Dynamic similarity and the peculiar allometry of maximum running speed

Abstract

Animal performance fundamentally influences behaviour, ecology, and evolution. It typically varies monotonously with size. A notable exception is maximum running speed; the fastest animals are of intermediate size. Here we show that this peculiar allometry results from the competition between two musculoskeletal constraints: the kinetic energy capacity, which dominates in small animals, and the work capacity, which reigns supreme in large animals. The ratio of both capacities defines the physiological similarity index Γ, a dimensionless number akin to the Reynolds number in fluid mechanics. The scaling of Γ indicates a transition from a dominance of muscle forces to a dominance of inertial forces as animals grow in size; its magnitude defines conditions of “dynamic similarity“ that enable comparison and estimates of locomotor performance across extant and extinct animals; and the physical parameters that define it highlight opportunities for adaptations in musculoskeletal “design” that depart from the eternal null hypothesis of geometric similarity. The physiological similarity index challenges the Froude number as prevailing dynamic similarity condition, reveals that the differential growth of muscle and weight forces central to classic scaling theory is of secondary importance for the majority of terrestrial animals, and suggests avenues for comparative analyses of locomotor systems.

Optimising the flow of mechanical energy in musculoskeletal systems through gearing

“…This result is both elemental and crucial— muscle has not one, but two characteristic energy capacities. The work capacity, W max , is joined by a “kinetic energy capacity” (Labonte 2023; Labonte et al 2024), The work and the kinetic energy capacity pose two hard, independent bounds on muscle work output, W m ≤ W max and W m ≤ K max , respectively. The maximum work muscle can deliver in a single contraction is consequently determined by whichever of the two limits is smaller (Labonte 2023) (Figure 2).…”

“…Crucially, and unlike W max , K max ∝ G − 2 depends explicitly on the mechanical advantage, and can thus be geared. As long as K max < W max , gearing can therefore alter muscle energy output, and so affect the output speed that results from concentric muscle contractions (Labonte 2023; Labonte et al 2024). In this regime, force-velocity trade-offs thus apply both instantaneously and ultimately, as argued by Arnold et al (Arnold et al 2011).…”

“…As a consequence, the work demanded to pay for the change in gravitational potential energy during the acceleration is larger in (a), W e = mgδ max /G a , so that less work is available to increase the kinetic energy K = W max − W e (chart inset). Consequently, animal (b) achieves a higher takeoff speed u b at the same takeoff height, and thus jumps higher above standing: a higher mechanical advantage results in a larger average net force, and a larger take-off velocity, in defiance of a force-velocity trade-off (see also Labonte 2023; Labonte et al 2024; McHenry 2012; Scholz et al 2006). This particular effect of G may be generalised for any external force through introduction of the reduced parasitic energy , a dimensionless number which directly quantifies the fraction of muscle work lost to parasitic forces (see text for a mathematical definition).…”

“…To investigate what a focus on mechanical energy reveals about Borelli’s riddle, we will estimate the optimal gear ratio for the simplest case: a muscle contraction against an inertial load, i. e. all external forces are of negligible magnitude. Although this may sound artificial, it is likely applicable to a large range of musculoskeletal systems, for the muscle force is much larger than the weight force in all but the heaviest vertebrate animals (Alexander 1985; Labonte 2023; Labonte et al 2024).…”

“…McHenry then went one step further, and demonstrated more formally what was already suggested in the discussion of the initial commentary: the velocity resulting from a contraction can vary with gear ratio even when muscle supplies a constant amount of energy , because gearing controls the “transmission efficiency” of the energy conversion, i. e. how much of each unit of muscle work serves to increase kinetic energy (McHenry 2012, This is unpacked more formally below). Using forward dynamic simulations, McHenry showed elegantly that variations in gear ratio can leave the resulting speed unaffected, decrease it, or increase it, depending on the “mechanical environment”; there exists no one-to-one mapping between gear ratio and output speed, and variations in G are neither necessary nor sufficient to change the maximum speed a musculoskeletal system can impart (see also Coombs 1978; Labonte 2023; Labonte et al 2024; Lee et al 2009; Nagano and Komura 2003; Olberding et al 2019; Osgood et al 2021; Richards 2011; Richards and Clemente 2013a, for related results).…”