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Eloy Gálvez-López, Department of Animal Biology, University of Barcelona, Av. Diagonal 645, Barcelona 08028, Spain. T: +34 93 402 14 56; F: +34 93 403 57 40; E: firstname.lastname@example.org
Six mechanically significant skeletal variables were measured on the calcanei from 60 Felidae specimens (22 species) to determine whether these variables were scaled to body mass, and to assess whether differential scaling exists. The power equation (y = a · xb) was used to analyse the scaling of the six variables to body mass; we compared traditional regression methods (standardised major axis) to phylogenetically independent contrasts. In agreement with previous studies that compared these methodologies, we found no significant differences between methods in the allometric coefficients (b) obtained. Overall, the scaling pattern of the felid calcaneus conformed to the predictions of the geometric similarity hypothesis, but not entirely to those of the elastic similarity hypothesis. We found that the moment arm of the ankle extensors scaled to body mass with an exponent not significantly different from 0.40. This indicated that the tuber calcanei scaled to body mass faster than calcaneus total length. This explained why the effective mechanical advantage of the ankle extensors increased with body mass, despite the fact that limb posture does not change in felid species. Furthermore, this finding was consistent with the hypothesis of the isometric scaling of ground reaction forces. No evidence for differential scaling was found in any of the variables studied. We propose that this reflected the similar locomotor pattern of all felid species. Thus, our results suggested that the differences in allometric coefficients for ‘large’ and ‘small’ mammals were in fact caused by different types of locomotion among the species included in each category.
The calcaneus is the largest tarsal bone in mammals; it consists of an anterior portion, where the astragalus articulates, and a posterior portion, the tuber calcanei, where the Achilles tendon inserts (Lessertisseur & Saban, 1967). The calcaneus forms a lever for the calf muscles because the Achilles tendon is shared by the gastrocnemius and soleus, the ankle extensors. The length of this lever arm determines the moment of the force produced by the limbs as they push against the ground, which causes the body to rise and advance during forward locomotion (Alexander, 1983). Furthermore, the length of the tuber calcanei is related to the muscle mechanical advantage at the ankle, which counteracts the moment exerted on the joint by the ground reaction force (Biewener, 1989, 2003).
The shape of the calcaneus is variable in mammals. It has been proposed that, given its important role in the mechanics of locomotion, this variability would probably be related to locomotor specialisation (Lessertisseur & Saban, 1967). However, size is another factor that must be taken into account (i.e. scaling). The main biomechanical consequences of scaling have been described in broadly comparative studies, and several hypotheses have been proposed to understand how increasing size affects animal design (Schmidt-Nielsen, 1984; Alexander, 2002; Biewener, 2005). These hypotheses, often referred to as similarity hypotheses, have been used to predict how anatomical structures and locomotion patterns would be affected by increasing body size. The hypothesis of geometric similarity, already supported by Hill (1950), states that two organisms are geometrically similar if their linear dimensions can be made equal by multiplying those of one of them by a constant (c). Thus, their surfaces could be made equal by multiplying by c2, whereas volumes should be multiplied by c3. Assuming a constant density (ρ), which would be logical if both organisms are made of the same materials, body mass would also be proportional to c3 (Mb = ρV = ρL3). Then, geometrically similar animals made of the same materials should present linear dimensions proportional to body mass1/3. The hypothesis of elastic similarity, proposed by McMahon (1975), is based on the assumption that different-sized organisms have evolved to resist buckling and bending loads similarly (Schmidt-Nielsen, 1984). In order to maintain this similar elastic recovery, and assuming again a constant density, diameters must scale to body mass3/8, and lengths to body mass1/4. Nevertheless, none of those hypotheses appears to provide a universal explanation for the effects of size. For instance, mammalian linear dimensions typically conform to geometric similarity (Alexander et al. 1979) but, in Bovidae, limb bone lengths appear to follow elastic similarity (McMahon, 1975). Another important point in scaling studies is whether general allometric calculations are applicable to a large range of variations in body size. Some studies on the scaling of skeletal elements appear to indicate otherwise. Economos (1983) predicted that, because volume increases faster than surface area, the pattern for scaling cross-sectional bone areas in large mammals (over 20 kg of body mass) should be different from that used in small mammals. This hypothesis of differential scaling was somewhat confirmed on mammalian long bones, mainly in carnivores (Bertram & Biewener, 1990; Christiansen, 1999a,b), and on mammalian body length (Silva, 1998). Thus, the first aim of this study was to determine the scaling pattern of the calcaneus bone, and to assess whether differential scaling could be found in this pattern.
