INTRODUCTION

TOP ABSTRACT INTRODUCTION A MECHANOCHEMICAL MODEL OF... RESULTS SUMMARY AND DISCUSSION NOTES APPENDIX A APPENDIX B APPENDIX C APPENDIX D APPENDIX E APPENDIX F APPENDIX G APPENDIX H APPENDIX I APPENDIX J REFERENCES

One of the defining characteristics of intracellular membrane compartments is the difference between their lumenal pH and that of the bulk cytoplasm. This pH differential is maintained principally by the vacuolar ATPases. These proteins comprise a class of proton pumps that bear a high degree of structural homology to the F-ATPases that manufacture ATP using the proton-motive force across the mitochondrial inner membrane (Engelbrecht and Junge, 1997; Finbow and Harrison, 1997; Kibak et al., 1992; Nelson, 1992; Taiz and Nelson, 1996). Indeed, bacterial F-ATPases can, under anaerobic conditions, reverse themselves to function as proton pumps. Building on this structural similarity, we present here a quantitative model for the V-ATPase proton pump, and demonstrate that it can reproduce many of the experimentally measured properties.

Recently there has been a flood of new information on the structure, mechanics, and biochemistry of the F-ATPases. The crystal structure of F₁ determined by John Walker's laboratory lent strong support to Boyer's "binding change" model that posited that the three catalytic sites alternated their activity in a rotational sequence (Abrahams et al., 1994; Boyer, 1997). This was dramatically confirmed by the startling visual demonstration that F₁ was indeed a rotary motor (Kinosita et al., 1999; Noji et al., 1997). The Japanese group also determined that the rotation of the F₁ motor advanced in three steps per revolution, with each step requiring the hydrolysis of one ATP, and that the motor developed an average torque of over 40 pN · nm. Amazingly, the work done against the average drag torque in a single rotation was nearly equal to the free energy drop accompanying the hydrolysis of three ATPs, indicating that the F₁ motor operates at nearly 100% efficiency. Wang and Oster combined the structural, biochemical, and mechanical data into a unified mechanochemical model for the F₁ motor (Wang and Oster, 1998).

Because it is largely a transmembrane structure, information on the F_o portion of ATP synthase has been more difficult to obtain. Recently, Sambongi et al. (1999) provided visual proof that the c-subunit oligomer rotated along with the -subunit shaft that connects it to the F₁ hexamer. Structures for the c-subunit oligomer have been deduced from NMR (Rastogi and Girvin, 1999) and x-ray crystallography studies (Stock et al., 1999). While they do not agree in all details, it is clear that the F_o motor consists of 10 or 12 subunits with rotational symmetry, as deduced from a variety of studies from several laboratories (Fillingame et al., 1999; Long et al., 1998; Nakamoto et al., 1999). To counter the large torque developed by the F₁ motor in hydrolysis mode, the F_o motor must generate an even larger torque in the opposite direction to synthesize ATP. It does this by converting the transmembrane proton-motive force into rotary motion. Various qualitative models for this energy transduction have been proposed (Junge et al., 1997; Vik and Antonio, 1994), and Elston et al. (1998) formulated a quantitative model that explained the torque generation mechanism as an "electrostatic Brownian ratchet."

The model for the V-ATPase we propose herein is built on the assumption that the structural similarities between the F- and V-ATPases reflect a mechanistic similarity in their operation. Indeed, in their model for the F_o motor, Elston et al. (1998) showed that, when driven in reverse, the F_o motor could indeed function as a proton pump, albeit not a very effective one. Here we shall show that by modifying that model in accordance with the known structural properties of the V-ATPases, we can construct a mechanochemical model for the vacuolar proton pumps that is in quantitative agreement with a variety of experimental observations. First, however, we briefly review the similarities between the two protein classes to buttress this fundamental assumption.

Both the V- and F-ATPase protein families consist of two major portions: a membrane-spanning portion (V_o, F_o) containing the proton channel, and a soluble portion (V₁, F₁) containing the nucleotide binding sites. Fig. 1 illustrates the overall geometry of the V-ATPase proton pumps and summarizes the subunit notation for both protein families (Stevens and Forgac, 1997). The soluble portions of both ATPases have a sixfold symmetry created by the alternation of A and B subunits in V₁ and the  and  subunits in F₁. A and B share 25% sequence identity with  and , respectively (Bowman et al., 1988a, b; Zimniak et al., 1988). In addition, each hexamer contains three catalytic sites capable of ATP hydrolysis (Arai et al., 1988). An important difference between the F- and V-ATPases lies in the structure of the c-subunits of (V_o, F_o)). In the F-ATPases 10-12 c-subunits are arranged in a ring; each c-subunit consists of a double -helix; -helix 2 of each subunit contains an acidic proton binding site (Asp-61, Escherichia coli numbering). The corresponding structure in the V-ATPases comprises six c-subunits; each c-subunit consists of four transmembrane -helices; helix 4 has an acidic proton binding site (Glu-140, Saccharomyces cerevisiae numbering). Each of the acidic sites on V_o binds a proton in an electrostatic well whose depth is related to its in situ pK_a values by V/k_BT = 2.3 pK_a (k_B is Boltzmann's constant and T the absolute temperature). Thus the F-ATPases have 12 proton carriers, while the V-ATPases have 6. This has important consequences for the pump's behavior, as we shall see (Arai et al., 1988; Dmitriev et al., 1995; Groth and Walker, 1997; Holzenburg et al., 1993). The strong resemblance of the two enzymes in all areas but the c-subunit stoichiometry has led to the hypothesis that V_o evolved by gene fusion and mutation in the c-subunits of F_o, resulting in the loss of one of the two initial carboxyl residues involved in transport (Cross and Taiz, 1990). This occurred without major change in the soluble ATP binding hexamer.

