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Genetic identity and Kinship
Genetic Identity
 G1 and G2 are identical by descent (i.b.d) if they
are physical copies of the same ancestor, or one of
the other.
 G1 and G2 are identical by state (i.b.s) if they
represent the same allele.
 The kinship between two relatives fij is the
probability that random gene from autosomal loci in
I and j are i.b.d.
 The interbreeding coefficient is the probability that
his or her two genes from autosomal loci are i.b.d.
Mutation occurred once
•Every mutation creates a new allele
•Identity in state = identity by descent (IBD)
A1A1
A1A2
A1A1
A1A2
A1A2
A2A2
The same mutation arises
independently
A1A1
A1A2
A1A2
A1A1
A1A1
A1A2
A1A2
A2A2
A1A2
A1A2
A2A2
A2 A2 IBD
A2 A2 IBD
A2 A2
alike in state (AIS)
not identical by descent
A1A1
Identity by descent
A - B
C - D
|
|
P
Q
|
X
Let fAC be the coancestry of A with C etc., i.e. the probability of
2 gametes taken at random, 1 from A and one from B, being
IBD.
Probability of taking two gametes, 1 from P and one from Q, as
IBD, FX
1
1
1
1
FX  f PQ  f AD  f AC  f BC  f BD
4
4
4
4
Identity by descent
 Example, imagine a full-sib mating
A - B
/
\
P - Q
|
X
 Indv. X has 2 alleles, what is the probability of IBD?
1
1
1
1
FX  f PQ  f AD  f AC  f BC  f BD
4
4
4
4
1
1  1 1  1
 2 f AB  f AA  f BB   0   
4
4  2 2  4
Identity by descent
 Example, imagine a half-sib mating
A - B - C
|
|
P - Q
|
X
1
1
1
1
FX  fPQ  fAD  fAC  fBC  fBD
4
4
4
4
1
1 
1  1
 2 fAB  fAC  fBC  fBB  0 0 0 
4
4 
2  8
Kinship and Interbreeding
 fii=0.5(1+fi)
 fi=fkl, where k and l are the parents of i.
 If fi>0 then i is said to be inbred.
 The question is how to compute kinship,
given a pedigree.
 Let us look for example at brothers and
sisters:
Pedigree
 Let us compute the
kinship coefficients
for all member of this
pedigree. We assume
that 1 and 2 are not
inbred and unrelated,
we start counting from
the oldest generation.
 We will develop an
algorithm to compute
fij for all members of
this pedigree.
1
2
3
4
5
6
Kinship coefficient algorithm
1
3
2
4
1
2
1
1/2
0
2
0
1/2
3
5
6
4
5
6
3
4
5
6
Kinship coefficient algorithm II
1
3
2
4
1
2
1
1/2
0
2
0
1/2
3
5
6
4
5
6
3
4
5
6
Kinship coefficient algorithm
III
1
3
5
1
2
1
1/2
0
2
0
1/2
2
3
4
4
6
3
4
If i originates
5
from k and l
Fii= ½+ Fkl 6
1/2
1/2
5
6
Kinship coefficient algorithm
III
1
3
5
1
2
3
4
1
½
0
¼
¼
2
0
½
¼
¼
3
¼
¼
½
¼
4
¼
¼
¼
½
2
4
6
If i originates
from k and l
Fij= Fji
= ½(Fjk + Fjl)
5
6
5
6
Kinship coefficient algorithm
IV
1
3
5
1
2
3
4
1
½
0
¼
¼
2
0
½
¼
¼
3
¼
¼
½
¼
4
¼
¼
¼
½
2
4
6
One can now
reapply the
algorithm on
the next
generation
5
6
5
6
Kinship coefficient algorithm V
1
3
5
The final
result is:
1
2
3
4
5
6
1
½
0
¼
¼
¼
¼
2
0
½
¼
¼
¼
¼
3
¼
¼
½
¼
3/8
3/8
4
¼
¼
¼
½
3/8
3/8
5
¼
¼
3/8
3/8
5/8
3/8
6
¼
¼
3/8
3/8
3/8
5/8
2
4
6
Identity coefficients:
 We can now
summarize
the kinship
coefficient of
some basic
family
relations:
Relation
f
Parent-Offspring
¼
Half Sibling
1/8
Full Sibling
¼
First Cousins
1/16
Double First Cousins
1/8
Second Cousins
1/64
Uncle-Nephew
1/8
Detailed Identity States – I
Allele 1
I
J
Allele 2
Detailed Identity States – I I
Detailed Identity States – I I I
Summary
 Many of these
relations are
redundant. If I is not
inbred 1,2,3 and 4 will
be zero.
 One can define kinship
in a condensed mater,
if we can interchange
the maternal and
paternal genes.
Condensed Identity States
Condensed Identity States
 S3=S*2S*2
 S5=S*4S*5
 S7=S*9S*12
 S8=S*10S*11
 S*13S*14
Condensed Identity States II
 D1 ,D2 ,D3 ,D4 are 0,
when i is not inbred.
 D1 ,D2 ,D5 ,D6 are 0,
when j is not inbred.
 D1 ,D3 ,D5 ,D7 and D8
are 0, when i and j are
unrelated.
 Fji=D1+1/2( D3 D5
D7)+1/4 D8
D1
D2
D3
D4
D5
D6
D7
D8
Kinship and identity
coefficients
Relation
Parent-Offspring
Half Sibling
Full Sibling
First Cousins
Double First Cousins
Second Cousins
Uncle-Nephew
D7
D8
D9
0
1
0
0
½
½
1/4 ½
¼
0
¼
¾
1/16 6/16 9/16
0 1/16 15/16
0
½
½
f
¼
1/8
¼
1/16
1/8
1/64
1/8
Genotype prediction.
 What is the probability that i has a given genotype,
given the genotype of j ?
 For example, If my uncle has a genetic disease,
what is the probability that I will also have it?
 What are the probabilities of brothers from inbred
parents to be homozygous for a disease causing
gene?
 ……
Genotype prediction.
If I is heterozygous, with an inbreeding coefficient fi
9
Pr( j  m / n | i  k / l )   Pr( j  m / n | S r , i  k / l )
r 1
* Pr( S r | i  k / l )
0
r4
Pr( S r , i  k / l )  D 2 p p
Pr( S r | i  k / l ) 