As pointed out by Bou et al. (1987), similarity hypotheses imply adaptive neutrality, or at least independence of the locomotor type of the species that are compared. Therefore, samples with extreme locomotor patterns should show large deviations from predicted relationships. In fact, as stated by Day & Jayne (2007), phylogenetic diversity among different-sized samples might obscure the effect of size alone. To avoid this problem, we chose Felidae as our study group, because they comprise a well-defined, phylogenetically narrow clade (Mattern & McLennan, 2000; Johnson et al. 2006) with substantial differences in body size (Wilson & Mittermeier, 2009). The sizes of different Felidae species span two orders of magnitude and bracket the suggested 20-kg body mass change point for allometric relationships (Economos, 1983). Furthermore, they have similar locomotor patterns (Day & Jayne, 2007; Wilson & Mittermeier, 2009).
We also wondered whether the scaling pattern of the felid calcaneus would be influenced by ankle mechanics. On one hand, broadly comparative studies have shown that larger animals tend to run with more erect limb postures; this reduces the magnitude of the joint moments produced by the ground reaction force (Fg) and, thus, reduces the stresses acting on the bones (for a review, see Biewener, 2005). Consequently, the effective mechanical advantage (EMA), defined as the ratio of the extensor muscle moment arm to the Fg moment arm (Fig. 1a), scales to body mass with positive allometry (e.g. EMAankle = Mb0.169 for a large sample of mammals; Biewener, 1989). On the other hand, Day & Jayne (2007) showed that large felids do not have more upright limbs than small felids. Thus, the angle of the ankle at footfall or midstance was not significantly correlated to body mass in felids. Although they could not definitively exclude the possibility that EMA increased with size in felids, the authors suggested that it would be very unlikely, because the Fg orientation changed very little, even among phylogenetically diverse taxa (Biewener, 2005). Therefore, to support that theory, the muscle moment arms would have to increase with strong positive allometry in felids. Nevertheless, Alexander et al. (1981) have shown that muscle moment arms in mammals scaled to body mass with an exponent of 0.40 (Mb0.40); this value was substantially higher than the exponents proposed for length scaling by similarity hypotheses (geometric similarity: Mb0.33; elastic similarity: Mb0.25). The muscle moment arm scaling factor (b = 0.40) was later supported by the work of Castiella & Casinos (1990) in insectivores and rodents. Thus, our second aim was to determine whether the moment arm of ankle extensors in felids scaled to body mass with the expected value of 0.40; this would provide evidence that EMA increased with body size in felids even though limb posture remained more or less constant. We chose the calcaneus bone, because it was assumed to have high mechanical significance.
Materials and methods
We studied 60 calcanei from 22 species of Felidae (Table 1) by measuring the total length (L), the moment arm of the ankle extensors (r), and the sagittal and transverse diameters (ds and dt, respectively) just distal to the calcaneus–astragalus articulation (Fig. 1b,c). The moment arm of a muscle is defined as the perpendicular distance from the centre of rotation of the joint to the line of action of the muscle (Fig. 1a); thus, it depends upon the configuration of the limb segments. As proposed by Biewener (1989), the distance from the midpoint of the calcaneus–astragalus articulation to the posterior end of the tuber calcanei was taken as an approximation of the moment arm of the calf muscles (r; Fig. 1c). In the case of the generalised carnivore standing limb posture, this approximation will not diverge substantially from the actual moment arm. Furthermore, it was previously demonstrated that limb posture in Felidae was not affected by size (Day & Jayne, 2007). This study included specimens that belonged to collections housed in the Muséum National d’Histoire Naturelle of Paris (the former laboratories of Anatomie Comparée and Mammalogie), the Museu de Ciències Naturals de la Ciutadella of Barcelona, the Museo Nacional de Ciencias Naturales of Madrid, the Museo Argentino de Ciencias Naturales ‘Bernadino Rivadavia’ of Buenos Aires, and the Museo de La Plata.
