Abstract
1) The behaviour of chromosome segments under continued self-fertilization was discussed from the stand-point that a chromosome composed of innumerable segments rather than particle genes, is divisible by crossing-over.
2) Let AA' be the initial chromosome pair whose genetical length is 100x0 units, and designate the length of homozygous segments derived from A and A' as 100LA and 100LA' units respectively and that of heterozygous segment as 100Lh units, so that at any generation LA+LA'+Lh=x0.
3) In the n times self-fertilized populations, the frequency of AA' pair is
(1-x0)2n/2n,
and the frequency (fn(x)dx) in which Lh lies between x and x+dx (0<x< x0) is given as the solution of the following equation, the initial condition of which is f1(x)=2-x0,
fn(x)=(1-x)2/2fn-1(x)+∫x0x(2-ξ)fn-1(ξ)dξ+(2-x0)(1-x0)2(n-1)/2n-1
4) The frequency of heterozygous pairs at the nth generation,
Hn=∫x00fn(ξ)dξ+(1-x0)2n/2n,
is approximately
1+2nx0/2n,
when x0 is small, and will be approximate to
nx0/2n-1,
when n is large.
5) In table 1 the figures of Hn are given for the initial 20 generations assuming that the chromosome length is 100 units. From these values, the curves showing the decrease of heterozygosis for plants with m pairs of such chromosomes can be easily constructed (Fig. 2). If m=7 the frequency of heterozygous plants is less than 3 in 10000.
6) After sufficient generations have elapsed and all the chromosome segments reached the state of fixation, the population contains three kinds of pairs, namely AA, A'A' and the recombined homozygote.
The frequency of AA and A'A' are both equal to
1/2e2-x0,
and in the recombined homozygotes the frequency in which LA'x0 lies between t and t+dt is
φ(t)dt=x0e-2x0{2I0(4x0√<t(1-t)>)+I1(4x0√<t(1-t)>)/√<t(1-t)>}dt,
where I0 and I1 are Bessel functions.
In figure 4 the frequency distribution is given by means of histogram for the chromosome whose genetical length is 100 units.
