Biological rules
A biological rule or biological law is a generalized law, principle,
or rule of thumb formulated to describe patterns observed in living
organisms. Biological rules and laws are often developed as succinct,
broadly applicable ways to explain complex phenomena or salient
observations about the ecology and biogeographical distributions of
plant and animal species around the world, though they have been
proposed for or extended to all types of organisms. Many of these
regularities of ecology and biogeography are named after the
biologists who first described them, though some obey Stigler’s law
of eponymy, stating that a scientific discovery is not always named
after is discoverer. Though the 3 fundamental rules of biology are
that: A biological and living system must follow all the laws of
thermodynamics;
 A biological and living system must be composed of membraneenclosed cells containing genetic material;
 A biological and living system must arise from a constant and
long-term evolutionary process;
From the birth of their science, biologists have sought to explain
apparent regularities in observational data through other, well-known
laws especially observed in the majority of the species. In his
biology, Aristotle inferred rules governing differences between livebearing tetrapods (in modern terms, terrestrial placental mammals).
Among his rules were that brood size decreases with adult body mass,
while lifespan increases with gestation period and with body mass,
and fecundity decreases with lifespan. Thus, for example, elephants
have smaller and fewer broods than mice, but longer lifespan and
gestation. Rules like these concisely organized the sum of knowledge
obtained by early scientific measurements of the natural world, and
could be used as models to predict future observations. Among the
earliest biological rules in modern times are those of Karl Ernst von
Baer (from 1828 onwards) on embryonic development, and
of Constantin W.L. Gloger on animal pigmentation, in 1833. There is
some scepticism among biogeographers about the usefulness of
general rules. For example, Briggs commented that while Willi
Hennig's rules on cladistics "have generally been helpful", his
progression rule is "suspect".
List of biological rules
I.
II.
III.
IV.
V.
VI.
Allen's rule states that the body shapes and proportions of
endotherms vary by climatic temperature by either
minimizing exposed surface area to minimize heat loss in
cold climates or maximizing exposed surface area to
maximize heat loss in hot climates. It is named after Joel A.
Allen who described it in 1877.
Bateson's rule states that extra legs are mirror-symmetric
with their neighbours, such as when an extra leg appears in
an insect's leg socket. It is named after the pioneering
geneticist William Bateson who observed it in 1894. It
appears to be caused by the leaking of positional signals
across the limb-limb interface, so that the extra limb's
polarity is reversed.
Bergmann's rule states that within a broadly distributed
taxonomic clade, populations and species of larger size are
found in colder environments, and species of smaller size are
found in warmer regions. It applies with exceptions to many
mammals and birds. It was named after Carl Bergmann who
described it in 1847.
Cope's rule states that animal population lineages tend to
increase in body size over evolutionary time. The rule is
named for the palaeontologist Edward D. Cope.
Dollo's law of irreversibility, proposed in 1893 by FrenchBelgian paleontologist Louis Dollo states that an organism
never returns exactly to a former state, even if it finds itself
placed in conditions of existence identical to those in which it
has previously lived, always keeping some trace of the
intermediate stages through which it has passed.
Eichler's rule states that the taxonomic diversity of parasites
co-varies with the diversity of their hosts. It was observed in
1942 by W. Eichler, and is named for him.
VII.
VIII.
IX.
X.
XI.
XII.
XIII.
Emery's rule, noticed by Carlo Emery, states that
insect social parasites are often closely related to their hosts,
such as being in the same genus.
Foster's rule, the island rule, or the island effect states that
members of a species get smaller or bigger depending on the
resources available in the environment. The rule was first
stated by J. B. Foster in 1964.
Gause's law or the competitive exclusion principle, named
for Georgy Gause, states that two species competing for the
same resource cannot coexist at constant population values.
The competition leads either to the extinction of the weaker
competitor or to an evolutionary or behavioral shift toward a
different ecological niche. It is seemingly an extension of
Darwin’s theory of natural selection.
Gloger’s rule states that within a species of endotherms,
more heavily pigmented forms tend to be found in
more humid environments, e.g. near the equator. It was
named after the zoologist Constantin W. L. Gloger, who
described it in 1833.
Haldane's rule states that if in a species hybrid, or the F1
offspring of two different species, one sex is sterile or absent,
that sex is usually the heterogametic sex. The heterogametic
sex is the one with two different sex chromosomes; in
mammals, this is the male, with XY chromosomes. It is
named after J.B.S. Haldane.
Hamilton's rule states that genes should increase in
frequency when the relatedness of a recipient to an actor,
multiplied by the benefit to the recipient, exceeds the
reproductive cost to the actor. This is a prediction from the
theory of kin selection formulated by W. D. Hamilton.
Harrison's rule states that parasite body sizes co-vary with
those of their hosts. He proposed the rule for lice, but later
authors have shown that it works equally well for many other
groups of parasite including barnacles, nematodes, fleas,
flies, mites, and ticks, and for the analogous case of small
herbivores on large plants.
XIV.
XV.
