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Home » Biology Articles » Biophysics » Medical Biophysics » The New Unified Theory of ATP Synthesis/Hydrolysis and Muscle Contraction, Its Manifold Fundamental Consequences and Mechanistic Implications and Its Applications in Health and Disease » Unified Theory of the ATP Cycle in ATP Synthesis/Hydrolysis and Muscle Contraction

Unified Theory of the ATP Cycle in ATP Synthesis/Hydrolysis and Muscle Contraction
- The New Unified Theory of ATP Synthesis/Hydrolysis and Muscle Contraction, Its Manifold Fundamental Consequences and Mechanistic Implications and Its Applications in Health and Disease

2.1. Specific Details of Energy Distribution among the Elementary Steps of Binding, Bond Cleavage, and Product Release in the ATP Synthesis/Hydrolysis Catalytic Cycle

All the membrane-bound/organized biological molecular machines that we are concerned with here, such as the F1FO-ATP synthase, the P- and V-type ATPases, muscle myosin II, the cargo-carrying kinesins and unconventional myosins etc., synthesize or utilize ATP. Hence it would be valuable for a unified theory if we can understand the quantum of energy required/released in each elementary step of the ATP cycle in synthesis/hydrolysis. This by itself would be an all-consuming exercise, but it can be made tractable for the present work by selecting as an appropriate starting point the section on physical, chemical and biological implications of Ref. [10], especially column 2 on p. 2232 [10]. As emphasized there, we can have different theories depending on how we redistribute the standard state Gibbs free energy of the ATP-ADP couple among the various elementary steps of binding, bond formation/breakage, and release. As clearly mentioned there (in the context of the rotation of the central γ-subunit in F1-ATPase/rotation of the myosin head in muscle myosin), “whether ATP binding energy, or ATP hydrolysis, or both cause rotation in the hydrolysis mode does not present problems for the torsional or RUT mechanism.” This was because the bond cleavage step immediately followed the ATP binding step on the enzyme. To this statement we may also add the step of Pi release, i.e., whether the ATP binding step, the Pβ-O-Pγ γ-phosphorus–oxygen bond cleavage step, or the step of release of Pi, any two of these steps, or all three steps in succession cause rotation of the γ-subunit in F1/F1FO in the hydrolysis mode, or of the myosin head in myosin II/energy storage in the S-2 coiled coil of myosin II does not present problems for the torsional or RUT mechanism. Since each of these elementary steps of Pi binding, Pβ-O-Pγ bond formation, and ATP unbinding and release require energy as repeatedly enunciated by the torsional mechanism [1, 2, 10, 13, 17], the most interesting case for physiologically important ATP-hydrolyzing molecular machines such as conventional muscle myosin II (Sections 2-4), unconventional myosins, kinesins, and ncd (Section 6) is one where the reverse of each of these steps releases free energy in succession. It only remains to specify the quantum of energy released in each step. Here, I propose that this distribution of free energy released among the steps of ATP binding, Pβ-O-Pγ (γ-phosphorus–oxygen) bond cleavage, and Pi release is ~9, ~9, and ~18 kJ/mol respectively, taking the standard Gibbs free energy of ATP hydrolysis at operating conditions of pH, pMg, and ionic strength, ΔG0′ as ~36 kJ/mol. (The free energy released in each of these steps can be readily distributed in the same proportions if other values of ΔG0′ are assumed, and the unified theory is robust and does not require the particular value employed here; however, it is required of other values of ΔG0′ to ensure, after distribution of the ΔG0′ energy among the various steps, that energy competency for each step of the mechanism is satisfied). ADP release has the same triggering function in various physiological ATP-hydrolyzing molecular motors of the myosin and kinesin family (e.g. it unblocks the kink/localized strain at the S-1–S-2 hinge of myosin II, and triggers the coiling back of the first few N-terminal heptads of S-2, which enables utilization of the stored energy of S-2 and subsequently leads to the initiation of the power strokes of myosin on actin about the S-1–S-2 hinge of myosin as the fulcrum), as specified earlier in minute detail [10]. In general, it can be said that energy has to be supplied from another source or some other interaction is required (e.g. with another subunit or agent) to unbind the bound MgADP from the enzyme. However, it should be understood that in the case of in vitro ATP hydrolysis by F1-ATPase, when the special phenomenon of nucleotide exchange of bound ADP with medium ATP in site 2 (L-site) is operative (Section 2.4.3), or if ATP binds to one catalytic site and ADP is released from another catalytic site in the α3β3γ subcomplex of F1 then, since ADP release can cause distortions in the catalytic site during its release [10] and since there are interactions of the catalytic site with γ, there is no reason why ADP release cannot cause some rotation of the γ-subunit. In other words, the ADP-ATP nucleotide exchange from a single catalytic site, or the ADP release from a catalytic site, together with ATP binding to a different catalytic site, can drive the ~80o sub-step of γ-rotation, in agreement with a recent proposal from sophisticated single molecule experiments on the α3β3γ subcomplex of F1 [30]. It should also be clearly understood that the binding energy of MgATP to myosin head is not ~9 kJ/mol but rather it is ~9 kJ/mol larger than the interaction energy of the actomyosin bond after the power stroke, because the MgATP binding energy to the myosin head is used to break the myosin-actin bond and the surplus (balance) of ~9 kJ/mol is released and is available to cause conformational changes in myosin II, or be stored. The part of the free energy of binding of Mg-nucleotide to the enzyme that helps break the actin-myosin/F1 β-ε interactions, when added to the standard free energy change upon ATP hydrolysis of ~36 kJ/mol, yields the total ΔG′ change of ~55 to ~60 kJ/mol for the entire cycle. Ultimately, it is imperative that this ~55 to ~60 kJ/mol per ATP synthesized be provided by the redox machinery in mitochondria during oxidative phosphorylation or by light energy in chloroplasts during photophosphorylation. It should also be noted that such a redistribution of free energies and such a mechanistic description in the ATP hydrolysis mode by the RUT energy storage mechanism is consistent with the description in the ATP synthesis mode by the torsional mechanism.

2.2. Contradictory Assumptions and Gross Inconsistencies among Previous Models as Seen from the Viewpoint of the Unified Theory

Thus, to arrive at a unified thermodynamic theory of the ATP catalytic cycle, it was necessary to specify the energetics of each step of binding, bond breaking and ligand release. This has been specified in Section 2.1. Of course, such an apportioning of energy release among the steps of ATP binding, hydrolysis and inorganic phosphate release (from being bound on the enzyme and removal to infinity, e.g. by release to the medium) contradicts existing theories in bioenergetics such as the binding change mechanism, which claims that the useful work is due to the substrate ATP binding energy released in the binding step. It should also be observed that theories in motility, such as the lever arm mechanism, attribute the power stroke to be primarily due to product phosphate/ADP release, which is completely different from that postulated by the binding change mechanism, as stressed earlier [10]. Why have there been these gross inconsistencies among previous models and how can the unified theory remove them?

2.3. Further Fundamental Differences between the Torsional Mechanism of ATP Synthesis and the Binding Change Mechanism

A major responsibility that has led to this state of affairs has to be taken by the binding change mechanism beginning ~1970, which claimed as a central tenet that the actual chemical synthesis step in the making of ATP required no external energy, was in fact, gratis, and focused almost exclusively on binding energy of the MgADP/MgATP substrate and its transmission from one catalytic site to other catalytic sites of the enzyme (cooperativity) where it was hypothesized to be utilized for other functions (e.g. product release). We have repeatedly pointed out the falsity of this and several other tenets of the binding change mechanism [1, 2, 10, 11, 13, 16, 17, 21, 29]. The mechanism is inconsistent with basic biophysical and structural data, such as the observation of tri-site catalysis in F1-ATPase by tryptophan fluorescence [31, 32], and the tri-site occupancy seen in the transition state by the Leslie-Walker structure of 2001 [33], as has already been discussed at great length [1, 2]. (Other discrepancies with basic biochemical data will be pointed out in Section 3.1). Firstly, such a tenet is contrary to the known principles of chemistry. For instance, in phosphate chemistry, hydrolysis of the terminal phosphate bond is known to have a standard Gibbs free energy of reaction of ~ -9 to -10 kJ/mol (and not zero). This is one reason why it was stated that in the binding change mechanism, the chemistry of the ATP hydrolysis/cleavage reaction and chemical reaction-linked conformational changes have not been given the importance they deserve and that the proposal had not been cast in detailed molecular terms to permit a proper evaluation [10]. This is not to say that the binding change mechanism has incorporated the physics of bringing charged moieties together correctly. In fact, after bond cleavage, if only one of the products is charged, as in the well-known case of hydrolysis of phosphoglycerate, then the value of ΔG0′ remains ~ -9 to -10 kJ/mol. However, if both the products are charged, as in the MgADP-Pi case, then the standard free energy change rises to ~ -36 kJ/mol, because of the Coulombic repulsion between the charged moieties, in our case, MgADP and inorganic phosphate, and this potential energy can be stored/can perform useful work. Hence, in the reverse process of bringing one of these charged moieties from infinity to the final bond distance in ATP of approximately 0.3 nm, we need to expend ~36 kJ/mol, as per the known principles of electromagnetic theory. This then is the estimate of potential energy/stored energy between the charges MgADP and Pi considered by the torsional mechanism.

