Whole-limb scaling of muscle mass and force-generating capacity in amniotes

Dataset

The present study is founded upon muscle architecture data derived from dissections (Table 1); all raw data used are provided in Table S1. The sources of the data are described in the following two subsections. Although this study addresses ‘whole limb’ scaling, it is restricted in scope to the musculature crossing the three primary joints of the limbs (shoulder/hip, elbow/knee and wrist/ankle), ignoring the intrinsic musculature of the manus and pes. This was necessitated by the practical difficulties of accurately dissecting and measuring the latter muscles (particularly for small species), such that data are very rarely reported in the comparative literature. Ignoring manual and pedal muscles is also justified given that they comprise a small fraction of total limb musculature, and presumably contribute only a small fraction towards limb support and propulsion during locomotor activities. Although extrinsic shoulder muscles attaching to the scapula can be important for locomotion in therian mammals (Hudson et al., 2011a; Jenkins & Weijs, 1979), these were excluded from consideration to facilitate a fair comparison across all species; only muscles that explicitly attached to the humerus or more distal forelimb skeleton were included. The forelimb dataset comprised architectural measurements for a total of 912 muscles in 31 species (21 mammals and 10 non-avian saurians, the latter hereafter referred to as ‘reptiles’ for simplicity) spanning more than four orders of magnitude in body mass, and the hindlimb dataset comprised measurements for a total of 1,181 muscles in 36 species (19 mammals, 12 reptiles and five birds) spanning almost five orders of magnitude in body mass. In the hindlimb dataset, birds and humans were collectively treated as a single group, ‘bipeds’, such that ‘mammals’ herein refer to all mammals except humans. Given the small sample size (and taxonomic skew) of birds in the current dataset, and the broad aims of the study, it was not considered justifiable to analyze birds and humans as separate entities here. Furthermore, as the present study concerns terrestrial amniotes, the forelimb of bipeds was not investigated.

For each muscle, its physiological cross-sectional area (PCSA) was calculated as

(1) $$\mathrm{P}\mathrm{C}\mathrm{S}\mathrm{A}={\displaystyle \frac{m_{\mathrm{m}\mathrm{u}\mathrm{s}\mathrm{c}\mathrm{l}\mathrm{e}}\cdot \cos ({\alpha }_{\mathrm{O}})}{\rho \cdot {\ell }_{\mathrm{O}}},}$$where m_muscle is belly mass, α_o is pennation angle, ℓ_o is fascicle (or ‘fibre’) length and ρ is muscle tissue density, the latter taken to be constant for vertebrate skeletal muscle at 1,060 kg/m³ (Hutchinson et al., 2015; Mendez & Keys, 1960). It is important to note that this equation assumes that all fascicles of a muscle act in parallel in generating force, allowing their individual cross-sectional areas to be summed (see also Sacks & Roy, 1982). In reality, the constituent fibres of a given fascicle are often shorter than the fascicle itself, wherein their ends overlap or interdigitate (Gaunt & Gans, 1992; Infantolino, Neuberger & Challis, 2012), although they may still activate simultaneously with one another, thus functioning as a single fibre (Bodine et al., 1982). Additionally, as per common practice, it was assumed that measurements of fascicle length and pennation angle corresponded to the muscle’s optimal fibre length and optimal pennation angle, respectively (Zajac, 1989).

Anatomical comparisons

Four main, and two subsidiary, comparisons were undertaken for both fore- and hindlimbs, as described below. For each comparison, patterns were examined for the limb as a whole, as well as just the proximal and distal limb muscles separately, given that some previous studies have noted scaling differences between the proximal and distal limb for individual muscles or functional groups (Alexander et al., 1981; Cuff et al., 2016a, 2016b; Dick & Clemente, 2016; Eng et al., 2008). ‘Proximal’ muscles were classified as those in which the majority of their bulk resides proximal to the elbow (forelimb) or knee (hindlimb) joints, whereas ‘distal’ muscles have the majority of their bulk residing distal to those joints. This volume-based approach is more relevant to the ‘fascicle packing problem’, and avoids the complications caused by flexor or extensor muscles crossing the joints involved. Furthermore, given that a substantial component of hindlimb locomotor muscle in reptiles is the caudofemoralis longus (CFL), which chiefly resides in the proximal tail, whole hindlimb and proximal hindlimb analyses were also re-run with this muscle excluded from the reptile sample. In addition to providing more nuanced insight into questions of scaling and fascicle packing, these additional variations provide greater sophistication to predictive relationships derived from the data. Ultimately, eight different sub-analyses were performed for each anatomical comparison (48 in total).

