Kinetic modeling of ATP synthesis by ATP synthase and its mechanistic implications

Field guide to Nath's research work on ATP synthesis and hydrolysis

Adenosine triphosphate (ATP) is the universal biological energy source that participates in the most prevalent chemical reactions in all cell life through the vital processes of oxidative phosphorylation (OXPHOS) and photosynthesis. Its synthesis and utilisation is an area of basic research that has seen significant progress over the last three decades. A series of Nath's publications in the 1990s culminated in a detailed description of the molecular mechanism of ATP synthesis in the FO and F1 portions of FOF1-ATP synthase in which energy from transmembrane ion gradients in FO are converted into chemical energy of ATP in F1. Subsequent papers provided a thorough theoretical basis and exploration of the validity of the new theory-named by other authors as Nath's torsional mechanism of energy transduction and ATP synthesis and Nath's two-ion theory of energy coupling. Violation of several physical laws by previous theories have been dealt with in considerable detail. In particular he has reevaluated the extensive literature on ATP hydrolysis and provides a rigorously argued tri-site molecular mechanism involving the three filled β-catalytic sites during hydrolysis by FO F1/F1-ATPase. Numerous applications have been proposed throughout his work that has resulted in four substantial publications dealing with re-interpretation of the Warburg Effect in cancer cells and a trilogy of papers dealing with biological thermodynamics of ATP synthesis applied to problems in comparative physiology, biochemistry and ecology. Finally strict mathematical methods have opened up new approaches to validate mechanistic events in ATP synthesis/hydrolysis. Here we provide a field guide for easy access to the different aspects of this body of work.

Symmetry breaking and mismatch in the torsional mechanism of ATP synthesis by FOF1-ATP synthase: mathematical number theory proof and its chemical and biological implications

Can mathematical proofs be employed for the solution of fundamental molecular-level problems in biology? Recently, I mathematically tackled complex mechanistic problems arising during the synthesis of the universal biological currency, adenosine triphosphate (ATP) by the FOF1-ATP synthase, nature's smallest rotary molecular motor, using graph-theoretical and combinatorial approaches for the membrane-bound FO and water-soluble F1 domains of this fascinating molecule (see Nath in Theory Biosci 141:249‒260, 2022 and Theory Biosci 143:217‒227, 2024). In the third part of this trilogy, I investigate another critical aspect of the molecular mechanism-that of coupling between the FO and F1 domains of the ATP synthase mediated by the central γ-subunit of ∼ 1 nanometer diameter. According to Nath's torsional mechanism of energy transduction and ATP synthesis the γ-subunit twists during ATP synthesis and the release of stored torsional energy in the central γ-stalk causes conformational changes in the catalytic sites that lead to ATP synthesis, with 1 ATP molecule synthesized per discrete 120° rotation. The twisted γ-subunit breaks the symmetry of the molecule, and its residual torsional strain is shown to readily accommodate any symmetry mismatch existing between FO and F1. A mathematical number theory proof is developed to quantify the extent of symmetry mismatch at any angular position during rotation and derive the conditions for the regaining of symmetry at the end of a 360° rotation. The many chemical and biological implications of the mechanism and the mathematical proof are discussed in detail. Finally, suggestions for further mathematical development of the subject based on ideas from symmetry and group theory have been made. In sum, the answer to the question posed at the beginning of the Abstract is a resounding YES. There exists new, relatively unexplored territory at the interface of mathematics and molecular biology, especially at the level of molecular mechanism. It is hoped that more mathematicians and scientists interested in interdisciplinary work are encouraged to include in their research program approaches of this type-a mathematical proofs-inspired molecular biology-that have the power to lead to new vistas. Such molecular-scale mechanistic problems in biology have proved extraordinarily difficult to solve definitively using conventional experimental, theoretical, and computational approaches.

2-Site versus 3-site models of ATP hydrolysis by F1-ATPase: definitive mathematical proof using combinatorics and conservation equations