View larger version (108K): [in this window] [in a new window]   FIGURE 1   Schematic structure of the V-ATPase proton pump. The correspondence between the F and V-ATPase notation is summarized in the table below. Those connections that are more speculative are accompanied by a question mark. Molecular weights are given in parentheses for Neurospora crassa (Margolles-Clark et al., 1999). The V1 portion consists of subunits (A3, B3, C, D, E, F, G2, H). The (AB)3 hexamer contains the ATP binding sites. The Vo portion consists of subunits (a, d, c6, c', c"). The subunits c' and c" are not represented in this picture since their role and stoichiometry are presently unclear (Margolles-Clark et al., 1999). The proton channel is formed in part by the a-subunit, which is composed of approximately nine transmembrane -helices. The key charges that participate in proton transport are Lys-593, His-743, and Glu-789 on a-subunits and the six proton binding sites (Glu-140) on the c-subunits, each of which consists of four transmembrane -helices. The counter-rotating subunits are denoted by convention as the rotor (c6CDE) and stator (A3B3aG2). Hydrolysis in V1 drives the rotation of the c subunit with respect to the a-subunit (indicated by the arrow). This drives protons from the basic reservoir (cytoplasm) to the acidic reservoir (lumen). V-ATPase a (98) A (67) B (57) F (13) D (28), E (26) c (16) G (13) F-ATPase (E. coli) a?      ?  ? c b

The a subunit in F_o is likely composed of five transmembrane -helices, and contains a critical basic residue (Arg-210) on helix 4 (Valiyaveetil and Fillingame, 1997). In addition to forming a path for protons across the membrane, subunit a is thought to interact electrostatically with protons bound to subunit c (Fillingame, 1990). The VPH1 gene encodes the a subunit of V_o (called the Vph1p subunit in yeast). This polypeptide contains three amino acids, Lys-593, His-743, and Glu-789, in putative membrane-spanning domains that when mutated result in reduced proton transport and ATPase activity (Leng et al., 1996, 1999). This has prompted researchers to suggest that the charged residues play a fundamental role in the mechanism of ion translocation, and that the a-subunit forms part of the proton-conducting pathway through the membrane (Finbow and Harrison, 1997; Margolles-Clark et al., 1999). We adopt this structural interpretation and assume that the a-subunit serves the same function in both V- and F-ATPase.

	A MECHANOCHEMICAL MODEL OF THE V-ATPase PROTON PUMP

TOP ABSTRACT INTRODUCTION A MECHANOCHEMICAL MODEL OF... RESULTS SUMMARY AND DISCUSSION NOTES APPENDIX A APPENDIX B APPENDIX C APPENDIX D APPENDIX E APPENDIX F APPENDIX G APPENDIX H APPENDIX I APPENDIX J REFERENCES

In this section we present a quantitative model for the V-ATPase proton pump. We base our analysis on the close structural similarities between the F- and V-ATPases. The conventional paradigm for active transmembrane ion transport is the "alternating access" mechanism: ions are bound tightly on the low concentration side and a conformational change exposes them to the high concentration side, weakening their binding affinity so that they dissociate. The pump then resets its conformation to repeat the cycle (Alberts et al., 1994; Eisenberg and Hill, 1985). The energy to drive the cycle is supplied by nucleotide hydrolysis or a complementary ion gradient. The structural basis of the conformational change that implements the alternating access has not been worked out for any ion pump. Here we construct a model of the V-ATPase proton pumps assuming the conformational change is a simple rotation.

Model geometry

Like the F-ATPases, it is believed the V-ATPase structure can be subdivided into a counter-rotating "stator" and "rotor" (Boekema et al., 1997; Elston et al., 1998; Fillingame, 1996; Forgac, 2000; Harrison et al., 1997; Junge et al., 1997; Vik and Antonio, 1994). The rotor is likely to consist of the c-, D-, and E-subunits. The stator is thought to be composed of the a-subunit, the (AB)₃ hexamer, a connecting unit analogous to the b₂ proteins in F-ATPase, and possibly H (Boekema et al., 1997; Graham and Stevens, 1999; Stevens and Forgac, 1997). Physically, a connecting unit, speculated to be subunit G, is essential to prevent the (AB)₃ hexamer from turning with the rotor. However, G lacks a membrane-spanning domain, which is present in subunit b, and it is likely present in two separate copies per enzyme (Boekema et al., 1999). It has been suggested that G binds tightly to a membrane-bound subunit, since it lacks this anchoring span (Margolles-Clark et al., 1999; Superkova et al., 1995).

