# Elastic coiled-coils act as energy buffers in the ATP synthase

Sébastien Neukirch, Alain Goriely, Andrew Hausrath

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Sébastien Neukirch, Alain Goriely, Andrew Hausrath. Elastic coiled-coils act as energy buffers in the ATP synthase. International Journal of Non-Linear Mechanics, 2008, 43 (10), pp.1064-1073. 10.1016/j.ijnonlinmec.2008.06.008. hal-05388735

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Elastic coiled-coils act as energy buffers in the
ATP synthase

Sébastien Neukirch*, Alain Goriely,† and Andrew C. Hausrath‡

*Laboratoire de Modélisation en Mécanique,
CNRS & Université Pierre et Marie Curie, Paris, France.

† Program in Applied Mathematics, BIO5 Institute, and Department of Mathematics,

‡Department of Biochemistry and Molecular Biophysics,
University of Arizona, Tucson, AZ 85721

E-mail: sebastien.neukirch@lmm.jussieu.fr,

goriely@math.arizona.edu

hausrath@email.arizona.edu

Corresponding author: Andrew C. Hausrath

■ Abstract

In the ATP synthase, transmission of energy from the membrane-embedded F0 sector to the catalytic
F1 sector is accomplished by two stalks composed of coiled-coils. The great efficiency of the enzyme,
despite the presence of a symmetry mismatch between the F1 and F0 sectors, suggests the involvement
of elastic elements that store energy during the catalytic cycle. Here, the stalk subunits γ and b are
investigated as the source of this elastic compliance using a continuum mechanical model of coiled-coils
and energy arguments. The analysis shows that the compliance of both subunits is required for efficient
energy transmission between F0 and F1. In addition, the predicted mechanical properties of coiled-coils
in the ATP synthase suggest mechanisms whereby regulatory subunits influence the enzyme activity.
# 1 Introduction

The F1F0 ATP synthase is a multi-subunit enzyme complex that transduces energy stored as a transmembrane proton gradient into chemical energy in the form of adenosine triphosphate (ATP). A wealth of structural, biochemical and biophysical data are available for this remarkable enzyme [1, 2, 3]. The general structure of the bacterial enzyme is illustrated in Figure 1. The membrane-embedded F0 sector is comprised of subunits a, b, and c, and the catalytic F1 sector is comprised of subunits α, β, γ, δ, and ε. Energy for ATP synthesis is obtained by translocating protons through a channel located at the interface of the a and c subunits in the F0 sector. Proton passage induces a rotation of the c-ring by one subunit thereby transducing the transmembrane proton gradient energy into rotary work. The γ subunit then transmits this rotary work to the F1 sector. As the γ subunit rotates, it interacts with the three catalytic β subunits, yielding the release of newly synthesized ATP molecules. That is, the rotary work is converted into chemical potential energy [4]. The crystallographic structure of the γ subunit shows that this mechanical linkage between F0 and F1 is accomplished through a left-handed coiled-coil [5].

The classic mechanical view of the enzyme presents the c-ring, the γ-shaft, and the ε subunit as parts of a rotor subcomplex which rotates relative to the *ab*2α3β3δ subcomplex. The a, b, and δ subunits act as a stator which enables the rotor to exert torque on the catalytic β subunits. No high-resolution structure is available for the complete F1F0 complex so the detailed molecular arrangement in the stator is not known. However, biochemical evidence shows that the two b subunits form a coiled-coil [7, 8]. Therefore both linkages between the two energy-transducing components of this molecular machine appear to be composed of coiled-coils which, as described below, have mechanical properties exceptionally well-suited to this role.

The stoichiometry of proton usage in ATP synthesis has been a long-standing question [9]. For a mechanism where proton transport is strictly coupled to ATP synthesis, this question could be answered by determining the relative numbers of β and c subunits within an active complex, as a single proton is passed by each c subunit and one ATP is produced by each β subunit. The F1 sector contains 3 β subunits and thus 3 active sites, but paradoxically the number of c subunits varies between organisms [10, 11, 12, 13]. Crosslinking studies indicate that the *E. coli* c-ring contains 10 subunits [10], and since a proton passage causes the c-ring to rotate by one unit, this single translocation event is associated with a rotation of $\frac{2\pi}{10}$ radians. This is the figure
used for the rest of this paper. In a bacterium with a typical transmembrane proton-motive
potential of 150 mV [14], each proton passage could supply as much as 150 meV. Under typical
cellular ATP and ADP concentrations, ATP synthesis requires 12 kcal/mol, equivalent to ~ 19
$kT$ or ~ 520 meV. Under these conditions, 3 proton translocations would not be sufficient to
supply the necessary energy while 4 protons would supply 80 meV extra energy. Irrespective of
the nature of catalytic events themselves, the enzyme must store and accumulate energy<fnref n="1" /> from
each individual proton passage until sufficient work has been done by the rotor to provide the
necessary energy for ATP synthesis [18, 19, 20, 21, 22, 23, 24].

A possible mechanism for energy storage is through the deformation of the coiled-coil stalk subunits $\gamma$ and $b$. The mechanical response of these subunits to extension and twisting can be described quantitatively within a continuum elastic theory of protein coiled-coils [25]. We assume here that the elastic power transmission between the 2 motors of the ATP synthase is via the $\gamma$ and $b$ coiled-coils. We are aware there are complications but, using this simple theory, we consider three scenarios for storing energy in F1F0:

H1. The *b* subunit is rigid, and energy is stored by coiling the $\gamma$ subunit alone.

H2. The *b* subunit is elastic and is anchored laterally on the $\alpha_3\beta_3$ subcomplex. Energy is stored by twisting $\gamma$ and stretching *b*.

H3. The *b* subunit is elastic and is anchored on the top of the $\alpha_3\beta_3$ subcomplex. Energy is stored by coiling both $\gamma$ and *b*: both subunits twist.

To test these hypotheses, we use the detailed mechanical, structural, and chemical information
available and we analyze the energy balance of ATP synthase using a mechano-elastic description
of the behavior of the $\gamma$ and $b_2$ coiled-coils.

