From Wikipedia, the free encyclopedia
Allometry is a well-known study, particularly in
statistical shape analysis for its theoretical developments, as well as in
biology
for practical applications to the differential growth rates of the
parts of a living organism's body. One application is in the study of
various
insect species (e.g.,
Hercules beetles),
where a small change in overall body size can lead to an enormous and
disproportionate increase in the dimensions of appendages such as legs,
antennae, or horns The relationship between the two measured quantities is often expressed as a
power law equation which expresses a remarkable scale symmetry:
or in a logarithmic form:
where
is the
scaling exponent of the law. Methods for estimating this exponent from data can use type-2 regressions, such as
major axis regression or
reduced major axis regression, as these account for the variation in both variables, contrary to
least squares regression, which does not account for error variance in the independent variable (e.g., log body mass). Other methods include
measurement-error models and a particular kind of
principal component analysis.
Allometry often studies shape differences in terms of
ratios
of the objects' dimensions. Two objects of different size, but common
shape, will have their dimensions in the same ratio. Take, for example, a
biological object that grows as it matures. Its size changes with age,
but the shapes are similar. Studies of ontogenetic allometry often use
lizards or
snakes as model organisms both because they lack
parental care after
birth or hatching and because they exhibit a large range of body sizes between the
juvenile and
adult stage. Lizards often exhibit allometric changes during their
ontogeny.
In addition to studies that focus on growth, allometry also
examines shape variation among individuals of a given age (and sex),
which is referred to as static allometry. Comparisons of species are
used to examine interspecific or evolutionary allometry.
Isometric scaling and geometric similarity
Scaling range for different organisms
| Group
|
Factor
|
Length range
|
| Insects
|
1000 |
10-4 to 10-1 m
|
| Fish
|
1000 |
10-2 to 10+1 m
|
| Mammals
|
1000 |
10-1 to 10+2 m
|
| Vascular plants
|
10,000 |
10-2 to 10+2 m
|
| Algae
|
100,000 |
10-5 to 100 m
|
Isometric scaling happens when proportional relationships are
preserved as size changes during growth or over evolutionary time. An
example is found in frogs — aside from a brief period during the few
weeks after metamorphosis, frogs grow isometrically.
Therefore, a frog whose legs are as long as its body will retain that
relationship throughout its life, even if the frog itself increases in
size tremendously.
Isometric scaling is governed by the
square-cube law.
An organism which doubles in length isometrically will find that the
surface area available to it will increase fourfold, while its volume
and mass will increase by a factor of eight. This can present problems
for organisms. In the case of above, the animal now has eight times the
biologically active tissue to support, but the surface area of its
respiratory organs has only increased fourfold, creating a mismatch
between scaling and physical demands. Similarly, the organism in the
above example now has eight times the mass to support on its legs, but
the strength of its bones and muscles is dependent upon their
cross-sectional area, which has only increased fourfold. Therefore,
this hypothetical organism would experience twice the bone and muscle
loads of its smaller version. This mismatch can be avoided either by
being "overbuilt" when small or by changing proportions during growth,
called allometry.
Isometric scaling is often used as a null hypothesis in scaling
studies, with 'deviations from isometry' considered evidence of
physiological factors forcing allometric growth.
Allometric scaling
Allometric scaling is any change that deviates from
isometry. A classic example discussed by
Galileo in his
Dialogues Concerning Two New Sciences
is the skeleton of mammals. The skeletal structure becomes much
stronger and more robust relative to the size of the body as the body
size increases. Allometry is often expressed in terms of a scaling exponent based on
body mass, or body length (Snout-vent length, total length etc.). A
perfectly isometrically scaling organism would see all volume-based
properties change proportionally to the body mass, all surface
area-based properties change with mass to the power of 2/3, and all
length-based properties change with mass to the power of 1/3. If, after
statistical analyses, for example, a volume-based property was found to
scale to mass to the 0.9th power, then this would be called "negative
allometry", as the values are smaller than predicted by isometry.
Conversely, if a surface area-based property scales to mass to the 0.8th
power, the values are higher than predicted by isometry and the
organism is said to show "positive allometry". One example of positive
allometry occurs among species of monitor lizards (family
Varanidae), in which the limbs are relatively longer in larger-bodied species. The same is true for some fish, e.g. the
muskellunge, the weight of which grows with about the power of 3.325 of its length.
A 30-inch (76 cm) muskellunge will weigh about 8 pounds (3.6 kg), while
a 40-inch (100 cm) muskellunge will weigh about 18 pounds (8.2 kg), so
33% longer length will more than double the weight.
