Specimens
All species included in this study were macropodiforms; the bone measurement dataset encompassed all extant families and subfamilies of Macropodiformes, and several major extinct lineages (Sthenurinae, Balbaridae, Protemnodon, the giant Macropus species). 179 specimens were measured in total, across 63 species and 25 genera. Of these, 139 specimens were modern, and 40 fossil. Many specimens had some missing data, and so were not included in all analyses (Table S1). For the analyses shown in the main body of this paper, 134 specimens (94 modern, 40 fossil) were used in total, with 89 specimens (65 modern, 24 fossil) being used to evaluate hypothesis one (Fig. 3), and 46 specimens (30 modern, 16 fossil) used in the final evaluation of hypothesis two (Fig. 5). The remainder are only referred to in the supplementary material. This was necessary as the main analyses, especially of hypothesis two, require a variety of measurements from articulated specimens of the kangaroo pes, which are relatively rare among fossils particularly. Body masses were gathered from the literature (16,22,36,42,43,44,45,46,47,48,49 for details, see Table S1). Where possible, the mass of the individual was used, but where this was not available, the mean body mass, corresponding to either the sex of the individual (in strongly dimorphic species), or the species as a whole, was used instead. An attempt was made to extrapolate body mass from calcaneal measurements instead, following Prideaux and Warburton (2023), on the basis that this uses direct evidence from the individual specimens used rather than species means. However, due to the disproportionate shortening and broadening of the calcanea in the giant kangaroos—see later discussion—this produced implausibly low estimates of body mass for all giant kangaroos.
All PCSA data used in this paper is from the previously published paper by McGowan et al.10, and as such is not separately published here.
Morphological data
Articular lengths of key hindlimb bones (the femur, tibia, fourth metatarsal, fourth proximal phalanx, and calcaneum) were collected, as well as antero-posterior and medio-lateral midshaft widths of the fourth metatarsal and width of the calcaneal tuberosity, where available (Fig. 2). Some of these measurements were taken from the literature16,22 and private correspondence (n = 317, nspecies = 65); others were collected for this study by the authors (n = 65, nspecies = 38). Details of specimens, including specimen numbers, and sources of body masses and bone dimensions can be found in Table S1. For some of these specimens, an additional set of calcaneal dimensions (31 specimens: 11 fossil, 20 modern) were collected to facilitate interpretation of the second hypothesis results (Table S1; Fig. 2). For each specimen, digital callipers were used to measure the width of the calcaneal tuberosity at its widest point, the calcaneal length (taken along the mediolateral centre of the bone), and the mediolateral and dorsoventral widths of the calcaneum, taken halfway along the length of the bone. Where available, the length of the associated fourth metatarsal was also measured. Measurements were taken to the nearest 0.01 mm. For all data collected for this study, see Table S1.
Ankle moments when hopping
To test our hypotheses, we first needed to estimate the moments experienced around the ankle joint of each specimen when hopping (Fig. 6). Kangaroo joint angles can differ among species and with hopping speed14. However, limited data are available, and while joint angles do vary, this variation is relatively small, as demonstrated by the constant effective mechanical advantage at the ankle joint among species25, and at different speeds within a species12. Thus, the joint angles at midstance to the nearest 5 degrees for Notamacropus eugenii (see Fig. 3 of50), are here taken as representative for all species. This species was used as it provides the best currently available data on joint angles throughout a hopping cycle, and as a midsized wallaby, it is a reasonable choice for a representative species. “Midstance” was defined as the point of peak ankle flexion during the stance phase. The mean angle derived from three stance phases gave a metatarsophalangeal joint angle of 1.95 radians (112°), and an ankle joint angle of 1.60 radians (92°). As a recent study found that joint angles can vary somewhat by body mass51, a sensitivity analysis was also performed, varying each of these joint angles by 10% towards a more or less crouched posture (Fig. S3) to see if this affected the final conclusions. The conclusions of neither hypothesis were affected by this change, so we retain these assumed joint angles for the remainder of the study. From the metatarsophalangeal joint angle (=1.95), and the length of the fourth metatarsal (LMt), the moment arm (R) of the ground reaction force at midstance was calculated:
Fig. 5
(a) Predicted gastrocnemius tendon widths, compared against actual calcaneal tuberosity widths. Shading around regression lines indicates 95% confidence intervals. Grey highlighting indicates approximate regions of implausible hopping, either due to tendons being too narrow to resist ground reaction forces (lower region), or due to the tendons being wider than the available insertion space (upper region). (b-d) Violin plots of ratio of predicted tendon width to calcaneal tuberosity width, in modern vs. fossil individuals, with region indicating tendons larger than available insertion area highlighted in grey. Tendon widths predicted based on (b) moment calculations, (c) scaling of gastrocnemius muscle, and (d) scaling of gastrocnemius tendon. n = 46; n = 43 for the moment-based calculation of tendon width. Calcaneum outline by MJ; Hindlimb image by MJ, based on10,25,59.
