Models and the scaling of energy costs for locomotion
School of Biology, University of Leeds, Leeds LS2 9JT, UK
e-mail: r.m.alexander{at}leeds.ac.uk
Accepted 28 December 2004
Summary
To achieve the required generality, models designed to predict scaling relationships for diverse groups of animals generally need to be simple. An argument based on considerations of dynamic similarity predicts correctly that the mechanical cost of transport for running [power/(body mass x speed)] will be independent of body mass; but measurements of oxygen consumption for running birds and mammals show that the metabolic cost of transport is proportional to (body mass)-0.32. Thus the leg muscles seem to work more efficiently in larger animals. A model that treats birds as fixed wing aircraft predicts that the mechanical power required for flight at the maximum range speed will be proportional to (body mass)1.02, but the metabolic power is found to be proportional to (body mass)0.83; again, larger animals seem to have more efficient muscles. A model that treats hovering hummingbirds and insects as helicopters predicts mechanical power to be approximately proportional to body mass, but measurements of oxygen consumption once again show efficiency increasing with body mass. A model of swimming fish as rigid submarines predicts power to be proportional to (body mass)0.5x(speed)2.5 or to (body mass)0.6x(speed)2.8, depending on whether flow in the boundary layer is laminar or turbulent. Unfortunately, this prediction cannot easily be compared with available compilations of metabolic data. The finding that efficiency seems to increase with body mass, at least in running and flight, is discussed in relation to the metabolic energy costs of muscular work and force.
Key words: cost of transport, running, flight, swimming, efficiency, body mass
Introduction
In this paper I look at the scaling of the energy costs of locomotion, and ask whether we can explain what we observe. The explanations must depend on mathematical models. If we cannot formulate a convincing model that predicts a scaling rule reasonably accurately, we have failed to explain the rule. The reverse is unfortunately not true; a model that predicts a scaling rule correctly does not guarantee that our explanation is correct, because several models may predict the same rule.
My interest here is in widely applicable scaling rules; for example in
rules that will predict the scaling of running over the range from small
rodents to elephants, or of flight from sparrows to swans. Mice are not scale
models of elephants, and do not move like tiny elephants, and sparrows are not
miniature swans. The models will have to be very general, incorporating little
specific anatomical or kinematic detail. Conveniently, this implies that they
will be simple. Many simple models of running, swimming and flight were
presented at the first Scaling Conference
(Pedley, 1977), on which this
paper builds.
In the decades preceding the first Scaling Conference, measurements of
metabolic rate during locomotion had been greatly facilitated by the
introduction of methods using treadmills
(Taylor et al., 1970), wind
tunnels (Tucker, 1968
) and
water tunnels (Brett, 1964). Allometric exponents relating the measured energy
cost of locomotion to body mass had been calculated by Taylor et al.
(1970
) for running; by Tucker
(1970
) for running and flight;
and for swimming by Schmidt-Nielsen
(1972
). Allometric equations
in more recent papers are referred to in later sections of this one.
Because muscles do not work with uniform efficiency, it is much more difficult to devise a model that predicts the metabolic energy cost of locomotion than one that predicts mechanical work. In contrast, oxygen consumption (and hence metabolic power) can be measured directly, whereas determination of mechanical work in locomotion generally involves calculations subject to a good deal of uncertainty. Thus comparisons between theoretical and observed energy costs are not easy. It is the metabolic cost of locomotion, rather than the mechanical work, which is important for the animal's energy budget.
Dynamic similarity
I will refer frequently to the concept of dynamic similarity (see, for
example, Alexander, 2003). Two
bodies are geometrically similar if one could be made identical to the other
by multiplying all its linear dimensions by the same factor
. By an
extension of the same idea, two motions are dynamically similar if they could
be made identical by multiplying all linear dimensions by a factor
,
all times by a factor
, and all forces by a factor
. For example,
the motions of two pendulums of different lengths, swinging through the same
angle, are dynamically similar. Strict dynamic similarity requires geometric
similarity.
Two systems can only have dynamically similar motion in particular
circumstances. If gravitational forces are important, ratios of (gravitational
force/inertial force) must be the same for the two systems, at corresponding
stages of their motions. For this to be possible, the systems must be moving
with equal Froude numbers [speed2/(gravitational acceleration
x length)]. A fuller explanation of this point can be found in Alexander
(2003). In calculating a Froude
number, any length characteristic of the systems may be used; for example, leg
length is generally used in discussions of running. If viscous forces are
important, dynamic similarity is conditional on equality of ratios of viscous
forces to inertial forces, which requires equal values of the Reynolds number
(speed x length x fluid density/viscosity). For dynamic similarity
of vibrating systems, the Strouhal numbers (frequency x length/speed)
must be equal. Froude, Reynolds and Strouhal numbers are dimensionless. Other
dimensionless numbers define conditions for dynamic similarity, in systems for
which other kinds of forces are important.