The F1-ATPase enzyme is the smallest-known molecular motor that rotates in 120° steps, driven by the hydrolysis of ATP. It is a multi-subunit enzyme that contains three catalytic sites. A central question is how the elementary chemical reactions that occur in the three sites are coupled to mechanical rotation. Various models and coupling schemes have been formulated in an attempt to answer this question. They can be classified as 2-site (bi-site) models, exemplified by Boyer's binding change mechanism first proposed 50 years ago, and 3-site (tri-site) models such as Nath's torsional mechanism, first postulated 25 years ago and embellished 1 year back. Experimental data collated using diverse approaches have conclusively shown that steady-state ATP hydrolysis by F1-ATPase occurs in tri-site mode. Hence older models have been continually modified to make them conform to the new facts. Here, we have developed a pure mathematical approach based on combinatorics and conservation laws to test if proposed models are 2-site or 3-site. Based on this novel combinatorial approach, we have proved that older and modified models are effectively bi‒site models in that catalysis and rotation in F1-ATPase occurs in these models with only two catalytic sites occupied by bound nucleotide. Hence these models contradict consensus experimental data. The recent 2023 model of ATP hydrolysis by F1-ATPase has been proved to be a true tri-site model based on our novel mathematical approach. Such pure mathematical proofs constitute an important step forward for ATP mechanism. However, in what must be considered an aspect with great scientific potential, the power of such mathematical proofs has not been fully exploited to solve molecular biological problems, in our opinion. We believe that the creative application of pure mathematical proofs (for another example see Nath in Theory Biosci 141:249-260, 2022) can help resolve with finality various longstanding molecular-level issues that arise as a matter of course in the analysis of fundamental biological problems. Such issues have proved extraordinarily difficult to resolve by standard experimental, theoretical, or computational approaches.

Phosphorus Chemistry at the Roots of Bioenergetics: Ligand Permutation as the Molecular Basis of the Mechanism of ATP Synthesis/Hydrolysis by FOF1-ATP Synthase

The integration of phosphorus chemistry with the mechanism of ATP synthesis/hydrolysis requires dynamical information during ATP turnover and catalysis. Oxygen exchange reactions occurring at β-catalytic sites of the FOF1-ATP synthase/F1-ATPase imprint a unique record of molecular events during the catalytic cycle of ATP synthesis/hydrolysis. They have been shown to provide valuable time-resolved information on enzyme catalysis during ATP synthesis and ATP hydrolysis. The present work conducts new experiments on oxygen exchange catalyzed by submitochondrial particles designed to (i) measure the relative rates of Pi-ATP, Pi-HOH, and ATP-HOH isotope exchanges; (ii) probe the effect of ADP removal on the extent of inhibition of the exchanges, and (iii) test their uncoupler sensitivity/resistance. The objectives have been realized based on new experiments on submitochondrial particles, which show that both the Pi-HOH and ATP-HOH exchanges occur at a considerably higher rate relative to the Pi-ATP exchange, an observation that cannot be explained by previous mechanisms. A unifying explanation of the kinetic data that rationalizes these observations is given. The experimental results in (ii) show that ADP removal does not inhibit the intermediate Pi-HOH exchange when ATP and submitochondrial particles are incubated, and that the nucleotide requirement of the intermediate Pi-HOH exchange is adequately met by ATP, but not by ADP. These results contradicts the central postulate in Boyer's binding change mechanism of reversible catalysis at a F1 catalytic site with Keq~1 that predicts an absolute requirement of ADP for the occurrence of the Pi-HOH exchange. The prominent intermediate Pi-HOH exchange occurring under hydrolytic conditions is shown to be best explained by Nath's torsional mechanism of energy transduction and ATP synthesis/hydrolysis, which postulates an essentially irreversible cleavage of ATP by mitochondria/particles, independent from a reversible formation of ATP from ADP and Pi. The explanation within the torsional mechanism is also shown to rationalize the relative insensitivity of the intermediate Pi-HOH exchange to uncouplers observed in the experiments in (iii) compared to the Pi-ATP and ATP-HOH exchanges. This is shown to lead to new concepts and perspectives based on ligand displacement/substitution and ligand permutation for the elucidation of the oxygen exchange reactions within the framework of fundamental phosphorus chemistry. Fast mechanisms that realize the rotation/twist, tilt, permutation and switch of ligands, as well as inversion at the γ-phosphorus synchronously and simultaneously and in a concerted manner, have been proposed, and their stereochemical consequences have been analyzed. These considerations take us beyond the binding change mechanism of ATP synthesis/hydrolysis in bioenergetics.

Elucidating Events within the Black Box of Enzyme Catalysis in Energy Metabolism: Insights into the Molecular Mechanism of ATP Hydrolysis by F1-ATPase

