The articular facets of interosseous joints must transmit forces while maintaining relatively low stresses. To prevent overloading, joints that transmit higher forces should therefore have larger facet areas. The relative contributions of body mass and muscle-induced forces to joint stress are unclear, but generate opposing hypotheses. If mass-induced forces dominate, facet area should scale with positive allometry to body mass. Alternatively, muscle-induced forces should cause facets to scale isometrically with body mass. Within primates, both scaling patterns have been reported for articular surfaces of the femoral and humeral heads, but more distal elements are less well studied. Additionally, examination of complex articular surfaces has largely been limited to linear measurements, so that ‘true area’ remains poorly assessed. To re-assess these scaling relationships, we examine the relationship between body size and articular surface areas of the talus. Area measurements were taken from microCT scan-generated surfaces of all talar facets from a comprehensive sample of extant euarchontan taxa (primates, treeshrews, and colugos). Log-transformed data were regressed on literature-derived log-body mass using reduced major axis and phylogenetic least squares regressions. We examine the scaling patterns of muscle mass and physiological cross-sectional area (PCSA) to body mass, as these relationships may complicate each model. Finally, we examine the scaling pattern of hindlimb muscle PCSA to talar articular surface area, a direct test of the effect of mass-induced forces on joint surfaces. Among most groups, there is an overall trend toward positive allometry for articular surfaces. The ectal (= posterior calcaneal) facet scales with positive allometry among all groups except ‘sundatherians’, strepsirrhines, galagids, and lorisids. The medial tibial facet scales isometrically among all groups except lemuroids. Scaling coefficients are not correlated with sample size, clade inclusivity or behavioral diversity of the sample. Muscle mass scales with slight positive allometry to body mass, and PCSA scales at isometry to body mass. PCSA generally scales with negative allometry to articular surface area, which indicates joint surfaces increase faster than muscles' ability to generate force. We suggest a synthetic model to explain the complex patterns observed for talar articular surface area scaling: whether ‘muscles or mass’ drive articular facet scaling is probably dependent on the body size range of the sample and the biological role of the facet. The relationship between ‘muscle vs. mass’ dominance is likely bone- and facet-specific, meaning that some facets should respond primarily to stresses induced by larger body mass, whereas others primarily reflect muscle forces.
As the point of contact between bones of the skeleton, articular facets of interosseous joints (primarily those of the synovial type, but also certain fibrous joints such as the intercentral joints, pubic symphysis, and iliosacral joints) transmit forces throughout the skeletal system. Thus, these articulations are of particular interest in studies of locomotion (Jungers, 1984; Ruff, 1988; Godfrey et al. 1995; Hamrick, 1996; Parr et al. 2011) and mastication (Smith et al. 1983; Bouvier, 1986; Taylor, 2006; Terhune, 2013), activities that may demand the transmission of large forces between bones. To prevent overloading and to maintain relatively low stresses, facet areas should scale in proportion to the forces they transmit (i.e. large facets should experience higher forces than small ones). More specifically, Alexander (1980:96) suggested that if animals are geometrically similar, then ‘maximum joint stresses may have the same order of magnitude in mammals of all sizes’.
Assuming this proposition is correct, two primary models have been proposed to explain the variance in force magnitude that drives synovial joint construction. Alexander (1980) postulated a model in which muscle-induced forces dominated joint construction. The ultimate prediction of this model is that facet area will scale isometrically with muscle physiological cross-sectional area (PCSA), the variable that most directly correlates with potential muscle force (Bodine et al. 1982; Perry & Wall, 2008; Myatt et al. 2011). Unfortunately, data on PCSAs are extremely rare. However, assuming (i) an isometric relationship between PCSA and muscle mass and, in turn (ii), an isometric relationship between muscle mass and body mass, a more easily testable prediction avails: the areas of articular surfaces and the ability to resist joint forces should scale isometrically with body mass (i.e. in an area–volume relationship, the scaling coefficient should be equal to 0.66). Despite the required assumptions, these two variables (body mass and facet area) have been most frequently utilized to address this hypothesis by previous authors (Jungers, 1988; Ruff, 1988; Swartz, 1989; Godfrey et al. 1991; Jungers, 1991; Ruff & Runestad, 1992; Williams et al. 1992; Godfrey et al. 1995; Hamrick, 1996; Parr et al. 2011).
The major alternative to Alexander's hypothesis of muscle-induced forces dominating joint formation is the hypothesis by Biewener (1982; Biewener & Taylor, 1986) that mass-induced forces dictate joint construction. Given two objects of the same shape and density but different absolute sizes, the larger one will have a higher mass/surface area ratio due to the greater dimensionality of mass. Increasing the size of the object will result in increasing stress, as mass scales to area with an exponent of 1.5. Because bone has the same physical properties regardless of body size (Biewener, 1982; Currey, 1984), larger animals operate closer to the safety limit of their skeletal elements. To compensate for this reduced safety margin, facet areas in large animals may be greater than predicted by isometric scaling. Thus, if mass-induced forces dominate joint construction, articular surface areas should scale with positive allometry (i.e. scaling coefficient greater than 0.66). Due to the different predictions for the relationship between facet area and body mass, Alexander's (1980) hypothesis of muscle-dominated forces is an isometric or geometrically similar model, whereas Biewener's (1982) hypothesis of mass-induced forces is an allometric model.
Several previous studies have examined the relationship between facet areas and body mass with conflicting results (Jungers, 1988; Ruff, 1988; Swartz, 1989; Godfrey et al. 1991; Jungers, 1991; Ruff & Runestad, 1992; Williams et al. 1992; Godfrey et al. 1995; Hamrick, 1996; Parr et al. 2011). Swartz (1989) suggested positive allometric scaling for the facet areas in 11 of 12 limb joints among anthropoids, highlighting the role of mass-induced forces and requisite compensation for a reduced safety factors among larger animals. These results were contested by Godfrey et al. (1991), who noted that Swartz's sample contained two distinct groups at opposite ends of the body mass spectrum: small-bodied monkeys and large-bodied apes. Although a regression line generated for both groups indicates positive allometry, isometry is maintained if both groups are analyzed separately (as shown by Ruff & Runestad, 1992). The larger-bodied suspensory hominoids have the same scaling coefficient, but the higher intercept for the group (interpreted as indicating a greater mobility at the shoulder) inflates the slope for the total sample.
Jungers (1991) examined linear dimensions of articular surfaces in hominoid limb joints, finding that five of 16 joints displayed positive allometry, and although isometry could not be ruled out for the other joints, there was a positive allometric trend among them as well. Jungers suggested these results may require ‘qualifications to Alexander's model’ (Jungers 1991:397), but did not invoke Biewener's model as a possible explanation.
Recently, Parr et al. (2011) examined scaling relationships for talar facet areas among hominoids. They expected to confirm the generality of isometric scaling, and allowed for the possibility that convex (male-type) facets would scale allometrically depending on locomotor roles. However, their results show only the ectal (or posterior calcaneal facet), a concave (female-type) facet, scaled with positive allometry. Among the other facets, although confidence intervals do not rule out isometry, there is a trend toward positive allometry in the relationship between facet surface area and body mass (Parr et al. 2011).
It is important to reiterate that the relationship between body mass and facet area is only a proxy for what Alexander's model actually posits. Isometric scaling between body mass and articular surface area is not necessarily expected if isometry does not hold for the scaling relationships of body mass, muscle mass, and muscle PCSA. For instance, if muscle mass increases with positive allometry relative to body mass, then we might expect the capacity for muscle force also to scale allometrically. Under this scenario, positive allometry between facet area and body mass would not refute Alexander's hypothesis because facet area might still be scaling isometrically with respect to muscle force. Alexander et al. (1981) addressed this particular assumption by showing an isometric relationship between body mass and muscle mass of distal leg muscles in a wide sample of mammals, including primates. More recently, Muchlinski et al. (2012) reported that primate muscle mass exhibits slight (but significant) positive allometry with body mass, bringing into question the validity of testing joint construction hypotheses by looking at scaling between total body mass and facet area. However, neither study utilized phylogenetic regression methods in their analyses, introducing the possibility of data distortion similar to Swartz (1989).
