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Modeling Torque Versus Speed, Shot Noise, and Rotational

Diffusion of the Bacterial Flagellar Motor

Thierry Mora1, Howard Yu2, and Ned S. Wingreen3

1Lewis-Sigler Institute for Integrative Genomics, Princeton University, Princeton, New Jersey, USA

2Joseph Henry Laboratories of Physics, Princeton University, Princeton, New Jersey, USA

3Department of Molecular Biology, Princeton University, Princeton, New Jersey, USA

Abstract

We present a minimal physical model for the flagellar motor that enables bacteria to swim. Our model

explains the experimentally measured torque-speed relationship of the proton-driven E. coli motor

at various pH and temperature conditions. In particular, the dramatic drop of torque at high rotation

speeds (the “knee”) is shown to arise from saturation of the proton flux. Moreover, we show that

shot noise in the proton current dominates the diffusion of motor rotation at low loads. This suggests

a new way to probe the discreteness of the energy source, analogous to measurements of charge

quantization in superconducting tunnel junctions.

The bacterial flagellar motor is a molecular machine that rotates a helical filament and thereby

powers the swimming of bacteria like E. coli [1]. Motor rotation is typically driven by H+ ions

that generate torque by passing into the cell via the motor, down an electrochemical gradient

called the proton motive force (PMF). Although much work has been devoted to understanding

proton translocation and its coupling to torque generation, biochemical details are lacking and

many questions remain unanswered. An important one is whether ion translocation is

cooperative, i.e., whether protons translocate individually or in groups. Here, we present a

minimal physical model for torque generation (Fig. 1) that not only explains a variety of

previous experimental observations, but also suggests a way to measure the cooperativity of

proton translocation. Specifically, the model predicts that at low loads, motor diffusion is

dominated by proton shot noise with a strong (quadratic) dependence on proton cooperativity.

The flagellar motor operates with near-perfect efficiency at low speeds [2]. As the speed is

increased, e.g., by reducing the load, the torque and efficiency initially remain high—the

“plateau” of the torque-speed relationship (TSR)—and then drop abruptly at a “knee” (cf. Fig.

2). This knee occurs at higher speeds as temperature is increased. Despite much experimental

[3–6] and modeling progress [6–11], the origin of the knee is still poorly understood. In [11,

12], the cause of the knee was argued to be the gating of proton translocation by the relative

position between stator and rotor. In [9], a detailed model of motor kinetics was proposed in

which the knee arises from the crossover between two time scales, one governing mechanical

relaxation and the other proton translocation. In our model, proton translocation, which is

assumed to be the rate-limiting step, is modeled by a barrier crossing event as in [11], with the

difference that the barrier height depends on the mechanical tension between stator and rotor.

The knee in the TSR then arises from the kinetically limited rate of proton translocation.

Importantly, our model fully incorporates proton thermodynamics and yields the separate

dependence of the TSR on the electrical and chemical parts of the PMF.

Three ingredients underlie our model. (i) Each torque-generating unit (MotA/B stator)

contributes independently and additively to the total torque, in agreement with experimental

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Published in final edited form as:

Phys Rev Lett. 2009 December 11; 103(24): 248102.

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observations 2 [13]. (ii) The torque from each MotA/B stator is applied to the rotor by a protein

spring. Proton translocation into the cell causes the stretching of a protein spring to its next

attachment site (Fig. 1) [10,14]. This assumption enforces the tight coupling between proton

current and rotation speed [2]. (iii) Assuming a cooperativity index of n, translocation occurs

through three reversible steps: first n protons load into an external gate, then all n cross an

energy barrier to an internal gate, and finally all n are released into the cell. The barrier crossing

event is the rate-limiting step. The external and internal gates are necessary to explain the

nonlinear dependence of the TSR on proton concentrations [Fig. 2]. (We define the distance

between two attachment sites as nδθ so that the average displacement per proton is δθ.)

