Salient features in locomotor evolutionary adaptations of proboscideans
revealed via the differential scaling of limb long bones

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By Valery B. Kokshenev and Per Christiansen

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Submitted to the Journal of the royal Society Interface 03 June 2009

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Abstract. The standard differential scaling of proportions in limb long bones
(length against circumference) is applied to a phylogenetically wide sample of
the Proboscidea, Elephantidae and the Asian (Elephas maximus) and African
elephant (Loxodonta africana). In order to investigate allometric patterns in
proboscideans and terrestrial mammals with parasagittal limb kinematics, the
computed slopes (slenderness exponents) are compared with published values for
mammals and studied within a framework of theoretical models of long bone
scaling under gravity and muscle forces. Limb bone allometry in E. maximus and
the Elephantidae are congruent with adaptation to bending and/or torsion
induced by muscular forces during fast locomotion, as in other mammals,
whereas limb bones in L. africana appear adapted for coping with the
compressive forces of gravity. Consequently, hindlimb bones are expected to be
more compliant than forelimb bones in accordance with in vivo studies on
elephant locomotory kinetics and kinematics, and the resultant negative limb
compliance gradient in extinct and extant elephants, which contrasts to other
mammals, suggests an important locomotory constraint preventing achievement of
a full-body aerial phase during locomotion. Differences in ecology may be
responsible for the subtle differences observed between African and Asian
elephant locomotion, and the more pronounced differences in allometric and
mechanical patterns established in this study.

Key words: long bone scaling models; standard differential scaling; limb
gradient functions; proboscideans; extinct and extant elephants.

##  I Introduction

Differential scaling of the proportions of the limb long bones in terrestrial
mammals has been studied by many researchers (most often bone length $L$ and
circumference $C$ or diameter $D$) with the aim to establish correlations
between design and posture of mammalian limbs coping with support of mass and
locomotion in the gravitational field. Many attempts to formulate generally
applicable allometric power laws ($L\varpropto M^{l}$ and $C=\pi D\varpropto
M^{d}$, where $M$ is body mass) have been the subject of a long standing
debate and controversy (e.g., McMahon 1973, 1975a, b; Alexander 1977;
Alexander et al. 1979a; Biewener 1983, 2005; Economos 1983; Bertram & Biewener
1990; Christiansen 1999a, b, 2002, 2007; Kokshenev, 2003, 2007; Kokshenev et
al. 2003). Among the proposed theoretical frameworks, the three most common of
which are: the _geometric_ (or isometric) _similarity model_ (GSM, with
$l_{0}=d_{0}=1/3$); the _elastic similarity model_ (ESM, with
$l_{0}=1/4,d_{0}=3/8$); and the static _stress similarity model_ (SSM, with
$l_{0}=1/5,d_{0}=2/5$). Subsequently, these similarity models have typically
been explored in analysis of allometric power laws using measured bone lengths
and diameters resulted in allometric exponents ($l$ and $d$), when body masses
are available. In cases where body masses are unknown, a _slenderness
exponent_ $\lambda$ can be computed (via $L\varpropto C^{\lambda}$) and
compared with that predicted as $\lambda_{0}=l_{0}/d_{0}=1$, $2/3$, or $1/2$
by the GSM, ESM, and SSM, respectively (McMahon 1975a).

Originally proposed to explain animal design to experience similar elastic
forces and stresses under gravity, the corresponding ESM (or ”buckling” model)
and SSM (McMahon 1973, 1975a, b) were found not to apply to terrestrial
mammals as a group, neither overall body proportions (Economos 1983; Silva
1998), nor allometric scaling of long bones (Alexander et al. 1979a; Biewener
1983, 2005; Economos 1983; Christiansen 1999a, b; Kokshenev et al. 2003).
Instead, body proportions and long bones were found closer to isometry, i.e.,
to GSM with $\lambda_{0}=1$, and in large mammals and in large mammals they
both become progressively more robust with increments in body size. Testing
McMahon’s models, many studies indicated that muscle forces, showing size-
dependent fluctuations among terrestrial mammals, are highly important (e.g.,
Alexander 1985, Biewener 1983, 1989, 1990, 2005). Direct evidences of muscle
forces affecting proportions of the allometric scaling of the major long bones
in Artiodactyla were provided by Selker and Carter (1989). It was analytically
shown that the failure of any one predicted power law was shown not caused by
the failure of underlying elastic force patterns but due to McMahon’s
simplifications for evolutionary adaptive properties for maintaining similar
skeletal functional stresses apart of muscles and under the dominating
influence of gravity (Kokshenev et al. 2003). Then, using the overall-bone
slenderness exponent $\lambda_{\exp}^{(mam)}=0.80\pm 0.02$, resulting from re-
analysis of bone proportions in a wide-ranging sample of terrestrial mammals
from Christiansen (1999a), also not matching any of McMahon’s original
predictions, the dominating role of muscle forces in long-bone scaling was
demonstrated from first physical principles (Kokshenev 2003).