XVI.
Hennig's progression rule states that when considering a
group of species in cladistics, the species with the most
primitive characters are found within the earliest part of the
area, which will be the center of origin of that group. It is
named after the father of cladistics, Willi Hennig, who
devised the rule.
Hesse's rule, also known as the heart–weight rule, states that
species inhabiting colder climates have a larger heart in
relation to body weight than closely related species
inhabiting warmer climates. It was proposed by German
ecologist R. Hesse as an extension of Bergmann’s rule.
Jordan's rule states that there is an inverse
relationship between water temperature
and meristic characteristics such as the number of fin rays,
vertebrae, or scale numbers, which are seen to increase with
decreasing temperature. It is named after the father of
American ichthyology, David Starr Jordan.
XVII.
Kleiber's law, named after Max Kleiber for his biology work
in the early 1930s, is the observation that, for the vast
majority of animals, an animal's metabolic rate scales to
the 3 4th power of the animal's mass. Symbolically: if q0 is
the animal's metabolic rate, and M the animal's mass, then
3
Kleiber's law states that q0 ~ 𝑴 4 .
XVIII.
Lack’s principle, proposed by David Lack, states that the
clutch size of each species of bird has been adapted
by natural selection to correspond with the largest number of
young for which the parents can, on average, provide enough
food.
XIX.
Moseley’s rule, noted in 1880 by Henry N. Moseley, states
that deep-sea animals are larger than their shallow-water
counterparts. In the case of marine crustaceans, it has been
proposed that the increase in size with depth occurs for the
same reason as the increase in size with latitude (Bergmann's
rule): both trends involve increasing size with decreasing
temperature.
XX.
Rapoport's rule states that the latitudinal ranges
of plants and animals are generally smaller at lower latitudes
than at higher latitudes. It was named after Eduardo H.
Rapoport by G. C. Stevens in 1989.
XXI.
Rensch's rule states that, across animal species within a
lineage, sexual size dimorphism increases with body size
when the male is the larger sex, and decreases as body size
increases when the female is the larger sex. The rule applies
in primates, pinnipeds and even-toed ungulates (artiodactyls,
such as cattle and deer). It is named after Bernhard Rensch,
who proposed it in 1950.
XXII.
Rubner’s law, or the rate-of-living theory, postulates that the
faster an organism’s metabolism, the shorter its lifespan. The
theory was originally created by Max Rubner in 1908 after
his observation that larger animals outlived smaller ones, and
that the larger animals had slower metabolisms.
Schmalhausen's law, named after Ivan Schmalhausen, states
that a population at the extreme limit of its tolerance in any
one aspect is more vulnerable to small differences in any
other aspect. Therefore, the variance of data is not simply
noise-interfering with the detection of so-called "main
effects", but also an indicator of stressful conditions leading
to greater vulnerability.
Thorson's rule states that benthic marine invertebrates at low
latitudes tend to produce large numbers of eggs developing
to pelagic (often planktotrophic [plankton-feeding]) and
widely dispersing larvae, whereas at high latitudes such
organisms tend to produce fewer and larger lecithotrophic
(yolk-feeding) eggs and larger offspring, often
by viviparity or ovoviviparity, which are often brooded. It
was named after Gunnar Thorson by S. A. Mileikovsky in
1971.
Van Valen's law states that the probability of extinction for
species and higher taxa (such as families and orders) is
constant for each group over time; groups grow neither more
resistant nor more vulnerable to extinction, however old their
XXIII.
XXIV.
XXV.
XXVI.
XXVII.
lineage is. It is named for the evolutionary biologist Leigh
Van Valen.
Von Baer's laws, discovered by Karl Ernst von Baer, state
that embryos start from a common form and develop into
increasingly specialised forms, so that the diversification of
embryonic form mirrors the taxonomic and phylogenetic tree.
Therefore, all animals in a phylum share a similar early
embryo; animals in smaller taxa (classes, orders, families,
genera, species) share later and later embryonic stages. This
was in sharp contrast to the recapitulation theory of Johann F.
Meckel (and later of Ernst Haeckel), which claimed that
embryos went through stages resembling adult organisms
from successive stages of the scala naturae from supposedly
lowest to highest levels of organisation.
Williston's law, first noticed by Samuel W. Williston, states
that meristic parts in an organism tend to become reduced in
number and greatly specialized in function. However,
empirical studies have not always confirmed this
generalization. For instance, a study of the evolution in the
number of branchiostegal rays in osteichthyans has failed to
support a generalized trend towards reduction. Instead, this
series of elements shows an early burst pattern (rapid
evolution early in the history of the group, followed by a
decrease in evolutionary rate).
Explanations of the rules
I.