Statistical analyses

All data processing and analyses were conducted in R v.4.1.0 (R Core Team, 2021), using the ‘caper’ (v.1.0.1; Orme et al., 2018), ‘evomap’ (v 0.0.9; Smaers & Rohlf, 2016) ‘phytools’ (v.0.7; Revell, 2012), ‘geiger’ (v.2.0.7; Harmon et al., 2008) and ‘nlme’ (v. 3.1; Pinheiro et al., 2021) packages; the full set of code and data used are provided in the Supplemental Information. Data were logarithmically transformed (base 10) prior to analysis, facilitating the use of linear statistical models. Phylogenetically informed statistical analyses were conducted using a single, fully resolved, time-calibrated phylogenetic tree of the study taxa (Fig. S1). The tree was generated using the TimeTree database (www.timetree.org; Hedges, Dudley & Kumar, 2006), which included all taxa except for Isoodon fusciventer; this was substituted with Isoodon obesulus (the only species of Isoodon in the database), which has no effect on divergence times with respect to other taxa in the present study.

For each of the above anatomical comparisons (for whole-limb and proximal and distal compartments), allometric scaling equations of the form log₁₀Y = log₁₀A + Blog₁₀X were derived for mammals, reptiles and bipeds separately, using phylogenetic generalized least squares (pGLS or ‘phylogenetic regression’; Smaers & Rohlf, 2016) to determine slopes (exponents, B) and intercepts (log₁₀A). This simultaneously estimated the λ parameter of Pagel (1999) using maximum likelihood, offering greater flexibility than a strict Brownian motion model of evolution in accounting for phylogenetic signal in the data. Additionally, the 95% confidence interval (CI) of the slope was determined, using the t-distribution, to facilitate comparison against the slope expected under isometry: if the expected exponent fell outside of the CI, the scaling was deemed to significantly depart from isometry at the P = 0.05 level. Subsequently, all groups were collated together and a new pGLS fit was computed to generate a ‘pan-amniote’ regression for a given anatomical comparison (again for whole-limb and proximal and distal compartments), thus deriving a statistical predictive framework that can be applied to extinct species. In addition to estimator coefficients, 95% CIs and prediction intervals were also derived (Smaers & Rohlf, 2016), which can provide error margins for future estimations.

To ascertain whether mammals, reptiles or bipeds exhibited different allometric trajectories for a given anatomical comparison, a phylogenetic analysis of covariance (pANCOVA) was performed, which tested for differences in slope and intercept between groups (both separately and together; Smaers & Rohlf, 2016). Here the off-diagonal elements of the variance–covariance matrix were scaled by the λ parameter estimated during calculation of the pan-amniote regression, to minimize false negatives caused by an overly conservative assumption of strict Brownian motion. The analyses were also run without accounting for phylogeny (ANCOVA, λ = 0), to evaluate the effect of phylogenetic relatedness on the results, given that the mammal, reptile and biped samples are nearly mutually exclusive phylogenetically (Fig. S1). Statistical significance was set at P = 0.05.

Total muscle mass v. body mass

Reptiles almost ubiquitously show negative allometry, whereas mammals and bipeds do not show significant departure from isometry (Fig. 2). The one exception to this generalization is negative allometry in the distal hindlimb of mammals: larger mammals tend to have relatively lighter (less muscled) distal hindlimbs. When the CFL is excluded, the values of the slopes (and CIs) for reptiles change minimally, but this is sufficient to render the revised scaling statistically indistinguishable from isometry (Fig. S4). Analyses of covariance (without accounting for phylogeny) indicate that mammals, reptiles and bipeds each exhibit significantly different allometric trajectories, in terms of both slope and intercept (Tables 2, Table S4). Reptiles have a lower slope and intercept compared to mammals, indicating that they have proportionately lower muscle mass, especially at large body sizes. In contrast, bipeds have proportionately greater hindlimb muscle mass than both mammals and reptiles, especially at larger body size and particularly in the distal limb. Many of these differences disappear (i.e., become statistically non-significant) once phylogeny is taken into consideration using the pANCOVA.