**2 Results**

**2.1 Testing H1: Rigid b subunit and energy storage in the γ subunit**

The first hypothesis for the mechanism by which energy storage is accomplished is by overwinding
of the $\gamma$ coiled-coil (see Fig. 2, top). Structural analyses have revealed that the $\gamma$ subunit coiled-

<footnote n="1">¹Unless the pore is leaky which is unlikely in view of the efficiency of the enzyme [15, 16, 17].</footnote>
coil can exist in different winding states [6, 26, 53] relative to that seen in the original structure [5]. The energy stored in the overwound $\gamma$ coiled-coil must approach 520 meV in order to have sufficient energy to synthesize one ATP. Applying an axial torque $M$ on an elastic filament induces a rotation of the top with respect to the bottom. The rotational deformation in the filament is given by the twist $\tau$, that is the axial rotation per unit length. For low torque, the filament behaves linearly: the torque is equal to the twist times the twist rigidity $C: M = C\tau$. A typical value for the twist rigidity of an $\alpha$-helix is computed in [27] and evaluated to be $C_{helix} = 100$ nm $kT$. We describe the mechanical response of a coiled-coil in terms of the mechanical properties of its constituent $\alpha$-helices: the twist rigidity $C$ of the coiled-coil depends on the twist rigidities $C_{helix}$ of its components. In the regime where a coiled-coil is comprised of two $\alpha$-helices close to their natural state, the twist rigidity of the coiled-coil $C$ is twice the twist rigidity of the constituent $\alpha$-helices: $C = 2C_{helix} = 200$ nm $kT$. Then, the torque $M$ and the elastic strain energy $E_{el}$ of a coiled-coil are

$$M = \nu\theta \quad \text{and} \quad E_{el} = \frac{1}{2}\nu\theta^2, \qquad (1)$$

where the linear coefficient of torque response $\nu$ for a coiled-coil of length $L$ is simply $\nu = C/L$. The total axial rotation imposed on the coiled-coil is $\theta = \tau L$, and $\theta = 0$ is the rest state. A coiled-coil can distribute deformation energy over its entire length. Equation (1) implies that a longer coiled-coil is more compliant (smaller $\nu$) and so rotates more (larger $\theta$) under a given applied torque $M$.

We now estimate the elastic energy stored in the overwound conformation of $\gamma$ and compare it to the mechanical work produced by the enzyme. This energy can be estimated from experiments on the F1 motor, where the $b$ subunit is not present and the observed torque is that exerted on $\gamma$ alone by the F1. The torque exerted during ATP hydrolysis by the isolated thermophilic F1 (TF1) reaches 40 pN nm under conditions where it appears that the enzyme is operating reversibly, at essentially 100 percent efficiency [28, 29]. Therefore the reverse reaction, ATP synthesis in the intact F1F0, driven by mechanical rotation of the $c$-ring, should also operate under these conditions [30, 17, 31]. We assume that the torque delivered during these experiments is an intrinsic property of the F1 motor and that it functions with the same mechanics in vivo. This mechanistic view neglects possible experimental effects which might
modify the torque values of the motor, such as attachment to a glass slide rather than to the stator.

The *E. coli* enzyme (EF1) appears to operate at a somewhat higher torque value of 63 pN nm [69]. In this organism, the *c*-ring consists of 10 units, so each proton passage induces a rotation of $\frac{2\pi}{10}$ radians which is transmitted to the $\gamma$-shaft. Energy is accumulated in the twisted shaft until enough is present to drive a catalytic event. At the other end of $\gamma$ there are 3 $\beta$ subunits in the F1 subcomplex, corresponding to a rotation of $2\pi/3$ radians for each ATP produced: when the $\gamma$-shaft is sufficiently twisted the active site is triggered: the shaft clicks round to the next active site, releasing enough energy to synthesize one ATP, and unwinding in the process.

The coiled-coil of $\gamma$ is left-handed and asymmetric. One strand spans residues 198-272 and the other approximately 2/3 of that, residues 1-54. We use as an initial estimate for the length of the coiled-coil, the length of the shorter helix: 8 nm (assuming a rise of $h = 1.53$ Å per residue). Using Eq. (1) with $L = 8$ nm, and $C = 200$ nm $kT$, we evaluate the linear coefficient $\nu$ to be $\nu = 25kT$. This implies a torque $M = 224$ pN nm, much larger than torques measured in single-molecule experiments [28, 69], and a stored energy $E_{el} = 55kT$, nearly triple that needed for ATP synthesis.

However, deletions of segments at both the N- and C-terminus of the $\gamma$ subunit [68, 43, 44] are tolerated by the F1 enzyme and suggest that the upper portion of the stalk may be elastically irrelevant, and that the elasticity may arise from that part of $\gamma$ between the chemical and electrical motors, which has a length of ~ 5 nm. Support for this conclusion is also provided by the finding that some conserved contacts with the $\beta$ subunit in this region do not appear to influence torque generation [70]. The structure also suggests that rotation of $\gamma$ is impeded by exerting torque on the lower domain of the $\beta$ subunit so that the upper part of $\gamma$ would be free-wheeling [46]. In this case, as the deformation would be distributed over a shorter elastic element ($L = 5$ nm), the compliance is decreased and the predicted torque would be increased to $M = 359$ pN nm, and the stored energy would be 88 $kT$, well over 4 times that needed.

We conclude that the $\gamma$ subunit alone is too stiff to explain the observed properties of the enzyme. Therefore an additional elastic element, or energy storage mechanism, must come into play. We propose that the additional compliance is provided by the *b* subunit, so that the $2\pi/3$ rotation of the *c*-ring induces a lower twist of the $\gamma$-shaft, yielding a lower torque value.
## 2.2 The elastic b subunit participates in energy storing

While the identity of the stator subunits is clear, and partial structures of both the *b* and *δ* subunits are available [32, 33, 34], the detailed positions and conformations of these elements within the ATP synthase complex are still a subject of active investigation. In particular the mode of attachment of the stator to the $\alpha_3\beta_3$ subcomplex is unknown. Two distinct major models have been proposed. One proposes that the point of attachment is on the side of the $\alpha_3\beta_3$ subcomplex while the other proposes that the mechanical linkage is at the top [35, 36, 37]. Here we discuss the advantage of each model in explaining the mechanical properties of the enzyme and we present evidence that the energetics is more compatible with the latter model.