Nonetheless, the relevance of allometry between muscle to body mass depends on the scaling relationship between body mass and PCSA: the most pertinent variable for determining the capacity of a muscle to generate force. PCSA is a function of the mass of the muscle belly, the muscle's density and the length of muscle fascicles, and can be calculated with the equation:
The maximum force a muscle can produce reflects the maximum number of sarcomeres contained within the muscle cross-sectional area (Bodine et al. 1982; Perry & Wall, 2008; Myatt et al. 2011); both a larger absolute area and an increased pennation angle can accommodate a greater number of sarcomeres. Therefore, a larger PCSA indicates a muscle's ability to generate larger absolute force.
If muscle-induced forces dominate joint construction, then PCSA should scale isometrically with articular facet area (area–area relationship, scaling coefficient = 1). Biewener's model for mass-induced forces influencing joint construction makes no predictions regarding the relationship between PCSA and facet area; neither positive nor negative allometry will refute the model. Comparing a broad range of mammalian species, both Alexander et al. (1979) and Pollock & Shadwick (1994) found positive allometric scaling in muscle cross-sectional area relative to body mass. However, analyzing a hominoid-only sample, Myatt et al. (2011) reported that scaling patterns of PCSA in most hindlimb muscles conformed to isometry. Once again, phylogenetically and functionally diverse groups have generated positively allometric slopes, whereas more restricted samples conform to isometry (Godfrey et al. 1991).
Here, we examine the scaling relationships between facets of the talus and body mass for euarchontans, the clade composed of primates, treeshrews, and colugos, as well as several smaller taxonomic and functional groups. The talus is well suited for this purpose, as it contains both convex (male-type) and concave (female-type) facets. Additionally, as no muscles attach directly to the bone, there is little chance that joint expansion has occurred to accommodate increased ligament attachment, as argued by Currey (1984). Recognizing that scaling between facet area and body mass is an indirect assessment of the competing hypotheses being evaluated, we also examine scaling relationships between muscle mass and body mass using phylogenetic generalized least squares regression, and determine the relationship between PCSA of the extrinsic muscles of the foot and talar facet areas.
We designed our approach to address key issues in assessing the alternative hypotheses of Alexander and Biewener on joint construction. Key elements of our approach were (i) to consider articular surfaces of distal limb elements; (ii) to measure actual facet areas rather than interpolating from linear measurements; (iii) to utilize phylogenetic generalized least squares regression to account for clade-level differences; and (iv) directly to test the relationship between muscle PCSA and facet areas. Previous studies have focused on proximal limb elements (Swartz, 1989; Godfrey et al. 1991; Jungers, 1991; Ruff & Runestad, 1992; Godfrey et al. 1995), particularly the expansion of the femoral head and its relation to increased bipedality (Jungers, 1988; Ruff, 1988). There are fewer analyses of more distal limb bones (Hamrick, 1996; Parr et al. 2011). As more distal elements have higher load-bearing requirements [i.e. they must support the axial mass and the mass of the limb of which they are a component (Demes & Gunther, 1989)], it is important to evaluate articular surfaces at interosseous joints throughout the body.
Additionally, as noted by Godfrey et al. (1991), most studies of articular surfaces have been limited to linear measurements to approximate true areas (Jungers, 1988, 1991; Ruff, 1988). This source of imprecision and random error can be somewhat alleviated by molding articular facets and flattening the three-dimensional mold to two dimensions (Swartz, 1989; Godfrey et al. 1995) but this method is difficult, time-consuming, and not ground-truthed. However, digital surface files, generated from surface laser scans (Parr et al. 2011) or microCT scans (this study), are highly amenable to surface area manipulations and measurements. These advances allow us to quantify accurately more complex shapes than previously afforded.
To address concerns highlighted by Godfrey et al. (1991, 1995), we employ phylogenetic generalized least squares regression (PGLS) for taxonomically diverse samples and reduced major axis regression (RMA) when phylogeny no longer provides meaningful information (i.e. λ is not significantly different from zero).
We recognize two hypotheses for joint construction: a strict isometric model of muscle-induced force (Alexander, 1980; Godfrey et al. 1991) and a more relaxed allometric model of mass-induced force (Biewener, 1983). Expectations for these models are presented graphically in Fig. 1. Under the strict model of geometric similarity, if muscle-induced forces dominate joint construction, then articular facet areas should scale isometrically with body mass (scaling coefficient = 0.66). Under the relaxed model, if mass-induced forces influence joint construction, then articular facet areas should scale with positive allometry relative to body mass (scaling coefficient > 0.66).
In this study, we accept Alexander's (1980) suggestion that facets must maintain similar stresses in animals of different size. Given this, if Alexander's (1980) hypothesis that muscle-induced forces dominate joint construction is correct, the following relationships can be expected to hold true when phylogenetic affinity and/or functional specializations are accounted for:
body mass and muscle mass scale with isometry (scaling coefficient = 1),
body mass and PCSA scale with isometry (scaling coefficient = 0.66),
body mass and facet areas scale with isometry (scaling coefficient = 0.66),
facet areas and PCSA scale with isometry (scaling coefficient = 1).
As listed, these relationships are progressively better (more direct) tests for evaluating the two models of joint construction we have presented here, culminating in Condition 4, which directly tests Alexander's postulate that muscle-induced forces dominate joint construction. Therefore, if taxonomically equivalent samples of all variables were available with equal sample sizes, it would be pointless to look at any relationship except that between PCSA and facet area. However, the rarity of PCSA data among extant taxa and its non-existence for fossils makes proxy methods critical for gaining knowledge. Furthermore, data on scaling of body mass and facet area can be applied to other important evolutionary questions as detailed in the discussion.
Materials and methods
Ten surface measurements were taken on the tali of 217 adult individuals representing 75 euarchontan species (Fig. 2). Specimens came from the American Museum of Natural History (AMNH), the Duke Primate Center (DPC), the Harvard Museum of Comparative Anatomy (MCZ), the US National Museum of Natural History (NMNH), and the osteological collection of Stony Brook University (SBU). Measurements are listed in Table 1 and include surface areas of the lateral tibial facet (LTFA), medial tibial facet (MTFA), fibular facet (FFA), ectal or posterior calcaneal facet (EFA), sustentacular facet (SFA), and navicular facet (NFA). Because the sustentacular and navicular facets are confluent within euarchontans, recognition of a division between these two facets would be relatively arbitrary and difficult to reproduce. Instead, values were summed in a composite variable termed ‘head articular surface’ (HEAD). All facet areas were summed to create the composite variable ‘articular surface area’ (ART). Values of the volume (VOLUME) and the total surface area of the entire talus (TOTAL) were also calculated.
Table 1. List of talar measurements.
