Transcript Lecture 3: Resemblance Between Relatives

Lecture 7:
QTL Mapping I:
Inbred Line Crosses
Experimental Design:
Crosses
P1 x P2
B1
Backcross design
F1 x F1
F2 F2
Fk
B2
Backcross design
F2 design
Advanced intercross
Design (AIC, AICk)
Experimental Designs:
Marker Analysis
Single marker analysis
Flanking marker analysis (interval mapping)
Composite interval mapping
Interval mapping plus additional markers
Multipoint mapping
Uses all markers on a chromosome simultaneously
Conditional Probabilities of
QTL Genotypes
The basic building block for all QTL methods is
Pr(Qk | Mj) --- the probability of QTL genotype
Qk given the marker genotype is Mj.
Pr(Qk M j )
Pr(Qk j M j ) =
Pr(M j )
Consider a QTL linked to a marker (recombination
Fraction = c). Cross MMQQ x mmqq. In the F1, all
gametes are MQ and mq
In the F2, freq(MQ) = freq(mq) = (1-c)/2,
freq(mQ) = freq(Mq) = c/2
Hence, Pr(MMQQ) = Pr(MQ)Pr(MQ) = (1-c)2/4
Pr(MMQq) = 2Pr(MQ)Pr(Mq) = 2c(1-c) /4
2 /4
Pr(MMqq)
=
Pr(Mq)Pr(Mq)
=
c
Why the 2? MQ from father, Mq from mother, OR
MQ from mother, Mq from father
Since Pr(MM) = 1/4, the conditional probabilities become
Pr(QQ | MM) = Pr(MMQQ)/Pr(MM) = (1-c)2
Pr(Qq | MM) = Pr(MMQq)/Pr(MM) = 2c(1-c)
Pr(qq | MM) = Pr(MMqq)/Pr(MM) = c2
2 Marker loci
Suppose the cross is M1M1QQM2M2 x m1m1qqm2m2
Genetic map
Q
M1
c1
M2
c2
c12
In F2, Pr(M
No interference:
c12 2=) c1 + c2 - 2c1c2
1QM2) = (1-c1)(1-c
Complete
interference:
c1 +1)c2c2
Pr(M1Qm
Pr(m1QMc212
) ==(1-c
2) = (1-c1) c2
Likewise, Pr(M1M2) = 1-c12 = 1- c1 + c2
A little bookkeeping gives
(1 °- c1 )2 (1 °- c2 ) 2
Pr(QQ j M 1M 1M 2 M 2 ) =
(1 °- c12 ) 2
2c1 c2 (1 °- c1)(1 -° c2 )
Pr(Qq j M 1M 1M 2 M 2 ) =
(1 °- c12) 2
c21 c22
Pr(qqj M 1M 1M 2 M 2 ) =
(1 °- c12 )2
Expected Marker Means
The expected trait mean for marker genotype Mj
is just
XN
πM j =
πQ k Pr( Qk j M j )
k= 1
For example, if QQ = 2a, Qa = a(1+k), qq = 0, then in
the F2 of an MMQQ/mmqq cross,
(πM M °- πm m )=2 = a(1 °- 2c)
• If the trait mean is significantly different for the
genotypes at a marker locus, it is linked to a QTL
• A small MM-mm difference could be (i) a tightly-linked
QTL of small effect or (ii) loose linkage to a large QTL
Hence, the use of single markers provides for
detection of a QTL. However, single marker means does
not allow separate estimation of a and c.