The *b* subunit of the *E. coli* enzyme is comprised of 156 residues. Subunit *b* is present in the F1F0 complex with a stoichiometry of 2 and when free in solution the membrane-extrinsic portion forms an elongated, α-helical dimer, strongly suggesting a parallel, right-handed dimeric coiled-coil arrangement [7, 38]. The $b_2$ coiled-coil spans the membrane, reaching from the bottom of the *c*-ring to the top of F1 [1, 35]. An analysis of the mechanical coupling between $b_2$ and the $\alpha_3\beta_3$ subcomplex is not straightforward. When $\alpha_3\beta_3$ attempts to rotate due to torque produced in the F0 sector, an attachment point at the side of $\alpha_3\beta_3$ would subject the $b_2$ coiled-coil primarily to mechanical elongation (axial force) due to the lever-arm effect of $\alpha_3\beta_3$, as well as some twisting. Conversely, an attachment point near the top would subject $b_2$ primarily to twisting, with a small stretching component. Contrasting these two extremes, we make the simplifying assumptions that an attachment point at the side of $\alpha_3\beta_3$ would simply stretch the $b_2$ coiled-coil (Hypothesis H2) while an attachment point at the top would simply twist the $b_2$ coiled-coil (Hypothesis H3). We analyze the energetics of each mechanism in terms of the experimentally measured mechano-chemistry of the ATP synthase.

The X-ray crystallographic structure of a construct consisting of *b* subunit residues 62-122 and subsequent biochemical studies led del Rizzo et al. to propose a parallel dimeric right-handed coiled-coil structure for the *b* subunit [33, 8]. Here we use this model to analyze the mechanical properties of the structure. The portion of the molecule sufficient to mediate dimerization, residues 53-122 [39, 7] is also assumed to be the coiled-coil segment dominating the mechanical properties of the stator. These 70 residues span approximately *L* = 10 nm (assuming a rise of *h* = 1.53 Å per residue). The structure of a coiled-coil has a non-linear influence on its elastic
properties [25]. The presence of an undecad repeat in the sequence of the b subunit, and detailed biochemical studies [7, 8] indicate $b_2$ is a right-handed coiled-coil. This does not affect the twist stiffness of the shaft: $C_γ \simeq C_{b_2}$. Nevertheless, since $b_2$ is longer than $γ$, $b_2$ is more compliant than $γ$. That is, imposing a given rotation $θ$ on $b_2$ is easier (i.e. achieved with lower torque $M$) than imposing the same rotation $θ$ on $γ$, see Eq. (1). A coiled-coil responds to a stretching force by increasing its pitch. Consequently, $b_2$ being less steeply wound than $γ$ (i.e. having larger pitch), its propensity to extend is less than $γ$'s. In conclusion $b_2$ is easier to rotate than $γ$, but harder to extend. As the mechanical responses of the two coiled-coils is different, their combined influence could lead to asymmetric responses in F1F0.

## 2.3 Testing H2: Energy storage shared by the $γ$ and b subunits, $γ$ winds and b stretches

In the case of a lateral attachment of the stator on the side of the $α_3β_3$ subcomplex, the $b_2$ coiled-coil responds to rotation mostly by stretching. The $α_3β_3$ subcomplex is schematically viewed as a sphere with radius $R = 5$ nm and the $b_2$ coiled-coil is supposed as attached on its equator (see Fig. 2, middle). Depending on the length of the b subunit and on the height of the attachment point, the b subunit will stretch in different ways. Under application of a tensile force $F$, a coiled-coil stretches from its rest length $L$ to a length $L'$. We use a linear relation between the tensile force $F$ and the fractional extension $ϵ = (L' - L)/L$:

$$F = \muϵ \quad \text{and} \quad E = \frac{1}{2}\mu Lϵ^2, \qquad (2)$$

where $μ$ is the stretching modulus, the linear coefficient of force response, which is taken to be $μ = 1$ nN (see molecular mechanics simulations in [40]). The tensile force $F$ induces a torque (along the vertical) $M_b = R F$ on the $α_3β_3$ subcomplex. This torque equilibrates with the torque of the $γ$-shaft. When the c-ring rotates (due to proton flux) through an angle $θ$, this rotation is shared by the $γ$-shaft (which rotates by an angle $θ_1$) and the $α_3β_3$ subcomplex (which rotates by an angle $θ_2$), with $θ = θ_1 + θ_2$. The respective values of $θ_1$ and $θ_2$ are selected by the equilibrium relation $M_γ = M_b$ where $M_γ = νθ_1$ and $M_b = R F = Rμϵ$. The fractional extension $ϵ$ is a function of $θ_2$: $ϵ = ϵ(θ_2)$. This function is not easy to evaluate and we give two boundary scenarios in the Appendix. Nevertheless the angles $θ_1$ and $θ_2$ are solutions of the two equations
$\theta = \theta_1 + \theta_2$ and $\nu\theta_1 = R\mu\epsilon(\theta_2)$ which reduce to:

$$ \theta = \frac{R\mu}{\nu}\epsilon(\theta_2) + \theta_2. \qquad (3) $$

We want to solve (3) for $\theta_2$ when $\theta = 2\pi/3$. The magnitude of the factor $R\mu/\nu \simeq 30$ (with $\nu = 40kT$) implies that $\theta_2$ is small (see the Appendix for examples). This means that the rotation $\theta$ is mainly distributed to $\theta_1$ which implies a torque $M_\gamma = \nu\theta_1$ on the $\gamma$ subunit (exerted by $b_2$) of approximately the same magnitude as in Hypothesis H1. Such a torque is too large compared to experimental measurements in $\gamma$ [28, 29, 69], as we saw in Section 2.1. We conclude that, due to the high stretching rigidity of the $b$ subunit, the enzyme in this proposed arrangement is not compliant enough to match the observed mechanical properties.