Lateral tibial facet area (LTFA)
Medial tibial facet area (MTFA)
Fibular facet area (FFA)
Ectal/posterior calcaneal facet area (EFA)
Sustentacular/anterior calcaneal facet area (SFA)
Navicular facet area (NFA)
Head articular surface (HEAD)
Total articular surface area (ART)
LTFA, MTFA, FFA, EFA, SFA, NFA
Total surface area (TOTAL)
All measurements were taken on microCT images of specimens scanned at the Microscopy and Imaging Lab at the American Museum of Natural History or at the imaging facility of Stony Brook University Center for Biotechnology. Scans were reconstructed using avizo 6.0 (Visualization Sciences Group, 2009), and a four-step process was employed to isolate articular surfaces. This process and the measured articular surfaces are depicted in Fig. 3. First, the element was aligned with the desired facet oriented obliquely to the viewing surface. Next, the selection tool of avizo 6.0 was used to highlight the desired area. The selection was then inverted and the edges of area inspected to ensure appropriate boundaries. Finally, the inverted selection was deleted, leaving only the articular surface. The cropped facet was saved as a separate file to allow secondary verification and for future processing. The surface area surface was calculated using the measurement tool in avizo. Raw measures for all examined taxa are provided in the Supporting Information (Appendix S1).
An error test was conducted to insure the reliability of the measured surface areas. For two specimens not otherwise included in this study, Saimiri boliviensis (AMNH 211613) and Lemur catta (AMNH 150039), all six facets were cropped and measured six times on separate days in avizo. Percentage error (PE) for each measurement was calculated with the following formula:
Percentage error = |Measurement−Mean|/Mean*100
Measurements with a PE of less than 5% were deemed reliable (White & Folkens, 2005). PE for these individuals was also compared with the PE within a species and within a sex for individuals included in the study. We expect lower PE in multiple measures of the same individual than either intra-species or intra-sex samples. PEs are presented in Supporting Information Appendix S3.
Primate body masses were collected from Smith & Jungers (1997). Whenever possible, mean values derived from the largest sample of non-captive populations were chosen. Male values were used for all specimens that lacked a sex, a potential source of bias. Non-primate body masses were collected from the Animal Ageing and Longevity Database (Tacutu et al. 2013). This database does not record sex-specific weights, so mean values were estimated from given ranges. This is another potential source of bias, but is likely insubstantial given the low proportion of non-primates in the sample. Although using body mass estimates from published sources is not as desirable as knowing the weight of the individual specimen used in the study, museum collections frequently lack that information, and this likely does not systematically bias analyses (Godfrey et al. 1991). Furthermore, as individual adult weights can fluctuate greatly seasonally and over a lifetime, there is no guarantee that the weight recorded at death is more representative than a species mean.
Each measurement was log-transformed and plotted against log-transformed body masses. Both RMA and OLS regressions were calculated in past (Hammer et al. 2001), while the r package ‘caper’ was used for PGLS (Orme et al. 2012). Each specimen was treated as an individual data point for both RMA and OLS regressions, and species means were used for PGLS regressions. Primate phylogenetic trees were downloaded from 10kTrees (Arnold et al. 2010) and edited in mesquite (Maddison & Maddison, 2011) to include scandentians and dermopterans. Branch lengths for the non-primate euarchontans were taken from Janecka et al. (2007) and Roberts et al. (2011). The resulting phylogeny is shown in Fig. 2.
Based on phylogenetic or functional considerations, smaller samples were analyzed separately. These restricted samples include ‘sundatherians’ (sensu Bloch et al. 2007) as well as most of the major radiations of living primates: strepsirrhines, galagids, lorisids, lemuroids, haplorhines, platyrrhines, catarrhines, hominoids, and cercopithecoids. One functional group, ‘vertical clingers and leapers’ (sensu Napier & Walker, 1967), was also analyzed separately.
There is considerable debate concerning the appropriate method for assessing relationships between biological data that include some amount of natural variation (McArdle, 1988; Martin & Barbour, 1989; Jolicoeur, 1990; Sokal & Rohlf, 1995; Carroll & Ruppert, 1996; see Warton et al. 2006 for review). Ordinary least squares methods assume no error in the x-axis variable, and report a shallower slope than reduced major axis as the correlation coefficient diverges from 1.0 (a perfect correlation). Feasibly, the use of different methods could generate opposing results from the same dataset. Here we favor PGLS whenever phylogeny contributes significant information (i.e. maximum likelihood λ does not equal zero). Among the phylogenetically or functionally restricted samples, λ often does not differ significantly from zero. In these instances, RMA regressions are more appropriate (Martin & Barbour, 1989; Warton et al. 2006; Smith, 2009). Using these guidelines (PGLS with significant λ, otherwise RMA), summary statistics are reported in Tables 2 and 3 for each variable and each sample.
Table 2. Summary statistics for articular surface–body mass regressions for the ectal, fibular, medial tibial, and lateral tibial facets. When λ differs significantly from zero, PGLS scaling coefficients are reported. In all other cases, RMA scaling coefficients are given.
Table 3. Summary statistics for articular surface–body mass regressions for total volume and head, articular, and total surface areas. When λ differs significantly from zero, PGLS scaling coefficients are reported. In all other cases, RMA scaling coefficients are given.
Although OLS regressions are not appropriate for these analyses, we report regression parameters generated using this method as Supporting Information (Appendix S2). Given the high correlation coefficients for all facet area–body mass regressions, there is little difference between RMA and OLS scaling coefficients.
To examine differences in slopes and intercepts within primates, within haplorhines, within catarrhines, within strepsirrhines and within lorisiforms, analyses of covariance were conducted using log-transformed data with body mass as a covariate. ancovas were conducted in spss 19.0 (IBM Corporation, 2010). A test for homogeneity of slopes was first performed by testing the significance of the interaction between the variable and clade of interest. Heterogeneous slopes render ancovas inappropriate (Engqvist, 2005) but they were recovered in a relatively few number of comparisons (12 of 48 tests). Summary statistics for ancovas are reported in Table 4.
Table 4. ancova results. Bold text indicates violations of ancova assumptions.
Recently, Muchlinski et al. (2012) published regressions for muscle mass to body mass relationships among mammals. We have used this data and replicated RMA and OLS regressions, as well as a PGLS regression (Appendix S3). The last of these methods is important to consider, as the largest primates in the Muchlinski et al. (2012) sample are all apes, whereas the majority of the smallest are strepsirrhines. Without consideration of phylogeny, coefficients may not reflect actual scaling relationships as accurately (Godfrey et al. 1991).
Physiological cross-sectional area scaling
For non-masticatory muscles, data for muscle physiological cross-sectional areas (PCSA) among primates are limited. As this study focuses on articulations at the talocrural and subtalar joints, we have compiled PCSA data for extrinsic muscles of the foot from several sources (Thorpe et al. 1999; Carlson, 2006; Payne et al. 2006; Channon et al. 2009; Oishi et al. 2009, 2012; Myatt et al. 2011). Together, these studies report PCSA for extrinsic muscles of the foot for five gorillas (Gorilla sp.), two bonobos (Pan paniscus), four chimpanzees (Pan troglodytes), three Bornean orangutans (Pongo pygmaeus), eight gibbons (four Hylobates lar, two Hylobates pileatus, two Hylobates moloch), and four siamangs (Symphalangus syndactylus). Data were available for the following eight hindlimb muscles: m. tibialis posterior, m. tibialis anterior, m. flexor fibularis, m. flexor tibialis, m. extensor hallucis longus, m. extensor digitorum longus, m. peroneus longus, and m. peroneus brevis. Log-transformed PCSAs were regressed against log-transformed body masses (from Smith & Jungers, 1997); results are reported in Appendix S3.
Talar articular surface area (ART) and PCSA for each muscle were averaged by species and log-transformed. The scaling relationships were evaluated using RMA regressions. For this area–area comparison, an isometric relationship would generate a slope equal to 1.0, with negative and positive allometry exhibiting slopes less and greater than one, respectively. Slopes, intercepts, confidence intervals, and correlation coefficients are reported for each muscle (Table 5).
Table 5. Summary statistics for muscle PCSA to talar articular surface (ART) regressions.