Oxygen exchange reactions occurring at β-catalytic sites of the FOF1-ATP synthase/F1-ATPase imprint a unique record of molecular events during the catalytic cycle of ATP synthesis/hydrolysis. This work presents a new theory of oxygen exchange and tests it on oxygen exchange data recorded on ATP hydrolysis by mitochondrial F1-ATPase (MF1). The apparent rate constant of oxygen exchange governing the intermediate Pi-HOH exchange accompanying ATP hydrolysis is determined by kinetic analysis over a ~50,000-fold range of substrate ATP concentration (0.1-5000 μM) and a corresponding ~200-fold range of reaction velocity (3.5-650 [moles of Pi/{moles of F1-ATPase}-1 s-1]). Isotopomer distributions of [18O]Pi species containing 0, 1, 2, and 3 labeled oxygen atoms predicted by the theory have been quantified and shown to be in perfect agreement with the experimental distributions over the entire range of medium ATP concentrations without employing adjustable parameters. A novel molecular mechanism of steady-state multisite ATP hydrolysis by the F1-ATPase has been proposed. Our results show that steady-state ATP hydrolysis by F1-ATPase occurs with all three sites occupied by Mg-nucleotide. The various implications arising from models of energy coupling in ATP synthesis/hydrolysis by the ATP synthase/F1-ATPase have been discussed. Current models of ATP hydrolysis by F1-ATPase, including those postulated from single-molecule data, are shown to be effectively bisite models that contradict the data. The trisite catalysis formulated by Nath's torsional mechanism of energy transduction and ATP synthesis/hydrolysis since its first appearance 25 years ago is shown to be in better accord with the experimental record. The total biochemical information on ATP hydrolysis is integrated into a consistent model by the torsional mechanism of ATP synthesis/hydrolysis and shown to elucidate the elementary chemical and mechanical events within the black box of enzyme catalysis in energy metabolism by F1-ATPase.

Beyond binding change: the molecular mechanism of ATP hydrolysis by F1-ATPase and its biochemical consequences

F1-ATPase is a universal multisubunit enzyme and the smallest-known motor that, fueled by the process of ATP hydrolysis, rotates in 120o steps. A central question is how the elementary chemical steps occurring in the three catalytic sites are coupled to the mechanical rotation. Here, we performed cold chase promotion experiments and measured the rates and extents of hydrolysis of preloaded bound ATP and promoter ATP bound in the catalytic sites. We found that rotation was caused by the electrostatic free energy change associated with the ATP cleavage reaction followed by Pi release. The combination of these two processes occurs sequentially in two different catalytic sites on the enzyme, thereby driving the two rotational sub-steps of the 120o rotation. The mechanistic implications of this finding are discussed based on the overall energy balance of the system. General principles of free energy transduction are formulated, and their important physical and biochemical consequences are analyzed. In particular, how exactly ATP performs useful external work in biomolecular systems is discussed. A molecular mechanism of steady-state, trisite ATP hydrolysis by F1-ATPase, consistent with physical laws and principles and the consolidated body of available biochemical information, is developed. Taken together with previous results, this mechanism essentially completes the coupling scheme. Discrete snapshots seen in high-resolution X-ray structures are assigned to specific intermediate stages in the 120o hydrolysis cycle, and reasons for the necessity of these conformations are readily understood. The major roles played by the "minor" subunits of ATP synthase in enabling physiological energy coupling and catalysis, first predicted by Nath's torsional mechanism of energy transduction and ATP synthesis 25 years ago, are now revealed with great clarity. The working of nine-stepped (bMF1, hMF1), six-stepped (TF1, EF1), and three-stepped (PdF1) F1 motors and of the α3β3γ subcomplex of F1 is explained by the same unified mechanism without invoking additional assumptions or postulating different mechanochemical coupling schemes. Some novel predictions of the unified theory on the mode of action of F1 inhibitors, such as sodium azide, of great pharmaceutical importance, and on more exotic artificial or hybrid/chimera F1 motors have been made and analyzed mathematically. The detailed ATP hydrolysis cycle for the enzyme as a whole is shown to provide a biochemical basis for a theory of "unisite" and steady-state multisite catalysis by F1-ATPase that had remained elusive for a very long time. The theory is supported by a probability-based calculation of enzyme species distributions and analysis of catalytic site occupancies by Mg-nucleotides and the activity of F1-ATPase. A new concept of energy coupling in ATP synthesis/hydrolysis based on fundamental ligand substitution chemistry has been advanced, which offers a deeper understanding, elucidates enzyme activation and catalysis in a better way, and provides a unified molecular explanation of elementary chemical events occurring at enzyme catalytic sites. As such, these developments take us beyond binding change mechanisms of ATP synthesis/hydrolysis proposed for oxidative phosphorylation and photophosphorylation in bioenergetics.