## 2.4 H3: Energy storage shared by the $\gamma$ and $b$ subunits: both the $\gamma$ and $b$ subunits twist.

We now consider the hypothesis that the rotational work done by the c-ring is distributed between the $\gamma$ and the $b$ subunits, with overwinding the $\gamma$ subunit but underwinding the $b_2$ coiled-coil (see Fig. 2 bottom). In the linear elastic regime, the effects of the two coiled-coils are additive and we can consider $b_2$ and $\gamma$ as a single coiled-coil of length $L = 15$nm, yielding a linear coefficient $\nu = 13.3kT$. The torque needed to achieve a $2\pi/3$ rotation on such a coiled-coil is $M = 120$ pN nm and the stored energy is $E = 29kT$. This torque estimate is near the maximal measured torque (133 pN nm) in EF1 [69] and the energy is more than sufficient for ATP synthesis, suggesting that both coiled-coils are involved in the mechanical function of the enzyme and that an axial attachment model is favored energetically as it avoids the large energetic penalty for extension of the stator. The axial attachment also appears to be consistent with recent electron microscopy data (see Figure 1 of [1]). (These estimates are somewhat higher than the values observed for TF1, but are still closer than that predicted for the arrangements described in sections 2.1 and 2.3.)

The final model for the coupling in the ATP synthase is a pair of elastic filaments with torsional elastic constants $\nu_\gamma$ and $\nu_b$ connected in series (Figure 3), with the $\alpha_3\beta_3$ complex in between them. Under proton passage, the c-ring and the $\gamma$-shaft rotate. The $\alpha_3\beta_3$ complex, which grips $\gamma$, is taken along and the rotation is eventually transmitted to $b_2$. Therefore the work
done by proton passage is distributed between $\gamma$ and $b_2$. We define the angle $\psi$ as the overall twist in the combined element. It is the rotation of the c-ring relative to the a subunit (the assumed anchor point of the $b_2$ element). The angle $\phi$ is the overall twist in $\gamma$, the rotation of the attachment point of $\gamma$ on $\alpha_3\beta_3$ relative to the c-ring. The overall twist in $b_2$ is the difference $\psi - \phi$. The total energy in the combined elastic system is obtained as the sum of the energies in the individual elements. Using Equation 1, we have:

$$E_{tot} = E_{\gamma} + E_{b} = \frac{1}{2}\nu_{\gamma}\phi^{2} + \frac{1}{2}\nu_{b}(\psi - \phi)^{2} \quad (4)$$

For fixed $\psi$, equilibrium requires the energy to be at a minimum:

$$\frac{dE_{tot}}{d\phi} = 0 \quad \text{so that} \quad \phi = \frac{\nu_{b}}{\nu_{b} + \nu_{\gamma}} \psi \quad \text{and then} \quad \psi - \phi = \frac{\nu_{\gamma}}{\nu_{b} + \nu_{\gamma}} \psi. \quad (5)$$

The partitioning of stored work between the two elastic elements is thus determined. The effective elastic constant of the two combined elements is

$$\nu_{tot} = \frac{\nu_{b}\nu_{\gamma}}{\nu_{b} + \nu_{\gamma}} \quad \text{so that} \quad E_{tot} = \frac{1}{2}\nu_{tot}\psi^{2}. \quad (6)$$

The behavior of the effective elastic element and its components is displayed in Figure 3B.

The model posits that the rotation of the c-ring stores energy in this combined element until sufficient energy is available for the synthesis of ATP. This work is distributed between the $\gamma$ and b coiled-coils according to their individual compliances. When the stored energy is sufficient, the $\alpha_3\beta_3$ complex extracts the necessary amount from this energy buffer, decreasing the stored energy and allowing the top of $\gamma$ to rotate by $2\pi/3$ to the next active site, and concomitantly releasing ATP. The bottom of $\gamma$ is attached to the c-ring and does not rotate at the same time. Since the amount of energy in the buffer has changed, it then redistributes between the b and $\gamma$ subunits. Further rotation of the c-ring therefore begins building up the stored energy from the residual left over from the previous catalytic event. The stored energy need not return to the same starting value each time. Therefore some catalytic events may require more additional energy input than others. Because the rotation of the c-ring takes place in discrete steps, this means that the stoichiometry of proton requirements per individual ATP is variable, although over many catalytic events the average must respect the conservation of energy.
## 2.5 The attachment point of the b subunit

Our model differs from that of del Rizzo et al. [8] in that we attribute the torsional strain to underwinding of the $b_2$ coiled-coil whereas in their model, torsional strain is assumed to be due to overwinding. In the linear regime, the magnitude of the deformation energy will be the same for over or underwinding, and so cannot be distinguished on these grounds. The models differ on the position of attachment point for the C-terminus of the b subunits: in [8] $b_2$ is anchored on the equator of the $\alpha_3\beta_3$ subcomplex ($\alpha = 0$ in Fig. 4) and in the present model $b_2$ is attached at the pole ($\alpha = \pi/2$ in Fig. 4). Consequently there is a latitude value, $0 < \alpha_0 < \pi/2$, at which rotation of the $\alpha_3\beta_3$ subcomplex will induce zero winding in the $b_2$ coiled-coil and thus the two models can be seen as limiting cases of a more general mechanism in which the attachment point would be at an intermediate latitude between the equator (hypothesis H2) and the pole (hypothesis H3). We have argued that the known energetics are consistent with the latter arrangement. But consideration of the mechanics in this hybrid model suggest some additional properties which are accessible to experiment. The geometry of the lateral attachment of the b subunit induces a stretching force that has two components: a horizontal that builds up the torque on the $\alpha_3\beta_3$ subcomplex and a vertical component that tends to tilt the $\alpha_3\beta_3$ subcomplex. This tilt would tend to bend the $\gamma$-shaft.

