Making ATP

INAUGURAL ARTICLE

Making ATP Jianhua Xing, Jung-Chi Liao, and George Oster* Departments of Molecular Cell Biology and Environmental Science, Policy, and Management, University of California, Berkeley, CA 94720-1132 This contribution is part of the special series of Inaugural Articles by members of the National Academy of Sciences elected on April 20, 2004.

We present a mesoscopic model for ATP synthesis by F1Fo ATPase. The model combines the existing experimental knowledge of the F1 enzyme into a consistent mathematical model that illuminates how the stages in synthesis are related to the protein structure. For example, the model illuminates how specific interactions between the ␥, ␧, and ␣3␤3 subunits couple the Fo motor to events at the catalytic sites. The model also elucidates the origin of ADP inhibition of F1 in its hydrolysis mode. The methodology we develop for constructing the structure-based model should prove useful in modeling other protein motors. ATP synthase 兩 ATP synthesis 兩 F1Fo ATPase

M

echanochemical proteins convert the energy of chemical bonds into mechanical forces and, in the case of one ancient protein, vice versa. ATP synthase (also called F1Fo ATPase) uses the energy stored in a transmembrane electrochemical gradient to synthesize ATP. ATP synthase is unique in combining two rotary molecular motors in one protein so that the assembly can either synthesize ATP or pump ions across a membrane, depending on which motor dominates. The F1Fo system has been studied extensively both experimentally and theoretically, so the basic operating principles of both the Fo and F1 motors are understood at near molecular detail (e.g. refs. 1–5). Modeling of mechanoenzymes takes three general approaches. Molecular dynamics (MD) attempts to account for the motion of every atom, and sometimes the surrounding water and other chemical species. Kinetic models represent the transitions between small numbers of chemical (Markov) states. Markov–Fokker– Planck (MFP) models lie somewhere between the atomic detail of MD and the phenomenology of kinetic models (6). These models describe the geometric motion along a few ‘‘collective coordinates’’ that describe the protein’s major conformational movements. Motion is driven by forces derived from a set of potential functions assigned to each coordinate, with Markov jumps between the potentials corresponding to chemical transitions. MFP models generalize kinetic models by replacing the discrete kinetic states with potential functions representing collective spatial coordinates. The distinctions between, and relative advantages of, each of these models have been discussed in detail elsewhere (6). A prominent feature of MFP models is that structural and dynamical information can be included in the models without the computational load of MD models. In this work, we take this approach. We emphasize that MFP models complement MD and kinetic models; each has its proper place in understanding a mechanochemical system as complex as a protein motor. In constructing an MFP model, the central step is to identify the important degrees of freedom that must be treated explicitly, and construct a set of free energy potential surfaces with these degrees of freedom as geometrical coordinates. Identification of the collective coordinates can often be inferred from the protein structure and its biochemistry (7). Normal mode analysis also can provide insight into the collective modes relevant to protein functions (8–17). Recently, we have used the Fo motor as an example to show how one can construct low-resolution free energy potential functions from various experimental findings, even when precise atomic structures are not available (18). In this work, we will apply this www.pnas.org兾cgi兾doi兾10.1073兾pnas.0507207102

approach to model the F1Fo ATPase in its synthesis mode by carefully choosing physically meaningful collective coordinates. ATP synthase is composed of two parts: F1 (subunits ␣3␤3␥␧␦) and Fo (subunits ab2c10–14); here, subscripts denote the stoichiometry. The soluble F1 enzyme catalyzes ATP synthesis by a rotary mechanism. Synthesis is driven by the Fo motor that converts the transmembrane ionmotive force into mechanical torque. This torque drives the rotation of an elastic shaft consisting of subunits ␥ and ␧ that couples Fo with F1 (18). Fig. 1 shows the essential geometry. The F1 subunit can also operate in reverse by hydrolyzing ATP and driving the Fo motor in reverse so that it becomes an ion pump; the mechanism is described in ref. 22. The F1 subunit consists of a hexamer of alternating ␣ and ␤ subunits whose interfaces harbor six nucleotide-binding sites. Three catalytic sites alternate with noncatalytic sites. Rotation of the coiled-coil ␥ shaft provides the energy to release newly synthesized ATP from the catalytic sites. The ␦ subunit sitting atop the ␣3␤3 hexamer also links to the Fo motor via the b2 subunit. Thus, the Fo motor is built from two counterrotating assemblies, conventionally called the ‘‘rotor’’ and ‘‘stator.’’ The rotor consists of the ␥␧-shaft and a ring-shaped array of 10–14 double-helical c-subunits, where the number of subunits depends on the species. The stator comprises the subunits ab2 of Fo along with subunits ␦␣3␤3 of F1. The a subunit consists of five or six membrane-spanning ␣-helices. The b2 and ␥ subunits are elongated structures that provide the opposing elastic linkages between the rotor and stator so that each motor can exert torque on the other. Voluminous experimental and theoretical studies have been summarized in several comprehensive reviews (e.g., refs. 1–5 and 23). This information serves as the basis for the model we present here, which combines the two motors to construct a mechanochemical model of the complete ATP synthesizing enzyme. Construction of the Model Supporting Information. For further details, see Supporting Text,

Tables 1–3, Figs. 8 and 9, and Movies 1 and 2, which are published as supporting information on the PNAS web site. Space and Time Scales Determine the Collective Coordinates. The first step in constructing the model is to identify collective coordinates that are adequate for describing the major conformational motions of the F1 hexamer, and the reaction coordinates that suffice to describe the steps in the synthesis cycle (24); details are given in Supporting Text. We will focus on motions that take place on the nanometer spatial scale and a time scale of milliseconds. These are the size and time scales on which the behavior of proteins can be investigated with current single molecule techniques. Thus, we average out processes that occur on smaller spatial and temporal scales, leaving molecular dynamics and mixed quantum兾molecular mechanics studies to address the atomic details of how the hydrolysis and synthesis reactions take place (e.g., refs. 25 and 26) and the angstrom level motions at the catalytic site during nucleotide binding and product release (e.g., refs. 27 and 28).