Molecular-level understanding of biological energy coupling and transduction: Response to "Chemiosmotic misunderstandings"

In a recent paper entitled "Chemiosmotic misunderstandings", it is claimed that "enough shortcomings in Mitchell's chemiosmotic theory have not been found and that a novel paradigm that offers at least as much explanatory power as chemiosmosis is not ready." This view is refuted by a wealth of molecular-level experimental data and strong new theoretical and computational evidence. It is shown that the chemiosmotic theory was beset with a large number of major shortcomings ever since the time when it was first proposed in the 1960s. These multiple shortcomings and flaws of chemiosmosis were repeatedly pointed out in incisive critiques by biochemical authorities of the late 20th century. All the shortcomings and flaws have been shown to be rectified by a quantitative, unified molecular-level theory that leads to a deeper and far more accurate understanding of biological energy coupling and ATP synthesis. The new theory is shown to be consistent with pioneering X-ray and cryo-EM structures and validated by state-of-the-art single-molecule techniques. Several new biochemical experimental tests are proposed and constructive ways for providing a revitalizing conceptual background and theory for integration of the available experimental information are suggested.

A Novel Conceptual Model for the Dual Role of FOF1-ATP Synthase in Cell Life and Cell Death

The mitochondrial permeability transition (MPT) has been one of the longstanding enigmas in biology. Its cause is currently at the center of an extensive scientific debate, and several hypotheses on its molecular nature have been put forward. The present view holds that the transition arises from the opening of a high-conductance channel in the energy-transducing membrane, the permeability transition pore (PTP), also called the mitochondrial megachannel or the multiconductance channel (MMC). Here, the novel hypothesis is proposed that the aqueous access channels at the interface of the c-ring and the a-subunit of FO in the FOF1-ATP synthase are repurposed during induction of apoptosis and constitute the elusive PTP/ MMC. A unifying principle based on regulation by local potentials is advanced to rationalize the action of the myriad structurally and chemically diverse inducers and inhibitors of PTP/MMC. Experimental evidence in favor of the hypothesis and its differences from current models of PTP/MMC are summarized. The hypothesis explains in considerable detail how the binding of Ca2+ to a β-catalytic site (site 3) in the F1 portion of ATP synthase triggers the opening of the PTP/MMC. It is also shown to connect to longstanding proposals within Nath's torsional mechanism of energy transduction and ATP synthesis as to how the binding of MgADP to site 3 does not induce PTP/MMC, but instead catalyzes physiological ATP synthesis in cell life. In the author's knowledge, this is the first model that explains how Ca2+ transforms the FOF1-ATP synthase from an exquisite energy-conserving enzyme in cell life into an energy-dissipating structure that promotes cell death. This has major implications for basic as well as for clinical research, such as for the development of drugs that target the MPT, given the established role of PTP/MMC dysregulation in cancer, ischemia, cardiac hypertrophy, and various neurodegenerative diseases.

Time-Resolved Oxygen Exchange Measurements Offer Novel Mechanistic Insights into Enzyme-Catalyzed ATP Synthesis during Photophosphorylation

Techniques to probe molecular mechanistic events occurring at a single catalytic site of multi-subunit enzymes in real time are few and are still under development. Here time-resolved information is extracted from measurements of the extensive oxygen exchange that occurs at an intermediate stage of adenosine triphosphate (ATP) synthesis during photophosphorylation by chloroplast thylakoids. A stochastic process-based approach for modeling exchange reactions is formulated that provides a physical basis for the kinetic theory. Compatible with the assumptions made in such a model of randomness, the formulation is shown to lead to a Poisson-type theory that enables kinetic analysis of oxygen-exchange data and offers novel physical insights. Parameters such as the apparent rate constant of exchange and the average lifetime of the exchanging intermediates during the synthesis of ATP by the chloroplast F₁F_O-ATP synthase have been determined over a 5000-fold range of ADP concentration. Experimental isotopomer distributions of [¹⁸O]ATP at high (0.5 mM), intermediate (10 μM), and low (0.2 μM) ADP concentrations have been quantified and compared to expected distributions from the theory. The observed distributions are shown to closely match the predicted distributions. A wealth of novel mechanistic insights such as the number of sites/pathways of oxygen exchange, the order of substrate binding steps at the enzyme catalytic site, and regulation of the process of energy coupling have been deduced, and the results are interpreted with the help of available high-resolution X-ray structures. The various biological implications for models of energy coupling have been discussed. Permutation of oxygen ligands about the phosphorus center is proposed as a possible and general but not well-recognized mechanism for oxygen exchange that is consistent with the principal results of this work, and several suggestions for future research are offered.

