Scale Effects between Body Size and Limb Design in Quadrupedal Mammals

Recently the metabolic cost of swinging the limbs has been found to be much greater than previously thought, raising the possibility that limb rotational inertia influences the energetics of locomotion. Larger mammals have a lower mass-specific cost of transport than smaller mammals. The scaling of the mass-specific cost of transport is partly explained by decreasing stride frequency with increasing body size; however, it is unknown if limb rotational inertia also influences the mass-specific cost of transport. Limb length and inertial properties – limb mass, center of mass (COM) position, moment of inertia, radius of gyration, and natural frequency – were measured in 44 species of terrestrial mammals, spanning eight taxonomic orders. Limb length increases disproportionately with body mass via positive allometry (length ∝ body mass0.40); the positive allometry of limb length may help explain the scaling of the metabolic cost of transport. When scaled against body mass, forelimb inertial properties, apart from mass, scale with positive allometry. Fore- and hindlimb mass scale according to geometric similarity (limb mass ∝ body mass1.0), as do the remaining hindlimb inertial properties. The positive allometry of limb length is largely the result of absolute differences in limb inertial properties between mammalian subgroups. Though likely detrimental to locomotor costs in large mammals, scale effects in limb inertial properties appear to be concomitant with scale effects in sensorimotor control and locomotor ability in terrestrial mammals. Across mammals, the forelimb's potential for angular acceleration scales according to geometric similarity, whereas the hindlimb's potential for angular acceleration scales with positive allometry.

the incisions necessary to remove the limb. Forelimbs were removed by making a sagittal incision along the length of the sternum, followed by a posterior incision from the caudal angle of the scapula to the sternal incision. The posterior incision was just caudal to the axilla of the forelimb so as to include the whole of the forelimb. We next made an incision from the caudal angle of the scapula along the dorsal and cranial margins of the scapula to the glenoid joint and a final incision from the glenoid joint to the cranial end of the manubrium. To free the limb from the torso, we dissected the pectoralis muscle group from the body wall and shaved the serratus ventralis from the medial surface of the scapula. For species possessing clavicles, the clavicle was included as part of the forelimb by severing the ligaments between the clavicle and the manubrium.
We removed hindlimbs by making a vertical incision just cranial to the crest of the ilium. A second incision was made starting at the previous incision and the dorsal border of the ilium and continuing to the caudal-most extent of the ischiadic tuberosity, parallel to the neural spines of the sacral vertebrae. These incisions exposed the anterior and dorsal edge of ilium and ischium. We then proceeded to shave the limb muscles off the lateral surface of the ilium, cutting through the joint capsule of the hip and the ligament of the femoral head. After shaving the limb muscles off the ilium and freeing the femoral head from the acetabulum, we then shaved along the ventrolateral face of the pelvis down to pelvic symphysis, freeing the whole limb from the pelvis.