Determining if a system is scaling with allometry
To
determine whether isometry or allometry is present, an expected
relationship between variables needs to be determined to compare data
to. This is important in determining if the scaling relationship in a
dataset deviates from an expected relationship (such as those that
follow isometry). The use of tools such as dimensional analysis is very
helpful in determining expected slope.
This ‘expected’ slope, as it is known, is essential for detecting
allometry because scaling variables are comparisons to other things.
Saying that mass scales with a slope of 5 in relation to length doesn’t
have much meaning unless knowing the isometric slope is 3, meaning in
this case, the mass is increasing extremely fast. For example,
different sized frogs should be able to jump the same distance according
to the geometric similarity model proposed by Hill 1950 and interpreted by Wilson 2000, but in actuality larger frogs do jump longer distances.
Dimensional analysis is extremely useful for balancing units in an equation or in this case, determining expected slope.
A few dimensional examples follow (M=Mass, L=Length, V=Volume, which is also L cubed because a volume is merely length cubed):
To find the expected slope for the relationship between mass and the
characteristic length of an animal (see figure), the units of mass (M=L
3,
because mass is a volume; volumes are lengths cubed) from the Y-axis
are divided by the units of the X-axis (in this case, L). The expected
slope on a double-logarithmic plot of L
3/ L
1 in this case is 3 (log
10(L
3)/log
10(L
1)=3).
This is the slope of a straight line, but most data gathered in science
do not fall neatly in a straight line, so data transformations are
useful.
It is also important to keep in mind what is being compared in the data. Comparing a characteristic such as head length to head width might
yield different results from comparing head length to body length. That
is, different characteristics may scale differently.
A common way to analyze data such as those collected in scaling is to use
log-transformation.
There are two reasons for log transformation - a biological reason and a
statistical reason. Biologically, log-log transformation places numbers
into a geometric domain so that proportional deviations are represented
consistently, independent of the scale and units of measurement. In
biology this is appropriate because many biological phenomena (e.g.
growth, reproduction, metabolism, sensation) are fundamentally
multiplicative.
Statistically, it is beneficial to transform both axes using
logarithms and then perform a linear regression. This will normalize the
data set and make it easier to analyze trends using the slope of the
line. Before analyzing data though, it is important to have a predicted slope of the line to compare the analysis to.
After data are log-transformed and linearly regressed, comparisons can then use
least squares regression with 95% confidence intervals or
reduced major axis analysis.
Sometimes the two analyses can yield different results, but often they
do not. If the expected slope is outside the confidence intervals, then
there is allometry present. If mass in this imaginary animal scaled with
a slope of 5 and this was a statistically significant value, then mass
would scale very fast in this animal versus the expected value. It would
scale with positive allometry. If the expected slope were 3 and in
reality in a certain organism mass scaled with 1 (assuming this slope is
statistically significant), then it would be negatively allometric.
Another example: Force is dependent on the cross-sectional area of muscle (CSA), which is L
2.
If comparing force to a length, then the expected slope is 2.
Alternatively, this analysis may be accomplished with a power
regression. Plot the relationship between the data onto a graph. Fit
this to a power curve (depending on the stats program, this can be done
multiple ways), and it will give an equation with the form:
y=
Zxn, where
n
is the number. That “number” is the relationship between the data
points. The downside, to this form of analysis, is that it makes it a
little more difficult to do statistical analyses.
Physiological scaling
Many
physiological and biochemical processes (such as heart rate,
respiration rate or the maximum reproduction rate) show scaling, mostly
associated with the ratio between surface area and mass (or volume) of
the animal. The
metabolic rate of an individual animal is also subject to scaling.
Metabolic rate and body mass
In plotting an animal's
basal metabolic rate (BMR) against the animal's own body mass, a logarithmic straight line is obtained, indicating a
power-law
dependence. Overall metabolic rate in animals is generally accepted to
show negative allometry, scaling to mass to a power of ≈ 0.75, known as
Kleiber's law,
1932. This means that larger-bodied species (e.g., elephants) have
lower mass-specific metabolic rates and lower heart rates, as compared
with smaller-bodied species (e.g., mice). The straight line generated
from a double logarithmic scale of metabolic rate in relation to body
mass is known as the "mouse-to-elephant curve".
These relationships of metabolic rates, times, and internal structure
have been explained as, "an elephant is approximately a blown-up
gorilla, which is itself a blown-up mouse."
Max Kleiber contributed the following allometric equation for relating the BMR to the body mass of an animal. Statistical analysis of the intercept did not vary from 70 and the slope was not varied from 0.75, thus:
(although the universality of this relation has been disputed both empirically and theoretically)
where
is body mass, and metabolic rate is measured in
kcal per day.