Consolidation of Nath's torsional mechanism of ATP synthesis and two-ion theory of energy coupling in oxidative phosphorylation and photophosphorylation

In a recent publication, Manoj raises criticisms against consensus views on the ATP synthase. The radical statements and assertions are shown to contradict a vast body of available knowledge that includes i) pioneering single-molecule biochemical and biophysical studies from the respected experimental groups of Kinosita, Yoshida, Noji, Börsch, Dunn, Gräber, Frasch, and Dimroth etc., ii) state-of-the-art X-ray and EM/cryo-EM structural information garnered over the decades by the expert groups of Leslie-Walker, Kühlbrandt, Mueller, Meier, Rubinstein, Sazanov, Duncan, and Pedersen on ATP synthase, iii) the pioneering energy-based computer simulations of Warshel, and iv) the novel theoretical and experimental works of Nath. Valid objections against Mitchell's chemiosmotic theory and Boyer's binding change mechanism put forth by Manoj have been addressed satisfactorily by Nath's torsional mechanism of ATP synthesis and two-ion theory of energy coupling and published 10 to 20 years ago, but these papers are not cited by him. This communication shows conclusively and in great detail that none of his objections apply to Nath's mechanism/theory. Nath's theory is further consolidated based on its previous predictive record, its consistency with biochemical evidence, its unified nature, its application to other related energy transductions and to disease, and finally its ability to guide the design of new experiments. Some constructive suggestions for high-resolution structural experiments that have the power to delve into the heart of the matter and throw unprecedented light on the nature of coupled ion translocation in the membrane-bound F_O portion of F₁F_O-ATP synthase are made.

Entropy Production and Its Application to the Coupled Nonequilibrium Processes of ATP Synthesis

Starting from the universal concept of entropy production, a large number of new results are obtained and a wealth of novel thermodynamic, kinetic, and molecular mechanistic insights are provided into the coupling of oxidation and ATP synthesis in the vital process of oxidative phosphorylation (OX PHOS). The total dissipation, Φ, in OX PHOS with succinate as respiratory substrate is quantified from measurements, and the partitioning of Φ into the elementary components of ATP synthesis, leak, slip, and other losses is evaluated for the first time. The thermodynamic efficiency, η, of the coupled process is calculated from the data on Φ and shown to agree well with linear nonequilibrium thermodynamic calculations. Equations for the P/O ratio based on total oxygen consumed and extra oxygen consumed are derived from first principles and the source of basal (state 4) mitochondrial respiration is postulated from molecular mechanistic considerations based on Nath’s two-ion theory of energy coupling within the torsional mechanism of energy transduction and ATP synthesis. The degree of coupling, q, between oxidation and ATP synthesis is determined from the experimental data and the irreversible thermodynamics analysis. The optimality of biological free energy converters is explored in considerable detail based on (i) the standard biothermodynamic approach, and (ii) a new biothermokinetic approach developed in this work, and an effective solution that is shown to arise from consideration of the molecular aspects in Nath’s theory is formulated. New experimental data in state 4 with uncouplers and redox inhibitors of OX PHOS and on respiratory control in the physiological state 3 with ADP and uncouplers are presented. These experimental observations are shown to be incompatible with Mitchell’s chemiosmotic theory. A novel scheme of coupling based on Nath’s two-ion theory of energy coupling within the torsional mechanism is proposed and shown to explain the data and also pass the test of consistency with the thermodynamics, taking us beyond the chemiosmotic theory. It is concluded that, twenty years since its first proposal, Nath’s torsional mechanism of energy transduction and ATP synthesis is now well poised to catalyze the progress of experimental and theoretical research in this interdisciplinary field.

Integration of demand and supply sides in the ATP energy economics of cells

The central aspects of the energy economics of living cells revolve around the synthesis and utilization of molecules of adenosine triphosphate (ATP). Current descriptions of cell metabolism and its regulation in most textbooks of biochemistry assume that enzymes and transporters behave in the same way in isolation and in a cell. Calculations of the mechanistic or maximal P/O ratios in oxidative phosphorylation by mammalian cells generally consider only the supply side of the problem without linking to ATP-demand processes. The purpose of this article is to calculate the mechanistic P/O ratio by integration of the supply and demand sides of ATP reactions. The mechanistic stoichiometry calculated from an integrated approach is compared with that obtained from the standard model that considers only ATP supply. After accounting for leaks, slips, and other losses, the actual or operative P/O calculated by the integrated method is found to be in good agreement with the experimental values of the P/O ratio determined in mitochondria for both succinate and NADH-linked respiratory substrates. The thermodynamic consequences of these results and the biological implications are discussed. An integrated model of oxidative phosphorylation that goes beyond the chemiosmotic theory is presented, and a solution to the longstanding fundamental problem of respiratory control is found.

