Allometric genetics III: A universal mathematical solution for the calculation of recombination values

Since the report "Allometric genetics" in MNL 55:18-19 we have been trying to unify allometric and genetic parameters in one body of theory and practical use. In that first report a graphic interpretation of gene effects was made in terms of a, the coefficient of allometry. Also it was shown how it could be used in quantitative genetics. In a preceding report in this volume a mathematical demonstration was done, completing the graphic one. It remains now to show how to use it in Mendelian genetics, and this is the objective of the present report.

For generations biometrists have calculated from a 2x2 table four parameters-a mean, a linear effect for the lines, another for the rows, and the interaction between them. Mendelian geneticists compute only one parameter, p, the recombination value. This seriously limits the range of work to near perfect segregations, 1:1:1:1, 3:1, etc. Segregations giving significant decimal digits are ignored. As a result, this field of work comprises only a small sample of biological variability. The usual method used is the Maximum Likelihood (K. Mather, "The measurement of linkage in heredity"). For more disturbed segregations, the Minimum Moment of the Products Method, also known as the Product Ratio, is used. We have never found a derivation, and for the particular case of the backcross we will show also how it can be done, as one more proof of the mathematical precision of the allometric solution proposed.

Looking at the most upward diagonal in MNL 55:18-19, we can imagine that the effect of a genetic factor A is + a, and of a, -a. Taking a second pair of allelles, B is + b, and b is -b. The combinations of these allometric coefficients are put in the theoretical frequency expectations, and equated to the observed ones. AA is a, Ab is b, aB is c and bb is d, and n is the family size as usual.

If we make

we have the exact numerical solution for the product moment method which is the solution of (1-p)2/p2 = (a x d) / (b x c), although nobody ever used it but us.

By the maximum likelihood method we take from the 2 x 2 table

Deriving in relation to p, after algebraic manipulation we arrive at

And the value of p is expressed by the equation

The second derivative is

And the variance of p is

The computation of the p value is cumbersome, but it can be arrived at by iterative methods and the logical values used compared with the results obtained by the classical methods.

For any other type of family to get the solution for p it is necessary just to add the appropriate a and b to the solution arrived at by the maximum likelihood expression.

Luiz Eugenio Coelho de Miranda and Luiz Torres de Miranda

Please Note: Notes submitted to the Maize Genetics Cooperation Newsletter may be cited only with consent of the authors.