Abbreviations: MD, molecular dynamics; MFP, Markov–Fokker–Planck; pmf, proton motive force. *To whom correspondence should be addressed. E-mail: [email protected]. © 2005 by The National Academy of Sciences of the USA

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Contributed by George Oster, August 23, 2005

can be used as ‘‘markers’’ for the reaction coordinates that are taking place on an angstrom scale inside the catalytic site. With current single molecule techniques, the only conformational motion of the protein that can be observed in experiments is the rotation of the ␥-subunit. However, previous experimental and theoretical studies have established that, in hydrolysis mode, the rotation of the ␥ shaft is driven by sequential hinge-bending motions of the ␤-subunits (7, 31–33). The bending motions of the ␤-subunits are currently accessible by fluorescence measurements (34). However, as revealed by the crystal structures, and discussed by Menz et al. (35), their trajectories likely involve motion along the axis of the ␣3␤3 hexamer, as well as motion orthogonal to the axis. Significantly, the trajectory must be cyclic, as discussed below.

Fig. 1. Structure of the F1Fo ATPase. The membrane-spanning Fo motor drives the rotation of the ␥兾␧-shaft by using the energy of the transmembrane ionmotive force (18). The F1 hexamer contains three catalytic sites alternating with three noncatalytic sites. The shaft is eccentric, so that its rotation sequentially stresses the catalytic sites to release ATP (see text). Details of the structure and Boyer’s ‘‘binding change’’ model can be found in refs. 20 and 21. [Reproduced with permission from ref. 19 (Copyright 2000, Springer兾Kluwer Academic, Dordrecht, The Netherlands).]

On the millisecond time scale, the only aspect of the hydrolysis or the synthesis cycle that can be observed is the occupancy state of the catalytic site. Usually, the cycle is described by four occupancy states: Empty (E) 7 ADP (D) 7 ADP䡠Pi (DP) 7 ATP (T) (see also Supporting Text). On this time scale, the transitions between these states can be regarded as instantaneous; thus, the steps in the kinetic cycle can be treated as Markov transitions between discrete reaction coordinates (29, 30). However, for motor proteins, the power stroke is a mechanical motion that is directly affected by external mechanical forces: the torque from Fo, in this case. Mechanical motions are generally coupled to chemical transitions, and require explicit treatment (6). In its hydrolysis mode, the power stroke of the F1 motor is driven by the annealing of the nucleotide into the catalytic site as the hydrogen bonds are formed, the so-called ‘‘binding zipper’’ (31). Conversely, in synthesis mode, the rotation of the ␥ shaft unzips the bonds holding newly synthesized ATP to the catalytic residues. Both of these processes take place on the millisecond time scale of ␥ shaft rotation. Therefore, when we use the catalytic site occupancies as reaction coordinates, this means that the processes of binding and unbinding from the catalytic site are described by motions along the collective spatial coordinates. This is certainly an approximation because the process of binding a nucleotide to the catalytic site involves only angstrom level motions between the nucleotide and the elements of the catalytic site (e.g., the P-loop and arginine finger). Thus, modeling the steps of the hydrolysis or synthesis cycles as kinetic jumps between occupancy states carries the implicit assumption that the actual molecular motions involved in each transition are ‘‘levered up’’ to nanometer size motions of the protein at positions distant (i.e., nanometers) from the catalytic site. These larger scale motions

Qualitative Considerations. Before commencing to construct the driving potentials, we examine the relationship between events at the catalytic site of a ␤-subunit and the potentials of mean force used to describe them. Our discussion will be qualitative, intending to introduce the quantitative modeling pursued below. Although this paper focuses on the synthesis direction, we will also cite results for the hydrolysis direction because this has been studied more extensively. Events in the catalytic site. The events that transpire at the catalytic site have been investigated by biochemical, genetic, crystallographic (23, 35), and theoretical (7, 25–29, 31, 36) studies. The nucleotide in the catalytic site is held by a network of hydrogen bonds to the Walker A and B motifs on the ␤-subunit and the arginine ‘‘finger’’ projecting from the neighboring ␣-subunit (35, 37). These catalytic residues emanate from loops joining the strands of the central ␤-sheet. The picture that emerges is that the hinge-bending motion of the ␤-subunit, which constitutes the power stroke in the hydrolysis direction, is driven by the progressive annealing of hydrogen bonds between the catalytic site and the nucleotide (31, 38, 39). Theory and experiment suggest that the resulting motor torque is nearly constant during a power stroke in the hydrolysis direction (31, 40–42). The binding zipper model describes how motions at the nanometer length scale generate stress at the catalytic site. In the synthesis direction, the process is reversed: the torque from Fo forces the catalytic site opens, unzipping the hydrogen bonds so that the nucleotide can be dislodged by thermal fluctuations. Structural (35) and MD (27) studies have addressed the location and sequence of hydrogen bonds during synthesis. The catalytic site must have a mechanism to distinguish between the ATP hydrolysis and synthesis cycles. During the hydrolysis cycle, one open catalytic site binds ATP molecule, closes, and then hydrolyzes ATP. Phosphate and ADP are released on the way back to the open conformation. However, during the ATP synthesis cycle, an open catalytic site binds phosphate and ADP, while closing, synthesizes ATP, then releases ATP while resuming the open conformation. The cycle is generally written as shown in Fig. 2. Although the ATP hydrolysis process and ATP synthesis process appear to be the reciprocal of each other, the ␤-subunit closing and ␤-subunit opening in the hydrolysis direction are not reciprocal of each other. The opening of the ␤-subunit associated with product release in hydrolysis is not simply the reverse of the closing of ␤ associated with ATP binding. (Although the opening of ␤ associated with ATP release in synthesis may be the reversal of the closing of ␤ associated with ATP binding in hydrolysis, at