Molecular mechanistic insights into uncoupling of ion transport from ATP synthesis

A procedure is evolved to assess the maximum uncoupling activity of the classical unsubstituted phenolic uncouplers of mitochondrial oxidative phosphorylation (OX PHOS) 2,4-dinitrophenol and 2,6-dinitrophenol. The uncoupler concentrations, C, required for maximum uncoupling efficacy are found to be a strong function of the pH, and a linear relationship of pC with pH is obtained between pH 5 to pH 9. The slopes of the uncoupler concentrations in the aqueous and lipid phases as a function of pH have been estimated. It is shown that the experimental results can be derived from first principles by an enzyme kinetic model for uncoupling that is based on the same equations as formulated for the coupling of ion transport to ATP synthesis in a companion paper after imposition of the special conditions arising from the uncoupling process. The results reveal the catalysis of a reaction that involves both the anionic and protonated forms of the phenolic uncouplers in the vicinity of their binding sites in a non-aqueous region of the cristae membranes of mitochondria. The rate-limiting step in the overall process of uncoupling has been identified based on the uncoupling data. The data cannot be explained by a simple conduction of protons by uncouplers from one bulk aqueous phase to another as postulated by Mitchell's chemiosmotic theory. It is shown that Nath's two-ion theory of energy coupling/uncoupling in ATP synthase is consistent with the results. A molecular mechanism for uncoupling of ATP synthesis by the dinitrophenols is presented and the chief differences between coupling and uncoupling in ATP catalysis are summarized. The pharmacological consequences of our analysis of uncoupling are discussed, with particular reference to the mode of action of the anti-tuberculosis drug bedaquiline that specifically targets the c-subunit of the F₁F_O-ATP synthase and uncouples respiration from ATP synthesis in Mycobacterium tuberculosis. Hence the work is shown to be important both from the point of view of fundamental biology and is also pregnant with possibilities for practical pharmaceutical applications.

On the dynamics of the adenylate energy system: homeorhesis vs homeostasis

Biochemical energy is the fundamental element that maintains both the adequate turnover of the biomolecular structures and the functional metabolic viability of unicellular organisms. The levels of ATP, ADP and AMP reflect roughly the energetic status of the cell, and a precise ratio relating them was proposed by Atkinson as the adenylate energy charge (AEC). Under growth-phase conditions, cells maintain the AEC within narrow physiological values, despite extremely large fluctuations in the adenine nucleotides concentration. Intensive experimental studies have shown that these AEC values are preserved in a wide variety of organisms, both eukaryotes and prokaryotes. Here, to understand some of the functional elements involved in the cellular energy status, we present a computational model conformed by some key essential parts of the adenylate energy system. Specifically, we have considered (I) the main synthesis process of ATP from ADP, (II) the main catalyzed phosphotransfer reaction for interconversion of ATP, ADP and AMP, (III) the enzymatic hydrolysis of ATP yielding ADP, and (IV) the enzymatic hydrolysis of ATP providing AMP. This leads to a dynamic metabolic model (with the form of a delayed differential system) in which the enzymatic rate equations and all the physiological kinetic parameters have been explicitly considered and experimentally tested in vitro. Our central hypothesis is that cells are characterized by changing energy dynamics (homeorhesis). The results show that the AEC presents stable transitions between steady states and periodic oscillations and, in agreement with experimental data these oscillations range within the narrow AEC window. Furthermore, the model shows sustained oscillations in the Gibbs free energy and in the total nucleotide pool. The present study provides a step forward towards the understanding of the fundamental principles and quantitative laws governing the adenylate energy system, which is a fundamental element for unveiling the dynamics of cellular life.

New perspectives on photosynthetic phosphorylation in the light of a torsional mechanism of energy transduction and ATP synthesis

New perspectives on photophosphorylation have been offered from the standpoint of the torsional mechanism of energy transduction and ATP synthesis. New experimental data on the involvement of malate anions in ATP synthesis in an acid-base malate bath procedure has been reported on spinach chloroplast thylakoids as the model system. The data cannot be reconciled with the chemiosmotic theory but has been shown to be naturally explained by the torsional mechanism. The path of malic acid in the acid and base stages of the experiment has been traced, offering further strong support to the new paradigm. Classical observations in the field have been re-interpreted in the light of these findings. A new concept of ion translocation, energy transduction and coupling at the overall physiological level in photophosphorylation has been presented and a large number of novel experimentally testable predictions have been made and shown to arise as logical consequences of the new perspectives.