Because animals living in cold climates need to conserve as much
heat as possible, Allen's rule predicts that they should have evolved
comparatively low surface area-to-volume ratios to minimize the
surface area by which they dissipate heat, allowing them to retain
more heat. For animals living in warm climates, Allen's rule
predicts the opposite: that they should have comparatively high
ratios of surface area to volume. Because animals with low surface
area-to-volume ratios would overheat quickly, animals in warm
climates should, according to the rule, have high surface area-tovolume ratios to maximize the surface area through which they
dissipate heat. Though there are numerous exceptions and this has
been shown to be a “poor ecological tenet derived from a single
species” in rare cases, many animal populations appear to conform
to the predictions of Allen's rule. The polar bear has stocky limbs
and very short ears that are in accordance with the predictions of
Allen's rule. And also, this rule can be applied to some ectotherms
as well, who derive their body temperatures from their
surroundings. A contributing factor to Allen's rule
in vertebrates may be that the growth of cartilage is at least partly
dependent on temperature. Temperature can directly affect the
growth of cartilage, providing a proximate biological explanation
for this rule.
III.
The earliest explanation, given by Bergmann when originally
formulating the rule, is that larger animals have a lower surface
area to volume ratio than smaller animals, so they radiate less body
heat per unit of mass, and therefore stay warmer in cold climates.
Warmer climates impose the opposite problem: body heat
generated by metabolism needs to be dissipated quickly rather than
stored within. Thus, the higher surface area-to-volume ratio of
smaller animals in hot and dry climates facilitates heat loss through
the skin and helps cool the body. It is important to note that when
analyzing Bergmann's Rule in the field that groups of populations
being studied are of different thermal environments, and also have
been separated long enough to genetically differentiate in response
to these thermal conditions. But, this rule does not apply
everywhere as the correlation with temperature is spurious, latitude
being a poor predictor of mass and size; instead, body size is
proportional to the duration of the annual productivity pulse, or
food availability per animal during the growing season. Resource
availability is a major constraint on the overall success of many
organisms. Resource scarcity can limit the total number of
organisms in a habitat, and over time can also cause organisms to
adapt by becoming smaller in body size. Resource availability thus
becomes a modifying restraint on Bergmann's Rule.
IV.
Directional selection (negative selection type where an extreme
phenotypical trait is chosen over other traits, resulting in the allele
frequency to become shifted towards that trait over time) appears
to act on organisms' size, whereas it exhibits a far smaller effect on
other morphological traits, though it is possible that this perception
may be a result of sample bias. This selectional pressure can be
explained by a number of advantages, both in terms of mating
success and survival rate. For example, larger organisms find it
easier to avoid or fight off predators and capture prey, to
reproduce, to kill competitors, to survive temporary lean times, and
to resist rapid climatic changes. They may also potentially benefit
from better thermal efficiency, increased intelligence, and a longer
lifespan. Offsetting these advantages, larger organisms require
more food and water, and shift from r to K-selection (r/K selection
theory relates to the selection of combinations of traits in an
organism that trade-off between quantity and quality of offspring).
Their longer generation time means a longer period of reliance on
the mother, and on a macroevolutionary scale restricts the clade's
ability to evolve rapidly in response to changing environments.
Left unfettered, the trend of ever-larger size would produce
organisms of gargantuan proportions. Therefore, some factors must
limit this process. At one level, it is possible that the clade's
increased vulnerability to extinction, as its members become larger,
means that no taxon survives long enough for individuals to reach
huge sizes. There are probably also physically imposed limits to
the size of some organisms; for instance, insects must be small
enough for oxygen to diffuse to all parts of their bodies, flying
birds must be light enough to fly, and the length of giraffes' necks
may be limited by the blood pressure it is possible for their hearts
to generate. Finally, there may be a competitive element, in that
changes in size are necessarily accompanied by changes in
ecological niche. For example, terrestrial carnivores over 21 kg
almost always prey on organisms larger, not smaller, than
themselves. If such a niche is already occupied, competitive
pressure may oppose the directional selection. The Canidae,
especially, shows a trend towards larger size. Moreover, from a
palaeontological view, many palaeobiologists are sceptical of the
validity of Cope's rule, which may merely represent a statistical
artefact. Purported examples of Cope's rule often assume that the
stratigraphic age of fossils is proportional to their "clade rank", a
measure of how derived they are from an ancestral state; this
relationship is in fact quite weak. Counterexamples to Cope's rule
are common throughout geological time; although size increase
does occur more often than not, it is by no means universal. For
example, among genera of Cretaceous molluscs, an increase in size
is no more common than stasis or a decrease. In many cases,
Cope's rule only operates at certain taxonomic levels (for example,
an order may obey Cope's rule, while its constituent families do
not), or more generally, it may apply to only some clades of a
taxon. Despite many counter-examples, Cope's rule is supported in
many instances. For example, all marine invertebrate phyla except
the molluscs show a size increase between the Cambrian and
Permian. Collectively, dinosaurs exhibit an increase in body length
over their evolution. Cope's rule also appears to hold in clades
where a constraint on size is expected. For instance, one may
expect the size of birds to be constrained, as larger masses mean
more energy must be expended in flight. Birds have been suggested
to follow Cope's law, although a subsequent reanalysis of the same
data suggested otherwise.
VII.