The percentage errors (PE) for surface area measurements were all relatively low, with a mean error across facets of 1.86%. This is well below the oft-stated desirable rate of 5% (White & Folkens, 2005). The highest error was obtained in the fibular (2.81%) and medial tibial facets (2.75%), whereas the lowest occurred in the composite articular surface area (0.84%). The higher PEs observed in the MTFA and FFA are likely the result of poorly defined posterior margins, but all rates are so low that they are unlikely to generate systematic bias in the results. Comparing the facet-wise error of Saimiri and Lemur, it is notable that although the error rates are similar, PE magnitudes are reversed in the medial tibial and fibular facets. This likely has to do with the well recognized morphological differences between strepsirrhines and haplorhines (Gebo, 1986, 1988, 1993; Dagosto, 1993; Gebo et al. 2000; Seiffert & Simons, 2001; Boyer et al. 2010; Boyer & Seiffert, 2013). Although the confluent sustentacular and navicular facets do not have the highest observed error rates (1.97 and 1.36%, respectively), the composite head articular surface area has a lower error rate than either facet individually (1.06%), justifying our approach combining them in our analyses. As predicted, both individuals had lower PEs at each facet than those observed in intra-species and intra-sex comparisons (Appendix S3).
Slopes of both regression methods indicate that scaling patterns are complex and vary by facet and taxonomic group. Under PGLS, phylogeny ceases to explain trait variation at high taxonomic levels, i.e. more exclusive groups (maximum likelihood λ = 0). Phylogenetic auto correlation is significant only among euarchontans, primates, and strepsirrhines, but not among haplorhines or higher taxonomic levels. Regressions for primates as a whole are presented in Fig. 4, with strepsirrhine and anthropoid clades presented in Figs 5 and 6, respectively. Correlation coefficients are uniformly high (> 0.93 across all groups except lorisids). Summary statistics for these regressions are presented in Tables 2 and 3ancova results comparing taxonomic subsets are presented in Table 4.
Under Alexander's model (1980), facet area should scale isometrically with body mass. Conversely, in Biewener's model (1983), facet area should scale with positive allometry. Our data reveal a strong trend toward positive allometry among all facets, although the confidence intervals for smaller groups do not rule out isometry. Using the guidelines for choosing a regression method outlined above (PGLS with significant λ, otherwise RMA), mean slopes for 78 of 104 variables (75%) scale with positive allometry, whereas 17 of 104 (16%) scale with negative allometry; the remainder (9 of 104, 9%) scale at isometry. Considering 95% confidence intervals, the majority of facets exhibit isometry (58 of 104, 56%), whereas a sizable percentage scale with positive allometry (44 of 104, 42%), and only two scale with negative allometry (2%; the lateral tibial facet within galagids and the ectal facet among vertical clingers and leapers). For groups that include more than 50 specimens (euarchontans, primates, strepsirrhines, haplorhines, platyrrhines), 22 of 40 variables scale with positive allometry (55%), and the remainder scale at isometry (18 of 40, 45%). Given the trend of most regressions is towards positive allometry, and that increasing the sample size (and statistical power) tends to increase the significance of these allometric trends, it is possible that allometric relationships are under-reported due to lack of statistical power. It should be noted, however, that the regressions with greater sample sizes also include a larger range of body sizes, and a greater range of phylogenetic and behavioral diversity.
Considering particular facets reveals consistent patterns as well. The ectal facet (Table 2) scales with positive allometry among all groups except ‘sundatherians’, strepsirrhines, lorisids, and galagids. The fibular facet (Table 2) scales with positive allometry among most groups except ‘sundatherians’, strepsirrhines, lorisids, galagids, hominoids, and vertical clingers and leapers. With the exception of lemuroids, the medial tibial facet (Table 2) and head articular surface area (Table 3) conform to isometry. For lemuroids, these results remain the same when the largest species (Propithecus sp. and Indri) are left out of the regression. Thus, there is no evidence the specialized leaping adaptations of the large-bodied indriids disproportionately affect lemuroid scaling coefficients. Total surface area of the talus, talar volume, and total articular surface area of the talus scale with positive allometry in most groups (Table 3). Within vertical clingers and leapers, all facets scale with isometry except the ectal facet, which is negatively allometric.
Even without the largest indriids, lemuroids span a greater range of body sizes than either galagids or lorisids, which may contribute to the consistently positively allometric scaling coefficients. To test this possibility, RMA regressions were conducted on a subset of lemuroids spanning a similar range of body masses as galagids and lorisids (~900 g). For this subset, all variables remained positively allometric except the fibular facet [slope = 0.71; 95%CI (0.78, 0.64)] and the head articular surface [slope = 0.70; 95%CI (90.74, 0.66)].
Among ‘sundatherians’, only the lateral tibial facet is positively allometric [slope = 0.78; 95%CI (0.88, 0.68)]; all other variables scale isometrically. This is not the case for primates, for which talar volume, total surface area, fibular, and ectal facets scale with positive allometry. ancovas reveal significantly different slopes for the fibular and ectal facet regressions, and intercepts are significantly different for all other variables. Intercepts are higher among primates for the talar volume, total surface area, lateral tibial facet, and articular surface area. However, they are higher among ‘sundatherians’ for the medial tibial facet and head articular surfaces.
Across most variables analyzed here, there is little difference between the scaling coefficients of haplorhines and strepsirrhines. The most notable differences between the two clades occur in the fibular and medial tibial facets (Fig. 4). For haplorhines, the fibular facet is positively allometric [slope = 0.76; 95%CI (0.78, 0.74)], whereas in strepsirrhines the fibular facet is isometric [slope = 0.67; 95%CI (0.76, 0.59)]. Testing for homogeneity of slopes reveals significant differences between the slopes of haplorhines and strepsirrhines (Table 4). For the medial tibial facet (Fig. 4G), the pattern of scaling relationships is reversed: haplorhines scale isometrically [slope = 0.68; 95%CI (0.71, 0.65)], whereas the mean slope for strepsirrhines is allometric (slope = 0.73), although the 95% confidence interval does not exclude isometry (0.84, 0.61). Using ancova, the MTFA regression coefficients of haplorhines and strepsirrhines were not found to be significantly different, but the intercepts were [F (df) = 13.237 (1,202), r2 = 0.934, P < 0.001]. The lateral tibial facet, ectal facet, and head articular surfaces also had significantly different intercepts with homogeneous slopes (Table 4). Regression parameters for the total volume, total surface area, and articular surface area did not differ significantly between the two groups.
Within strepsirrhines, lemuroids are highly positively allometric at the medial tibial facet [slope = 0.81; 95%CI (0.85, 0.77)], whereas the slow-climbing lorisids and leaping galagids are both isometric (Fig. 5G). The slopes and intercepts within various strepsirrhine groups are also informative. For all examined variables, lemuroids exhibit significant positive allometry. This pattern holds true even when the largest species are left out of the regression (Propithecus sp. and Indri). Phylogenetic autocorrelation is not significant in this case (λ does not differ significantly from zero). ancovas reveal that lemuriforms and lorisiforms (galagids and lorisids) have heterogeneous regression coefficients for all variables examined (Table 4).
Galagids and lorisids are similar across nearly all variables, consistently exhibiting isometric scaling patterns. The only difference between the two clades is seen in the lateral tibial facet, which scales with negative allometry within galagids, but is isometric in lorisids (Fig. 5D). The most striking difference between the two groups can be seen in the difference of their intercepts. Galagos have significantly higher intercepts than lorisids for all variables, which can clearly be seen in Fig. 5 and demonstrated through ancova. For each examined variable, galagids and lorisids share homogeneous scaling coefficients but significantly different intercepts (Table 4), with galagids, having relatively larger facet areas at any given body mass. These intercept differences are likely the result of higher peak forces sustained at talar joints by the leaping galagids from muscle-induced forces.