The above analysis implies that hydrolysis of the terminal phosphate-oxygen bond in ATP plus the electrostatic repulsion energy of the charged MgADP and Pi products works out to be ~9 + ~27 = ~36 kJ/mol. In the reverse ATP synthesis mode, this ~36 kJ/mol has to be transmitted to the catalytic sites from a source of potential/stored energy within a subunit of the F1 portion of the ATP synthase, which in turn is transduced (and stored) from the electrochemical gradients of ions, which itself is derived from redox/light energy. According to the torsional mechanism, the source for bringing (or rather forcing) the negatively charged MgADP and inorganic phosphate to a final P-P distance in ATP of ~0.3 nm from infinity is the stored torsional energy within the γ-subunit of ATP synthase. The free rotation of γ envisaged by the binding change mechanism is simply powerless to directly perform this key function. In fact, the envisaged free rotation of the older mechanism is properly classified as a (rotational) kinetic energy. The torsional strain in γ postulated by the torsional mechanism, which arises from the theory of elasticity, is completely different, and is properly classified as a form of (storable) potential energy, or more precisely, as elastic strain energy, or the energy of elastic (and in this particular case, torsional) deformation, which, being conservative in nature, can be conserved and stored, and, ideally, is capable of being fully recovered later. Thus the difference between the two mechanisms is absolutely fundamental. As has been recently pointed out on p. 2232 of Ref. [10], when the γ shaft (whose movement is not continuous but discrete) slows down, the rotational kinetic energy will be thermalized and dissipated as heat, and not converted to useful external work or stored energy, i.e., the rotational kinetic energy of the binding change mechanism, or of the so-called “rotational catalysis” has a dissipative character. Other fundamental differences have already been summarized in a tabular column on pages 132-133 of Ref. [2].

A major achievement of the torsional mechanism of ATP synthesis appears to lie in the fact that it dealt with and proposed unprecedented details of energy transduction in the ATP synthesis mode as far back as ten years ago [1, 2, 10-29], even though the vast majority of the experimental data then was (and still continues to be) in the hydrolysis mode collected without the membrane-bound FO portion of the synthase and in the absence of the electrochemical ion gradients that drive the ATP synthesis process. Another key difference right from inception is between cooperativity, fundamental to the binding change mechanism, and asymmetry, which is fundamental to the torsional mechanism [1, 2, 13, 16-18]. The fact that the first molecule of substrate binds very tightly and with the highest affinity to a catalytic site of F1, while the second molecule of substrate binds with lowered affinity to a second catalytic site, and the third substrate molecule only binds with a very low affinity to a third catalytic site is attributed to a negative cooperativity of binding by the binding change mechanism. The torsional mechanism explains this experimental finding as arising from the asymmetric interactions of the catalytic sites with the single copy γ- and ε-subunits [1, 2, 16-19]. Thus, one of the catalytic sites interacts strongly with the ε-subunit, another catalytic site interacts with the γ-subunit while a third catalytic site interacts neither with γ nor with ε. Thus, according to the torsional mechanism, the catalytic site interacting neither with γ nor with ε binds substrate most tightly (site 1), the site interacting with γ binds substrate with intermediate affinity (site 2), and the site interacting strongly with the ε-subunit is the site of lowest affinity (site 3). Thus the observation of differing nucleotide affinities of the three catalytic sites does not have anything to do with a negative cooperativity of binding but, in stark contrast, arises from the asymmetric interactions of the catalytic sites with the single copy subunits of the F1 portion of the ATP synthase [1, 2, 16-19, 29]. According to positive catalytic cooperativity in the binding change mechanism, binding of the first molecule of substrate to the catalytic site yields only a slow uni-site catalysis rate of ATP synthesis/hydrolysis and increased substrate concentration leads to occupation of a second site which increases the catalysis rate constant and the enzyme reaches maximal activity due to a positive catalytic cooperativity among catalytic sites (which is further hypothesized by the binding change mechanism to operate simultaneously with the negative cooperativity of binding among the catalytic sites). However, according to the torsional mechanism, uni-site catalysis is a non-physiological mode of operation of the synthase, and the enzyme conformation with two sites filled is the resting (ground) state of the enzyme. Rotation and physiological steady-state ATP synthesis then only occurs when all three catalytic sites are occupied by bound Mg-nucleotide. (Thus, no continuous rotation or steady-state ATP synthesis or hydrolysis occurs in uni-site or bi-site modes of catalysis by F1FO and the enzyme only functions in a steady state in the single mode of tri-site catalysis when all three catalytic sites are filled with bound Mg-nucleotide). Since the enzyme functions only in the tri-site mode, each substrate molecule enters and binds to the enzyme in the same (unchanging) state in each catalytic cycle, and hence there can be no question of change in catalysis rate constant with substrate concentration (and transition in the number of sites filled from one to two) in the physiological steady-state mode of operation, and hence we could not conceive of positive catalytic cooperativity in this mode of operation. These mechanistic aspects have been analyzed earlier [13, 16-19] and also discussed in previous reviews [1, 2, 29].

Finally, analysis of a general kinetic scheme of steady-state ATP hydrolysis by F1/F1FO (Section 2.6) suggests the occurrence of competitive inhibition in F1-ATPase by MgADP as the inhibitor in the hydrolysis mode, which is consistent with the known property of MgADP as a competitive inhibitor in ATP hydrolysis (and also with the occurrence of “MgADP inhibition” due to its trapping in a catalytic site) [1-3]. This has important mechanistic implications because it imposes an order on binding and release events taking place on the enzyme, i.e. product MgADP release must precede substrate MgATP binding during steady-state hydrolysis, in contrast to the binding change mechanism which requires that substrate MgATP binding precede product MgADP release or be simultaneous with it in the hydrolysis mode in order that cooperativity and signal transmission from one β catalytic site (via an intervening α-site) to another β catalytic site can occur. Since the analysis (Section 2.6) is based on a general kinetic scheme that is applicable to all mechanisms, the above implication is valid irrespective of the specific mechanism.

2.4. Further Development of the Torsional Mechanism of ATP Synthesis/Hydrolysis, the Rotation- Uncoiling-Tilt (RUT) Energy Storage Mechanism of Muscle Contraction and the Unified Theory

2.4.1. Complete Details of Quantized Release and Utilization of Energy in the Synthesis Mode and its Mechanistic Implications

