Fertility Selection, Genetic Selection, and Evolution
R. B. Campbell
Department of Mathematics
University of Northern Iowa
Cedar Falls IA 506140506
campbell@math.uni.edu
http://www.math.uni.edu/ ~ campbell
(319) 2732447
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
 (y_{i}  
_ y

)(x_{i}  
_ x

) 
s_{Y} s_{X}

= r, 

where x_{i} is the number of progeny of the i^{th} individual, y_{i} is
the number of progeny of the parent of the i^{th} individual (different
y_{i} 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) nonepistasis 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(unpublished).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 10^{5} 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)s^{2} x(1x)
(1+sx)^{2} (1+2sx+s^{2}x)

, 

where x is the frequency and 1+s is the relative viability of the favored
allele. This provides a cumulative correlation of
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
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 nonepistasis 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 nonepistatsis 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 nonepistatic 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 31:3950.
Caballero, A. 1994. Heredity 73:657679.
Campbell, R. B. 1999. Theoretical Population Biology 55:(in press).
Huestis, R. R. and A. Maxwell. 1932. J. Hered. 23:7779.
Imaizumi, Y., M. Nei, and F. Toshiyuki. 1970. Ann. Hum. Genet.
(London) 33:251259.
Kimura, M. 1968. Nature 217:624626.
King, J. L. and T. H. Jukes. 1969. Science 164:188198.
Nei, M. and M. Murata. 1966. Genet. Res. (Camb.) 8:257260.
Pearson, K. 1899. Phil. Trans. A 192:257278.
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