Biomedical Engineering Reference
In-Depth Information
The multi-locus computation of sib-pair genotypes depends on the genotypes of par-
ents. Parents with unknown genotypes are called founders and the offspring with
known genotypes are called nonfounders.
Consider an arbitrary family (known as a pedigree) with
f
founders and
n
nonfounders. (A sib-pair pedigree is a special case with two founders and two non-
founders). The length of the inheritance vector is now 2
n
(paternal and maternal alleles
of each nonfounder), and the number of possible vectors also called possible states is
2
2
n
. Denote by
a
l
=
(a
l
,1
,
...
,
a
l
,2
f
)
the vector of alleles assigned to the founders at
locus. The formulation of the computation of the probability of the observed marker
genotypes
m
l
given the inheritance vector
v
l
at marker locus
l
is
a
l
P
P
{
m
l
|
v
l
}=
{
m
l
,
a
l
|
v
l
}
.
The IBD probabilities at locus
l
is the summation of
P
{
m
l
|
v
l
}
over
v
l
corresponding
to IBD
=
0,1or2.
1.1.8 Pedigree Probabilities
For a pedigree composed of
k
persons, each person
K
has an ordered geno-
type relative to many locations (called loci):
X
k
=
(x
km
,
x
kp
,
x
km
,
x
kp
,
...
,
x
km
,
x
kp
)
where
x
km
is the allele inherited from the mother at locus
i
and
x
kp
is the allele inherited from the father at locus
i
. The problem is to calculate
the probability of phenotypes (unknown genotype) of all persons of a family, when
knowing the genotype at different positions.
Another way of defining the problem, given the family with many loci in an order
and the phenotypes of members of the pedigree, given also the recombination's frac-
tions between consecutive loci
θ
i
, the problem will be to reconstruct the best genetic
map by maximizing the chance of having data presented and to replace the old recom-
bination fractions by the new one calculated by the algorithm. Genetic mapping is
placing genes and other genetic markers on chromosomes, ordering them in relation
to one another and other landmarks (such as centromeres), and assigning genetic
distance to adjacent pairs.
(x
k
,
x
k
,
...
,
x
k
)
=
1.1.8.1 Probability Types
We distinguish three types of probabilities:
1.
Founder Probabilities
: If the person
K
is a founder, the probability of the
ordered genotype across all loci is equal to the multiplication of all allele
frequencies at different loci.
P
x
k
]∗
x
k
]∗ ··· ∗
x
k
]
[
X
k
]=
P
[
P
[
P
[
.
P
l
3
. Thus, the
allele frequency at locus l is the multiplication of population frequency of each
allele at the specific locus.
2.
Transmission Probabilities
: This is the probability that a child inherits a par-
ticular genotype from his
/
her parents.
P
X
k
=
P
l
1
∗
If
X
k
has as alleles 1 and 3 at locus
l
→
P
[
1, 3
]=
[
X
k
=
x
k
|
X
m
=
x
m
,
X
p
=
x
p
]
, then all
genotypes are ordered
P
[
X
k
|
x
m
,
x
p
]=
P
[
X
km
|
x
m
]∗
P
[
X
kp
|
x
p
]
.
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