CHANGES IN MEAN AND VARIANCE UPON SELFING IN AN IDEALIZED BREEDING NURSERY The F2 Generation  1. F2 Population Mean and Variance (p = q = 0.5)  Genotype BB Bb bb Genotypic frequency 0.25 0.5 0.25 Genotypic value (bu ac-1) 24 24 16 Population mean (frequency x value):  0.25(24) + 0.5(24) + 0.25(16) = 22 

Population mean : 0.25(24) + 0.5(24) + 0.25(16) = 22 The F2 Generation  1. F2 Population Mean and Variance (p = q = 0.5)  Genotype BB Bb bb Genotypic frequency 0.25 0.5 0.25 Genotypic value (bu ac-1) 24 24 16 Population mean : 0.25(24) + 0.5(24) + 0.25(16) = 22 Total population variance (frequency x value2 - mean2):  0.25 (24)2 + 0.5 (24)2 + 0.25 (16)2 - (22)2 = 12

Genetic Variance 2G = 2A + 2D 2A,  or additive genetic variance is the variance of the effects of the genes 2D or dominance variance, is the variance due to interaction of alleles

The total genetic variance at a single locus comprises additive genetic variance (2A) plus dominance genetic variance (2D).    when p = q = 0.5, as in this example, then,  2G = 2A + 2D.  Thus in an F2 or S0 population where allele frequencies are = 0.5, we designate, by convention, the total genetic variance as equal to 2A + 2D, in the absence of epistasis. Many studies utilize F2 or S0 populations as a baseline, or a starting point – remember the F2 is the perfect HW pop. As we will illustrate below, the changes in 2A and 2D that occur during the inbreeding generation following the F2 are expressed relative to the values of 2A and 2D in the original F2 population.

F3 Population Mean and Variance (p = q = 0.5) The F3 Generation   F3 Population Mean and Variance (p = q = 0.5)  Genotype BB Bb bb Genotypic frequency 0.375 0.25 0.375 Genotypic value (bu ac-1) 24 24 16   F3 population mean :  0.375 (24) + 0.25 (24) + 0.375 (16) = 21  Total F3 population variance :  0.375 (24)2 + 0.25 (24)2 + 0.375 (16)2 - (21)2 = 15. = 3/2 2A + 3/4 2D

Total F3 population variance :  = 3/2 2A + 3/4 2D   = 15 Remember, Total F2 population variance = 2A + 2D = 12 

Changes in the mean Note the change in population mean. In the F2 it was 22. In the F3 it was 21. This reflects a slow regression back to the homozygote midparent value (20). This is due to the decrease in heterozygotes with genotypic value = 24. Upon complete inbreeding the frequency of Bb is zero, and the mean is equal to the midparent value. The F2 Generation  1. F2 Population Mean and Variance (p = q = 0.5)  Genotype BB Bb bb Genotypic frequency 0.25 0.5 0.25 Genotypic value (bu ac-1) 24 24 16

Inbreeding Depression Homozygosity increases the frequency of the loci with alleles identical by descent. This will include unfavorable recessive alleles whose genotypic inferiority is not masked by the presence of an alternate dominant type. Thus, the overall desirability of individuals decrease. This is referred to as inbreeding depression

Changes in the Mean Of course, if there is no dominance, the population mean is equal to the midparent value in the F2, and the F3, and the F4, etc., until the population is completely inbred.

Changes in the variance There is one-half again the additive genetic variance expressed in the F3 population as in the F2 population This is a direct result of the increased frequency of homozygotes which are tending to polarize the population into two opposite categories--homozygous dominants (BB at a frequency of 0.375) and homozygous recessives (bb at a frequency of 0.375).

Changes in Variance In fact, when dominance is complete, the major noticeable phenotypic change from the F2 to F3 would be the increased frequency from 25 to 38% of recessives (bb) yielding only 16 bu ac-1. Notice that dominance variance has decreased as a result of the decrease in heterozygotes.

The Underlying Family Structure of the F3 Population: Variance Among and Within F2:3-Derived Lines  Unlike the F2 or S0 which contained no family structure, the F3 generation in this example has a defined family structure. F2:3-derived lines were developed by harvesting F2 plants individually and keeping the seed separate. As a result, the total population genetic variance can be partitioned into A) Variance among F2:3 lines and B) Variance within F2:3 lines.

A) Variance Among F2:3 Lines F2 plant source of F2:3 line BB Bb bb Frequency of F2:3 line 0.25 0.5 0.25 Mean value of F2:3 line   24 22 16 Note: Pay particular attention to the mean genotypic values of the F2:3 lines. Lines derived from BB individuals will have the mean value 24. Lines derived from bb individuals will have the mean value 16. And (this one stumps students--but just think about it a minute), lines derived from Bb individuals will have the mean value equivalent to an F2 population--i.e., 22 Why? Because Bb is equivalent to an F1 heterozygote.

2G among F2:3 lines (frequency x value2 - mean2): = 0.25 (24)2 + 0.5 (22)2 + 0.25 (16)2 - (21)2   = 2A + 1/4 2D. = = 9.   In addition, pay particular attention to the correction for the mean--i.e., (21)2. F2:3 lines are in the F3 generation--thus the F3 population mean is appropriate for the correction factor.

B) Variance within F2:3 lines F2 plant source of F2:3 line BB Bb bb Frequency of F2:3 line 0.25 0.5 0.25 Variance within F2:3 line 0 .25(24)2+ .25(24)2+ .25(16)2 0   - (22)2 Note: There will be no variance within an F2:3 line derived from either a BB or bb F2 plant. The variance within an F2:3-line derived from a Bb plant will have within-line variance equivalent to an F2 population.