Running
Because gravitational forces are important, dynamic similarity in walking
and running is possible only between animals travelling with equal Froude
numbers. Alexander (1977)
plotted relative stride length (stride length/leg length) against the square
root of Froude number, for ostriches and various mammals, and found that all
the points lay near a single line. Alexander and Jayes
(1983
) showed in more detail
that mammals of different sizes, running with equal Froude numbers, tend to
dynamic similarity: they use the same gait, similar relative stride lengths
and duty factors, and exert similar patterns of force on the ground. There are
some discrepancies (notably, rodents and other small mammals, which run with
their legs more bent than larger mammals, and take relatively longer strides
at the same Froude number), but the predictions of dynamic similarity hold
reasonably well. Biewener
(1989
) pointed out that larger
mammals need to run on straighter legs than small ones, to avoid excessive
bone and muscle stresses. Birds of different sizes, running at equal Froude
numbers, also tend to move in dynamically similar fashion, with discrepancies
due to the largest birds keeping their legs straighter
(Gatesy and Biewener,
1991
).
Some of the energy that mammals and birds would otherwise need for running
is saved by tendons that store and then return elastic strain energy, in the
course of a step. Alexander
(1988) pointed out that, for
dynamic similarity in running, animals should be elastically similar; in other
words, forces proportional to their body weights should cause equal strains
(fractional length changes). Bullimore and Burn
(2004
) showed that this
presents a problem, because tendon has the same elastic modulus in mammals of
all sizes. Elastically similar structures undergo equal strains (change of
height/height) when loaded with their own weight. Structures with equal
elastic moduli loaded with their own weight, however, undergo strains in
proportion to the stress, which is (weight/cross sectional area). If they are
geometrically similar, their weights are proportional to (length)3
and their cross-sectional areas to (length)2. Therefore elastic
similarity, between mammals of different sizes whose tendons have equal
elastic moduli, is inconsistent with strict dynamic similarity. Bullimore and
Burn (2004
) went on to show
that the size-related changes in posture that Biewener
(1989
) had shown to be
necessary to avoid excessive stresses in large mammals, also made approximate
elastic similarity possible.
For animals running in dynamically similar fashion, all forces are
proportional to body weight and all velocities to the speed of running. Thus
mechanical power is proportional to (weight x speed), and the mechanical
cost of transport [power/(mass x speed)] is independent of body mass.
Alexander (1977) showed that
cost of transport was independent of mass for specific models of walking and
running.
Taylor et al. (1982) showed
for a wide range of mammals and birds that the metabolic power required for
running was linearly related to speed. They subtracted the intercept at zero
speed (representing the metabolic rate while standing still) to obtain the net
power required for running. They found that the net metabolic cost of
transport was proportional to body mass
(Mb)-0.32.
Fig. 1 shows that the same
relationship also fits data from reptiles, amphibians and arthropods
(Full, 1989
). Calculations
based on force plate records and films of a smaller sample of birds, mammals
and arthropods showed, however, that the mechanical cost of transport was
independent of body mass (Heglund et al.,
1982
; Full and Tu,
1991
), as predicted by the dynamic similarity model.
|
The efficiency with which the muscles perform the work required for running is the work divided by the metabolic energy cost. Fig. 1 seems to show that large runners are more efficient than small ones; that the efficiency of running is approximately proportional to Mb0.3. The mechanical costs shown in Fig. 1, however, ignore the savings made by elastic mechanisms, leading to the impossible prediction (by extrapolation) of efficiencies greater than 100% for animals of the size of elephants. The apparent increase of efficiency with increasing body size could be misleading if savings by elastic mechanisms are proportionately larger in larger animals. Apparent efficiencies calculated from the data of Fig. 1 are 17 times greater for 100 kg runners than for 10 g runners. It seems most unlikely that any difference in the effectiveness of elastic mechanisms, between runners of different sizes, is large enough to account for that. We must conclude that the muscles of larger runners do indeed perform work with higher efficiency. Before discussing this further, we will ask whether the same is true for other modes of locomotion.
