on being the right size

Allometry [uh-lom-i-tree] is the study of the relationship of body size to shape. In particular, it refers to the rate of growth of one part of the body compared to other parts. In most cases, the relative size of body parts changes as the body grows. Most allometric relationships are adaptive.

For example, organs which depend on their surface area (such as the intestine) grow faster as the body weight increases. Also, there are changes in allometry in the evolution of a clade (branches on the tree of life).

Allometry is an important way to describe changes in gross morphology (body shape) during evolution. Changes in time of development in an evolutionary series or clade are very common. The trend is known as heterochrony. British geneticist JBS Haldane’s 1926 essay ‘On being the right size’ gives an overview of the way size interacts with body structure. Haldane’s thesis is that sheer size very often defines what bodily equipment an animal must have: ‘Insects, being so small, do not have oxygen-carrying bloodstreams.

What little oxygen their cells require can be absorbed by simple diffusion of air through their bodies. But being larger means an animal must take on complicated oxygen pumping and distributing systems to reach all the cells.’ Many of his examples are based on the square-cube law (as a shape grows in size, its volume grows faster than its surface area). If an animal’s length is doubled, its surface area will be squared and its weight cubed. This alone causes allometric changes in any evolutionary lineage where successive species get larger of smaller.

If an animal were scaled up by a considerable amount, its relative muscular strength would be severely reduced, since the cross section of its muscles would increase by the square of the scaling factor while its mass would increase by the cube of the scaling factor. As a result of this, cardiovascular and respiratory functions would be severely burdened. In the case of flying animals, the wing loading would be increased if they were scaled up, and they would therefore have to fly faster to gain the same amount of lift. Air resistance per unit mass is also higher for smaller animals, which is why a small animal like an ant cannot be crushed by falling from any height.

Large animals do not look like small animals: an elephant cannot be mistaken for a mouse scaled up in size. The bones of an elephant are necessarily proportionately much larger than the bones of a mouse, because they must carry proportionately higher weight. As was elucidated by J. B. S. Haldane, ‘…consider a man 60 feet high…Giant Pope and Giant Pagan in the illustrated ‘Pilgrim’s Progress’…. These monsters…weighed 1000 times as much as Christian. Every square inch of a giant bone had to support 10 times the weight borne by a square inch of human bone. As the human thigh-bone breaks under about 10 times the human weight, Pope and Pagan would have broken their thighs every time they took a step.’

The giant animals seen in horror movies (e.g., Godzilla or King Kong) are also unrealistic, as their sheer size would force them to collapse. However, it’s no coincidence that the largest animals in existence today are aquatic animals, because the buoyancy of water negates to some extent the effects of gravity. Therefore, sea creatures can grow to very large sizes without the same musculoskeletal structures that would be required of similarly sized land creatures.

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. 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 because they lack parental care after birth or hatching and because they exhibit a large range of body size between the juvenile and adult stage. Lizards often exhibit allometric changes during their ontogeny (development). 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 occurs when changes in size (during growth or over evolutionary time) do not lead to changes in proportion. 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. Therefore, 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.

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. 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.’ 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.

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.

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.’ 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.

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. 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. Lung volume is also directly related to body mass in mammals. The lung has a volume of 63 ml for every kg of body mass. 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.

This metabolic advantage that larger animals have make it possible for larger marine mammals to dive for longer durations than their smaller counterparts. The fact that the heart rate is lower means that larger animals can carry more blood, which carries more oxygen. Traveling long distances and deep dives are a combination of good stamina and moving in an efficient way to create laminar flow, reducing drag and turbulence. In sea water as the fluid, large mammals, such as whales, are aided 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.

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.

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. 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.

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.

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 physiological design (e.g. animals with a closed circulatory system are larger than animals with open or no circulatory systems) and mechanical design (e.g. animals with tubular endoskeletons tend to be larger than animals with exoskeletons or hydrostatic skeletons).

An animal’s habitat throughout its evolution is also 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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