r
k l
r4
Pr(i  k / l )
 (1  f i )2 pk pl
r4
0

Dr

r4
 (1  f i )
Genotype prediction II
If I is homozygous, with an inbreeding coefficient fi
D r pk

Pr( S r , i  k / k )  f i pk  (1  f i ) pk2
Pr( S r | i  k / k ) 

2
D
p
Pr(i  k / k )
r k

 f i pk  (1  f i ) pk2
Dr

r4
 f  (1  f ) p
i
k
 i
2
D r pk

r4
 f i  (1  f i ) pk
r4
r4
Genotype prediction III
Pr( j  m / n | S r , i  k / l ) 
j i
S  1,7
S  2,4,6,9 j is independen t of i


j shares one gene with i
S  3,8
S  5
j is either k/k or l/l
When j is independent of i, it only follows the H,W equilibrium.
When j is equivalent to i, the probability is one if m/n=k/l
and zero otherwise.
When j shares one allele with I, m/n and k/l must overlap with
one allele and the other one has H.W distribution.
Example
 What is the blood type of non-inbred
siblings?
1
Pr( j  A / B | i  A / B)  Pr( j  A / B | S 7 , i  A / B)
4
1
1
 Pr( j  A / B | S8 , i  A / B)  Pr( j  A / B | S9 , i  A / B)
2
4
1
1 1
1
1
 *1  ( p A  pB )  2 p A pB
4
2 2
2
4
Example I
 What is the blood type of non-inbred
siblings?
Pr( j  A | i  O / O)  Pr( j  A / O | i  O / O)
 Pr( j  A / A | i  O / O)
1
Pr( j  A / O | S 7 , i  O / O)
4
1
1
 Pr( j  A / O | S8 , i  O / O)  Pr( j  A / O | S 9 , i  O / O )
2
4
 .......................

1
1
1 2 1
1
 * 0  * 0  2 p A  p A  2 p A po
4
2
4
2
4
Risk Ratios and Genetic Model
Discrimination.
 Let us assume that each person in the population is
assigned a factor of X=1 if he/she is affected by a
condition and X=0 otherwise.
 The Prevalence of the condition is K=E(X).
 Given two non-inbred relatives i and j and given
that i is affected, what is the probability that J is
affected?
 KR=P(Xj=1|Xi=1(
 P(Xj=1,Xi=1) = P(Xj=1|Xi=1(P(Xi=1) = KRK =
E(XiXj)
Risk Ratios and Genetic Model
Discrimination.
 P(Xj=1|Xi=1) = E(XiXj)/K = (cov(Xi,Xj)+K2)/K =
cov(Xi,Xj)/K+K
 This result simply represents the fact that the extra
risk for j results from the covariance of X between i
and j.
 The risk ratio can thus be defined as:
 lR= cov(Xi,Xj)/K2
 Let us compute this covariance, and following it
the risk ratio.
Covariance
 In a more general way, let us assume that i
and j are non-inbred relatives. The
covariance between their genes is defined
only by condensed identity states 7,8 and 9
D1
D2
D3
D4
D5
D6
D7
D8
Covariance
 Let us assume that a given property is defined by a
single gene with multiple alleles.
E ( X )    kl pk pl
k
l
For the sake of simplicity let us normalize E(x)=0, and
divide:
 kl   k   l   kl ;  k pk  0

k
k
kl
pk  0
Covariance
E ( X i X j )    mn  kl p (m, n | k , l ) pk pl
k
l
m
n
 D 7 ij  ( k   l   kl ) 2 pk pl  D 8ij  ( k   l   kl )( k   m   km ) pk pl pm
k
l
k
l
k
l
m
 D 9ij  ( k   l   kl )( m   n   mn ) pk pl pm pn
m
n




2
2
2
 D 7 ij 2  k pk    kl pk pl   D 8ij 2  k pk 
l
k

 k

 k
1

1
2
2
 2 D 7 ij  D 8ij 2  k pk  D 7 ij   kl pk pl
4
 k
2
l
k
 2fij a2  D 7 ij d2
Risk Ratio
R
Relative Type
Risk Ratio
M
Identical Twin
a2/K2 +d2/K2
S
Sibling
a2/2K2 +d2/4K2
1
First Degree
a2/2K2
2
Second Degree
a2/4K2
3
Third Degree
a2/8K2
Summary
Concepts
Kinship coefficient.
Inbreeding coefficient
Identity states
Condensed identity states
Identity coefficients
Genotype prediction