Fig. 2. 16540 兩 www.pnas.org兾cgi兾doi兾10.1073兾pnas.0507207102

Xing et al.

Kinematics of coupling between the catalytic site and the ␥-subunit: The tight-coupling approximation. In hydrolysis mode, the eccentric shape

of the ␥ shaft converts the hinge-bending motions of the ␤-subunits into a rotary motion analogous to turning the crank on an automobile jack (7, 39). Conversely, in synthesis mode, the rotation of the ␥ shaft drives the ␤-subunits through their hinge-bending cycle. Several authors have found evidence that the mechanical coupling between the ␤ and ␥ subunits is tightened by a circumferential ‘‘track’’ of hydrophobic and positively charged residues around the ␥ shaft that guides the tips of the ␤-subunits (43, 44). This track is shown in Fig. 3a. Fig. 3b unwraps the ␥ shaft and plots these residues on ␥ in the ␪-z plane, where ␪ is the rotational angle of the ␥ shaft and z is the vertical coordinate along the rotation axis. Providing that the tip of each ␤-subunit remains in contact with the ␥ shaft, each rotation of the ␥ shaft drives every ␤ tip along a cyclic circumferential path. Here we will assume that the ␤-tips follow this path so that the rotation of the ␥ shaft is tightly coupled to the bending of the ␤-subunits.† The track on ␥ defines a one-to-one map between the orbit of the catalytic residues in Rn (denoted by CCS) and the curve on the ␥ shaft traced out by the tip of a ␤-subunit, (denoted by C␥); this is summarized by the cartoon in Fig. 3c showing the mechanical coupling between the ␤-subunit and the ␥ shaft. In the supporting information we illustrate the periodic motions of the catalytic residues in a movie based on the interpolated crystal structures. Finally, the contact curve on the ␥ shaft also projects to the rotational angle, ␪, of the ␥-subunit. Thus, the curve C␪ can be used as a surrogate for the configurational cycle, CCS, at the catalytic site. This identification depends on the tight mechanical coupling between the bending of the ␤-subunit and the sliding contact with the ␥ shaft along the contact strip.‡ Whether this is an adequate approximation cannot be settled theoretically; however, we will see that it permits us to fit the experimental data quite accurately. With this construct, we can express the potentials that drive rotation of the ␥ shaft in terms of the rotational angle, ␪. The approach follows our previous modeling of the F1 motor operating in the hydrolysis mode (7, 28, 31). We describe construction of the potentials in detail below. It will turn out that the structure-based potentials derived for synthesis reproduce our previous work in the hydrolysis direction as well. Dynamics: Mesoscopic forces derived from free energy potentials. Fig. 4 illustrates our graphical representation of free energy potentials for a generic mechanoenzyme traversing the reaction cycle: E (Empty) 7 S (Substrate) 7 P (Product) 7 E. In Fig. 4a, each chemical occupancy state is represented as a contour map of the free energy potential whose slope gives the driving force. The vertical axis connecting the energy landscapes is the reaction coordinate, ␰, so that transitions between potential surfaces represent the chemical

Step-by-Step Construction of the Synthesis Driving Potentials Along ␪.

In what follows, we base our considerations on the F1-ATPase of Escherichia coli. Construction of the synthesis potentials requires that we identify the crucial interactions between the ␤ and ␥ subunits and specify the forces opposing rotation of the ␥ shaft in the synthesis direction. Some of these forces originate from stress generated by ligand binding. These stresses radiate outward from the catalytic site, through the central ␤-sheet, and oppose the hinge-bending motion of the ␤-subunits that are driven by rotation of the eccentric ␥ shaft. Because we assume that the motion of the ␤-subunits are tightly coupled to the rotation of the ␥ shaft, the potentials that drive the motion of the ␤-subunits can be mapped onto the ␪ representation according to Fig. 4; this produces a set of effective driving potentials seen by the ␥ shaft, denoted Gi(␪), where i ⫽ (E, D, DP, T), the occupancy states of the catalytic sites. The rotational symmetry of the system requires that all of the potentials be 360° periodic. In Supporting Text, we give a detailed discussion of the procedure for constructing the free energy potential as a function of ␪ in each occupancy state. Here we shall only list the central ideas. One of the main contributions to the synthesis potentials arises from deformation of the ␤-subunit. The generic shapes of the potentials shown in Fig. 5a are based on the following considerations; additional discussion is given in Supporting Text. 1. All potentials are 2␲ periodic. 2. When driven by the Fo torque, all potentials are ‘tilted’ by adding the work done by the Fo torque: Gi(␪) 哫 Gi(␪) ⫹ t䡠q, i ⫽ E, T, DP, D. In addition, each potential has the following characteristics. Y

Y

†This

tight-coupling assumption can be slightly relaxed without affecting the following discussions. For a given ␥ position, the ␤-subunit can fluctuate over a certain range. Then the contribution of the ␤-subunit to the free energies should be understood as averaged over fluctuations (the adiabatic approximation).