Flight
Flying insects, birds and bats use their wings to drive air downward,
exerting on the air a downward force that balances body weight. In fast
forward flight, the speeds of the beating wings relative to the body are low
compared to the speed of the body relative to the air. In slow and hovering
flight, the reverse is true. We can make rough estimates of mechanical energy
costs by modelling fast fliers as fixed-wing aircraft (following
Pennycuick, 1969, with
modifications) and slow fliers as helicopters
(Weis-Fogh, 1973
). The
cruising flight of birds, bats and large insects such as locusts is fast in
this sense. Hummingbirds and many insects hover, and even in cruising flight
the wings of small insects move much faster than their bodies; their flight is
slow, in the sense used in this paragraph.
Flow over the wings of aeroplanes and the rotors of helicopters is steady,
in the sense that velocities remain constant. In contrast, flow over a
flapping wing is unsteady. Aerodynamic forces acting in unsteady systems
cannot be predicted accurately by equations for steady flow. Consequently,
calculations based on steady aerodynamics, of the power required for flapping
flight, are subject to error (Ellington,
1995; Rayner,
1995b
). The greater the distance travelled by the wing in a single
beat, expressed as a multiple of its chord length, the less serious are these
errors likely to be. Thus they are likely to be less serious in fast flight,
than in slow flight or hovering. In this paper, which requires only rough
answers, I tolerate the errors for the sake of simplicity.
I will consider fast flight first. The power P required for flight by a fixed-wing aircraft is the sum of two components. The induced power is required to give kinetic energy to the air that is driven downwards to counteract gravity. The profile power is required to overcome the drag due to the viscosity of the air in the boundary layer, and to give kinetic energy to the air that is drawn forward in the wake. The power needed to overcome the drag on the body (parasite power) is sometimes calculated separately, but here I include it in the profile power:
Total power = profile power + induced power
![]() | (1) |
(see, for example, Alexander,
2003). In this equation,
is the density of the air,
v is the speed, C0 is the zero-lift drag
coefficient, A is the wing area, k is the induced drag
factor, Mb is body mass, g is the
gravitational acceleration and
is the aspect ratio (the ratio of wing
span to the mean chord; the chord is the distance between the front and rear
edges of the wing). As speed increases, the induced power falls and the
profile power increases.
Two optimum speeds can be defined: the minimum power speed at which the
total power P is least, and the maximum range speed at which the
energy required to travel unit distance (P/v) is least.
These speeds can be obtained from Equation 1 by calculus (see, for example,
Alexander, 1996). Textbooks of
aerodynamics usually treat the zero-lift drag as constant, as it would be if
the flight of different-sized animals were dynamically similar. Dynamic
similarity would, however, require animals to fly at equal Reynolds numbers,
that is at speeds inversely proportional to their linear dimensions; a moth
with a 10 mm wing chord would have to fly ten times as fast as a bird with a
100 mm chord. That would generally not be the case, so we must take account of
differences in C0. All but the largest and fastest birds
fly with Reynolds numbers Re below 106, at which
C0 is expected to be approximately proportional to
1/
Re and so to 1/
(chord x speed). For example,
pigeon wings have a chord of about 12 cm. When the bird flies fast at 15 m
s-1, their Reynolds number is only 120 000.
A calculation of the maximum range speed, taking account of this dependence
of C0 on Re, yields:
![]() | (2) |
and
![]() | (3) |
Rayner (1988) found that
for birds, excluding hummingbirds, wing area is approximately proportional to
Mb0.72 and wing span to
Mb0.39. Hence chord (=area/span) is
proportional to Mb0.33 and aspect ratio to
Mb0.07. By substituting these proportionalities
into Equations 2 and 3 we obtain:
![]() | (4) |
and
![]() | (5) |
Rayner's calculations (Rayner,
1995a), which took account of the vortex structure of the wake,
gave maximum range speed proportional to Mb0.14
and power to Mb1.10. Observed cruising speeds
of birds increase with body mass
(Pennycuick, 1997
), but the
correlation is too weak for a quantitative comparison with the calculated
exponents for maximum range speed. These exponents for estimated mechanical
power are very different from the exponent for measured metabolic power;
Rayner (1995a
) found that the
metabolic power used in bird flight was proportional to
Mb0.83. As for running, the efficiency of
flight increases with body mass (Fig.
2).
|
Until recently, calculations of the mechanical power required for animal
flight depended on mathematical models (often more sophisticated than those
presented here). There were no more direct estimates until Biewener et al.
(1992) measured the forces
exerted by the flight muscles of flying birds, using a strain gauge bonded to
the humerus. Powers determined by this method (see especially
Dial et al., 1997
) agree
reasonably well with the results of mathematical modelling
(Alexander, 1997b
).