Assuming that proton translocation involves a rotation of V_o in the membrane, there are several structural features that must be present for pumping to occur. Within the context of our model, we now discuss the biochemical and structural environment that the proton must experience for translocation. This discussion has little experimental support since it involves membrane proteins for which high-resolution structures are difficult to obtain. Notwithstanding, physical principles tell us that the true description must be similar to one of the two following scenarios.

Two models have been suggested for the rotor-stator assemblies, a two-channel model and a one-channel model (Dimroth et al., 1999; Elston et al., 1998; Junge et al., 1997; Vik and Antonio, 1994). Fig. 2 shows a face-on view of both models. In both, hydrolysis of ATP in V₁ by the (AB)₃ hexamer provides the torque that turns the rotor to the left (i.e., clockwise viewed from the cytoplasm). We shall not address the mechanism by which V₁ generates torque; for the purposes of this study we can represent V₁ by its load-velocity characteristic. To compute this we shall assume that the V₁ motor works the same way as the F₁-ATPase described in Oster and Wang (2000) and Wang and Oster (1998). As the rotor turns, protons are pumped from the basic reservoir (top) through the stator to the acid reservoir (bottom). Both of the configurations shown in Fig. 2 operate on the same "alternating access" principle discussed below (Alberts et al., 1994). The primary difference lies in the path the protons take in traversing the stator (shown by the dashed lines in Fig. 2). However, both pump models perform almost identically.

The interface between the rotor and stator must present a hydrophobic barrier to prevent proton leakage between the reservoirs; thus, there can be no direct proton path through the stator. The one- and two-channel models shown in Fig. 2 accomplish this somewhat differently. In the one-channel model, the six rotor acidic sites are located above the level of the membrane so that they are always in equilibrium with the basic reservoir outside of the stator. By contrast, the acidic sites in the two-channel model are located near the midplane of the membrane. In this position a rotor site can only be protonated from the basic reservoir when it enters the input channel within the rotor-stator interface; at all other positions rotor sites are buried in the lipid membrane and cannot communicate with either reservoir. In both models the output (acidic) channel extends into the stator to the level of the rotor proton binding sites. Thus the path of a proton in the one-channel model is short, while in the two-channel model a proton must board the rotor and make a complete circuit as the rotor turns before exiting the output channel.

The hydrophobic interface prevents unprotonated rotor sites from rotating to the left out of contact with the cytoplasm (see Eq. 21 in Appendix D) while a protonated rotor site can pass into the hydrophobic rotor-stator interface and enter the output channel. However, the rotor site will not release its proton unless its pK_a is reduced considerably. This is accomplished by the trio of stator charges (Lys-593, His-743, and Glu-789) since we assume that they lie near the path of the rotor sites. As a protonated site approaches the stator charges its pK_a is reduced, forcing it to relinquish its proton to the output channel. The trio of stator charges can be modeled by a single "equivalent" charge (see Eq. 20 in Appendix D). A polar hydrophilic strip connects the output channel to the basic reservoir, as shown in Fig. 2, to permit the passage of unprotonated rotor sites. The stator charge blocks proton leakage along this path.

Since the structure of the a-subunit is not well characterized, it is difficult to determine which stator assembly, if either, is correct. The one channel model describes experiments performed on the sodium F-ATPase; however, it requires exposing hydrophobic sections of the c-subunits to water (Dimroth et al., 1999). The more widely accepted two-channel model that places the rotor sites in the lipid bilayer cannot explain the sodium exchange experiments in the sodium F-ATPase (Kaim and Dimroth, 1999). In Dimroth et al. (1999) we have shown that the operating principle of the F_o motor is the same for both geometries. Similarly, the operating principle of the V_o pump is the same for both geometries, and so we have adopted the more widely accepted two-channel geometry shown in Fig. 2b.

