Biology Reference
In-Depth Information
the tip of the zygomatic arch to the first molar, and thus measures the lengthening of the
zygomatic spine as well as facial lengthening. For the cotton rat, the most pronounced
positive allometries are for v2 and v10, the most anterior premaxilla and that distance
between zygomatic spine and molar. Thus, these coefficients convey what we know to be
general features of mammalian development
the skull generally lengthens relative to its
width, the face lengthens even more so, and widens relative to the width of the braincase.
The interpretation of b(or log b) is less straightforward, and there has been some
controversy about its biological meaning. One reason for doubting that b has any general
biological significance is that its value depends on the units of measurement; unlike k, b is
not a dimensionless quantity. However, a more important one is that log(b) is the value of
log(Y) when log(X) is zero, a size at which Y might not yet exist. For example, when
the body is 1 mm long, the dorsal fin might not have developed yet so it cannot have a
meaningful size. Additionally, log(b) is estimated under the assumption that k is constant
from log(X)
0, not just that it is constant over the range of values actually sampled.
Under one condition, b does have a simple interpretation: when species do not differ in
k. In that case, differences in b will persist throughout the entire ontogeny. Although we
might reasonably hesitate to infer a value for log(Y) when log(X)
5
0, the difference
between species at any point in ontogeny will be invariant over ontogeny. Although we
might also be hesitant to infer that species have diverged either when log(X)
5
0 or before,
we could conclude that the difference arose prior to the stage when we first observe
them and persists throughout the rest of ontogeny. That difference in b says how those
populations will differ at any given value of X. Under other conditions, b can be viewed as
just a parameter needed to predict Y at a given value of X. To determine whether species
differ at a particular age of interest (such as birth, or the transition from larval to juvenile
growth, or at weaning), we can use the regression equation to determine the predicted
values for the dependent variables at the relevant value for X.
5
The Developmental Meaning of
and
k
Most of the literature on ontogenetic allometry has focused on the developmental
meaning of k because b is static
b
it is not a descriptor of development but rather of where
the regression line intersects the Y-axis. At the heart of the literature is the view of growth
as a multiplicative process. This was the rationale given by
Huxley (1932)
for the power
law, and it is the basis for cellular models of allometric growth (
Katz, 1980
). Within that
context, the meaning of k has been viewed from both spatial and temporal perspectives.
Huxley (1932)
emphasized the spatial interpretation of k, proposing that differences in
k over the organism indicate spatially organized “growth intensities”. He noted that values
of k tend to be spatially coherent, rising and falling in organized patterns across the body.
To help visualize spatial patterns in k, we can first put the coefficients on the organism
rather than in a table (
Figure 11.8
). We can see that they increase from the head to the
middle of the body, then fall towards the tail, although not to a level as low as found in
the head. This is (approximately) an inverted U-shaped gradient, which is interesting
because it is the inverse of the gradient found in several teleost larvae (
Fuiman, 1983
).
This suggests that
the allometry of
juvenile growth compensates for that of
larval