Now we have to consider hovering. The rotating blades of a helicopter, and
the beating wings of a hovering animal, drive air downwards. The power
required for this can be estimated as:
![]() | (6) |
where r is the radius of the rotor (equivalent to wing length in
animals) and the other symbols have the same meanings as before (see
Alexander, 2003). Wing length
is proportional to Mb0.53 in hummingbirds
(Rayner, 1988
) and to
Mb0.42 in euglossine bees
(Casey et al., 1985
). In both
cases, larger animals are not geometrically similar to small ones, but have
relatively longer wings. With these exponents we get:
![]() |
and
![]() | (7) |
for hummingbirds and bees, respectively. Rayner's vortex theory of hovering
(Rayner, 1979) treats the
aerodynamics more realistically than the simple helicopter approximation, but
gives similar predictions of induced power.
As for fixed wing aircraft, the power requirements of helicopters include profile power as well as induced power. For well-designed rotors, profile power is expected to be about proportional to induced power, so the exponents in Equation 7 should apply to total aerodynamic power as well as to induced power.
Hovering animals may incur yet another power requirement, which does not
arise for helicopters. This is known as the inertial power, to distinguish it
from the aerodynamic (induced plus profile) power. At the beginning of each
stroke the wings and an `added mass' of air that moves with them are given
kinetic energy, which they lose at the end of the stroke. If this energy were
small compared to the kinetic energy given to the air, it could be transferred
to the air in the later part of the stroke, while the wings were decelerating.
In fact, for hovering hummingbirds and insects, the inertial power is commonly
as large or larger than the aerodynamic power (see
Alexander, 2003). In many
insects, some or all of the kinetic energy may be stored in elastic structures
at the end of the stroke, and recovered in the next stroke. Elastic structures
of optimal stiffness could in principle supply all the inertial power, leaving
the muscles to supply only the induced and profile power.
The kinetic energy given to two wings at the start of a stroke is
I2, where I is the moment of inertia of
one wing. If the wing beat frequency is f, the mean power required to
supply this energy (twice in each wing beat cycle) is
2fI
2. I will assume that each wing beats through an
angle
during a half cycle of duration 1/2f, making the mean
angular velocity 2
f. For simplicity, I assume that the stroke is
made with constant angular velocity. Thus the inertial power is given by:
![]() | (8) |
Each wing beat of a hovering animal adds a vortex ring to the wake of
moving air below it. For dynamic similarity between animals of different
sizes, the spacing of the rings should be proportional to wing length.
Alexander (2000) pointed out
that this implies equal Strouhal numbers, which in turn implies that the wing
beat frequency should be proportional to
Mb/r2. I showed that it is
indeed approximately proportional to
Mb/r2, both for hummingbirds
and for euglossine bees. Thus:
![]() | (9) |
Weis-Fogh's data for Lepidoptera, Hymenoptera and Diptera
(Weis-Fogh, 1973) show that
wing moment of inertia is approximately proportional to
Mb2, and wing length to
Mb0.4. With these proportionalities, Equation 9
gives inertial power proportional to Mb1.1. The
exponent is approximately the same as the one estimated above for aerodynamic
power for bees. Weis-Fogh's moments of inertia refer only to the wing itself,
ignoring the added mass of air, but this has probably had little effect on our
calculated exponent. Ellington
(1984
) found that the added
mass was about the same proportion of wing mass (0.4) for a bee, a wasp and a
moth.
Casey (1981) compared
metabolic rates of hovering sphinx moths with model-based estimates of
mechanical power. The models gave induced power proportional to
Mb1.07 and profile power proportional to
Mb1.08. Measured metabolic powers were
proportional to Mb0.77. If inertial power
requirements were taken care of by elastic storage, muscle efficiency was
approximately proportional to Mb0.3. Casey and
Ellington (1989
) made similar
comparisons for hovering euglossine bees. They found that metabolic power was
proportional to Mb0.58, and efficiency to
Mb0.51 or
Mb0.47, for no elastic storage and perfect
elastic storage, respectively. These data for moths and bees refer to fairly
narrow ranges of body mass (about one order of magnitude in each case), so the
exponents could not be determined very precisely, but it seems clear that for
hovering insects, as for running animals and flying birds, efficiency
increases with body mass.
Swimming
In this simple analysis, I will estimate the power required for swimming by
fish as if they were rigid submarines. This approach underestimates the power
by a factor of about 3, for reasons discussed by Webb
(1992). This factor changes
only a little with Reynolds number, in experiments with the same fish swimming
over a wide range of speeds. As the Strouhal numbers of swimming fish vary
little with size and speed (see Alexander,
2003
), the factor can be expected to be the about same for
different-sized fish swimming at the same Reynolds number. The same conclusion
seems to follow from the computational fluid dynamics model of Schultz and
Webb (2002
). The following
discussion is concerned with the scaling of power rather than with absolute
values, so would not be affected by a size-independent factor.