Naturally, bone shape is a hereditary property, but bone is a phenotypically
plastic tissue, capable of reacting powerfully to its mechanical environment.
In the growing fetus, bone shape, size and position is initially determined by
the early cartilaginous anlagen during embryonic skeletogenesis, which are
subsequently gradually replaced by endochondral ossification during ontogeny
(Favier & Dollé 1997; Currey 2003; Provet & Schipani 2005). However, bone
shape is heavily influenced by a mechanical response to the environment during
ontogeny and throughout an animal’s life. It has been demonstrated that strain
rate and magnitude, surrounding tissue formation, and fetal muscle
contractions are prerequisites for normal bone formation during ontogeny
(Rodriguez et al. 1992; Mosley et al. 1997; Mosley & Lanyon 1998; Lamb et al.
2003). In post-natal and adult mammals, bone is capable of reacting to changes
in mechanical stresses enforced by physical activity and muscle mechanics with
rapid alterations of size and shape (Biewener 1983, 1989, 1990; Carrano &
Biewener 1999; Curry 2003; Firth et al. 2005; Warden et al. 2005; Franklyn et
al. 2008), and hereditary properties determined by the genome appear primarily
responsible for bone patterning during fetal ontogeny and is less relevant for
bone size and shape in the adult animal (Mariani & Martin 2003). These factors
are seemingly beyond macroscopic elastic theories for formulating allometric
power laws for bone scaling. Moreover, the concept of uniform elastic
similarity also seems to be inconsistent with a diversity of functional local
elastic forces and stresses, which are not constant in bones during locomotion
and therefore are not likely to be reflected in scaling analysis of external
bone dimensions (Doube et al. 2009). Nevertheless, the authors hopefully
believe that scaling predictions arising from the macroscopic spatial
continuous mechanics applied to bone tissue under broad spectrum of loading
conditions (Kokshenev et al. 2003), and therefore reflecting most general
trends in proportional limb bone adaptations to environmental conditions, can
be reliably verified at least by the overall limb and bone allometric data.

The above bone scaling studies and other statistical and experimental studies
(e.g., Biewener et al. 1983a, b, 1989; Rubin & Lanyon 1984; Selker & Carter,
1989; Streicher & Muller 1992; Carrano & Biewener 1999) have stimulated
formulation of novel theoretical concepts in light of dynamic bone strain
similarity (Rubin & Lanyon, 1984) or mechanical (strain and stress) similarity
(Kokshenev 2007). Basic conceptions of McMahon’s (1973, 1975a) elastic
similarity hypothesis have also been reconsidered (Kokshenev et. al., 2003,
Kokshenev 2003). The framework of bone scaling is, by default, limited to long
bones approximated by cylinders with $L\gg D$ justified by the ratio
$L/D\backsim 10$, at least for mammalian humerus, radius, ulna, femur, and
tibia. Moreover, the justification for application of the elastic theory
patterns established for arbitrary loaded long solid cylinders (Kokshenev
2007) is generally based on the assumption that the long bones play the
primary role in body support. Consequently, the positive allometry of long
bone structure in relation to body mass observed in regression analysis may be
expected to be better understood by biomechanical adaptation of bones to
maximal external loads emerging during fast locomotion and studied via bone-
reaction elastic forces and stresses, whereas non-mechanical ontogeny effects
of limb bone adaptation associated with a small Prange’s index are relatively
small (Kokshenev 2007).

In the present study we analyze the surprisingly varied differential scaling
of the limb long bones in a taxonomically narrow clade of mammals, the extant
proboscideans (Proboscidea, Elephantidae); the Asian elephant (Elephas
maximus), and African savannah elephant (Loxodonta africana). These gigantic
land mammals have a more upright limb posture, notably much more upright
propodials and different locomotor mechanics from other terrestrial mammals in
that fast locomotion is ambling with no suspended phase in the stride, but
with duty factors $\beta>0.5$ (Gambaryan 1974; Alexander et al. 1979b;
Hutchinson et al. 2003, 2006). We compare theoretical predictions with the
data from a phylogenetically wide sample of extinct proboscideans from
Christiansen (2007) completed here by the Elephantidae family, as well as
allometry results from scaling studies of running mammals with parasagittal
limb kinematics, in the hope of establishing generic allometric patterns
distinguishing limb postures characteristic of high-power locomotion in
proboscideans and mammals.

Modern elephants are characteristic in having very long limb bones for their
body size (Christiansen 2002), and a very upright, though not strictly
columnar (Ren et al. 2007), limb posture, in which the two propodial bones
(femur and in particular humerus) are kept at a distinctly greater angle
compared to the ground than is the case in other large, quadrupedal mammals.
The steeply inclined propodials imply that during standing and at low speeds,
the primary forces affecting the limb bones will be axially compressive.
However, with increments in speeds, joint flexion increases during the support
phase, and during the recovery phase, joint flexion can be high (Ren et al.
2007). Although there is no difference in joint flexion between juvenile and
adults or between Asian and African elephants, the propodial bones are still
markedly more inclined compared to horizontal even during the fast locomotion
than is in the case for other quadrupedal running mammals. During the support
phase in locomotion, the ankle, unlike the more mobile wrist, displays spring-
like mechanical properties, reminiscent of, albeit less than in quadrupedal
running mammals, matching the more compliant hind limbs during locomotion.
This also is consistent with the tendons of the hind-foot scaling with
positive allometry during ontogeny, whereas those of the forelimb scale with
negative allometry, and thus become progressively more gracile (Miller et al.
2008), thereby supporting the observation that the hindlimbs are more
compliant with bouncing kinematics than the stiffer, vaulting forelimbs,
during fast locomotion. In view of the progress in application of bone scaling
models (Kokshenev 2003, 2007), a general problem arises whether kinematic
evidences on the mechanical influence on bone ontogeny can be independently
revealed by the differential limb bone scaling?