Mean size-normalized isometric strength ( F_max*) v. body mass

Both mammals and reptiles generally do not show significant departure from isometry, although reptiles exhibit negative allometry in the distal hindlimb (Fig. 3). When the CFL is excluded, the values of the slopes for reptiles change minimally, but narrower CIs render the revised scaling significantly negatively allometric, for both the whole and proximal hindlimb (Fig. S5). Notably, bipeds exhibit strong positive allometry throughout the hindlimb, particularly in the distal limb (Fig. 3); notwithstanding the small sample size, mean size-normalized force-generating capacity appears to be scale-invariant. Analyses of covariance without accounting for phylogeny indicate that mammals, reptiles and bipeds frequently exhibit significantly different allometric trajectories, especially in terms of slope (Table 3, Table S5). Differences in intercept were mostly detected for the forelimb, where reptiles exhibit a markedly lower intercept than mammals, indicating lower mean force-generating capacity regardless of body size. Most of the differences between mammals and reptiles disappear once phylogeny is taken into consideration using pANCOVA; in contrast, most differences between mammals and bipeds, and reptiles and bipeds, were retained following phylogenetic correction, attesting to strong allometric deviation in the biped sample.

Characteristic fascicle length (L*) v. body mass

In the forelimb, both mammals and reptiles exhibit negative allometry, although in reptiles this is not statistically significant in the distal limb, due to wide CIs in that instance (Fig. 4). Mammals and bipeds also exhibit negative allometry in the hindlimb, although in mammals this is driven by the distal limb only; mirroring patterns noted above, bipeds show a stronger departure from isometric scaling in the distal limb. Reptiles do not show any significant departure from isometry in the hindlimb, a result that remains unaltered when the CFL is excluded from the dataset (Fig. S8). In stark contrast to the previous comparisons, ANCOVA (without accounting for phylogeny) revealed almost no significant difference among mammals, reptiles or bipeds (Table 4, Table S8). The only difference detected (which disappeared following phylogenetic correction using pANCOVA) was that mammals have a lower intercept in the distal hindlimb compared to both reptiles and bipeds, indicating that, overall, the distal hindlimb of mammals possesses proportionately shorter muscle fascicles.

Total PCSA v. total muscle mass

Mammals and bipeds almost ubiquitously exhibit positive allometry, whereas reptiles generally do not exhibit statistically significant departures from isometry (Fig. 5). Two exceptions to this are mammals not showing a significant departure from isometry in the distal forelimb, and reptiles exhibiting positive allometry in the forelimb as a whole. Excluding the CFL does not significantly alter the reptile hindlimb scaling patterns (Fig. S9). As for L*, ANCOVA (without accounting for phylogeny) notably revealed almost no significant difference among mammals, reptiles or bipeds (Tables 5, Table S9). Mammals have a lower slope in the hindlimb overall compared to bipeds, although this cannot be attributed to differences within the proximal or distal compartments. Additionally, mammals have a higher intercept in the distal hindlimb compared to reptiles (a result which disappeared following phylogenetic correction using pANCOVA), indicating that the distal limb of mammals possesses relatively greater fascicle area.

Pan-amniote regression

Coefficients for the pan-amniote regressions (Table 6) provide an empirical basis for estimating some important measures of muscle mass and force-generating capacity in extinct terrestrial amniotes. Note that the coefficients reported here were computed to the exclusion of bipeds, given that bipeds have been shown above to frequently differ from quadrupeds, and that the majority of species throughout tetrapod history were quadrupedal. The code provided in the Supplemental Information enables for predictive relationships to be derived that includes bipeds in the dataset. In addition to the coefficients, mean percent prediction error (%PE) is also reported, expressed in terms of the original, non-log-transformed dimensions of the data (Smith, 1980); this provides an alternative to prediction intervals (see SI code) as a way of deriving upper and lower estimates for a given taxon. Coefficients computed when the CFL was excluded from the reptile dataset (Table S10) are largely similar to those reported in Table 6, although notably mean %PE is generally higher.

Mass and force-generating capacity scaling across terrestrial amniotes

In terms of total muscle mass (Fig. 2), the present study found that reptile (i.e., non-avian saurian) limbs typically show negative allometry with respect to body mass (exponents 0.884 to 0.939), whereas mammals (exponents 0.989 to 1.019) and bipeds (exponents 1.104 to 1.109, but wide CIs) typically displayed isometry. This is in partial agreement with prior studies, where weak negative to modest positive allometry has been recovered (exponents 0.9 to 1.15 across all amniotes; see references in Introduction). The differences in findings may be due to multiple factors, including variation in sample sizes affecting CI calculation, the line-fitting approach used and even the species that contribute to the underlying datasets. It may also reflect a genuine biological phenomenon, where allometric patterns observed for a select few muscles (i.e., those which extend the range of exponents recovered by previous studies) are ‘cancelled out’ by those of many other muscles, when considered together at the level of the whole limb. Irrespective of the proximate cause(s), it is evident that reptiles have proportionately less muscle mass than mammals or bipeds, and bipeds have proportionately greater hindlimb muscle mass, especially in the distal limb. Neither result is hardly surprising, given that the reptiles sampled have short limbs and a long, massive tail that contributes substantially toward total body mass, and that bipeds (by virtue of being bipeds) ought to invest a greater fraction of body mass into longer, more heavily muscled hindlimbs.

