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| {{Multiple image|align=right|direction=vertical|width=300|image1=Elephant_skeleton.jpg |caption1=Skeleton of an [[elephant]]|image2=Dasyurus_maculatus_skeleton.jpg|caption2=Skeleton of a [[Tiger Quoll]] ''(Dasyurus maculatus)''.<br><br>Note the proportionately thicker bones in the elephant, an example of allometric scaling}}
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| '''Allometry''' is the study of the relationship of body size to [[shape]],<ref>{{cite book|first=Christopher G. |last=Small |title=The Statistical Theory of Shape |year=1996 |publisher=Springer |isbn=0-387-94729-9 |page=4}}</ref> [[anatomy]], [[physiology]] and finally behaviour,<ref>{{cite journal |author=Damuth J |title=Scaling of growth: plants and animals are not so different |journal=Proc. Natl. Acad. Sci. U.S.A. |volume=98 |issue=5 |pages=2113–4 |date=February 2001 |pmid=11226197 |pmc=33381 |doi=10.1073/pnas.051011198 |url=http://www.pnas.org/cgi/pmidlookup?view=long&pmid=11226197|bibcode = 2001PNAS...98.2113D }}</ref> first outlined by [[Otto Snell]] in 1892,<ref>{{cite journal |author=Otto Snell |title=Die Abhängigkeit des Hirngewichts von dem Körpergewicht und den geistigen Fähigkeiten |journal=Arch. Psychiatr. |volume=23 |issue= 2|pages=436–446 |year=1892 |doi=10.1007/BF01843462}}</ref> [[D'Arcy Thompson]] in 1917<ref>{{cite book| first=D'Arcy W |last=Thompson |title=On Growth and Form |edition=Canto |year=1992 |publisher=Cambridge University Press |isbn=978-0-521-43776-9 |url=http://books.google.com/books?id=7_F4OUJmLFcC
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| }}</ref> and [[Julian Huxley]] in 1932.<ref>{{cite book |first=Julian S. |last=Huxley |title=Problems of Relative Growth |edition=2nd |year=1972 |publisher=Dover |location=New York |isbn=0-486-61114-0}}</ref> 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., the [[Hercules Beetle]]), 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]]:
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| : <math>y = kx^{a} \,\!</math> or in a logarithmic form: <math>\log y = a \log x + \log k\,\!</math>
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| where <math>a</math> is the '''scaling exponent''' of the law. Methods for estimating this exponent from data 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 [[Power_law#Power-law_functions|principal component analysis]].
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| Allometry often studies shape differences in terms of [[ratio]]s 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 because they lack [[parental care]] after [[birth]] or hatching and because they exhibit a large range of body size between the [[Juvenile (organism)|juvenile]] and [[adult]] stage. Lizards often exhibit allometric changes during their [[ontogeny]].<ref>{{cite journal |last= [[Theodore Garland, Jr.|Garland]]| first=T., Jr. |coauthors=P. L. Else |date=March 1987 |title=Seasonal, sexual, and individual variation in endurance and activity metabolism in lizards | url=http://www.biology.ucr.edu/people/faculty/Garland/GarlEl87.pdf |journal=Am J Physiol. |volume=252 |issue=3 Pt 2 |pages=R439–49 |pmid=3826408 }}</ref>
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| 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]]).
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| ==Isometric scaling and geometric similarity==
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| Isometric scaling occurs 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.<ref>{{cite journal |author=Emerson SB |title=Allometry and Jumping in Frogs: Helping the Twain to Meet |journal=Evolution |volume=32 |issue=3 |pages=551–564 |date=September 1978 |jstor=2407721 }}</ref> 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.
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| 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.
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| Isometric scaling is often used as a null hypothesis in scaling studies, with 'deviations from isometry' considered evidence of physiological factors forcing allometric growth.
