Scaling of rotational inertia in murine rodents and two species of lizard
Department of Biology, 201 South Biology Building, University of Utah, Salt Lake City, UT 84112, USA
* e-mail: walter{at}biology.utah.edu
Accepted 3 May 2002
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Summary |
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Key words: agility, manoeuvrability, moment of inertia, locomotion, lizard, rat, mouse, Mus domesticus, Rattus norvegicus, Iguana iguana, Varanus exanthematicus
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Introduction |
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A possible locomotor advantage of some body forms and limb configurations
is an increase in the ability to turn sharply and to maneuver through
non-uniform terrain. Turning is often important in predatorprey
interactions, intraspecific competition and movements through the complex
terrain that most animals inhabit, so it seems likely that selection would act
to improve this parameter of locomotion. For example, Djawdan and Garland
(1988) point out that the
highly erratic zig-zagging predator-avoidance behavior of kangaroo rats
Dipodomys might allow them to inhabit more open microhabitats, where
the enhanced risk of predation is prohibitive for similar species with more
predictable escape behaviors. Although animals use different turning
strategies depending on their body configuration
(Eilam, 1994
;
Kafkafi and Golani, 1998
;
Jindrich and Full, 1999
),
certain phenotypes would probably result in greater turning performance.
One factor that has been shown to have a strong influence on turning
ability is rotational inertia (Lee et al.,
2001). Rotational inertia (I) is a body's resistance to
rotation about an axis. It is defined as the differential elements of a body's
mass (m) multiplied by the square of their distances from the
rotational axis (r) (I=
mr2)
(Halliday et al., 1993
).
Because the distance of the mass from the rotational axis is squared, bodies
whose mass is distributed far from the axis of rotation have very high
rotational inertia. The torque (
) required to achieve a given magnitude
of rotational acceleration (
) about an axis is directly proportional to
the body's rotational inertia (
=I
) about the axis. Hence,
one might expect natural selection to favor characters that reduce the mass of
elements far from the axis of rotation or reposition body mass closer to the
rotational axis. For example, the reduction in trunk and tail lengths that
occurred during the evolution of basal synapsids to cynodonts and mammals
clearly must have significantly reduced rotational inertia.
Although it is obvious that modern mammals have greatly reduced rotational
inertia compared with basal synapsids (Fig.
1), the magnitude of this decrease is not clear. The present study
attempts to estimate this reduction by comparing the rotational inertia of
murine rodents (similar in body form to the early mammal
Megazostrodon) with that of iguanid and varanid lizards (resembling
basal synapsids such as pelycosaurs). Skeletal restorations of pelycosaurs
such as caseids, ophiacodontids, edaphosaurids and sphenacodontids show
`lizard-like' body forms (Carroll,
1997; Fig. 1).
These animals had a long trunk and tail and a sprawled limb posture. Although
pelycosaurs were much larger than the lizards being used in this study, with
some caseids approaching 3 m in length
(Benton, 1997
), this size
difference can be addressed with the assumption of geometric similarity.
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A further consideration of the present study is the scaling of angular
acceleration with body size. If rotational inertia increases more quickly with
body size than the ability to apply torque, then larger animals will be poorer
turners than smaller animals. In geometrically similar animals, rotational
inertia scales with mass1.67 (I=mr2,
where r is a length along the body and scales with
mass0.33). The ability to apply torque, however, is directly
proportional to mass (=Fd, where muscle force, F,
scales with mass0.67 and distance of application, d,
scales with mass0.33). This means that, in geometrically similar
animals, angular acceleration (
=
/I) decreases with
increasing mass (
m-0.67), and one would
expect poorer turning performance in larger geometrically similar animals than
in smaller ones (Carrier et al.,
2001
). For this reason, one might expect natural selection to have
resulted in lower rotational inertia in large species and growth stages than
would be predicted by geometric similarity.
Here, we test the hypothesis that rotational inertia exhibits negative allometry by measuring the rotational inertia of a size range of murine rodents (Mus domesticus and Rattus norvegicus) and lizards (Iguana iguana and Varanus exanthematicus). These measurements also allow us to estimate the decrease in rotational inertia that occurred during the evolution of mammals from the `lizard-like' basal synapsids.
