Optimal Gearing of Musculoskeletal Systems

Movement is integral to animal life, and most animal movement is actuated by the same engine: striated muscle. Muscle input is typically mediated by skeletal elements, resulting in musculoskeletal systems that are geared: at any instant, the muscle force and velocity are related to the output force and velocity only via a proportionality constant G, the "mechanical advantage". The functional analysis of such "simple machines" has traditionally centered around this instantaneous interpretation, such that a small vs large G is thought to reflect a fast vs forceful system, respectively. But evidence is mounting that a comprehensive analysis ought to also consider the mechanical energy output of a complete contraction. Here, we approach this task systematically, and deploy the theory of physiological similarity to study how gearing affects the flow of mechanical energy in a minimalist model of a musculoskeletal system. Gearing influences the flow of mechanical energy in two key ways: it can curtail muscle work output, because it determines the ratio between the characteristic muscle kinetic energy and work capacity; and it defines how each unit of muscle work is partitioned into different system energies, that is, into kinetic vs "parasitic" energy such as heat. As a consequence of both effects, delivering maximum work in minimum time and with maximum output speed generally requires a mechanical advantage of intermediate magnitude. This optimality condition can be expressed in terms of two dimensionless numbers that reflect the key geometric, physiological, and physical properties of the interrogated musculoskeletal system, and the environment in which the contraction takes place. Illustrative application to exemplar musculoskeletal systems predicts plausible mechanical advantages in disparate biomechanical scenarios, yields a speculative explanation for why gearing is typically used to attenuate the instantaneous force output ($G_{\text{opt}} \lt 1)$, and predicts how G needs to vary systematically with animal size to optimize the delivery of mechanical energy, in superficial agreement with empirical observations. A many-to-one mapping from musculoskeletal geometry to mechanical performance is identified, such that differences in G alone do not provide a reliable indicator for specialization for force vs speed-neither instantaneously, nor in terms of mechanical energy output. The energy framework presented here can be used to estimate an optimal mechanical advantage across variable muscle physiology, anatomy, mechanical environment, and animal size, and so facilitates investigation of the extent to which selection has made efficient use of gearing as a degree of freedom in musculoskeletal "design."