View larger version (37K): [in this window] [in a new window]   FIGURE 2   Perspective and face-on views of the rotor-stator assembly for the one- and two-channel models showing the paths protons follow in moving from the cytoplasm (top) into the lumen (bottom). Torque supplied by V1 from ATP hydrolysis turns the rotor to the left (clockwise viewed from the cytoplasm in Fig. 1). (a) The two-channel model for the a-subunit. Two half-channels penetrate the stator to the level of the rotor sites; all other parts of the rotor-stator interface are hydrophobic. A horizontal polar strip connects the channels to allow the passage of an unprotonated site, but protons are blocked from leaking by the stator charge. Protons enter the input (basic) channel and bind to a rotor site, largely neutralizing it. Rotation carries the protonated site one complete revolution (to the left) where the site enters the output (acidic) channel (from the right). The stator charge forces the site to relinquish its proton into the lumen. Note that the sizes of the rotor and stator are such that two rotor sites cannot fit in the rotor-stator interface at once. (b) The one-channel model for the a-subunit. The rotor sites now lie above the level of the membrane and are in equilibrium with the cytoplasm when not in the stator. A single channel penetrates the stator to the level of the rotor sites; all other parts of the rotor-stator interface are hydrophobic. A horizontal polar strip connects the channel to the cytoplasm to allow passage of an unloaded site. The stator charge blocks passage of protons through the strip. Rotation brings a protonated site into the output channel where the stator charge forces it to release its proton to the lumen. The unloaded site then rotates through the polar strip, past the stator charge, exiting the rotor-stator interface to the left.

The two-channel proton pump model works according to the following sequence of events shown in Fig. 3. The torque generated by V₁ moves a protonated rotor site out of the lipid bilayer and across the hydrophobic interface into the acidic channel. If the site enters the polar strip still protonated, the stator charge will reduce its pK_a, forcing it to relinquish its proton to the acidic reservoir. The site is then captured by the stator charge. The torque from V₁ is sufficient to surmount the electrostatic attraction between the rotor and stator charges and rotate the empty site into the basic reservoir. Until a rotor site is protonated, the torque generated by V₁ is insufficient to force it through the hydrophobic barrier at the channel stator interface. However, once neutralized by protonation, the site can be rotated across the barrier and into the bilayer (see Note 1 at end of text preceding appendices). While this scenario sounds reasonable, the only way to verify that the pump will operate as described is to construct a quantitative model, which we now proceed to do.

View larger version (50K): [in this window] [in a new window]   FIGURE 3   (a) A face-on cartoon of the stator geometry corresponding to the events in (b) and (c). (b) A typical sequence of events following a site as it passes through the rotor-stator interface. (1  2): The V1 motor rotates a protonated site out of the membrane, across the hydrophobic interface, and into the acidic channel. In the channel the site has a high probability of staying protonated since kon is large. However, when the protonated site rotates close to the stator charge, the rotor site pKa decreases so that koff increases until the proton is relinquished to the acidic reservoir (2  3). The empty site is then driven through the hydrophilic strip, past the stator charge, and into the basic reservoir (3  4). An empty site in the basic channel is reflected by the dielectric boundary until a proton binds, neutralizing the site (4  5). The torque from V1 can then rotate the site through the hydrophobic barrier (where koff ~ 0) into the membrane region. The protonated site continues to rotate one full turn back to position (1) where the cycle repeats. (c) The free energy of a site as it passes through the rotor-stator interface. The free energy scale corresponds to a rotor site with pKa = 5.4 pumping from pH = 7 to pH = 4. The potential curve is tilted due to the 45 pN · nm torque from V1 driving the rotor to the left. (1  2): the site is driven out of the lipid bilayer into the acidic channel and approaches the stator charge. As the pKa of the rotor sitedecreases, the free energy of the site increases until the proton is driven off the site into the acidic channel (2  3). (3  4): the V1 motor must supply sufficient torque to pull the rotor site out of the electrostatic grasp of the stator charge and into the basic reservoir. However, it cannot surmount the hydrophobic barrier at the channel boundary. Eventually, the site will protonate (4  5) whereupon it can be driven through the hydrophobic rotor-stator interface into the membrane domain.

Mathematical formulation of the model

We describe the rotor position by its rotation angle, (t). A rotor site can be empty (charged) or occupied (nearly neutral). Thus the chemical state of a rotor site can be described by the binary variable s_i = (Empty, Full), i = 1, ... , 6. However, we need only keep track of those sites that interact with the stator. Since the rotor charges are spaced  = 2/6 apart, the stator spans at most two sites; thus it is sufficient to follow the protonation status of four sites spanning the stator interface. Then the rotor state, denoted s, is determined by the states of these four sites, and can take on 2⁴ = 16 values. Since the association and dissociation of protons from rotor sites is much faster than the motion of the rotor, we can model the transitions between rotor ionization states as a Markov chain. As it passes through the stator, a rotor site encounters different physical environments: the polar strip, output channel, hydrophilic interface, and basic reservoir. Therefore, the binding and dissociation of protons from a rotor site depend on its angular position, ; it can be described by the Markov equation

(1)

Here K() is the transition matrix between chemical states, i.e., the proton association and dissociation rates from the rotor sites. A full description of K() and its elements is given in Appendix C.

The motion of the rotor can be computed from a force balance equating the viscous drag on the rotor to the torques that act on the rotor and the Brownian force modeling the rotor's thermal fluctuations, (i.e., Langevin's equation) (Risken, 1989):

(2)

The terms in Eq. 2 are as follows; the explicit forms are given in Appendix D.