For an attachment point P of the b subunit near the equator of the $\alpha_3\beta_3$ subcomplex, the b subunit is bent when no rotation is present (see Fig. 4). The energy associated with this bending would oppose reversal of the enzyme: this arrangement implies an “energy activation barrier” for switching between ATP synthesis and proton pumping modes, as explained in Section 3.4. To estimate the magnitude of this barrier, note that the length $d(AP)$ depends on the height $H$ of the equator of the $\alpha_3\beta_3$ subcomplex and on the latitude $\alpha$ of the attachment point P. Simple geometry yields $d(AP)^2 = (H + R\sin\alpha)^2 + R^2(1 - \cos\alpha)^2$, see Fig. 4, left. The shape of a slightly bent elastic filament compressed at both ends can be approximated by the arc of a circle in which case the bending energy is $E_b = (1/2) BL\kappa^2$ where $\kappa$ is the curvature of the bent filament and B is its bending rigidity. For the $b_2$ coiled-coil, the bending rigidity B is taken as approximately equal to the twist rigidity $B = C = 200 \text{ nm kT}$. The curvature $\kappa$ of an arc of
circle of length L and chord AP is the solution of equations:

$$ \sin \omega = \kappa d(AP)/2 \quad \text{and} \quad \omega = \kappa L/2. \qquad (7) $$

When the curvature is small ($\kappa L \ll 1$), we have $\kappa^2 L^2 = 24 (1 - d(AP)/L)$. As an example, in the case $R = 5$ nm, $L = 10$ nm, $H = 6$ nm, and $\alpha = 45^\circ$ the value of the bending energy is $E_b = 8.5 kT$. This energy barrier has to be overcome when the enzyme reverses. Note that the $\gamma$ coiled-coil could bend also, so as to partly relieve the bending of the b coiled-coil during this transition between modes, but the energy barrier would still be present.

# 3 Discussion

## 3.1 Role of Elasticity in the Efficiency of the ATP Synthase

In the model presented here, the b subunit is considered as a compliant element rather than as a rigid stator of the molecular motor. The true stator part is the a subunit which is anchored in the membrane. The complementarity between the two coiled-coils during the loading process can help account for the efficiency of the enzyme. Under conditions where proton passage results in energy beyond that needed for synthesis, or in the case of an non-integer ratio between the number of c and $\beta$ subunits [22], the excess (of rotation or energy) can be stored as torsional strain in the coiled-coils. However deformation of the $\gamma$-shaft alone could cause an angular mismatch between the rotation of the c-ring and the positions of the catalytic sites. Our hypothesis H3 suggests that the torsional compliance of the b subunit coiled-coil enables it to store (the major) part of the excess energy, allowing the $\gamma$-shaft to relax and restore the optimal angular relationship for synthesis. The b subunit acts as an energy and torque buffer, maintaining a net torque on the F1 subcomplex, so that subsequent proton passages do not have to restore the applied torque from scratch to the level needed for synthesis. The activity of the ATP synthase is tolerant to changes in length of the b subunit [41, 42], which, in our model, corresponds to changes in the elastic properties of the $b_2$ coiled-coil (Eqs. (1) and (2)).

Since the isolated F1 appears to be a constant-torque motor, it is necessary that other parts of the synthase be able to deliver the appropriate torque. Functioning as energy accumulators, $b_2$ coiled-coils of different lengths will eventually achieve the torque needed for the
mechanochemistry to proceed but would do so after different angular rotations of the *c*-ring.

In our model, the $\gamma$ subunit functions as an accumulator as well. As the function of an accumulator is predicted to be robust to changes in length, making insertions or deletions in the $\gamma$ coiled-coil would be an experimental means to test the proposed model. However, $\gamma$ must make interactions with both the *c*-ring at one end and $\alpha_3\beta_3$ at the other, and so much of $\gamma$ may be under additional functional constraints. Therefore insertions in the $\gamma$ stalk just below the portion contacting the $\alpha$ and $\beta$ subunits would be the best candidate locations for such a test. In contrast with the parallel coiled-coil in the *b*-stalk, which because of its dimeric nature can be extended with a single insertion, $\gamma$ contains an antiparallel coiled-coil and so an extension would require equal insertions in both helices, which would also have to be cleverly devised to maintain as much of the coiled-coil interface as possible. To our knowledge, experiments which change the length of the coiled-coil of $\gamma$ in this way have not been performed. As described above, deletions at the C-terminus of $\gamma$ (in the portion which extends past the coiled-coil) are tolerated and have minimal effect on torque [43, 44]. This is likely because the upper portion is not subject to torsional strain, and is consistent with the notion that the accumulator function of $\gamma$ is localized within the lower portion.

Other sources of inefficiency could be unproductive proton transport or loss of energy during catalysis. The permanent presence of a torsional strain applied on the rotor would have the effect of suppressing rotational diffusion of the *c*-ring which could cause proton leakage. An additional consideration is that the torque-utilizing step in the ATP synthesis reaction might itself have an activation barrier. For such a mechanism the presence of an energy storage mechanism is necessary since there would have to be more energy in the rotor/stator combination than eventually used in the synthesis reaction. The height of this activation barrier could vary with environmental conditions (e.g. pH) independently of the mechanical constraints acting on the mechanism. Therefore the presence of such an activation barrier is a further requirement for an energy storage mechanism.

## 3.2 Comparison with other models

Detailed kinetic models of the ATP synthase mechanism have been devised. They accomplish the remarkable feat of reconciling the large body of experimental details about affinities, rates,
torques, transport properties, and energetics which have been determined for this very complex
enzyme (e.g. [45, 46, 47, 18, 19, 20, 48, 49]). Most of these treatments employ elastic elements to
treat the symmetry mismatch between F1 and F0. For example, the models in [18, 19] employ a
postulated elastic spring identified with the γ subunit and are able to reproduce measured rates
of ATP synthesis and hydrolysis and proton transport over a wide pH range with impressive
accuracy. The authors noted that the elasticity in their model could have contributions from
other subunits including *b* and *β*. Other more structurally detailed models have analysed the
particular structural transitions taking place in the rotary mechanism in terms of rigid body
domain motions connected by elastic hinges [46, 49]. Again, these models have great explanatory
power and can reconcile many diverse experimental details. But the source of the elasticity in
these works was not treated explicitly. Our analysis seeks to understand the source of the
assumed elastic properties in mechanical terms specific to the known structures. We find that
an elastic treatment of deformations of the two coiled-coils in the two stalks of the ATP synthase
is able to account for many of the mechanical properties of the enzyme.

