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Single Marker Analysis Using F2 Progeny
Reference: Ben Hui Lui. Statistical Genomics: Linkage,
Mapping and QTL Analysis. pgs. 402-405.
In this situation we have one marker linked to a QTL.
We do not have a marker on either side of the QTL, which
would be the case for flanking markers. Marker genotypes
are MM, Mm, mm and the QTL genotypes are QQ, Qq, and qq.
The marker and the QTL are in coupling phase linkage.
In the F2 population the relative frequencies
and means of the three marker classes are:
| Marker
Class |
Relative
Frequency |
Marker
Mean |
| MM |
0.25 |
u1 |
| Mm |
0.5 |
u2 |
| mm |
0.25 |
u3 |
In this same F2 population the relative frequencies
and means of the three QTL classes are:
| QTL
Class |
Relative
Frequency |
QTL
Mean |
| QQ |
0.25 |
m
+ a |
| Qq |
0.5 |
m
+ d |
| qq |
0.25 |
m
- a |
The observed recombination fraction between the marker
and the QTL is ‘r'. The relative frequency of crossovers
events between the marker and QTL loci is 2r. This is
because for each crossover event only two out of four
strands are involved in recombination. The gametic relative
frequencies for each marker-QTL combination are:
| Type
of Gamete |
Relative
Frequency |
| QM |
(1
- r)/2 |
| qm |
(1
- r)/2 |
| Qm |
r/2 |
| qM |
r/2 |
A Punnett square shows the genotypes in the F2
population.
(1 - r)/2
(1 - r)/2 r/2
r/2
| |
|
QM |
qm |
Qm |
qM |
| (1
- r)/2 |
QM |
QQMM |
QqMm |
QQMm |
QqMM |
| (1
- r)/2 |
qm |
QqMm |
qqmm |
Qqmm |
qqMm |
| r/2 |
Qm |
QQMm |
Qqmm |
QQmm |
QqMm |
| r/2 |
qM |
QqMM |
qqMm |
QqMm |
qqMM |
We can collect like genotypes to derive the relative frequencies
of each class.
Genotypes Relative
Frequency
| QQMM |
(1
- r)2/4 |
| QqMM |
(r
- r2)/2 |
| qqMM |
r2/4 |
| QQMm |
(r
- r2)/2 |
| QqMm |
(1
- r)2/2 + r2/2 |
| qqMm |
(r
- r2)/2 |
| QQmm |
(r/2)2 |
| Qqmm |
(r
- r2)/2 |
| qqmm |
(1
- r)2/4 |
We can rearrange the results from the above table into
a 3 X 3 table.
|
QQ |
Qq |
qq |
| MM |
(1
- r)2/4 |
(r
- r2)/2 |
r2/4 |
| Mm |
(r
- r2)/2 |
{(1
- r)2 + r2}/2 |
(r
- r2)/2 |
| mm |
r2/4 |
(r
- r2)/2 |
(1
- r)2/4 |
The marginal frequency for the MM marker class is the
relative frequency of this marker genotype averaged across
the three QTL genotypes:
| (1-r)2 |
+ |
(r-r2) |
+ |
r 2 |
=(1/4)[1-2r+r2+2r+2r2+r2]=1/4 |
| 4 |
2 |
4 |
The marginal frequency for the Mm marker class is:
| (r-r2) |
+ |
[(1-r2) + r2] |
+ |
(r-r2) |
=(1/2){r-r2+1-2r+r2+r2+r-r2}=1/2 |
| 2 |
2 |
2 |
The marginal frequency for the mm marker class is:
| r2 |
+ |
(r-r2) |
+ |
(1-r2) |
=(1/4){r2+2r-2r2+1-2r+r2}=1/4 |
| 4 |
2 |
4 |
The MM marker class can be identified with banding patterns
and includes QTL genotypes QQ, Qq, and qq. The MM marker
class is a mixture of QTL classes due to recombination
between the marker and the QTL. We can only identify the
means of each marker class, because we cannot identify
the QTL genotypes associated with a given marker in an
F2 plant. We can only estimate the means of
each marker class.
To determine the means of each marker class we must first
find the conditional probability of each QTL class for
a given marker class:
| P(Qj/Mi) = |
P(QjnMi) |
| P(Mi) |
The conditional probability of the QQ genotype given that
the MM marker genotype is present in that F2
is:
| P(QQ/MM)= |
P(QQMM) |
= |
{(1-r)2/4} |
=(1-r)2 |
| P(MM) |
4 |
We are dividing the probability of each marker-QTL genotype
by the probability of the marker genotype. We can develop
the following table:
|
QQ |
Qq |
qq |
| MM
1/4 |
(1
- r)2 |
2r(1
- r) |
r2 |
| Mm
1/2 |
(r
- r2) |
{(1
- r)2 + r2} |
(r
- r2) |
| mm
1/4 |
r2 |
2r(1
- r) |
(1
- r)2 |
Now we can calculate the mean of each marker class. The
markers do not contribute to the mean value. Only the
QTL loci contribute to the mean value.