• The left-hand side is the viscous drag torque on the rotor. d/dt is the rotational velocity.  is the drag coefficient of the rotor treated as a cylinder rotating in the plane of the membrane;
• _Q(, s) is the torque due to the electrostatic interaction between the stator charge and the rotor sites that are within the hydrophilic rotor-stator strip. Because of the high dielectric constant in the aqueous environment of the cytoplasm and the proton channels, this interaction is negligible except when the rotor site is within the polar strip. The charged (unoccupied) site will be attracted by the stator charge according to Coulomb's law corresponding to the dielectric and shielding environment of the stator;
• (, s) is the electrostatic torque on unoccupied rotor sites due to the membrane potential drop across the horizontal segment between the acidic channel and the basic reservoir;
• (, s) is the electrostatic barrier that opposes the entry of a charged site into the hydrophobic rotor-stator interface. The height of this barrier is given approximately by ~200 (1/_s  1/_c)  45 k_BT  27 kcal/mol (k_B is Boltzmann's constant, and T the absolute temperature), where _c ~ 80 and _s ~ 4 are the dielectric constants of the cytoplasm and the rotor-stator interface, respectively (Israelachvili, 1992);
• _D() is the torque exerted on the rotor from V₁ as it hydrolyzes ATP. This torque is transmitted via the rotating shaft (D, E) to the rotor while the stator is connected through the G subunit to the (AB)₃ hexamer;
• _B(t) is the random Brownian torque due to the thermal fluctuations of the rotor.

As indicated by their dependence on s, the electrostatic torques depend on the chemical state of the rotor site; that is, whether the site is charged (unoccupied) or uncharged (occupied). The model equations were solved numerically in the Fokker-Planck representation; the mathematical details are given in Appendix E. In the next section we present the results of these simulations.

Protons can "slip" past the stator

At modest transmembrane pH differences, each ATP hydrolyzed transports two protons across the membrane. That is, one revolution of the rotor consumes three ATPs and delivers six protons from the cytoplasm to the lumen. The coupling ratio, , is defined as the ratio of protons transported per ATP consumed:

(3)

where J_H is the proton flux and J_ATP is the ATP hydrolysis rate. The pH gradient and the membrane potential are thought to affect the value of  (Davies et al., 1994; Muller et al., 1996). This is generally interpreted as evidence of some sort of "slip" within the pump mechanism. We can formulate this notion within the context of the model as follows.

To define the slip coefficient we first introduce some notation (see Fig. 4):

• J_In = flux of protons that enter the stator hydrophobic interface from the basic reservoir (i.e., from the right in Fig. 4);
• J_Out = flux of protons that enter the stator hydrophobic interface and successfully dissociate into the output channel;
• J_S1 = flux of protons that enter the stator hydrophobic interface from the right but pass the stator charge and dissociate back into the basic reservoir. This is part of the slip flux;
• J_S2 = leak flux of protons shuttled by rotor sites that fluctuate between the output channel and input channel;
• J_In = J_Out + J_S1. Since we assume the rotation of V₁ is tightly coupled to ATP hydrolysis, we have J_In = 2 · J_ATP;
• The proton pumping rate is J_Out  J_S2.

With these definitions, the slip coefficient, , can be defined as

(4)

The first part of the slip coefficient measures the fraction of protons that enter the stator but do not successfully dissociate into the output channel. J_S1 is dominant, except near stall, where J_S2 becomes important. Slip is related to the coupling ratio by:

(5)

Proton slip is affected by any mechanism that changes the binding or dissociation rates of protons to the rotor site in the proton channels. Increasing the membrane potential or decreasing lumenal pH increases the proton concentration adjacent to the rotor sites facing the output channel, thus increasing the proton binding rate. This increases the probability that a protonated site will rotate out of the output channel carrying a proton into the input channel.

View larger version (64K): [in this window] [in a new window]   FIGURE 4   The possible pathways for protons passing through the stator. A fraction of the protons that rotate into the output channel "slip" past the stator charge and ride on the rotor site out of the stator, where they dissociate back into the basic reservoir. The proton flux, JIN, enters from the cytoplasmic channel, makes one complete revolution (to the left) and reenters the stator on the right. A fraction, JOUT, dissociates into the output channel, and a fraction JS1, slips past the stator charge and leaves the input channel; thus JS1 = JIN  JOUT. Actually, a site traversing the output channel will bind and dissociate a proton ~104-105 times (see Appendix C), so the proton that "slips" back into the basic reservoir is unlikely to be the same one that initially entered on the rotor site. Additionally, the back-and-forth diffusive motion of the rotor may shuttle protons from the output channel to the basic reservoir, creating a second contribution to the slip flux, JS2. Thus the total slip flux is JSLIP = JS1 + JS2. This second component of the slip flux is only important when the V1 motor is nearly at stall, i.e., when the rotation rate of Vo is small.