The significance and general relevance of this pattern are still a
matter of some debate, as a great many exceptions exist, though a
common explanation for the phenomenon when it occurs is that the
parasites may have started as facultative parasites within the host
species itself (such forms of intraspecific parasitism are wellknown, even in some species of bees), but later became
reproductively isolated and split off from the ancestral species, a
form of sympatric speciation (the evolution of a new species from a
surviving ancestral species while both continue to inhabit the same
geographic region). When a parasitic species is phylogenetically a
sister taxon to its host, the relationship is considered to be in
"strict" adherence to Emery's rule. When the parasite is a close
relative of the host but not its sister species, the relationship is in
"loose" adherence to the rule.
VIII.
Foster’s rule is part of the general Island Syndrome, that describes
the differences between insular species and their mainland
counterparts in morphology, physiology, behaviour, etc. This is
because island ecosystems cannot support a sufficient biomass of
prey in order to accommodate large predators. This largely relieves
prey species of the risk of predation, which mostly removes the
selection pressure for morphologies, ecologies and behaviours that
help to evade large predators. Insular ecosystems tend to comprise
large populations of a limited number of species (a state
termed density compensation) and therefore, they exhibit
low biodiversity. This results in reduced interspecific
competition and increased intraspecific competition. There is also
reduced sexual selection in insular species, which is especially
prominent in birds which lose their sexually dimorphic plumage
used in sexual displays. Finally, there is reduced parasite diversity
in insular ecosystems which reduces the level of selection acting on
immune-related genes — all of which is further driven by the
generally mild, insular climate. And resultantly, there also exists an
inverse form of this rule, termed island gigantism, the underlying
statement of which is that due to the lack of predators and various
empty niches, different species tend to evolve adaptive traits over
time to occupy them, ensuring plentiful resources for every species
and thus resulting in bigger size. This is commonly seen in Pacific
islands like the Hawai’i and the Galapagos which sit at the
confluence of different species (especially birds) coming from
almost every biogeographical realm that border the ocean. Birds in
this case, especially those growing larger, also tend to become
flightless due to the said reasons.
IX.
Gause’s law is based on Darwin’s theory of natural selection. This
rule of competitive exclusion is best predicted by mathematical and
theoretical models like the generalized Lotka–Volterra equations
showing trophic relations:where x is the number of prey (for
example, rabbits); y is the number of
𝑑𝑥
𝑑𝑦
some predator (for example, foxes);
and
𝑑𝑡
𝑑𝑡
represent the instantaneous growth rates of the
two populations and are proportional to the body size of the
respective species; t represents time; α, β, γ, δ are positive
real parameters describing the interaction of the two species.
Or, for 2 species competing for some common resource, the
competitive Lotka-Volterra model can be used, where —
and
And for N species,
Here, x is the size of the population at a given time, r is inherent
per-capita growth rate, and K the carrying capacity of the
environment, or the maximum population size of a
biological species that can be sustained by that specific
environment, given the food, habitat, water, and other resources are
available. α12 represents the effect species 2 has on the population
of species 1 and α21 represents the effect species 1 has on the
population of species 2. These values do not have to be equal.
Because this is the competitive version of the model, all
interactions must be harmful (competition) and therefore all αvalues are positive. Also, each species can have its own growth rate
and carrying capacity. K However, for poorly understood reasons,
competitive exclusion is rarely observed in natural ecosystems, and
many biological communities appear to violate Gause's law, and
the aforementioned equations apply in situations provided only
that: The prey population gets ample food at all times;
 The predator population has limitless appetite and it depends
on the prey population entirely for food;
 There does not occur any environmental or bodily change
like mutation or evolution in favour of any species.
The best-known example illustrating the drawbacks of this law is
the so-called "paradox of the plankton": all plankton species live
on a very limited number of resources, primarily solar energy and
minerals dissolved in the water. According to the competitive
exclusion principle, only a small number of plankton species
should be able to coexist on these resources. Nevertheless, large
numbers of plankton species coexist within small regions of open
sea. Some communities that appear to uphold the competitive
exclusion principle are MacArthur's warblers and Darwin's finches,
which live in the same ecosystem but have developed different
adaptations and traits to occupy different niches and thus remove
competition, though the latter still overlap ecologically very
strongly, being only affected negatively by competition under
extreme conditions.
X.
One explanation of Gloger's rule in the case of birds appears to be
the increased resistance of dark feathers to feather- or hairdegrading bacteria such as Bacillus licheniformis. Feathers in
humid environments have a greater bacterial load, and humid
environments are more suitable for microbial growth; dark feathers
or hair are more difficult to break down. More
resilient eumelanins (dark brown to black) are deposited in hot and
humid regions, whereas in arid regions, pheomelanins (reddish to
sandy color) predominate due to the benefit of crypsis.
Among mammals, there is a marked tendency in equatorial
and tropical regions to have a darker skin color than poleward
relatives. In this case, the underlying cause is probably the need to
better protect against the more intense solar UV radiation at lower
latitudes.
