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Lecture 6:Inbreeding and Heterosis
Inbreeding• Inbreeding = mating of related individuals• Often results in a change in the mean of a trait• Inbreeding is intentionally practiced to:
– create genetic uniformity of laboratory stocks – produce stocks for crossing (animal and plant
breeding)• Inbreeding is unintentionally generated:
– by keeping small populations (such as is found at zoos)
– during selection
Genotype frequencies under inbreeding
• The inbreeding coefficient, F• F = Prob(the two alleles within an
individual are IBD) -- identical by descent
• Hence, with probability F both alleles in an individual are identical, and hence a homozygote
• With probability 1-F, the alleles are combined at random
Genotype Alleles IBD Alleles not IBD frequency
A1A1 Fp (1-F)p2 p2 + Fpq
A2A1 0 (1-F)2pq (1-F)2pq
A2A2 Fq (1-F)q2 q2 + Fpq
p A1
A2q
F
F
A1A1
A2A2
1-F
1-Fp
p A1A1
A2 A1
q
A2A1q
A2A2
Alleles IBDRandom mating
Changes in the mean under inbreeding
F = 0 - 2Fpqd
Using the genotypic frequencies under inbreeding, the population mean F under a level of inbreeding F isrelated to the mean 0 under random mating by
Genotypes A1A1 A1A2 A2A2
0 a+d 2a
freq(A1) = p, freq(A2) = q
πF=π0°2FkXi=1piqidi=π0°BFFor k loci, the change in mean isB=2XpiqidiHere B is the reduction in mean under complete inbreeding (F=1) , where
• There will be a change of mean value dominance is present (d not zero)
• For a single locus, if d > 0, inbreeding will decrease the mean value of the trait. If d < 0, inbreeding will increase the mean
• For multiple loci, a decrease (inbreeding depression) requires directional dominance --- dominance effects di tending to be positive.
• The magnitude of the change of mean on inbreeding depends on gene frequency, and is greatest when p = q = 0.5
Inbreeding Depression and Fitness traits
Inbred Outbred
Define ID = 1-F/0 = 1-(0-B)/0 = B/0
Drosophila Trait Lab-measured ID = B/0
Viability 0.442 (0.66, 0.57, 0.48, 0.44, 0.06)
Female fertility 0.417 (0.81, 0.35, 0.18)
Female reproductive rate 0.603 (0.96, 0.57, 0.56, 0.32)
Male mating ability 0.773 (0.92, 0.76, 0.52)
Competitive ability 0.905 (0.97, 0.84)
Male fertility 0.11 (0.22, 0)
Male longevity 0.18
Male weight 0.085 (0.1, 0.07)
Female weight -0.10
Abdominal bristles 0.077 (0.06, 0.05, 0)
Sternopleural bristles -.005 (-0.001, 0)
Wing length 0.02 (0.03, 0.01)
Thorax length 0.02
Why do traits associated with fitness show inbreeding depression?
• Two competing hypotheses:– Overdominance Hypothesis: Genetic variance for fitness is
caused by loci at which heterozygotes are more fit than both homozygotes. Inbreeding decreases the frequency of heterozygotes, increases the frequency of homozygotes, so fitness is reduced.
– Dominance Hypothesis: Genetic variance for fitness is caused by rare deleterious alleles that are recessive or partly recessive; such alleles persist in populations because of recurrent mutation. Most copies of deleterious alleles in the base population are in heterozygotes. Inbreeding increases the frequency of homozygotes for deleterious alleles, so fitness is reduced.
Estimating BIn many cases, lines cannot be completely inbred due to either time constraints and/or because in many species lines near complete inbreeding are nonviable
In such cases, estimate B from the regression of F on F,
F = 0 - BF
0
0
1
0 - B
If epistasis is present, this regression is non-linear, with CkFk for k-th order epistasis
F
F
F
F
Minimizing the Rate of Inbreeding
• Avoid mating of relatives• Maximum effective population size Ne
• Ne maximized with equal representation – Contribution (number of sibs) from each parent as equal
as possible – Sex ratio as close to 1:1 as possible– When sex ratio skewed (r dams/sires ), every male
should contribute (exactly) one son and r daughters, while every female should leave one daughter and also with probability 1/r contribute a son
•
Line Crosses: Heterosis
P1 P2
F1
x
F2HF1=πF1°πP1+πP22When inbred lines are crossed, the progeny show an increase in mean
for characters that previously suffered a reduction from inbreeding.
This increase in the mean over the average value of theparents is called hybrid vigor or heterosis
A cross is said to show heterosis if H > 0, so that the F1 mean is average than the average of both parents.
Expected levels of heterosis
If pi denotes the frequency of Qi in line 1, let pi + i denotethe frequency of Qi in line 2.HF1=nXi=1(±pi)2diThe expected amount of heterosis becomes
• Heterosis depends on dominance: d = 0 = no inbreeding depression and no.heterosis as with inbreeding depression, directional dominance is required for heterosis.
• H is proportional to the square of the difference in gene frequency Between populations. H is greatest when alleles are fixed in one population andlost in the other (so that | i| = 1). H = 0 if = 0.
• H is specific to each particular cross. H must be determined empirically,since we do not know the relevant loci nor their gene frequencies.
Heterosis declines in the F2HF2=πF2°πP1+πP22=(±p)2d2=HF12In the F1, all offspring are heterozygotes. In the F2,
Random mating has occurred, reducing the frequency of heterozygotes.
As a result, there is a reduction of the amount of heterosis in the F2 relative to the F1,
Since random mating occurs in the F2 and subsequentgenerations, the level of heterosis stays at the F2 level.
Agricultural importance of heterosis
Crop % planted as hybrids
% yield advantage
Annual added
yield: %
Annual added
yield: tons
Annual land
savings
Maize 65 15 10 55 x 106 13 x 106
ha
Sorghum 48 40 19 13 x 106 9 x 106 ha
Sunflower 60 50 30 7 x 106 6 x 106 ha
Rice 12 30 4 15 x 106 6 x 106 ha
Crosses often show high-parent heterosis, wherein the F1 not only beats the average of the two parents (mid-parent heterosis), it exceeds the best parent.
Variance Changes Under Inbreeding
F = 0F = 1/4F = 3/4F = 1
Inbreeding reduces variation within each populationInbreeding increases the variation between populations(i.e., variation in the means of the populations)
Variance Changes Under Inbreeding
General F = 1 F = 0Between lines
2FVA 2VA 0
Within Lines (1-F) VA 0 VA
Total (1+F) VA
2VA VA
Mutation and Inbreeding• As lines lose genetic variation from drift, mutationintroduces new variation• Eventually these two forces balance, leading toan equilibrium level of genetic variance reflectingthe balance between loss from drift, gain from mutation
Symmetrical distribution of mutational effects
Assuming:
Strictly neutral mutations
Strictly additive mutationsVA=VG=2NeVMVM = new mutation variation each generation, typicallyVM = 10-3 VE
Between-line DivergenceVB=2VM[t°2Ne(1°e°t=2Ne)]The between-line variance in the mean (VB) in generationt is
For large t, the asymptotic rate is 2VMt