The mechanism by which the electrochemical gradient of protons and anions is transduced to the torsional energy in the γ-subunit of ATP synthase has already been elaborated within the torsional mechanism [1, 2, 10-13, 15, 19]. I now further propose that of the total torsional energy stored in γ (estimated as ~54 kJ/mol in the entire process, though not all the ~54 kJ/mol is present as stored torsional energy at an instant of time, as described further) the required quanta of energy are released in stages, distributed and used. First, upon ion translocation in the membrane-bound FO portion of ATP synthase, as the bottom of γ attempts to rotate counterclockwise (viewed from the F1 side) while the top of γ is stationary, ~9 kJ/mol torsional energy of γ is used to help break the ε-βE interactions by adding to the binding energy of MgADP in the βE/C catalytic site in the F1 portion of ATP synthase. Thus, in the O (open and empty) site, MgADP binds with a binding energy of ~27 kJ/mol and as the site closes, a new intermediate closed site, C is created where the MgADP binds tighter, with a binding energy of ~35 kJ/mol. Thus C contains tightly bound MgADP. These changes occur during the 0-30o counterclockwise rotation (viewed from F1) of the bottom of the γ-subunit [1, 2, 17, 18] (taking the number of subunits in the c-oligomer as 12 primarily for pedagogical reasons and for ease of explanation of a complex mechanism, but the molecular mechanism readily works for other numbers, including 10, the number employed for thermodynamic analysis in Section 4.3; in general the angle rotated in each step would be 360o/n, where n is the number of c-subunits). Another ~9 kJ/mol of torsional energy of γ is released and used to create the site for inorganic phosphate binding. This happens during the 30-60o counterclockwise rotation step of the bottom of the γ-subunit, with the top of γ stationary. Between 60-90o counterclockwise rotation of the bottom of γ while the top of γ is stationary and does not rotate, Pi binds in the newly created site/pocket and now the occupancy of the site, let us call it C′, is MgADP + Pi. In the C′ conformation of the catalytic site, both the substrates MgADP and Pi are bound; however, the β-phosphate of MgADP and the inorganic phosphate are too far apart to interact and form the terminal phosphorus-oxygen bond, as clearly inferred from the structure of the half-closed conformation with a sulfate group mimicking the phosphate [33], i.e. they are not in an activated state for nucleophilic attack and subsequent bond formation. Incidentally, it should be noted that the binding change mechanism only incorporates O (open), L (loose) and T (tight) conformations of the β catalytic sites, and such intermediate C (closed) or C′ (closed) conformations of the catalytic sites between the O and L conformations appear nowhere in the mechanism, as pointed out earlier as a defect [1, 2, 17]. However, a C′-site has been clearly visualized in the X-ray structure of 2001, as mentioned above [33], and designated HC (for half-closed), but this HC nomenclature is with reference to the open O-site, and a quarter-closed, half-closed, or three-quarter-closed etc. site can always be termed a closed (C) site with respect to the open (O) site. Furthermore, in the torsional mechanism, we had predicted the occurrence of the new intermediate closed (C) site with respect to the open or empty (O or E) site [16-19] several years before the solution of the crystal structure [33]. To continue, Pi binding releases ~9 kJ/mol energy, which affects (lowers) the threshold torsional strain in γ at ~90o angular position, and thus helps the top of γ to rotate and release the stored torsional energy in the next step. Thus, finally, during the 90-120o step of the counterclockwise rotation of the bottom of γ, the accumulated torsional strain in γ crosses the threshold and the constraints present at the top of the γ-shaft (due to the interactions of the top of the γ-subunit with the β catalytic sites) are broken and the top of the γ-subunit now rotates counterclockwise (seen from the F1 side) in a single step from 0-120o, and the remaining ~36 kJ/mol of torsional energy stored in γ is released, transmitted to the β catalytic sites and used to convert the C′-site to the L-site and the L-site to the T-site [1, 2, 10- 13, 16-18, 29]. The L-site contains bound MgADP.Pi with the MgADP–O- and the HPO4 2- now activated for nucleophilic attack, which occurs in the next conformational change (L to T), leading to the formation of the transition state and further to terminal phosphorus-oxygen bond formation in ATP, and the T-site contains tightly bound MgATP waiting to be unbound and released during the subsequent T → O transition of the catalytic site due to interaction of the ε-subunit with the T-site, converting it to an O-site. During the C′ → L transition, MgADP concomitantly binds in L with a reduced binding energy (by ~9 kJ/mol), compared to its binding energy in the C′-site, such that MgADP is bound in L with a binding energy of ~27 kJ/mol, while MgATP concomitantly binds tighter in T by ~9 kJ/mol (compared to the binding of MgADP.Pi in L) during the L → T transition, and the binding energy of the MgATP in the T-site works out to be ~45 kJ/mol. The critical role of Mg2+ in this catalysis has already been described in detail [1, 2, 17-19]. Approximately 18 kJ/mol is distributed and employed for the C′ to L transition and the activation process (the P-P distance is reduced from infinity earlier, before phosphate binds, to ~0.6 nm in L), and another ~18 kJ/mol is utilized to force the L to T transition (there is a reduction in the P-P distance from ~0.6 nm in the L-site (βTP-like) to the transition state distance of ~0.4 nm, and then a compression of the transition state to a P-P distance of ~0.3 nm in the T-site (βDP-like), the bond distance in ATP). During the final bond formation step, as the P-P distance reduces from ~0.4 to ~0.3 nm, the MgATP concomitantly becomes more tightly bound in the βDP-like catalytic site by ~9 kJ/mol than MgADP.Pi was bound in the βTP-like catalytic site. We have thus arrived at a diametrically opposite view from the binding change mechanism in which the making of the ATP was a trivial thing that merited no description, and the energy quantum required for the process was dismissed as inconsequential. In contrast, according to the torsional mechanism, the chemical synthesis of ATP by conformational changes is an exquisite process in which everything cannot be performed in one step but requires a number of ordered, sequential steps with quantization of energy as detailed above.

Before concluding this section, it should be re-emphasized that the torsional mechanism of ATP synthesis readily works for other values of H+/ATP stoichiometries and for different numbers of csubunits in the c-oligomer of ATP synthase, for instance, ten, the number favored in Section 4.3 (Figures 1, 2). For a c10 ring, three ATP molecules will be synthesized in each 360o revolution of the c10 oligomer, and ten protons will be required, i.e. an average of 3.33 protons per ATP synthesized; thus the c10 stoichiometry of the c-ring dictates a nonintegral H+/ATP ratio. Each step of rotation in FO will then measure 36o and each sub-step, 18o (Figure 1). This means that, starting from a strain-free resting state at an angular position of 0o, the c-oligomer and the bottom of the γ-subunit will rotate in 36o steps (instead of the 30o steps used in our simplified analysis in this section in order to facilitate easier understanding of a complex and dynamic mechanism), and the short pauses or dwells will be at 36o, 72o, 108o and 144o (= 24o) (for synthesizing the first ATP molecule), at 144o (= 24o), 180o (= 60o), 216o (= 96o) and 252o (= 132o = 12o) (for synthesizing the second ATP molecule), and 252o (= 132o = 12o), 288o (= 48o), 324o (= 84o) and 360o (= 120o = 0o, back to the resting state after a full revolution) (for synthesizing the third ATP molecule). Thus, four protons will be required to make the first ATP molecule and three protons each to synthesize the second and the third ATP molecules in one complete 360o revolution, i.e. on the average, 3.33 protons will be needed to synthesize one ATP molecule, or H+/ATP = 3.33. For a 10-subunit c-ring, ~180 meV of energy from the elementary ion translocation events in the FO portion of ATP synthase produces each rotation step of 36o {i.e., each elementary act of proton/anion binding and unbinding creates a local electrical potential of ~45 mV in the access halfchannels at the a-c interface in FO (Sections 5.1-5.5)} and stores ~16.2-16.5 kJ/mol as torsional energy in γ. An input of ~9 kJ/mol from the torsional energy stored in the γ-subunit is required to help MgADP bind tightly in the C conformation (this section), and according to the torsional mechanism [1, 2, 16-19], because the top of γ always rotates in 120o steps (irrespective of the number of c-subunits in the c-oligomer), the γ-subunit possesses ~12 kJ/mol torsional energy at the pause or dwell at an angular position of 144o (= 24o), sufficient to help MgADP to bind tightly, and ~6 kJ/mol at the pause or dwell at 252o (= 132o = 12o), which can be readily increased upon γ rotation and storage of torsional energy and made sufficient (to ~9 kJ/mol) and then donated to enable tight MgADP binding in C, as described in this section and earlier [1, 2, 16-19]. Further, Pi can readily bind following the formation of a Pi-binding pocket by γ rotation after a single step of 36o. Hence three steps in the catalytic cycle at the bottom of the γ shaft at the end of which MgADP binds, Pi binds, and the contacts at the top of the γ shaft are broken (following which the top of γ rotates in a single 120o step and ~36 kJ/mol torsional energy is released) respectively are minimally required to allow all the events described in this section to occur. In Figure 1, the torsional mechanism with symmetry mismatch and sub-steps of 18o for a cring with ten c subunits is illustrated. Thus, in conclusion, the mechanism works whether there symmetry or there is symmetry mismatch between F1 and FO. It is often believed that symmetry mismatch between the catalytic and rotor domains is a fundamental intrinsic feature of F-type ATPases [33] and that elasticity in the central or peripheral stalk is present only to permit a mismatched symmetry to operate. On the other hand, according to the torsional mechanism, symmetry mismatch between F1/FO is not obligatory, and the property of torsional elasticity of the central stalk is a fundamental intrinsic structural and mechanistic feature of F-type ATPases and torsional energy storage in the γ-subunit of ATP synthases has a central role in function, i.e., in synthesizing ATP, irrespective of whether there is F1/FO symmetry mismatch or no mismatch. This is also evidenced from the history of the development of the torsional mechanism [1, 2, 10-29], which was proposed before the presence of a mismatched symmetry in ATP synthases from certain sources was revealed by structural studies (see Section 4.3). This central role of the torsional strain in the γ-subunit of F1FOATP synthase has been highlighted and repeatedly emphasized, and has been captured in the name of the mechanism itself.

Finally, the stoichiometry of the c-ring is not a variable but is an intrinsic, structural property of the c-subunits of the F1FO-ATP synthase from a particular source. The exact value of the local electrical potential created by ion translocation at the sites in FO will depend on the local electrical conductivity in the FO portion of the ATP synthase in the energy-transducing membrane. Since the energy required to synthesize a molecule of ATP is constant (in contrast to the view originating from observed c-ring stoichiometries that the energy required to make an ATP molecule is variable [33]), the lower the value of the local electrical potential, the higher the number of c-subunits required in the c-oligomer has to be, and the higher will be the H+/ATP ratio (and the lower the efficiency (Section 4.3), other things being the same). The variability in the stoichiometry of ATP synthases from different sources can then be readily explained as arising from the different electrical properties of the energytransducing membranes of synthases from different sources. Thus, the variable number of c-subunits in the c-oligomer can be viewed as adaptation mechanisms and have fascinating evolutionary implications, discussion of which however is beyond the scope of this paper.