Mb, average body mass for the indicated species; n, number of specimens measured.
Acinonyx jubatus (Aju)
Caracal caracal (Cca)
Felis silvestris (Fsi)
Leopardus colocolo (Lco)
Leopardus geoffroyi (Lge)
Leopardus pardalis (Lpa)
Leopardus tigrinus (Lti)
Leopardus wiedii (Lwi)
Leptailurus serval (Lse)
Lynx canadensis (Lca)
Lynx lynx (Lly)
Lynx pardinus (Lpd)
Lynx rufus (Lru)
Panthera leo (Ple)
Panthera onca (Pon)
Panthera pardus (Ppa)
Panthera tigris (Pti)
Panthera uncia (Pun)
Prionailurus viverrinus (Pvv)
Profelis aurata (Pau)
Puma concolor (Pco)
Puma yagouaroundi (Pya)
The corresponding transverse second moment of area (I) was calculated from the diameters measured with the following formula (Alexander, 1983):
This formula assumed that the sagittal plane was the major axis of flexion during quadruped locomotion (Cubo & Casinos, 1998a).
The ratio r/L was also calculated for each specimen. This non-dimensional index reflected the relative length of the calf moment arm with respect to the total length of the calcaneus. Non-dimensional indexes are typically independent of body size, which allows comparisons among specimens independent of scale. Nevertheless, we expected r to be proportional to Mb0.40, which is a higher value than that expected for the scaling of L (Mb0.33 or Mb0.25, according to geometric and elastic similarity, respectively). Therefore, this index should scale with positive allometry to body mass (i.e. Mb0.40–0.33 = Mb0.07 or Mb0.40–0.25 = Mb0.15, for geometric and elastic similarity, respectively).
Because the number of specimens per species was diverse (Table 1), and we used a standard body mass for each species (based on values obtained from the literature), we used average values for variables other than body mass for every species.
We used regression methods to relate the following variables to body mass (Mb): L, r, ds, dt, r/L and I. All regressions were calculated with the standardised major axis method (SMA), because we were primarily interested in the regression slopes. In contrast, common least squares regression methods tend to underestimate the slope of the line-of-best-fit, because it is calculated to fit the predicted y-values as closely as possible to the observed y-values (Warton et al. 2006). We assumed the power equation:
and 95% confidence intervals were calculated for both a and b.
Many studies (e.g. Felsenstein, 1985; Grafen, 1989; Harvey & Pagel, 1991; Christiansen, 1999a,b, 2002a,b) have discussed that, in interspecific analyses, the error terms are correlated, because species are not independent of each other, but rather can be arranged in a hierarchical sequence (phylogenetic tree). Thus, a phylogenetic signal is introduced into the analysis, and the individual points cannot be considered truly independent. Alternatively, the method of phylogenetically independent contrasts (PIC; Felsenstein, 1985) takes into account this phylogenetic signal in regressions on interspecific data; therefore, we also calculated SMA regression slopes for PIC with the PDAP: PDTREE module of Mesquite (Maddison & Maddison, 2010; Midford et al. 2010). These PIC slopes were then compared with those obtained by traditional regression analysis with an F-test (α < 0.01) to assess whether this phylogenetic signal had any effect on our results. The structure of the phylogenetic tree for the included species was that described by Johnson et al. (2006), and is shown in Fig. 2.