A role for anions in ATP synthesis and its molecular mechanistic interpretation

ATP, the 'universal biological energy currency', is synthesized by utilizing energy either from oxidation of fuels or from light, via the process of oxidative and photo-phosphorylation respectively. The process is mediated by the enzyme F(1)F(0)-ATP synthase, using the free energy of ion gradients in the final energy catalyzing step, i.e., the synthesis of ATP from ADP and inorganic phosphate (P(i)). The details of the molecular mechanism of ATP synthesis are among the most important fundamental issues in biology and hence need to be properly understood. In this work, a role for anions in making ATP has been found. New experimental data has been reported on the inhibition of ATP synthesis at nanomolar concentrations by the potent, specific anion channel blockers 4,4'-diisothiocyanostilbene-2, 2'-disulphonic acid (DIDS) and tributyltin chloride (TBTCl). Based on these inhibition studies, attention has been drawn to anion translocation (in addition to proton translocation) as a requirement for ATP synthesis. The type of inhibition has been quantified and an overall kinetic scheme for mixed inhibition that explains the data has been evolved. The experimental data and the type of inhibition found have been interpreted in the light of the torsional mechanism of energy transduction and ATP synthesis (Nath J Bioenerg Biomembr 42:293-300, 2010a; J Bioenerg Biomembr 42:301-309, 2010b). This detailed and unified mechanism resolves long-standing problems and inconsistencies in the first theories (Slater Nature 172:975-978, 1953; Williams J Theor Biol 1:1-17, 1961; Mitchell Nature 191:144-148, 1961; Mitchell Biol Rev 41:445-502, 1966), makes several novel predictions that are experimentally verifiable (Nath Biophys J 90:8-21, 2006a; Process Biochem 41:2218-2235, 2006b), and provides us with a new and fruitful paradigm in bioenergetics. The interpretation presented here provides intelligent answers to the unexplained existing results in the literature. It is shown that mechanistic interpretation of the experimental data requires substantial addition to available conceptual foundations such that present concepts, theories, and mechanisms must be revised.

Beyond the chemiosmotic theory: analysis of key fundamental aspects of energy coupling in oxidative phosphorylation in the light of a torsional mechanism of energy transduction and ATP synthesis--invited review part 1

In Part 1 of this invited article, we consider the fundamental aspects of energy coupling in oxidative phosphorylation. The central concepts of the chemiosmotic theory are re-examined and the major problems with its experimental verification are analyzed and reassessed from first principles. Several of its assumptions and interpretations (with regard, for instance, to consideration of the membrane as an inert barrier, the occurrence of energy transduction at thermodynamic equilibrium, the completely delocalized nature of the protonmotive force, and the notion of indirect coupling) are shown to be questionable. Important biological implications of this analysis for molecular mechanisms of biological energy transduction are enumerated. A fresh molecular mechanism of the uncoupling of oxidative phosphorylation by classical weak acid anion uncouplers and an adequate explanation for the existence of uncoupler-resistant mutants (which until now has remained a mystery) has been proposed based on novel insights arising from a new torsional mechanism of energy transduction and ATP synthesis.

Energy transfer from adenosine triphosphate: quantitative analysis and mechanistic insights

The ATP-ADP thermodynamic cycle is the fundamental mode of energy exchange in oxidative phosphorylation, photophosphorylation, muscle contraction, and intracellular transport by various molecular motors and is therefore of vital importance in biological energy transduction and storage. Following a recent suggestion in the pages of this journal (Ross, J. J. Phys. Chem. B 2006, 110, 6987-6990), we have carried out a simple quantitative analysis of a direct molecular mechanism of energy transfer from adenosine triphosphate (ATP). The simulation provides new insights into the mechanistic events following terminal phosphorus-oxygen bond cleavage during ATP hydrolysis. This approach also allows for the division of the energy-transfer process into elementary steps and for the prediction of the distribution of the standard-state Gibbs free energy of the overall ATP hydrolysis process among the various steps of substrate binding, bond cleavage, and product release in the enzymatic cycle, which had proved very difficult to specify previously. These predictions are consistent with available experimental data on ATP hydrolysis by protein biomolecular machines. The fundamental biological implications arising from our results are also discussed in detail. The aspects considered in this work enable us to look at the entire process of ATP synthesis/hydrolysis and energy transduction and storage in various biological molecular machines in a logically consistent and unified way.

