Allometry

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 (allometric equation) which expresses a remarkable scale symmetry: : y = k x^a, or in a logarithmic form, : \log y = a \log x + \log k, or similarly, : \ln y = a \ln x + \ln k, where a 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. The allometric equation can also be acquired as a solution of the differential equation : \frac{dy}{y} = a \frac{dx}{x}. 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 (and sex), which is referred to as static allometry. Comparisons of species are used to examine interspecific or evolutionary allometry (see also Phylogenetic comparative methods). ==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 (snout–vent length, total length, etc.). 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 (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 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 (such as those that follow isometry). 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 (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, 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, 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 (assuming this slope is statistically significant), it would be negatively allometric. Examples axes To find the expected slope for the relationship between mass and the characteristic length of an animal (see figure), the units of mass (M) from the y-axis are divided by the units of the x-axis, Length (L). The expected slope on a double-logarithmic plot of L3 / L is 3 (\frac{\log_{10}\mathrm{L}^3}{\log_{10}\mathrm{L}}=3). This is the slope of a straight line. Another example: Force is dependent on the cross-sectional area of muscle (CSA), 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 (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. 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: :\text{Metabolic rate} = 70 M^{0.75} (although the universality of this relation has been disputed both empirically and theoretically) where M 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, 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). : \mathrm{frequency} = \frac{1}{\mathrm{mass}^{1/3}} 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. The allometric study of these systems has fostered a better understanding of why spring mechanisms are so common, Hu and Hayton in 2001 discussed whether the basal metabolic rate scale is a or power of body mass. The exponent of might be used for substances that are eliminated mainly by metabolism, or by metabolism and excretion combined, while 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 organism uses flagella and can effectively move through a fluid it is suspended in, while on the other end of the scale, a blue whale is much more massive and dense relative to 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: laminar flow, where the fluid is relatively uninterrupted after the organism moves through it, and turbulent flow, where the fluid moves roughly around an organism, creating 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 keep up velocity through momentum. The rate of oxygen consumption per gram body size decreases consistently with increasing body 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, scale with city population. This phenomenon goes under the name of urban scaling. Theoretical explanations for the presence of allometry in cities propose different mechanisms. Bettencourt’s model suggests that superlinear scaling arises from the quadratic growth of social interactions with population size under budget constraints. A different mechanism was proposed by Gomez-Lievano et al. in which superlinear scaling is linked to the exponential growth in outputs resulting from the combination of diverse, complementary factors (or capabilities) found in cities, which scale logarithmically with city size. ==Examples==

Some examples of allometric laws: • Kleiber's law, metabolic rate q_0 is proportional to body mass M raised to the 3/4 power: :: q_{0} \sim M^{\frac 3 4} • breathing and heart rate t are both inversely proportional to body mass M raised to the 1/4 power: :: t \sim M^{-\frac{1}{4}} • mass transfer contact area A and body mass M: :: A \sim M^{\frac 7 8} • the proportionality between the optimal cruising speed V_{opt} of flying bodies (insects, birds, airplanes) and body mass M raised to the power 1/6: :: V_\text{opt} \sim M^{\frac 1 6} ==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: • Physiological design : 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. 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. ==See also==