Similar questions motivated a study by Sun et al. [24], following the work of Wang and Oster
on the F1 ATPase [46], where a critical role for elasticity was demonstrated. It was determined
by Sun et al. that the molecular basis of the elastic properties of F1 could be well-described in
terms of deformation of the β-sheet in the β subunit. Their model suggests that ATP binding to
an open β subunit provides the energy to close the β-subunit, concomitantly deforming the β-
sheet and driving the γ subunit to the next position. In turn, γ opens the subsequent β subunit
and releases the bound ADP, assisted by the transition of the β-sheet from the deformed closed
position to the relaxed open conformation. (This is consistent with the open conformations of
the β subunits in the structure of the nucleotide-free α₃β₃ complex [50].) In the ATP synthesis
reaction our coiled-coil theory complements the work of Sun et al. The energy to close the
β subunit is supplied by binding ADP and Pi. In the bound form these reactants equilibrate
rapidly with ATP [51]. The pre-strained β subunit is already closed and so lacks a degree of
freedom in which to store energy in excess of that required to achieve the closed conformation.
So when torque is supplied to γ, strain accumulates *in the stalk* until a point is reached where
the β subunit opens, ATP is released, and γ is able to progress to the next position. In this
picture, the accumulation of strain during ATP synthesis is not in the β subunit but rather in
the stalk subunits.

3.3 Structural Evidence for Elastic Behaviour

Elasticity is introduced in the present mechanical model of the enzyme through the linearly-
elastic response of two coiled-coils (γ and b₂) present in the structure. The presence of two
elastic shafts provides a decoupling between the rotation of the c-ring and the rotation of the
α₃β₃ subcomplex, decreasing the importance of the relative stoichiometry of the c and β subunits.
Evidence for the elastic behaviour of the γ-shaft is provided by crystal structures showing differ-
ent winding states for the γ subunit, supporting the notion of torsional elasticity [52, 6, 26, 53].
In addition, other F1 crystal structures in which γ is partially disordered support the conclusion
that the native state of this subunit has intrinsic flexibility [5, 54, 55]. A monomeric crystal
structure of the soluble portion of the b subunit revealed a straight helix [33]. Detailed bio-
physical and biochemical investigations proved that the solution conformation is a coiled-coil,
documenting the elastic nature of the b subunit and showing that it also can flex while remaining
in the folded conformation [33, 8].

The proposed elastic energy buffer mechanism depends on geometrical parameters which
can vary smoothly rather than in discrete jumps, so as to accommodate the different mechanical
constraints on the model. It was observed by electron microscopy [56, 57] that the distance
between F1 and F0, parameter *H* in our model, could change by 1-2 nm under different solution
conditions which may reflect different modes of operation. These observations provide strong
evidence that the stalk subunits are flexible, but the structural nature of the deformations are
not clear at this resolution. The observed changes in *H* may be associated with the bending of
γ and/or b₂. In our model, a change in the location of the attachment point of the *b* subunit on
the α₃β₃ subcomplex (selected by the parameters *H* and *α*) induces different responses of the
rotor. Additionally, a more detailed analysis of coiled-coil mechanical properties shows there is
a coupling between twisting and extension [25, 58]. The observed changes in distance between
F1 and F0 may reflect contributions from this effect.

The model describes the mechanical behavior of the stalk subunits in terms of elastic filaments with an assumed elastic constant. In this description the elastic response is homogeneously distributed throughout the filament and does not take into account local variations. Rather, it
uses an overall resultant flexibility which results from the combined influence of the many local
interactions contributing to the stability of the structure. We have used an elastic constant value
of $C_{helix}$ = 100nm kT which is an estimate for the flexibility of a generic α-helix [27]. However
there are additional interactions particular to both the γ and b subunits which are not taken
into account when using this value. For example, the elasticity of γ may be due in part to the
presence of proline 44 (using *E. coli* numbering) which introduces a kink in the N-terminal helix
and could introduce additional flexibility to this helix of the coiled-coil. If so, the effect of this
proline residue within the model would be to reduce the elastic constant. Also present in the γ
subunit are various hydrogen bonding and salt bridge interactions, some of which stabilize the
more tightly wound conformations, and others which stabilize the loosely wound conformations.
The effect of such interactions on the elastic constant depends on the relative stabilization of
the over and underwound conformations. If the more tightly wound states form interactions
not present in the loosely wound states, the conformations at higher twist would be stabilized.
Such a coiled-coil would less effectively oppose twist, and would be described with a lower elastic
constant. A lower elastic constant in γ would decrease the need for the additional elastic ele-
ment as described in section 2.1. Although we have inferred the elastic properties indirectly in
the present work, with modern single molecule techniques, it might be possible to measure the
elastic response of the γ subunit directly using mutants in which rotation is prevented [71, 72].

In the *b* subunit, interactions with the side of α₃β₃ (independent of the point of mechanical
attachment) could stabilize either the over- and under-wound states of *b*₂ and so could potentially
either increase or decrease the elastic constant. Our objective here is not to provide a detailed
molecular description but rather examine how the overall architecture of the stalk subunits may
contribute to elastic properties. As attested by the tolerance to insertions in the *b* subunit,
it may be functionally less important what the particular resistance to torque is, and more
important what range of torque can be accomodated. On a more general level, a continuum
mechanical description of the energetics is akin to a thermodynamic description, in that much
can be learned knowing the magnitudes of the energies involved in a process, even in the absence
of a complete understanding of their microscopic origin.
## 3.4 Regulation and Reversal

The requirement for the presence of a stator indicates that the synthetic apparatus of F1F0 can only function against a particular range of torque. Additionally, the isolated F1-ATPase delivers a constant torque irrespective of the load [28]. Therefore the mechanochemistry in either direction requires a mechanism to ensure that the torque is in the proper range. Our model posits that strain builds up in the coiled-coils until this range of torque is reached. We now consider how this energy storage role offers insights into possible regulatory mechanisms.

Under conditions of greatly diminished proton-motive force, more protons are needed to accumulate sufficient energy for synthesis of ATP and so larger angular displacements of the *c*-ring will be required. The reciprocity between the stalks (*i.e.* the mechanism by which overwinding of $\gamma$ is opposed by underwinding of $b_2$) may result in a significant underwinding of the $b_2$ coiled-coil which could play a role in determining the mode of operation of the enzyme. Underwinding of $b_2$ which is anchored to the *a* subunit could change the disposition of the proton pore at the interface of the *a* and *c* subunits so as to drive rotation of the *c*-ring in the opposite sense (cf. [59], Fig. 14). In turn this would maintain the reversed conformation of the *b* subunit until the proton-motive force is restored and the enzyme can revert to ATP synthesis. Therefore the mechanical properties of coiled-coils provide a possible mechanism for adaptation to diminished membrane potential during ATP synthesis, as well as a mechanism for maintaining the proton gradient under a variety of steady state conditions.

