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.

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.

a Schematic of a minimalistic physical model of a musculoskeletal system, defined by the muscle work density, W_ρ, the muscle fascicle length, l_m, the maximum muscle strain rate, ${\stackrel{°}{\epsilon }}_{max}$, the muscle mass, m_m, 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, $\mathrm{\Gamma }~m{\left(l_m{\stackrel{°}{\epsilon }}_{max}\right)}^2{\left(W_{\rho }{m}_m\right)}^{-1}{G}^{-2}$, and the reduced parasitic energy, ${\kappa }_g~mg{\left(F_{max}G\right)}^{-1}$. 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 m_t (see (d)); if the muscle has force-velocity properties, it is more gradual (dot-dashed line). κ_g quantifies the fraction of muscle work which flows into kinetic vs gravitational potential energy. The energy demanded by gravity only becomes appreciable for large κ_g, eventually resulting in a sharp asymptote at a critical body mass, m_c, 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 ∝ m^2/3 and κ_g ∝ m^1/3). 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 κ_g, 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'', v_Hi ∝ m^1/3, v_Fr ∝ m^1/6 and v_Bo ∝ v_St ∝ m⁰ (see text). Source data for (d) are provided as a Source Data file.

Fig. 2. The variation of the musculoskeletal gear ratio, G, with size across 42 vertebrate species varying by five orders of magnitude in mass^–,.

In the initial across-clade analysis of the allometry of maximum running speed—which included invertebrates much smaller than 0.01 kg and vertebrates heavier than 2 t—an average gear ratio, G = 0.3 (solid line) was used, because it complies with the parsimonious assumption of geometric similarity, and because the gear ratio is confounded by evolutionary history, as evidenced by the different slopes for quadrupedal mammals vs. bipedal Macropodoidea, so that extrapolation bears significant risks [dashed vs. dotted line^,]. The assumption of a size-invariant gear ratio is subsequently relaxed for the size range for which experimental data are available, and the consequences of a size-variable gear ratio are discussed. Source data are provided as a Source Data file.

Fig. 3. Running animals accelerate over multiple steps, so that the absolute maximum speed they can achieve is larger than the maximum increment dictated by the Hill- or the Borelli-limit, which both define the limit for a single contraction.

A quantitative estimation of the maximum achievable speed thus requires an assumption about how speed increments accumulate over multiple steps. a If the maximum positive speed increment per contraction is v_i, and each step n loses a fixed fraction 1 − η of the current speed v, the predicted acceleration profile $v=v_i\left(1-{\eta }^n\right){\left(\left(1-\eta \right)\right)}^{-1}$ resembles empirical data [see e.g. refs. ^,], and the maximum speed is asymptotic to v_i(1 − η)⁻¹. b In order to estimate the effective coefficient of restitution η, we re-analysed data from refs. ^,, and extracted stride sequences over which the average speed remained approximately constant. Within these sequences, the maximum speed $v_{\max }$ and minimum speed $v_{\min }$ were extracted for each step, and the step with the maximal ratio $v_{\min }{v}_{\max }^{-1}$ was selected for further analysis, as it presents an upper bound for the minimal kinetic energy loss. Each data point in the plot represents a different individual (n = 52); the data represent experimental trials involving 9 species of birds and 13 species of lizards, varying between 8 g and 80 kg in body mass. η was estimated from these data via an ordinary least-squares regression forced through the origin (hence the unusual shape of the confidence bands), which yielded η = 0.89 (95% CI [0.88, 0.92]). Source data are provided as a Source Data file.