uMM = ( 1 - r)2u1 +
2r( 1 - r)u2 + r2u3
Now u1 = u + a; u2 = u + d; u3
= u - a;
uMM = (1 - r)2(u + a) + 2r(1 - r)(u
+ d) + r2 ( u - a)
= [(1 - r)2
+ r2 + 2r(1 - r)]u + (1 - r)2a -
r2a + 2r(1 - r)d
= (1 - 2r + r2
+ r2 + 2r - 2r2)u + (1 - r)2a
- r2a + 2r(1 - r)d
= u + [1 - 2r + r2
- r2]a + 2r(1 - r)d
= u + (1 - 2r)a +
2r(1 - r)d
We can show that
UMm = r(1 - r)(u + a) + [(1 - r)2
+ r2](u + d) + r(1 - r)( u - a)
= u + [(1 -
r)2 + r2]d
We can also show that
Umm = r2(u + a) + 2r(1 - r)(u +
d) + (1 - r)2(u - a)
= u - (1 - 2r)a +
2r(1 - r)d
We consolidate these results into a table.
Marker ni
pi
Marker mean value
| MM |
n1 |
0.25 |
UMM
= u + (1 - 2r)a + 2r(1 - r)d |
| Mm |
n2 |
0.50 |
UMm
= u + [(1 - r)2 + r2]d |
| mm |
n3 |
0.25 |
Umm
= u - (1 - 2r)a + 2r(1 - r)d |
We have four unknowns to estimate that include: u, a ,
d, and r. We only have the three equations above. With
three equations we cannot estimate four unknowns.
The genetic effects which include u, a, and d are confounded
with the estimate of the recombination fraction (r). We
cannot determine the recombination between the marker
locus and the QTL locus. A one-way analysis of variance
will only provide evidence that the marker locus is linked
to a QTL. The R2 is the amount of variation
explained by the marker, not the QTL.
The markers are the independent variable and the mean
of each marker class is the dependent variable. The marker
means are a mixture of QTL genotypes. For example, the
mean of the MM marker class is UMM = (1 - r)2u1
+ 2r(1 - r)u2 + r2u3.
This equation indicates that there is a proportion (1
- r)2 of the QQ genotype, 2r(1 - r) of the
Qq genotype, and a proportion r2 of the qq
genotype associated with the MM marker genotype.
| R2 = |
Sum of Squares due to regression |
| Sum of Squares Total |
The R2 does not provide information on the
amount of variation explained by the QTL locus.
We can also show that the additive and dominance effects
cannot be determined when we use data from a single marker-QTL
association. The linear contrasts for the additive and
dominance effects are so S
ci = 0. Where ci is the contrast
coefficient.
Contrast
Marker genotype
MM
Mm
mm
| additive |
1 |
0 |
-1 |
| dominance |
1 |
-2 |
1 |
The additive contrast is the difference in the means of
the two homozygous genotypes.
Additive contrast = SciYi
= (1)uAA + (-1)uaa
Where uAA = u1(1 - r)2 + 2u2 r(1 -
r) + u3 r2 and
uaa = u1 r2 + 2r(1 - r)u2 + (1 - r)2 u3.
The contrast is then uAA - uaa = u1(1 - 2r) + (2r -1)u3
= (u +a)
(1 - 2r) + (2r -1)(u - a) = 2( 1 - 2r)a.
This
shows that the additive effect cannot be separated from
the recombination (r) value. Now we will derive the dominance
contrast = SciYi
=(1)[u + (1 - 2r)a + 2r(1 - r)d]-(2){u + [(1 - r)2
+ r2]d}+(1)
[u - (1 - 2r)a + 2r(1 - r)d]
=
(-2)(1 - 2r)2d
This shows that the dominance effect cannot
be separated from the recombination (r) value. For both
the additive and dominance contrasts the genetic effects
are confounded with the recombination value. When we have
a marker located on only one side of the QTL, we cannot
determine the additive or dominance effects or the recombination
fraction between the marker and the QTL. Because of recombination
between the marker and the QTL, a marker may be associated
with either the + or - alleles of the QTL locus. Each
marker mean includes the effects of homozygous and heterozygous
QTL genotypes. Each marker class is a mixture of QTL genotypes.
Because we cannot determine the recombination between
the marker and QTL, the R2 for regression of marker genotype
on the mean of each marker class is only a measure of
the amount of variation explained by the marker. The amount
of variation explained by the QTL cannot be determined
for the case of a single marker-QTL association.
"The
hypothesis test based on the contrasts for the marker
genotypes corresponds to the marker and the QTL being
independent or no genetic effects can be detected for
the putative QTL, which is r = 0.5 or a = 0 and d=0."
From Ben Hui Lui (pg. 405).
Copyright
2000©, Ted Helms |
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