Elucidating a Complex Mechanism

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.

The Warburg Effect Reinterpreted 100 yr on: A First-Principles Stoichiometric Analysis and Interpretation from the Perspective of ATP Metabolism in Cancer Cells

The Warburg Effect is a longstanding enigma in cancer biology. Despite the passage of 100 yr since its discovery, and the accumulation of a vast body of research on the subject, no convincing biochemical explanation has been given for the original observations of aerobic glycolysis in cancer cell metabolism. Here, we have worked out a first-principles quantitative analysis of the problem from the principles of stoichiometry and available electron balance. The results have been interpreted using Nath's unified theory of energy coupling and adenosine triphosphate (ATP) synthesis, and the original data of Warburg and colleagues have been analyzed from this new perspective. Use of the biomass yield based on ATP per unit substrate consumed, [Formula: see text], or the Nath-Warburg number, NaWa has been shown to excellently model the original data on the Warburg Effect with very small standard deviation values, and without employing additional fitted or adjustable parameters. Based on the results of the quantitative analysis, a novel conservative mechanism of synthesis, utilization, and recycling of ATP and other key metabolites (eg, lactate) is proposed. The mechanism offers fresh insights into metabolic symbiosis and coupling within and/or among proliferating cells. The fundamental understanding gained using our approach should help in catalyzing the development of more efficient metabolism-targeting anticancer drugs.

Coupling and biological free-energy transduction processes as a bridge between physics and life: Molecular-level instantiation of Ervin Bauer's pioneering concepts in biological thermodynamics

The nonequilibrium coupled processes of oxidation and ATP synthesis in the biological process of oxidative phosphorylation (OXPHOS) are fundamental to all life on our planet. These steady-state energy transduction processes ‒ coupled by proton and anion/counter-cation concentration gradients in the OXPHOS pathway ‒ generate ∼95 % of the ATP requirement of aerobic systems for cellular function. The rapid energy cycling and homeostasis of metabolites involved in this coupling are shown to be responsible for maintenance and regulation of stable nonequilibrium states, the latter first postulated in pioneering biothermodynamics work by Ervin Bauer between 1920 and 1935. How exactly does this occur? This is shown to be answered by molecular considerations arising from Nath's torsional mechanism of ATP synthesis and two-ion theory of energy coupling developed in 25 years of research work on the subject. A fresh analysis of the biological thermodynamics of coupling that goes beyond the previous work of Stucki and others and shows how the system functions at the molecular level has been carried out. Thermodynamic parameters, such as the overall degree of coupling, q of the coupled system are evaluated for the state 4 to state 3 transition in animal mitochondria with succinate as substrate. The actual or operative P to O ratio, the efficiency of the coupled reactions, η, and the Gibbs energy dissipation, Φ have been calculated and shown to be in good agreement with experimental data. Novel mechanistic insights arising from the above have been discussed. A fourth law/principle of thermodynamics is formulated for a sub-class of physical and biological systems. The critical importance of constraints and time-varying boundary conditions for function and regulation is discussed in detail. Dynamic internal structural changes essential for torsional energy storage within the γ-subunit in a single molecule of the FOF1-ATP synthase and its transduction have been highlighted. These results provide a molecular-level instantiation of Ervin Bauer's pioneering concepts in biological thermodynamics.

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.