‡This

is clearly an approximation because a sliding contact generally is a nonholonomic constraint, which we are approximating as an (integrable) constraint curve (45).

Xing et al.

INAUGURAL ARTICLE

reaction steps. Motion along the reaction coordinate represents chemical bond rearrangements that take place on a spatial scale of angstroms and a time scale of nanoseconds or less, whereas motions along the spatial coordinate take place on a spatial scale of nanometers and a time scale of milliseconds, corresponding to protein conformational motions. If we consider only motions along the cyclic pathway, then we can simplify the representation by projecting onto a single periodic spatial coordinate, as shown in Fig. 4b, where the potentials are drawn on free energy (⌬G) vs. the spatial displacement coordinate, ␪. The chemical transitions between the potential surfaces generally can take place only in a restricted interval of configurations, shown shaded in Fig. 4b. Each transition region is a saddle-shaped free energy barrier over which the system must pass in moving from one occupation state to another. The transition state lies at the apex of the saddle path. Outside of this region, the transition barrier is assumed to be too high to allow chemical transitions to take place at rates appreciable compared to the time scale of rotation. The distance between the potentials in the transition region is the free energy drop for the transition. This free energy drop depends on the solution concentrations of reactants and products, whereas motion along a potential surface is independent of the solution concentrations. In this work, we will use the representation shown in Fig. 4b.

Y

E Potential: An empty ␤ subunit takes the open conformation; that is, the ‘‘rest state’’ of an isolated ␤-subunit is set at ␪ ⫽ 0 ⬅ 2␲, the potential minimum (46). T Potential: An ATP-bound ␤ subunit takes the closed conformation. The potential minimum of GT(␪) is set at ␪ ⫽ 150° (the closed ␤ conformation) to fit the observed substeps in hydrolysis model (ref. 28 and references therein, and ref. 31). DP Potential: The DP curve is similar to the T potential, except that it should rise more sharply from its minimum because DP can only exist near the closed conformation; the PNAS 兩 November 15, 2005 兩 vol. 102 兩 no. 46 兩 16541

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near equilibrium, it must be.) Thus, at the angstrom level, where the ␥-phosphate bond is either made or broken, the collective motions of the catalytic site atoms must be cyclic; this is illustrated in Fig. 3c. Thus, we can conclude that the positions of the catalytic residues must traverse a periodic geometrical orbit in opposite directions during hydrolysis or synthesis. (Of course, because of thermal fluctuations, these orbits reflect the average positions of the catalytic atoms.) Although this may be obvious, it is not obvious that this cyclic orbit is reflected in the observable nanometer level motions of the ␤-subunits as they undergo their cyclic hinge bending motions. To see this, we next discuss the constraints structure imposes on the motions of the ␤-subunits.

Fig. 4. Schematic representation of a set of potential surfaces for the generic reaction cycle: E (Empty) 7 S (Substrate) 7 P (Product) 7 E. (a) The potential of mean force for the E state as a function of the catalytic site conformation (here shown schematically with two degrees of freedom, x1 and x2). As the substrate binds, a chemical transition drops the state through a free energy ⌬GES to the S state. A further configurational energy drop takes place along the S potential surface until the reaction region for the E 3 P transition is reached, whereupon the system drops to the P surface, and so on, until the E surface is reached once again, completing the cycle, with a total free energy drop of ⌬GCycle. (b) The diagram can be simplified by showing only the free energy cross-sections along the cycle parametrized by ␪. The shaded regions are the saddle-shaped free energy reaction paths where chemical transitions are allowed.

Y

Fig. 3. Kinematics coordination between the catalytic site and the rotation of the ␥ shaft during the synthesis cycle. (a) The track on the ␥ shaft (colored green) that guides the hinge-bending motion of the ␤-subunits (43, 44). Two ␤-subunits are shown in closed (red) and open (blue) conformations. Motion along this track couples the hinge-bending motion of the ␤-subunit and the ␥ shaft rotation. (b) The ␥ shaft unrolled to show residues lining the circumferential track. Contact residues obtained from interpolated structures are marked within the band of the track. Residues that form the boundaries of this track are shown in colors (blue, positively charged; red, negatively charged; pink, polar) (43). Some other important residues are also shown: ␥R36 is the most eccentric residue, labeled MEP (i.e., furthest from the axis of rotation); ␥Q255 activates binding of nucleotide successively by interacting with the ␤-subunit; ␥R242 activates phosphate release in the hydrolysis direction by interacting with DELSEED; ␥M23 also interacts with DELSEED via chargehydrophobic interactions. Residue IDs follow bovine structures (see also Movie 1). (c) The coordinates of the catalytic residues execute a closed loop in their n-dimensional configuration space, Rn, during one synthesis cycle. The mechanical linkage between the catalytic site and the DELSEED motif at the C terminus of the ␤-subunit couples the motion of the catalytic site to the ␤ tip via its hinge-bending motion. The ␤ tip pushes on the eccentric ␥ shaft driving its rotation. As the ␥ shaft rotates, the ␤-subunit hinges between its open and closed positions. The contact between the ␤ and ␥ shaft is confined to the strip defined in b; the location of the tip on the ␥ shaft surface can be located by the bending angle, ␾, and the azimuthal angle, ␺. We will assume that the ␤ bending and the ␥ rotation is tightly coupled, so that the ␤ tip traces out a 16542 兩 www.pnas.org兾cgi兾doi兾10.1073兾pnas.0507207102