##  II Materials and Methods

###  II.1 Theoretical background

A theoretical analysis of non-critical elastic forces emerging in long bones
of adult mammals resulted in mode-independent relationships for bone scaling
exponents

$d=\frac{1}{3}+b\text{ and }l=\frac{1}{3}-b\text{, with
}\lambda(b)=\frac{l}{d}=\frac{1-3b}{1+3b}$ (1)

discussed in Eq. (8) in Kokshenev 2007. Here $b$ is Prange’s index scaling
bone mass to body mass. Since scaling index $b$ is consistently small for
mammals (see e.g. table 1 in Kokshenev 2003), equation (1) matches previous
observations mentioned in Introduction of the closeness of mammalian bone
allometry to isometry. Accordingly, small deviations from the force-isotopic
GSM are described via the directly observable model-independent index $b$. The
differential scaling data for mammals (Christiansen 1999a, 1999b) supports the
physically justified inequalities $d>1/3>l$ (Kokshenev 2007) providing the
constraint $\lambda(b)<1$, resulting in $b<1/6$ following from equation (1).
The empirically established constraint $b_{\exp}^{(mam)}=0.04\pm 0.01$, for
overall-averaged long bones in mammalian limbs (see table 1 in Kokshenev
2007), results in the _model-independent_ pattern, namely

$d_{pre}^{(mam)}=0.37\pm 0.01\text{ and }l_{pre}^{(mam)}=0.29\pm 0.01\text{,
or }\lambda_{pre}^{(mam)}=0.785\pm 0.025$ (2)

predicted for mammals capable of true running with a fully suspended aerial
phase in the stride $[1]$.11footnotetext: It seems to be interesting to
compare the proposed empirical exponent
$\lambda_{\operatorname{mod}}^{(mam)}=0.785$, substituting McMahon’s
$\lambda_{0}=1/2$, with the theoretical estimate
$\lambda_{\operatorname{mod}}=7/9$ ($\thickapprox 0.778$) obtained on the
basis of a pattern of non-axial elastic forces in bone resisting functionally
relevant limb muscles (Kokshenev 2008). The bone-muscle scaling theory will be
discussed elsewhere.

As for McMahon’s models discussed in Introduction and revised by Kokshenev
(2003, 2007) and Kokshenev et al. (2003), they can be broadly interpreted as
follows. If both gravitational and muscular competitive forces driven by
structural bone adaptation to complex (axial and non-axial) compression were
equally important in bone interspecific allometry, the observed statistically
overall-bone slenderness exponent $\lambda_{\exp}$ is expected to be nearly
isometric, i.e., close to maximum $\lambda_{0}=1$, following from the GSM. If,
however, gravitational forces were dominating, bone proportions could be
expected to become optimized for exploitation of long bone stiffness during
modes of fast locomotion in mammals with near parasagittal limb kinematics,
and would result in $\lambda_{0}=2/3$, as predicted by the ESM, as predicted
by the ESM, also known as the buckling model. Although the ESM was originally
introduced for the states of elastic instability (McMahon 1973) or
thermodynamic instability (Kokshenev et al. 2003), the domain of resulting
scaling relations extends far below critical amplitudes of forces (and
critical stresses and strains), which should still exclude non-axial elastic
forces (Kokshenev 2007). In contrast, the adaptation to faster forms of
locomotion via more compliant long bones subjected to the non-axial
compression could be observed via the exponent $\lambda_{0}=1/2$, predicted by
the SSM, treated as a ”bending-torsion” model (Kokshenev 2007). As an example
of theoretical rationalization underlying these two distinct patterns of
bipedal locomotion in the sagittal plane, a transition from stiff-limbged
slow-walking to the compliant-limbged fast walking following by running was
illuminated in terms of a dynamic instability of the trajectory of center of
gravity in humans (Kokshenev 2004).

When muscle forces play a dominating role in the formation of bone
proportions, the appropriate modified model for bone evolution (hereafter
termed SSMM) predicts $\lambda_{pre}^{(bend)}=0.80\pm 0.03$, as derived from
both allometric mammalian limb bone and muscle data (see equation (18) in
Kokshenev 2003). Since this prediction is congruent with the one made in
equation (2), we infer that mammalian long bones are designed to resist peak
bending and/or torsional bone compressions produced by muscles during fast
running modes, that was initially established via bending functional bone
stress (Rubin & Lanyon 1982) and then explained analytically (see figure 1 in
Kokshenev 2007). For mammals as group, we therefore use the SSMM estimate
$\lambda_{\operatorname{mod}}^{(mam)}=0.785$ predicted semi-empirically in
equation (2) and theoretically in $[1]$.

###  II.2 Materials and analysis

We used previously published data on external limb long bone dimensions from
$19$ species and $217$ specimens of proboscideans (Christiansen 2007), which
was supplemented by new data collected for the purpose of this study. We
compared this to published data from $79$ and $98$ species of running mammals
from Christiansen 1999a and 1999b, respectively. We conducted regression
analysis on $Log_{10}$ transformed external limb long bone articular lengths
and diaphysial diameters using the standard _least squares_ (LS) and _reduced
major axis_ (RMA) methods. All species with multiple specimens were averaged
prior to analysis. The significances of the regression parameters were
evaluated by computing the correlation coefficient, the standard error of the
estimate, and the F-statistic of the regressions, and the $95\%$ confidence
intervals for the regression intercepts and slopes (see online electronic
supplementary material). The regression analysis included the entire
Proboscidea; the Elephantidae (Elephas sp., Loxodonta africana, and Mammuthus
sp.). We computed separate regression analyses for the extant Asian elephant
(Elephas maximus) and African savannah elephant (Loxodonta africana), since
these are, by default, the only taxa for which locomotory information exists;
we included no data from the forest elephant (Loxodonta cyclotis) within the
African elephant, because this taxon most likely constitutes a separate
species (Barriel et al. 1999).