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Materials and methods |
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To obtain size ranges of iguanid and varanid lizards, we used ontogenetic series. This was performed mainly for practical reasons of specimen availability. However, given that these lizards must forage and escape predators independently from hatching, it is reasonable to expect selection on rotational inertia throughout ontogeny. In contrast, to obtain a size range of murine rodents, we used adult mice and late juvenile and adult rats. Being altricial, very young mice and rats do not actively forage or run from predators and, thus, are unlikely to experience similar selection for turning agility. Because the mice and rats are in the same sub-family (Murinae) and have similar life-styles and body morphologies, they were used to create a single scaling relationship, while the varanid and iguanid lizards were analyzed separately.
Rotational inertia was measured in 13 murine rodent carcasses ranging in mass from 25 to 546 g, including three adult mice and 10 juvenile and adult rats. For comparison, rotational inertia was measured in 17 green iguanas (I. iguana) ranging from 7.5 to 1664 g. Of these, 12 were carcasses, while five of the larger iguanas were anesthetized during measurements. Because small iguana specimens were readily available, rotational inertia values from two 9 g iguanas were averaged, as were those from four 10 g iguanas, to give a total of 13 data points. Rotational inertia was also measured in 11 savannah monitors (V. exanthematicus) ranging in mass from 18 to 2497 g. Three of the larger varanids were anesthetized with isoflurane; the others were frozen specimens.
Determination of center of mass and rotational inertia
Animals were weighed and then placed on a beam supported at one end by a
metal stand and at the other by the scale. Knowing the distance between the
scale and the metal stand (D1), the scale reading
(Sc) and the actual mass of the animal (m), we determined
the distance of the animal's center of mass from the stand
(D2) and thereby its position on the animal
(Nigg and Herzog, 1999):
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Murine rodent and lizard specimens were first frozen with their trunk and tail straight and their limbs in trotting postures, whereas live lizards were anesthetized during all measurements. This positioning with the trunk straight would tend slightly to overestimate the rotational inertia of a running animal because both lizards and rodents bend their trunk and tail during locomotion. Measurements consisted of locating the center of mass as described above and allowing specimens to oscillate as pendulums about two swing points (Fig. 2). Anesthetized lizards were weighed, and their center of mass was determined both before and after they had been braced in trotting postures using a light wooden support structure. The frame, to which the lizards were firmly taped (the tape peeled off their scales easily without causing apparent damage), weighed 30 g. Its rotational inertia was measured by oscillating it separately and was subtracted from the total rotational inertia.
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Swing points consisted of two holes drilled on a dorsoventral axis through each animal or through the support for anesthetized animals. Holes were placed along the central axis in either the head or tail region so as to locate them as far as possible from the center of mass, thereby improving the pendulum's swing. Animals were then allowed to oscillate as a pendulum about an axle through each of the holes. A reflective marking was placed as far as possible from the axis about which the animal oscillated, and the position of this marker was videotaped at 120 Hz for several oscillations (Camera 007086 Peak Systems, Inc). These positional data were used to determine the period of oscillation and thereby the rotational inertia about the axis of oscillation (IOA).
The parallel axis theorem (Halliday et
al., 1993):
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Statistical analysis
Rotational inertia about the center of mass was computed separately for
each of the two axes about which each animal oscillated, and the values were
averaged. A similar method of calculating moment of inertia from oscillations
about two axes, and averaging the moments for increased accuracy, was used by
Fedak et al. (1982). In cases
where the two values, which should theoretically be identical, differed by
more than 10 %, measurements were repeated.
Possible error in the measurement of rotational inertia could result from friction about the axis of oscillation. We attempted to minimize this by spraying the axle with WD-40. Measurement error could also result from partial thawing of the narrow tail of the mice and smaller lizards during oscillations. We attempted to minimize this error by keeping animals in the freezer until just prior to measurement and refreezing them between the two measurements when thawing was apparent.
The averaged rotational inertia values and the subject's masses were
log10-transformed. Allometric scaling relationships of mass
versus rotational inertia were calculated using reduced major axis
slopes (Sokal and Rohlf,
1997). To determine whether they differed significantly from
geometric similarity or from one another, 95 % confidence intervals for both
the slopes and the intercepts were computed.
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Results |
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Over the size range measured, the murine rodents had a 3- to 4.9-fold lower rotational inertia than the either of the lizard species (Fig. 3). The igaunas had a 1.2- to 1.3-fold greater rotational inertia than the varanids.