	RESULTS

TOP ABSTRACT INTRODUCTION A MECHANOCHEMICAL MODEL OF... RESULTS SUMMARY AND DISCUSSION NOTES APPENDIX A APPENDIX B APPENDIX C APPENDIX D APPENDIX E APPENDIX F APPENDIX G APPENDIX H APPENDIX I APPENDIX J REFERENCES

In this section we compute the behavior of the V-ATPase model and compare it to available experimental data. Tables 1 and 2 summarize the parameter values used to calculate the results of this section. We define the membrane potential as  =  (cytoplasm)   (lumen) and current flow into the compartment is taken as positive. Therefore, a positive membrane potential will drive positive current flow into the vacuole.

Current-voltage relationships

The basic operating characteristic of an electrogenic ion pump is its current voltage behavior. Such data on V-ATPase proton pumps is scarce, but three sets of data on plant vacuoles by Davies et al. (1994), Bentrup et al. (1986), and Gambale et al. (1994) are presented in Fig. 5. The model fits these data sets well using the parameter values shown in Tables 1 and 2. These three experiments were performed at saturating ATP concentrations where ATP binding was not a rate-limiting step. The load-velocity curve of the V₁ motor used to drive the pump was derived previously, and is discussed in Appendix F (Wang and Oster, 1998).

In Appendix J we show that the membrane potential is more effective in resisting the proton flux than the pH gradient (see Fig. 11 a). The reason for this is that the membrane potential drop over the polar strip tilts the electrostatic potential seen by the unprotonated site, thus biasing its escape from the central stator charge to the output channel (2  3 in Fig. 3 c). This escape requires a thermal fluctuation so that the probability of escape depends exponentially on the depth of the well. The imposed membrane potential lowers the height of this barrier in one direction. This strong influence on the dynamics is required to explain why the top two panels of Fig. 5 span two orders of magnitude in the proton pump rate over the range of imposed membrane potentials.

The strength of the electrostatic interaction between the stator and the rotor determines the amount of slip and the overall rotation rate of the enzyme. To compute this interaction involves accounting for the geometry and dielectric environment of the rotor and stator charges. Without a molecular structure this cannot be done realistically, and so we have represented this interaction strength by varying the size of the effective stator charge and the pK_a of the rotor sites. However, there are other contributing factors of equivalent weight that we hold constant, such as the distance between charges, the stator dielectric constant, and the screening length; details of the calculation are given in Appendix D. In general, a large stator charge reduces the rotation and pumping rate. However, if the stator charge is too small the proton flux may prematurely vanish near stall because of increased slip. The top panel in Fig. 5 exhibits this effect: the flux drops to zero long before the thermodynamic stall of ~300 mV, because the effective stator charge is too small.

To fit the data in Fig. 5 we must estimate the total number of active pumps. We deduced this by scaling the single pump rate to the observed experimental currents. This scaling parameter affects the height and slope of the current voltage curve. Appendix G describes how the pump densities for each system were determined from fitting the current-voltage curves.

View larger version (20K): [in this window] [in a new window]   FIGURE 5   V-ATPase current-voltage behavior. The left ordinate is current caused by one pump: protons/s/pump (positive is flow inward). The right ordinate is the total current by all pumps in pA = 0.62 × 107 H+/s. The abscissa is the imposed transmembrane voltage. The solid lines are computed from the model using the parameters in Tables 1 and 2. The most important parameters in fitting the data were the voltage drop across the polar strip, the effective strength of the stator charge, and the estimated number of pumps (Appendix G). The top panel shows the data (triangles) of Gambale et al. (1994) on the sugar beet. This is the ATP-dependent current for the enzyme pumping against no pH gradient. The middle panel shows the data (circles) of Davies et al. (1994) on the red beet (Beta vulgaris) pumping from pH 8 (cytoplasm) to pH 5.5 (lumen). The bottom panel shows the ATP-dependent current-voltage values (squares) recorded by Bentrup et al. (1986) from isolated Chenopodium rubrum L. vacuoles when pumping from pH 7 (cytoplasm) to pH 5 (lumen).

What limits the pH to which the V-ATPase can pump?

Under physiological conditions the free energy of ATP hydrolysis is ~21 k_BT, and the energy required to move a proton against a pure concentration gradient is 2.3 pH (in units of k_BT). If six protons are transported across the lumen at the expense of three ATPs, the maximum pH gradient is 6 × (2.3 pH_max) = 3 × (21 k_BT); thus the thermodynamic limiting pH gradient attainable is pH_max = (3 × 21)/(2.3 × 6) ~ 4.6. By comparison, the maximum pH the F-ATPase can achieve when operating in the reverse (pump) direction is pH_max = (3 × 21)/(2.3 × 12) ~ 2.3. The model provides a way to understand these thermodynamic limits in mechanistic terms.