Consequently, the body mass itself can explain the majority of
the variation in the BMR. After the body mass effect, the taxonomy of
the animal plays the next most significant role in the scaling of the
BMR. The further speculation that environmental conditions play a role
in BMR can only be properly investigated once the role of taxonomy is
established. The challenge with this lies in the fact that a shared
environment also indicates a common evolutionary history and thus a
close taxonomic relationship. There are strides currently in research to
overcome these hurdles; for example, an analysis in muroid rodents,
the mouse, hamster, and vole type, took into account taxonomy. Results
revealed the hamster (warm dry habitat) had lowest BMR and the mouse
(warm wet dense habitat) had the highest BMR. Larger organs could
explain the high BMR groups, along with their higher daily energy needs.
Analyses such as these demonstrate the physiological adaptations to
environmental changes that animals undergo.
Energy metabolism is subjected to the scaling of an animal and
can be overcome by an individual's body design. The metabolic scope for
an animal is the ratio of resting and maximum rate of metabolism for
that particular species as determined by oxygen consumption.
Oxygen consumption V
O2 and maximum oxygen consumption
VO2 max. Oxygen consumption in species that differ in body size and organ system dimensions show a similarity in their charted V
O2
distributions indicating that, despite the complexity of their systems,
there is a power law dependence of similarity; therefore, universal
patterns are observed in diverse animal taxonomy.
Across a broad range of species, allometric relations are not
necessarily linear on a log-log scale. For example, the maximal running
speeds of mammals show a complicated relationship with body mass, and
the fastest sprinters are of intermediate body size.
Allometric muscle characteristics
The
muscle
characteristics of animals are similar in a wide range of animal sizes,
though muscle sizes and shapes can and often do vary depending on
environmental constraints placed on them. The muscle tissue itself
maintains its contractile characteristics and does not vary depending on
the size of the animal. Physiological scaling in muscles affects the
number of muscle fibers and their intrinsic speed to determine the
maximum power and efficiency of movement in a given animal. The speed of
muscle recruitment varies roughly in inverse proportion to the cube
root of the animal’s weight (compare the intrinsic
frequency of the sparrow’s flight muscle to that of a stork).
For inter-species allometric relations related to such ecological
variables as maximal reproduction rate, attempts have been made to
explain scaling within the context of
dynamic energy budget theory and the
metabolic theory of ecology. However, such ideas have been less successful.
Allometry of legged locomotion
Methods of study
Allometry has been used to study patterns in locomotive principles across a broad range of species. Such research has been done in pursuit of a better understanding of
animal locomotion, including the factors that different gaits seek to
optimize.
Allometric trends observed in extant animals have even been combined
with evolutionary algorithms to form realistic hypotheses concerning the
locomotive patterns of extinct species.
These studies have been made possible by the remarkable similarities
among disparate species’ locomotive kinematics and dynamics, “despite
differences in morphology and size”.
Allometric study of locomotion involves the analysis of the
relative sizes, masses, and limb structures of similarly shaped animals
and how these features affect their movements at different speeds. Patterns are identified based on dimensionless
Froude numbers, which incorporate measures of animals’ leg lengths, speed or stride frequency, and weight.
Alexander incorporates Froude-number analysis into his “dynamic
similarity hypothesis” of gait patterns. Dynamically similar gaits are
those between which there are constant coefficients that can relate
linear dimensions, time intervals, and forces. In other words, given a
mathematical description of gait A and these three coefficients, one
could produce gait B, and vice versa. The hypothesis itself is as
follows: “animals of different sizes tend to move in dynamically similar
fashion whenever the ratio of their speed allows it.” While the dynamic
similarity hypothesis may not be a truly unifying principle of animal
gait patterns, it is a remarkably accurate heuristic.
It has also been shown that living organisms of all shapes and
sizes utilize spring mechanisms in their locomotive systems, probably in
order to minimize the energy cost of locomotion. The allometric study of these systems has fostered a better understanding of why spring mechanisms are so common, how limb compliance varies with body size and speed, and how these mechanisms affect general limb kinematics and dynamics.
Principles of legged locomotion identified through allometry
- Alexander found that animals of different sizes and masses traveling with the same Froude number consistently exhibit similar gait patterns.
- Duty factors—percentages of a stride during which a foot maintains
contact with the ground—remain relatively constant for different animals
moving with the same Froude number.
- The dynamic similarity hypothesis states that "animals of different
sizes tend to move in dynamically similar fashion whenever the ratio of
their speed allows it".
- Body mass has even more of an effect than speed on limb dynamics.
