In MNL 55:18-19 we presented a graphical interpretation of what we called allometric genetics. Here we present a mathematical demonstration. It is based on a derivation by I. L. Gridi-Papp (Dr. Thesis Agronomy School, Piracicaba, SP, Brazil, 1970) which, although correct, was not developed to our present full interpretations.
The velocity of growth of one certain character in a certain moment is a function of the dimension already attained (X), of time (t), and of a factor of proportionality that depends on the character and the genotype:
dX/dt = Kx f(X,t) (1)
Time (t) is considered as a measure not only from the organ but also from the variations of environment.
Formula (1) expresses a hypothetical relationship because the function RX,t) remains unknown. Nevertheless, because at time t = 0, X = 0, its primitive has the form:
X = Kx F(X,t) (2)
F (X,t) is also unknown. To get some information about this function it is necessary to consider within the same organ a second dimension, that which presents itself as the character most influenced by the size of the organ. If (Y) is this character it can be formulated:
Y = KY F(Yt) (3)
Thus equal values of (t) correspond to a (Y) and an (X).
The relative behaviour of two characters of the same organ such as (Y) and (X) has been the object of investigations of several authors in the past and was described by the so-called allometry law proposed by J.S. Huxley in 1932. This law is based on constant growth velocities, which can be written as d(log X)/dt = ad(log Y)/dt, where a is a constant. Integrating we get log X = a log Y + log A. We substitute A for C, the usual constant of integration, because A is really C and is the usual symbol used in allometry.
Log A is constant for a given pair of characters. It follows that:
X = AYa or X/Ya = A (4)
By (2) and (3) comes
X = KxF(Yt)
A comparison between (4) and (5) suggests choosing the constant A such that A = a(Kx/Ky), where a is a factor of proportionality, which means that X = a(Kx/Ky)Ya. AYa and that F(X,t)/F(Yt) = AYa-1 = X/Y, with X' = X/Y, then X' = AYa-1 and we have shown in our prior work that X' (100/Y)1-a. 1-a is the more nearly biologically correct value of half the difference between the different homozygotes for the same locus. 2(1-a) is the logarithm of A, the constant of integration of the "law of allometry".
Using a in the expectations of the appropriate classes in the observed phenotypes a mathematically exact solution can be derived. With only one type of family we must have also an allometric measure of the homozygotes, say BB against bb. With two different types of families a simultaneous solution can be derived for a and p, even if only by iterative methods, though not necessarily.
Luiz Torres de Miranda and Luiz Eugenio Coelho de Miranda
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