112 Utilisation and conservation of farm animal genetic resources
Herwin Eding and Jörn Bennewitz
Het
t
Het = 1 –
F where
Het
t
is heterozygosity in generation t and Het
in the founder generation, and F is the inbreeding coeicient relative to the founder generation. Kinship, also called
coancestry f , is used to calculate the inbreeding coeicient and F
X
= f
PQ
, where f
PQ
is the coancestry of the parents P and Q of individual X. Twice the kinship, the coeicient
of additive relationship is used to calculate the additive genetic variance σ²
A.
Because σ²
A
is proportional to heterozygosity, over t generations we have Falconer and Mackay, 1996; Gilligan
et al., 2005: σ²
A,t
σ²
A,0
= 1 – F
here are many diferent estimators for relatedness. here are coancestry based estimators Toro
et al., 2002; Eding and Meuwissen, 2001; Oliehoek et al., 2006 and two- and four-gene identity coeicient estimators Lynch and Ritland, 1999; Wang,
2002. Coancestry based estimators are more general in nature and perform well in a wide variety of population classes, while two- and four-gene identity coeicients show a
substantial loss of eiciency in non-random mating populations Oliehoek et al., 2006.
For these reasons we focus on coancestry based estimators of kinship.
2.3.1. Genetic similarities
he most basic measure of relatedness is the genetic similarity. Genetic similarities are also known as allele sharing cf. Lynch, 1988. Basically any pair of individuals within
or between populations is scored for common alleles for a number of loci. he total score is then averaged over loci to obtain the mean similarity between individuals.
Further averaging over pairs of individuals gives the mean similarity between or within populations.
here are two main methods to score genetic similarities with co-dominant polymorphic markers: the genic similarity Lynch, 1988 and the Malécot similarity Eding and
Meuwissen, 2001. he diference between these two can be found in scoring of similar genotypes Table 5.1. While the genic similarity is the number of alleles shared by two
individuals out of the total number of alleles e.g. 4 in diploid organisms, the Malécot similarity is the probability that an allele randomly drawn from one individual is the
same as an allele randomly drawn from the other individual Malécot, 1948. he latter is derived from the deinition of the coeicient of kinship. Hence, if we assume that
alleles can only be identical by descent that is: all similar alleles are copies from one and the same ancestral allele, the mean Malécot similarity calculated over multiple loci is
expected to be equal to the coeicient of kinship.
Utilisation and conservation of farm animal genetic resources 113
Chapter 5. Measuring genetic diversity in farm animals
In formula form the Malécot similarity for two individuals with genotypes ab and cd at locus k is written as:
bd bc
ad ac
k xy,
I +
I +
I +
I =
S 4
1
¦
where I
xy
is an indicator variable that is 1 when allele x and y are similar, otherwise I
xy
is zero. hus the similarity can have values 0, ¼, ½ and 1. he Malécot similarity is advantageous over other similarity measures because it can be
calculated directly from allele frequencies Eding and Meuwissen, 2001. For a locus with M alleles the similarity between populations
I and J is:
¦
M =
m m
J, m
I, IJ
p p
= S
1
Table 5.1. Scoring of similar genotypes in the genic and Malecot genetic similarity and probabilities in populations at Hardy-Weinberg equilibrium.
genotype genic
Malecot Probability
AA-AA 1
1
Table 5.1.
genotype genic Malecot Probability
¦
M =
m m
J, m
I,
p p
1 2
2
¦
¦
¦¦
z
¦¦ ¦¦
z z
AA-AB ½
½
genotype genic Malecot Probability
¦
¦
M =
m m
J, m
J, m
I,
p p
p
1 2
1 2
¦
¦¦
z
¦¦ ¦¦
z z
AB-AA ½
½
genotype genic Malecot Probability
¦
¦
¦
M =
m m
J, m
I, m
I,
p p
p
1 2
1 2
¦¦
z
¦¦ ¦¦
z z
AB-AB 1
½
genotype genic Malecot Probability
¦
¦
¦
¦¦
z M
= m
M m
n n
J, n
I, m
J, m
I,
p p
p p
1
2
¦¦ ¦¦
z z
AB-AC ½
¼
genotype genic Malecot Probability
¦
¦
¦
¦¦
z
¦¦ ¦¦
z z
M =
m M
m n
n J,
n I,
m I,
m J,
m I,
M =
m M
m n
n J,
m J,
n I,
m J,
m I,
p p
p p
p +
p p
p p
p
1 1
1 2
1 2
114 Utilisation and conservation of farm animal genetic resources
Herwin Eding and Jörn Bennewitz
he expression for population similarity can be found in diferent guises in diferent genetic diversity measures, showing that there is a strong connection between genetic
diversity and kinship Box 5.3.
2.3.2. Correction for alleles alike in state
Technically indistinguishable alleles are either identical by descent IBD, two alleles are
copies of the same ancestral allele due to kinship, or alike in state AIS, two alleles are
indistinguishable from one another, but are not IBD. he probability of alleles AIS is indicated with the symbol
s. he mean expected similarity
S
ijl
between two individuals i and j for a locus l is a
function of both the kinship between i and j f
ij
and s
l
at this locus Lynch, 1988:
l ij
l l
ij ij
ijl
s +
f s
= s
f +
f =
S 1
1
Rearrangement of the equation, substituting expected similarities with observed similarities and averaging over loci gives an estimator of
f
ij
for L loci:
¦
L =
l l
l ijl
ij
s s
S L
= f
1
1 1
ˆ
= 0 + 1 – 0
¦
hus, to estimate kinships between individuals or populations some value of s
l
must be assumed or estimated.
If we assume a model with an ininite number of alleles in the founder population, s
l
equals zero and f
ij
is expected to be equal to S
ijl
for all loci . However, when s
l
is non-zero, a founder population, in which all individuals are assumed unrelated
f = 0, will have S
l
= 0 + 1 – 0 s = s
l
. Hence deinition of s
l
implicitly deines the founder population.
2.3.3. Molecular coancestry
Toro et al., 2003 estimates coeicient of kinships under the assumption that s
l
= 0 for all loci. Between and within populations the molecular coancestry is the average over
loci of the Malécot similarity. Between individuals, this similarity can be expressed as:
¦
ˆ
= 0 + 1 – 0
l bd
bc ad
ac L
= l
ij ij
M,
I +
I +
I +
I L
= S
= f
¦
1
4 1
1