2.4.2. Quantized Release and Utilization of Energy in the Hydrolysis Mode and Its Mechanistic Implications for Muscle Contraction Once ATP synthesis has been solved in detail, as given above, the ATP hydrolysis case can be readily worked out. From the above discussion, the energy transduction does indeed occur at the hydrolysis bond cleavage step [10], but whether all or only part of the post-hydrolysis Coulombic repulsion energy between the products is released at the hydrolysis step itself or remains stored in the form of potential energy and is only released at a later step (e.g. upon release of phosphate to the medium) depends upon whether the γ-phosphate can move away from the MgADP to infinity at the hydrolysis bond cleavage step, as tacitly assumed [2, 10, 11], or not. The former assumption need not be true, as pointed out in a recent prescient and most valuable suggestion by Ross [34]. Thus, if MgATP is tightly bound to the catalytic site of the enzyme, ATP hydrolysis can take place on the enzyme, but the Pi cannot move away from the MgADP, even though there exists Coulombic charge repulsion between the two species. Hence, reduction in Coulombic repulsion cannot take place, and the potential energy between the charges cannot be converted into mechanical work, but remains stored as potential energy until the binding is reduced. This reduction of binding can occur at a later step (e.g., of Pi release) and the electrostatic repulsion energy of hydrolysis can thus be stored/used for performing work at a later instant of time. Hence the source of energy is indeed the hydrolysis step, but the release of energy is distributed over several steps of the enzymatic cycle. This change is readily made on pages 2223 and 2228 of Ref. [10] by substituting the word “during ATP hydrolysis” with “during ATP hydrolysis and later” or more specifically in the context of the present paper by “during ATP hydrolysis and subsequently upon Pi release.” Added to this the fact that the available balance of MgATP binding free energy to myosin (estimated as ~9 kJ/mol for muscle myosin) over the interaction energy of actin-myosin also causes rotation of myosin head and can be stored in S-2 implies that the binding step, hydrolysis step per se, and the potential energy between the charged products upon ATP hydrolysis that is released upon phosphate release to the medium (i.e. to infinity), all contribute to the high-energy state of the S-2 coiled coil of myosin II, and thus the free energy release is distributed over these three steps of the ATP cycle (~9 + ~9 + 18 kJ/mol respectively) and stored/locked in a high free energy (by ~36 kJ/mol) uncoiled nonequilibrium conformational state of the S-2 coiled coil of myosin II and is subsequently used to produce the power strokes of the two myosin heads on actin about the S-1–S-2 hinge as the fulcrum as described earlier [2, 10, 11, 14]. The paragraph in column 2 of p. 2229 of Ref. [10] needs to be modified in the light of the insights of this section, which had also been alluded to on p. 2232 of Ref. [10] and further in Section 2.1 of this paper.

These ideas take us beyond the lever arm model of muscle contraction from the point of view of the driving forces for the power strokes, the pressing need to consider both S-1 and S-2 together as forming the muscle “crossbridge” (unlike S-1 alone currently in the lever arm model), the crucial function of energy storage in the S-2 coiled coil, and the key role of hydrophobic interactions (the hydrophobic residues of the coiled coil are forced to intrude into a region of liquid water, thereby storing the free energy of mechanical torque/elastic deformation in a high free energy state of the submolecular coiled coil element of myosin II, with the thermodynamic propensity of water to repel the hydrophobic residues of S-2 and regain the stabler low free energy resting state of the S-2 coiled coil a primary driving force for muscle contraction). Finally, it should be noted that the fulcrum of the power stroke was postulated to be the actin-myosin attachment site by the swinging crossbridge model, and this view prevailed for almost three decades from ~1955 - ~1985. The lever arm model postulated the further distal movement of the fulcrum from the actomyosin site to the junction of the catalytic and regulatory domains within the myosin head, and this view has held sway for the past two decades from ~1985 to date. According to the RUT energy storage mechanism of muscle contraction, there is a twoway communication of energy along the helical backbone structure of myosin II (from S-1 to S-2 and then back again from S-2 to the actomyosin) and the S-1–S-2 hinge is the true fulcrum of the power strokes of myosin heads on actin executed during the backward release of stored energy. Hence the location of the hinge about which the power strokes occur needs to be moved further distally once more, and it can be stated with confidence that here, after five decades, it will finally have found its abode of peace. Other generic differences between the RUT energy storage mechanism and other models of muscle contraction have been covered in much detail in earlier papers (especially pp. 2227- 2229 of Ref. [10], pp. 160-164 of Ref. [2], and p. 82 of Ref. [14]).

2.4.3. Complete Details of Quantized Release and Utilization of Energy in the Hydrolysis Mode and its Mechanistic Implications for F1-ATPase After the above analysis, complete details of the discrete, quantized nature of energy release in the ATP hydrolysis process and especially by F1-ATPase in vitro can be better understood. Upon cleavage of the terminal phosphorus-oxygen bond in ATP, due to the T → L conformational transition of the β catalytic site caused by ~80-90o rotation of the γ-subunit, there is a change in the P-P distance from ~0.3 nm to a distance of ~0.4 nm in the transition state, and ~9 kJ/mol of free energy is released. (Simultaneously, upon crossing the transition state, the binding energy of the MgADP in the new L conformation is reduced by ~ 9 kJ/mol compared to the binding energy of MgATP in the T conformation). The free energy released is used to weaken the binding of Pi to the enzyme catalytic site by ~9 kJ/mol. This reduction of binding of Pi leads to a reduction of Coulombic repulsion between MgADP and Pi, and the phosphate moves apart from MgADP to an intermediate separation between them, which is predicted to be ~0.6 nm, and involves a Coulombic energy change of ~9 kJ/mol. This ~9 kJ/mol energy released can be stored (e.g. in the S-2 region of myosin), or be used to weaken some other interaction (e.g. ε-βE in F1-ATPase), or can cause other conformational changes. Upon actin/microtubule interaction or interaction of the γ-subunit with the catalytic site in F1-ATPase, Pi is released to the medium (this occurs after clockwise rotation of γ (looking from the F1 side) by ~80-90o during the Pi-release waiting dwell of ~1 ms duration which follows the ~1 ms long catalytic dwell in which hydrolysis occurs [30] and, due to the further reduction in electrostatic potential as the distance between the charges increases from ~0.6 nm to infinity, a further free energy release of ~18 kJ/mol takes place. (All of these discrete, quantized energy release events occur in a medium dielectric constant of ~25 typically found in a non-membranous protein fold). The ~18 kJ/mol released is further stored as an uncoiled state of S-2 in myosin II, or causes clockwise rotation of γ-subunit (looking from the F1-side) by approximately 40o (for which it has the requisite energy competency). Thus, previous work (p. 138 and the entry in the right hand side column on page 133 of Ref. [2], and p. 12 and the Legend to Figure 8 on p. 13 of Ref. [11]) needs to be modified to ascribe only the ~40o sub-step to the ATP hydrolysis energy, and not the entire 120o, and the ~80o sub-step to another source (given in the next few lines). I further propose for the first time (and this is genuinely and startlingly novel) that the sub-step of ~80o clockwise rotation (looking from F1) of the γ-subunit in the in vitro experiments on F1-ATPase or F1FO occurs due to ADP-ATP nucleotide exchange (i.e. exchange of the bound ADP in the catalytic site with medium ATP) in site 2 (βTP). This ADP-ATP exchange in site 2 (L-site or βTP), i.e. the release of ADP and the binding of ATP to the site with intermediate affinity in F1, contributes a free energy of at least ~35 kJ/mol in the catalytic site which is energetically competent to cause the ~80o rotation of the γ-subunit. It must be especially emphasized that in the ATP synthesis mode in vivo, at the physiological nucleotide concentrations in the matrix/stroma in mitochondria/chloroplasts, such a nucleotide exchange mechanism from site 2, or for that matter from any of the catalytic sites of the ATP synthase enzyme, does not occur. This is thus a major difference between the in vivo synthesis and laboratory in vitro hydrolysis modes of functioning of the enzyme. This is also a major reason why the molecular mechanism of one mode is not an exact reversal of the other mode. Further, the MgADP inhibition that has been routinely observed and documented in Vmax ATP hydrolysis has never been observed during physiological ATP synthesis, again pointing to a lack of exact reversal of the two modes. As the unified theory shows, the dwells and sub-steps of ATP synthesis caused by the ion gradients are different from, and do not mirror, the dwells and sub-steps of ATP hydrolysis by F1- ATPase. This then is the answer to a question posed recently [35]. (It has been repeatedly stressed that the mechanisms, though macroscopically irreversible [1, 2, 12, 13, 29], are microscopically reversible [2, 11]). In fact, I suggest that, once such a nucleotide exchange from the loose site (site 2) is not operative in F1-ATPase, since the tight site (site 1) contains tightly bound, non-exchangeable MgATP and the open site (site 3), which is interacting with the ε-subunit according to a key proposal of the torsional mechanism [1, 2, 16-19], is empty of bound nucleotide in the resting state (true ground state) of the enzyme [1, 17-19], or contains bound Mg-nucleotide whose binding energy is not even sufficient to break the ε-βE interactions, let alone break the interactions and also rotate γ, there would be no agent to drive the ~80o rotation sub-step of γ. This ~80o rotation of the γ-subunit is the initiation step and a prerequisite for causing the change in conformation of the tight site to the loose site that enables the bound MgATP to hydrolyze to MgADP.Pi in the new loose site and hence ATP hydrolysis cannot occur. Thus, during ATP synthesis under physiological conditions in mitochondria and chloroplasts in vivo, ATP hydrolysis cannot take place. Finally, in the absence of the ε-subunit and owing to the crucial role of this subunit in energy coupling according to the torsional mechanism [1, 2, 16-19], experiments with the α3β3γ subcomplex of F1, popularly used in single molecule hydrolysis studies (and inappropriately called “F1” [30]) can lead to artefactual mechanistic results since it is a different system from the complete F1/F1FO (containing ε). For instance, in the α3β3γ subcomplex of F1, even the identity of catalytic sites is altered compared to intact F1 or F1FO, and further, the kinetic/steric and energetic barrier due to ε-βE interactions (which is a key novel proposal of the torsional mechanism) is removed due to the absence of the ε-subunit itself in the subcomplex. Thus, in the absence of the ε-subunit, the O-site exhibits properties, especially of nucleotide binding affinity, akin to that of the C-site and is now the new site 2, intermediate in affinity between L and T. In the absence of ε, binding of MgATP to such a C-site together with MgADP release from the L-site can drive rotation of the γ-subunit by ~80o. Hence, in the α3β3γ (minus ε) subcomplex of F1, the hydrolysis mechanism is indeed bi-site, as proposed recently [30]. Thus, in the α3β3γ subcomplex of F1, due to the absence of the ε-subunit, the rotation of γ takes place with only two catalytic sites occupied by bound nucleotide and does not require filling of the third catalytic site, and the MgATP bound in C at 0o will be released as MgADP from L, i.e. upon the sequential C → T and T → L transitions, after (120o + 120o) = 240o angular rotation of γ. However, the fact that the catalysis is bi-site in experiments on α3β3γ has no bearing on, and cannot be extrapolated to what it would be in hydrolysis by F1 (especially given the crucial role of the ε-subunit discussed above and in earlier works [1, 2, 16-19]) let alone synthesis by the entire F1FO, which become hierarchical super-systems. This answers yet another question posed recently in the literature [35]. In contrast, a true understanding of the molecular mechanism of the physiologically meaningful ATP synthesis process by F1FO-ATP synthase enables us to understand also the mechanism of the ATP hydrolysis phenomenon by F1, or even by α3β3γ, as these are all sub-systems of F1FO.