Muscle force-generating capacity in the present study was expressed in a more intuitive fashion by normalizing to units of body weight. Thus, to compare exponents derived here to those of prior studies that have examined PCSA directly, this requires adding or subtracting 1.0 to the exponent. Three different measures of normalized force-generating capacity (F_max*) were explored here—mean (Fig. 3), median (Fig. S2) and total (Fig. S3)—which showed a consistent overarching set of patterns with respect to body mass. Mammals and reptiles overall exhibit isometric scaling, although in some circumstances mammals tended towards positive allometry (total range of exponents −0.30 to −0.245) whereas reptiles tended towards negative allometry (total range of exponents −0.465 to −0.302). This is partially consistent with the results of prior studies (corrected exponents −0.31 to −0.09; see references in Introduction). Again, discrepancies in findings may belie the different spatial scales concerned; for example, the high end of previously reported exponents derive from the plantaris muscle of the mammalian hindlimb (Pollock & Shadwick, 1994) and the proximal forelimb of felids (Cuff et al., 2016a). Of note, reptiles often displayed a markedly lower intercept than mammals or bipeds, indicative of reduced relative force-generating capacity and again consistent with possessing short limbs and massive tails. It therefore follows that how ‘body mass’ is gauged (total body mass, body mass excluding tail, etc.) will influence interpretations of interspecific differences in force-generating capacity, although from a purely mechanical perspective total body mass is appropriate in the context of terrestrial locomotion, since ultimately the limbs must support and propel all of it.

Bipeds were found to almost ubiquitously display positive allometry in muscle force-generating capacity, and were frequently found to significantly differ from mammals and reptiles even after phylogeny was accounted for (in 13 of 15 comparisons with mammals, all comparisons with reptiles); again, this can be related to the greater investment of biomass in the hindlimbs of these species. Notably, positive allometry was exceedingly strong in the distal hindlimb (total range of exponents −0.016 to 0.025), to the point that relative force-generating ability may increase with increasing body size. This is comparable to exponents reported for bipedal hopping macropods (corrected exponents −0.22 to 0.33; Bennett & Taylor, 1995; McGowan, Skinner & Biewener, 2008), although intriguingly is higher than the exponents recovered by Maloiy et al. (1979) for the ankle extensors of a selection of ground-dwelling birds (corrected exponent −0.24 to −0.19). These results should be viewed with some caution, given that only six species were examined in the present study. Nonetheless, they raise the interesting question of whether extreme positive allometry would persist at larger body sizes beyond the largest extant biped, the ostrich (Struthio camelus). For example, how would muscle strength fare in multi-tonne non-avian theropod dinosaurs, and what implications may this have for locomotor performance? These are questions that must await future investigation, but the comparative dataset marshalled here can provide an empirical basis for such efforts.

In lieu of investigating fascicle length per se, this study examined an equivalent region-level metric, characteristic fascicle length (L*), which tended to exhibit negative allometry with respect to body mass in both mammals and bipeds, as well as the forelimb of reptiles. The exponents obtained (0.171 to 0.324) fell within the range reported previously for actual fascicle length in individual muscles or functional groups (0.14 to 0.5; see references in Introduction). Negative allometry indicates that muscle fascicles are on the whole proportionately shorter in larger species. This is especially true in the mammalian distal hindlimb, which also exhibits a lower intercept (Fig. 4); shorter muscle fascicles in the distal hindlimb of mammals in turn translate to relatively greater PCSA per unit mass in this compartment (Fig. 5). All else being equal, proportionately shorter muscle fascicles in larger species will translate to reduced joint excursion, and at least for mammals and birds this correlates with the habitual use of more extended limb postures in larger species (Biewener, 1989; Biewener, 2005; Bishop et al., 2018; Gatesy & Biewener, 1991).