##  III Results

Searching for generic morphometric patterns in long bones via external
dimensions, to which the similarity models (McMahon 1973, 1975a; Kokshenev
2003, 2007; Kokshenev et al. 2003) are broadly addressed, we study the
slenderness bone exponent $\lambda$ as directly observed by the slopes in
plots $Log_{10}L$ vs $Log_{10}C$. In table 1, the results of bone-size
regression analysis in species-averaged specimens of proboscideans are
compared with those for mammals.

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Species | Ele | phant | idae | Pro | bosci | deans | Ma | mmals |
---|---|---|---|---|---|---|---|---|---
Limb bones | $N$ | $\lambda$ | $r$ | $N$ | $\lambda$ | $r$ | $N$ | $\lambda$ | $r$
Humerus | 7 | 0.912 | 0.990 | 16 | 1.134 | 0.831 | 189 | 0.7631 | 0.9738
Radius | 6 | 0.813 | 0.853 | 10 | 1.078 | 0.878 | 189 | 0.7530 | 0.9957
Ulna | 6 | 0.727 | 0.888 | 14 | 0.929 | 0.866 | 189 | 0.849∗ | 0.9600
Femur | 7 | 0.747 | 0.966 | 14 | 0.802 | 0.816 | 189 | 0.8431 | 0.9763
Tibia | 6 | 0.751 | 0.925 | 11 | 0.772 | 0.857 | 188 | 0.7641 | 0.9499
Limb bone, LS | 6 | 0.790 | 0.924 | 13 | 0.943 | 0.850 | 189 | 0.795 | 0.971
Limb bone, RMA | 6 | 0.856 | 0.924 | 13 | 1.165 | 0.850 | 189 | 0.778 | 0.971

Table 1. The statistical data on the slenderness of individual and effective
limb bones in animals. The data for Elephantidae and Proboscidea are shown on
the basis of regression data provided in online electronic supplementary
material, extending table 2 in Christiansen 2007, and these for mammals are
taken from table 2 in Christiansen 1999b. The mean _slenderness exponents_
$\lambda=dLog_{10}L/dLog_{10}C$ presented by the slopes ($\lambda$)
exemplified in figure 1 are observed in $N$ _species_ through the LS
regression of with the _correlation coefficient_ $r$. The limb bone LS
characterization corresponding to the overall-bone mean data is introduced by
the standard mean over all 5 bones. The RMA data in the last row are shown
only for the resulting limb bone data. The bold numbers are the data used
below in figures. The italic numbers indicate the slope data contrasting to
mammalian data with $\lambda<1$ (see also discussion in section 2.1 in the
Methods). ∗) The data estimated with the help of ratio $l/d$ for the ulna
allometric exponents taken from table 2 in Christiansen 1999a.

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In figure 1 and table 2, we analyze modern elephants.

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Place figure 1

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Species | E. | maxi | mus | Loxo | donta |  | afri | cana
---|---|---|---|---|---|---|---|---
Limb bones | $n$ | $\lambda$ | $r$ | $n$ | $\lambda$ | $r$ | $p_{\min}$ | $p_{\max}$
Humerus | 22 | 0.754 | 0.985 | 14 | 0.616 | 0.978 | 0.01 | 0.02
Radius | 19 | 1.014 | 0.987 | 8 | 0.675 | 0.994 | — | 0.001
Ulna | 20 | 0.818 | 0.981 | 11 | 0.644 | 0.979 | 0.01 | 0.02
Femur | 25 | 0.758 | 0.972 | 13 | 0.618 | 0.986 | 0.01 | 0.02
Tibia | 20 | 0.913 | 0.970 | 10 | 0.688 | 0.939 | 0.02 | 0.05
Forelimb bone | 20 | 0.862 | 0.984 | 11 | 0.645 | 0.984 | 0.01 | 0.02
Hindlimb bone | 23 | 0.836 | 0.971 | 12 | 0.653 | 0.963 | 0.015 | 0.035
Limb bone, LS | 21 | 0.851 | 0.979 | 11 | 0.648 | 0.975 | 0.013 | 0.028
Limb bone, RMA | 21 | 0.869 | 0.979 | 11 | 0.665 | 0.975 | 0.013 | 0.028

Table 2. The statistical data on $Log_{10}L$ vs $Log_{10}C$ regression for
individual limb bones in Elephas maximus and Loxodonta africana. Notations of
table 1 are extended by the t-test comparisons of slopes in $n$ _specimens_
shown by $p_{min}<p<p_{max}$ (for details see the electronic supplementary
material). Characteristic of a given group of species _forelimb_ and _hindlimb
bones_ are introduced through the fore-bone (humerus, radius, and ulna) and
hind-bone (femur and tibia) standard means, respectively. The effective _limb
bone_ is determined by the overall (5-bone) standard mean. Other notations are
the same as in table 1.

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The main results obtained by the LS regression in tables 1 and 2 are displayed
and analyzed in figure 2.