The external and internal gates are assumed to be in fast equilibrium with the external and

internal proton concentrations, respectively. Their dissociation constants are denoted by Kext

and Kint so that the occupancy of the external gate is

concentration outside the cell, and similarly for the internal gate. The energy difference

between the internal and external gates is n[eΔψ + τδθ + kBT log(Kint/Kext)]. Δψ is the

transmembrane electric potential (ψint − ψext), and τ δθ is the work necessary to stretch the

protein spring. When Δψ = 0 and τ = 0, the energy barrier per proton to inward translocation

is

, and the barrier to outward translocation is

general, some fraction α of the electric potential and some fraction β of the work contribute to

the inward barrier so that the barriers per proton to inward and outward translocations are,

respectively,

we have neglected any dependence of Kint,ext on Δψ and τ. Then the rate of inward proton

translocations is

, where Hext is the proton

, with

. In

. For simplicity,

(1)

J0 is a kinetic constant (in Hz), and the other prefactors represent the occupancies of the external

and internal gates. The outward rate Jout is given by a similar expression so that the net inward

proton flux is

(2)

Δp := Δψ + (kBT/e) log(Hint/Hext) is the PMF composed of the electrical and chemical potential

differences. Its value is approximately −150 mV in normal conditions. To account for the data,

we assume that the height of the barrier may depend on temperature, and we expand the

prefactor to linear order in temperature:

as a reference temperature.

, where T0 = 17.7 °C, is chosen

Rotation is then described by coupled stochastic equations for the angular position of the rotor

θ and the stretching of the protein springs i = 1, …, N, where N is the number of stators, each

exerting a torque τi on the rotor

(3)

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(4)

ν is the frictional drag coefficient of the load, and k(τ) is the spring constant of the (possibly

non-Hookean) protein springs. The spring constant need not be specified as none of the

observables computed below depend on it. In the second equation, each spring relaxes as the

rotor moves (backreaction of the rotor onto stators, −dθ/dt), but gets restretched by proton

translocations (Ω(τi) := (Jin − Jout)δθ). ξ(t) is the thermal noise on the load and satisfies the

Einstein relation: 〈ξ(t)ξ(t′)〉 = 2(kBT/ν)δ(t − t′). is the shot noise at each stator i, due to the

randomness of proton translocation events. To obtain the average speed ω and torque per stator

τ = 〈τi〉, we solve Eqs. (3) and (4), at steady state (dτi/dt = 0) with no noise, yielding: ω = dθ/

dt = Nτ/ν and τi = τ, with Ω(τ) = ω. Along with Eq. (2), this gives a closed set of equations

from which we obtain the TSR. Note that the resulting expression depends separately on the

electrical and chemical potential for protons. At stall (ν→∞), the system is in equilibrium. The

energy necessary for a protein spring to move to its next attachment site, which is proportional

to the torque τ it exerts on the rotor, is matched by the PMF, τδθ + eΔp = 0. Consequently, the

total torque grows linearly with the PMF, Nτ = −NeΔp/δθ, and the efficiency near stall is

≈100%, in agreement with experiments [15,16].

Our model with no cooperativity (n = 1) can fit all existing measured TSR of the E. coli motor.

Some our model’s parameters are fixed properties of the motor and thus are fit by single values:

(δθ = 4.6°, Kint = 1.2 × 10−8, Kext = 2 × 10−7, J͂0 = 670 Hz, α = 0.2, β = 0.078, η = 0.11 K−1)

while others depend on conditions (T, Hint,ext, Δψ, number of stators N), and may or may not

have been measured in the experiments.

Figure 2 shows fits of TSRs measured under various pH conditions [5]. The electric potential

Δψ was not measured and so was used as a fitting parameter for each set of pH conditions. Our

fit indicates that |Δψ| increases with pH (Fig. S1 [17]), consistently with previous measurements

[18,19].

