration of the response regulator CheY-P and responds by varying motor’s ultrasensitive response to CheY-P. Our choice of this the bias between its counterclockwise (CCW) and clockwise (CW) particular system is motivated by the recent experimental mea- rotational states. The response is ultrasensitive with a large Hill surements of detailed single motor switching statistics (13). coefficient (Ϸ10). Recently, the detailed distribution functions of the CW and the CCW dwell times have been measured for different Results CW biases. Based on a general result on the properties of the Nonexponential Dwell-Time Statistics and the Breakdown of Detailed dwell-time statistics for all equilibrium models, we show that the Balance. The observable state of the flagellar motor is repre- observed dwell-time statistics imply that the flagellar motor switch sented by a binary variable s: s ϭ 0, 1 corresponds to the CW and operates out of equilibrium, with energy dissipation. We propose CCW rotational states of the motor. The internal state of the a dissipative allosteric model that generates dwell-time statistics switch is described by an integer variable n: n ϭ 0,1,2,...,N consistent with the experimental results. Our model reveals a corresponds to the CheY-P occupancy among the N(ϭ 34) FliM general nonequilibrium mechanism for ultrasensitivity wherein monomers. The stochastic switching kinetics is determined by the switch operates with a small energy expenditure to create high the transition probability rates between these 2 ϫ (n ϩ 1) states as sensitivity. In contrast to the conventional equilibrium models, this illustrated in Fig. 1. The CheY-P binding and unbinding rates for a Ϫ ϩ mechanism does not require one to assume that CheY-P binds to STATISTICS given state (s, n) are ks (n) and ks (n) , which depends on the the CCW and CW states with different affinities. The estimated CheY-P concentration [Y]. The switching rate from the s statetothe energy consumption by the flagellar motor switch suggests that (1 Ϫ s) state for a given FliM occupancy n is ␻s(n). Given these the transmembrane proton motive force, which drives the motor’s transition rates, the steady state probability p(0)(n)instate(s, n), the rotation, may also power its switching. The existence of net s CW (CCW) bias BCW (BCCW), and the distribution function Ps(␶) transitional fluxes between microscopic states of the switch is for dwell time ␶ in the s state can be determined either by predicted, measurement of these fluxes can test the nonequilib- numerical simulation or by solving the master equations (see rium model directly. Both the results on the general properties of Materials and Methods for details). the dwell-time statistics and the mechanism for ultrasensitivity BIOPHYSICS For equilibrium systems, detailed balance is satisfied between should be useful for understanding a diverse class of physical and pairs of states, e.g., ␻ (n)p(0)(n) ϭ ␻ (n)p(0)(n), and, equiva- biological systems. 0 0 1 1 lently, the transition rates obey thermodynamic relations, e.g., ϩ Ϫ Ϫ ϩ k (n)␻ (n ϩ 1)k (n ϩ 1)␻ (n) ϭ k (n ϩ 1)␻ (n ϩ 1)k (n)␻ (n). signal transduction ͉ dissipative system ͉ dwell-time statistics ͉ 0 0 1 1 0 1 1 0 energy consumption ͉ flagellar motor By using these relations in analyzing the master equations for Ps(␶), we discover that the dwell-time distribution function can be expressed as a sum of exponential decay functions with igh sensitivity has been observed in many signaling systems positive definite coefficients [see supporting information (SI) Hin biology ranging from calcium signaling in skeletal muscle Text for details of the proof]: (1) to chemotaxis response in Escherichia coli (2, 3). One of the most studied systems is the flagellar motor switch, where the key N component of the switching complex is a ring of around 34 P ͑␶͒ ϭ ͸ c exp͑Ϫ␭ ␶͒, ␭ Ͼ 0, c Ն 0. [1] identical FliM proteins (4). The motor switches stochastically s j j j j ϭ between the CCW and the CW states, with a bias affected by the j 0 binding of CheY-P to FliM. The conventional picture described We emphasize that this result is valid for any equilibrium model the switch by an equilibrium two-state model where the free with detailed balance, including the Ising-type model (see SI Text energies of the two states depend on the CheY-P concentration, and Fig. S1 for details). From