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| ==Allometric scaling==
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| Allometric scaling is any change that deviates from [[isometry]]. A classic example is the skeleton of mammals, which becomes much more robust and massive relative to the size of the body as the body size increases.<ref name=Schmidt-Nielsen>{{harvnb|Schmidt-Nielsen|1984}}</ref> 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 2/3, and all length-based properties change with mass to the 1/3 power. If, after statistical analyses, for example, a volume-based property was found to scale to mass to the 0.9 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 0.8 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.<ref name="Christian_and_Garland_1996">{{cite journal |author = Christian, A. |coauthors = [[Theodore Garland, Jr.|Garland, T., Jr.]] |year= 1996 |title= Scaling of limb proportions in monitor lizards (Squamata: Varanidae) |journal= Journal of Herpetology |volume= 30 |pages= 219–230 |url= http://www.biology.ucr.edu/people/faculty/Garland/ChriGa96.pdf |format=PDF |issue = 2 |jstor = 1565513 }}</ref>
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| ==Determining if a system is scaling with allometry==
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| 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.<ref>{{cite book|last=Pennycuick|first=Colin J.|title=Newton Rules Biology|year=1992|publisher=Oxford University Press |isbn=0-19-854021-3 |page=111}}</ref><ref>{{harvnb|Schmidt-Nielsen|1984|p=237}}</ref><ref>{{cite book |first=J.C. |last=Gibbings |title=Dimensional Analysis |url=http://books.google.com/books?id=Q6iflrgVaWcC |year=2011 |publisher=Springer |isbn=978-1-84996-317-6}}</ref> 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 you know 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<ref name="Hill_1950">{{cite journal|last=Hill|first=A.V.|title=The dimensions of animals and their muscular dynamics|journal=Science Progress|date=April 1950|volume=150|bibcode=1949Natur.164R.820|page=820|doi=10.1038/164820b0|issue=4176}}</ref> and interpreted by Wilson 2000,<ref>{{cite journal |author=Wilson RS, Franklin CE, James RS |title=Allometric scaling relationships of jumping performance in the striped marsh frog ''Limnodynastes peronii'' |journal=J. Exp. Biol. |volume=203 |issue=Pt 12 |pages=1937–46 |date=June 2000 |pmid=10821750 |url=http://jeb.biologists.org/cgi/pmidlookup?view=long&pmid=10821750}}</ref> but in actuality larger frogs do jump longer distances.
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| [[Dimensional analysis]] is extremely useful for balancing units in an equation or in our case, determining expected slope. A few dimensional examples below:
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| M-Mass
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| L-Length
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| V-Volume (which is also L cubed because a volume is merely length cubed)
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| [[File:Scalingfigure.jpg|thumb|Allometric relations show as straight lines when plotted on [[Logarithmic scale|double-logarithmic]] axes]]
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| If trying to find the expected slope for the relationship between mass and the characteristic length of an animal (see figure) you would take the units of mass (M=L<sup>3</sup>, because mass is a volume-volumes are lengths cubed) from the Y axis and divide them by the X axis (in this case L). Your expected slope on a double-logarithmic plot of L<sup>3</sup>/ L<sup>1</sup> in this case is 3 (log<sub>10</sub>(L<sup>3</sup>)/log<sub>10</sub>(L<sup>1</sup>)=3). This is the slope of a straight line, but most data gathered in science does not automatically fall neatly in a straight line, so data transformations are useful.
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| It is also important to keep in mind what you are comparing in your data. Comparing head length to head width can yield different results than comparing to body length. Sometimes a characteristic length such as head length may scale differently to its width than body length.<ref>{{cite journal|last=Robinson|first=Michael|first2=Philip |last2=Motta|title=Patterns of growth and the effects of scale on the feeding kinematics of the nurse shark (''Ginglymostoma cirratum'')|journal=Journal of Zoology, London|year=2002|volume=256|pages=449–462|doi=10.1017/S0952836902000493|issue=4}}</ref>
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| A common way to analyze data such as those collected in scaling is to use [[Log-log plot|log-transformation]]. 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.<ref>{{cite journal |author=O’Hara, R.B.; Kotze, D.J. |title=Do not log-transform count data |journal=Methods in Ecology and Evolution |volume=1 |pages=118–122 |year=2010 |doi=10.1111/j.2041-210X.2010.00021.x |issue=2 }}</ref> Before analyzing data though, it is important to have a predicted slope of the line to compare your analysis to. After data are log transformed and linearly regressed. You can then use [[least squares regression]] with 95% confidence intervals or [http://www.bio.sdsu.edu/pub/andy/RMA.html 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 our 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 is 3 and in reality in a certain organism mass scaled with 1 (assuming this slope is statistically significant), then it would be negatively allometric.