Animals develop the thrust required to propel them through water by driving
some of the water backwards. The power required for swimming is the sum of
parasite power, required to overcome the drag on the body; induced power,
required to give kinetic energy to the water driven backwards; and inertial
power, required to give kinetic energy to body parts that are accelerated at
the beginning of each stroke. For an animal or other body travelling at speed
v through water of density :
![]() | (10) |
where A is the total surface area of the body and
CD is the drag coefficient based on total area. For
well-streamlined bodies such as most fish, whales and squid, the drag
coefficient is proportional to Re-0.5 at Reynolds numbers
(Re) up to about 106, and to Re-0.2 at
higher Reynolds numbers. The change in exponent is due to flow in the boundary
layer changing from laminar to turbulent as the Reynolds number passes
106. Small fish swim in the laminar range, and whales generally in
the turbulent range. Medium-sized fish span the transition; for example, a 0.5
m fish swimming at 2 m s-1 would have a Reynolds number of
106. Reynolds number is proportional to vl, where
l is the length of the body. Thus for laminar flow:
![]() |
and for turbulent flow:
![]() | (11) |
Cetaceans ranging from small dolphins to blue whales have lengths
proportional to Mb0.34, very close to geometric
similarity (Economos, 1983).
Fish of different sizes also tend to be close to geometric similarity
(Peters, 1983
). For
geometrically similar animals, A is proportional to
Mb0.67 and l to 0.33, so that for
laminar flow:
![]() |
and for turbulent flow:
![]() | (12) |
Swimmers can develop the thrust they need by driving small volumes of water
backward at high speed, or large volumes at low speed. The latter requires
less induced power. The ratio [parasite power/(parasite power + induced
power)] is known as the Froude efficiency. It is close to 1.0 for two of the
species studied by Wardle et al.
(1995), and 0.75 for the third
(see Alexander, 2003
, for the
method of calculation). For these fish, and presumably also for other fish and
for whales, induced power is much smaller than parasite power and can be
ignored in this simple analysis. Squids swimming by jet propulsion operate at
lower Froude efficiencies, so induced power could not be ignored in
discussions of them. Inertial power may be substantial in swimming dolphins
and tunas, mainly due to the mass of water that oscillates with the tail
rather than to the mass of the tail itself, but may be supplied in part by
tendon elasticity rather than muscle action
(Blickhan and Cheng, 1994
;
Alexander, 1997a
).
Yates (1983) defined a
coefficient of thrust:
![]() | (13) |
If Equation 10 held and muscle efficiency were constant, the coefficient of
thrust would be a constant multiple of the coefficient of drag which, as we
have seen, is a function of Reynolds number. Yates analysed Brett's data for
salmon (Brett, 1965), plotting
thrust coefficient against Reynolds number for fish of different sizes. Large
salmon swim at higher ranges of Reynolds number than small ones but, if
efficiency had been constant, the data for all sizes of salmon should have
lain on the same line on this graph. They did not lie on the same line; larger
fish had higher thrust coefficients at the same Reynolds number, seeming to
show that larger fish were less efficient. This contrasts with the finding
that larger running and flying animals are more efficient. Yates
(1983
) recognised, however,
that the data probably suffered from two sources of error. At low speeds the
fish may have behaved erratically, using more energy than would have been
needed for steady swimming. At high speeds, some of their metabolism was
probably anaerobic, and would have been missed by the measurements of oxygen
consumption. Thus the thrust coefficients for slow swimming were probably
misleadingly high, and those for fast swimming too low, for each fish.
Correction for this would make the thrust coefficients lie more nearly along a
single line. We cannot exclude the possibility that the swimming muscles of
different-sized fish work with different efficiencies, but there is no
evidence of larger fish being more efficient.
Efficiency
Previous sections have shown that, at least for running and flight, muscles
do the work of locomotion more efficiently in larger animals. Kram and Taylor
(1990) tried to explain this
observation for running. They ignored the metabolic cost of doing work, and
considered only the cost of exerting the force required to counteract gravity.
They assumed that muscles work over the same ranges of the force-velocity
relationship, irrespective of speed and body size; faster muscle fibres would
be recruited at higher running speeds, and smaller animals (whose feet remain
on the ground for shorter times) would need faster muscles than large ones.
