Let p be the frequency of a more favorable gene in tolerance to heat,
say lte1; 1p the frequency of a less favorable gene, lte1, allelic to
lte1; a, half the difference between the two homozygous genotypes; and
d the degree of dominance; with K being the average value of the less favorable
gene. The conventional representation of the basic genetic parameters is
represented as follows:
Genotypic value  
Genotype  Frequency.  Number of favorable genes  Uncoded W'  Coded W=W'Ka 
lte1/lte1  p2  2  K+2a  a 
Lte1/lte1  2p(1p)  1  K+a+da  da 
Lte1/Lte1  (1p)2  0  K  a 
The problem is a. Suppose a represents the effect
of one gene on survival, or fitness. In percent under no stress, a would
be zero, and it could happen in the other extreme that one genotype survives
100%, the other 0%. In this case a would be 50%. In this model a is one
point in spacetime. The definition of a new a which takes into account
spacetime must be in terms of an angle. In the relationship of allometry,
being X and Y two measures, they are usually related by X=AY^{a}
(J. S. Huxley, Problems of Relative Growth, 1932), really related XY with
Y. Relating X÷Y with Y leads to the allometric relationships shown
in Fig. 1. It shows that a
and A are redundant, A being the antilog 2(1a)
and the simplest representation being X' = (100÷Y)1a.
The conventional interpretation is a point in those lines. Substituting
1a and d
for d in the conventional representation we have a genetic interpretation
of the allometric relationships as follows:


Genotype  Frequency  Number of favorable genes  Uncoded
W' 
Coded
W=W'K(1a) 
lte1/lte1  p^{2}  2  K+2(1a)  1a 
Lte1/lte1  2p(1p)  1  K+(1a)+d(1a)  d(1a) 
Lte1/Lte1  (1p)^{2}  0  K  a1* 
*See in Fig. 1 that (a1) = 1a
This also opens the way to analyze, and to map, single genes with quantitative effects including survival or fitness. It should be warned that the best fit is gotten calculating by the minimum products method derived by G. Teissier (Biometrics No. 1, 4:1453, 1948). The minimum squares method underestimates the absolute value of the coefficients. It is interesting to point out that L. I. GripiPapp (Dr. Thesis, ESALQ. Univ. of Sao Paulo, 1970) in a demonstration of X=AY^{a}arrived first at the form X'=AY^{a1.}
Luiz Torres de Miranda
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