The Recombination of Chromosome Segments Through Continued Self-Fertilization

抄録

1. In this paper the process of recombination of chromosome segments through continued self-fertilization has been studied mathematically by deriving formulae for the rate of attaining homozygosis, the frequency distribution of the length of heterozygous segments, the number of effective factors, and the distribution of segments to be finally attained in the population.
2. Let AA' be the initial chromosome pair whose genetical length is 100x₀ units, and let us designate the length of homozygous segments derived from A and A' as 100L_A, and 100L_A', units respectively and that of heterozygous segment as 100L_n, units, so that at any generation L_A+L_A'+L_n=x₀.
In the nth generation of a self-fertilizing jopulation, the relative frequency of AA' pair becomes
(1-x₀)²ⁿ/2ⁿ
The frequency (f_n(x)dx) in which L_n lies between x and x+dx(0<x<x₀) may be given by the solution of the following equation, assuming the initial condition as f₁(x)=2-x₀,
f_n(x)=(1-x)²/2f_n-1(x)+∫_x^x₀(2-ξ)f_n-1(ξ)dξ+(2-x₀)(1-x₀)^2(n-1)/2^n-1.
The frequency of heterozygous pairs at the nth generation,
H_n=∫₀^x₀f_n(ξ)dξ+(1-x₀)²ⁿ/2ⁿ,
may be approximately
1+2nx₀/2ⁿ,
when x₀ is small, and
nx₀/2^n-1,
when n is large.
3. In Table 1 the values of H_n 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 constructed (Fig. 2). If m=7, the relative frequency of heterozygous plants is less than 3/10000.
4. After a sufficient number of generations and after all the chromosome segments have reached the state of fixation, the population consists of the following three types of chromosome pairs, namely AA, A'A' and recombined homozygotes. In such population the frequency of AA and A'A' are both equal to
e^-2x₀
In the recombined homozygotes the relative frequency in which L_A/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 Fig. 4 the frequency distribution for the chromosome having 100 genetical length units is given by a histogram.
5. As regards the quantitative inheritance, if infinitesimal segments with equal and plus genetic effect construct chromosome A, while such segments with minus effect construct chromosome A', the number of effective factors (cf. Mather 1949) will amount to
1/(1-2/3x₀), under the assumption that neither dominance nor epistasis is present, and only single crossing-over is possible. Thus if x₀ is 0.5, which corresponds to the one chiasma length, the number of effective factors will be 1.5.