XI.
Many different hypotheses have been advanced to address the
evolutionary mechanisms to produce Haldane's rule. Currently, the
most popular explanation for Haldane's rule is the composite
hypothesis, which divides Haldane's rule into multiple
subdivisions, including sterility, inviability, male heterogamety,
and female heterogamety. The composite hypothesis states that
Haldane's rule in different subdivisions has different causes.
Individual genetic mechanisms may not be mutually exclusive, and
these mechanisms may act together to cause Haldane's rule in any
given subdivision. In contrast to these views that emphasize
genetic mechanisms, another view hypothesizes that population
dynamics during population divergence may cause Haldane's rule.
The main genetic hypotheses are that:
 Heterogametic hybrids are affected by all X-linked alleles (be
they recessive or dominant) causing incompatibilities due to
divergent alleles being brought together. However,
homogametic hybrids are only affected by dominant deleterious
X-linked alleles. Heterogametic hybrids, which carry only a
single copy of a given X-linked gene, will be affected by
mutations regardless of dominance. Thus, an X-linked
incompatibility between diverging populations is more likely to
be expressed in the heterogametic sex than in the homogametic
sex.
 Male genes are thought to evolve faster due to sexual selection.
As a result, male sterility becomes more evident in male
heterogametic taxa (XY sex determination). This hypothesis
conflicts with Haldane's rule in male homogametic taxa, in
which females are more affected by hybrid inferiority. It
therefore only applies to male sterility in taxa with XY sex
determination, according to the composite theory.
 In hybrid populations, selfish genetic elements inactivate sperm
cells (i.e.: an X-linked drive factor inactivates a Y-bearing
sperm and vice versa).
 Genes on hemizygous chromosomes may evolve more quickly
by enhancing selection on possible recessive alleles causing a
larger effect in reproductive isolation.
 Hybrid incompatibilities affecting the heterogametic sex and
homogametic sex are fundamentally different isolating
mechanisms, which makes heterogametic inferiority
(sterility/inviability) more visible or preserved in nature.
XII.
Formally, genes should increase in frequency when rB > C,
where,
r = the genetic relatedness of the recipient to the actor, often
defined as the probability that a gene picked randomly from
each at the same locus is identical by descent.
B = the additional reproductive benefit gained by the recipient of
the altruistic act,
C = the reproductive cost to the individual performing the act.
This inequality is called Hamilton's rule, part of the general
phenomenon of kin selection (the evolutionary strategy that
favours the reproductive success of an organism's relatives, even at
a cost to the organism's own survival and reproduction. Kin
altruism can look like altruistic behaviour whose evolution is
driven by kin selection. Kin selection is an instance of inclusive
fitness — personal fitness is the number of offspring an individual
begets — which combines the number of offspring produced with
the number an individual can ensure the production of by
supporting others, such as siblings), and named after W. D.
Hamilton who in 1964 published its first formal quantitative
treatment. The relatedness parameter (r) in Hamilton's rule was
introduced in 1922 by Sewall Wright as a coefficient of
relationship that gives the probability that at a random locus,
the alleles there will be identical by descent. According to
Hamilton, the selective advantage which makes behaviour
conditional in the right sense on the discrimination of factors which
correlate with the relationship of the individual concerned is
therefore obvious. It may be, for instance, that in respect of a
certain social action performed towards neighbours
indiscriminately, an individual is only just breaking even in terms
of inclusive fitness. If he could learn to recognise those of his
neighbours who really were close relatives and could devote his
beneficial actions to them alone an advantage to inclusive
fitness would at once appear. Thus, a mutation causing such
discriminatory behaviour itself benefits inclusive fitness and would
be selected. In fact, the individual may not need to perform any
discrimination so sophisticated as we suggest here; a difference in
the generosity of his behaviour according to whether the situations
evoking it were encountered near to, or far from, his own home
might occasion an advantage of a similar kind.
XIII.
Poulin and Harrison observed that in comparisons across species,
the variability of parasite body size also increases with host body
size. It is self-evident that we expect greater variation coming
together with greater mean body sizes due to an allometric power
law scaling effect. However, they referred to parasites' increasing
body size variability due to biological reasons, thus we expect an
XVIII.
increase greater than that caused by a scaling effect.
The allometry between host and parasite body sizes constitutes an
evident aspect of host–parasite coevolution. The slope of this
relationship is a taxon-specific character. Parasites' body size is
known to covary positively with fecundity and thus it likely affects
the virulence of parasitic infections as well.