2.5. Removal of the Various Inconsistencies in Previous Models by the Unified Theory

2.5.1. Models of Muscle Contraction and Motility

Now the various apparently contradictory assumptions and inconsistencies between the majority views of researchers in bioenergetics and motility can be readily reconciled. In current models of muscle contraction such as the lever arm model, force generation and movement is considered to be directly coupled to the release of bound ligands/nucleotides (e.g. “phosphate/ADP release does work”) while according to the bioenergetics literature, release of bound nucleotides requires redox/light energy or the energy of ion gradients, which is quite the opposite assumption. First of all, production of the elementary contractile force and movement is not directly coupled to release of ligands, because the energy of MgATP binding, MgATP hydrolysis and Pi release has to be first stored in a nonequilibrium conformational state in a localized region of myosin before it can be released and lead to force production and movement. But such statements are routinely made because in the conventional view, it is not deemed necessary to include energy storage in muscle myosin, which according to the RUT energy storage mechanism of muscle contraction is a central aspect of the problem. Secondly, it is not as if kicking off phosphate/ADP does work, which may be misleading (in fact, MgADP release does indeed require energy from another source). As shown by the reactions of phosphate hydrolysis (e.g. phosphoglycerate mentioned above), if only one of the products is charged and the other is not, then release of the ligands cannot perform useful work. In reality, the energy is still the energy of ATP hydrolysis, which remained stored as potential energy (i.e. the energy of Coulombic repulsion) between the two charged products, ADP and phosphate, until reduction of binding, which allowed the phosphate to move away from the ADP. It is, in fact, ~half of the total ATP hydrolysis energy, whose release has been delayed in time and allowed to be distributed in another elementary step. For conventional muscle myosin, since all the ~36 kJ/mol is stored in a single coiled coil, this distribution of the energy over other elementary steps of the catalytic cycle may appear puzzling, but such distribution of free energy over other elementary steps of the cycle is absolutely necessary for proper functioning of other large classes of molecular machines which are doubleheaded and require processive motion on a microtubule/actin track, such as kinesins and unconventional myosins (Section 6). This is because ~18 kJ/mol has to be made available after a time lag (upon phosphate release, which ensures that the first and now leading head which had stepped forward by ~9 kJ/mol of MgATP binding energy + ~9 kJ/mol of hydrolysis energy is bound tightly to the track and hence cannot be released except by MgATP binding at a subsequent stage in the enzymatic cycle) and is used to move the second rear head, which is bound loosely to the track, forward past the tightly bound front head as per the rotation-twist energy storage mechanism for processive molecular motors [10, 11]. If the entire energy had been released at the time of hydrolysis, then the first head would move double the distance, or the energy would be dissipated; in any case (even if somehow this energy were conserved), there would be no energy source left to move the second head forward in an asymmetric way on the other side of the track (compared to the first head), and the processive movement of the molecular machine on its microtubule/actin track would be arrested. Moreover such discrete, step-wise energy release is at the heart of biologically designed molecular machines due to the quantized release process of ATP hydrolysis described here. Only through an in-depth analysis, and a true understanding of the entire process of energy transduction, accumulation, storage, release, transmission, distribution and utilization in a unified way, as in this work and in our earlier works [1, 2, 10-29], are we able to evaluate the various statements made, and lend them a logical and intellectually satisfying interpretation.

2.5.2. Models in Bioenergetics

This is not to say that researchers in bioenergetics have not committed errors. A cornerstone of the binding change mechanism is the postulate that ATP forms reversibly and spontaneously from ADP and Pi in a catalytic site with an equilibrium constant value, Keq ~1, i.e., ΔG0~0 [7]. In fact, according to Boyerean dynamic reversibility, one can go back and forth between ADP.Pi and ATP (i.e. form the bond and break it) repeatedly, without requiring any energy input, and that this occurs as many as 400 times before tightly bound ATP is released from the catalytic site. This concept is false, as repeatedly point out by us [1, 2, 11, 13, 17], and arose from an inept interpretation of oxygen exchange data, in which the fundamental error was made that the timescale on which oxygen exchange occurred was ignored (and any change in the extent of oxygen exchange was attributed instead to an alteration in the value of the exchange rate constant, and not to the time available for exchange, which should have changed with substrate concentration, but unfortunately this was not considered in the Boyerean equations), as revealed by a kinetic analysis from first principles [11]. Such a kinetic analysis revealed a perfect constancy of the rate constant of exchange over five decades of substrate (MgATP) concentration for mitochondrial F1-ATPase (Figure 7 of [11]). Even in an irreversible situation, when there is an energy trade-off or exchange, a redistribution such that free energy lost by one entity/subunit equals the free energy gained by another entity/subunit, there is a net free energy change of zero; but then this is not an equilibrium situation with Keq= 1. For illustration, let us apply the unified theory to the situation upon cleavage of the terminal P-O bond in ATP. The standard free energy change of reaction is ~ -9 kJ/mol, and is conventionally given a negative sign. This energy is immediately used to weaken Pi binding to the site, i.e. the binding free energy of Pi to its site is now less negative (i.e. less stable) by ~ +9 kJ/mol. The net change in standard state free energy is thus zero due to the free energy trade-off in the site, but this does not mean that it is a dynamic equilibrium situation and that the changes can move back and forth; in fact the free energy flow is strictly one-way during the hydrolysis process. During steady-state synthesis, the flow would be one-way in the reverse direction, and ~ +9 kJ/mol of external energy will concomitantly lead to a tighter binding of MgATP to the site, i.e. a change of ~ -9 kJ/mol, with a net standard free energy change of zero again. But this does not mean that “no external energy was required” (in fact 9 kJ/mol of external energy has been used), nor that this is an “equilibrium situation with Keq= 1.” This was a false interpretation of the binding change mechanism, and certainly it was not the only way to look at the energy transfer process, as explicitly revealed by the torsional mechanism and the unified theory, and the important arguments on p. 73 of Ref. [1] still retain their validity. Moreover, thermodynamic calculations based on experimentally measured site dissociation constants for MgADP/MgATP can further help adjudicate on this point (Section 3.1). The torsional mechanism of ATP synthesis has directed attention to an irreversible mode of operation of the ATP synthase [1, 2, 10-29]. Thus, a discrete, unidirectional rotation of the γ-subunit during steady-state synthesis or hydrolysis by ATP synthase is only possible due to the presence of a driving force in one direction. The driving forces in the synthesis mode are the concentration gradients of membrane-permeant ions, while ATP and its cleavage products and their release are the energy sources for reverse rotation in the hydrolysis mode, i.e. the agents/energy sources driving rotation of the γ-subunit are different in the two modes, and both driving forces or modes do not operate simultaneously in the same enzyme molecule, as conceived by Boyerean dynamic reversibility in particular, and by reversible catalysis in general. Thus, in steady-state operation, a single enzyme molecule either functions in the synthesis mode and the γ-subunit rotates in the counterclockwise sense (viewed from F1) or works in the hydrolysis mode in which the γ-subunit rotates continuously in the clockwise sense (viewed from F1), in agreement with single molecule experiments, and does not alternate from one mode to another as conceived by the binding change mechanism (where such a reversal is presumed to occur as many as 400 times before a single product molecule is released into the medium). This is due to the fact that the initial and boundary conditions of the macroscopic system especially in terms of permeant ion concentrations and substrate MgADP/MgATP concentrations are different in the two modes. Thus, ion binding and unbinding processes to/from their binding sites in the membrane-bound FO portion of the ATP synthase along their concentration gradients drive steady-state ATP synthesis in counterclockwise ~15o/~30o sub-steps of the rotating elements, while concentration gradients of ADP (from a catalytic binding site in F1 to the medium) and ATP (from the medium to a catalytic binding site in F1) and subsequent ATP hydrolysis and Pi release to the medium drive the rotating elements in clockwise ~80o/~40o sub-steps (with analogs of ATP) or ~90o/~30o sub-steps (with ATP) during steady-state hydrolysis, as addressed in consummate detail in the unified theory.