- Leg stiffness,
, is proportional to 
, where is body mass.
- Peak force experienced throughout a stride is proportional to
.
- The amount by which a leg shortens during a stride (i.e. its peak displacement) is proportional to
.
- The angle swept by a leg during a stride is proportional to
.
- The mass-specific work rate of a limb is proportional to
.
Drug dose scaling
The physiological effect of drugs and other substances in many cases scales allometrically.
West, Brown, and Enquist in 1997 derived a hydrodynamic theory to
explain the universal fact that metabolic rate scales as the ¾ power
with body weight. They also showed why lifespan scales as the +¼ power
and heart rate as the -¼ power. Blood flow (+¾) and resistance (-¾)
scale in the same way, leading to blood pressure being constant across
species.
Hu and Hayton in 2001 discussed whether the basal metabolic rate
scale is a ⅔ or ¾ power of body mass. The exponent of ¾ might be used
for substances that are eliminated mainly by metabolism, or by
metabolism and excretion combined, while ⅔ might apply for drugs that
are eliminated mainly by renal excretion.
An online allometric scaler of drug doses based on the above work is available.
The US
Food and Drug Administration
(FDA) published guidance in 2005 giving a flow chart that presents the
decisions and calculations used to generate the maximum recommended
starting dose in drug
clinical trials from animal data.
Allometric scaling in fluid locomotion
The mass and density of an organism have a large effect on the
organism's locomotion through a fluid. For example, a tiny organisms
uses flagella and can effectively move through a fluid it is suspended
in. Then on the other scale a blue whale that is much more massive and
dense in comparison with the viscosity of the fluid, compared to a
bacterium in the same medium. The way in which the fluid interacts with
the external boundaries of the organism is important with locomotion
through the fluid. For streamlined swimmers the resistance or drag
determines the performance of the organism. This drag or resistance can
be seen in two distinct flow patterns. There is Laminar Flow where the
fluid is relatively uninterrupted after the organism moves through it.
Turbulent flow is the opposite, where the fluid moves roughly around an
organisms that creates vortices that absorb energy from the propulsion
or momentum of the organism. Scaling also affects locomotion through a
fluid because of the energy needed to propel an organism and to keep up
velocity through momentum. The rate of oxygen consumption per gram body
size decreases consistently with increasing body size.
In general, smaller, more streamlined organisms create laminar flow (
R < 0.5x106), whereas larger, less streamlined organisms produce turbulent flow (
R > 2.0×106).
Also, increase in velocity (V) increases turbulence, which can be
proved using the Reynolds equation. In nature however, organisms such
as a 6‘-6” dolphin moving at 15 knots does not have the appropriate
Reynolds numbers for laminar flow
R = 107, but exhibit it in
nature. Mr. G.A Steven observed and documented dolphins moving at 15
knots alongside his ship leaving a single trail of light when
phosphorescent activity in the sea was high. The factors that contribute
are:
- Surface area of the organism and its effect on the fluid in
which the organism lives is very important in determining the parameters
of locomotion.
- The Velocity of an organism through fluid changes the dynamic of the
flow around that organism and as velocity increases the shape of the
organism becomes more important for laminar flow.
- Density and viscosity of fluid.
- Length of the organism is factored into the equation because the
surface area of just the front 2/3 of the organism has an effect on the
drag
The resistance to the motion of an approximately stream-lined solid through a fluid can be expressed by the formula: Cfρ(total surface)V2/2 V = velocity
- ρ = density of fluid
- Cf = 1.33R − 1 (laminar flow) R = Reynolds number
- Reynolds number [R] = VL/ν
- V = velocity
- L = axial length of organism
- ν = kinematic viscosity (viscosity/density)
Notable Reynolds numbers:
- R < 0.5x106 = laminar flow threshold
- R > 2.0x106 = turbulent flow threshold
Scaling also has an effect on the performance of organisms in fluid.
This is extremely important for marine mammals and other marine
organisms that rely on atmospheric oxygen to survive and carry out
respiration. This can affect how fast an organism can propel itself
efficiently and more importantly how long it can dive, or how long and
how deep an organism can stay underwater. Heart mass and lung volume
are important in determining how scaling can affect metabolic function
and efficiency. Aquatic mammals, like other mammals, have the same size
heart proportional to their bodies.
Mammals have a heart that is about 0.6% of the total body mass
across the board from a small mouse to a large Blue Whale. It can be
expressed as: Heart Weight = 0.006Mb1.0, where Mb is the body mass of
the individual.
Lung volume is also directly related to body mass in mammals (slope =
1.02). The lung has a volume of 63 ml for every kg of body mass. In
addition, the tidal volume at rest in an individual is 1/10 the lung
volume. Also respiration costs with respect to oxygen consumption is
scaled in the order of Mb.75.
