Size matters in metabolic scaling: Critical role of the thermodynamic efficiency of ATP synthesis and its dependence on mitochondrial H+ leak across mammalian species

In this last article of the trilogy, the unified biothermokinetic theory of ATP synthesis developed in the previous two papers is applied to a major problem in comparative physiology, biochemistry, and ecology—that of metabolic scaling as a function of body mass across species. A clear distinction is made between intraspecific and interspecific relationships in energy metabolism, clearing up confusion that had existed from the very beginning since Kleiber first proposed his mouse-to-elephant rule almost a century ago. It is shown that the overall mass exponent of basal/standard metabolic rate in the allometric relationship $P=P_0{M}^{b^{\prime }+b}$ is composed of two parts, one emerging from the relative intraspecific constancy of the slope ($b$), and the other ($b^{\prime }$) arising from the interspecific variation of the mass coefficient, $a\left(M\right)$ with body size. Quantitative analysis is shown to reveal the hidden underlying relationship followed by the interspecific mass coefficient, $a\left(M\right)=P_0{M}^{0.10}$, and a universal value of $P_0=3.23$ watts, $W$ is derived from empirical data on mammals from mouse to cattle. The above relationship is shown to be understood only within an evolutionary biological context, and provides a physiological explanation for Cope's rule. The analysis also helps in fundamentally understanding how variability and a diversity of scaling exponents arises in allometric relations in biology and ecology. Next, a molecular-level understanding of the scaling of metabolism across mammalian species is shown to be obtained by consideration of the thermodynamic efficiency of ATP synthesis η, taking mitochondrial proton leak as a major determinant of basal metabolic rate in biosystems. An iterative solution is obtained by solving the mathematical equations of the biothermokinetic ATP theory, and the key thermodynamic parameters, e.g. the degree of coupling $q$, the operative $P/O$ ratio, and the metabolic efficiency of ATP synthesis η are quantitatively evaluated for mammals from rat to cattle. Increases in η (by $\sim 15\%$) over a $2000-$fold body size range from rat to cattle, primarily arising from an $\sim 3-$fold decrease in the mitochondrial $H^{+}$ leak rate are quantified by the unified ATP theory. Biochemical and mechanistic consequences for the interpretation of basal metabolism, and the various molecular implications arising are discussed in detail. The results are extended to maximum metabolic rate, and interpreted mathematically as a limiting case of the general ATP theory. The limitations of the analysis are pointed out. In sum, a comprehensive quantitative analysis based on the unified biothermokinetic theory of ATP synthesis is shown to solve a central problem in biology, physiology, and ecology on the scaling of energy metabolism with body size.