The mechanism of switching between ATP synthesis and hydrolysis may also be analyzed based on this mechanical picture. As explained in Section 2.5, if the attachment point of the *b* subunit is placed somewhere in between the equator and the pole (see Fig. 4) the rotation of the $\alpha_3\beta_3$ subcomplex will induce both coiling and stretching of the $b_2$ coiled-coil which would work as described in Scenario B of the Appendix. In the forward gear, when producing ATP, the *b* subunit would work as in Fig. 6, right. When working in the reverse direction the points P and P' would be on the other side of the $\alpha_3\beta_3$ subcomplex. It is then easy to see that to go from one position to the other, the *b* subunit has to pass through a conformation where it is bent (Fig. 6, left or Fig. 4). Ultimately the ATP synthase is an enzyme which equilibrates ATP levels with the proton gradient. Thus switching to proton pumping mode should be favored when ATP concentrations are relatively high but the transmembrane voltage is low. When this
occurs, binding of ATP to the α₃β₃ subcomplex applies torque to γ and induces it to rotate, but
the energy and torque made available have to first overcome the bending energy barrier of the
b subunit, before subsequent ATP turnovers can be utilized for proton transport. The bending
energy in the intermediate configuration acts as an energy barrier preventing the enzyme from
reversing erratically. Indeed, depending on the value of α and H, this energy barrier can vary
from zero to few kT. Note that in the eukaryotic ATP synthases, there are additional regulatory
subunits associated with the stalk, which may act to modulate this energy barrier by binding
and stabilizing either the straight or bent conformation of the stalk [60].

A similar mechanism may also occur in the central stalk. The regulatory ϵ subunit has long
been known as an inhibitor of ATPase activity [61, 62, 57], which was difficult to reconcile with
its location at the base of the central stalk, over 50 Å distant from the catalytic centers in
F1 [52, 6]. Subsequent structural and biochemical work revealed a dramatic conformational
rearrangement of its C-terminal helical hairpin domain [63, 64, 65] which reaches up from the
base of the γ coiled-coil to interact with the α₃β₃ hexamer, possibly influencing the catalytic
properties of the active sites. In this conformation ϵ also serves to prevent backward rotation
and ATP hydrolysis [66, 67], but the mechanism by which this is accomplished is less clear. It
is interesting to note that before these studies were available, based purely on kinetic grounds
Cherepanov et al. delineated the mechanical prerequisite that there be asymmetry between
the synthesis and hydrolysis directions, postulating that "the averaged elastic state during the
turnover of ATP synthase is expected to differ from that of ATPase." [18]. We conclude with
the proposition that under synthesis conditions the extended C-terminal helix of ϵ incorporates
into the γ coiled coil, forming a 3-stranded γ/ϵ coiled-coil. ATP synthesis occurs at higher values
of the proton-motive force. Therefore optimal mechanical functions require a stiffer coupling
to transmit higher torque available from a single proton passage. Mechanically, a 3-stranded
coiled-coil is stiffer than a 2-coiled-coil. Therefore, the switch between 2 and 3 strands would
significantly stiffen the central stalk and introduce the necessary mechanical asymmetry.

**4 Appendix : Two scenarios for the stretching of b**

Depending on the length *L* of the *b* subunit and on the height *H* of the equator of the α₃β₃ subcomplex, the *b* subunit will stretch in different ways. We study two scenarios where *b* is
attached on the equator of $\alpha_3\beta_3$, seen as a sphere of radius $R = 5$ nm. The two scenarios differ in the height $H$.

## 4.1 Scenario A

Here the height of the equator of the $\alpha_3\beta_3$ subcomplex is equal to the length of the b subunit ($H = L$). Consequently when $\alpha_3\beta_3$ is not rotated, b is straight. A rotation, of an angle $\theta$ around the vertical, of the c-ring induces a torsion in the $\gamma$-shaft (of angle $\theta_1$) and a rotation (of angle $\theta_2$) of the $\alpha_3\beta_3$ subcomplex. Consequently the attachment point P of the b subunit moves, thereby stretching it from length $L$ to length $L'$. The two rotations $\theta_1$ and $\theta_2$ add up to $\theta$ the rotation of the c-ring : $\theta = \theta_1 + \theta_2$. The torque in the $\gamma$-shaft is $M_\gamma = \nu\theta_1$ and the torque exerted by the stretching force $F$ in the b subunit is $M_b = RF \sin\theta_3$ (where $\theta_3$ is the tilt of the $b_2$ coiled-coil). To relate the extension $\epsilon$ of the b subunit to the rotation $\theta_2$ of the $\alpha_3\beta_3$ subcomplex, note that $R\theta_2 \simeq L\theta_3$ and $L' = L/\cos\theta_3$, see Fig.5, right. This implies a fractional extension:

$$ \epsilon(\theta_2) \stackrel{\text{def}}{=} \frac{L'}{L} - 1 = \frac{1}{\cos\theta_3} - 1 \left( \simeq \frac{1}{2} \left( \frac{R}{L} \right)^2 \theta_2^2 \right) \quad (8) $$

and the equilibrium equation $M_b = M_\gamma$ reads:

$$ \theta = \frac{R\mu}{\nu} \left\{ \tan \left( \frac{R}{L} \theta_2 \right) - \sin \left( \frac{R}{L} \theta_2 \right) \right\} + \theta_2 \quad (9) $$

which is solved to yield $\theta_2 = 0.86$ rad with $\theta = 2\pi/3$, $R\mu/\nu = 30$ and $L = 10$ nm. This implies $\theta_1 = 1.24$ rad which yields a torque in the $\gamma$-shaft $M_\gamma = 206$ pN nm and stored energies of $E_b = \frac{1}{2}\mu L\epsilon^2 = 11.5$ kT and $E_\gamma = \frac{1}{2}\nu\theta_1^2 = 30$ kT.