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| Another example: Force is dependent on the cross-sectional area of muscle (CSA), which is L<sup>2</sup>. If comparing force to a length, then your expected slope is 2.
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| Alternatively, this analysis may be accomplished with a power regression. Plot the relationship between your data onto a graph. Fit this to a power curve (depending on your stats program, this can be done multiple ways) and it will give you an equation with the form y=Zx^number. That “number” is the relationship between your data points. The downside to this form of analysis is that it makes it a little more difficult to do statistical analyses.
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| ==Physiological scaling==
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| 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.
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| ===Metabolic rate and body mass===
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| 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|power-law]] dependence. Overall metabolic rate in animals is generally accepted to show negative allometry, scaling to mass to a power ≈ 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), this straight line 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."<ref name=bettencourt>{{cite journal |author=Bettencourt LM, Lobo J, Helbing D, Kühnert C, West GB |title=Growth, innovation, scaling, and the pace of life in cities |journal=Proc. Natl. Acad. Sci. U.S.A. |volume=104 |issue=17 |pages=7301–6 |date=April 2007 |pmid=17438298 |pmc=1852329 |doi=10.1073/pnas.0610172104 |url= http://www.pnas.org/content/104/17/7301.long|bibcode = 2007PNAS..104.7301B }}</ref>
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| Max Kleiber contributed the following allometric equation for relating the BMR to the body mass of an animal.<ref name="Willmer_2009">{{cite book|last=Willmer|first=Pat|title=Environmental Physiology of Animals|year=2009|publisher=Wiley-Blackwell}}</ref> Statistical analysis of the intercept did not vary from 70 and the slope was not varied from 0.75, thus:
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| :<math>\mathrm{Metabolic\ Rate} = 70 M^{0.75}</math>
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| where <math>M</math> is body mass, and metabolic rate is measured in [[Calorie|kcal]] per day.
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| Consequently the body mass itself can explain the majority of the variation in the BMR. In removing the concept of body mass, the taxonomy of the animal assumes a major 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,<ref name=Willmer_2009 /> 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 this demonstrate the physiological adaptations to environmental changes that animals undergo.
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| 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.
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| Oxygen consumption V<sub>O</sub><sub>2</sub> 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<sub>O</sub><sub>2</sub> distributions indicating that despite the complexity in their systems, there is a power law dependence of similarity; therefore, universal patterns are observed in diverse animal taxonomy.<ref>{{cite journal |author=Labra FA, Marquet PA, Bozinovic F |title=Scaling metabolic rate fluctuations |journal=Proc. Natl. Acad. Sci. U.S.A. |volume=104 |issue=26 |pages=10900–3 |date=June 2007 |pmid=17578913 |pmc=1904129 |doi=10.1073/pnas.0704108104 |url=http://www.pnas.org/cgi/pmidlookup?view=long&pmid=17578913|bibcode = 2007PNAS..10410900L }}</ref>
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| 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.<ref name='Garland_1983_JZL'>{{cite journal|last=[[Theodore Garland, Jr.|Garland]] Jr.|first=T. |year=1983 |title=The relation between maximal running speed and body mass in terrestrial mammals| url=http://www.biology.ucr.edu/people/faculty/Garland/Garl1983_JZL.pdf |journal=Journal of Zoology, London |volume=199 |issue= 2|pages=157–170 |doi=10.1111/j.1469-7998.1983.tb02087.x |format=PDF}}</ref><ref name='Chappell_1989'>{{cite journal|last=Chappell|first=R. |year=1989 |title=Fitting bent lines to data, with applications to allometry |journal=Journal of Theoretical Biology |volume=138 |issue= 2|pages=235–256 |doi=10.1016/S0022-5193(89)80141-9|pmid=2607772 }}</ref>
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| ===Allometric muscle characteristics===
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| 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, such as the intrinsic [[frequency]] of the sparrow’s flight muscle compared to that of a stork’s.
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| : <math>\mathrm{freq} = \frac{1}{\mathrm{mass}^{1/3}}</math>
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| 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.