As discussed above, the torsional mechanism of energy transduction and ATP synthesis, the RUT energy storage mechanism of muscle contraction, and the unified theory have directed attention to the importance of an irreversible mode of operation of the energy-transducing enzyme [1, 2, 10-29]. This does not mean that the enzyme cannot be reversed (in this sense of usage of the term “reversibility,” the ATP synthase enzyme is indeed reversible), but that due to the initial and boundary conditions prevalent in the in vivo system or imposed by the experimentalist on the system in vitro that ensure the presence of a driving force in one direction, the enzyme works in a single mode (synthesis or hydrolysis), until these initial and boundary conditions are altered.

In addition to the ~9 kJ/mol released upon cleavage of the terminal P-O bond in ATP (discussed above), there is furthermore ~9 + ~18 kJ/mol, the latter although it is released upon phosphate release, is, in reality, a part of the hydrolysis potential energy, as discussed in Section 2.4.2. In the reverse mode, during physiological ATP synthesis, energy has to be supplied by the torsional energy of γ and used to convert the C′ catalytic site containing MgADP + Pi to the loose site containing MgADP.Pi (~18 kJ/mol) and another ~9 kJ/mol + ~9 kJ/mol used during the L to T conformational transition of the catalytic site to form the transition state and compress the transition state respectively in order to reach the final P-P distance in ATP of ~0.3 nm.

Over time, attempts have been made to modify the binding change mechanism, for instance to try and include the experimental fact of tri-site catalysis (as opposed to the bi-site catalysis of the original binding change mechanism), but without altering other fundamental tenets of the mechanism. First of all it is a matter of kudos that in a subject where there is much variance, the assignment of the structurally tight site (βDP-like) and the structurally loose site (βTP or βTP-like) in the works of Walker and Leslie, Senior, and Nath are identical. It is a different matter that computational approaches like MDsimulations in the hands of several investigators and other methods have led to βTP being assigned as the tight site [36-38]. We completely disagree with this latter assignment as evident from close structural inspection, magnesium coordination chemistry considerations, and calculations of the buried surface area at the catalytic interfaces. The order of conformations that a catalytic site passes through during hydrolysis is postulated by Menz et al. [33] to be:

βE → ~βHC → βTP → βDP → βHC → βE (1)

where the ~ sign means that the state has not yet been structurally visualized. Note that the transition state has been excluded in the equation. Therefore, for synthesis of ATP they have to have:

βE → βHC → βDP → βTP → ~βHC → βE (2)

The torsional mechanism had made 15 predictions for ATP synthesis by F1FO-ATP synthase, which were listed in 2002 (pp. 79-80 of Ref. [1]; see also pp. 132-133 of Ref. [2]). We still stand by these predictions made for the synthesis mode. The order of conformations that we had predicted is in contradiction with the order postulated by Menz et al. (Eq. (2)), though our predictions of the order of conformations of a β catalytic site are in consonance with Senior [3]. However, Senior and colleague have suggested that Pi binding precedes ADP binding during ATP synthesis [31] and that two catalytic sites may carry out the ATP synthesis reaction simultaneously [3]; in contrast, according to the torsional mechanism, ADP binding precedes Pi binding during physiological ATP synthesis, and ATP is synthesized during the loose to tight conformational transition of a β catalytic site and only one of the three catalytic sites (the T-site), can contain the synthesized, tightly bound MgATP at one time [1, 2, 17]. From the unified theory above, after incorporating the C′ state, predictions 8 and 9 [1] yield the following sequence of conformations that a single catalytic site cycles through during physiological ATP synthesis:

βE → βC(MgADP) → βC′(MgADP+Pi) → βTP-like(MgADP.Pi) → βDP-like(MgATP) → βE (3)

i.e. we have five different conformational states (excluding the transition state) to describe the synthesis mechanism. Or, in words, the sequence of conformations that a particular β-subunit passes through during ATP synthesis is O (open, βE) to C (closed, βC) to C′ (closed, βC′) to L (loose, βTP-like) to T (tight, βDP-like) and back to O (open, βE) (Figure 2). The bound nucleotide occupancies of the catalytic sites during ATP synthesis are: no bound nucleotide in βE (open), MgADP in βC (closed), MgADP + Pi in βC′(closed), MgADP.Pi in βTP-like (loose), and MgATP in βDP-like (tight) (Figure 2). Extending the sequence of events at a single β catalytic site to the ATP synthase enzyme during ATP synthesis as a whole (Figure 7 of Ref. [1] is still very much apposite here), it is predicted that the order of the conformational changes of the catalytic sites is O → C → C′, followed by T → O, followed by L → T, and lastly, C′ → L. Note that due to the presence of torsion in the γ-shaft in accordance with a central tenet of the torsional mechanism of ATP synthesis, there is sufficient time (while the bottom of γ is rotating step-wise in ~15o/~30o intervals but the top of γ is stationary) for the formation of the intermediate conformations C and C′ in which MgADP and Pi can bind respectively. In the binding change mechanism, on the other hand, the postulate of free rotation of γ does not provide sufficient time for the binding of substrates, and this is another reason why the concept of free rotation, though a part of current scientific dogma, leads to acute mechanistic difficulties.

A major problem with the sequence for synthesis in Eq. (2) [33] is that it is enabling ADP to bind to an open site, Pi also to bind to that site, ADP and Pi to be activated for nucleophilic attack, transition state formation, and ATP synthesis, all in a single binding change. Since Pi binding and subsequent steps require energy, this is very difficult, if not impossible to conceive. The latest version of the binding change mechanism [39] has also incorporated this sequence of conformations, a major departure from its earlier stance of several decades [7], without any justification (except perhaps to bring it in-line with the proposal in Eq. (2) [33], though no such, or any other, explanation was offered [39]), which was criticized as even more problematic for it than earlier versions (see p. 74 of Ref. [1], and further related aspects on pp. 135-136 of Ref. [2]). If, indeed, all the above changes can take place in a single O → T binding change, as hypothesized, then what is the need to have three catalytic sites in the F1-portion of the enzyme, and what is the function of the L-site? Thus this basic structural feature of the F1-ATPase will then defy explanation. Note that none of these problems beset the torsional mechanism and the unified theory, because each of the above chemical steps takes place in a different conformation ([1, 2] and Eq. (3) of this work and the description below it). A possible reason for the structural interpretations could be that the mitochondrial F1 X-ray structures [33, 40] are of an “MgADP-inhibited state” (as mentioned in the original 1994 native mitochondrial F1 structure paper [40]), as they were carried out with various inhibitors. I propose that the state captured in the crystal structures mimics a metastable, post-hydrolysis, pre-product release state after ~80o rotation of the γ-subunit.

In the light of the unified theory and the above discussion, the sub-predictions in prediction numbers 8 and 9 for the hydrolysis mode (p. 80 of Ref. [1]) must be revised. Within the framework of that paper, we can now state that the order of conformations that an F1-ATPase catalytic site passes through during steady-state Vmax hydrolysis is O to T to L to C and back to O. The bound nucleotide occupancies of the catalytic sites during steady-state Vmax ATP hydrolysis by F1-ATPase are: no bound nucleotide in βE (open), MgATP in βDP-like (tight), MgADP.Pi in βTP (loose), and MgATP in βC (closed). Starting with the enzyme in the L, O, T resting state (as on page 79 of Ref. [1] seen from the F1 side, except that now γ will rotate clockwise) with L containing bound MgADP, O either containing no bound nucleotide or containing bound MgATP (depending on whether the MgATP bound to O before/during, or after the clockwise ~80-900 γ rotation sub-step, but since the ε-subunit is still interacting with this catalytic site (site 3), it is called O here), and T (i.e., site 1 which is βDP-like) containing bound MgATP. After ADP-ATP exchange has occurred in the L-site (Section 2.4.3), it contains bound MgATP. The top of γ (γt) rotates ~80-900 clockwise due to ADP-ATP nucleotide exchange occurring in the L-site. Upon γt movement clockwise, the L-site changes to the C conformation. The ε-subunit and the bottom of γ (γb) have not rotated yet, and the ε-subunit continues to interact with O (βE). After the first rotation sub-step, with γ paused at an angular position of ~80- 90o, T changes to the L conformation and ATP hydrolysis occurs during the catalytic dwell, and now the new L-site contains bound MgADP.Pi. This is the snapshot (in the overall structural sense, in terms of the angular position of the single-copy subunits γ and ε, and conformationally of the β-catalytic sites) of the native mitochondrial F1-ATPase structure of 1994 [40] in the so-called “MgADP-inhibited state,” though of course not in terms of nucleotide occupancies. The O-site (corresponding to βE in the structure) is either empty of bound nucleotide (as in the structure) or may contain bound MgATP, the L-site contains bound MgADP.Pi (and corresponds to βTP in the structure where it contains a bound MgAMP-PNP analog), and the C-site contains bound MgATP (equivalent to βDP in the structure where, however, it is occupied by bound MgADP). It should be carefully noted that this catalytic site, though structurally tight compared to the other two catalytic sites in the structure, is not the T-site (site 1). After release of ~9 kJ/mol of MgATP hydrolysis energy has helped to reduce Pi binding in the Lsite and the other ~9 kJ/mol has been funneled via ε-βTP interactions (through the interaction of the helix tip ending in Met-138 of the ε helix-turn-helix motif) to the O-site (which, together with the binding energy of MgATP binding in the site, helps to close it and break the ε-βE interaction occurring via ε-Ser-108), the ε and γb move away clockwise. Thus, now O has changed its conformation to T, and contains tightly bound MgATP. Meanwhile, ε and γb continue their clockwise rotation to an angular position of ~80-90o. Concomitantly, the phosphate is released from the L-site (which now contains bound MgADP) and the ~18 kJ/mol is used to rotate γ (both γt and γb) and ε clockwise from ~80/90o to 120o. The interaction of ε with the C-site changes it to an O-site, from which the bound MgATP is released, and thus the O-site is empty of bound nucleotide, and the resting/ground state of the enzyme from which we initiated the cycle has now been regained, but with a 120o clockwise shift (looking from the F1-side), and the new L-site is ready to release its ADP at the 120o angular position of γ, participate in the next round of ADP-ATP exchange, and thus the next one-third part of the hydrolysis catalytic cycle can begin afresh. A similar mechanism operates in F1FO [11] except that, after phosphate release from the L-site, γt rotates clockwise from ~80/90o to 120o while ε and γb rotate clockwise in ~15/~30o steps from 0o until they reach 120o.

