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 å (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

• Independent segregation between loci
• Additive (or multiplicative) non-epistasis between loci
• Variance of progeny distribution is equal to 1 (e.g., Poisson progeny distribution)

The Data

• The correlation between the number of progeny of parents and offspring is in the range 0.1 to 0.2. This range comes from studies (primarily between mothers and daughters) by Heustis and Maxwell (1932), Berent (1953), Nei and Murata (1966), Imaizumi, Nei, and Toshiyuki (1970), and Campbell (unpublished).
• The rate of evolution is one gene substitution every other generation. This is obtained assuming 10-9 substitutions per codon per year (e.g., hæ moglobin, King and Jukes 1969), 250 codons per locus, 20 years per generation, and 105 loci in the genome (we are studying humans).
• There are 100 locations in the genome which segregate independently. This is obtained from 23 chromosomes, with different arms segregating independently, and distant regions on the same arm also segregating independently.

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

• If correlation in progeny number between parents and offspring is due to genetic selection with no epistasis between loci, the selection differentials (s) at the loci undergoing substitution would need to equal approximately 0.2 (on average). If some substitutions are neutral, selection would need to be even stronger at the selected loci.

• Multiplicative non-epistasis between loci is the model which Kimura (1968) used to demonstrate the necessity of neutral evolution. In particular, selective differentials of 0.2 at 100 loci would require 82 million progeny per individual to allow selection (selection differentials of 0.1 at 50 loci would require over 100 progeny per individual). Hence correlation of progeny number cannot be due to genetic selection with multiplicative non-epistatsis between loci.
• If fertility selection is not due to genetic selection, it will have only a small effect on genetic evolution (Caballero 1994, Campbell 1999).
• This study is based on the model of non-epistatic interaction between loci. Other forms of genetic selection may provide different results. For example, truncation selection can explain the observed rate of evolution with only six progeny per individual, hence no need for neutral mutations (but such selection provides no correlation between parent and offspring progeny numbers).

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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