Fig. 1: Animals small and large move by using muscle as a motor, but the maximum running speed they can achieve varies non-monotonously with size: the fastest animals are of intermediate size.
From: Dynamic similarity and the peculiar allometry of maximum running speed
a Schematic of a minimalistic physical model of a musculoskeletal system, defined by the muscle work density, Wρ, the muscle fascicle length, lm, the maximum muscle strain rate, \({\dot{\varepsilon }}_{max}\), the muscle mass, mm, the gear ratio G and the mass m that is moved. Terrestrial locomotion also involves the gravitational acceleration g. b The performance space of this minimalistic system is fully characterised by two dimensionless numbers: the physiological similarity index, \(\Gamma \sim m{({l}_{m}{\dot{\varepsilon }}_{max})}^{2}{({W}_{\rho }{m}_{m})}^{-1}{G}^{-2}\), and the reduced parasitic energy, \({\kappa }_{g} \sim mg{({F}_{max}G)}^{-1}\). Γ quantifies the competition between the kinetic energy and work capacity of muscle: for Γ ≤ 1, the system can only deliver a fraction Γ of its maximum work capacity, and for Γ ≥ 1 it has access to its full work capacity (solid line). For a muscle force that is independent of muscle strain rate, the transition between these regimes is sharp and occurs at a body mass mt (see (d)); if the muscle has force-velocity properties, it is more gradual (dot-dashed line). κg quantifies the fraction of muscle work which flows into kinetic vs gravitational potential energy. The energy demanded by gravity only becomes appreciable for large κg, eventually resulting in a sharp asymptote at a critical body mass, mc, at which movement is no longer possible (grey dashed line, see (d)). c Both dimensionless numbers vary systematically with size for geometrically similar animals (Γ ∝ m2/3 and κg ∝ m1/3). As a consequence of the increase of Γ, larger animals have access to a larger fraction of their work capacity and are thus generally faster. However, due to the increase in κg, an increasingly larger share of this work has to pay for fluctuations in gravitational potential energy, eventually resulting in a reduction in speed (see (b)). d The combination of both effects results in the peculiar allometry of maximum running speed (n = 633); the black dashed line is a least-square fit of Eq. (5), leaving only a dimensionless scaling coefficient as free parameter (see text). Γ thus emerges as a fundamental dimensionless number for musculoskeletal dynamics, which may be used and interpreted akin to the Reynolds number (see discussion). The three short solid lines illustrate asymptotic scaling relations defined by three alternative indices of ‘dynamic similarity'', vHi ∝ m1/3, vFr ∝ m1/6 and vBo ∝ vSt ∝ m0 (see text). Source data for (d) are provided as a Source Data file.
