Inbreeding Effect on means and variances

1
InbreedingEffect on means and variances
1

• Defining a locus in terms of scale
• bb mp
  Bb BB

a
-a
d
2
Degree of dominance
2

• Ratio of d/a - degree of dominance
• d 0 no dominance
• d gt 0 lt a partial dominance
• d a complete dominance
• d gt a over dominance

3
Means and variances
3
Mean Variance covariance
Mean Freq valueVariance Freq value2 -
mean2
4
Mean of population frequency x value
4

• Genotype Frequency Value (scale)
• BB p2 .25 a 24 4
• Bb 2pq .50 d 24 4
• bb q2 .25 -a 16 - 4
• Mean p2a 2pqd - q2a
• simplifies to a(p-q) 2 pqd (-20)

5
Mean of an F2 population
5

• P1 (BB) x P2 (bb)
• F1 (Bb)
• F2 array
• BB Bb bb
• .25 .5 .25
• a d -a
• Mean a(p-q) 2pqd .5d 1/2d
• (note pq .5 in an F2 popn)

6
Variance of F2 population frequency x value2
- mean2
6

• Genotype Frequency Value (scale)
• BB p2 .25 a 24 4
• Bb 2pq .50 d 24 4
• bb q2 .25 - a 16 - 4
• Variance p2a2 2pqd2 - q2a2 - 1/2d 2
• simplifies to 1/2a2 1/4d2

7
Advancing the population by further inbreeding
(F2 F3)
7

• Genotypes BB Bb bb
• F2 Frequency 1/4 1/2 1/4
• Value a d -a
• F3 familes BB (1/4BB 1/2Bb1/4bb ) bb
• Value a 1/2d -a

8
F2 plants 1/4 1/2 1/4
BB Bb bb
8
Individuals

• BB BB BB bb
• BB Bb Bb bb
• BB Bb Bb bb
• BB bb bb bb

F3 Families
9
Partitioning F3 variancefreq value2 - mean
2
9

• Total variance
• 3/8(a)2 1/4(d) 2 3/8(-a) 2 - 1/4d 2
• 3/4(a)2 3/16(d) 2
• Among F3 families
• 1/4(a)2 1/2(1/2d) 2 1/4(-a) 2 - 1/4d 2
• 1/2(a)2 1/16(d) 2
• Within F3 families (by subtraction)
• 1/4(a)2 1/8(d) 2

10
Partitioning variance among and within families
10

• Variance
• Generation Mean Total Among Within
• F1 d 0 0 0
• F2 1/2 d 1/2a2 0 0
• F3 1/4 d 3/4a2 1/2a2 1/4a2
• F4 1/8 d 7/8a2 3/4a2 1/8a2
• F 0 a2 a2 0

11
Patitioning variance among and within families
11

• Change in mean value upon selfing is due to
  frequency of heterozygotes and value of d, if
  d0 then there will be no change in mean upon
  inbreeding
• At F all the variance is among families and
  variance within families is 0.
• At F variance among families doubles from
  1/2a2 to a2.
• At F variance due to dominance d2 is 0

12
Backcross generations
12

• P1 (BB) x P2 (bb)
• F1 (Bb) x P1 (BB)
• Backcross progeny
• BB Bb 1/2 1/2 a d Mean
  1/2a 1/2d
• Variance 1/4a2 1/4d2 - 1/2ad
• Note if d0, then Variance of BC is 1/2 that of
  F2