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)