affinity of Pi is very low for most catalytic site conformations, even when ADP is bound (5). D Potential. The ADP-bound ␤ subunit takes a partially closed conformation because it has fewer hydrogen bonds than ATP. There are two partially closed conformations during one cycle: one during closing, the other during opening. In hydrolysis model, ADP is known to bind to the catalytic site during its closing phase; therefore, the D potential has two minima during one cycle [0, 2␲]. This may be related to the phenomenon of ‘‘ADP inhibition’’ during hydrolysis.

Given only these properties, the potentials in Fig. 5a are not adequate to describe normal synthesis (as confirmed by simulations): the system slides over the potentials without many chemical transitions, so that ATP synthesis is very inefficient. To ensure the chemical transitions occur in the correct sequence, additional residue interactions must be included. By examining

closed curve, C␥, on the surface of the ␥-subunit that is confined within the track. Because points along the track C␥ can be mapped onto the rotational angle, ␪, the mechanical linkage defines the kinematic relationship between the configurational cycle at the catalytic site and the rotational position of the ␥ shaft. This mapping allows us to parametrize the potentials of mean force describing the conformational changes in the catalytic site by the rotation coordinate, ␪, of the ␥ shaft.

Xing et al.

INAUGURAL ARTICLE

Fig. 5. Step-by-step construction of the potentials in the synthesis direction for one ␤-subunit along the rotational coordinate, ␪. (a) Basic potentials due to interactions within a catalytic site. The shaded regions are the chemical transition windows. (b) Potentials modified by interactions between ␣3␤3 and ␥兾␧; the dashed lines show where the potentials in a have been modified. The motor is subjected to a constant 45兾3 pN䡠nm torque from the Fo motor, so the total torque acting on the three ␤-subunits is 45 pN䡠nm. During one rotation cycle, the catalytic site executes a conformational cycle: open 3 closed 3 open. In b, we have labeled features revealed by the model, such as pmf dependence, that are consistent with experimental observations. The substrate concentrations used in this figure are [ATP] ⫽ 1 mM, [ADP] ⫽ 0.1 mM, [Pi] ⫽ 1 mM. The numbers follow a typical cycle in the ATP synthesis direction (the ␧ subunit is presumably in the ‘‘up’’ conformation). 1 3 2 3 3, an open empty site closes sufficiently to bind an ADP molecule; 3 3 4 3 5, binding phosphate requires F1 being driven upward along the D curve by the torque from Fo (i.e., ion motive force is necessary for phosphate binding); 5 3 6 3 7, ATP is synthesized; 7 3 8, hydrogen-bond network is peeled off by the torque from Fo; 8 3 9, ATP molecule is released; 9 3 1, cycle repeats. (c) Here we use interactions between ␥M23 and the DELSEED motif of a ␤-subunit to illustrate

Xing et al.

the available crystal structures and various dynamical experiments, we can identify and propose structural correspondences of these modifications (see Supporting Text for detailed discussion). These modifications to the basic potentials require adding potential ‘‘bumps’’ shown in Fig. 5b that reflect specific electrostatic or steric interactions between ␥兾␧ and the ␣3␤3 complex. In the ATP hydrolysis direction, the bump on the E curve enforces the observed pause waiting for ATP to bind, and the bump centered around 330° on the D curve enforces the observed pause waiting for ADP release (47–49). A schematic illustration of the structural explanation of the bumps is given in Fig. 5 c and d. The potential bumps caused by interactions between ␣兾␤ and ␧兾␥ force each catalytic site to change its nucleotide-binding configuration in the correct sequence and timing. As suggested by Abrahams et al. (46), these interactions serve as checkpoints for the catalytic site nucleotide configurations. Fig. 5 focuses on the mechanochemical cycle corresponding to the synthesis of one ATP in a single ␤-subunit of an ␣3␤3 hexamer. The ␥ shaft mechanically couples the three ␤-subunits: as ␥ rotates, it drives the bending of each ␤ in strict sequence with a fixed phase difference. This coupling enforces the apparent

origins of the potential bumps introduced in b. With ADP bound in the catalytic site (the ␤-subunit shown in cyan), the DELSEED motif (shown in yellow) hits ␥M23 (shown in red), which produces the potential bump introduced on the D curve near 330°. Releasing ADP results in conformational changes of the ␤-subunit (shown in transparent orange) in directions orthogonal to the rotation direction, so DELSEED no longer interacts strongly with ␥M23. (d) Cartoon illustrating the effect of the modified potentials. PNAS 兩 November 15, 2005 兩 vol. 102 兩 no. 46 兩 16543