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Place figure 2

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In figure 2, the model predictions for bone slenderness exponent are compared
with those of the entire group of proboscideans, Elephantidae, mammals, and
modern elephants. As seen from the data presented by the bone-averaged
exponents (in table 2) resulting from the LS and RMA species-average
statistics (shown by bars), the limb bones of the family Elephantidae are
structurally designed likewise those in mammals. The data for extant Elephas
maximus are also quite similar to mammals, whereas the bone exponents in
Loxodonta africana are distinctly lower (see also figure 1). This implies that
the data for Elephas maximus as well as the family Elephantidae, are better
explained by adaptations to peak muscular forces during locomotion, whereas
the limb bones in Loxodonta africana are indicative of adaptation to cope with
the forces of gravity $[2]$.22footnotetext: Our sample of Elephas maximus has
more large juveniles included than in Loxodonta africana, and all resulting
slopes are generally higher than in Loxodonta africana (table 2). Comparing
only adult specimens, therefore ignoring ontogenetic adaptations, the radius
and femur slopes remain significantly lower in Loxodonta africana than Elephas
maximus, whereas the exponents for humerus, ulna and tibia become non-
significantly different (see online supplementary material). In Loxodonta
africana, all exponents for adult specimens only are similar to the full
sample including large juveniles, whereas the slopes in Elephas maximus become
significantly higher (femur) and lower (tibia). Overall, the LS-average
slenderness exponent in adult Elephas maximus ($\lambda_{\exp}=0.839$) remains
significantly higher ($p<0.05$) than in adult Loxodonta africana
($\lambda_{\exp}=0.662$). The Elephantidae and individual species within this
family also have thinner long bones than more primitive proboscideans (Haynes
1991; Christiansen 2007).

.

Place figure 3

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In figure 3, we examine the allometry exponents of individual bones, with the
aim to interpret their adaptation to the patterns of peak elastic forces and
stresses. Within the Elephantidae, the similarity in variation of bone
slenderness in those three groups is very evident from the parallel lines, as
shown in figure 3. When the mechanical origin of the model predictions
provided in Methods is taken in consideration, the limb bones in Elephas
maximus indicate adaptation to complex muscular and, in part, gravity
stresses.

It is well established in comparative zoology that the humerus of running
parasagittals is loaded differently owing to large muscle attachments and an
inclined angle compared to the epipodials, but in elephants the humerus and
femur are not very steeply inclined. This is broadly congruent with the
observation in figure 3 that the femur, which being almost vertical when the
animal stands motionless and much more inclined in running mammals than in
fast moving elephants, should involve lower bending and torsional moments in
Elephas maximus than in other mammals and much less muscular moments in
Loxodonta africana. Indeed, as predicted by the ESM beyond experimental error,
the limb bones in Loxodonta africana appear to be adapted for axially
compressive stress generated by gravity. Such a distinction in allometry
slopes induced by mechanical adaptation, clearly distinguishes Loxodonta
africana from Elephas maximus, which, in turn appears to bear a mechanical, if
not morphological similarity to the limb bones of mammals capable of true
running. Accordingly, the limbs of Elephas maximus would appear to be more
adapted for resisting the forces from limb muscles, which broadly is more
consistent with faster forms of locomotion.

As seen in figure 3, the radius and femur in modern elephants expose different
loading trends of those in mammals. A consequent distinct mechanical
characterization of the hindlimbs and forelimbs is displayed in figure 4.

Place figure 4

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In figure 4, the differently designed forelimb and hindlimb bones are
schematically shown through the hind-fore-limb bone vector, indicating the
existence of a gradient in the limb bone functions, likewise gradients in
muscle functions in mammals (Biewener et al. 2006) increasing the stability of
running (Daley et al. 2007). One can see that the Elephantidae contrasts to
mammals in limb bone functions.

##  IV Discussion

###  IV.1 Surveying kinematic empirical data

Recent studies have provided new insights and have significantly enhanced our
understanding of elephant locomotion. Typically called graviportal, elephants
are unable to run with a suspended phase in the stride, and even during fast
locomotion, one limb is placed firmly on the ground (Gambaryan 1974). New
studies have, however, indicated that fast moving elephants are not merely
walking, but that kinetics and kinematics during fast locomotion differ from
walking, and numerous locomotor parameters are similar to those of running
mammals, for instance offsetting the of phase of forelimb and hindlimb
footfalls; achieving the transient walk-run duty factors $0.5$; and
maintenance of pendular locomotor kinematics, typical of walking gaits, for
the forelimbs, whereas the hind limbs move with a more spring-like (bouncing)
action (Hutchinson et al. 2006; Ren & Hutchinson 2007). Accordingly, at high
speeds elephant locomotor kinematics are indicative of walking gaits whereas
kinetic analyses indicate running, as in other quadrupedal mammals. The
primary differences are that elephants never achieve a full-body aerial phase
during any form of locomotion, although at speeds of just over $\char
126\relax 2$ $ms^{-1}$, the hind limbs begin exhibiting an aerial phase with
bouncing kinetics, implying that the limbs become compliant, whereas the
forelimbs maintain a more straight morphology, consistent with more vaulting
mechanical properties (Ren & Hutchinson 2007). Another significant difference
from running, quadrupedal mammals, there is no marked change of gait at high
speeds, even at Froude numbers $Fr>3$, a value when other quadrupedal animals
have changed gaits to a bouncing, running gats with a full-body aerial phase
(Alexander 1983, 1989; Alexander & Jayes 1983).