Though there exists no complete and satisfactory explanation
behind Kleiber’s law, one lies in the difference between structural
and growth mass. Structural mass involves maintenance costs,
reserve mass does not. Hence, small adults of one species respire
more per unit of weight than large adults of another species
because a larger fraction of their body mass consists of structure
rather than reserve. Within each species, young (i.e., small)
organisms respire more per unit of weight than old (large) ones of
the same species because of the overhead costs of growth. So, over
the same timespan as an example, a cat having a mass 100 times
that of a mouse will consume only about 32 times the energy the
mouse uses. Explanations for 2⁄3-scaling tend to assume that
metabolic rates scale to avoid heat exhaustion. Because bodies lose
heat passively via their surface, but produce heat metabolically
throughout their mass, the metabolic rate must scale in such a way
as to counteract the square–cube law. The precise exponent to do
so is 2⁄3. Such an argument does not address the fact that different
organisms exhibit different shapes (and hence have
different surface-to-volume ratios, even when scaled to the same
size). Reasonable estimates for organisms' surface area do appear
to scale linearly with the metabolic rate. A model by
scientists West, Enquist, and Brown (hereafter WEB) suggests
that 3⁄4-scaling arises because of efficiency in nutrient distribution
and transport throughout an organism. In most organisms,
metabolism is supported by a circulatory system featuring
branching tubules (i.e., plant vascular systems, insect tracheae, or
the human cardiovascular system). WEB claim that (1) metabolism
should scale proportionally to nutrient flow (or, equivalently, total
fluid flow) in this circulatory system and (2) in order to minimize
the energy dissipated in transport, the volume of fluid used to
transport nutrients (i.e., blood volume) is a fixed fraction of body
mass. They then proceed by analyzing the consequences of these
two claims at the level of the smallest circulatory tubules
(capillaries, alveoli, etc.). Experimentally, the volume contained in
those smallest tubules is constant across a wide range of masses.
Because fluid flow through a tubule is determined by the volume
thereof, the total fluid flow is proportional to the total number of
smallest tubules. Thus, if B denotes the basal metabolic rate, Q the
total fluid flow, and N the number of minimal tubules,
B ∝ Q ∝ N.
Circulatory systems do not grow by simply scaling proportionally
larger; they become more deeply nested. The depth of nesting
depends on the self-similarity exponents of the tubule dimensions,
and the effects of that depth depend on how many "child" tubules
each branching produces. Connecting these values to macroscopic
quantities depends (very loosely) on a precise model of tubules.
WEB show that, if the tubules are well-approximated by rigid
cylinders, then, in order to prevent the fluid from "getting
clogged" in small cylinders, the total fluid volume V satisfies
N4 ∝ V3.
Because blood volume is a fixed fraction of body mass,
B∝𝑀
3
4.
Closer analysis suggests that Kleiber's law does not hold over a
wide variety of scales. Metabolic rates for smaller animals (birds
under 10 kg, or insects) typically fit to 2⁄3 much better than 3⁄4; for
larger animals, the reverse holds. As a result, log-log plots of
metabolic rate versus body mass appear to "curve" upward, and fit
better to quadratic models. In all cases, local fits exhibit exponents
in the [2⁄3,3⁄4] range. Adjustments to the WEB model that retain
assumptions of network shape predict larger scaling exponents,
worsening the discrepancy with observed data. But one can retain a
similar theory by relaxing WEB's assumption of a nutrient
transport network that is both fractal and circulatory. (WEB argued
that fractal circulatory networks would necessarily evolve to
minimize energy used for transport, but other researchers argue
that their derivation contains subtle errors. Different networks are
less efficient, in that they exhibit a lower scaling exponent, but a
metabolic rate determined by nutrient transport will always exhibit
scaling between 2⁄3 and 3⁄4. If larger metabolic rates are
evolutionarily favoured, then low-mass organisms will prefer to
arrange their networks to scale as 2⁄3, but large-mass organisms will
prefer to arrange their networks as 3⁄4, which produces the observed
curvature. An alternative model notes that metabolic rate does not
solely serve to generate heat. Metabolic rate contributing solely to
useful work should scale with power 1 (linearly), whereas
metabolic rate contributing to heat generation should be limited by
surface area and scale with power 2⁄3. Basal metabolic rate is then
the convex combination of these two effects: if the proportion of
useful work is f, then the basal metabolic rate should scale as
B = f · kM + (1 – f) · k′𝑀
2
3.
where k and k′ are constants of proportionality. k′ in particular
describes the surface area ratio of organisms and is
−2
approximately 0.1 kJ·h−1·𝑔 3 ; typical values for f are 1520%. The theoretical maximum value of f is 21%, because the
efficiency of glucose oxidation is only 42%, and half of the ATP so
produced is wasted. Analyses of variance for a variety of physical
variables suggest that although most variation in basal metabolic
rate is determined by mass, additional variables with significant
effects include body temperature and taxonomic order. A 1932
work calculated that the scaling was approximately 0.73. A 2004
analysis of field metabolic rates for mammals conclude that they
appear to scale with exponent 0.749. It has been shown that
attempts to explain Kleiber's law via any sort of limiting factor is
flawed, because metabolic rates vary by factors of 4-5 between rest
and activity. Hence any limits that affect the scaling
of basal metabolic rate would in fact make elevated metabolism —
and hence all animal activity — impossible. WEB conversely
argue that animals may well optimize for minimal transport energy
dissipation during rest, without abandoning the ability for less
efficient function at other times.
