FO-F1 coupling and symmetry mismatch in ATP synthase resolved in every FO rotation step

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.

The ATPase asymmetry: Novel computational insight into coupling diverse FO motors with tripartite F1

ATP synthase, a crucial enzyme for cellular bioenergetics, operates via the coordinated coupling of an FO motor, which presents variable symmetry, and a tripartite F1 motor. Despite extensive research, the understanding of their coupling dynamics, especially with non-10-fold symmetrical FO motors, remains incomplete. This study investigates the coupling patterns between eightfold and ninefold FO motors and the constant threefold F1 motor using coarse-grained molecular dynamics simulations. We unveil that in the case of a ninefold FO motor, a 3-3-3 motion is most likely to occur, whereas a 3-3-2 motion predominates with an eightfold FO motor. Furthermore, our findings propose a revised model for the coupling method, elucidating that the pathways' energy usage is primarily influenced by F1 rotation and conformational changes hindered by the b-subunits. Our results present a crucial step toward comprehending the energy landscape and mechanisms governing ATP synthase operation.

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.

Theoretical insights into rotary mechanism of MotAB in the bacterial flagellar motor

Many bacteria enable locomotion by rotating their flagellum. It has been suggested that this rotation is realized by the rotary motion of the stator unit, MotAB, which is driven by proton transfer across the membrane. Recent cryo-electron microscopy studies have revealed a 5:2 MotAB configuration, in which a MotB dimer is encircled by a ring-shaped MotA pentamer. Although the structure implicates the rotary motion of the MotA wheel around the MotB axle, the molecular mechanisms of rotary motion and how they are coupled with proton transfer across the membrane remain elusive. In this study, we built a structure-based computational model for Campylobacter jejuni MotAB, conducted comprehensive protonation-state-dependent molecular dynamics simulations, and revealed a plausible proton-transfer-coupled rotation pathway. The model assumes rotation-dependent proton transfer, in which proton uptake from the periplasmic side to the conserved aspartic acid in MotB is followed by proton hopping to the MotA proton-carrying site, followed by proton export to the CP. We suggest that, by maintaining two of the proton-carrying sites of MotA in the deprotonated state, the MotA pentamer robustly rotates by ∼36° per proton transfer across the membrane. Our results provide a structure-based mechanistic model of the rotary motion of MotAB in bacterial flagellar motors and provide insights into various ion-driven rotary molecular motors.

ATP binding by an F1Fo ATP synthase ε subunit is pH dependent, suggesting a diversity of ε subunit functional regulation in bacteria

It is a conjecture that the ε subunit regulates ATP hydrolytic function of the F₁F_o ATP synthase in bacteria. This has been proposed by the ε subunit taking an extended conformation, with a terminal helix probing into the central architecture of the hexameric catalytic domain, preventing ATP hydrolysis. The ε subunit takes a contracted conformation when bound to ATP, thus would not interfere with catalysis. A recent crystallographic study has disputed this; the Caldalkalibacillus thermarum TA2.A1 F₁F_o ATP synthase cannot natively hydrolyse ATP, yet studies have demonstrated that the loss of the ε subunit terminal helix results in an ATP synthase capable of ATP hydrolysis, supporting ε subunit function. Analysis of sequence and crystallographic data of the C. thermarum F₁F_o ATP synthase revealed two unique histidine residues. Molecular dynamics simulations suggested that the protonation state of these residues may influence ATP binding site stability. Yet these residues lie outside the ATP/Mg²⁺ binding site of the ε subunit. We then probed the effect of pH on the ATP binding affinity of the ε subunit from the C. thermarum F₁F_o ATP synthase at various physiologically relevant pH values. We show that binding affinity changes 5.9 fold between pH 7.0, where binding is weakest, to pH 8.5 where it is strongest. Since the C. thermarum cytoplasm is pH 8.0 when it grows optimally, this correlates to the ε subunit being down due to ATP/Mg²⁺ affinity, and not being involved in blocking ATP hydrolysis. Here, we have experimentally correlated that the pH of the bacterial cytoplasm is of critical importance for ε subunit ATP affinity regulated by second-shell residues thus the function of the ε subunit changes with growth conditions.