During progressively faster locomotion, elephants initially increase speeds
primarily by increments in stride frequency, but at high speeds, further speed
increase is facilitated primarily by increments in stride length (Hutchinson
et al. 2006). This slow-to-fast gait transition is similar to walking-trotting
transition in quadrupedal running mammals, where increments in speed is a
function of those in both stride length and frequency, whereas in the case of
running gaits with bouncing limb kinetics and a full-body aerial phase is
characterized primarily by increments in stride length (e.g., Heglund et al.
1974; Pennycuick 1975, Biewener 1983; Alexander 1983, 1989; Alexander & Jayes
1983). Locomotor kinematic parameters in Asian and African elephants are
broadly similar, but statistically significant differences exist, pertaining
to relative stride lengths, stride frequencies, stance phase, and duty factor
with speed. African elephants have higher duty factors, shorter stride lengths
and higher stride frequencies than Asian elephants (Hutchinson et al. 2006).
Interestingly, large elephants, such as full grown bulls, appear incapable of
reaching the same locomotor intensity as small elephants, and have duty
factors $\beta<0.5$, implying that no limb pair ever exhibits an aerial phase;
they are, in effect, only walking during fast locomotion. Consequently, this
observation may imply that large individuals may exploit the compliance of
bone tissue to a lesser extent.

###  IV.2 What does the overall-bone statistics tell us?

The theoretical concepts provided in the Methods section permits an
interpretation of the limb bone allometric data in terms of the adapted
loading patters. Beyond any modeling, the regression analysis illustrated in
figure 1 indicates that the individual limb bones in distinct extant elephants
are likely similar in bone ontogeny but they are evidently different in their
mechanical adaptation as revealed by distinct slopes in size proportions. When
known theoretical models are employed in terms of the overall-bone slenderness
exponent (in figure 2), one can see that the limbs in Elephas maximus, as well
as in Elephantidae, show a similarity to the limbs of other mammals, broadly
exploiting muscular forces during efficient locomotion. This observation
agrees with in vivo data on the elephant locomotor kinematics having generally
many patterns in common with typical tetrapods (e.g., Hutchinson at al. 2006).
In contrast, our analysis show that Loxodonta africana successfully employs
body gravitation and reaction gravitation forces for efficient walking. More
specifically, the analysis in figure 2 indicates that the overall-bone
averaged slenderness allometry exponent associated with the structure of an
effective limb bone (defined in table 2) can be understood in mammals and the
Elephantidae (but not intraspecifically in Loxodonta africana), by its
adaptation to peak muscular forces generated during fast locomotion, whereas
the limb bones in Loxodonta africana are indicative of adaptation to cope with
the forces of gravity more successfully exploiting in walking. This finding
also implies that both propodial and epipodial limb bones in mammals are
established to be adapted for peak functional bending and torsional stresses
(figure 3) by the exploiting of bone compliance, contrasting to more stiffer
limbs in Asiatic elephants, as predicted by the ESM (figure 4).

When Elephas maximus and Loxodonta africana are compared, no contrasting
adaptation of any individual limb bone is revealed in figure 3, because all
bone lines are parallel. This is not the case of lines lying between mammals
as group and elephants. This observation suggests a similarity between bone
joint angles generally established for modern elephants during fast locomotion
by direct observations, e.g., by Ren et al. (2007). Consequently, bone angles,
which are evidently similar in elephants, and distinct from other mammals,
indicate differing loading conditions related to limb postures in elephants,
extant and extinct, from those of other mammals. More specifically, the
epipodial femur, shows in figure 3 its adaptation to complex compression
(bending and/or torsion) during bouncing kinematics of the hindlimb involved
in fast locomotion, thereby exploiting rather the bone compliance, than the
bone stiffness associated with more isometric bone proportion scaling exposing
by the forelimb radius, in Elephas maximus and forelimb humerus in
Elephantidae, thereby contrasting to parasagittal femur and radius. Such
mechanical trends are consistent with the kinematic data (Hutchinson et al.
2006) that the hind limbs of modern elephants during fast locomotion are
broadly more compliant than the fore limbs. This empirical finding in modern
elephants is now generalized over extinct elephants.

The contrasting postures between mammals as group and elephants can be
understood by the different design of fore limbs and hind limbs revealed in
figure 4 by the opposed directions of the gradient of the limb functions in
locomotor kinematics. As for the small negative gradient in Loxodonta africana
with respect to Elephas maximus, it can be ignored, because the typical
statistical error (shown by the bars) exceeds the length of the limb bone
vector. In other words, the crossover of the bone lines between the modern
elephants can be referred to a small statistical uncertainty that can be
ignored, providing a qualitative agreement with the overall similarity in limb
postures revealed in figure 3. On the contrary, the observation in figure 4 of
the gradient of limb functions for the Elephantidae exceeding the statistical
error may suggest a trend for forelimb bones to be more isometric, and also
that the forelimb bones in Elephas maximus are therefore stiffer than bones in
the hind limbs. This also is congruent with the differences in limb locomotor
kinematics (Hutchinson et al. 2006), outlined above.