Overall, catarrhines and platyrrhines exhibit similar scaling patterns. Volume, total surface area, articular surface area, head articular surface area, and the ectal and fibular facets scale with positive allometry in both groups. The medial tibial facet scales isometrically in both (Fig. 6G). For catarrhines, the lateral tibial facet scales isometrically, whereas this facet scales with positive allometry in platyrrhines (Fig. 6D). The increased slope for the platyrrhine lateral tibial facet (0.78) is similar to that of lemuroids (0.76), and is significantly different from the slope seen in catarrhines (Table 4). ancovas reveal that all other slopes are homogeneous in catarrhine–platyrrhine comparisons. Intercepts are significantly different between the two groups for all variables except the fibular facet.
For hominoids, the scaling patterns reported here exactly match those in a similar study by Parr et al. (2011). In their study, all facets scaled isometrically except the ectal facet, which scaled with positive allometry. Our results replicate their findings. Analyses of covariance reveal there are no significant differences in the slopes or intercepts of hominoids and cercopithecoids (Table 4), although the facets of cercopithecoids generally scale with positive allometry (Tables 2 and 3).
Muscle mass scaling
Alexander's model of geometric similarity (and the importance of muscle-induced forces for joint construction) predicts an isometric relationship between muscle mass and body mass. Muchlinski et al. (2012) demonstrated this to be true when considering a wide range of mammals. But when evaluated alone, primates were found to exhibit slight but significant positive allometry (Appendix S3). The PGLS regression calculated here finds that within primates, muscle mass scales with isometry relative to body mass, although λ is not significantly different from zero. This makes it difficult to distinguish the appropriate regression method to assessing the scaling relationship. Here we favor the interpretation presented by Muchlinski et al. (2012): slight positive allometry in the muscle mass–body mass relationship within primates, with the caveat that an expanded sample may generate significant PGLS results.
A recent examination of scaling relationships of muscle physiological cross-sectional area and body mass (Myatt et al. 2011) found a consistent isometric signal among the hindlimb muscles in hominoids (isometry was maintained for 11 of their 14 variables). Among the distal muscles of the hindlimb, PCSAs of the knee flexors, plantarflexors, and everters of the foot all scaled isometrically, whereas the dorsiflexors scaled with positive allometry and the digital flexors with negative allometry. Overall, the mean scaling exponent for distal hindlimb muscles was 0.62 [CI (0.48, 0.76)]: a slight negative allometry with inclusion of isometry in the confidence interval.
We have expanded on the dataset provided by Myatt et al. (2011) and include hindlimb muscle PCSA from several additional sources (Carlson, 2006; Channon et al. 2009; Oishi et al. 2009, 2012). Regression results are presented in Appendix S3. Among hominoids, all muscles except m. tibialis anterior and m. extensor hallucis longus scale isometrically with body mass. M. tibialis anterior, a dorsiflexor of the foot, and m. extensor hallucis longus, an adductor of the hallux, scale with positive allometry. Regressions are significant for all muscles except m. flexor fibularis (P = 0.072), indicating only a weakly linear relationship between the PCSA of this digit flexor and body mass within hominoids. The results of Myatt et al. (2011) and our expanded regressions support geometric similarity between muscle PCSA and body mass among hominoids.
Summary statistics for scaling relationships between the talar articular surface area (ART) and physiological cross-sectional areas of several extrinsic muscles of the foot are presented in Table 5. Due to limited availability of PCSA data, firm conclusions about the nature of the scaling relationships can not be drawn. For all hindlimb muscles, the 95% confidence intervals are quite large and include isometry. However, there remains a consistent trend toward negative allometry among the articular surface area to PCSA regressions. For seven of the eight muscles examined here, the mean slope is less than 1; only the slope for the m. tibialis anterior is greater than 1.0. For two muscles, the m. tibialis posterior and the m. peroneus longus, the reported slope is close to isometric (0.98), but for all other muscles, the slope is appreciably different from that predicted by isometry. Given the small sample sizes, only five regressions have P-values less than 0.05; among these muscles, the m. tibialis anterior is positively allometric (slope = 1.05, r = 0.995), while the m. flexor tibialis (slope = 0.90, r = 0.996), m. extensor hallucis longus (slope = 0.92, r = 0.954), m. extensor digitorum longus (slope = 0.93, r = 0.959) and m. peroneus brevis (slope = 0.77, r = 0.969) are strongly negatively allometric.
Our results confirm previous studies which suggest scaling patterns are taxon- and context-specific (Godfrey et al. 1991; Parr et al. 2011) but with an overall trend toward positive allometry (Jungers, 1991). For the ectal and fibular facet, there is a strong positive allometric signal across nearly all groups. These patterns are consistent with Biewener's model of mass-induced forces. The medial tibial facet and head articular surface exhibit the shallowest slopes. Although mean slopes for these variables trend toward positive allometry, confidence intervals do not rule out isometry. They therefore conform to Alexander's model of muscle-induced forces.
Although the changing scaling patterns of our regressions show the importance of evaluating taxonomic groups independently, the infrequency of a strong phylogenetic signal is surprising. Alexander et al. (1981) give three reasons why geometric similarity may not hold: (i) different adaptations; (ii) different evolutionary histories; and (iii) geometric similarity may not be optimal at various body sizes. Further, Godfrey et al. (1991) note that the clade offsets in Swartz (1989) may be the result of either behavioral or phylogenetic differences. Our application of PGLS indicates that phylogenetic covariance has no significant effect above the suborder level, except among strepsirrhines. ancova results help to explain this pattern, showing persistent differences among the regression parameters at higher taxonomic levels within strepsirrhines (e.g. significant differences between galagids and lorisids), but not within haplorhines (regressions for cercopithecoids and hominoids are statistically identical). This result may be due to a high correlation between locomotor behavior and phylogeny within Strepsirrhini (e.g. galagids vs. lorisids). In cases where the phylogenetic and behavioral distance matrices approximate each other, and there is autocorrelation of a continuous variable with the phylogenetic distance matrix, it is ambiguous whether phylogenetic inertia or functional/adaptive response drive the correlations. Taken on the whole, this pattern may indicate that functional differences are more likely to confound true scaling relationships than phylogenetic divergence. The absence of a significant phylogenetic signal among prosimian vertical clingers and leapers, a phylogenetically diverse sample, reinforces this interpretation. These results also indicate the importance of behaviorally and phylogenetically specific regressions for predictive applications.
Testing assumptions of geometric similarity
The regressions conducted here permit assessment of Alexander (1980) and Biewener's (1982) proposed models of joint construction. As outlined in the introduction, the scaling relationships become progressively more direct tests of geometric similarity and its predictions regarding expansion of talar articular areas.
Condition 1: Body mass and muscle mass scale with isometry
As reported by Muchlinski et al. (2012), primate muscle mass scales with slight positive allometry to body mass. However, the scaling coefficients in their study were calculated using OLS and RMA regressions, which allows the possibility that phylogenetic interdependence and autocorrelation affected their findings (Godfrey et al. 1991). The PGLS regression calculated here finds that within primates, muscle mass scales with isometry relative to body mass, a result that agrees with Alexander's proposal of geometric similarity. However, λ is not significantly different from zero in this relationship, making it difficult to distinguish the appropriate methodology for assessing the scaling pattern. Here we favor the interpretation presented by Muchlinski et al. (2012): slight positive allometry in the muscle mass–body mass relationship within primates. This result violates a model of geometric similarity between muscle mass and body mass but is only a coarse assessment of the relationship in question.