Other researchers have also noted that this criticism of the law
tends to focus on precise structural details of the WEB circulatory
networks, but that the latter are not essential to the model. But
overall, it should be noted that Kleiber's law only appears when
studying animals as a whole; scaling exponents within taxonomic
subgroupings differ substantially.
XIX.
In crustaceans, it has been proposed that the explanation for the
increase in size with depth is similar to that for the increase in size
with latitude (Bergmann's rule): both trends involve increasing size
with decreasing temperature. The trend with latitude has been
observed in some of the same groups, both in comparisons of
related species, as well as within widely distributed species.
Decreasing temperature is thought to result in increased cell size
and increased life span (the latter also being associated with
delayed sexual maturity), both of which lead to an increase in
maximum body size (continued growth throughout life is
characteristic of crustaceans). In Arctic and Antarctic seas where
there is a reduced vertical temperature gradient, there is also a
reduced trend towards increased body size with depth, arguing
against hydrostatic pressure being an important parameter.
Temperature does not appear to have a similar role in influencing
the size of giant tube worms. Riftia pachyptila, which lives in
hydrothermal vent communities at ambient temperatures of 2–30
°C, reaches lengths of 2.7 m, comparable to those of
Lamellibrachia luymesi, which lives in cold seeps (areas of the
ocean floor where hydrogen sulphide, methane and other
hydrocarbon-rich fluid seepage occurs, often in the form of a brine
pool. Cold does not mean that the temperature of the seepage is
lower than that of the surrounding sea water. On the contrary, its
temperature is often slightly higher. The "cold" is relative to the
very warm (at least 60 °C or 140 °F) conditions of a hydrothermal
vent)-the former, however, has rapid growth rates and short life
spans of about 2 years, while the latter is slow growing and may
live over 250 years. Food scarcity at depths greater than 400 m is
also thought to be a factor, since larger body size can improve
ability to forage for widely scattered resources. In organisms with
planktonic eggs or larvae, another possible advantage is that larger
offspring, with greater initial stored food reserves, can drift for
greater distances. As an example of adaptations to this situation,
giant isopods gorge on food when available, distending their bodies
to the point of compromising ability to locomote; they can also
survive 5 years without food in captivity. According to Kleiber's
rule, the larger an animal gets, the more efficient its metabolism
becomes; i.e., an animal's metabolic rate scales to roughly the ¾
power of its mass. Under conditions of limited food supply, this
may provide additional benefit to large size. An additional possible
influence is reduced predation pressure in deeper waters. A study
of brachiopods found that predation was nearly an order of
magnitude less frequent at the greatest depths than in shallow
waters. Dissolved oxygen levels are also thought to play a role in
deep-sea gigantism. A 1999 study of benthic amphipod crustaceans
found that maximum potential organism size directly correlates
with increased dissolved oxygen levels of deeper waters. The
solubility of dissolved oxygen in the oceans is known to increase
with depth because of increasing pressure, decreasing salinity
levels and temperature. The proposed theory behind this trend is
that deep-sea gigantism could be an adaptive trait to combat
asphyxiation in ocean waters. Larger organisms are able to intake
more dissolved oxygen within the ocean, allowing for sufficient
respiration. However, this increased absorption of oxygen runs the
risk of toxicity poisoning where an organism can have oxygen
levels that are so high that they become harmful and poisonous.
XX.
The methods used to demonstrate the rule have been subject to
some controversy. Most commonly, authors plot means of
latitudinal ranges in a particular 5° latitudinal band against latitude,
although modal or median ranges have been used by some. In the
original paper by Stevens, all species occurring in each band were
counted, i.e., a species with a range of 50 degrees occurs in 10 or
11 bands. However, this may lead to an artificial inflation of
latitudinal ranges of species occurring at high latitudes, because
even a few tropical species with wide ranges will affect the means
of ranges at high latitudes, whereas the opposite effect due to high
latitude species extending into the tropics is negligible: species
diversity is much smaller at high than low latitudes. As an
alternative method the "midpoint method" has been proposed,
which avoids this problem. It counts only those species with the
midpoint of their ranges in a particular latitudinal band. An
additional complication in assessing Rapoport's rule for data based
on field sampling is the possibility of a spurious pattern driven by a
sample-size artifact. Equal sampling effort at species-rich and
species-poor localities tends to underestimate range size at the
richer localities relative to the poorer, when in fact range sizes
might not differ among localities. But some situations happen to
oppose the rule also — for instance, marine benthic invertebrates
and some parasites have been shown to have smaller dispersal
abilities in cold seas (Thorson's rule), which would counteract
Rapoport's rule. The tropics have far more uniform temperatures
over a far wider latitudinal range (about 45 degrees) than high
latitude species. As temperature is one of the most important (if not
the most important) factor determining geographical distribution,
wider latitudinal ranges in the tropics might therefore be expected.
XXI.
