Biology Reference
In-Depth Information
of allometry, it is nonetheless implicit. This becomes evident when considering why
the power law holds in the first place. As mentioned before, the primary biological
explanation for allometry is that growth is a multiplicative process. When analyzing the
relationship between size and time, the best-fitting models are usually not linear but rather
are sigmoidal in form. An important feature of these models is that growth rates decay
over time. Similarities in decay rates are interpreted by Laird (1965) as the explanation for
the linear relationship among log-transformed measurements. In effect, all measurements
follow the same growth curve; their differing values of k tell us how they are displaced
relative to each other in time
different parts of the body reach the same point on their
growth curves at different times. Laird et al. (1968) elaborated on this theory, stating the
relationship between k and lag time (
Δ
T) as:
1
α
Δ
T
ln ð k Þ
(11.3)
52
where
is the decay rate and k is an allometric coefficient.
To measure decay rates we need information about age, but we can use Equation 11.3
to understand the temporal relationships among growth curves even without known age
samples so long as we are willing to assume that all measurements whose logs are linearly
related have the same decay rates. Because growth rates decay over time, we would intuit
that a more negatively allometric part has decayed over a longer time, and that it has
decayed for longer because it began growing earlier. The increment of time by which we
need to shift one curve to match another that starts growing later is
α
Δ
T. Based on this
interpretation of allometric coefficients, we would conclude that those for piranha body
growth mean that the head and caudal peduncle develop before the midbody, that the eye
is the first structure to develop, and that the body elongates before it deepens. In the case
of the mammalian skull, we would interpret the allometric coefficients to mean that the
broad bulbous cranium of the neonate, resulting from the rapid prenatal growth
of the brain, becomes relatively narrowed over postnatal growth by the rapid elongation
of the skull as a whole and especially of the face, which also widens relative to the cranial
base, as part of the overall increase in facial dimensions relative to braincase and cranial
base.
The spatial and temporal perspectives on allometric coefficients are not antagonistic.
The spatial coherence noted by Huxley, interpreted within the temporal framework of
Laird, suggests that growth is spatiotemporally organized. There is no reason to think that
either space or time is primary. We do not need to adopt one view over the other they
are mutually consistent, and help explain each other. With increasing information
about the spatial determination of development, in conjunction with that on its temporal
organization, we can relate allometric coefficients to the underlying developmental
processes that explain them.
To interpret these coefficients in terms of both growth and function, we can apply theo-
ries about scaling relationships to growth. Applied to ontogenetic series, such theories
may explain ontogenetic allometry in terms of the ontogeny of function. For example,
in many larval teleosts the head and caudal region are highly positively allometric, which
is due to the early demands imposed by swimming, feeding and respiration (see, for
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