It should be noted that if there are high (~mM) concentrations of Mg2+ and ATP in the surrounding medium, then a favorable concentration gradient for release of the MgATP from the O-site into the medium may not exist, in which case the O-site will contain MgATP in it, and the mechanism will be tri-site. In any case, to obtain steady-state hydrolysis activity, i.e. if the rotation of γ and ε is to occur continuously over several cycles, and not stop after ~80o, then the O-site will have to be filled by medium MgATP, and the steady-state hydrolysis mechanism by F1-ATPase will have to be tri-site. This is because if the O-site is empty, then upon the O → T transition of the catalytic site (i.e., without requiring MgATP binding to the O-site if such a transition can occur at all, which itself is very unlikely in the first place), the T-site will be empty (instead of containing tightly bound MgATP), and hence no hydrolysis and therefore no phosphate release can take place after the T → L transition of the catalytic site upon ~80o rotation of the γ-subunit driven by ADP-ATP exchange in the L-site of F1- ATPase. Hence, in either case, the ~40o rotation of γ will not take place, and the F1 enzyme molecule will be trapped at an angular position of γ of ~80o, and no steady-state hydrolysis activity will be observed. Hence the criterion of observing steady-state rotation and obtaining steady-state ATPase activity in F1-ATPase imposes the requirement that all three catalytic sites be filled with bound Mgnucleotide, as discussed above. In other words, the mechanism of steady-state hydrolysis by F1- ATPase has to be tri-site, as found experimentally [3, 32]. However, in such a tri-site hydrolysis mechanism by F1-ATPase, the MgATP bound to the O-site will be released as MgADP from the L-site after ~240o clockwise rotation of the γ-subunit (looking from the F1 side) and not wait another ~120o (i.e. till ~360o) for release until the L → C → O transition has occurred (or be released as MgADP between 320-360o, as suggested [35]), because of the operation of the special phenomenon of ADPATP nucleotide exchange in the L-site (site 2) in the hydrolysis mode. Thus, at high (~mM) concentrations of Mg2+ and ATP in the medium, there exists a favorable concentration gradient for bound ADP in the catalytic site to be released into the medium and a favorable concentration gradient for entry and binding of ATP from the medium into the catalytic site. It should be noted that MgATP bound in catalytic sites of the enzyme will have very little tendency to be released into the medium because the concentration gradient favors the binding of MgATP into the catalytic sites rather than its release from the sites. Thus, the ATP which exchanged into the L-site for ADP may remain bound as MgATP or be released after 120o rotation of γ and ε, when the catalytic site changes conformation from L to C and then to an O-site upon interaction with the ε-subunit, depending on the local concentration gradient seen by the bound MgATP. If it is released, it will have to rebind in order to continue steady-state hydrolysis. However, as discussed above, at high medium Mg2+ and ATP concentrations of >1 mM that lead to maximal hydrolysis rates by F1-ATPase, release of bound MgATP from a catalytic site will be prohibited due to the presence of an adverse concentration gradient of Mg2+ and ATP. On the other hand, unbinding and release of ADP from a catalytic site and entry and binding of ATP into an unoccupied catalytic site will be strongly favored due to the presence of a downhill concentration gradient of ADP from the site to the medium and likewise a downhill concentration gradient of ATP from the medium to the site. Looked at in another way, exchange of bound ATP with medium ATP is not a real exchange; hence during hydrolysis under conditions of high medium ATP, nucleotide exchange is meaningful only from a catalytic site containing bound ADP, i.e. site 2 in F1-ATPase.

In the above mechanism for hydrolysis of ATP by F1 or F1FO, if a second, different definition of nsite is used, then the first initiation step of ~80o rotation of γ may be termed tri-site or bi-site respectively, depending on whether MgATP binds to site 3 (O-site) before the ~80o rotation sub-step due to ADP-ATP exchange in site 2 (L-site), or after the ~80o sub-step and before the ~40o sub-step due to the ATP hydrolysis energy released upon phosphate release in site 2 (L). The second sub-step of ~40o rotation occurs under tri-site conditions, because the second ~40o γ-ε sub-step has to wait for MgATP binding in the third unoccupied site (O), and until this happens, the enzyme will be trapped in the so-called “MgADP-inhibited state.” Besides, the open hinged-out O-site will offer steric hindrance to continual γ rotation, which cannot be relieved without MgATP binding to the O-site. Thus, leaving the O-site unfilled will not lead to steady-state turnover, and the rotation and hydrolysis will cease very soon, for example after ~80o rotation of the top of γ. Moreover, at high millimolar concentrations of Mg2+ and ATP in the medium, when there is more than sufficient MgATP present to fill all three catalytic sites, there is no reason why the MgATP will wait to bind to site 3 rather than taking the first available opportunity to bind to it (for example before the beginning of the ~80o sub-step of rotation itself). Finally, since, due to the absence of a driving force for release, there is no reason why MgATP bound in the O-site should unbind and be released when the medium itself contains high Mg2+ and ATP concentrations, it logically implies that MgATP binding to site 3 (O-site) does not occur every 120o, and therefore this substrate binding event to site 3 cannot be the energy source that causes complete or partial rotation of γ in each 120o cycle. In any case, an appropriate definition of n-site is the maximum number of β catalytic sites of the F1/F1FO enzyme that are required to be filled with Mgnucleotide in order to observe continuous steady-state γ or γ-ε rotation and/or obtain steady-state ATP hydrolysis rates. This is also consistent with the definition espoused earlier [p. 135 of Ref. 2]. Note also that the ADP itself cannot unbind and exit from site 2 until after the ~40o rotation sub-step has occurred: during the ~40o sub-step of rotation, between the angular positions of γ from ~80-90o to 120o, the binding of MgADP progressively starts loosening in its site, and it can exit from L at an angular γ position of 120o. Thus the mechanism of hydrolysis in a steady-state mode by F1 or F1FO is tri-site, and the fraction of the total enzyme molecules that will incur three-site filling (given the known nucleotide binding affinity (Kd) properties of the three F1 catalytic sites) and therefore exhibit steady rotation in a rotation assay and steady-state Vmax hydrolysis activity will depend on the experimental conditions (e.g. the Mg2+ and ATP concentrations) in the medium.

Finally, it should be noted that the hydrolysis mechanism postulated here is microscopically the reverse of the ATP synthesis mechanism postulated by the torsional mechanism of energy transduction and ATP synthesis, as can be clearly seen by carrying out the cycle of Figure 7 on page 79 of Ref. [1] in the reverse clockwise direction as per the torsional mechanism of ATP hydrolysis presented here. The fact that microscopic reversibility is satisfied by the molecular mechanisms of the synthesis mode and the hydrolysis mode, each of which was derived independently and at different times, is anothe important point lending great confidence to the mechanistic correctness of the proposals of the torsional mechanism and the unified theory.