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| ===Allometry of legged locomotion===
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| ====Methods of study====
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| Allometry has been used to study patterns in locomotive principles across a broad range of species.<ref name="Daley2010">{{Cite journal | title=Two explanations for the compliant running paradox: reduced work of bouncing viscera and increased stability in uneven terrain | journal=Biology Letters | volume=6 | pages=418–421 | year=2010 | first1=Monica A. | last1=Daley | first2=James R. | last2=Usherwood | doi=10.1098/rsbl.2010.0175 | issue=3 | pmid=20335198 | pmc=2880072}}</ref><ref name="Farley1993">{{cite journal |author=Farley CT, Glasheen J, McMahon TA |title=Running springs: speed and animal size |journal=J. Exp. Biol. |volume=185 |pages=71–86 |date=December 1993 |pmid=8294853 |url=http://jeb.biologists.org/cgi/pmidlookup?view=long&pmid=8294853 |issue=1}}</ref><ref name="Sellers2007">{{Cite journal | title=Estimating dinosaur maximum running speeds using evolutionary robotics | journal=Proceedings of the Royal Society B | volume=274 | pages=2711–6 | year=2007 | first1=William Irving | last1=Sellers | first2=Phillip Lars | last2=Manning | doi=10.1098/rspb.2007.0846 | pmid=17711833 | pmc=2279215 | issue=1626}}</ref><ref name="Alexander1984">{{Cite journal | title=The gaits of bipedal and quadrupedal animals | journal=The International Journal of Robotics Research | volume=3 | issue=2 | pages=49–59 | year=1984 | first1=R. McN. | last1=Alexander | doi=10.1177/027836498400300205}}</ref> Such research has been done in pursuit of a better understanding of animal locomotion, including the factors that different gaits seek to optimize.<ref name="Alexander1984" /> Allometric trends observed in extant animals have even been combined with evolutionary algorithms to form realistic hypotheses concerning the locomotive patterns of extinct species.<ref name="Sellers2007" /> These studies have been made possible by the remarkable similarities among disparate species’ locomotive kinematics and dynamics, “despite differences in morphology and size”.<ref name="Daley2010" />
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| 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.<ref name="Alexander1984" /> Patterns are identified based on dimensionless [[Froude number]]s, which incorporate measures of animals’ leg lengths, speed or stride frequency, and weight.<ref name="Sellers2007" /><ref name="Alexander1984" />
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| 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.<ref name="Alexander1984" />
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| 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.<ref name="Roberts2011">{{Cite journal | title=Fleximble mechanisms: the diverse roles of biological springs in vertebrate movement | journal=The Journal of Experimental Biology | volume=214 | pages=353–361 | year=2011 | first1=Thomas J. | last1=Roberts | first2=Emanuel | last2=Azizi | doi=10.1242/jeb.038588 | issue=3}}</ref> The allometric study of these systems has fostered a better understanding of why spring mechanisms are so common,<ref name="Roberts2011" /> how limb compliance varies with body size and speed,<ref name="Daley2010" /> and how these mechanisms affect general limb kinematics and dynamics.<ref name="Farley1993" />
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| ====Principles of legged locomotion identified through allometry====
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| * Alexander found that animals of different sizes and masses traveling with the same [[Froude number]] consistently exhibit similar gait patterns.<ref name="Alexander1984" />
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| * 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.<ref name="Alexander1984" />
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| * 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”.<ref name="Alexander1984" />
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| * Body mass has even more of an effect than speed on limb dynamics.<ref name="Farley1993" />
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| * Leg stiffness, <math>\mathrm{k_{leg}}=\frac{\mathrm{peak\ force}}{\mathrm{peak\ displacement}}</math>, is proportional to <math>M^{0.67}</math>, where <math>M</math> is body mass.<ref name="Farley1993" />
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| * Peak force experienced throughout a stride is proportional to <math>M^{0.97}</math>.<ref name="Farley1993" />
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| * The amount by which a leg shortens during a stride (i.e. its peak displacement) is proportional to <math>M^{0.30}</math>.<ref name="Farley1993" />
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| * The angle swept by a leg during a stride is proportional to <math>M^{-0.034}</math>.<ref name="Farley1993" />
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| * The mass-specific work rate of a limb is proportional to <math>M^{0.11}</math>.<ref name="Farley1993" />
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| ==Allometric scaling in fluid locomotion==
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| ''This technical section is not well-written and needs editing''.