Fig. 6
(a) Schematic drawing of the distal hindlimb bones of Macropus giganteus, adapted from25, with key measured bone lengths labelled, and (b) a free-body diagram illustrating the terms used in the text for forces and angles (black), as well as lever arms (blue). Red indicates the bones themselves. Abbreviations: FAE = force exerted by ankle extensors; GRF = ground reaction force; Lcalc = length of the calcaneum; LMt = length of the metatarsal; R = lever arm of GRF; r = lever arm of FAE; = metatarsophalangeal joint angle; = ankle joint angle. Hindlimb image by MJ, based on10,25,59.
$$R = L_{{Mt}} \cos (\pi – \theta )$$
(1)
The peak ground reaction force (GRF) acting on each individual hindlimb was assumed to be three times the weight (3 mg) of the animal, occurring at midstance and being oriented vertically52. Although a peak ground reaction force of 5 mg has been recorded in red kangaroos14, this seems to be a value for the whole animal (both hindlimbs), rather than for the hindlimbs considered individually, which would imply that each limb experienced ~ 2.5 mg of force. Thus, 3 mg was considered a conservative estimate for hopping animals, and this value was used here. The vertical orientation of GRF is in line with the results of a recent study which measured hopping kangaroos on a force plate51. From the peak GRF and the GRF moment arm R, the moment at the ankle joint was calculated as:
$$M_{{{\text{GRF}}}} = GRF \cdot R$$
(2)
GRF is here assumed to act at the metatarsophalangeal (MTP) joint. A sensitivity analysis was also run, comparing the results shown here to those found if GRF was assumed to act at the midpoint of the phalanges, assuming that the first phalanx represented 42% of the total phalanx length (a mean value derived from data provided by Christine Janis, pers. comm.). The results of this sensitivity analysis (Fig. S2) are consistent with our findings when GRF is assumed to act at the MTP joint, with fossil individuals falling within the safety factor ranges of modern kangaroos. However, it also predicts a safety factor of < 1 for many modern kangaroos, including those for which GRF values close to our assumed value have been recorded (see figure below). We know this to be inaccurate, as these species do hop without fracturing their metatarsals. Therefore, we conclude that our original assumption of GRF acting at the MTP joint is more likely to produce an accurate model of hopping abilities among extinct species in this case. Thus, we proceed with this assumption for the remainder of the study.
Hypothesis 1: metatarsal safety factors
For those specimens where the antero-posterior (AP) and medio-lateral (ML) diameters of the fourth metatarsal were known (n = 89), the second moment of area at the midshaft (I) was predicted as follows53:
$$I = (\pi\cdot{r_{ml}}\cdot{r_{ap}} ^{3} )/4$$
(3)
Where rml is the mediolateral radius, and rap is the anteroposterior radius. Then, the bending moment of the GRF at the midshaft (Mmid) was calculated:
$$M_{{mid}} = GRF \cdot 0.5L_{{Mt}} \cdot \cos (\pi – \theta )$$
(4)
Next, peak stress at the midshaft () was calculated based on these values for Mmid, = rap, and I (from53, p. 16]):
$$\sigma = (M_{{mid}} \cdot r_{{ap}}) /I$$
(5)
The safety factor of the metatarsal at peak stress was calculated by dividing the bending failure strength of mammalian bone by the peak stress recovered above. The failure strength of mammalian bone varies somewhat across species and bone type; for the sake of this study, we approximate it as 200 MPa in kangaroos. This value is the mean found for larger mammals in a study by Biewener21, and is within the range of values found for the kangaroo rat, the most comparable species in terms of locomotion included in this study.