This shows that mammals, regardless of size, have the same size
respiratory and cardiovascular systems and it turn have the same amount
of blood: About 5.5% of body mass. This means that for a similarly
designed marine mammals, the larger the individual the more efficiently
they can travel compared to a smaller individual. It takes the same
effort to move one body length whether the individual is one meter or
ten meters. This can explain why large whales can migrate far distance
in the oceans and not stop for rest. It is metabolically less expensive
to be larger in body size.
This goes for terrestrial and flying animals as well. In fact, for an
organism to move any distance, regardless of type from elephants to
centipedes, smaller animals consume more oxygen per unit body mass than
larger ones. This metabolic advantage that larger animals have makes it
possible for larger marine mammals to dive for longer durations of time
than their smaller counterparts. That the heart rate is lower means that
larger animals can carry more blood, which carries more oxygen. Then
in conjuncture with the fact that mammals reparation costs scales in the
order of Mb.75 shows how an advantage can be had in having a larger
body mass. More simply, a larger whale can hold more oxygen and at the
same time demand less metabolically than a smaller whale.
Traveling long distances and deep dives are a combination of good
stamina and also moving an efficient speed and in an efficient way to
create laminar flow, reducing drag and turbulence. In sea water as the
fluid, it traveling long distances in large mammals, such as whales, is
facilitated by their neutral buoyancy and have their mass completely
supported by the density of the sea water. On land, animals have to
expend a portion of their energy during locomotion to fight the effects
of gravity.
Flying organisms such as birds are also considered moving through
a fluid. In scaling birds of similar shape, it has also been seen that
larger individuals have less metabolic cost per kg than smaller species,
which would be expected because it holds true for every other form of
animal. Birds also have a variance in wing beat frequency. Even with
the compensation of larger wings per unit body mass, larger birds also
have a slower wing beat frequency, which allows larger birds to fly at
higher altitudes, longer distances, and faster absolute speeds than
smaller birds. Because of the dynamics of lift-based locomotion and the
fluid dynamics, birds have a U-shaped curve for metabolic cost and
velocity. Because flight, in air as the fluid, is metabolically more
costly at the lowest and the highest velocities. On the other end,
small organisms such as insects can make gain advantage from the
viscosity of the fluid (air) that they are moving in. A wing-beat timed
perfectly can effectively uptake energy from the previous stroke.
(Dickinson 2000) This form of wake capture allows an organism to recycle
energy from the fluid or vortices within that fluid created by the
organism itself. This same sort of wake capture occurs in aquatic
organisms as well, and for organisms of all sizes. This dynamic of fluid
locomotion allows smaller organisms to gain advantage because the
effect on them from the fluid is much greater because of their
relatively smaller size.
Allometric engineering
Allometric engineering is a method for manipulating allometric relationships within or among groups.
In characteristics of a city
Arguing
that there are a number of analogous concepts and mechanisms between
cities and biological entities, Bettencourt et al. showed a number of
scaling relationships between observable properties of a city and the
city size. GDP, "supercreative" employment, number of inventors, crime,
spread of disease, and even pedestrian walking speeds scale with city population.
Examples
Some examples of allometric laws:
- Kleiber's law, metabolic rate
is proportional to body mass raised to the 
power:
-
- breathing and heart rate
are both inversely proportional to body mass raised to the 
power:
-
- mass transfer contact area
and body mass :
-
- the proportionality between the optimal cruising speed
of flying bodies (insects, birds, airplanes) and body mass raised to the power 
:
-
Determinants of size in different species
Many
factors go into the determination of body mass and size for a given
animal. These factors often affect body size on an evolutionary scale,
but conditions such as availability of food and
habitat size can act much more quickly on a species. Other examples include the following:
- Basic physiological design plays a role in the size of a given
species. For example, animals with a closed circulatory system are
larger than animals with open or no circulatory systems.
- Mechanical design can also determine the maximum allowable size
for a species. Animals with tubular endoskeletons tend to be larger than
animals with exoskeletons or hydrostatic skeletons.
- An animal’s habitat throughout its evolution
is one of the largest determining factors in its size. On land, there
is a positive correlation between body mass of the top species in the
area and available land area.
However, there are a much greater number of “small” species in any
given area. This is most likely determined by ecological conditions,
evolutionary factors, and the availability of food; a small population
of large predators depend on a much greater population of small prey to
survive. In an aquatic environment, the largest animals can grow to have
a much greater body mass than land animals where gravitational weight
constraints are a factor.
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