CSS 650 Advanced Plant Breeding Module 2: Inbreeding Small Populations –Random drift –Changes in...
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Transcript of CSS 650 Advanced Plant Breeding Module 2: Inbreeding Small Populations –Random drift –Changes in...
CSS 650 Advanced Plant Breeding
Module 2: Inbreeding•Small Populations
–Random drift
–Changes in variance, genotypes
•Mating Systems–Inbreeding coefficient from pedigrees
–Coefficient of coancestry
–Regular systems of inbreeding
Population size
• Sampling can lead to changes in gene frequency in small populations
• Changes are random in direction (dispersive), but predictable in amount– random drift – accumulation of small changes due to sampling
over time
– differences among subgroups of the population increase over time
– increase in uniformity and level of homozygosity within subgroups (Wahlund effect)
• Two perspectives– changes in variances due to sampling
– changes in genotype frequencies due to inbreeding
Falconer, Chapt. 3
Dispersive process - idealized population
Base populationN =
2N 2N 2N 2N 2N 2N 2N gametes
N N N N N N N
2N 2N 2N 2N 2N 2N 2N
N N N N N N N
sub-populations
t=0
t=1
t=2
Idealized population assumptions
• Mating occurs within sub-populations
• Mating is at random (including self-fertilization)
• Sub-populations are equal in size
• Generations do not overlap
• No mutation, migration or selection
No change in the average gene frequency among sub-populations over generations
0qq
Random drift (genetic drift)
• Every generation, the sampling of gametes within each sub-population centers around a new allele frequency changes accumulate over time
• Changes occur at a faster rate in smaller populations
22 !Pr( ) 1
!(2 )!
N kkNk p p
k N k
probability of obtaining k copies of an allele with frequency p in the next generation
sampling process
Random drift (genetic drift)
• Gene frequencies in the sub-populations drift apart over time, until all frequencies become equally probable (steady state)
• Once the steady state is attained, the rate of fixation is 1/N in each generation
• The longterm effect of drift for a finite population is a loss of genetic variation
• Historical effects of drift are locked in (founder effect or bottleneck effect)
Buri, Peter. 1956. Gene frequency in small populations of mutant Drosophila. Evolution 10:367-402.
eye color in Drosophila105 populations, N=16at t=0 f(bw)=f(bw75)=0.5
Dispersive process – effects on variance
N2
qp 002q
2q Variance in gene frequency
among sub-populations at t=1
Variance among sub-populations increases in each generation. At time t:
20 0
11 1
2q
t
p qN
p0q0 at t =
Change in genotype frequency
Genotype Frequency across sub-populations
A1A1 2q
20 σp
A1A2 2q00 2σq2p
A2A2 2q
20 σq
• As gene frequencies become more dispersed towards the extremes
– there is an increase in homozygosity and decrease in heterozygosity within each sub-population
– genetic uniformity increases within sub-populations
Definition of inbreeding
inbreeding = mating of individuals that have common ancestors
• identical by descent (ibd) = alleles are direct descendents from a common ancestral allele (autozygous)
• identical in state = alleles have the same nucleotide sequence but descended from different ancestral alleles (allozygous)
• An individual is inbred if it has alleles that are identical by descent
Coefficient of inbreeding
• Probability that two alleles at any locus in an individual are ibd (also applies to alleles sampled at random from the population)
• Must be in relation to a base population
2N
1ΔF Change in inbreeding
in a single generation
Inbreeding at generation t1tt F
N2
11
N2
1F
new old
tF11Ft Recurrence equation
Inbreeding
N2
qp 002q
2q
Remember:For a single generation
2N
1ΔF
tF11Ft
20 0
11 1
2q
t
p qN
At time t
Fqp 002q
t002q Fqp
Genotype frequencies with inbreeding
Genotype Frequency across sub-populations
A1A1 2q
20 σp
A1A2 2q00 2σq2p
A2A2 2q
20 σq
Genotype Frequency across sub-populations
Showing origin
A1A1 Fqpp 0020 FpF1p 0
20
A1A2 Fq2pq2p 0000 F1q2p 00
A2A2 Fqpq 0020 FqF1q 0
20
What will genotype frequencies be when the sub-populations are completely inbred?
Calculation of F from population data
F can be viewed as the deficiency in observed heterozygotes relative to expectation:
Genotype Frequency
A1A1 FpF1p 020
A1A2 F1q2p 00
A2A2 FqF1q 020
e
e e
2pq 2pq 1 FH H H1 F
H 2pq H
H = observed frequency of heterozygotesHe = expected frequency of heterozygotes
F statistics – relative deficiency of heterozygotes
(1-FIT)=(1-FIS)(1-FST)
I = individual S = sub-population T = total
Base populationN =
2N 2N 2N
N N N
t=0
Generation t
1 2 3 4 5…..Individuals in a subpopulation
FIT
FIS
FST
I I S
T S T
H H HH H H
What population sizes are needed for breeding?