Measuring inertial properties
All limbs were fully thawed. After first weighing the limb, its length was measured. The length was taken after fully extending the limb and then releasing it to flex on its own accord. The distance from the limb's pivot to distal-most extreme of the limb was then taken as the limb's length. In the case of the hindlimb, the limb's pivot is the femoral head. For the entire forelimb (i.e., the scapula and all segments distal) the pivot was considered to be at the intersection of the scapular spine and the vertebral margin of the scapula [S1-S3]. For the forelimb with the scapula removed, the humeral head was considered the limb's pivot. The passively flexed length of the limb was taken as the We next attached the limb to a bar with known inertial properties (mass, center of mass position, moment of inertia, radius of gyration, and natural frequency) in the extended (i.e., straightened) position using cable ties. Limbs were attached to this bar 1) to prevent flexion of the limb during data collection and 2) to be able to mount the limb onto a pivot consisting of a rotary shaft passing through two bearings. A shaft clamp on the rear face of the bar coupled the bar and its attached limb to the shaft of the pivot. When attaching the limb to the bar, the limb's pivot point during locomotion was aligned with the shaft clamp, insuring that inertial properties were measured with respect to the limb's axis of rotation during locomotion. As the mass of the limbs varies greatly across species, several bars of differing length, mass, and moment of inertia were used for data collection to minimize differences between inertial properties of the bar and those of the limb.
To find limb COM position, we suspended the limb-bar combination from two spring scales. Using a tape measure, we then measured the distance from the limb's pivot to each spring scale. We calculated the COM position of the combined limb and bar using the following equation which balances the moments acting on the bar and limb: where COM total equals the COM position of the combined bar and limb and m total equals the total mass of both the bar and limb. For the spring scale proximal to the limb's pivot, s 1 and d 1 respectively correspond to the scale's reading and its distance to the limb's pivot. s 2 and d 2 likewise correspond to the second scale's reading and its distance to limb's pivot. Knowing the inertial properties of the bar, we used the following equation to find the limb's COM position: where COM limb is the limb's COM position, COM bar is the bar's COM position, m limb is limb mass, and m bar is bar mass. After finding limb COM position, we proceeded to determine the moment of inertia and natural frequency of the limb. Mounting the combined limb and bar onto a pivot via the bar's shaft clamp, we deflected the limb and attached bar from the their rest position by a small angle (~20-30°) and then released them to swing freely in a parasagittal plane. While the limb was swinging, we videotaped it at 30 fps for 15 cycles. For taxa with smaller limbs (e.g., Sciurus, Tamias, etc), we videotaped limbs at 90-120 fps. Using the video, we determined the natural period (the inverse of natural frequency) of the limb and bar, treating them as a single object. We were then able to calculate the moment of inertia of the limb by using the following equation from Myers & Steudel (1997): € MOI limb = [(m limb COM limb ) + (m bar COM bar )]g(t total 2π ) 2 − MOI bar (S3) where MOI limb is the moment of inertia for a limb, g is gravitational acceleration (= 981 cm/s 2 ), t total is the natural period of the bar, the limb, and the shaft of the pivot as one unit, and MOI bar is the moment of inertia of the bar attached to the shaft. In Equation S3, the variables m bar and COM bar , refer to the bar fixed to the shaft; m bar and COM bar were recalculated to account for the shaft's mass and its effect upon the bar's COM position, as the bar is securely attached to the shaft as it swings. In Equation S3, m bar is the total mass of the bar and the shaft, and COM bar is the COM position of the bar after the shaft has been secured to the bar using the shaft clamp. Now knowing the moment of inertia of the limb, I proceeded to calculate limb natural frequency. we calculated limb natural frequency as follows: with F limb being limb natural frequency. Also, using Equation 1, we determined limb radius of gyration by using the mass and moment of inertia of the limb. Species means for limb inertial properties of both fore-and hindlimbs are given in the Appendix A. Figure S1. Experimental setup to measure limb inertial properties. After dissecting the limb from the torso and weighing it, the limb is attached in its passively flexed position to a bar with known inertial properties, with the pivot of the limb (Point H) overlying the pivot of the bar (Point P) (A and inset). The position of the limb on the bar is shown in B. Next the combined limb and bar are suspended from two spring scales that are a known distance from the pivot of the limb and bar (C). Knowing the distance of each scale from the pivot (D 1 and D 2 ), the readings of the scale (S 1 and S 2 ), and the mass of the limb and bar, the limbs's COM position is determined using Equation S1 and then Equation S2. Next, the combined limb and bar are set upon a pivot (inset), and the limb is allowed to freely swing back and forth on the pivot while being videotaped (D). Knowing the number of frames it takes for the combined limb and bar to complete a full oscillation and the frame rate, the time required for the combined limb and bar to complete a full oscillation can be calculated (i.e., its natural period). Using Equations 1 (main text), S3, and S4, we then calculated the limb's MOI, radius of gyration, and natural frequency.

