Allometry
Allometry is the study of the relationship of body size to shape, anatomy, physiology and behaviour, first outlined by Otto Snell in 1892, by D'Arcy Thompson in 1917 in On Growth and Form and by Julian Huxley in 1932.
Overview
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, 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,
or similarly,
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. Other methods include measurement-error models and a particular kind of principal component analysis.
The allometric equation can also be acquired as a solution of the differential equation
Allometry often studies shape differences in terms of ratios of the objects' dimensions. Two objects of different size, but common shape, 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, which is referred to as static allometry. Comparisons of species are used to examine interspecific or evolutionary allometry.
Isometric scaling and geometric similarity
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. A perfectly allometrically 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, 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 muskellunge will weigh about, while a muskellunge will weigh about, 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. Using 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.Data gathered in science do not fall neatly in a straight line, so data transformations are useful. It is also important to remember 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 why logarithmic transformation should be used to study allometry —a biological reason and a statistical reason. 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 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, 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 . Sometimes, the two analyses can yield different results, but often they do not. If the expected slope is outside the confidence intervals, allometry is present. If the mass in this imaginary animal scaled with a slope of 5, which 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, it would be negatively allometric.
Examples
To find the expected slope for the relationship between mass and the characteristic length of an animal, the units of mass from the y-axis are divided by the units of the x-axis, Length. The expected slope on a double-logarithmic plot of L3 / L is 3. This is the slope of a straight line.Another example: Force is dependent on the cross-sectional area of muscle, which is L2. 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, 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 show scaling, mostly associated with the ratio between surface area and mass 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 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 have lower mass-specific metabolic rates and lower heart rates, as compared with smaller-bodied species. 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:
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 had lowest BMR and the mouse 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 VO2 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 VO2 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.