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 muscle's steady-state contraction and proper definitions of power p and efficiency eta, the behavior of these quantities is analysed in terms of the parameters that determine the energetics of the muscle, here denoted by s(o)and alpha. s(o)is proportional to the so-called maintenance heat, while alpha is the parameter that determines the curvature of the Hill's force-velocity curve. The dependence of the muscle's power and efficiency, averaged over the whole range of force the muscle can exert, on the parameters s(o)and alpha is studied. The average power p(avg)is a function only of alpha, and is a growing function that approaches 1/6 asymptotically as alpha goes to infinity. The average efficiency eta(avg)is a function of both alpha and s(o). With the value of s(o)fixed, the graph of the function eta(avg)(s(o), alpha) is a convex curve with a single maximum. The value and the position of this maximum point both depend on s(o). In the limit alpha-->0, s(o)-->0, eta(avg)tends to 1. The points (s(o), alpha(m)(s(o))), with alpha(m)(s(o)) the value of alpha that maximizes eta(avg)for a given s(o), are fitted by the curve alpha=s(o 1/2). 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 Huxley's cross-bridge model, show that the same kinetic parameters control both the maintenance heat (s(o)) and the muscle's power output (alpha), 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.