Inbreeding

GEST325 Lecture 22

Overview

• Relatonships between individuals! Inbreeding (F)! Coancestry (f)

• Effective population size (Ne) &F rate• Consequences of inbreeding• Balancing F and genetic merit

Coefficient of inbreeding (F)

Inbreeding is due to the mating of relatives

The coefficient of inbreeding (F) is the probability that two alleles at a randomly chosen locus are identical by descent (IBD)IBD = copies of same alleles from common ancestor

F ranges from 0 to 1

What is F of individual X?Recall: The coefficient of inbreeding (F) is the probability of the 2 alleles at a randomly chosen locus being identical by descent

Shortcut: n = 3 descendents (D, A, E) in the closed loop -FX = (½ )3 = 1/8

161

21

21

21

21

11 ====

==== xxxp AA

161

21

21

21

21

22 ====

==== xxxp AA

81====xF

AB C

D E

X1 2XX

A A1 2

What is F of individual X?Recall: The coefficient of inbreeding (F) is the probability of the 2 alleles at a randomly chosen locus being identical by descent

161

21

21

21

21

11 ====

==== xxxp AA

161

21

21

21

21

22 ====

==== xxxp AA

41====xF

A B

C D

X

A1A2 B1B2

X1X2 161

21

21

21

21

11 ====

==== xxxp BB

161

21

21

21

21

22 ====

==== xxxp BB

Shortcut: 2 closed loops (C,A,D n=3 & C,B,D n=3)FX = (½ )3+(½ ) = 1/43

Common ancestor X is inbred (F=0.25)

X Y

P Q

R

C DLoops are

P,X,Q (1/2)3 x (1+0.25) = 0.156

P,Y,Q (1/2)3 x (1+0) = 0.125

FR = 0.281

What is F of individual R?A B

Common ancestor is not inbred! sum over loops, where n is number of animals in

each loop excluding x itself

Common ancestor is inbred! sum over loops, but multiply by F of the common

ancestor (FA)

nXF )2

1(ΣΣΣΣ====

(((( ))))(((( ))))An

X FF ++++××××ΣΣΣΣ==== 1)21(

Question

Which alleles are IBD (identical by descent) and which are IBS (identical by state)?

A1A2 B1B2 B1B2 C1C2

A1B1 A1B2 B2C1

A1A1 B2B2

Another measure of relatedness

The coancestry (fij) is the probability that two gametes taken at random from two individuals (i & j) carry alleles that are IBD

f ranges from 0 to 1

What is f of individuals D,E?

The coancestry of i & j (fi,j) is equivalent to the F of their progeny

AB C

D E

Pick an allele in D and an allele in E. Chances of being IBD are 2/8 if an paternal allele was picked, and 0 otherwise.

Recall: The coancestry (f) is the probability of that 2 gamettes taken at random from 2 individuals carry alleles identical by descent

A1A2 C1C2B1B2

A1B1A1B2A2B1A2B2

A1C1A1C2A2C1A2C2

81====f

sire

Another measure of relatedness

Addditive genetic relationship (a) between 2 individuals reflects their proportion of genes in common

! e.g. between parent and offspring a=0.5! e.g. between two full-sibs a=0.5! e.g. between two half-sibs a = 0.25

Summarising

FX=(1/2)3+(1/2)3 = 0.25

fC,D=FX=0.25

aC,D=0.50

A B

C D

X

Ne

The size of the idealized population with the same inbreeding rate as the actual population

The idealized population has ‘simplified’ properties:! Closed! Random mating! Discrete generations! Constant population size! Equal sex ratios! Poisson distribution of family size

Calculating Ne

Accounting for unequal sex ratio! Ne = 4 Nm Nf / (Nm + Nf)! Effective pop’n size reduces towards sex with

fewer breeding individuals

472.719.57.97.34Ne100,000220205202224N99999200200200202Females / generation

1205222Males / generation

The rate of inbreeding

F can be calculated directly for progeny. This is useful when designing specific matings

For populations it is more relevant to calculate the inbreeding rate

t

t NeF

−−=

2111 where t is number

of generations

Note - this only holds for no selection and random mating

F rate

t

t NeF

−−−−−−−−====

2111

Generation 0

A1A2 A3A4 A5A6

Gene pool: 2N alleles

Generation 1

Sample with replacement. The probability of sampling 2 copies of the same allele is 1/(2N). The probability of sampling 2 different alleles is 1-1/(2N) (in later gen’s x by F as some alleles will be IBD)

