Allometry

When Allometrie (from Greek allos "different"; geometry "measure") is about the measurement and comparison of relationships between body size and their relationship to various biological variables. For example, the performance of an organism cannot simply be transferred from small to large. The proportions within a species cannot be implemented 1: 1 either. If, for example, the head increased from an infant to an adult in proportion to the total length of the body, that is, isometric, our head would have to be around 45 cm high.

Mathematical basics

The classic allometric formula

${\displaystyle y=a\cdot x^{b}}$

goes back to Otto Snell . is the body mass (or another reference measure ), the dependent quantity (organ mass, physiological size, etc.), and free parameters. The underlying differential equation is ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle a}$ ${\displaystyle b}$

${\displaystyle {\frac {dy}{y}}=b\cdot {\frac {dx}{x}}}$

Allometric shapes

The exponent is a measure of the ratio of the relative growth rates (absolute growth rate divided by the growth size at the same point in time). If one speaks of isometry , of negative allometry and of positive allometry . However, this limit only applies to dimensions of the same dimension. If the body mass is (3-dimensional) and a length (one-dimensional), then is isometric. In the double logarithmic coordinate system , the power function becomes a straight line ${\displaystyle b}$ ${\displaystyle b=1}$ ${\displaystyle b<1}$ ${\displaystyle b>1}$ ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle b={\tfrac {1}{3}}}$

${\displaystyle \log(y)=\log(a)+b\cdot \log(x)}$

and is her rise . If the slope of this straight line does not change, one speaks of a simple allometry (blue line). If the allometric exponent (i.e. the increase) changes, one speaks of a complex allometry (red line pair). ${\displaystyle b}$

In more recent studies, principal component analyzes are carried out for the allometric calculation of data sets with several variables .

Allometric shapes

Depending on the individuals used for the allometric comparison, a distinction is made between different variants.

• Ontogenetic allometry (growth allometry ): It compares e.g. B. Organ-body relations in growing individuals of a species.
• Intraspecific allometry : Here, biological variables in adult individuals of a species are compared with one another.
• Interspecific allometry : The interspecific allometry compares the behavior of measured variables in adult individuals of several, closely related species up to the taxon family.
• Phylogenetic allometry : Here, biological quantities in adult individuals of different taxa are compared with one another.

It should be noted that the allometry coefficients are not constant over these different levels and therefore cannot be compared with one another. For example, the allometric exponent for the relationship between brain and body mass in the mallard is 0.37, the intraspecific 0.27, the interspecific for the Anatinae 0.58, the phylogenetic for the birds 0.52.

Examples

Metabolism as a function of the size of organisms

The mean size of organisms is between 10 ⁻⁸  m for viruses and 30 m for whales . If one plots time constants of the energy metabolism over this quantity, one recognizes a connection in the form of a power function . In a double logarithmic diagram, the relationship appears linear and the allometric exponent is easily readable as the slope ¾. For example, the mouse-elephant diagram (see Figure 1), which shows the energy metabolism from the mouse to the elephant as a function of weight, is particularly well-known among mammals . The exponent is then 1/4, since the weight is already linked to the size via an exponent of 3.

Figure 1: Mouse-elephant diagram Three examples are shown: heartbeat, breathing and lifespan in a double logarithmic diagram; on the x-axis: body weight in grams; on the y-axis: cycle length in hours

These allometric ratios are also important for everyday life. For example, the dosage of medication is still often calculated based on body weight. The difference may be relatively small in adults. However, if one calculates the drug administration for children in a uniformly proportional ratio, this can have fatal consequences.

In 1960 experiments were carried out to test the effects of LSD on animals. An amount of approximately 290 mg was administered to an elephant. That number was arrived at by converting the amount a cat had romped about by the weight of the elephant. The elephant collapsed dead after five minutes. It was concluded that elephants are particularly sensitive to LSD. However, if one had taken into account its lower metabolic rate, it would have become clear that 80 mg would have been sufficient. An elephant breaks down the substance much more slowly, which increases concentration considerably.

The different metabolic turnover of different species can also be illustrated using the human-mouse example. A person consumes around one-fiftieth of their body weight in food (1.5 kg) per day, but a mouse 25–50% of its own weight. Their lives happen with greater "speed", they multiply faster, they age earlier than humans. The food consumption of 5,000 mice (which together make up approximately the weight of a human) is almost 17 times that of a human.

The most interesting cases for biologists are those that evade the three-quarters law. Especially with such deviating conditions, the question arises as to which mechanisms are behind it and how it comes about.

Brain allometry

The Gehirnallometrie deals with the ratio of brain mass to body mass and the mass ratio of the different brain regions to each other ( hippocampus , neocortex , cerebellum ). Studies have shown that the allometry factor in primates (0.92) is far greater than in other large mammals, such as the whale (0.46) and largest in humans (1.8). But to conclude from this that humans are the only creatures in whom the increase in the brain has exceeded the increase in body mass is wrong. Some mammalian tribe insectivores have an even higher allometric factor than humans. The same is true of some species of fish, such as the elephantnose fish . So brain allometry alone is not able to explain the human mind and its abilities.

Allometry of jewelry and weapons

Male sexual characteristics that play a role in mating behavior, be it to impress females or to deter or fight rivals, always show a positive allometry. The allometric exponents are typically in the range between 1.5 and 2.5 and are intraspecific somewhat higher than interspecific. This applies to such different characteristics as the weight / length of the antlers of deer , height of the back crest of pond newts , length of the swords of swordtails or the dorsal fins of sailfish , each related to the size of the entire animal measured in the same dimension. In various species of scarab beetles , the allometric exponent for the horn length in relation to the length of the pronotum is even in the range from 4 to 16.

See also

• Scale law
• Similarity theory in the context of physical processes
• Buckingham's Π-theorem
• Kleiber's law

swell

1. ↑ Otto Snell: The dependence of the brain weight on the body weight and mental abilities . Arch. Psychiatr. 23: 436-446 (1892).
2. ^ LJ West, CM Pierce, WD Thomas: Lysergic acid diethylamide: its effects on a male asiatic elephant . In: Science . 138, 1962, pp. 1100-1103. doi : 10.1126 / science.138.3545.1100 .
3. ^ Astrid Kodric-Brown, Richard M. Sibly, James H. Brown: The allometry of ornaments and weapons