These assumptions led them to the hypothesis:
![]() | (14) |
They measured foot contact time and oxygen consumption for mammals of a
wide range of sizes, running at a wide range of speeds, and found excellent
agreement with the hypothesis. Roberts et al.,
(1998) found similar agreement
for running birds. Herr et al.
(2002
) modelled a selection of
mammals, ranging from a chipmunk to a large horse. For each model they
simulated running at a range of speeds, and showed that Kram and Taylor's
hypothesis (Kram and Taylor,
1990
) successfully predicted the metabolic cost of transport of
the real animal. Because models of particular species were used, rather than a
general model of variable size, this study threw only limited light on scaling
principles.
Kram and Taylor did not discuss the physiological basis for their
hypothesis in detail. The metabolic rate of an active muscle is not a function
solely of the force it is currently exerting, but depends also on the rate at
which its length is changing; for any given force, the metabolic rate is
greater when the muscle is shortening and less when it is being stretched
(fig. 1b in
Alexander, 2002). In every
step, leg muscles are first stretched (doing negative work) and then shorten
(doing positive work). Suppose that an active leg muscle has a metabolic rate
x times the isometric rate. Then if the corresponding muscle in a
different-sized animal is working at the same point in its force-velocity
curve, it also is expected to have a metabolic rate x times its
isometric rate. If the leg muscles of different-sized animals work over the
same range of their force-velocity curves, as Kram and Taylor assumed, their
metabolic rates (averaged over a stride) may be equal multiples of the rate in
isometric contraction. It would be interesting to have experimental
confirmation of the assumption. Kram and Taylor
(1990
) also assumed that
strain rates of muscles should be inversely proportional to ground contact
time. This implies that equal strains should occur, in the leg muscles of
different-sized animals. Again, experimental confirmation would be
welcome.
The following calculation highlights a possible problem with the assumption
of equal strains. The mechanical cost of transport
[work/(Mb x distance)] is about the same for runners
of all sizes (Heglund et al.,
1982), and stride lengths in similar gaits are proportional to
Mb0.38
(Heglund et al., 1974
). Hence
the work required for a stride is proportional to
Mb1.38. The work performed in a stride is
(muscle stress x muscle strain x muscle volume). Muscle stress in
similar gaits is proportional to Mb-0.06, and
muscle mass and volume are proportional to
Mb1.03 (Biewener,
1989
,
1990
). If strains are
independent of body mass, the (positive and negative) work performed by the
muscles should be approximately proportional to
Mb0.97, which does not match the observed
proportionality of work to Mb1.38.
In flight, as in running, muscles have to exert forces to counteract
gravity. It is therefore reasonable to ask whether Kram and Taylor's
hypothesis successfully predicts metabolic power for flight. For foot contact
time we can substitute the duration of a wing beat, which is inversely
proportional to frequency:
![]() | (15) |
For forward flight, frequency is proportional to
Mb-0.26
(Rayner, 1995a), giving
predicted metabolic power proportional to
Mb0.74, not too far from the observed
proportionality of Mb0.83. For hovering
euglossine bees, wing beat frequency is proportional to
Mb-0.27, giving predicted metabolic rate
proportional to Mb0.73, which is higher than
the observed proportionality of (in this case)
Mb0.58
(Casey and Ellington, 1989
).
Kram and Taylor's hypothesis (Kram and
Taylor, 1990
) works reasonably well for bird flight, but less well
for bees.
Conclusions
I have based this paper on very simple, long-established models. There is scope for using more modern modelling approaches; for example, because fish of different sizes are often close to geometric similarity, it would not be too difficult to use computational fluid dynamics to investigate the scaling of swimming energetics. However sophisticated the methods, the models must be kept conceptually simple, and the need for anatomical and kinematic data must be kept to a minimum, if the generality required for broad scaling studies is to be preserved.
The most urgent need, however, in studies of the scaling of energy costs in
locomotion, is for better understanding of the relationship between mechanical
work and metabolism. Some studies of locomotion have attempted to model this
relationship (Minetti and Alexander,
1997; Anderson and Pandy,
2001
; Sellers et al.,
2003
), but this approach has not been used in scaling studies. It
might, indeed, be premature to apply this approach to the scaling of energy
costs, until it has a sounder foundation in muscle physiology. The problem is
that, as used in the cited papers, it uses data from isotonic contractions to
estimate energy costs in work loops. Hill's Equation, derived from isotonic
experiments, does not predict forces in work loops well
(Askew and Marsh, 1998
), and
we do not know whether the equations for metabolic rate work well for work
loops.