Variance within F2:3 lines F2 plant source of F2:3 line BB Bb bb Frequency of F2:3 line 0.25 0.5 0.25 Variance within F2:3 line 0 .25(24)2+ . 5(24)2+ .25(16)2 0   - (22)2 We can obtain a simple mean value of the variance within all the F2:3 lines. Thus, we utilize the simple (frequency x value) method.  Mean 2G within F2:3 lines (frequency x value):  = 0.25 (0) + 0.5 (.25(24)2+ .5(24)2+ .25(16)2- (22)2) + 0.25 (0) = 6.

Changes in Variance In developing F2:3-derived lines, the breeder has enhanced the chances of making superior selections. the total genetic variance has increased due to the increased frequencies of homozygotes. Increased additive genetic variance, decreased dominance variance.

Variances Partitioned The variances have been partitioned. 60% of the total variance is among the means of the F2:3 lines, and of more importance, two-thirds of the additive variance in the F3 generation is observed among the means of the F2:3 lines.

Additive Variance Additive variance is the chief determinant of the breeding value of an individual so production of F2:3 lines should help the breeder do a superior job during selection.

F4 Population Mean and Variance (p = q = 0.5) The F4 Generation  F4 Population Mean and Variance (p = q = 0.5)   Genotype BB Bb bb Genotypic frequency 0.437 0.125 0.437 Genotypic value (bu ac-1) 24 24 16 F4 population mean :  0.437 (24) + 0.125(24) + 0.437(16) = 20.5

Total F4 population variance :   0.437 (24)2 + 0.125 (24)2 + 0.437 (16)2 - (20.5)2 = 15.75 = 7/4 2A + 7/16 2D  

Variance among F3:4 lines   F3 plant source of F3:4 line BB Bb bb Frequency of F3:4 line 0.375 0.25 0.375 Mean value 24 22 16 2G among F3:4 lines (frequency x value2 - mean2):  = 0.375(24)2 + 0.25 (22) + 0.375 (16)2 - (20.5)2= = 3/2 2A + 3/16 2D   = 12.75

Variance within F3:4 lines F3 plant source of F3:4 line BB Bb bb Frequency of F3:4 line 0.375 0.25 0.375 Variance within F3:4 line 0 .25(24)2+ .25(24)2+ .25(16)2 0   - (22)2=12 Mean 2G within F3:4 lines (frequency x value):  = 0.375(0) + 0.25 (12) + 0.375(0).   1/4 2A + 1/4 2D = = 3 The simplest way to calculate variance within lines is by subtraction, : Total (15.75) - Among (12.75) = Within (3)

F = the probability that at a locus in an inbred individual I, the two General Formulas for Determining Additive and Dominance Variance Based on F, Wright’s Coefficient of Inbreeding F = the probability that at a locus in an inbred individual I, the two alleles are Identical By Descent In a population where p = q = 0.5, the formula simplify to  Total 2G = (1+F) 2A + (1-F2) 2D  If we assume that F = 0 in the F2 or S0, then  2G = (1+0) 2A + (1-02) 2D = 2A + 2D.

Additive Variance Among Derived Lines 2A = (1+Ft) 2A  where Ft = inbreeding coefficient of the generation from which individual plants were selected. For example, in the case of F2:3 lines, Ft = 0, but for F3:4 lines, then Ft = (1 + 1/2) = 3/2. Dominance Variance Among Derived Lines  2D = [(1+Ft)/(1-Ft)](1-F)2 2D. The simplest way to calculate 2A and 2D within lines is by subtraction, as follows:  Variance (within lines) = total variance - variance (among lines)  For example, for F2:3 lines,  Variance (within lines) = (3/2 2A + 3/4 2D) - (2A + 1/4 2D) = 1/2 2A + 1/2 2D.

Distribution of variances among and within lines under continuous selfing when p = q = 0.5 (after Hallauer and Miranda, 1981; Table 2:11).   Among lines Within lines Total Generation F 2A 2D 2A 2D 2A 2D  F2, S0 0 1 1 F2:3, S0:1 1/2 1 1/4 1/2 1/2 3/2 3/4 F3:4, S1:2 3/4 3/2 3/16 1/4 1/4 7/4 7/16 F4:5, S2:3 7/8 7/4 7/64 1/8 1/8 15/8 15/64 F5:6, S3:4 15/16 15/8 15/256 1/16 1/16 31/16 31/256 F:+1, S:+1 1 2 0 0 0 2 0

Impact of Inbreeding on Mean At the completely inbred generation we have  Genotype BB Bb bb Genotypic frequency 0.5 0 0.5 Genotypic value 24 24 16  Population mean =0.5 (24) + 0.5 (16) = 20 i.e., the population mean has regressed to the midparent value.

Impact of Inbreeding on Genetic Variance Total population variance = 0.5 (24)2 + 0.0 (24)2 + 0.5 (16)2 - (20)2 = 16 Recall the population variance in the F2 was 12; therefore the genetic variance has increased upon inbreeding. It has not doubled though? Why not? Because some of the initial genetic variance was due to dominance and it has “disappeared” as heterozygote frequency diminished.

Other Effects of Inbreeding Inbred lines, exhibit a greater sensitivity to environmental sources of variation than noninbred lines. This can interfere with experimental studies on changes in variation upon inbreeding. This may vary over successive inbreeding generations.   

Random mate the derived lines in all combinations Random mate the derived lines in all combinations. Provided there have been no changes in allele frequencies during inbreeding, variance among the random F1 hybrids will be equal to the variance in the original base population (F2 or S0). There will be no hybrid in the resulting population that could not have been found in an infinitely large base population.

Can you Calculate the Inbreeding Coefficient Of the Famous Shorthorn Bull Roan Gauntlet?