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Fig. 6. The overall potentials of wild-type E. coli. F1 ATPase, as a function of the rotational angle ␪ with the three catalytic sites tightly coupled by the rotary ␥␧ shaft. As the ␥ shaft rotates, the three catalytic sites move together with a fixed phase difference (heavy lines), resulting in multisite coupling. A typical pathway is shown by the heavy line. The free energy profiles shown correspond to substrate concentrations [ATP] ⫽ 1 mM, [ADP] ⫽ 0.1 mM, [Pi] ⫽ 1 mM. Also shown above the figure is the approximate correspondence between the catalytic site conformational states discussed by Menz et al. (using their labels) (35) and the continuum representation used here. Also shown are the regions where the torque from the ␥ shaft (i.e., the pmf) is strictly required to advance the cycle. The most probable angular regions for chemical transitions vary with substrate concentration. In a purely chemical kinetic model, the ‘‘dominant’’ reaction pathways would change with substrate concentrations. An animation of this figure showing the synthesis cycle is available as Movie 2.

Fig. 7. Fitting the data with the model. The calculated (solid lines) and experimentally measured steady state ATP synthesis rates (circles) as a function of ADP concentration (with fixed Pi concentrations as labeled) (a) and phosphate concentration (with fixed ADP concentrations as labeled) (b) (data are from ref. 52). Also shown are multiple reaction pathways calculated with the model. (c) Physiological conditions: [ATP] ⫽ 1 mM, [ADP] ⫽ 0.1 mM, and [Pi] ⫽ 1 mM. (d) Experimental condition: [ATP] ⫽ 1 ␮M, [ADP] ⫽ 1.3 mM, and [Pi] ⫽ 3.2 mM. To calculate the flux diagram 120° of the the rotational coordinate ␪ is divided into four equal regions. Fluxes among the 64 ⫻ 4 chemical–mechanical states are calculated; those shown in the figures contribute ⬎95% of the overall flux (defined as the total flux summed over the 64 chemical states at a given angular position). Because of the threefold rotational symmetry of the system, when rotating out of 0°, a state labeled as (s1, s2, s3) becomes (s2,s3,s1) entering from at 120°. Note that even though a state makes a large contribution to the flux, this does not necessarily mean that the state has a high probability of being observed.

cooperativity between the three catalytic sites, as illustrated in Fig. 6. The process is shown in Movie 2. These generic potentials are based on quite general considerations. Without any numerical calculations, one can already draw some conclusions about the features of ATP synthesis by F1, for example, because the ␤-subunits are tightly coupled to each other and to rotation of the ␥ shaft driven by Fo: Y

Y Y

ADP binding, formation of the phosphate binding pocket, and ATP release require the torque generated by the proton motive force (pmf). Substrate binding on one catalytic site can help substrate release on the other site (see discussion below). Because of its double-well potential, there exists an ADP inhibition state in hydrolysis mode (discussed below).

Fitting the Potentials into a Mathematical Model Because there are 3 catalytic sites in the F1 hexamer, and each site can be in 4 occupancy states, there are 43 ⫽ 64 chemical states. We index the chemical states as follows: S ⫽ (s1 ⫺ 1) ⫻ 16 ⫹ (s2 ⫺ 1) ⫻ 4 ⫹ s3, where si ⫽ 1, 2, 3, 4 represents the chemical states [E, T, DP, D] of each catalytic site, respectively. The overall potential of F1 resisting the Fo motor is obtained by summing over the potentials ˆ (␪,S) ⫽ Gs (␪) ⫹ Gs (␪ ⫹ 2␲兾3) ⫹ Gs (␪ ⫹ of each catalytic site: G 1 2 3 4␲兾3). We place our coordinate system on the ␥ shaft so that the angular position of the F1 hexamer is described by the rotational angle, ␪ (see Fig. 8). The dynamics is described by 64 Fokker–Planck equations for the probability densities ␳S(t, ␪) for finding F1 at angular position ␪ in chemical state S at time t: D ⭸ ⭸␳s ⫽ ⫺ ⭸t k BT ⭸ ␪

冢

ˆ 共 ␪ , S兲 ⭸G ⫺ ⭸␪ Ç load torque from Fi

⫹

␶ motor torque from Fo

冣

冘

⭸ 2␳ S K SS⬘ ⫻ ␳ S⬘, S ⫽ 4 3 ⫽ 64 chemical states. ⫹ D 2 ⫹ ⭸ ␪ s⬘ Ç Ç Diffusion

Kinetics

Here D ⬇ 1.5 ⫻ rad2兾s is the rotational diffusion constant of the ␥ shaft (7, 28). ␶ is the driving torque from Fo, which we approximate as a constant torque of 45 pN䡠nm, the minimum amount of free energy needed to synthesize three ATP molecules per cycle at physiological conditions. Eq. 1 is solved in the steady state, ⭸␳S兾⭸t ⫽ 0 to obtain the density functions, from which the synthesis rates are computed (See Supporting Text for details). The model parameters are given in Tables 1–3. Most of them are based on estimations from experiments, and the 12 parameters specifying chemical rate constants and potential heights were obtained by using an optimization algorithm based on experimental biochemical data. This technique was developed previously to compute the rate constants for other AAA ATPases; the technique is described briefly in Supporting Text (50). These parameters, obtained through optimization, are consistent with other independent measurements and studies. For example, the equilibrium constant for ATP formation within the catalytic site is ⬇1 (51), and the energetic cost to close an empty ␤-subunit is ⬇6–10 kBT (36). 104

Results Fig. 7 shows the model fit to the measured ATP synthesis rates (52). Here we discuss some important observations implied by the energy curves. The Equilibrium Constant for Synthesis ⬇1. Least square fits of the

␳s

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[1]

synthesis data results in T and DP curves that lie very close to one another within the transition region for ATP hydrolysis and synthesis (compare Fig. 6). This is equivalent to Keq ⬇ 1 for the hydrolysis and synthesis steps. This derived result of the model is Xing et al.