F₁ is a rotary motor driven by ATP hydrolysis. Under physiological conditions it develops a rotary torque of ~40 pN · nm (Kinosita et al., 1999; Wang and Oster, 1998). F_o is also a rotary motor that is driven in the opposite direction by a transmembrane ion-motive force (Dimroth et al., 1999; Elston et al., 1998). The two motors are connected by a common shaft so that the torque developed by each motor opposes the other. According to the binding change mechanism of Boyer, during ATP synthesis the F_o motor must develop a torque sufficient to free tightly bound ATP from the catalytic site (Boyer, 1997, 1998). That is, it must develop a counter-torque in excess of that developed by the F₁ motor. The mechanochemical theory for the F_o motor is given in Dimroth et al. (1999) and Elston et al. (1998). The important result of this analysis is that the F_o motor operates as a Brownian ratchet wherein the rotational diffusion of the rotor is rectified by the transmembrane proton-motive gradient (Peskin et al., 1993). This means that the F_o motor is a stochastic "stepper" with step size equal to 2R/12, where R ~ 3-4 nm is the rotor radius. The consequence of reducing the number of rotor charges from 12 to 6 is that the rotor is required to diffuse twice as far before being rectified. Since thermal fluctuations are Boltzmann distributed, under the load from V₁ such a fluctuation is exponentially more rare, and so the V_o motor is much less effective than the F_o motor. This gives a greater advantage to the V₁ motor, which can drive the rotor in reverse more effectively, and so the V_o V₁ assembly functions better as a pump; that is, it can achieve lower lumenal pH values, but it will take longer to pump down. Fig. 11 in Appendix J compares the pump performances of the V-ATPase with those of the F-ATPase based on this difference in the number of rotor charges.

Because of the ineffectiveness of the V_o as a motor, one expects that under normal conditions the V-ATPases do not reverse and synthesize ATP: the normal function of the F-ATPases. However, Yokoyama et al. (1998) have reported recently that proteoliposomes containing bacteriorhodopsin and the V-ATPase from Thermus thermophilus can synthesize ATP when the concentration of Mg-ADP and phosphate is high enough (see Note 2). Fig. 6 shows the range of proton-motive force where the V-ATPase is capable of synthesizing ATP from phosphate and ADP. The curve separating pumping from synthesis in this figure corresponds to the pmf required for the V-ATPase to produce the 45 pN · nm torque required for ATP synthesis. For a tightly coupled system (no slip), this curve can be determined from thermodynamics without appealing to a dynamic model. However, thermodynamics says nothing about the behavior of the enzyme away from the stall point. Fig. 6 shows that the model predicts that the rotation rate in the synthesis direction quickly increases to a point where ATP production can reach measurable quantities such as those seen in the experiments of Yokoyama et al. At physiological values of the vacuolar membrane potential, 10-30 mV, it would require a lumenal pH <2 in order to drive synthesis of ATP (this range of  is not plotted in Fig. 6).

View larger version (56K): [in this window] [in a new window]   FIGURE 6   ATP synthesis by the V-ATPase. With the cytoplasmic pH fixed at 7, the V-ATPase rotation rate is plotted as a function of lumenal pH and membrane potential. The shaded region corresponds to conditions when the enzyme operates in reverse to synthesize ATP. The experiments of Yokoyama et al. (1998) have shown ATP synthesis is possible in light-driven bacteriorhodopsin-VoV1-ATPase proteoliposomes. The coupling to V1 was simulated by providing a constant torque of 45 pN · nm, equivalent to that produced by hydrolyzing three ATPs per revolution (Elston et al., 1998; Wang and Oster, 1998). The approximate number of V-ATPases can be calculated from the amount of purified enzyme used in the experiment. Assuming 100% activity, a lower limit of 6-10 ATP/s per enzyme was calculated as an average synthesis rate over the course of the experiment. This rate corresponds to the 3 Hz curve shown on the figure.

Finally, it is puzzling to find situations in which cells can achieve a pH < 1, since this would appear to violate the above thermodynamic limits (Futai et al., 1998). However, recent structural studies on the F_o rotor suggest a theoretical mechanism. Rastogi and Girvin (1999) demonstrated that the proton binding c-subunit has a different conformation at acidic and basic conditions, with the protonable rotor site (Asp-61) rotating so that the site is buried within the rotor structure at pH ~ 5, but exposed at the rotor periphery at pH ~ 8. This suggests the following scenario for V- ATPases attempting to pump down to very low pH. If exposure to a very acidic lumenal environment titrates rotor sites such that they are withdrawn from participation in proton exchange, then the number of active rotor sites could be reduced from six to, say, three. This would make the back torque developed by the V_o motor even smaller in opposing the torque from V₁, and so the pump would pump slower, but be capable of achieving a pH ~ 9, which could account for the phenomenon of such acidic organelles. Of course, some other pump than the V-ATPases may be involved in pumping down to these unusual pH values. However, the ability to sense the lumenal pH is theoretically attractive since it would allow the pump to "change gears" and adapt to a steeper pH, analogous to a bicyclist pedaling up an increasing slope.