Finally, we tested for the presence of differential scaling in the felid calcaneus with the model proposed by Jolicoeur (1989). This model would detect the presence of complex allometry in our sample (i.e. variables that are not proportional to each other, as in simple allometry):
where A is a constant (corresponding to a in Eq. 2), C is the allometry exponent, xmax is the maximum observed value of the independent variable (i.e. body mass, Mb) and D is the exponent of complex allometry, a time-scale factor. In our case, D > 1 indicated faster relative growth in small individuals, and D < 1 indicated that relative growth increased with size. The complex allometry hypothesis was thus accepted when D was significantly different from 1 (P < 0.05). Equation 3 was fitted with SPSS for Windows (release 15.0.1 2006; SPSS, Chicago, IL, USA), and 95% confidence intervals were calculated for all parameters.
The coefficients for the allometric equations obtained with both traditional regression analysis and PIC are shown in Table 2. No branch length transformations were necessary for PIC regressions, except in the case of the ratio r/L. For all other variables, the absolute values of the standardised contrasts were not significantly correlated to the corresponding standard deviations (Fig. 3). Consequently, we used the Rho transformation proposed by Grafen (1989) in the case of r/L.
Table 2. Regression coefficients obtained from traditional regression analysis and from PIC.
All variables were plotted against body mass. The allometric coefficients (b) obtained with traditional regression analysis were not significantly different from those obtained with PIC (P > 0.01 for all comparisons). Values shown in italics indicate a non-significant regression.
95% CIa, 95% confidence interval for the coefficient (a); 95% CIb, 95% confidence interval for the allometric coefficient (b); ds, sagittal diameter; dt, transverse diameter; I, second moment of area; L, calcaneus total length; r, moment arm of the ankle extensors; R, correlation coefficient.
Overall, the correlation coefficients (R) from the PIC analysis were lower than those from traditional regression (Table 2). This was consistent with previous studies that indicated a higher risk of type I errors (i.e. indicating a significant correlation between two variables when there was none) when the correlation analysis neglected the effect of phylogeny (Grafen, 1989; Christiansen, 2002a). This could explain the different findings for the ratio r/L (traditional regression: R = 0.558; P = 0.011; PIC: R = 0.358; P = 0.132; Table 2). In all cases, zero was not included in the 95% confidence interval for the slope (b) (Table 2). However, in both methodologies, the value predicted by geometric similarity (Mb0.07) was included in the 95% confidence interval, but not the value predicted by elastic similarity (Mb0.15). Thus, although phylogeny appeared to account for most of the correlation between variables, the ratio r/L showed, as expected, a positive allometry to body mass. This indicated that the moment arm of the ankle extensors (r) provided stronger scaling than calcaneus length (L) (see below).
For variables other than the ratio r/L, the allometric coefficients (b) obtained with traditional regression analysis were not significantly different from those obtained with PIC (Table 2; Fig. 4). This was consistent with previous studies that compared these methodologies (Christiansen, 1999a,b, 2002b). This indicated that the scaling of our variables with body mass was not dependent on the phylogenetic relationships within our sample. We also found that the 95% confidence interval for the allometric coefficient (b) of the regressions for calcaneus total length (L) included 0.33, the expected value for geometrically similar animals, but not 0.25, the value proposed for elastic similarity (Schmidt-Nielsen, 1984; Table 2). On the other hand, for both diameters (dt, ds), the 95% confidence intervals included the values predicted by both geometric (0.33) and elastic (0.375) similarities. As expected, neither method produced a slope for the moment arm of the ankle extensors (r) that was significantly different from the predicted value for muscle moment arms (0.40; Alexander et al. 1981; Castiella & Casinos, 1990). Finally, the scaling exponent of the second moment of area (I) was not significantly different from either 1.33 or 1.50, the values expected in geometrically and elastically similar animals, respectively (Cubo & Casinos, 1998b). Overall, the scaling pattern of the felid calcaneus conformed to the predictions of geometric similarity, but not entirely to those of elastic similarity.