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| The mass/density of organism has a large effect on organisms locomotion through a fluid. For example tiny organisms use 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.<ref name="Schmidt-Nielsen97"/> (Knut Schmidt-Nielson 2004)
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| In general, smaller, more streamlined organisms create laminar flow (R<0.5x106), whereas larger, less streamlined organisms produce turbulent flow (R>2.0×106).<ref name=Hill_1950 /> 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:
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| * 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.
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| * 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.
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| * Density and viscosity of fluid.
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| * 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
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| '''The resistance to the motion of an approximately stream-lined solid through a fluid can be expressed by the formula: <big>C<sub>fρ</sub>(total surface)V2/2</big> <ref name=Hill_1950 /> <br />'''V = velocity<br />
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| ρ = density of fluid<br />
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| C<sub>f</sub> = 1.33R-1 (laminar flow) R= Reynolds number
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| Reynolds number <big>[R]=VL/ν</big><br />
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| V = velocity<br />
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| L = Axial length of organism<br />
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| ν = kinematic viscosity (viscosity/density)<br />
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|
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| Notable Reynolds numbers:<br />
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| R<0.5x106 = Laminar Flow threshold<br />
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| R>2.0x106 = Turbulent Flow threshold
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| 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.
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| 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.<ref name="Schmidt-Nielsen97"/> 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.<ref name="Schmidt-Nielsen97"/> 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.<ref name="Schmidt-Nielsen97"/> 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. This metabolic advantage that larger animals have make it possible for larger marine mammals to dive for longer durations of time than their smaller counterparts. The fact 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.
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| 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 the fact that they are neutrally buoyant 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.
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| It should be mentioned that 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. This 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. This 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 gain advantage because the effect on them from the fluid is much greater because of there relatively smaller size.
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| <ref name="Schmidt-Nielsen97">{{cite book |first=Knut |last=Schmidt-Nielsen |title=Animal Physiology: Adaptation and Environment |url=http://books.google.com/books?id=Af7IwQWJoCMC |date=10 April 1997 |publisher=Cambridge University Press |isbn=978-0-521-57098-5 |edition=5th |ref=harv}}</ref>
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| <ref>{{cite journal |author=Dickinson MH, Farley CT, Full RJ, Koehl MA, Kram R, Lehman S |title=How animals move: an integrative view |journal=Science |volume=288 |issue=5463 |pages=100–6 |date=April 2000 |pmid=10753108 |bibcode=2000Sci...288..100D |doi=10.1126/science.288.5463.100 }}</ref>
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| ==Allometric engineering==
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| [[Allometric engineering]] is a method for manipulating allometric relationships within or among groups.<ref name=sinervo>{{cite journal|author=Sinervo, B.; Huey, R.|year=1990|title=Allometric Engineering: An Experimental Test of the Causes of Interpopulational Differences in Performance|pages=1106–9|journal= [[Science (journal)|Science]]|volume=248|doi=10.1126/science.248.4959.1106|url=http://faculty.washington.edu/hueyrb/pdfs/SinervoScience1.pdf|issue=4959 |format=PDF|bibcode = 1990Sci...248.1106S }}</ref>
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| ==Allometry in characteristics of a city==
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| 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,<ref name=bettencourt/> and even pedestrian walking speeds<ref>{{cite journal |author=Bornstein MH, Bornstein HG |title=The Pace of Life |journal=Nature |volume=259 |issue=5544 |pages=557–9 |date=19 February 1976 |doi=10.1038/259557a0 |url=http://www.nature.com/nature/journal/v259/n5544/abs/259557a0.html|bibcode = 1976Natur.259..557B }}</ref> scale with city population.