1. Calculate the population size needed to have the expectation of obtaining one ideal genotype
For a trait controlled by 10 unlinked loci:
(1/4)10 in an F2, so N = 410 = 1,048,576
(1/2)10 in an inbred line, so N = 210 = 1024
2. Consider how to stabilize variance of allele frequencies
Bernardo, Chapt. 2
0.00
0.05
0.10
0.15
0.20
0.25
0 50 100 150 200 250
Population Size (N)
Sta
nd
ard
err
or
of
q Would be more critical for a long-term recurrent selection program than for a particular F2 population
Effective population size
Number of individuals that would give rise to the calculated sampling variance, or rate of inbreeding, if the conditions of an idealized population were true
F2
1Ne
eN2
1F
Falconer, Chapt.4
Effective population size
• unequal numbers in successive generations
– effects of a bottleneck persist over time
• different numbers of males and females
1 2 3
1 1 1 1 1 1....
e tN t N N N N
4 m fe
m f
N NN
N N
Falconer, Chapt.4
harmonic mean
Half-sib recurrent selection in meadowfoam
Year 1 – create half-sib families
500 spaced plants in nursery
outcrosshalf-sibs familiesselfS1 families
Year 2 – evaluate families in replicated trials
Year 3 - Should I go back to remnant half-sib seed of selected families or use the selfed seed for recombination?
Migration
• How many new introductions do I need in my breeding program to counteract the loss of genetic diversity due to inbreeding (genetic drift)?
m is the migration rate (frequency)
Nem is the number of individuals introduced each generation
A few new introductions each generation can have a large impact on diversity in a breeding population
STe
1F
4N 1
m
Inbreeding coefficients from pedigrees
AB AC BX CX Prob.
a1 a1 a1 a1 (½)4
a2 a2 a2 a2 (½)4
a1 a2 a1 a2 (½)4
a2 a1 a2 a1 (½)4
A
B C
X
x
a1a2
FX=2*(½)4+2*(½)4*FA
=(½)3+(½)3FA= (½)3(1+FA)
An
21
X F1F n = number of individuals in path
including common ancestor
Falconer Chapt. 5; Lynch and Walsh pgs 131-141
Inbreeding coefficients from pedigrees
B
E
D
G
H
C
J
A Paths of Relationship n
F of common ancestor
Contribution to FJ
EBACH 5 0 (1/2)5
EBADGH 6 0 (1/2)6
EBCH 4 0 (1/2)4
ECADGH 6 0 (1/2)6
ECBADGH 7 0 (1/2)7
ECH 3 1/4 (1/2)3*(1+0.25)
FJ= 0.2891
• E is inbred but this does not contribute to FJ
• No individual can appear twice in the same path• Path must represent potential for gene
transmission (BCA is not valid, for example)
Coefficient of coancestry
identical by descent (ibd) = alleles descended from a common ancestral allele
ABC θF
CF inbreeding coefficient = probability that alleles in C are ibd
ABθ coefficient of coancestry• probability that alleles in A are ibd with alleles in B• aka coefficient of kinship, parentage or consanguinity
A B
C
x
Note: AB = fAB in Bernardo’s text
Coefficient of coancestry
A B
C
x
ABθ • alleles received by A and B• alleles sampled from A and B (to go to offspring)
CF • alleles received by C
ccθ • alleles sampled from C (to go to offspring)
Formal calculation of coancestry
A B
C
xa1a2 b1b2
a b
c1c2
),, 1111 babba(a PθF ABC
),, 2121 babba(a P),, 1212 babba(a P),, 2222 babba(a P
)()()() 22122111 bababab(a4
1 PPPPθAB
Rules of coancestry
BDBCADACEG θθθθ41θ
BCACEC θθ2
θ1
EDECEG θθ2
θ1
HHH F12
θ1
A x B
E
x
C x D
G
H
Coancestry: selfing
AAA X A
1 1 1F F 1 Fθ
2 2 2
Aa1a2
x Aa1a2
X¼ a1a1
½ a1a2
¼ a2a2
XXX
11 Fθ
2
Derivation of the rules: another example
A x B
E
x
C x D
G
H
Alleles from E Alleles from C
1/4 a1 1/2 c1 AC
1/4 a1 1/2 c2 AC
1/4 a2 1/2 c1 AC
1/4 a2 1/2 c2 AC
1/4 b1 1/2 c1 BC
1/4 b1 1/2 c2 BC
1/4 b2 1/2 c1 BC
1/4 b2 1/2 c2 BC
BCACBCACEC θθ2
θ44θ8
θ11
Coancestry of full sibs
A x B
C
x
A x B
D
E
A
C
x B
D
E
BABA
BBABAA
BBBAABAACD
F12
1θ2F1
2
1
4
θθ2θ4
θθθθ4
θ
1
1
1
42
1
2
1
4θ
11CD
with no prior inbreeding
Note: could get same resultby calculating FE
Tabular method for calculating coancestries
A B
C D
E F
G
• Can accommodate different levels of inbreeding in parents
• Can incorporate information from molecular markers about the contribution of parents to offspring (may vary from 0.5 due to segregation during inbreeding)
• Can be automated
CFGFCEGECG θθθ
contribution of E to G = 0.5
Excel
Regular systems of inbreeding
• Same mating system applied each generation
• All individuals in each generation have the same level of inbreeding
• Purpose is to achieve rapid inbreeding
• Develop recurrence equations to predict changes over time
A
B
AAAB F12
θF1
1-tt F12
F1
Example: repeated selfing
Regular systems of inbreeding
A CB
D E G
H J
Mating system CoancestryNo prior
inbreedingRecurrence
equation
full sibs EG=(1/4)(2BC+BB+CC) 1/4 Ft=(1/4)(1+2Ft-1+Ft-2)
half sibs DE=(1/4)(AB+AC+BB+BC) 1/8 Ft=(1/8)(1+6Ft-1+Ft-2)
parent-offspring AD=(1/2)(AAAB) 1/4 Ft=(1/2)(1+Ft-2)
backcrossing FH=BD=(1/2)(BB+AB) 1/4 Ft=(1/4)(1+FB+2Ft-1)
selfing BB=(1/2)(1+FB) 1/2 Ft=(1/2)(1+Ft-1)