Sources of methodological error
We should address two potential sources of error lie in our methodology. The first source is our identification of the scapula's point of rotation. The scapula's contribution to forelimb kinematics is undeniable [S4]; however, the exact location of the pivot during both stance and swing phases remains unknown for the bulk of species sampled in our study, as well how the pivot's position varies with size across species. Previous studies have found that the scapular pivot lies within the proximal third of the scapula where the scapular spine crosses the vertebral margin of the scapula [S1-S2]. We therefore measure forelimb inertial properties with respect to the intersection of the scapular spine and scapula's dorsal edge. By measuring forelimb COM position, MOI, radius of gyration, and natural frequency all with respect to the same anatomical landmark for all individuals, the data can be corrected as more detailed information on scapular kinematics becomes available for a greater diversity of species.
The second potential source for error is the extent to which the limb is flexed/extended while collecting data. As limbs were not frozen at the time of study, minor amounts of joint flexion may have been added inadvertently when mounting the limb for study. To test the effect that different degrees of limb flexion may have had upon the measurement of limb inertial properties, limb inertial properties were measured for a single hindlimb of a grey wolf (Canis lupus) over five separate trials (Table S1). The coefficient of variation is less than 5.0% for all the variables measured over these five trials. Such low values for the coefficient of variation indicate that the data are only weakly affected by differences in limb flexion between studied limbs.  Table S1. Estimation of measurement error due to differences in limb flexion between measured limbs. Limb inertial properties were measured for a single hindlimb of Canis lupus over five separate trials. CV refers to the coefficient of variation.

Null model predictions
Central to geometric similarity are explicit relationships between length and diameter. According to the model [S5-S6], linear dimensions remain in direct proportions with one another as body size increases. Providing us with the relationship: S5) where l and d are length and diameter, respectively. Therefore to increase the size of an organism by a factor of 2, all linear dimensions would only have to be multiplied by a factor of 2. This would be the case for any factor n.
Mass and length. Of primary interest for the scaling of limb inertial properties is how limb length relates to mass, regardless of whether it is body mass (M) or limb mass (m) under examination. Mass is the product of density (ρ) and volume (V) and volume is a product of length and area (π(d/2) 2 ), we can write the following equation: € M = ρV = ρπ(d 2) 2 l (S6) and from Equation S6 we can write the following proportionality: € M ∝ ld 2 (S7) Given proportionalities S5 and S7, the following proportionality can be written between length and mass for geometric similarity: Mass distribution and MOI. Center of mass (COM) position was taken as the length from the limb's COM to the limb's axis of rotation, and the radius of gyration (r) is the length between the centroid of the moment of inertia (MOI) and the limb's axis of rotation. Thus COM position and radius of gyration scale with mass according to the following proportionalities for the null model: MOI is a product of limb mass and the square of the radius of gyration: MOI = m•r 2 . It is important here to note that geometric similarity also predicts that changes in mass are also directly proportional, such that M ∝ m [S7-S8]. By using proportionality S10, the following relationships between limb MOI and mass can be derived: Natural frequency. The formula for calculating the natural frequency of a gravitational pendulum is: where F is the natural frequency and g is the gravitational constant. From S11 and S12, the following proportionality can be written: The mass terms cancel out in S13, and, recognizing that COM ∝ r from Equations S9 and S10, proportionality S14 is produced: € F ∝ r −0.5 (S14) By substituting the relationships for the radius of gyration to mass from S10 into S14, we can predict how limb natural frequency scales to body mass under geometrically similar scaling.

Comparative Methods
The taxa included in this study span eight taxonomic orders. In order to perform comparative methods, a composite phylogeny had to be constructed (Fig. 1). Relationships between mammalian orders and families were based upon Meredith et al [S9]. Topologies for mammalian orders and families were based off published phylogenies for artiodactyls [S10-S12], carnivorans [S13-S16], rodents [S17-S18], and perissodactyls [S19]. In order to scale branch lengths to divergence time across these different phylogenies, divergence time branch lengths for all phylogenies cited were scaled to the divergence times of Meredith et al. [S9] by means of common, dated nodes shared by Meredith et al. [S9] and the remaining phylogenies.  Table S2. Phylogenies used to scale divergence times within the composite phylogeny. Node names refer to those referenced in Meredith et al. [S9].  Table S5. Results of PGLS regressions that simultaneously estimate λ along with regression coefficients [S20]. 'C. L.' denotes 95% confidence intervals. (Table SA1) and forelimbs (Table SA2).