Now consider using interval mapping (flanking markers)
π
M 1 M 1M 2 M 2
∂
1 ° c1 ° c2
= a- - 1 ° c1 ° c2 + 2c1 c2
' a (1 °- 2c1 c2 )
∂
1
πM 1 M 1This
-° πmis1 m
essentially a for
2a modest linkage
even
∂
πM 1 M 1 ° πm 1 m 1
πM 1 M 1 M 2 M 2 -° πm 1 m 1 m 2 m 2
°- πm 1 m 1 m 2 m 2
2
1µ c1 =
1°
2
µ
1
'
1 °2
(
(
µ
)
)
Hence, a and c can be estimated from the mean values of
flanking marker genotypes
Linear Models for QTL Detection
The use of differences in the mean trait value
for different marker genotypes to detect a QTL
and estimate its effects is a use of linear models.
One-way ANOVA.
zi k = π + bi + ei k
Value ofEffect
trait
inofkth
individual
of marker
genotype
Detection:
a QTL
marker
is linked
genotype
to the
marker
i on trait
if at
value
least
type
one
ofi the bi is significantly different from zero
Estimation (QTL effect and position): This requires
relating the bi to the QTL effects and map position
Detecting epistasis
One major advantage of linear models is their
flexibility. To test for epistasis between two QTLs,
used an ANOVA with an interaction term
z = π + ai + bk + di k + e
Effect
Effect
from marker
from
marker
genotype
genotype
at genotypes
firstat second
Interaction
between
marker
i in 1st
• At least
one
of (can
theset
abe
different
marker
marker
set
>k 1inloci)
i significantly
marker
set
and
2nd marker
set from 0
---- QTL linked to first marker set
• At least one of the bk significantly different from 0
---- QTL linked to second marker set
• At least one of the dik significantly different from 0
---- interactions between QTL in sets 1 and two
Maximum Likelihood Methods
ML methods use the entire distribution of the data, not
just the marker genotype means.
More powerful that linear models, but not as flexible
in extending solutions (new analysis required for each model)
Basic likelihood function:
`(z j M j ) =
XN
k= 1
' (z; πQ k ; æ2 ) Pr( Qk j M j )
Trait
Distribution
value
Probability
givenofmarker
trait
of QTL
value
genotype
genotype
givenisQTL
type
k given
genotype
j marker
is kgenotype
Sum over the N possible linked QTL genotypes
j --- with
genetic
map
linkage
phase
entire
here
is normal
mean
mQkand
. (QTL
effects
enter
here)
This is a mixture model
ML methods combine both detection and estimation
Of QTL effects/position.
Test for a linked QTL given from the LR test
max ` r (z)
LR = ° 2ln
max `(z)
Maximum of the likelihood under a no-linked QTL
Themodel
LR score is often
Maximum
plotted
of the
by trying
full likelihood
different locations
for the QTL (I.e., values of c) and computing a LOD score
for each
LOD(c) = °- log10
∑
∏
max ` r (z)
LR(c)
LR(c)
=
'
max `(z; c)
2 ln 10
4:61
{
}
A typical QTL map from a likelihood analysis
Estimated QTL location
Support interval
Significance
Threshold
Interval Mapping with
Marker Cofactors
Consider interval mapping using the markers i and i+1.
Now
wethese
also add
the two
markers
the
QTLssuppose
linked to
markers,
but
outsideflanking
this
interval
i+2)
interval,(i-1
canand
contribute
(falsely) to estimation of
CIM also (potentially) includes unlinked markers to
QTL position and effect
account for QTL on other chromosomes.
i-1
i
i+1
i+2
Inclusion
Interval
of
being
i-1cofactors
mapped
and
fully
Interval
However,
mapping
still
domarkers
not
+ marker
account
fori+2
QTLs
is called
inaccount
the areas
for anyInterval
linked QTLs
to the
left of i-1 and the
Composite
Mapping
(CIM)
right of i+2
CIM works by adding an additional term to the
linear model ,
X
bk x k j
k6
= i ;i + 1
Power and Repeatability:
The Beavis Effect
QTLs with low power of detection tend to have their
effects overestimated, often very dramatically
As power of detection increases, the overestimation
of detected QTLs becomes far less serious
This is often called the Beavis Effect, after Bill
Beavis who first noticed this in simulation studies