Symmetry breaking and mismatch in the torsional mechanism of ATP synthesis by F_OF₁-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 F_OF₁-ATP synthase, nature's smallest rotary molecular motor, using graph-theoretical and combinatorial approaches for the membrane-bound F_O 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 F_O and F₁ domains of the ATP synthase mediated by the central γ-subunit of $\sim 1$ 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 F_O 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.