Fertility Selection, Genetic Selection, and Evolution

R. B. Campbell
Department of Mathematics
University of Northern Iowa
Cedar Falls IA 50614-0506
campbell@math.uni.edu
http://www.math.uni.edu/ ~ campbell
(319) 273-2447

Introduction
Evolution entails differential reproductive success sustained over several generations. One way to measure this is with the correlation between the number of progeny of parents and offspring (Pearson 1899)

1
N

(yi - _
y
 
)(xi - _
x
 
)

sY sX
= r,
where xi is the number of progeny of the ith individual, yi is the number of progeny of the parent of the ith individual (different yi may refer to the same parent, with the different subscripts identifying different progeny of that parent), and the overscores and sigmas denote the means and standard deviations of the distribution. This measure of selection, sometimes called fertility selection, predates the rediscovery of Mendelian genetics in the early 1900's. The purpose of this investigation is to reconcile this mode of selection with selection based on the genetic composition of individuals. The conclusion is that fertility selection cannot be explained by genetic selection, but fertility selection does have a small impact on genetic selection.

The Model

The Data

Analysis

In order to investigate whether the observed correlation in progeny number can be explained by genetic selection, the correlation due to genetic selection is calculated. The correlation in progeny number due to selection at a single locus is given by

(1+s)s2 x(1-x)
(1+sx)2 (1+2sx+s2x)
,
where x is the frequency and 1+s is the relative viability of the favored allele. This provides a cumulative correlation of
ln(1+s)
during the course of fixation. Assuming that s is the same at all loci, summing across loci in a single generation should provide the same quantity, modified by a factor of 0.5 reflecting that there is a fixation event every other generation. Hence
0.5 ×ln(1+s) = 0.1,
where 0.1 is the correlation of progeny number, which is the same every generation. This requires that s \doteq 0.2 to account for the observed correlation (if some substitutions are near neutral, selection must be stronger at the other loci).

This is a haploid model, but the diploid model reduces the correlation by a factor of 2, hence doubles the necessary magnitude of s.

The deterministic time until fixation of a selected allele is approximately (1+(2/s)) ln2N generations. If s = 0.2 and N = 40,000,000, this will provide 200 generations until fixation, hence with a gene substitution every other generation, approximately 100 loci should be segregating at a time.

Discussion

Literature Cited
Berent, J. 1953. Milbank Memorial Fund Quarterly 31:39-50.
Caballero, A. 1994. Heredity 73:657-679.
Campbell, R. B. 1999. Theoretical Population Biology 55:(in press).
Huestis, R. R. and A. Maxwell. 1932. J. Hered. 23:77-79.
Imaizumi, Y., M. Nei, and F. Toshiyuki. 1970. Ann. Hum. Genet. (London) 33:251-259.
Kimura, M. 1968. Nature 217:624-626.
King, J. L. and T. H. Jukes. 1969. Science 164:188-198.
Nei, M. and M. Murata. 1966. Genet. Res. (Camb.) 8:257-260.
Pearson, K. 1899. Phil. Trans. A 192:257-278.


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