References
Alexander, R. McN. (1977). Mechanics and scaling of terrestrial locomotion. In Scale Effects in Animal Locomotion (ed. T. J. Pedley). London: Academic Press.
Alexander, R. McN. (1988). Elastic Mechanisms in Animal Movement. Cambridge: Cambridge University Press.
Alexander, R. McN. (1995). Energy for Animal Life. Oxford: Oxford University Press.
Alexander, R. McN. (1996). Optima for Animals edn. 2. Princeton: Princeton University Press.
Alexander, R. McN. (1997a). Optimum muscle design for oscillatory movement. J. Theor. Biol. 184,253 -259.[CrossRef]
Alexander, R. McN. (1997b). The U, J and L of bird flight. Nature 390,13 .[CrossRef]
Alexander, R. McN. (2000). Hovering and jumping: contrasting problems in scaling. In Scaling in Biology (ed. J. H. Brown and G. B. West), pp.37 -50. Oxford, Oxford University Press.
Alexander, R. McN. (2002). Tendon elasticity and muscle function. Comp. Biochem. Physiol. 133A,1001 -1011.
Alexander, R. McN. (2003). Principles of Animal Locomotion. Princeton: Princeton University Press.
Alexander, R. McN. and Jayes, A. S. (1983). A dynamic similarity hypothesis for the gaits of quadrupedal mammals. J. Zool. 201,135 -152.
Anderson, F. and Pandy, M. (2001). Dynamic optimization of human walking. J. Biomech. Eng. 123,381 -390.[CrossRef][Medline]
Askew, G. N. and Marsh, R. L. (1998). Optimal
shortening velocity (V/Vmax) of skeletal muscle during
cyclical contractions: length-force effects and velocity-dependent activation
and deactivation. J. Exp. Biol.
201,1527
-1540.
Biewener, A. A. (1989). Scaling body support in mammals: limb posture and muscle mechanics. Science 245, 45-48.[Medline]
Biewener, A. A. (1990). Biomechanics of mammalian terrestrial locomotion. Science 250,1097 -1103.[Medline]
Biewener, A. A., Dial, K. P. and Goslow, G. E. (1992). Pectoralis muscle force and power output during flight in the starling. J. Exp. Biol. 164, 1-18.
Blickhan, R. and Cheng, J.-Y. (1994). Energy storage by elastic mechanisms in the tail of large swimmers - a re-evaluation. J. Theor. Biol. 168,315 -321.[CrossRef]
Brett, J. R. (1965). The relation of size to the rate of oxygen consumption and sustained swimming speeds of sockeye salmon (Onchorhynchus nerka). J. Fish. Res. Bd Can. 22,1491 -1501.
Bullimore, S. R. and Burn, J. F. (2004). Distorting limb design for dynamically similar locomotion. Proc. R. Soc. B 271,285 -289.[CrossRef]
Casey, T. M. (1981). A comparison of mechanical and energetic estimates of flight costs for hovering sphinx moths. J. Exp. Biol. 91,117 -129.
Casey, T. M. and Ellington, C. P. (1989). Energetics of insect flight. In Energy Transformations in Cells and Organisms (ed. W. Wieser and E. Gnaiger), pp.200 -210. Stuttgart: Thieme.
Casey, T. M., May, M. L. and Morgan, K. R. (1985). Flight energetics of euglossine bees in relation to morphology and to wing stroke frequency. J. Exp. Biol. 116,271 -289.
Dial, K. P., Biewener, A. A., Tobalske, B. W. and Warrick, D. R. (1997). Mechanical power output of bird flight. Nature 390,67 -70.[CrossRef]
Economos, A. C. (1983). Elastic and/or geometric similarity in mammalian design? J. Theor. Biol. 103,167 -172.[CrossRef][Medline]
Ellington, C. P. (1984). The aerodynamics of hovering insect flight. II. Morphological parameters. Phil. Trans. R. Soc. B 305,17 -40.
Ellington, C. P. (1995). Unsteady aerodynamics of insect flight. In Biological Fluid Dynamics (ed. C. P. Ellington and T. J. Pedley), pp. 109-129. Cambridge: The Company of Biologists.
Full, R. J. (1989). Mechanics and energetics of terrestrial locomotion: bipeds to polypeds. In Energy Transformations in Cells and Organisms (ed. W. Wieser and E. Gnaiger), pp. 175-182. Stuttgart: Thieme.
Full, R. J. and Tu, M. S. (1991). Mechanics of a rapid running insect: two-, four- and six-legged locomotion. J. Exp. Biol. 156,215 -231.[Abstract]
Gatesy, S. M. and Biewener, A. A. (1991). Bipedal locomotion: effects of speed, size and limb posture in birds and humans. J. Zool. 224,127 -147.