Condition 2: Body mass and PCSA scale with isometry
Data from Myatt et al. (2011) provide a finer test of Alexander's model, examining the scaling relationship between muscle PCSA and body mass. In hominoids, the distal muscles of the hindlimb, including extrinsic muscles of the foot, exhibit a consistent isometric signal. PCSAs of the knee flexors, plantar flexors, and everters of the foot all scale isometrically, whereas the dorsiflexors scale with positive allometry, and the digital flexors with negative allometry. Thus, geometric similarity is supported in this scaling relationship, indicating that muscle-induced forces dominate joint construction. However, two more predictions must be confirmed before this conclusion is secure.
Significance of facet area to body mass scaling patterns
Condition 3: Body mass and facet areas scale with isometry
The majority of this study focuses on the scaling relationship between body mass and talar articular surface areas. Across all groups and all variables, the majority of mean slopes scale with positive allometry. When considering 95% confidence intervals, an appreciable number of articular facets still scale with positive allometry relative to body mass (44 of 104, 42%). Although certain taxonomic groups consistently conform to isometry (particularly those with a small range of body masses, such as lorisids and galagids), there are also groups that consistently scale with positive allometry (lemuroids, platyrrhines, and cercopithecoids).
Individual facets also have regular scaling patterns across taxonomic groups. In particular, the ectal and fibular facets most frequently scale with positive allometry, whereas the medial tibial facet and head articular surface scale isometrically. As the ectal facet is located on the plantar aspect of the talus, it is likely responsible for transmitting large amounts of mass-induced force; its scaling pattern is therefore quite intuitive. The complexity of scaling patterns in this relationship does not allow for simple adherence to either the muscle or mass-induced force models, but does clearly indicate that a strict application of geometric similarity is insufficient.
A direct test of Alexander's model
Condition 4: PCSA and facet areas scale with isometry
For four of the five muscles with the highest correlation coefficients (m. flexor tibialis, m. extensor hallucis longus, m. extensor digitorum longus, and m. peroneus brevis), the relationship between muscle PCSA and facet area is strongly negatively allometric. However, due to very small sample sizes (n = 4), confidence intervals do not rule out the possibility of isometric scaling. These findings indicate that the ability of a muscle to generate force increases more slowly than the joint's ability to transmit force (as indicated by the area of the articular surface). Clearly, expanded samples of muscle PCSA are required to evaluate properly the scaling patterns between articular surfaces and muscle PCSA. However, using the available data, a strict application of geometric similarity is inappropriate for determining which forces dominate joint construction. If talar articular surface area increases more rapidly than hindlimb muscle PCSA, then mass-induced forces must play a role in joint construction.
Differences between proximal and distal joint surfaces
Most previous studies examining the relationship between body mass and facet area, with the exception of Parr et al. (2011), have focused on articular surfaces of proximal joints (Swartz, 1989; Godfrey et al. 1991; Jungers, 1991; Ruff & Runestad, 1992; Godfrey et al. 1995). These studies generally recovered isometric scaling between body mass and facet area, indicating muscle-induced forces as the primary forces for joint construction. Our results indicate a mixture of mass- and muscle-induced forces. The differences between previous studies and our results may be the due to the differences between proximal and distal joints. Distal joints must transmit a higher proportion of mass-induced force than proximal ones. Demes & Gunther (1989) note ‘70% of the total body weight are located above the hip joint, and already 90% above the knee’ (p. 130). The proportion would be even higher for the talocrural and subtalar joints. Thus, as more distal joints must transmit a greater proportion of mass-induced forces, it follows that Biewener's model becomes increasingly more applicable. Additionally, because the number of joints separating the body from the point of contact with the ground increases distally to proximally, the number of possible solutions for modifying the stress environment at a given joint also increases. There are fewer alternative strategies (expansion of articular surfaces, changes in limb posture, etc.) for reducing stresses at more distal joints than more proximal ones. Geometric similarity and isometric scaling may well hold true for more proximal joints, as reported by multiple previous studies, although a study taking an approach similar to that used here should be conducted to evaluate this possibility.
Joint type and load-bearing properties
Of the two articulating surfaces in a synovial joint, the smaller of the facets should be more informative regarding load-bearing at that joint (Williams et al. 1992; Godfrey et al. 1995; Hamrick, 1996). Further, as noted by MacConaill & Basmajian (1969), convex or male facets of the ovoid type are always larger than concave (female) facets. Most facets of the talus (lateral tibial, sustentacular, and navicular facets) are modified male ovoid surfaces (sensu MacConaill, 1973). The fibular facet is a modified sellar articular surface (concave dorsoplantarly but often convex anterio-posteriorly). The medial tibial facet is a modified female ovoid among strepsirrhines (particularly lemuroids), but is flatter and more sellar among most anthropoids (especially posteriorly). Only the ectal facet is a modified female ovoid across all taxa observed in this study, and comparisons with the calcaneal ectal facet for 124 specimens (data from Gladman & Boyer, 2012), support the contention of MacConaill & Basmajian (1969). The average ratio of talar ectal facet to calcaneal ectal facet was 0.95 (SD = 0.14). Seventy-eight of 124 specimens had ratios lower than 1, indicating the concave ectal facet of the talus is consistently smaller than the convex ectal facet of calcaneus.
Given these joint type classifications, the consistency of positively allometric scaling for the ectal facet is particularly notable. As the smaller of the two articulating surfaces, the talar ectal facet should give the clearest signal for fitting muscle- or mass-induced models. In lemuroids, the concave medial tibial facet is also strongly positively allometric. This may be indicative of more mass-induced force being transferred through the medial malleoleus and the medial tibial facet during inverted foot postures. However, this hypothesis should be regarded as merely tentative; scaling coefficients can not indicate the amount of force being transferred through a particular joint, merely that the articular surface increases in size more rapidly than body mass.
Articular surface area could also expand to increase the range of excursion and increase mobility at the joint (Ruff, 1988; Hamrick, 1996). As a general rule, however, Biewener (1983, 1989) observed that larger animals have smaller ranges of excursion than smaller animals, which would counteract possible expansion for increased mobility. This negatively allometric relationship has been noted in comparisons among galagids, tarsiers, and indriids (Demes et al. 1998); larger-bodied indriids have a smaller range of movement at the ankle joint than the smaller-bodied galagids and tarsiers.
As detailed above, convex modified ovoid facets are the most reliable indicators of mobility (Hamrick, 1996). Our results show that both the lateral tibial facet and head articular surface scale isometrically, except among lemuroids, cercopithecoids, and platyrrhines, for which the facets scale with positive allometry. In these cases, the positive allometric relationship may be interpreted as increasing the mobility of the ankle joint at larger body sizes, rather than increasing the safety factor of the joint. However, given that larger-bodied animals generally have smaller excursion angles (Biewener, 1983, 1989), it seems more appropriate to interpret these relationships as support for the mass-induced model of facet scaling.
Results are unique among galagids, where the lateral tibial facet scales with negative allometry. This indicates a decrease in the range of excursion at the talocrural joint, as seen in Biewener's studies (1983, 1989). Within Galagidae, the mean scaling coefficients for six of the eight variables are negatively allometric (although confidence intervals included isometry for all except the lateral tibial facet). It is possible that allometrically decreasing mobility at the talocrural and subtalar joints has a strong enough effect among galagos as to obscure the scaling patterns of load-bearing. Boyer et al. (2013) estimate the ancestral galagid body mass at 250 g, although this is probably a bit of an overestimation. This means at least a fourfold increase in mass between the ancestral state and the largest galagos. According to Boyer et al. (2013), despite allometry in limb segment length proportions among galagos, 1000 g otolemurs remain much more geometrically similar to small galagos than would be predicted to result from selection for leaping in a 1000-g generalist ancestor. Therefore, the persistence of geometric similarity for a highly specialized leaping-adapted bauplan over a wide body range may make stronger demands to maintain dynamic similarity in other ways (such as limiting joint flexion). A comparison of joint angles during leaping in galagos of a range of body-sizes could test this idea.
