A Thermodynamic Optimization Analysis of a Possible Relation between the Parameters that Determine the Energetics of Muscle Contraction in Steady State

Given the phenomenological relations for muscleʹs steady-state contraction and proper definitions of powerpand efficiencyη, the behavior of these quantities is analysed in terms of the parameters that determine the energetics of the muscle, here denoted bysoandα.sois proportional to the so-called maintenance heat, while α is the parameter that determines the curvature of the Hillʹs force–velocity curve. The dependence of the muscleʹs power and efficiency, averaged over the whole range of force the muscle can exert, on the parameterssoandαis studied. The average powerpavgis a function only ofα, and is a growing function that approaches 1/6 asymptotically asαgoes to infinity. The average efficiencyηavgis a function of bothαandso. With the value ofsofixed, the graph of the functionηavg(so,α) is a convex curve with a single maximum. The value and the position of this maximum point both depend onso. In the limitα→0,so→0,ηavgtends to 1. The points (so,αm(so)), withαm(so) the value ofαthat maximizesηavgfor a givenso, are fitted by the curveα=so1/2. This relation was experimentally found by A.V. Hill in his early studies of muscle energetics. Other experimental data are found to qualitatively satisfy the same relation. Although some dynamical microscopic models for muscle contraction, based upon Huxleyʹs cross-bridge model, show that the same kinetic parameters control both the maintenance heat (so) and the muscleʹs power output (α), we suggest that the exact relation between them has been reached due to the evolutive stresses that made individuals with equally powerful and more efficient muscles more suitable to reproduce.