Results of the tests for complex allometry are shown in Table 3. In all cases, the 95% confidence interval for the exponent of complex allometry (D) included 1. Thus, no evidence for differential scaling was found in any of our variables.
Table 3. Results of the complex allometry test.
95% CIln A
None of the exponents of complex allometry (D) deviated from 1; thus, none of our variables deviated from simple allometry. 95% CIC, 95% confidence interval for the coefficient (C); 95% CID, 95% confidence interval for the exponent of complex allometry (D); 95% CIln A, 95% confidence interval for ln A; ds, sagittal diameter; dt, transverse diameter; I, second moment of area; L, calcaneus total length; r, moment arm of the ankle extensors; R, correlation coefficient.
−0.326 to −0.165
−0.046 to 0.084
−0.885 to 3.719
Our results showed that the scaling pattern of the felid calcaneus fit the predictions of the geometric similarity hypothesis better than the elastic similarity hypothesis. This finding was consistent with previous studies on the scaling of long bone dimensions in carnivores (Bertram & Biewener, 1990; Christiansen, 1999a). The work of Silva (1998) on the scaling of body length on a big sample of mammals also reported similar results, especially when considering terrestrial non-volant mammals, terrestrial carnivores and felids. Furthermore, our findings supported previous studies that described elastic similarity as an atypical scaling pattern found mostly in large bovids (Alexander et al. 1977, 1979; Biewener, 1983; Christiansen, 1999b; Rocha-Barbosa & Casinos, 2011). Also, consistent with previous studies (Christiansen, 1999a,b, 2002b), the phylogenetic signal had no significant effect on the scaling pattern, because we obtained similar values for the allometric coefficients (b), regardless of whether phylogeny was taken into account.
Biomechanical consequences of moment arm scaling
Our results supported the notion that muscle moment arms (r) scale to body mass as Mb0.40 (Alexander et al. 1981; Castiella & Casinos, 1990). This indicated that a larger body mass corresponded to a longer tuber calcanei (relative to the total length of the calcaneus, which scaled to body mass as L ∝ Mb0.33). In turn, this allows the EMA of the ankle extensors to increase with body mass (Biewener, 1989) without requiring a change in the limb posture of felid species (Day & Jayne, 2007), given that the segment lengths (i.e. distances between joints) and joint angles remain unaffected by the length of the muscle moment arms.
As mentioned above, the EMA is defined as the ratio of the extensor muscle moment arm (r) to the moment arm of the ground reaction force (R). This assumes that the force exerted by the extensor muscles (Fm) confers a mechanical advantage that counteracts the mechanical moment exerted by the ground reaction force (Fg; Biewener, 2003). Biewener (1989) found that, for mammalian ankle extensors, the EMA scaled to body mass with an exponent of 0.169 (± 0.046):
When r in Eq. (4) is substituted with the assumed proportionality for muscle moment arms (r ∝ Mb0.40), we can derive a hypothesis about the scaling of the moment arm of the ground reaction force (R):
Under equilibrium conditions, the moments acting on a joint must be balanced; that is, the moments of the muscle forces acting on the joint must equal the moment of the ground reaction force:
where S is the cross-sectional area of the muscle and σ is the maximum isometric stress (250–300 kPa). Because σ is a constant, the following equation holds:
where V is the volume of the muscle, lf the mean fibre length, m the muscle mass and ρ the muscle density (1060 kg m−3). In the case of pinnated muscles, a correction factor equal to the cosine of the pinnation angle should be added to lf, but this angle can be assumed to be constant for each muscle; thus, we can disregard it with the other constants in this proportionality. Once constants have been removed, Eq. 8 can be written as:
Castiella & Casinos (1990) found that, for a large sample of mammals, muscle mass scaled to body mass as m ∝ Mb1.06, and the mean fibre length scaled to body mass as lf ∝ Mb0.20. By substituting these values in Eq. 9, the scaling of muscle force to body mass can be hypothesised:
This supports the hypothesis proposed by Alexander et al. (1977) that the scaling of the ground reaction force is isometric, as the derived exponent of 1.029 is not significantly different from 1.