Differences in intercepts
Both regression analyses (Figs 4-6) and ancovas (Table 4) reveal differences in the intercepts for multiple groups. This is particularly pronounced when comparing galagids and lorisids, and catarrhines and platyrrhines. Among the lorisiformes, galagids have higher intercepts for all examined variables: at a given body mass, they have larger values than expected from the lorisid regression. In these cases, the significant differences in the intercepts of the groups' regressions likely reflect higher peak forces experienced at that joint in galagids during leaping (Demes & Gunther, 1989). Directional anisotropy (DA) of trabecular bone in the talus may also reflect higher peak forces in galagos. By itself, data showing differences in DA only imply differences in force environment variability. However, pairing this information with data on bone volume fraction can allow inferences of higher density of trabeculae in a particular direction. In turn, this unidirectionality can be expected to correlate with greater magnitudes of stress (Barak et al. 2011). Therefore, when comparing two taxa, if one has both equal or greater DA and bone volume fraction to the other, higher peak forces can reasonably be inferred. Among galagids, trabecular bone is strongly unidirectionally oriented in the femoral head and neck (MacLatchy & Müller, 2002; Ryan & Ketcham, 2002) and talar body (Hébert et al. 2012), especially when compared with lorisids. Galagids, however, are not exceptional in bone volume fraction when compared with lorisids or other primates (Hébert et al. 2012). Thus, it is likely the intercept differences observed in galagids and lorisids reflect differences in peak loadings.
Among haplorhines, platyrrhines have higher intercepts than catarrhines for six of the eight variables (volume, total surface area, medial tibial facet, ectal facet, head articular surface, articular surface area). In general, platyrrhines are a more arboreal clade than catarrhines; it is possible that the demands of a highly arboreal lifestyle (i.e. climbing, leaping, and landing) result in higher peak forces. Platyrrhines display levels of DA in talar trabecular bone that are similar to tarsiiformes (Hébert et al. 2012), indicating a high degree of directional stress experienced in the talus. None of the mean DA values reported for platyrrhines in Hébert et al.'s study is as low as values reported for four non-human catarrhine species by DeSilva & Devlin (2012). However, bone volume fractions are higher among catarrhines. We are cautious about comparing results from two different studies, but these results suggest peak loadings may be more consistently oriented within platyrrhines than in catarrhines. They do not, however, permit the assertion that peak loadings are relatively greater among platyrrhines.
Different habitual postures
One of the most well recognized morphological differences between haplorhines and strepsirrhines is the shape and orientation of the fibular facet (Gebo, 1986, 1988, 1993; Dagosto, 1993; Gebo et al. 2000; Seiffert & Simons, 2001; Boyer et al. 2010; Boyer & Seiffert, 2013) and the relative size of the medial tibial facet (Gebo, 1986, 1988, 1993; Dagosto, 1993; Gebo et al. 2000; Seiffert & Simons, 2001; Boyer et al. 2010). In strepsirrhines, the fibular facet slopes gradually laterally, compared with the vertical orientation seen among haplorhines. Evaluating the regression parameters of the fibular facet with ancova shows that the scaling coefficients of haplorhines and strepsirrhines are significantly different, with haplorhines having a greater and positively allometric slope [0.76; 95%CI (0.78, 0.74)]. Among strepsirrhines, the relationship of the fibular facet to body mass is isometric [0.67; 95%CI (0.76, 0.59)]. In contrast, the medial tibial facet of haplorhines scales isometrically [0.68; 95%CI (0.71, 0.65)], whereas MTFA is close to positive allometry among strepsirrhines [0.73; 95%CI (0.84, 0.61)]. Because the slope differences are not significant, ancovas were used and indicate a significant difference in the intercepts of these two clades, with strepsirrhines exhibiting a lower intercept for the medial tibial facet compared with haplorhines. In contrast, the intercept for the fibular facet is higher for strepsirrhines, but the significance of this difference could not be tested due to heterogeneous slopes.
These opposing patterns, particularly the contrast of intercepts, could indicate that mass-induced forces are more important on the lateral margin of the talus among haplorhines, but mass-induced forces more heavily influence joint construction on the talar medial margin among strepsirrhines. However, it is difficult to explain such differences. It could be argued that strepsirrhines' habitual use of small diameter supports and inverted foot postures orients the fibular facet of the talus so that it is nearly parallel to, and therefore shielded from, the gravity vector. On the other hand, some researchers (Gebo, 2011; Boyer & Seiffert, 2013) have argued the strepsirrhine fibular facet is sloped in order to augment the lateral tibial facet in supporting body mass. An apparently higher intercept for the strepsirrhine line would be consistent with a greater load-bearing role in strepsirrhines; however, slope differences preclude statistically meaningful statements here.
Apparently more problematic is the statistical difference in intercept demonstrable for the medial tibial facet. Explaining facet differences between strepsirrhines and haplorhines as consequences of habitual postural differences leads to the expectation that the medial tibial facet should also be larger in strepsirrhines than haplorhines at a given size. Instead, the opposite would appear to be true here, as slopes are not significantly different and the intercept is greater among haplorhines.
It is worth examining Fig. 4G further before taking these results at face value. Note that for the sampled range, strepsirrhines actually tend to have larger medial tibial facet areas for body mass, and a correspondingly elevated line. Furthermore, a t-test shows that strepsirrhines have significantly greater average residual (mean = 0.23) from the haplorhine line than do haplorhines (for which the mean residual is closer to zero, 0.06) (Levene's test for equal variance: P = 0.0004; t (for unequal variances) = 3.81, P (for unequal variances) = 0.0002). How then can we reconcile these observations with the results from ancova? It is important to remember that non-significant differences in slope can result from insufficient power as well as true similarity. In such cases, the analysis will force lines through the data that are not actually lines of best fit, yet nonetheless are centered so the average deviations are 0. If the ranges of the covariates for the two different test groups are significantly different in position or range, this could lead to inaccurate positioning of the lines. This appears to be the case in the comparison of haplorhines and strepsirrhines with ancova. A corollary of this argument is that the slopes of these two distributions probably are actually significantly different, with strepsirrhines exhibiting a steeper slope, as expected if mass-induced forces are more important. Therefore, careful consideration of these results suggests that both predictions of greater weight-bearing in the medial tibia facet of strepsirrhines are upheld. Regardless, these unexpected findings warrant further study of the morphological differences of strepsirrhines and haplorhines to re-assess the possible functional explanations.
A synthetic model
Based on this and previous studies, we propose the following synthesis of the allometric context for mass, muscle, force, and joint scaling among non-aquatic mammals. First, the stress environment is determined by two distinct classes of behaviors: positional and locomotor. During positional behaviors, the stresses experienced can be expected to change with increasing body mass only as a result of the decreasing body mass to surface area ratio. Therefore, to maintain equivalent joint stresses experienced during positional behaviors, joint surface area should scale with strong positive allometry relative to body mass (close to slope = 1, the line of equivalence in Fig. 1). On the other hand, stresses induced by locomotor behaviors are determined by both body mass and acceleration of that mass by muscle force. Most likely, the larger the animal, the less influence accelerations due to muscle force have on joint surface area, since the ratio of muscle force (PCSA) to body mass decreases with increasing size. Starting from a model that assumes a perfect allometric response to maintain complete stress equivalence due to mass, we can add the effect of accelerations due to muscle force in locomotor behaviors. The effects of muscle-induced forces should be more important for small animals (for whom the muscle force is large relative to body mass), and consequently dominate joint construction. Regressions through a wide sample of body ranges would yield relationships similar to those in this study (Fig. 7A): a general trend toward positive allometric scaling, but not enough to keep stresses similar if they were solely mass-induced. Regressions through a sample that does not include very large animals would support Alexander's model (1980) with an isometric slope (Fig. 7B).