This means that when the c-ring rotates by an angle $\theta = 2\pi/3$, $\gamma$ rotates by an angle $\theta_1 = 1.24$ rad and the torque $M_\gamma$ created by this rotation is above the maximal measured value [69]. In conclusion we see that the compliance due to the b subunit (by comparison with a totally rigid b subunit, see section 2.1) is not sufficient to lower the torque present in the $\gamma$ shaft so as to match the experimentally observed values.
## 4.2 Scenario B

Here the b subunit is attached in the same manner as in the previous case but the height H of the equator is reduced in such a way that when $\alpha_3\beta_3$ is not rotated with regard to the a subunit, the b subunit is bent (see Fig. 6, left). A rotation of the $\alpha_3\beta_3$ subcomplex is needed to straighten the b subunit. We define this point as the rest point and consider the effect of stretching b from this point.

In this case a rotation $\theta_2$ of the $\alpha_3\beta_3$ subcomplex will induce a larger stretch of the b subunit than in the previous scenario : for example if the height H is small compared to L, a rotation $\theta_2$ implies $L' - L \simeq R\theta_2$ and $\epsilon(\theta_2) \simeq (R/L)\theta_2$, larger than (8). The equilibrium equation reads:

$$ \theta \approx \frac{R\mu R}{\nu L} \theta_2 + \theta_2 \approx 16\theta_2 \qquad (10) $$

which yields $\theta_2 = 0.13$ rad. A refined (though not exact) geometrical picture (see Fig. 6, right) yields the following equation:

$$ \theta = \frac{R\mu}{\nu} \left( \frac{1}{L} - \frac{1}{\sqrt{R^2\theta_2^2 + L^2 + 2R\theta_2\sqrt{L^2 - H^2}}} \right) (R\theta_2 + \sqrt{L^2 - H^2}) + \theta_2 \qquad (11) $$

which solves to $\theta_2 = 0.19$ rad for $H = 6$ nm. In this case $\theta_1 = 1.91$ rad and $M_\gamma = 318$ pN nm which is too large compared to the maximal experimental value reported in [69]. The energies stored in the two elastic units are : $E_\gamma = \frac{1}{2}\nu\theta_1^2 = 71kT$ and $E_b = \frac{1}{2}\mu L\epsilon^2 = 6.8kT$. We conclude that this setup does not produce the appropriate compliance.
5 Acknowledgments

A.H. and A.G. acknowledge support for this work through a grant from the Bio5 Institute,
and A.H. also thanks the Department of Biochemistry and Molecular Biophysics, University of
Arizona. This material is based in part upon work supported by the National Science Foundation
under Grant No. DMS-0604704 and DMS-IGMS-0623989 to A.G.
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Figure Legends

Figure 1. **Cutaway schematic view of the F1F0 ATP synthase.** The *b* subunit comprising the stator is present in 2 copies in the complex and they are depicted here as a right-handed coiled-coil (red and yellow). (The structure of the *b* subunit in the intact complex is not known in detail, and the pitch and length of the *b*₂ coiled-coil portion have been exaggerated in this representation.) The *γ* subunit (orange) forms a left-handed coiled-coil. The F0 sector transduces energy from proton translocation into rotational work which is used to drive catalysis in the F1 sector (from [6]).

Figure 2. **Three different possibilities for rotor-stator energy storage.** During proton translocation, in H1 (top) the rotor γ-shaft winds; in H2 (middle) the peripheral attachment for the *b* subunit primarily results in extensional deformation; in H3 (bottom) the axial attachment for the *b* subunit primarily results in torsional deformation. In H2 and H3, both rotor and stator act in concert.

Figure 3. **Continuum model of power transmission** A: Mechanical schematic diagram with the elastic element representing $\gamma$ in orange and $b_2$ in yellow. The $\alpha_3\beta_3$ is between the two. The total twist in the combined element is $\psi$. The twist in $\gamma$ is $\phi$ and the twist in $b_2$ is $\psi - \phi$. B: Work done on the combined element (black) as a function of the twist angle $\psi$. The energy stored in $\gamma$ and $b$ are in red and blue respectively. The response of $\gamma$ alone (red dashed line), $b_2$ alone (blue dashed line), to the equivalent twist shows that the combined element is more compliant than either component in isolation. The effect of a decreased elastic constant is to broaden the energy well. C: Energy stored in the combined elastic element vs protons utilized. Energy stored in the combined element increases until sufficient work is done (blue horizontal line) to allow the synthesis of ATP. Top and bottom display the response where proton passage delivers 150 meV or 100 meV respectively.

Figure 4. **Transverse bending of the stator.** Left: The location of the attachment point *P* of the *b* subunit on the $\alpha_3\beta_3$ subcomplex varies with the height *H* of the equator of the subcomplex and the latitude $\alpha$ of *P*. Right: when no rotation is present, the *b* subunit, which has length *L* > *d(AP)*, is bent. This predicts the
presence of an energy barrier for the reversal of the enzyme (see text).

Figure 5. **Stator attachment scenario A** Conformation of the enzyme in the case where the length *L* of the *b* subunit is equal to the height of the attachment point *P*. Left: when no rotation is present. Right (side view): when the *c*-ring and $\alpha_3\beta_3$ subcomplex rotate, the *b* subunit is stretched to a length *L*′.

Figure 6. **Stator attachment scenario B** Conformation of the enzyme in the case where the length *L* of the *b* subunit is larger than the height *H* of the equator of the $\alpha_3\beta_3$ subcomplex. Left: when no rotation is present the *b* subunit is bent. Right (side view): when the *c*-ring and $\alpha_3\beta_3$ subcomplex rotates, the *b* subunit is stretched to a length *L*′.
<figure><img src="image_2.png" /><figcaption>Figure 1:</figcaption></figure>
<figure><img src="image_3.png" /><figcaption>Figure 2:</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>Figure 3:</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>Figure 4:</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>Figure 5:</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Figure 6:</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>Figure 7: For Table of Contents Use Only Manuscript Title: Elastic coiled-coils act as energy buffers in the ATP synthase Authors: Sebastien Neukirch, Alain Goriely, and Andrew C. Hausrath</figcaption></figure>