Allometric genetics V: Completing the 5th theorem by the calculus of a and b variances

In MNL 61:32-34 we presented an universal mathematical solution for the calculus of p, recombination value, and a and b, the allometric coefficients of the effects of genes A and B. The p variance was also calculated. The purpose of this report is to obtain the variance of a and b.

Only the signs of the 3rd and 4th terms change. Deriving again, these same terms change sign again and we are at a loss to decide which means which, on the inequality. A solution is found for a similar problem by R.A. Fisher (Ann. Eug. 9:50, 1939), and we reproduce his work with the adaptations required for our case.

That is, the variance of a or b is the square root of the total sum of the inverse each squared first, four terms. It is another inequality to extend the Cramer-Rao one. Note in Table 1 that by allometry the value is usually a little smaller. The discrepancies are mostly with the markers B, Su, and Pl due to values of p > 0.5 which is not congruent with the mathematical models. Comparing with p measures several-fold bigger populations are needed to measure effects of a and b.

Only the Ts5 Tr a effect approaches significance. It becomes more clear that the expression of pd and tr is due to a much more complex system of epistatic effects, since with so many duplicated genes it expresses itself as being always a single pair of recessive alleles with disturbed segregation, which once attained, blocks the effects of the other loci, because we always have a little less than one fourth tr or pd in F2 segregation. Within cultivated maize, these numerous genes at different loci act in a more quantitative way, the dominants adding their effects to achieve higher kernel row numbers.

Luiz Torres de Miranda, Luiz Eugenio Coelho de Miranda and Toshio Igue


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