Simple muscle-lever systems are not so simple: The need for dynamic analyses to predict lever mechanics that maximize speed

Abstract

Levers impose a force-velocity trade-off. In static conditions, a larger moment arm increases a muscle's force capacity, and a smaller moment arm amplifies output velocity. However, muscle force is influenced by contractile velocity and fiber length, while contractile velocity is influenced by the inertial properties of the lever system. We hypothesize that these dynamic effects constrain the functional output of a muscle-lever system. We predict that there is an optimal moment arm to maximize output velocity for any given muscle-lever configuration. Here we test this hypothesis by computationally building and systematically modifying a simple lever system. We generated 3600 modifications of this model with muscles with varying optimal fiber lengths, moment arms and starting normalized muscle lengths. For each model we simulated the motion that results from 100% activation and extracted the maximum output lever velocity. In contrast to a tradeoff between force and velocity in a lever system, we found that there was, instead, an optimal moment arm which maximized both velocity and total impulse. Increasing output velocity always required increasing output force. From this we conclude that in a dynamic lever system where muscle activation is held constant, there is no tradeoff between force and velocity.

Optimising the flow of mechanical energy in musculoskeletal systems through gearing

“…Alternative but not mutually exclusive suggestions posit that the magnitude of G influences the efficiency of locomotion (Reilly et al 2007;Ren et al 2010;Usherwood 2013); that systematic variations in G with animal body size help ensure that static equilibrium can be maintained with equal muscle effort despite the systematic variation of the muscle to weight force ratio (Biewener 1989a;Dick and Clemente 2017); or that an appropriate choice and dynamic variation of G can optimise muscle mechanical performance during running (Carrier et al 1998b(Carrier et al , 1994, and maximise the mechanical energy than can be stored in elastic elements to enhance performance during explosive movements (Olberding et al 2019;Roberts and Marsh 2003). Dissenting interpretations, questioning whether G encodes a universal force-velocity trade-off, have existed since at least the late 1970s (Coombs 1978), and have grown both louder and more numerous in the last decade (McHenry and Summers 2011;McHenry 2010McHenry , 2012Olberding et al 2019;Osgood et al 2021). What is the origin of the disagreement?…”

“…", as supposed by the instantaneous perspective, an energy perspective on gearing asks: "How does gearing influence the ability of muscle to do work, and how does it control the 'transmission efficiency' of muscle work into the kinetic energy?" (McHenry and Summers 2011;McHenry 2010McHenry , 2012Olberding et al 2019;Osgood et al 2021). In other words, instantaneous quantities, such as force and velocity, are replaced with integral quantities, such as work or impulse; the focus lies no longer on a specific instant in time during a contraction, but on the contraction outcome.…”

“…For now, the influence of G on the flow of mechanical energy provides at least one further hypothesis not only for why larger animals should operate with larger G (Biewener 1989b), but also for why smaller animals should operate with smaller G (Usherwood 2013). By demonstrating that a variation in G is neither necessary nor sufficient to alter the maximum speed a musculoskeletal system can impart, McHenry cast doubt on the widespread practice to use G as a proxy for functional specialisation for force vs velocity (McHenry and Summers 2011;McHenry 2012)an objection that has been echoed by others since (Osgood et al 2021). In the forward simulations that supported the argument, the energy input was fixed, so that, without internal or external dissipative forces, force-velocity trade-offs could only ever apply instantaneously.…”

“…The instantaneous interpretation of G implies that if the muscle in both systems exerts an instantaneous force F i and shortens with an instantaneous velocity v , then the instantaneous output force and velocity differ by factors of two as well– the system with the larger gear ratio exerts a larger net force, F o = 2 GF i , but moves with a smaller velocity, u = v (2 G ) − 1 and vice versa ; varying G trades instantaneous output force against instantaneous output velocity. This interpretation is certainly instantaneously correct (Arnold et al 2011), and is not called into question (previous misunderstandings partially arose from the use of incorrect terminology, see below and Arnold et al 2011; McHenry and Summers 2011; McHenry 2010; Osgood et al 2021). But how did the muscle in both systems come to achieve the shortening velocity v ?…”

“…Heron, in his Mechanics , was perhaps the first Western scholar to note that the decrease in required forces comes with a cost Gatto (2017), but it was Galileo who firmly established this insight through the introduction of the concepts of power and work: instead of in “a certain manner deceiving nature”, gearing may reduce the force required to move an object, but it cannot alter the work a machine can do. It is this focus on muscle work, the integral of force with respect to displacement, that has called the universality of force-velocity trade-offs into question (McHenry and Summers 2011; McHenry 2010, 2012; Olberding et al 2019; Osgood et al 2021). Instead of asking “For a given instantaneous muscle force F i , shortening speed v and gear ratio G , what are the instantaneous output force F o and velocity u ?”, as supposed by the instantaneous perspective, an energy perspective on gearing asks: “How does gearing influence the ability of muscle to do work, and how does it control the ‘transmission efficiency’ of muscle work into the kinetic energy?” (McHenry and Summers 2011; McHenry 2010, 2012; Olberding et al 2019; Osgood et al 2021).…”