N21

N211 −−−−

A4A1A1 A3 A5 A3 A4A6

A2A2 A5

A2A2 A6A3

1211

21

−−−−

−−−−++++==== tt F

NNF

A6

F rate & Ne

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50

Generation

F

Ne=5Ne=25Ne=50Ne=500Ne=5000

Example calculation! F after 20 years when breeding with age structure

• Lm=2.5 years, Lf = 4 years, L=3.25 years

• Number of animals entering flock per generation

Nm=1 male / year x 3.25 years / gen = 3.25 male / gen

Nf=20 fem. / year x 3.25 years / gen = 65 fem. / gen

• Ne = (4NmNf)/ (Nm+Nf) = (4x3.25x65)/(3.25+65)=12.38

Age 2 3 4 5 6

Males 1 1

Females 20 20 20 20 20

23.038.122

11125.3/20

25.3/20 ====

−−−−−−−−====

xF

t

t NeF

−−=

2111

f rate

At equilibrium,under consistent selection and with random mating, ∆F is equal to ∆f

Constraining ∆f is likely the most effective way to constrain ∆F! i.e. select parents such that the mean f (of all

mating pairs) is restricted to an acceptable level

Questions

! In breeding programs recording pedigree are absolute values of F usually known?

! How can F within a population be reduced within the short-term?

! Should a desired long-term rate of inbreeding be included in the breeding objective?

Consequences of inbreeding

Inbreeding can result in:! Increased expression of recessive genes! Loss of performance & viability termed inbreeding

depression! Loss of genetic variation within the inbred population

Inbreeding increases expression of recessive alleles

Genotype frequencies! Non-inbred: q2 2pq p2

! Inbred: q2+pqF 2pq-2pqF p2+pqF

Example, q=0.02 (2%)

10.2 in 1000

0.50

5.3 in 10002.9 in 10000.4 in 1000Prob. aa

0.250.1250F

Change in genotype frequenciesin response to inbreeding

For example, p=q=0.5

Note that allele frequencies do not change

0.500.5At F=1.0

0.3750.250.375At F=0.5

0.250.500.25At F=0

p2+pqF2pq-2pqFq2+pqFFrequency

AAAaaaGenotype

Inbreeding depression reduces productivity & viability

Inbreeding depression! Due to increased homozygosity, in relation to traits

that show dominance! Most notable effect is on reproductive fitness

Inbreeding depression is typically greater in the wild than in captivity! Trait depression variable, often 2-20% per 10% F

Inbreeding depression

For example Zebra fish, survival to 30 days was depressed by 43% per 25% F

020406080

100

0 0.1 0.2 0.3

F

% S

urvi

val

Inbreeding reduces genetic variance

As individuals become more alike, the withinpopulation genetic variance decreases

VA is additive genetic variance

VA (with inbreeding) = (1-F) VA (without inbreeding)

Questions

! A recessive allele (called non-Agouti) causes uniformly black sheep. Would this be more of a problem in a lowly or highly inbred population?

! If a trait is controlled by purely additive genes (i.e. no dominance) will expression of the trait be effected by inbreeding depression?

! Why is it a concern that inbreeding reduces genetic variance?

Genetic gain and inbreeding

Select few individuals

high genetic gainbut

low Ne and high F

Need to balance rates of F and genetic gain

Select many individuals

low genetic gainbut

high Ne and low F

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

mea

n EB

V

coancestry

The relationship between merit and relatedness

Unlimited reproductivity

Restricted reproductivity

Mean coancestry of selected parents

Mer

it

Optimal contribution theory

Aimed at maximising long-term response by weighting genetic response against future contributions to inbreeding

Maximise an objective function of:

where G is genetic merit, b is a weight, and f is coancestry

fbG ∆∆∆∆−−−−∆∆∆∆

Balancing merit and diversity

‘Total Genetic Resource Management’ (TGRM)! a software program (Brian Kinghorn)! determines which animals to select for breeding,

and how these animals should be mated to each other ‘mate selection’

! balances genetic merit & f, but also handles logistical and other constraints

! driven by an evolutionary algorithm

Possible outcomes

Judgement

Data construction:Pedigree

Trait records

TGRM Control CenterInternet driven

Action : A mating list

Opportunities ConstraintsAttitudesIssuesCosts

TGRM Server

TGRM interface

Summary and useful concepts

• Inbreeding is due to mating of relatives, and may have undesirable consequences

• The inbreeding coefficient is relative to some base population, and thus it is more useful to consider the rate of F rather than the absolute value

• Theories exist to balance rates of F and genetic gain