Hypothesis 2 preparation: ankle extensor muscle physiological Cross-sectional areas (PCSAs)
To test the second hypothesis, we must estimate the force the ankle extensor muscles exert on their tendons, in order to estimate the requisite tendon cross-sectional area, and thus width, to resist this force. We employ two different methods to make this estimate.
The first method predicts ankle extensor muscle PCSAs in the giant kangaroos from extrapolated allometric scaling patterns. The measured PCSAs of ankle extensor muscles for a variety of modern macropodoids were collected from the literature (10, provided by Craig McGowan, Pers. Comm.), including values for the gastrocnemius (GAS), plantaris (PL), and flexor digitorum longus (FDL). The PCSA values of these three muscles were summed to produce a total ankle extensor muscle PCSA. Linear ordinary least squares regressions were then performed on the log10-transformed PCSA and body mass data for three datasets: (1) the PCSAs estimated from ankle moments; (2) the summed measured ankle extensor PCSAs; and (3) the measured gastrocnemius PCSAs (Fig. 4). It is worth noting that the FDL possesses a reduced lever arm, relative to the other ankle extensor muscles, as it passes closer to the rotational centre of the ankle joint, meaning that it contributes less to the effective PCSA required to balance ground reaction forces at the ankle. Since we do not have specific data on the moment arm of the FDL, we disregard this muscle and the plantaris in the subsequent calculations of ankle extensor tendon width, in favour focussing on the gastrocnemius muscle. The gastrocnemius tendon is also the only one which inserts directly on the calcaneal tuberosity, meaning that this is the tendon which determines if adequate insertion area is available on the calcaneal tuberosity.
This method does rely on extrapolation beyond the mass range of living species, which, as previously discussed, is not ideal, and means that these particular estimates are subject to the same issues mentioned for previous studies. However, there are no available osteological indicators of the size of the extensor muscles in giant kangaroos. The scaling relationships for ankle extensor muscles among modern kangaroos are hyper-allometric, with PCSA ∝ Mb (Fig. 4), whereas based on isometry, the only other option we have for estimating PCSA from body mass, we would expect PCSA ∝ Mb 2/3. Therefore, it is likely that if this extrapolation from living species is inaccurate, it is an overestimate of the PCSA available for the extinct species, if they did not hop, and is thus a conservative estimate relative to our hypothesis.
Our second method, however, does not rely on allometric extrapolation at all, instead using our estimate of peak ground reaction force (GRF) and measured bone lengths to calculate the minimum force the ankle extensor muscles must produce to resist GRF. It thus avoids the problems which come with allometric extrapolation. For this method, the amount of force the ankle extensor muscle-tendon units (MTUs) were required to produce (FAE) to balance the moment of the GRF at the ankle joint was calculated as:
$$F_{{{\text{AE}}}} = M_{{{\text{GRF}}}} /r$$
(6)
where r is the moment arm of the ankle extensor MTUs. To find this moment arm, both the length of the calcaneum and the angle between the calcaneum and the ankle extensor MTUs needed to be known. The length of the calcaneum in each case was already in our measured dataset. Meanwhile, the line of action of the MTUs was assumed to run parallel to the tibia, and the calcaneum parallel to the metatarsal, meaning that the angle between the two is the same as the ankle joint angle (\(\phi\)) (Fig. 6). Thus, r was calculated as:
$$r = L_{{{\text{calc}}}} \sin \phi$$
(7)
where Lcalc is the length of the calcaneum.
From the calculated ankle extensor force, the required total ankle extensor muscle PCSA (in m2) was calculated by dividing FAE by 3,000,000—since the maximal isometric stress of the muscles was assumed to be 0.3 MPa, following McGowan et al.10 for consistency with prior studies. This calculation provides a measure of the minimum ankle extensor muscle PCSA required to balance the moments involved in hopping.