It is important to note that different facets may exhibit different scaling patterns due to their role in transmitting mass-induced forces in the skeleton. If a facet transmits a greater proportion of mass-induced force relative to muscle-induced force, it should exhibit a higher scaling coefficient (Fig. 7C). However, sampling a small range of body masses can mitigate this effect (Fig. 7D). In the present study, the talar ectal facet is the most consistently positively allometric facet. Given its orientation on the plantar aspect of the talus and the fact that it is the only female ovoid facet on the talus, the high scaling coefficients observed for the ectal facet conform to our model's predictions. The transition point (open circle in Fig. 7) from muscle-induced to mass-induced forces as the primary factor in joint construction is not known. Given the myriad ways in which animals can deal with increasing joint stresses (changing limb postures, increasing duty factors, decreasing locomotor performance, etc.), it likely encompasses a larger region than depicted in Fig. 7.
Nonetheless, as animals become larger, the ratio of muscle-induced to mass-induced force experienced at a particular joint becomes progressively smaller, and could potentially reverse depending on the functional role of the joint. This implies that there may be differential scaling patterns at small and large body masses. Such a transition was predicted by Economos (1983), who stated ‘until a certain size was reached…gravitational loading did not present a structural problem’ (p. 170). Smaller animals will scale close to geometric similarity, as predicted by the importance of muscle-induced forces in Alexander's model (1980). For larger animals, scaling patterns will track the increasing importance of mass-induced forces. Differential scaling has been suggested for the long bones of insectivores and rodents (Bou et al. 1987), and demonstrated in the long bones of terrestrial carnivores (Bertram & Biewener, 1990) and a broader sample of mammals (Christiansen, 1999). All three studies have noted the insufficiency of linear regressions for describing their observations (but see Gálvez-López & Casinos, 2012).
This model predicts differential or complex scaling at smaller and larger body masses, contingent on the changing proportions of muscle-induced to mass-induced forces experienced at that joint and the range of body masses sampled (Fig. 7). Without knowing what this proportion is for a particular joint, it is impossible to determine at what body mass the transition will occur. However, we tested the differential scaling model with two methods: (i) fitting a second order polynomial to the data (as in Bertram & Biewener, 1990) and (ii) fitting a nonlinear equation proposed by Jolicoeur (1989). Results of these tests are presented in Table 6.
Table 6. Tests of differential scaling. Fit of second-order polynomial equations and Jolicoeur's (1989) complex allometry model.
If the joint transmits a large amount of mass-induced force, the predicted transition should occur at a lower body mass, and the polynomial should fit the data significantly better than a linear regression. In such instances, our model predicts that the second-order variable will be positive, indicating an increasing slope as the size increases (i.e. the curve will be upwardly concave). We tested polynomial goodness-of-fit (McDonald, 2009) in the four samples that spanned the largest body ranges (increasing the probability that the transition point had been reached): euarchontans, primates, strepsirrhines, and haplorhines. Of the 14 positively allometric regressions, second-order polynomials significantly improved fit in seven of them. A total of 13 regressions were improved by fitting a polynomial curve to them. In each case, the second-order variable was positive, revealing an upwardly concave curve, in support of our proposed model.
A more conservative approach for detecting differential scaling, Jolicoeur's (1989) complex allometry model uses the equation:
in which A is the intercept, C the scaling coefficient and D the ‘exponent of complex allometry’. If D is not significantly different from 1, then a complex allometry model can be rejected. If D <1, there is a larger scaling coefficient at larger body masses, whereas if D >1, there is a larger scaling coefficient at smaller body masses. Our model predicts D <1 for those facets that are more sensitive to mass-induced forces (particularly the ectal and fibular facets). Jolicoeur's model reveals significant differences in D (Table 6) for the ectal facet of euarchontans, the fibular facet of primates, the ectal facet of haplorhines, and several variables among strepsirrhines (talar volume, total surface area, medial tibial facet, lateral tibial facet, head articular surface, ectal facet, and articular surface area). These results also lend support to our proposed model.
Summary and conclusions
A strict application of the Alexander's (1980) geometric similarity model does not hold for scaling patterns among talar facets of euarchontans. Both muscle-induced and mass-induced forces must impact the size of joint articular surfaces in the talus. The various tests conducted here show (i) muscle mass scales with slight positive allometry to body mass among primates; (ii) muscle PCSA scales isometrically with body mass; (iii) certain talar facets scale with positive allometry relative to body mass; and (iv) muscle PCSAs generally scale with negative allometry relative to articular surface area (although confidence intervals do not rule out isometry). These results lend support to Biewener's model (1982) of mass-induced forces impacting articular surface area. Body mass does play a role in joint construction, although that role can be variable within lineages and depend on other behavioral adaptations.
Unfortunately, only a relatively small amount of data is available for evaluating the most appropriate test of both models: the scaling pattern between facet area and muscle PCSA. The initial results reported here indicate articular surfaces increase at a faster rate than a muscle's ability to generate force. For three of the four muscles with the highest correlation coefficients, the slope is negatively allometric. However, this sample is too small to generate meaningful confidence intervals. More data are needed to assess true scaling relationship between these two variables. As this relationship is a direct test of the muscle- and mass-induced models, collecting this data should prove to be quite beneficial.
Our proposed model relies on two factors to explain the observed variability in scaling patterns: the range of the body masses included in the sample and the functional role of the articular surface in question. A wide range of body masses that includes large taxa (or only large taxa) will exhibit dominance of mass-induced forces experienced at the joint, and will result in strong positive allometric scaling. In contrast, samples with small body mass averages and ranges should have a high ratio of muscle/mass-induced forces experienced at the joint and will exhibit isometric scaling. Data should also become more curvilinear with increasing body mass range and a decreasing ratio of muscle/mass-induced force transmitted through the joint.
In accordance with these findings, it is possible that certain facets are more or less ‘mass-sensitive’ and conform more strictly to Biewener's model than others. These facets should therefore be more reliable estimators of body mass in fossil species, and body mass estimates inferred from these facets should have smaller confidence intervals than those that are more impacted by muscle-induced forces. In the case of the talus, both the ectal and fibular facets are strong candidates to generate reliable body masses for fossil taxa.
We would like to express our gratitude to the staff at the American Museum of Natural History, Duke Primate Center, National Museum of Natural History, and Stony Brook University for use of collection materials. Additionally, we would like to thank Stephen Chester for providing access to the Ptilocercus specimens, and Biren Patel and Caley Orr for supplying many of the catarrhine specimens. Justin Gladman was kind enough to supply measurements for calcaneal ectal facets. Ian Wallace executed and processed scans at Stony Brook University's Center for Biotechnology. Jared Butler, Alex Garberg, and Jonathan LoVoi helped generate surface files from CT data. Chris Wall, Andrea Taylor and three anonymous reviewers provided very helpful comments on the project. This research was supported by BCS 1317525 (formerly BCS 1125507), an American Association of Physical Anthropologists Professional Development Grant to D.M.B., and NSF DDIG SBE 1028505 to S. Chester and E. J. Sargis. The authors have no conflict of interest to declare.
G.S.Y. designed the study, cropped and measured specimens, performed analyses, and prepared the manuscript. D.M.B. designed the study, scanned specimens and prepared the manuscript. Both authors approved the final article.