Electrorotation experiments [3,7] have been used to apply an external torque on the load via

an oscillating field. When the motor is driven backwards (upper-left quadrant of the TSR), the

internal torque is approximately equal to its stall value up to speeds of−100 Hz [7]. When the

motor is driven to speeds larger then the maximum operating speed (lower-right quadrant of

the TSR), the motor resists rotation, resulting in a negative internal torque. In this regime, the

slope of the TSR remains approximately the same as for positive torques beyond the knee

[3]. Our model agrees with measurements in both regimes. The absence of a barrier to backward

rotation follows from the reversibility of proton translocation. For negative torques, the model

predicts an inflection of the TSR, as seen for the red curve in Fig. 2.

Within our model, the proton flux is limited by the loading of the external and internal gates

and by the barrier crossing. This limitation on flux accounts for the knee of the TSR. In Fig.

2, the position of the knee strongly depends on pH values: as the internal proton concentration

increases, the internal gate gets saturated, preventing protons from translocating inwards, and

thus limiting the stretching of stator springs and the applied torque.

Our model also accounts quantitatively for measurements of the TSR at different temperatures

(Fig. S2 [17]) [4], as well as for measurements with different numbers of stators [Inset of Fig.

2] [6]. We compared the model to earlier measurements at different temperatures [3] (data

reported in [4]), with no additional fitting parameters, and found excellent agreement (Fig. S3

[17]).

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Experiments by Gabel and Berg (Fig. 3) [20] have been interpreted to imply that the rotation

speed is proportional to PMF, even at high speeds beyond the knee of the TSR. Our model

predicts that speed is proportional to PMF at low speeds, in the plateau region of the TSR.

However, at high speeds, the torque is limited by the proton flux, and therefore both torque

and speed grow sublinearly with PMF. Nevertheless, our model (with n = 1) is fully consistent

with the measurements reported in [20]. Experimentally, speeds were simultaneously recorded

for two motors of the same cell, one rotating the cell itself (high load) and the other rotating a

small polystyrene bead (low load), as shown schematically in the lower inset of Fig. 3. These

two loads correspond to the two dashed lines in the upper inset of Fig. 3. As cells were

deenergized by the introduction of a respiratory poison, the PMF Δp regressed from −150 mV

to 0, and the motors slowed, with the two speeds approximately proportional to each other,

even at low temperature where the low-load, high-speed motor was in the kinetically limited

regime. We fitted the data for each cell, using N/νcell and N/νbead as free parameters, and

assuming that during deenergization the electric and chemical parts of the PMF regressed in

fixed ratio to each other. The model fits are consistent with the data, both at 24 °C (Fig. S6

[17]) and at 16.2 °C (Fig. 3). However, a further reduction of the low load would be predicted

to lead to a strong deviation from proportionality (dotted line in Fig. 3).

Adding cooperativity (n>1) to the model still captures the general shape of the TSR, as well

as its temperature dependence (Fig. S4 [17]), but agrees poorly with the observed pH

dependence (Fig. S5 [17]), as it implies a stronger dependence of proton current on ion

concentrations. The model with n>1 is consistent with the Gabel-Berg data for most cells (Fig.

S7 [17]), although it breaks down for the two cells with the largest high speeds.

Although n = 1 appears to best explain the data, measurements of average rotation speeds do

not allow us to discriminate with certainty the proton cooperativity. In contrast, our model

predicts that diffusion of the rotor angle at low loads should depend very strongly on proton

cooperativity. The discrete nature of proton translocations implies the existence of shot noise

in the stretching of the protein spring at each stator i. We can approximate the shot noise

as Gaussian white noise:

approximation is valid for times much larger than

including shot noise and thermal noise ξ(t), we find an exact expression for the effective

diffusion coefficient of the rotor angle:

, with Dshot = (1/2)(Jin + Jout)nδθ2 (this

). Solving Eqs. (3) and (4),

(5)

where N is the number of stators and µ(τ) = −(dΩ/dτ)−1 is minus the local slope of the TSR.