To date, most studies on differential scaling have focused on comparing a sample of ‘large’ mammals with a sample of ‘small’ mammals (e.g. Economos, 1983; Christiansen, 1999b, 2002a). One problem with that approach is that it depends on a ‘threshold’ body mass value that is rather arbitrarily chosen for separating ‘large’ from ‘small’ mammals. Furthermore, this threshold varies depending on the group under consideration (i.e. 20 kg might be appropriate for mammals as a whole, but not for scaling among bovids). Two alternate solutions to this problem have been proposed: first, a quadratic regression can be used to test for non-linear trends in log-transformed data (Bertram & Biewener, 1990); or second, a Gompertz-derived model can be fit to bivariate data in order to quantify the deviation from simple allometry (Jolicoeur, 1989). Both methodologies can determine whether relative growth increases or decreases with size. We chose the model proposed by Jolicoeur (1989), because it was equivalent to the power equation (used to describe simple allometry) when D was not different from 1, and it was equivalent to quadratic regression (used by Bertram & Biewener, 1990) when D was not different from 2.
As mentioned above, we found no evidence of differential scaling in the felid calcaneus, despite the wide range of body masses that spanned two orders of magnitude. This result suggested that, at least in the particular case of Felidae, similarity in allometric scaling was a consequence of the similar locomotor requirements of all felid species (Day & Jayne, 2007; Wilson & Mittermeier, 2009). Another possible explanation would be a phylogenetic constraint; however, this seems unlikely, because the recent origin of this family has not prevented wide variations in felid size, for example, from the tiny Felis nigripes Burchell 1824 (about 1.5 kg) to well over 250 kg in the tiger (Panthera tigris Linnaeus, 1758) and in other species evidenced in fossils. Assuming that the similarity in allometric scaling was a consequence of the similarity in locomotor requirements, we would expect that other skeletal variables with mechanical significance would also show similar allometric scaling among felids. To investigate this, we revisited the data of previous scaling studies that included felid species and tested for complex allometry in skeletal variables. In particular, we reanalysed the data of Bertram & Biewener (1990) and that of Christiansen (1999b). As expected, we found no evidence for complex allometry in the scaling of sagittal diameter, transverse diameter or bone circumference to bone length, or in the scaling of those four variables to body mass. This was consistent for all the long bones measured (humerus, radius, femur, tibia; Tables S1 and S2). Nevertheless, like in the original studies (Bertram & Biewener, 1990; Christiansen, 1999b) and others dealing with differential scaling (Economos, 1983; Silva, 1998), we found evidence for complex allometry when we included a large sample of carnivores in the analysis, and when we included the whole sample studied by Christiansen (1999b), which included species from several orders of mammals (Tables S1 and S2). In those cases, the samples included species with different types of locomotion (Van Valkenburgh, 1985, 1987; Wilson & Mittermeier, 2009). In light of these results, we propose that the differences found in allometric coefficients (b) between ‘large’ and ‘small’ mammals of different species (i.e. differential scaling) must be more related to differences in locomotor requirements, rather than differences in body mass. This hypothesis requires further scaling studies to investigate whether there are grade shifts (different slopes) that correspond to different types of locomotion among different species.
We would like to thank the following organisations for partially funding this research: la Caixa; Deutscher Akademischer Austausch Dienst (DAAD); the University of Barcelona (UB); Agència de Gestió d’Ajuts Universitaris i de Recerca (AGAUR); Departament d’Innovació, Universitats i Empresa de la Generalitat de Catalunya; and the European Social Fund (ESF). Finally, this work was completed with the assistance of funds from research grants CGL2005-04402/BOS and CGL2008-00832/BOS from the Ministerio de Educación y Ciencia (MEC) of Spain.