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-14) The frequency of heterozygous pairs at the nth generation, Hn=∫x00fn(ξ)dξ+(1-x0)2n/2n, is approximately1+2nx0/2n, when x0 is small, and will be approximate tonx0/2n-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 to1/2e2-x0, and in the recombined homozygotes the frequency in which LA'x0 lies between t and t+dt isφ(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.
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Claims 6
Claim · model-written summary
Under continued self-fertilization, the frequency of heterozygous pairs at generation n is approximately 1+2nx0/2n when x0 is small.
- Taxon
- plantsViridiplantaee2
- Outcome
- frequency of heterozygous pairs at the nth generatione1
- Value
- 1+2nx0/2ne1
- Qualifier
- when x0 is smalle1
Not reportedDirection
Verbatim
e1“The frequency of heterozygous pairs at the nth generation, Hn=∫x00fn(ξ)dξ+(1-x0)2n/2n, is approximately1+2nx0/2n, when x0 is small, and will be approximate tonx0/2n-1, when n is large”
e2“the curves showing the decrease of heterozygosis for plants with m pairs of such chromosomes”
Claim · model-written summary
Under continued self-fertilization, the frequency of heterozygous pairs is approximately nx0/2^(n-1) when n is large.
- Taxon
- plantsViridiplantaee2
- Outcome
- frequency of heterozygous pairse1
- Value
- nx0/2n-1e1
- Qualifier
- when n is largee1
Not reportedDirection
Verbatim
e1“The frequency of heterozygous pairs at the nth generation, Hn=∫x00fn(ξ)dξ+(1-x0)2n/2n, is approximately1+2nx0/2n, when x0 is small, and will be approximate tonx0/2n-1, when n is large”
e2“the curves showing the decrease of heterozygosis for plants with m pairs of such chromosomes”
Claim · model-written summary
With seven pairs of such chromosomes, the frequency of heterozygous plants is less than 3 in 10000.
Verbatim
e3“If m=7 the frequency of heterozygous plants is less than 3 in 10000”
Claim · model-written summary
The frequency of the heterozygous segment length Lh between x and x+dx is given as the solution of a recurrence equation over self-fertilized generations.
- Taxon
- plantsViridiplantaee2
- Outcome
- the frequency (fn(x)dx) in which Lh lies between x and x+dxe4
Not reportedDirection
Verbatim
e4“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”
e2“the curves showing the decrease of heterozygosis for plants with m pairs of such chromosomes”
Claim · model-written summary
After sufficient generations, AA and A'A' pairs each have frequency 1/2e2-x0.
- Taxon
- plantsViridiplantaee2
- Outcome
- frequency of AA and A'A'e5
- Value
- 1/2e2-x0e5
Not reportedDirection
Verbatim
e5“The frequency of AA and A'A' are both equal to1/2e2-x0”
e2“the curves showing the decrease of heterozygosis for plants with m pairs of such chromosomes”
Claim · model-written summary
In recombined homozygotes, the frequency in which LA'x0 lies between t and t+dt is given by a Bessel-function distribution.
- Taxon
- plantsViridiplantaee2
- Outcome
- the frequency in which LA'x0 lies between t and t+dte6
- Qualifier
- in the recombined homozygotese6
Not reportedDirection
Verbatim
e6“in the recombined homozygotes the frequency in which LA'x0 lies between t and t+dt isφ(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”
e2“the curves showing the decrease of heterozygosis for plants with m pairs of such chromosomes”