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Discussion |
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As the size range of rats represents an ontogenetic series, and mammals are
known to exhibit allometric growth (Gould,
1977; Carrier,
1996
), we would not necessarily expect to see geometric similarity
in the scaling of either rotational inertia or torque production. Studies
showing that juveniles of some species have relatively longer limbs and
greater acceleration than adults (e.g. Carrier,
1995
,
1996
) suggest relatively
greater ability to produce torque in the juveniles of these species. Thus,
using geometric scaling for ability to apply torque represents a conservative
argument; angular acceleration may decrease more quickly with increased mass
in some species than predicted above.
Given that agility and maneuverability on variable terrain often appear to
be important for survival, we had originally expected rotational inertia to
scale with significant negative allometry. Since body shape varies greatly
among mammalian species, it seems unlikely that developmental constraints
restrict murine rodents to geometric similarity in rotational inertia.
Nevertheless, several explanations can be proposed for the
higher-than-predicted scaling of rotational inertia in murine rodents. First,
competing selective factors affecting body shape might prevent a negative
departure from geometric similarity. For instance, as R. norvegicus
originally inhabited burrows with long branching tunnels
(Nowak and Paradiso, 1983),
the need to maintain a low and narrow profile to move easily through these
tunnels may have prevented the evolution of body allometries that reduce
rotational inertia in larger individuals. Second, because the rodents used in
this study were relatively small in comparison with many terrestrial mammals,
rotational inertia may not increase enough over this size range to be a
limiting factor in running turns. An animal's ability to turn sharply while
running is dependent upon the rate at which it can decelerate in the direction
of its initial heading and accelerate in the new direction and on the rate at
which it can rotate to face a new direction
(Jindrich and Full, 1999
). It
is not yet known which of these factors, linear acceleration and deceleration
or rotational acceleration, limits turning performance in quadrupeds. Because
angular acceleration is expected to decrease more quickly with increasing mass
(scaling as mass-2/3) than does linear acceleration (scaling as
mass-1/3), it seems likely that angular acceleration, and thus
rotational inertia, would become more of a limiting factor in turning agility
in larger animals. Even the largest rats measured were quite small compared
with the full size range of terrestrial mammals. It may be that, in the
measured size range, linear acceleration and deceleration abilities place
stricter limitations on running turns than does rotational acceleration, while
in much larger mammals, in which the exponential term has a greater impact on
acceleration, angular acceleration acts as the limiting factor.
Scaling of rotational inertia in iguanid and varanid lizards
In lizards, scaling relationships have been calculated for a variety of
morphometric parameters including snoutvent length, limb lengths and
limb diameters (White and Anderson,
1994; Kramer,
1951
; Marsh, 1988
;
Laerm, 1974
;
Dodson, 1975
;
Pounds et al., 1983
;
Christian and Garland, 1996
).
These studies show that, while lizards exhibit growth that is much more
isometric than that of mammals, their growth is allometric. Therefore, it is
not immediately apparent that rotational inertia should scale with geometric
similarity. Further, these studies show much interspecific variation in
scaling relationships even within families such as Iguanidae
(Pounds et al., 1983
).
While further experiments and modeling would be necessary to determine
exactly how various reallocations of mass throughout growth would affect the
scaling of rotational inertia, some predictions can be made. For instance, if
muscle mass increased relative to trunk and appendage length during growth,
then one might expect the decreased rotational inertia of the relatively
shorter and stouter body forms of larger lizards to yield negative allometry.
Longer or more massive forelimbs should increase rotational inertia about the
lizards' relatively caudal center of mass, whereas increasing the length of
the hindlimbs (which are much closer to the center of mass), as seen in larger
species of Sceloporus and Varanus
(Dodson, 1975;
Christian and Garland, 1996
),
should have less effect on rotational inertia. Increasing hindlimb diameter
(rather than locating the mass elsewhere), as in Diposaurus dorsalis
and Varanus spp. (Marsh,
1988
; Christian and Garland,
1996
), might decrease rotational inertia.
Another possible effect of the scaling of limb lengths and diameters on the
scaling of maximal angular acceleration is the relative ability of various
limb morphologies to apply torque. Applied torque, which is equal to the force
times the lever arm at which it is applied, would be relatively greater in
larger lizards if both limb length and muscle cross-sectional area scaled with
positive allometry. Christian and Garland
(1996) found such positive
allometry in a comparison of 22 species of adult varanid. Although savannah
monitors have a relatively short and stout body form for a varanid lizard, if
the ontogenetic scaling of their limb lengths and diameters were similar to
the interspecific scaling in adult varanids, then angular acceleration would
decrease much less dramatically with mass than predicted by geometric
similarity.