Eq. 1, Ps(␶) satisfies a set of and the transition is driven by thermal fluctuation (5, 6). The constraints: large Hill coefficient in the motor CheY-P response curve (7) is m ͑␶͒ then explained in terms of cooperative interactions between the d Ps FliM molecules. Models falling within this equilibrium frame- ͑Ϫ1͒m Ͼ 0, ᭙ ␶ Ͼ 0, m ϭ 1, 2, 3, . , [2] d␶m work include the classical Monod–Wyman–Changeux (MWC) allosteric model (8–10) and the Ising-type model (11) with in particular, it should be monotonically decreasing (m ϭ 1) and nearest-neighbor interactions. Indeed, cooperative protein in- convex (m ϭ 2). For nonequilibrium systems, some of the teraction has been proposed as a general mechanism for under- standing ultrasensitivity in signaling (12). However, most bio- logical complexes, such as the FliM ring, are embedded in and Author contributions: Y.T. designed research, performed research, analyzed data, and could strongly interact with other components in the system. A wrote the paper. fundamental question therefore arises on how good an approx- The author declares no conflict of interest. imation these equilibrium models are in describing the under- *E-mail: [email protected]. lying biological processes. Are there any relevant nonequilibrium This article contains supporting information online at www.pnas.org/cgi/content/full/ effects? If so, how can they be detected and characterized? In 0804641105/DCSupplemental. this article, we try to address these general questions while also © 2008 by The National Academy of Sciences of the USA www.pnas.org͞cgi͞doi͞10.1073͞pnas.0804641105 PNAS ͉ August 19, 2008 ͉ vol. 105 ͉ no. 33 ͉ 11737–11741 Downloaded by guest on September 26, 2021 kn+ (1)− kn+ () CCW 1 1 A B − − (s=1) kn1 () kn1 (1)+ 1.0 100 ω1 ()n ω0 ()n + + CW kn0 (1)− kn0 () (s=0) kn− () kn− (1)+ p 0 0 ) τ ( 0 (1)n − n (1n + ) N 1 γ=1 P 4 γ=2 τ> & τ < CheY-P occupancy τ> 8 < γ=2 10 −1 st() γ =0 τ ()CCW 0.0 1 0.0 0.2 0.4 0.6 0.8 1.0 0.8 0.9 1.0 1.1 1.2 τ/<τ> [Y]/Kd 0 Fig. 2. The behaviors of the dwell-time distribution and its dependence on t τ ()CW ␥ and [Y]. (A) System starting from being exponential for the equilibrium model (black line) with ␥ ϭ 1, P (␶) shows progressively stronger nonequilib- Fig. 1. Illustration of the general kinetic model for flagellar motor switching 1 rium signatures as ␥ increases: concavity for ␥ ϭ 16 (dotted red line), non- in the state space spanned by two state variables s and n, which represent the monotonicity for ␥ ϭ 256,ϱ (dashed and solid red lines). The solid red line is for rotational state and the CheY-P occupancy of the switch, respectively. A the limit ␥ arrow ϱ and ␻៮ arrow 0 with ␻៮ ␥ ϭ 12.8, which corresponds to the stochastic switching time series is shown for the observable state variable s(t). unidirectional case where at least one of the transitional rates ␻0(n)or␻1(n) is zero for any given n: ␻0(n)␻1(n) ϭ 0. Only in this unidirectional case, P1(␶) vanishes at ␶ ϭ 0. All of the curves correspond to [Y]/Kd ϭ 1 (or BCW ϭ 0.5), other coefficients cj can become negative and consequently some of ϭ ϭ ϭ ␻៮ ϭ ␶ parameters used here are nL 12, nU 22, C 1, 0.05 (except for the solid the constraints for Ps( ) can be violated. red line). (B)The average dwell time ͗␶͘ (solid lines) and the peak time ␶p dotted This general property of the dwell-time statistics provides a lines) versus CheY-P concentration [Y] from our model with ␥ ϭ 256 and other powerful tool in identifying possible nonequilibrium effects. parameters the same as in A, the CCW state (blue lines) and the CW state Violation of any constraints in Eq. 2 is sufficient (but not (green lines) show opposite dependence on [Y]. necessary) to imply breakdown of detailed balance in the underlying kinetics. For the E. coli flagellar motor, detailed statistics of the single motor switching kinetics were measured different model parameters. In Fig. 2A, we show the changes of ␶ ␥ recently by the Cluzel laboratory (13), for a series of given CW the CCW dwell-time distribution functions P1( )as increases. ␥ ϭ ␶ biases and at a higher resolution than previous experiments (14). For 1 (the equilibrium model), P1( ) follows an exponential- The distribution functions of the CW and CCW dwell times are like distribution. As ␥ increases, nonequilibrium characteristics, found to have a distinctive peak at a finite value.