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| ==Examples==
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| [[File:Allometric Law of Body Mass vs Cruising Speed in Constructal Theory.JPEG|thumb|300px|right|Allometric law of cruising speed versus body mass]]Some examples of allometric laws:
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| * [[Kleiber's law]], metabolic rate <math>q_{0}</math> is proportional to body mass <math>M</math> raised to the <math>3/4</math> power:
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| : <math>q_{0} \sim M^{\frac 3 4}</math>
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| * breathing and heart rate <math>t</math> are both proportional to body mass <math>M</math> raised to the <math>1/4</math> power:
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| : <math>t \sim M^{\frac 1 4}</math>
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| * mass transfer contact area <math>A</math> and body mass <math>M</math>:
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| : <math>A \sim M^{\frac 7 8}</math>
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| * the proportionality between the optimal [[cruising speed]] <math>V_{opt}</math> of flying bodies (insects, birds, airplanes) and body mass <math>M</math> raised to the power <math>1/6</math>:
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| : <math>V_{opt} \sim 30 \cdot M^{\frac 1 6} [m \cdot s^{-1}]</math>
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| ==Determinants of size in different species==
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| 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:
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| *Physiological design
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| : 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.<ref name=Willmer_2009 />
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| *Mechanical design
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| : 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.<ref name=Willmer_2009 />
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| *Habitat
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| : 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.<ref name="Burness_2001">{{cite journal|last1=Burness|first1=G. P.|last2=Diamond|first2=Jared|last3=Flannery|first3=Timothy|title=Dinosaurs, dragons, and dwarfs: The evolution of maximal body size|journal=Proc. Natl. Acad. Sci. U.S.A. |year=2001 |volume=98 |issue=25|pages=14518–23|doi=10.1073/pnas.251548698 |pmid=11724953 |pmc=64714|bibcode = 2001PNAS...9814518B }}</ref> 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.<ref name=Hill_1950 />
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| ==See also==
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| {|
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| |-
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| * [[Biomechanics]]
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| * [[Comparative physiology]]
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| * [[Constructal theory]]
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| * [[Evolutionary physiology]]
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| * [[Metabolic theory of ecology]]
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| * [[Phylogenetic comparative methods]]
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| * [[Power law]] (also known as a [[scaling law]])
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| * [[Rensch's rule]]
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| |}
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| ==References==
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| {{reflist|2}}
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| ==Further reading==
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| {{refbegin}}
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| *{{cite book |author=Calder |first=W. A. |title=Size, function and life history |publisher=Harvard University Press |year=1984 |isbn=0-674-81070-8 }}
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| *{{cite book |author=McMahon |first=T. A. |first2=J. T. |last2=Bonner |title=On Size and Life |publisher=Scientific American Library |year=1983 |isbn=0-7167-5000-7 }}
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| *{{cite book |author=Niklas |first=K. J. |title=Plant allometry: The scaling of form and process |publisher=University of Chicago Press |year=1994 |isbn=0-226-58081-4 |url=http://books.google.com/books?id=2th19CVNWtcC}}
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| *{{cite book |author=Peters |first=R. H. |title=The ecological implications of body size |publisher=Cambridge University Press |year=1983 |isbn=0-521-28886-X |url=http://books.google.com/books?id=OYVxiZgTXWsC}}
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| *{{cite book |author=Reiss |first=M. J. |title=The allometry of growth and reproduction |publisher=Cambridge University Press |year=1989 |isbn=0-521-42358-9 |url=http://books.google.com/books?id=SkguxvCDDFcC}}
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| *{{cite book |last=Schmidt-Nielsen |first=K. |authorlink=Knut Schmidt-Nielsen |title=Scaling: why is animal size so important? |publisher=Cambridge University Press |location=Cambridge |year=1984 |isbn=0-521-31987-0 |ref=harv |url=http://books.google.com/books?id=8WkjD3L_avQC}}
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| *{{cite book |first=Thomas T. |last=Samaras |title=Human body size and the laws of scaling: physiological, performance, growth, longevity and ecological ramifications |publisher=Nova Publishers |year=2007 |isbn=1-60021-408-8 |url=http://books.google.com/books?id=PCU0RwDI6c4C}}
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| {{refend}}
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| == External links ==
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| * [http://www.fda.gov/cber/gdlns/dose.htm FDA Guidance for Estimating Human Equivalent Dose] (For "first in human" clinical trials of new drugs)
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| {{anatomy}}
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| [[Category:Branches of biology]]
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| [[Category:Physiology]]
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| [[Category:Ecological experiments]]
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| [[Category:Scales]]
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