Cooperation Among Catalytic Sites. There is a great deal of experimental data on positive and negative cooperativity between catalytic sites, much of it analyzed by using Michaelis–Menten kinetic schemes. However, deeper insight can be obtained from the potential curves shown in Fig. 6. Because the hinge bending of the ␤s are tightly coupled to rotation of ␥, each catalytic site can influence the dynamics of other sites via the torsional stress transmitted through the ␥ shaft. Therefore, multisite coupling via the ␥ shaft is automatically included within the Fokker–Planck framework. For example, an ADP bound on site 2 moves downhill along the D curve, which helps the ␥ shaft rotate uphill along the T curve aiding site 1 to release ATP, and site 3 to bind Pi as it moves along the D curve (see Fig. 6 and Movie 2). The Role of ␥Met-23. To fit the synthesis data a bump on the D

Concentration-Dependent Multiple Reaction Pathways. Because

there are 64 occupancy states of the F1 hexamer, there are many possible reaction paths connecting them. Which pathways are actually traversed depend on the substrate and product concentrations because vertical distances between potential curves in Fig. 6 are affected by substrate concentrations (see Supporting Text). Because transitions with negative free energy changes take place spontaneously, the major contributing reaction pathways (and the average occupancy of the catalytic sites) are concentrationdependent. Fig. 7 shows the reaction pathways computed for physiological and experimental conditions. The relative flux strengths are indicated by the relative thicknesses of the transition paths. The important lesson contained in these flux diagrams is that under most experimental situations, there are multiple reaction pathways whose fluxes may be comparable. For example, one can see from the diagram that bi- and trisite fluxes may be appreciable at higher substrate concentrations. The existence of multiple-pathways is automatically included in a MFP formulation, as mentioned in refs. 6 and 7, and was also recognized by some kinetic model studies as well (50, 76). Only at very low substrate and product concentrations will the highest-affinity pathway be dominant. Thus, fitting data with a single a priori reaction path is likely to give a misleading picture of the biochemistry of synthesis.

potential must be introduced (see Fig. 5). This potential bump is presumably due to unfavorable interactions between the negatively charged DELSEED motif of ␤ and the hydrophobic ␥Met-23 residue. This interaction should also exist in the hydrolysis direction. The potential bump forces an ADP bound catalytic site to release its ADP as it approaches ␥Met-23. According to this explanation, we expect that the bump to be unaffected by mutating ␥Met-23 with hydrophobic or negatively charged residues. However, if ␥Met-23 is mutated to a positively charged residue, a potential well is formed instead of a potential bump, and the motor function should be affected. This potential well is exactly what Nakamoto and coworkers (3, 61, 62) found: mutation of the residue to a positively charged lysine or arginine keeps ATPase activity nearly the same as that of the wild type, but the resultant electrochemical gradient of protons is largely reduced. On the other hand, the energy coupling efficiency is not affected by substitutions by other type of amino acids (include ␥23Met3Asp or Glu) (3, 61, 62). The model explains the loss of coupling efficiency with positive charged residue mutations. The overall potential governing the ␥ shaft motion is a sum of the interactions between ␥兾␧ and the three catalytic sites offset by 120°. A direct consequence of the mutation is that a potential well by site 2 partially compensates the potential bump around 150° by site 3, which in the wild-type enzyme prevents the ␥ shaft from rotating backward. The overall potential after mutation is much flatter, so that thermal fluctuations allow the ␥ shaft to slip backward in the observed temperature-sensitive manner (3).

have modeled the F1 ATPase operating as a rotary motor in hydrolysis mode (7, 28, 31). Slight modifications of the potentials derived here for F1 operating in synthesis mode (to account for the conformational changes of the ␧ subunit) reproduce all of the previous results when operated in hydrolysis mode.

The ␧ Subunit Ensures Energy Coupling in the Synthesis Direction.

Relationship with Other Models. Our model treats the ␥ rotation as

Experiments show that, in the synthesis direction, ␥ shaft rotation is tightly coupled to synthesis of ATP. The potential bumps on the E and D curves force the catalytic site to make the correct transition sequence E 3 D 3 DP 3 T to proceed in the synthesis direction. Without the ␧ subunit, we suggest that the potential bumps would be reduced, and the resultant energy surface cannot effectively prevent rotation without ATP synthesis. We propose that these bumps are related to the residues identified by cross-linking experiments, e.g., ␧S108, which may interact with ␤E381 and ␣S411 (63–65) (see Supporting Text). Further discussion on the dynamics effect of ␧ in the hydrolysis mode can be found in Supporting Text. The Mg䡠ADP Inhibition State. Normally, a catalytic site should be

mostly occupied by ATP within the angular range 0 ⱕ ␪ ⱕ 2␲兾3 (the ATP-binding region in Fig. 6). However, ADP competes with ATP, and there is some probability that ADP is bound instead, which traps the system in the D curve potential well within that angular range. In the hydrolysis direction, the system can proceed only if it releases ADP and binds ATP. This state may correspond to the Xing et al.