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

Complete details of the thermodynamics and molecular mechanisms of ATP synthesis/hydrolysis and muscle contraction are offered from the standpoint of the torsional mechanism of energy transduction and ATP synthesis and the rotation-uncoiling-tilt (RUT) energy storage mechanism of muscle contraction. The manifold fundamental consequences and mechanistic implications of the unified theory for oxidative phosphorylation and muscle contraction are explained. The consistency of current mechanisms of ATP synthesis and muscle contraction with experiment is assessed, and the novel insights of the unified theory are shown to take us beyond the binding change mechanism, the chemiosmotic theory and the lever arm model. It is shown from first principles how previous theories of ATP synthesis and muscle contraction violate both the first and second laws of thermodynamics, necessitating their revision. It is concluded that the new paradigm, ten years after making its first appearance, is now perfectly poised to replace the older theories. Finally, applications of the unified theory in cell life and cell death are outlined and prospects for future research are explored. While it is impossible to cover each and every specific aspect of the above, an attempt has been made here to address all the pertinent details and what is presented should be sufficient to convince the reader of the novelty, originality, breakthrough nature and power of the unified theory, its manifold fundamental consequences and mechanistic implications, and its applications in health and disease.

Making ATP

We present a mesoscopic model for ATP synthesis by F(1)F(o) ATPase. The model combines the existing experimental knowledge of the F(1) enzyme into a consistent mathematical model that illuminates how the stages in synthesis are related to the protein structure. For example, the model illuminates how specific interactions between the gamma, epsilon, and alpha(3)beta(3) subunits couple the F(o) motor to events at the catalytic sites. The model also elucidates the origin of ADP inhibition of F(1) in its hydrolysis mode. The methodology we develop for constructing the structure-based model should prove useful in modeling other protein motors.

Molecular mechanisms of energy transduction in cells: engineering applications and biological implications

The synthesis of ATP from ADP and inorganic phosphate by F1F0-ATP synthase, the universal enzyme in biological energy conversion, using the energy of a transmembrane gradient of ions, and the use of ATP by the myosin-actin system to cause muscular contraction are among the most fundamental processes in biology. Both the ATP synthase and the myosin-actin may be looked upon as molecular machines. A detailed analysis of the molecular mechanisms of energy transduction by these molecular machines has been carried out in order to understand the means by which living cells produce and consume energy. These mechanisms have been compared with each other and their biological implications have been discussed. The thermodynamics of energy coupling in the oxidative phosphorylation process has been developed and the consistency of the mechanisms with the thermodynamics has been explored. Novel engineering applications that can result have been discussed in detail and several directions for future work have been pointed out.

The molecular mechanism of ATP synthesis by F1F0-ATP synthase: a scrutiny of the major possibilities

A critical goal of metabolism in living cells is the synthesis of adenosine triphosphate (ATP). ATP is synthesized by the enzyme F1F0-ATP synthase. This enzyme, the smallest-known molecular machine, couples proton translocation through its membrane-embedded, hydrophobic domain, F0, to the synthesis of ATP from adenosine diphosphate (ADP) and inorganic phosphate (Pi) in its soluble, hydrophilic headpiece, F1. Animals, plants and microorganisms all capture and utilize energy by this important chemical reaction. How does it occur? The binding change mechanism and the torsional mechanism of energy transduction and ATP synthesis are two mechanisms that have been proposed in the literature. According to the binding change mechanism (which considers reversible catalysis and site-site cooperativity), energy is required primarily for release of synthesized ATP, but not for its synthesis. On the other hand, according to the torsional mechanism (which considers an irreversible mode of catalysis and absence of cooperativity), all the elementary steps require energy, and the ion-protein interaction energy obtained from the ion gradients is used to synthesize ATP, for Pi binding, and for straining the beta-epsilon bond in order to enable ADP to bind. The energy to release preformed ATP from the tight catalytic site (betaDP) is provided by the formation of the beta-epsilon ester linkage. First, the central features of these mechanisms are clearly delineated. Then, a critical scrutiny of these mechanisms is undertaken. The predictions of the torsional mechanism are listed. In particular, how the torsional mechanism deals with the specific difficulties associated with other mechanisms, and how it seeks to explain a wealth of structural, spectroscopic, and biochemical data is discussed in detail. Recent experimental data in support of the mechanism are presented. Finally, in view of the molecular machine nature of energy transduction, the indispensability of applying engineering tools at the molecular level is highlighted. This paves the way for the development of a new field: Molecular Physiological Engineering.

Cardiac energy metabolism: models of cellular respiration

The heart requires a large amount of energy to sustain both ionic homeostasis and contraction. Under normal conditions, adenosine triphosphate (ATP) production meets this demand. Hence, there is a complex regulatory system that adjusts energy production to meet this demand. However, the mechanisms for this control are a topic of active debate. Energy metabolism can be divided into three main stages: substrate delivery to the tricarboxylic acid (TCA) cycle, the TCA cycle, and oxidative phosphorylation. Each of these processes has multiple control points and exerts control over the other stages. This review discusses the basic stages of energy metabolism, mechanisms of control, and the mathematical and computational models that have been used to study these mechanisms.