###  IV.3 Positive gradient of limb stiffness as major locomotor constraint in
elephants

Our study of the allometry of individual limb bones reveals different patterns
in limb mechanical adaptation in proboscideans and begs the question how they
correlate with kinematic patterns characteristic of modern elephants? Such a
correlation is expected, since the mean data on duty factor $\beta$ in limbs
of modern elephants (Hutchinson et al. 2006) are underlaid by the overall-bone
(and overall-muscle) limb characterization described here though the limb bone
slenderness exponent $\lambda.$

Near the walk-run transition in mammals, with the Froude number
$Fr\thickapprox 1$, the scaling predictions for the forelimb duty factor
$\beta_{FL}=0.52$ and the hindlimb duty factor $\beta_{HL}=0.53$ (estimated on
the basis of empirical scaling relations by Alexander & Jayes 1983), provide
negative _limb duty factor gradient_ $\Delta\beta$
($\equiv\beta_{FL}-\beta_{HL}$) for mammals, contrasting with the positive
gradient $\Delta\beta_{ele}$ for elephants (see table 6 in Hutchinson et al.
2006). These data can be related to our LS bone exponent $\lambda_{FB}=0.788$
in mammalian forelimb and $\lambda_{HB}=0.804$ in mammalian hindlimb,
corresponding to the mammalian overall-bone exponent
$\lambda_{\exp}^{(mam)}=0.795$ (table 1) and the negative _limb bone gradient_
$\Delta\lambda_{\exp}^{(mam)}=-0.016$ (with,
$\Delta\lambda\equiv\lambda_{FL}-\lambda_{HL}$). For elephants, the duty
factor gradient $\Delta\beta>0$ also correlates to the bone gradient
$\Delta\lambda>0$ (figure 4). Indeed, as follows from tables 1 and 2,
$\Delta\lambda_{\exp}^{(ele)}=0.026$, for Elephas maximus, and
$\Delta\lambda_{ele}=0.068$, for Elephantidae. As for the discrepancy in signs
between $\Delta\beta_{\exp}^{(ele)}=0.026$ (table 4 by Hutchinson et al. 2006)
and $\Delta\lambda_{\exp}^{(ele)}=-0.008$ for the Loxodonta africana, it was
referred above to the statistical error.

Employing the similarity in limb functions and their gradients observed
directly (kinematically) in modern elephants and indirectly (allometrically)
via limb bones of extinct and extant elephants, we develop a simple linear
model predicting limb duty factors for Elephantidae, which includes more
primitive groups of proboscideans (see the electronic supplementary material).
The linearization procedure of the mean limb data and their gradients known
for living elephants (shown, respectively, by the dashed lines in figure 5 and
its inset) results in the duty factor predictions for Elephantidae, as
explained in figure 5.

.

Place figure 5

.

The geometrical visualization of locomotory constraints imposed on animal
limbs in a certain locomotion mode can be displayed on a $\lambda$-$\beta$
diagram presented by figure 5. Such a characterization makes a link between
the limb bone and limb bone gradient bone proportions (figure 4) and the limb
and limb gradient kinematics (figure 4 by Hutchinson et al. 2006).

The major limb functional difference in mammals as a group and elephants
indicated by different orientation of the characteristic vectors shown in
figure 5 is due to the difference in signs of the limb stiffness-compliant
gradient transferred between fore and hind limbs during the animal’s forward
propulsion of the body. Running mammals, having hindlimbs which are stiffer
than the forelimbs and therefore transfer the positive compliance limb
gradient (or negative stiff limb gradient), but they are able to achieve a
full-body aerial phase during fast locomotion. In contrast, elephants,
transferring negative limb compliance gradient (hindlimb bones are more
compliant than forelimb ones) do not achieve a full-body aerial phase during
any form of locomotion, though are able to sufficiently reduce the positive
stiff limb bone gradient by limb muscles when showing a negative limb duty
factor gradient $\Delta\beta$ during both slow and fast walking gaits (see
figure 4B by Hutchinson et al. 2006). However, it is not enough for changing
of the positive direction of the limb gradient vector in the $\lambda$-$\beta$
diagram, contrasting to mammals (figure 5), since elephants, being naturally
constrained in limb bone proportions, are not able to change sign of the
positive gradient in limb bone slenderness $\Delta\lambda$. Hence, the
preserved excessive positive hindlimb-forelimb stiffness gradient ensured by
the corresponding mass-independent and speed-independent positive limb bone
slenderness, explains the inability of elephants to perform true running with
a full-body aerial phase discussed by Hutchinson et al. 2003. Consequently, in
order to move fast they increase stride frequency, mostly exploiting forelimb
stiffness, instead of compliance, and in order to increase the stride length,
they are forced to use hindlimb bone compliance, instead of stiffness.

###  IV.4 Asian compared with African elephants

Being similar in fast locomotion gaits with respect to the lateral sequence
footfall pattern, the Asian, African and most likely extinct elephants (as
predicted figure 5) are found to differ in limb bone and perhaps also muscle
constraints. All having similar projections of the characteristic vector in
the $\lambda$-$\beta$ diagram, the limb bone stress indicated by the bone
slenderness for the African elephant is significantly distinct from that in
other elephants. According to the SSMM, in the Elephantidae and in particular
in Elephas maximus, the bone off-axial external muscle forces generated during
fast locomotion, broadly exceeding body weight and causing a complex bending-
torsion elastic bone stress, provide a relatively high level of limb
compliance conducted by the structurally adapted limb long bones. In contrast,
limb bones in Loxodonta africana generally adapted for axial bone compression,
are most likely tuned by limb muscles to employ better gravitation reaction
forces, in accord with relatively low bone slenderness, explained by the ESM.

Having long bones designed to maintain axial stress and avoiding bending and
torsion, African elephants can be expected to exhibit shorter stride lengths
and therefore to use higher stride frequencies than Asian elephants, at
increased locomotor speeds. From an energetic point of view, this implies the
higher energy cost of the Asian elephant locomotion, whereas higher duty
factors characteristic of African elephants (figure 5) indicate less bending
moments about the joints accommodated ground reaction forces. We infer that
Asian elephants, having more compliant limb bones than African elephants, are
broadly able to maintain higher speeds more easily that is statistically
supported by the observation (via $\beta<0.5$) of Asian elephants (figure 4A
by Hutchinson et al. 2006). Even the the limb duty factors in Asian and
African elephants may achieve those in mammals, the negative gradient for the
limb bone compliance limits the maximum stride length and therefore the
maximal running speed with respect to mammals.