The hydrolysis mechanism above is different from and simpler than that proposed by Senior and colleagues (reviewed in Ref. [3]), as seen for example from the number of steps required for the interconversion of sites during Vmax hydrolysis in the two mechanisms. A further difference is that in our mechanism, as in our earlier scheme [1, 2], two ATP molecules and one ADP molecule are bound in time average at the catalytic sites during steady-state hydrolysis, unlike two ADP and one ATP in Senior’s mechanism [3]. Moreover, the hydrolysis bond cleavage step occurs before the completion of the ~80o sub-step of γ rotation in Senior’s model [3] as well as in the Walker-Leslie model [33] and both ATP binding to the lowest affinity site and ATP hydrolysis at the highest affinity site acting in sequence drive the ~80o sub-step of γ rotation, and in fact, the hydrolysis step is held directly responsible for the major fraction of the 120o γ rotation in these models [3, 33]. However, in the torsional mechanism of ATP hydrolysis and the unified theory, the chemical hydrolysis step occurs during the catalytic dwell and follows the completion of the ~80o sub-step of γ rotation and hence is not linked to the ~80o sub-step of γ rotation, in agreement with single molecule experiments [30]. It should also be pointed out that the ATP hydrolysis mechanism postulated by Menz et al. [33] incorporates five different conformational states, while only four different conformations are required in our torsional mechanism of ATP hydrolysis presented in this section, and hence our molecular mechanism offers the minimum number of different conformational states required for steady-state ATP hydrolysis, because three conformational states (O, L and T), as envisaged by the binding change mechanism [7], are not sufficient, as discussed earlier in some detail [1, 2, 16-19]. Moreover, the mechanistic scheme for ATP hydrolysis given in Menz et al. [33] has additional states such as L′ and T* which have not been observed by the leading structural group in the field despite the passage of almost a decade, and hence, for all the reasons given above, is neither the simplest nor the shortest pathway that explains the observations. It is not clear why a longer pathway is being proposed, rather than the simpler and shorter pathway offered by the torsional mechanism. In fact, the pathway postulated by the torsional mechanism is consistent with all the structural observations. Further, the mechanism in Figure 4 of Ref. [33] can be reduced to a bi-site mechanism. (Note that the L-site simply contains ATP in all the four diagrams of Figure 4 [33]). Thus, the mechanism appears tri-site only because of the labeling. It has no real useful information above a bi-site mechanism. The objections of Berden [41, 42] to certain mechanistic schemes may also be pertinent in this context. On the other hand, the torsional mechanism is a true tri-site mechanism, as repeatedly emphasized [1, 2, 16-18], and also explains why the ATP synthesis mechanism should be tri-site.

2.6. Kinetic Analysis of ATP Hydrolysis by F1-ATPase

A kinetic scheme based on a general sequence of events leading to ATP hydrolysis which considers irreversibility of the catalysis steps (Section 2.5.2), as proposed earlier by us [1, 13, 15, 16] was developed, as depicted in Figure 3. In this kinetic scheme, E represents the F1-ATPase enzyme molecule, e.g. after MgADP has been released from the L-site, E.ATP the enzyme-ATP complex, e.g. containing MgATP bound in the T-site after the O → T conformational transition of the catalytic site, E.ADP.Pi the enzyme-ADP-inorganic phosphate complex, e.g. the catalytic site occupancy in the L-site after ATP hydrolysis has occurred during the catalytic dwell, and E.ADP the enzyme-ADP complex, e.g. after phosphate release from the L-site during the Pi-release waiting dwell (Sections 2.4.3 and 2.5.2). K1 and K2 denote the dissociation constants of the corresponding elementary steps (Figure 3). kr denotes the rate constant for conversion of E.ATP to E.ADP.Pi, and kr' the rate constant for conversion of E.ADP.Pi to E.ADP. (It should of course be understood that in the kinetic scheme of Figure 3, the shorter symbols of ATP, ADP.Pi and ADP stand for the MgATP, MgADP.Pi and MgADP, as the true substrate of the enzyme is the Mg-nucleotide). kt stands for the constant of proportionality relating the rate of transport of adenine nucleotides by the adenine nucleotide transporter (ANT) for the case in which the operation of the ATP synthase has been reversed in vivo (e.g. in necrosis and other diseases) to the corresponding adenine nucleotide concentration gradients (Figure 3). The subscript cy represents the concentration in the cytoplasm of the cell and the subscript m denotes the concentration in the organelle (e.g. mitochondrion) or the medium in vitro in which the hydrolysis reaction takes place.

Now a mathematical analysis of the kinetic scheme of Figure 3 will be carried out. For steady-state operation, the rate of ATP hydrolysis and rates of transport, binding and dissociation of ATP and ADP are equal. Thus vhyd, the rate of ATP hydrolysis can be written as


If E0 represents the total enzyme concentration, then, from a material balance on E, we have






Combining Eqs. (5) and (10), we have



Eq. (11) can also be re-written as



For in vitro hydrolysis, Eq. (11) or Eq. (12) is sufficient, but for the general case in the presence of transport steps, we can write ATP and ADP concentrations in the organelle in terms of the ATP and DP concentrations in the cytoplasm of the cell. This results in


For the case of fast diffusion/exchange of adenine nucleotides into and out of the F1-ATPase (e.g. by the action of the adenine nucleotide transporter in mitochondria), i.e. for large kt, we finally obtain




with the enzymological parameters


Eq. (14), or equivalently Eq. (15) – Eq. (18) constitute the final result of the mathematical analysis of the kinetic scheme for the general case (with transport steps). For the in vitro case, Eq. (12) is itself the principal result, where the subscript m refers to the ATP and ADP concentrations in the medium.
Comparing Eq. (12) with an equation similar to Eq. (15) in which the subscript cy is replaced by the subscript m yields the enzymological parameters for the in vitro case, i.e.


The in vitro result (Eqs. (19)-(21)) is a special case of the general result in the presence of transport steps, as can be readily seen by putting kt very large (i.e. kt → ∞) in Eqs. (17) and (18). Thus the general result reduces to the in vitro result when there are no transport barriers and ATPcy = ATPm and ADPcy = ADPm, as it should, based on physical intuition. Thus, the principal results of our model show hyperbolic Michaelis-Menten kinetics with respect to ATP in the hydrolysis mode at different ADP concentrations in the medium and show the occurrence of competitive inhibition of F1-ATPase by MgADP as the inhibitor in the steady-state hydrolysis mode. These results are consistent with the known experimental fact that MgADP is a competitive inhibitor of ATP hydrolysis activity. This implies that the MgADP competes with the substrate MgATP, or the bound MgADP changes the conformation of the site meant for substrate MgATP binding, thereby not allowing the MgATP to bind to the site. Hence, unless product MgADP is released from the catalytic site, binding of substrate MgATP cannot occur. Thus, an important consequence of the competitive inhibition is the order imposed on binding and release events during steady-state hydrolysis, i.e., product MgADP release must precede substrate MgATP binding. As the analysis is based on a general kinetic scheme that is applicable to all mechanisms, the above implication is valid irrespective of the specific mechanism.

As discussed above, the above mathematical analysis of the general kinetic scheme provides deep insights into the sequence of events in steady-state Vmax ATP hydrolysis and has important biological implications for hydrolysis mechanisms because it shows that product release must precede substrate binding. This contradicts all current mechanisms and models (except Nath’s mechanisms) because in all of these, substrate binding is postulated to precede product release or be simultaneous with it during steady-state hydrolysis. This also suggests that if, in certain instances, e.g. at high medium Mg2+ and ATP concentrations, MgATP binding to the O-site along its concentration gradient (from medium to site) were to precede MgADP release down its own concentration gradient (site to medium) from the L-site (this could happen since the events of MgADP release from L and MgATP binding to O occur due to their own, independent driving forces, as discussed in Section 2.5.2, and one does not influence the other) it will not violate the principal result of the kinetic model of this section if substrate binding to the O-site is not the event that donates energy and helps cause the partial or complete ~80o rotation of the γ-subunit and catalysis by F1-ATPase. However, if MgADP release from L precedes MgATP binding to O in F1-ATPase, then it is far more likely that MgATP binds to L before it binds to O, because of the much higher nucleotide binding affinity of the L-site over the O-site. (Otherwise, the mechanism would suffer from the same flaws pointed out by us earlier in relation to the binding change mechanism [1], for example of relying on unlikely and improbable events such as filling the lowest-affinity site 3, yet keeping the higher-affinity site 2 empty of bound nucleotide). Hence our mechanistic interpretation of the above is that MgATP needs to bind to site 3, and thereafter an MgATP molecule has to either stay bound in site 3 after the L to C to O conformational transition of a catalytic site, or if it is released from site 3 after the L to C to O conformational change of the catalytic site during activity, it has to rebind to site 3, in order to ensure steady-state hydrolysis, as discussed in detail in Section 2.5.2. However, whether MgATP binds to site 3 before, after, or simultaneously with MgADP release from site 2 is immaterial to the process, because substrate binding to site 3 is not the event that donates energy to cause rotation of the γ-subunit and hence the MgATP that binds to site 3 is not the true substrate molecule (i.e. the E.ATP in the kinetic equations, e.g. in Eq. (5) and Eq. (8) of this section) that causes rotation and catalysis in F1-ATPase. In fact, in the absence of a driving force for release of bound MgATP in site 3 into a medium itself containing high Mg2+ and high ATP concentrations, MgATP binding to site 3 is not expected to occur every 120o and hence for all the reasons given above, it cannot be the binding event that causes complete or partial γ rotation in each 120o cycle during ATP hydrolysis. On the other hand, MgADP release from the L-site and subsequent MgATP binding to the L-side, i.e. the process of ADP-ATP nucleotide exchange in site 2 has sufficient energy to rotate the γ-subunit by ~80o and is also in agreement with the order of binding and release events imposed on all hydrolysis mechanisms by kinetic analysis.

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