Hypothesis 2: ankle extensor tendon width
To test our second hypothesis, the muscle PCSAs calculated in the previous section were used to predict the minimum tendon diameter required to maintain a tendon safety factor above one when hopping. From the PCSA of a muscle, the theoretical maximum force can be calculated; from this the minimum cross-sectional area (CSA), and then the tendon diameter needed to withstand this force can be derived. To accommodate hopping without tendon rupture, the calcaneal tuberosity width, a proxy for the maximum possible diameter of the tendon, must exceed this minimum required tendon diameter.
Three sets of predicted tendon diameters were created. The first was derived from the moment-based estimation of the ankle extensor muscle PCSA created in the section above, and represents the absolute minimum tendon size required to prevent rupture during hopping. The second was derived from the gastrocnemius PCSA regression equation calculated from measured PCSAs in modern kangaroos10 in the section above, and represents the tendon size if we assume similar muscle scaling to living species. The PCSA estimates from the first two methods were used to predict minimum tendon CSA as follows:
The maximum stress experienced by a tendon (σt) is equal to the maximum isometric stress which can be exerted by the muscle—assumed to be 0.3 MPa—multiplied by the ratio of muscle physiological cross-sectional area (Am) to tendon cross-sectional area (At)10:
$$\sigma _{t} = 0.30(A_{m} /A_{t} )$$
(8)
The safety factor of the tendon can be calculated by dividing the failure strength of the tendon—assumed to be 100 MPa, once again following the methods of McGowan et al. (2008)—by σt.
$$SF_{t} = 100/\sigma _{t}$$
(9)
If we assume a safety factor of 1, then using the above equations, we find that:
$$A_{m} /A_{t} = 333.3$$
(10)
A safety factor of one is lower than would be acceptable in real life, given that a safety factor of < 1 would indicate tendon rupture. However, this value is used here to represent the absolute lower limit of tendon safety factors. Equation 10 was used to calculate the minimum tendon cross-sectional area (CSA) for all modern and fossil individuals where calcaneal measurements and Mb values were available, based on the two muscle PCSAs described above.
A third estimate of tendon diameter was derived from an existing regression equation for tendon CSA against mass10. While this approach relies entirely upon extrapolation from modern data, it was included for comparison to the previous two approaches, and allows us to assess the sensitivity of our conclusions to changing the method for estimating tendon CSA in extinct species.
The gastrocnemius tendon diameter was calculated from all three sets of tendon CSA predictions. To do this, a reasonable estimate of tendon ellipticity at insertion is required. To our knowledge there are no published data on the major vs. minor axis dimensions of kangaroo hindlimb tendons. The wider literature on mammal gastrocnemius tendons is likewise limited. Peterson et al.54 state that, across mammals, the anteroposterior and mediolateral widths of the tendon “are rarely very different because the tendon is quite round at its thinnest point”. This is not necessarily reflective of the cross-sectional shape at insertion, however, and they do not provide raw data, so the details cannot be judged. However, Obst et al.55 find that, in humans (who may be more comparable to kangaroos than to most mammals, as kangaroos are bipedal), the Achilles tendon is highly elliptical at insertion on the calcaneum. Raw data for this study is likewise not available, but based on values attained from digitising Fig. 3 using WebPlotDigitizer56, the major axis is 4.80 times greater than the minor axis at rest, and 4.94 times greater at maximal contraction. With this degree of ellipticity assumed, our projected required tendon widths are increased by a factor of 2.24, relative to a circular cross-section. If the resulting minimum tendon diameter exceeds the measured calcaneal width, then tendon rupture would be likely during hopping locomotion and it can be ruled infeasible.
The resulting tendon width predictions were compared to each other, and to the measured widths of the calcaneal tuberosities for the same species, to see if the tendons would fit the calcanea observed in the fossil record. We do not suggest that there is a predictable relationship between calcaneum width and tendon size, as the tendon may not insert on the entire width of the calcaneal tuberosity. However, we calculate the ratio of the three sets of predicted tendon sizes to measured calcaneal width for modern and fossil kangaroos, to see whether there is any evidence that the fossil specimens were closer to being unable to accommodate the tendons required for hopping than any of their living relatives.
All statistics performed in this study were linear least-squares regressions performed in base R v.4.4.157, with plots produced using the package ggplot258.