It is traditionally believed that mechanical differences must be large to
produce differences in bone morphology (Frost 1990), but more recent studies
have demonstrated that temporal continuous stimulation three orders of
magnitude below the maximal peak forces characteristic of fast locomotion (see
Rubin & Lanyon 1982) is likely sufficient to produce significant changes in
bone morphology (Rubin et al. 2001). African elephants are typically found in
open environments and routinely undertake long-distance seasonal migrations at
leisurely paces, whereas Asian elephants are mostly found in topologically
more heterogeneous, forested environments and appear to undertake fewer and
shorter, if any, seasonal migrations (Sikes 1971; McKay 1973; Laws et al.
1975; Sukumar 1991, 1992). Potentially, this could imply subtle differences
even in low-force every-day locomotor mechanics imposed by the structure of
the environment. Thus, differences in ecology and migratory activity may
conceivably be responsible for the subtle differences in locomotor mechanics
between African and Asian elephants, as observed by Hutchinson et al. (2006).
In this study, more pronounced differences in allometric and mechanical
patterns are demonstrated for Elephas maximus, which appear more similar to
those in other mammals, and Loxodonta africana, which is divergent from both
and also from the Elephantidae.

On the other hand, neither allometric nor kinematic or kinetic studies deal
with forces and bone stresses directly. It remains therefore a challenge to
further analyze the reactive-force elastic-stress patterns revealed for limb
bones in extant elephants. Nevertheless, there is another difference between
the two species of extant elephants congruent with our findings. Because the
limb bone pattern for African elephants indicates axial bone stress, not
increasing with body mass (Rubin & Lanyon 1982, 1984) and therefore would
constitute non-critical stress (Kokshenev 2007), both the mean and maximal
body masses for Asian elephants are expected to be below of those for African
elephants. Indeed, African elephants appear to be larger on average than Asian
elephants; African elephant large bulls routinely weight $5-7$ tons, whereas
$4-5$ tons is more common for Asian elephant bulls (Wood 1976; Shoshani 1991).
Tentative maximal size of African elephant also appears to be distinctly
larger. World recorded bulls are as close to or even exceeding 4 m in standing
shoulder height, and estimated body masses of $10-12$ tons (Wood 1976;
McFarlan 1992), whereas Asian elephants are estimated at $3.3-3.4$ m in
shoulder height and around 8 tons (Pilla 1941; Wood 1976; Blashford-Snell &
Lenska 1996).

.

Acknowledgments

One of the authors (V.B.K.) acknowledges financial support by the CNPq.

.

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Figure Captions

.

Figure 1. Long bone articular lengths against diaphysial least circumferences
in extant elephants. A, humerus; B, ulna; C, femur; D, tibia. Closed squares
are Elephas maximus; open squares are Loxodonta africana. Regression
coefficients are shown in table 2.

.

Figure 2. A comparison of the predictions by the theory of similarity with the
bone slenderness exponents observed via the regression slopes in different
groups of proboscideans. Notations: open circles are McMahon’s predictions for
bones adapted for the influence of gravity ($\lambda_{0}=1,2/3$, and $1/2$ due
to GSM, ESM, and SSM); the closed circle shows mean data
$\lambda_{pred}=0.785$ predicted for the limb bone, which is primarily adapted
for resisting peak muscle forces during locomotion [discussed in equation (2)
]. The bars are the mean LS and RMA data for the limb bone characteristic of
Proboscidea, Elephantidae, and mammals taken from table 1 and these for
Elephas maximus and Loxodonta africana, taken from table 2. The error bars
show statistical variations between the means of regression data.

.

Figure 3. The observation of trends in limb bone mechanical adaptation in
different species. The bars show maximal variations of the mean exponents of
individual limb bones. The slenderness exponents in humerus (H), radius (R),
ulna (U), femur (F), and tibia (T) are analyzed in view of elastic similarity
models. The notations on the symmetry of bone compression follow from the
models described in Methods. Other notations correspond to those in figure 2.

.

Figure 4. Observation of elastic similarity in the effective forelimb bone
(humerus, radius, and ulna) and hindlimb bone (femur and tibia). The arrows
indicate deviations in the trends of adaptation for forelimb and hindlimb
mechanical functions. The bar shows statistical error. Other notations
correspond to those in fig. 3.

.

Figure 5. Limb bone scaling in mammals and elephants against limb kinematics
in fast walking. The vector positions and magnitudes indicate the slenderness
exponent and duty factor and the vector directions indicate their forelimb-
hindlimb gradients. The dashed vector position predicts the limb duty factor
$\beta_{pre}=0.58$ consistent with $\lambda_{\exp}=0.790$ for Elephantidae
obtained by liner interpolation between the kinematic data for Asian and
African elephants, as shown by the thin dashed line. The corresponding the
model duty factor gradient $\Delta\beta_{\operatorname{mod}}=0.043$ is found
in the inset, through the linear extrapolation (shown by the dashed line) of
the gradient known for Elephas maximus (blue point) and
$\Delta\lambda_{mod}=0.008$ adopted for the Loxodonta africana (green point).
Other data are provided above and/or taken from table 4 by Hutchinson et al.
2006.