Gray, J. (1936). Studies in animal locomotion VI. The propulsive powers of the dolphin. J. Exp. Biol. 13,192 -199.
Heglund, N. C., Fedak, M. A., Taylor, C. R. and Cavagna, G. A. (1982). Energetics and mechanics of terrestrial locomotion. IV. Total mechanical energy changes as a function of speed and body size in birds and mammals. J. Exp. Biol. 97, 57-66.[Abstract]
Heglund, N. C., Taylor, C. R. and McMahon, T. A. (1974). Scaling stride frequency and gait to animal size: mice to horses. Science 186,1112 -1113.[Medline]
Herr, H. M., Huang, G. T. and McMahon, T. A.
(2002). A model of scale effects in mammalian quadrupedal
running. J. Exp. Biol.
205,959
-967.
Kram, R. and Taylor, R. C. (1990). Energetics of running: a new perspective. Nature 346,265 -267[CrossRef][Medline]
Minetti, A. E. and Alexander, R. McN. (1997). A theory of metabolic costs for bipedal gaits. J. Theor. Biol. 186,467 -476.[CrossRef][Medline]
Pedley, T. J. (1977). Scale Effects in Animal Locomotion. London: Academic Press.
Pennycuick, C. J. (1969). The mechanics of bird migration. Ibis 111,525 -556.
Pennycuick, C. J. (1997). Actual and `optimal'
flight speeds: field data reassessed. J. Exp. Biol.
200,2355
-2361.
Peters, R. H. (1983) The Ecological Implications of Body Size. Cambridge: Cambridge University Press.
Rayner, J. M. V. (1979). A new approach to animal flight mechanics. J. Exp. Biol. 80, 17-54.
Rayner, J. M. V. (1988). Form and function in avian flight. Curr. Ornithol. 5, 1-66.
Rayner, J. M. V. (1995a). Flight mechanics and constraints on flight performance. Israel J. Zool. 41,321 -342.
Rayner, J. M. V. (1995b). Dynamics of the vortex wakes of flying and swimming vertebrates. In Biological Fluid Dynamics (ed. C. P. Ellington and T. J. Pedley), pp.131 -155. Cambridge: The Company of Biologists.
Roberts, T. J., Kram, R., Weyand, P. G. and Taylor, C. R.
(1998). Energetics of bipedal running. I. Metabolic cost of
generating force. J. Exp. Biol.
201,2745
-2751.
Schmidt-Nielsen, K. (1972). Locomotion: energy cost of swimming, flying and running. Science 177,222 -228.[Medline]
Schultz, W. W. and Webb, P. W. (2002). Power requirements of swimming: do new methods resolve old questions? Integr. Comp. Biol. 42,1018 -1025.
Sellers, W. I., Dennis, L. A. and Crompton, R. H. (2003). Predicting the energy costs of bipedalism using evolutionary robotics. J. Exp. Biol. 201,2745 -2751.
Taylor, C. R., Schmidt-Nielsen, K. and Raab, J. L.
(1970). Scaling of energetic cost of running to body size in
mammals. Am. J. Physiol.
219,1104
-1107.
Taylor, C. R., Heglund, N. C. and Maloiy, G. M. O. (1982). Energetics and mechanics of terrestrial locomotion. I. Metabolic energy consumption as a function of speed and body size in birds and mammals. J. Exp. Biol. 97, 1-21.[Abstract]
Tucker, V. A. (1968). Respiratory exchange and evaporative water loss in the flying budgerigar. J. Exp. Biol. 48,67 -87.
Tucker, V. A. (1970). Energetic cost of locomotion in animals. Comp. Biochem. Physiol. 34,841 -846.[CrossRef][Medline]
Wardle, C. S., Videler, J. J. and Altringham, J. J. (1995). Tuning in to fish body waves: body form, swimming mode and muscle function. J. Exp. Biol. 198,1629 -1636.[Medline]
Webb, P. W. (1992). Is the high cost of body/caudal fin undulatory swimming due to increased friction drag or inertial recoil? J. Exp. Biol. 162,157 -166.
Weis-Fogh, T. (1973). Quick estimates of flight fitness in hovering animals, including novel mechanisms for lift production. J. Exp. Biol. 59,169 -230.
Yates, G. T. (1983). Hydromechanics of body and caudal fin propulsion. In Fish Biomechanics (ed. P. W. Webb and D. Weihs), pp. 177-213. New York: Praeger.
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