Scaling relationships of hindlimb parameters vary among species of iguanid
lizard. In groups such as Diposaurus dorsalis
(Marsh, 1988) and in two
species of Basiliscus (Laerm,
1974
), in which hindlimb lengths scale with negative allometry,
the ability to apply torque would scale much lower than in the varanids
(closer to or less than mass1.0). Thus, the increase in rotational
inertia with increasing mass would not be partially compensated for by
positive allometry in ability to apply torque. It is interesting that, in
I. iguana, rotational inertia does scale with slight, but
significant, negative allometry, unlike in V. exanthematicus. Still,
a much greater negative allometry in the scaling of rotational inertia would
be necessary to prevent a decrease in angular acceleration as iguanas increase
in mass. Other factors could prevent modifications that reduce rotational
inertia in larger iguanas. For instance, iguanas use their tail for swimming
and as a weapon (Schmidt and Inger,
1957
). Thus, while decreasing tail size would probably improve
turning ability, it might have greater costs in other aspects of
performance.
In summary, it seems that, although V. exanthematicus and I.
iguana both deviate from isometric scaling in ways that ameliorate the
expected reduction in angular acceleration with increased size, they
accomplish this quite differently. Varanids show positive allometry in torque
production through positive allometry in limb length and muscle
cross-sectional area (Christian and
Garland, 1996), while iguanas show negative allometry in
rotational inertia. However, both groups are still expected to show a decline
in rotational acceleration with increasing mass. Therefore, we would predict
that the agility of both groups will decrease with increasing size. While no
scaling studies have been carried out on the turning abilities of iguanid or
varanid lizards, White and Anderson
(1994
) noted that, in
macroteiid lizards, the smaller Callopistes flaviipunctatus
demonstrated greater agility in escape maneuvers than two species of
Tupinambis, which were 1.3-7.8 times larger.
Evolutionary significance
Selection for improved turning performance may have contributed to the
major tail reduction that occurred independently in the lineages that gave
rise to birds (Carrier et al.,
2001) and mammals, two groups known for their exceptional
locomotor agility. This tail reduction is surprising when one considers the
ancestral condition of these groups, in which the major hindlimb retractor
muscle and, thus, the primary element of forward propulsion was located in the
tail (Gatesy, 1990
;
Carroll, 1997
). In fact, faster
sprint speeds in lizards have been correlated with distal expansion of the
caudofemoralis and a more distal break-point for caudal autotomy
(Zani, 1996
;
Russell and Bauer, 1992
). Much
of the tail mass in lizards is composed of the caudofemoralis and the epaxial
and hypaxial muscles necessary to brace the tail when the caudofemoralis
contracts. Thus, associated with the tail reduction that occurred during the
evolution of synapsids was a change in the muscles that retract the hindlimb
from the caudofemoralis to the hamstrings. This seems disadvantageous
energetically because it locates the hindlimb retractor mass in the thigh,
where it must be accelerated and decelerated with each locomotor cycle.
However, this change would improve turning abilities by greatly reducing the
rotational inertia of the body and thereby increasing angular acceleration
(
=
/I).
The loss of the large and heavy tail may also have affected maneuverability by moving the center of mass cranially. Many quadrupedal mammals, including mice and dogs, perform high-speed galloping turns by planting their forelimbs and doing a semi-handstand while swinging the hindlimbs around. If the force applied by the forelimbs in this `handstand' were directed in front of the center of mass, it would cause a pitching moment propelling the back end of the body downwards. Consequently, for this `handstand' turn to work, the force applied by the forelimbs must be directed through or close to the center of mass. Although it is not known whether this `handstand' turn increases maneuverability, it would probably have been facilitated by the cranial migration of the center of mass. In short, tail reduction in the synapsid lineage may have improved turning abilities both by reducing rotational inertia and, thus, the torque required to make the turn and by moving the center of mass cranially, making the mammalian `handstand' turn possible during galloping.
Carroll (1997) describes the
body configuration of pelycosaurs, early synapsids, as being best represented
among extant taxa by varanid or iguanid lizards. The present study has shown
that these lizards have a 3.0 to 4.9-fold greater rotational inertia than
similar-sized mammals because of their elongated trunk and long heavy tail. In
the evolution of therapsids and early mammals, loss of the large heavy tail
and replacement of the caudofemoralis with the hamstrings as the major
hindlimb retractor gave rise to animals with greatly reduced rotational
inertia and, therefore, probably much greater agility.
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Acknowledgments |
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