Fascicle packing

Analysis of covariance of muscle mass and force-generating capacity identified numerous differences in scaling patterns between mammals, reptiles and bipeds, but when phylogeny was taken into account the majority of these became statistically non-significant (Tables 2, 3, Tables S2, S3). Hence there are genuine physical differences in mass or force-generating capacity between each group (which is biomechanically relevant), and these largely exist due to each group having evolved along different phylogenetic trajectories. However, when total muscle mass was factored into consideration, either directly (comparison #4: ΣPCSA v. Σm_muscle) or indirectly (comparison #3: L* v. m_body, wherein Σm_muscle is used to compute L*), ANCOVA recovered almost no statistically significant difference between any group; this was true regardless of whether phylogeny was accounted for or not, and was observed for the limb as a whole as well as proximal and distal compartments separately (Tables 4, 5). That is, the fore- and hindlimbs of reptiles, mammals and bipeds almost ubiquitously follow a single common scaling pattern¹ . This in turn implies that observed differences in whole-limb force-generating capacity between groups are principally driven by differences in limb muscle mass, not internal architecture: one group does not ‘pack their fascicles better’ for a given volume of limb muscle, they just invest a greater fraction of total biomass into limb muscle in the first instance. The only possible exception to this generality is the distal hindlimb of mammals, which at larger body sizes at least do indeed appear to ‘pack their fascicles better’, with a greater PCSA (and in turn, force-generating capacity) per unit muscle mass, correlating with proportionately shorter fascicles for a given body size. All else being equal, this result suggests that larger species may benefit from greater elastic energy storage in the distal hindlimb tendons during locomotion (cf. Pollock & Shadwick, 1994), although it should nevertheless be treated with caution, since a large proportion of the currently sampled mammals are ‘cursorial’ taxa, which may bias the analysis.

The existence of a single overarching pattern, across a diverse array of terrestrial amniotes that span more than four orders of magnitude in body mass, is remarkable. It implies the tendency towards some adaptive optimum in organismal ‘design’ or, more likely, the presence of one or more constraints that prevent significant and systematic deviation from a common pattern. These constraints may be functional in nature, such as trade-offs that could occur between conflicting requirements of individual muscles in the execution of disparate tasks (see Introduction), or may have a developmental basis (e.g., Evans et al., 2021). Given the multidimensional and nonlinear aspects of muscle architecture and function, and terrestrial locomotor biomechanics in general, it would be naïve to suggest that a bivariate statistical model such as those derived here can sufficiently represent the mechanical phenomena involved (Taylor & Thomas, 2014). Deciphering the proximate underlying cause(s) of this strong consistency must therefore await future study. It is also important to recognize that there is still scope for (non-systematic) variation; for example, the greyhound (Canis familiaris, 27 kg) and snow leopard (Panthera uncia, 36 kg) have near-identical total hindlimb muscle mass (~1.64 kg), yet the greyhound—which is selectively bred for high-speed running—has approximately double the PCSA of the snow leopard (0.028 v. 0.014 m²; Table S1). Important insight into the fascicle packing problem can therefore be gained by better understanding the reasons for variation about the common pan-amniote pattern, through exploring variables that were not investigated in the current analyses, such as segment lengths, muscle moment arms, joint mobility, posture and locomotor ecology.

Considerations for future studies

A number of other points are worth noting in the context of future comparative investigations of muscle architecture. Firstly, despite a superficially broad taxonomic coverage, the dataset of the present study is still biased, with few birds, marsupials or non-varanid squamates, numerous ‘cursorial’ eutherians and not one testudine; future studies should therefore target currently under- or unsampled parts of amniote phylogeny. Increased diversity will be especially useful for elucidating the true extent, and underlying cause(s) of, conservatism in whole-limb fascicle packing. Furthermore, it is recommended that as much architectural data be collected and reported as possible, even if it may not all be immediately used in a given study, since this will maximize its potential utility for later studies. One reason why several prior datasets were excluded from the present study was their omission of pennation angle from the set of reported (or even collected) measurements. A second point worth considering is that, out of necessity, the present study ignored the intrinsic musculature of the manus and pes; if further datasets for these muscles become available in the future, they should be analysed, since this may reveal important differences between plantigrade and digitigrade species.

One final consideration is that the present study found numerous differences between mammals, reptiles and bipeds in the allometry of muscle mass and force-generating capacity with respect to body mass. This is hardly surprising, but it has an important implication for how comparative studies should undertake their analyses. In order to facilitate comparison within or across species, previous studies that have explored muscle architecture and function typically normalize raw architectural measurements by body mass, or the dimensionally appropriate exponent of body mass such as m_body^⅓ for normalized fascicle length (Allen et al., 2010; Bates & Schachner, 2012; Dick & Clemente, 2016; Fahn-Lai, Biewener & Pierce, 2020; Martin et al., 2019; Regnault et al., 2020). This may be acceptable for comparisons of closely related species, but becomes questionable when divergent allometries with body mass are involved. A more sensible approach for future studies would be to normalize against more directly relevant parameters, such as limb or limb segment length, or alternatively by the appropriate clade-specific scaling exponents. Exactly which is the most appropriate normalizing metric to use may vary depending on a given study’s core questions, and deserves further scrutiny.