Figure 4 shows the effective diffusion coefficient as a function of motor speed for different

proton cooperativities. Parameters were chosen so that the speed at zero torque is 200 Hz.

At high loads (ν ≫ µ), diffusion is entirely due to thermal noise: Deff ≈ kBT/ν. However, at low

loads (ν ≪ µ), diffusion is dominated by shot noise: Deff ≈ Dshot/N. In fact, the thermal noise

is completely suppressed in the low-load limit: e.g., a small thermally induced backward jump

in rotor angle causes the stretching of all springs, which then rapidly pull the rotor forward,

thus canceling the jump. Notice that in the low-load limit, the shot-noise contribution is

inversely proportional to the number of stators. Intuitively, a small jump in the angular stretch

Δθi of one protein spring ultimately only causes the rotor to move Δθi/N because the rotor is

equally coupled to all N stators. The variance per jump is therefore (Δθi/N)2, and with N

independent stators, the resulting diffusion scales as 1/N.

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Equation (5) could be used to infer n experimentally. In the low-load limit, we have Jin ≫

Jout, and therefore n ≈ 2Deff/(ωδθ). δθ can be determined by measuring the torque per stator

and the PMF at stall: δθ = −eΔp/τ.

Our analysis of rotor diffusion suggests a novel experimental test to investigate the

cooperativity of proton translocation. Some rotational diffusion measurements have already

been made [21,22], but not in the regime of very low load, where shot noise is expected to

dominate. Although we have derived the expression for diffusion in the specific framework of

our minimal spring model, the same approach is generalizable to more detailed models of the

bacterial flagellar motor.

Acknowledgments

We thank Avidgor Eldar, Yigal Meir, and Anirvan Sengupta for helpful suggestions. T. M. was supported by the

Human Frontier Science Program and N. S.W. by National Institutes of Health Grant No. R01 GM082938.

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FIG. 1.

(color). Schematic model of the bacterial flagellar motor. Left: The passage of a proton, or

possibly a group of protons, through a torque-generating unit (a MotA/B stator—only one

stator of about 10 is shown) causes a protein spring to stretch to its next attachment site,

represented by circles, on the rotor. Right: To translocate, a proton must pass through an

external gate, over a barrier, and finally through an internal gate, with all the steps assumed to

be reversible. The net energy difference driving proton translocation is the electrical potential

energy, −eΔψ, minus the work, τδθ, necessary to stretch the protein spring by δθ, where τ is

the torque applied by the spring to the rotor.

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FIG. 2.

(color). The torque-speed relationship (TSR) of the E. coli flagellar motor. Rotation speed is

measured with beads of various loads (data points) attached to a flagellar stub, under different

pH conditions (colors) [5]; solid curves: model fits. Inset: Total torque vs speed from [6],

normalized by the number of stators. Data collapse indicates that stators contribute

independently and additively to the total torque. Solid curve: model TSR for a single stator

using the same parameters as in the main figure and Fig. S2 [17], with the temperature fit as

21 °C and the stall torque fit as 300 pN · nm.

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FIG. 3.

(color). Relationship between the high-speed (low-load) and low-speed (high-load) regimes of

the E. coli flagellar motor, as the PMF varies from −150 mV to 0, at temperature T = 16.2 °C.

Symbols: Experimental data from individual cells [20]. Solid curves: model fits with the same

parameters as in Fig. 2. Dotted curve: the model predicts loss of proportionality at very low

loads. Lower inset: schematic of the experiment [20]. Upper inset: model TSR for PMFs of

−150, −100, and −50 mV; the dashed lines show the low- and high-load lines of the cell

represented by ◊ in the main panel, and the dotted line is the load line corresponding to the

dotted curve in the main panel.

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FIG. 4.

(color). Effective diffusion as a function of rotation speed for a single stator (N = 1), with

different ion translocation cooperativities n = 1, … , 4 (bottom to top). Inset: schematic of

cooperative ion translocation.

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