	SUMMARY AND DISCUSSION

TOP ABSTRACT INTRODUCTION A MECHANOCHEMICAL MODEL OF... RESULTS SUMMARY AND DISCUSSION NOTES APPENDIX A APPENDIX B APPENDIX C APPENDIX D APPENDIX E APPENDIX F APPENDIX G APPENDIX H APPENDIX I APPENDIX J REFERENCES

Because of the lack of structural information, models of ion pumps have necessarily been formulated as kinetic equations describing the proposed chemical steps involved in ion translocation (e.g., Lauger, 1991). This approach omits all mechanical details of how ion translocation actually takes place and how the energy derived from nucleotide hydrolysis is stoichiometrically coupled to translocation. Here we have exploited the close structural similarities between the F and V-ATPases to propose a mechanochemical model of the V-ATPase proton pump. The advantages of this formulation over kinetic models are manifold. First, the model provides a mechanical interpretation of the pump operation that can quantitatively reproduce a variety of experimental observations. Second, by delineating precisely how the energy derived from ATP hydrolysis is used to pump up a transmembrane ion gradient it offers a mechanistic explanation of what is meant by "osmotic work," "slip," and how non-integer coupling ratios arise. Third, the model gives insight into the physics of possible regulatory mechanisms. Finally, the model makes it possible to compute a "pumping surface," like the one pictured in Fig. 7, giving the proton flux as a function of pH and membrane potential. This surface has been partially calibrated by patch clamp studies; however, more thorough experiments must be carried out to construct an accurate pump response to the two components of the proton-motive force. With a reliable pumping surface one can construct a model compartment that incorporates a variety of other proteins that affect lumenal pH, including proton and ion leaks, Donnan potentials, and other electrogenic pumps such as the sodium/potassium ATPase. Such a model could be an important exploratory and explanatory tool for understanding intracellular pH regulation. Since these compartments are not always in thermodynamic equilibrium with each other, a nonequilibrium model is essential (Farinas and Verkman, 1999; Wu et al., 1999). We shall present such a model in a subsequent publication.

View larger version (48K): [in this window] [in a new window]   FIGURE 7   The performance surface for a single, tightly coupled V-ATPase. The proton pumping rate is plotted as a function of membrane potential and pH gradient. The cytoplasmic pH is 7 and the free energy of ATP hydrolysis is 21 kBT. The parameters used are identical to those found in Table 4 for six sites.

The structure of the V-ATPases is not known with the fidelity of the F-ATPases, and so there is more uncertainty about the details of their operation. However, we suggest that the principle of operation we have proposed will find concrete interpretations in the molecular structures that will surely be forthcoming in the near future. Our analysis of the rotor-stator interactions reveals that electrostatics plays the central role in regulating proton flow. Finally, we believe that certain aspects of the operating principles governing the V-ATPases will apply in modified form to the P-type ATPase pumps. Both operate on the "alternating access" principle (Alberts et al., 1994; Jencks, 1989) whereby a conformational change moves a tightly bound ion exposed to the basic side of the membrane to a position where it faces the acidic side of the membrane while simultaneously reducing its binding affinity so that it can dissociate. The conformational changes driving translocation in the P-type pumps are not rotations in the plane of the membrane, but are likely to involve small motions perpendicular to the membrane. Moreover, the ATPase driving these motions appears to be more intimately associated with the translocation machinery than they are in the V-ATPases (Scarborough, 1999). However, we believe that the mechanism by which the V-ATPases reduce the ion-binding affinity of the carrier sites by forcing them into proximity to oppositely charged fixed sites will find an analog process in the P-ATPases. Perhaps even the basic mechanisms by which mechanical forces are driven by nucleotide hydrolysis will prove universal (Oster and Wang, 2000; Wang and Oster, 1998).

	NOTES

TOP ABSTRACT INTRODUCTION A MECHANOCHEMICAL MODEL OF... RESULTS SUMMARY AND DISCUSSION NOTES APPENDIX A APPENDIX B APPENDIX C APPENDIX D APPENDIX E APPENDIX F APPENDIX G APPENDIX H APPENDIX I APPENDIX J REFERENCES

1.  The equivalent sequence of events for the one channel model is as follows. Rotor sites are in equilibrium with the basic reservoir. Until a rotor site is protonated, the torque generated by V₁ cannot force it over the hydrophobic barrier. However, once neutralized by protonation, the site is driven across the barrier into the stator output channel. A protonated site that rotates out of the channel into the polar strip interacts electrostatically with the stator charge, which reduces its pK_a, forcing it to relinquish its proton to the acidic reservoir. Further rotation of the rotor carries the rotor site out of the stator interface. After that, the rotor site is once again in equilibrium with the basic reservoir. 2.  The proteolipid subunits have a molecular weight of 13 (Yokoyama et al., 1994), comparable to the c-subunits of eucaryotic V-ATPases (~14-16 kDa) consistent with the existence of only six binding sites.