After controlling for confounding factors such as evolutionary
history, an increase in average body size makes the difference in
body size larger if the species has larger males, and smaller if it has
larger females. Some studies propose that this is due to sexual
bimaturism, which causes male traits to diverge faster and develop
for a longer period of time. The correlation between sexual size
dimorphism and body size is hypothesized to be a result of an
increase in male-male competition in larger species, a result of
limited environmental resources, fuelling aggression between
males over access to breeding territories and mating partners.
Phylogenetic lineages that appear to follow this rule
include primates, pinnipeds, and artiodactyls. This rule has rarely
been tested on parasites. A 2019 study showed that
ectoparasitic philopterid and menoponid lice comply with it,
while ricinid lice exhibit a reversed pattern.
XXII.
Mechanistic evidence for this law was provided by Harman's free
radical theory of aging, created in the 1950s, stating that organisms
age over time due to the accumulation of damage from free radicals
in the body. It also showed that metabolic processes, specifically
the mitochondria, are prominent producers of free radicals. This
provided a mechanistic link between Rubner's initial observations
of decreased lifespan in conjunction with increased metabolism.
Support for this theory has been bolstered by studies linking a
lower basal metabolic rate (evident with a lowered heartbeat) to
increased life expectancy. This has been proposed by some to be
the key to why animals like the Giant Tortoise can live over 150
years. However, the ratio of resting metabolic rate to total
daily energy expenditure can vary between 1.6 and 8.0 between
species of mammals. Animals also vary in the degree of coupling
between oxidative phosphorylation and ATP production, the
amount of saturated fat in mitochondrial membranes, the amount
of DNA repair, and many other factors that affect maximum life
span. Furthermore, a number of species with high metabolic rate,
like bats and birds, are long-lived. In a 2007 analysis it was shown
that, when modern statistical methods for correcting for the effects
of body size and phylogeny are employed, metabolic rate does not
correlate with longevity in mammals or birds.
XXIV.
Several explanations of the rule have been given, including that:
1. Because of the reduced speed of development at low
temperatures, most species cannot complete development
during the short time of phytoplankton bloom, on which
planktotrophic species depend.
2. Most species cannot synchronize hatching with the
phytoplankton bloom.
3. Slower development increases the risk of predation on
pelagic larvae.
4. Non-pelagic larvae can settle close to the parent, i.e., in a
favourable environment.
5. Small pelagic larvae may have osmotic difficulties in Arctic
and Antarctic summers, due to the melting ice.
6. Small larvae may not be able to survive at very low
temperatures.
7. Cold temperature may select for large size at the beginning of
development, resulting in non-pelagic larvae.
8. In cold waters it is more difficult to precipitate dissolved
calcium, which results in reduced body size of animals
supported by calcium skeletons, leading to viviparity.
Most of these explanations can be excluded for the Monogenean
flatworms, whose larvae are never planktotrophic (therefore
eliminating explanations 1 and 2), their larvae are always shortlived (3), gyrodactylid flatworms are most common not only close
to melting ice but in cold seas generally (5). Explanation 6 is
unlikely, because small organisms are common in cold seas,
Gyrodactylidae are among the smallest Monogenea (7), and
Monogenea do not possess calcareous skeletons (8). The
conclusion is that the most likely explanation for the Monogenea
(and by implication for other groups) is that small larvae cannot
locate suitable habitats at low temperatures, where physiological
including sensory processes are slowed, and/or that low
temperatures prevent the production of sufficient numbers of
pelagic larvae, which would be necessary to find suitable habitats
in the vast oceanic spaces.
XXVI.
The Von Baer laws are a series of statements generally summarised
into four points: The more general characters of a large group appear earlier in
the embryo than the more special characters.
 From the most general forms the less general are developed, and
so on, until finally the most special arises.
 Every embryo of a given animal form, instead of passing
through the other forms, rather becomes separated from them.
 The embryo of a higher form never resembles any other form,
but only its embryo.
To explain these, Von Baer discovered the blastula (the early
hollow ball stage of an embryo) and the development of
the notochord (the stiffening rod along the back of all chordates,
that forms after the blastula and gastrula, the cup-shaped stage with
trilayered cells coming after the blastula stage). From his
observations of these stages in different vertebrates, he realised
that Johann Meckel's recapitulation theory must be wrong. For
example, he noticed that the yolk sac is found in birds, but not
in frogs. According to the recapitulation theory, such structures
should invariably be present in frogs because they were assumed to
be at a lower level in the evolutionary tree. Von Baer concluded
that while structures like the notochord are recapitulated during
embryogenesis, whole organisms are not. He asserted that the
embryo successively adds the organs that characterize the animal
classes in the ascending scale. When the human embryo, for
instance, is but a simple vesicle, it is an infusorian; when it has
gained a liver, it is a mussel; with the appearance of the osseous
system, it enters the class of fishes; and so forth, until it becomes a
mammal and then a human being. In terms of taxonomic hierarchy,
according to von Baer, characters in the embryo are formed in topto-bottom sequence, first from those of the largest and oldest taxon,
the phylum, then in turn class, order, family, genus, and finally
species.