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 100
x0 units, and let us designate the length of homozygous segments derived from
A and
A' as 100
LA, and 100
LA', units respectively and that of heterozygous segment as 100
Ln, units, so that at any generation
LA+
LA'+
Ln=
x0.
In the
nth generation of a self-fertilizing jopulation, the relative frequency of
AA' pair becomes
(1-
x0)
2n/2
nThe frequency (
fn(
x)
dx) in which
Ln 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,
fn(
x)=(1-
x)
2/2
fn-1(
x)+∫
xx0(2-ξ)
fn-1(ξ)
dξ+(2-
x0)(1-
x0)
2(n-1)/2
n-1.
The frequency of heterozygous pairs at the
nth generation,
Hn=∫
0x0fn(ξ)
dξ+(1-
x0)
2n/2
n,
may be approximately
1+2
nx0/2
n,
when
x0 is small, and
nx0/2
n-1,
when
n is large.
3. In Table 1 the values 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 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-2x0In the recombined homozygotes the relative frequency in which
LA/
x0, lies between
t and
t+
dt is
φ(
t)
dt=
x0e-2x0{2
I0(4
x0√
t(1-
t)+
I1(4
x0√
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/(1-2/3
x0), 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.
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