INAUGURAL ARTICLE

so-called ‘‘Mg䡠ADP inhibition state’’ (66–71). The existence of the Mg䡠ADP inhibition state is a natural consequence of the doublewell shaped D curve, which is necessary to ensure coupling between rotation and chemical transitions in both the ATP synthesis and hydrolysis directions. Structurally, it may correspond to a rather closed catalytic site, and may be related to two crystal structures, where the catalytic site that normally binds ATP binds ADP instead, as suggested by Hirono-Hara et al. (71–73). Referring to Fig. 6, we see that the activation of the inhibited F1 motor by external force observed by Hirono-Hara et al. is evident in our model because the force helps ADP release (71). Indeed, their result can be used to refine the potential curves. Mg䡠ADP inhibition is not a serious problem in ATP synthesis direction because the catalytic site is always occupied by ATP before entering this angular region (74, 75).

The Model Works in Hydrolysis Mode. In previous publications, we

a continuous variable. If the rotational degree of freedom is discretized, one can recover features of qualitative models proposed by others. In Fig. 6, we also showed the approximate correspondence between the catalytic site conformations discussed by Menz et al. (using their labels; ref. 35) and the continuum representation used here. With the methodology described here, one can construct a quantitative, detailed, and systematic treatment of the catalytic site conformational changes. Discussion In this study, we have presented a mathematical model for ATP synthase operating in its ATP synthesis mode. The model is ‘‘mesoscopic’’; that is, we describe the mechanochemistry on the scale at which single molecule studies can be carried out: milliseconds and nanometers. We show that the rotational angle of the ␥ shaft can be used as a collective coordinate, a surrogate for the high-dimensional periodic trajectory executed by the catalytic residues during the synthesis cycle. We used experimental information from crystal structures, biochemical measurements, and mutation PNAS 兩 November 15, 2005 兩 vol. 102 兩 no. 46 兩 16545

BIOPHYSICS

consistent with the results of the 18O isotope exchange experiments under unisite conditions (1, 53, 54). At higher nucleotide concentrations, the 18O isotope exchange decreases quickly (55–60). This observation is explained in the model because rotation of ␥ quickly pushes a catalytic site out of the chemical transition window.

studies to construct a set of free energy potential surfaces that capture the important mechanochemical interactions between the subunits. We emphasize that many of our conclusions, such as the pmf dependence and the existence of the Mg䡠ADP inhibition state, are consequences of the generic potential shapes, and so are independent of the model parameters used in the quantitative calculations. We find that the model can fit the measured rates of ATP synthesis under several substrate conditions. Furthermore, we suggest functional roles for several structural features and residues found essential for synthesis. We believe that the methodology we used in constructing the model provides an appropriate framework to unify a large body of experimental observations and suggest further experiments. The main results of this work are the set of free energy potential surfaces from which the dynamical behavior of the protein in both synthesis and hydrolysis modes can be inferred. In principle, the potential surfaces can be tested experimentally; for example, by single-molecule force measurements at various substrate concentrations. One of the lessons of our simulation is that there is not a single ‘‘main kinetic pathway,’’ except under very low substrate兾 product concentrations when only the highest affinity states are populated (50). Under physiological (and most experimental) conditions, there are multiple pathways, several of which may have appreciable flows. Thus, attempting to fit data with an a priori selected biochemical path will likely prove fruitless or misleading. One notable prediction of the synthesis model is an explanation for the familiar phenomenon of ‘‘ADP inhibition’’ of F1 in its

hydrolysis mode. Because of the periodic motion of the catalytic residues during the hydrolysis cycle, the ADP potential curve must have two potential minima separated by a moderately high free energy barrier. This requirement does not cause serious problem in synthesis mode, but tends to trap ADP in hydrolysis mode; this is hardly surprising because F1 evolved to synthesize, and only under laboratory conditions does the eukaryotic F1 operate in hydrolysis mode. The bacterial F1 can operate as an ion pump under anerobic conditions, and one might expect that it could be less prone to ADP inhibition. The V1 motor of the vacuolar ATPase, being designed for ion pumping, may have avoided ADP inhibition by the evolution of additional subunits. This prediction remains to be tested experimentally. The single molecule experiment performed by Imamura et al. does show different behaviors between V1 and F1 motors (H. Imamura, M. Takeda, S. Funamoto, K. Shimabukuro, M. Yoshida, and K. Yokoyama, unpublished data). Future studies of the F1 motor will certainly improve our understanding of the motor. More experiments are necessary to reduce uncertainties in model parameters. However, we believe that the basic physics revealed by this model will be upheld in future experiments, although some details such as the exact location of each potential minimum, may need some adjustment. In this effort, single molecule measurements will surely prove helpful (71).

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We thank Paul Boyer, Hongyun Wang, and Hiromi Imamura for helpful suggestions that greatly clarified the exposition and William Brusilow for data used in Fig. 7. The financial support was provided by National Science Foundation Grant DMS 0414039.

Xing et al.