qtl mapping in animals. it works qtl mapping in animals it works it’s cheap

QTL mapping in animals

QTL mapping in animals

• It works

QTL mapping in animals

• It works

• It’s cheap

QTL mapping in animals

• It works

• It’s cheap

• It’s relevant to human studies

Genomic resource

Nature December 5 2002

No more crosses?

In silico mapping

Method

We wanted to determine whether chromosomal regions regulating quantitative traits (QTL intervals) could be computationally predicted with the use of the mSNP database and available phenotypic information on inbred strains.

Method

Using the allelic distributions across inbred strains contained in the mSNP database, the computational method calculates genotypic distances between loci for a pair of mouse strains. These genotypic distances are then compared with phenotypic differences between the two mouse strains. The process is repeated for all mouse strain pairs for which phenotypic information is available. Lastly, a correlation value is derived using linear regression on the phenotypic and genotypic distances for each genomic locus.

Recombinant InbredsF0 Parental Generation

F1 Generation

F2 Generation

Interbreeding for approximately 20 generations to produce recombinant inbreds

RI strain phenotypes

RI strain genotypesChr1 D1Byu4 B B D B B D D BChr1 D1Rik100 B B D B B D D BChr1 D1Rik101 B B D . D . B BChr1 D1Rik102 B B D D D B D DChr1 D1Rik103 B B D D B B D BChr1 D1Rik104 B B D B B D D BChr1 D1Rik86 B B D B B . D BChr1 D1Rik87 B B D B B D D BChr1 D1Rik88 B B D . B D D BChr1 D1Rik89 B B D B B D D BChr1 D1Rik90 B B D D B B D BChr1 D1Rik91 B B D D B B D BChr1 D1Rik92 B B D D B B D BChr1 D1Rik94 B B D D B B D BChr1 D1Rik95 B B D D B B D BChr1 D1Rik96 B B D D B B D BChr1 D1Rik97 B B D B B D D BChr1 D1Rik98 B B D B B D D BChr1 D1Rik99 B B D B B D D BChr1 D1Hgu1 B B D D D B B DChr1 Ugt1a1-rs1B B D D D B B ?Chr1 D1Mit294 B B D D D B D DChr1 D1Mit1 . . . D D B D DChr1 D1Mit67 B B B D D B D DChr1 D1Rp2 B B B D D B D DChr1 D1Mit231 D B B D D B D DChr1 Odc-rs10 B B* D D D B D DChr1 D1J2 D B B D D B D DChr1 D1Mit211 D B B D D B D DChr1 D1Nds4 D B B D D B D D

QTL for airway responsiveness

Power

n -2 = (t + t)2/(s2QTL/s2RES)

t and t are values on the t distribution corresponding to the desired value

s2QTL is the phenotypic variance explained by a QTL

s2RES the unexplained variance.

Number Power QTL Effect10 90 6110 50 674 90 884 50 83

Experimentally verified QTL for airway responsiveness

Chromosome LOD %Varexp

9 2.5 5.2

10 3.8 8.3

11 3.65 7.5

17 2.1 4.4

Zhang, Y. et al. A genome-wide screen for asthma-associated quantitative trait loci in a mouse model of allergic asthma. Hum. Mol. Genet. 8, 601-605 (1999).

Inbred Strain Cross

Quantitative Trait Locus Detection

Marker QTL

M

m

Q

q

r

Marker QTL

M

m

Q

q

r

MM QQ Qq qq

Mm QQ Qq qq

mm QQ Qq qq

Marker QTL

MM QQ

Mm QQ

Mm QQ

P (QQ | MM) = (1-r)2

P (Qq | MM) = 2r(1-r)

P (qq | MM) = r2

(1-r)2 + 2r(1-r) + r2

QTL Genotypic values

Alleles at the QTL: q and QAdditive value: aDegree of dominance: d

QQ = + 2aQq = + a(1+d)qq =

Mean values for marker genotypes

Marker alleles: M and m Recombination frequency between QTL and marker: r MM = + 2a(1-r)2 + 2r (1- r)(1+d)a

Mean values for marker genotypes

Marker alleles: M and m Recombination frequency between QTL and marker: r MM = + 2a(1-r)2 + 2r (1- r)(1+d)a Mm = + 2ar(1-r) +(1-2r(1- r))(1+d)a mm = + 2ar2 + 2r (1- r)(1 + d)a

Two things follow

• Contrasts of single marker means can be used to detect QTL

r = 0.1 (1-r)2

+ 2r (1- r)MM = + 2a * 0.81 0.18 (1+d) * a

QTLeffects.xls

Example

r = 0.1 (1-r)2 + 2r (1- r)MM = + 2a * 0.81 0.18 (1+d) * a

r(1-r) + (1-2r(1- r))Mm = + 2a * 0.09 0.82 (1+d) * a

r2 + 2r(1-r)mm = + 2a * 0.01 0.18 (1+d) * a

r = 0.5 (1-r)2 + 2r (1- r)MM = + 2a * 0.25 0.5 (1+d) * a

r(1-r) + (1-2r(1- r))Mm = + 2a * 0.25 0.5 (1+d) * a

r2 + 2r(1-r)mm = + 2a * 0.25 0.5 (1+d) * a

Example

REAL_DATA/Real data.xls

Two things follow

• Contrasts of single marker means can be used to detect QTL

• Estimates of position and effect are confounded

Additive and dominance estimates

Additive effect (MM -mm)/2 = (1-2r) * a Dominance effect Mm – (MM + mm)/2) / ((MM - mm)/2) = d * (1-2r)

Flanking markers

M1

m1

M2

m2

Flanking markersM1

m1

M2

m2

M1M1 M2M2M1M1 M2m2M1M1 m2m2

M1m1 M2M2M1m1 M2m2M1m1 m2m2

m1m1 M2M2m1m1 M2m2m1m1 m2m2

Interval mapping

M1

m1

M2

m2

Q

q

r1 r2

r12

Interval mapping

M1

m1

M2

m2

Q

q

r1 r2

r12

r2 =( r12 – r1)/(1-2r1) No interference

r2 = r12- r1 Complete interference

Interval mapping

M1M1 M2M2

M1

m1

M2

m2

Q

q

r1 r2

r12

p(M1QM2 | M1QM2) = ((1-r1) (1-r2)/2)2

Interval mapping

M1

m1

M2

m2

Q

q

r1 r2

r12

p(QQ|M1M1M2M2) = ((1-r1) 2(1-r2)2)/(1-r12)2

p(Qq|M1M1M2M2) = (2r1r2(1-r1) (1-r2) )/(1-r12)2

p(qq|M1M1M2M2) = (r1 2r22)/(1-r12)2

Significance thresholds

Permutation tests to establish thresholds

Empirical threshold values for quantitative trait mappingGA Churchill and RW Doerge

Genetics, 138, 963-971 1994

An empirical method is described, based on the concept of a permutation test, for estimating threshold values that are tailored to the experimental data at hand.

Permutation tests

Trait values are randomly reassigned to genotypes

10,000 re-samplings for 1% value

Permutation tests

• Robust to departures from normality

• Robust to missing or erroneous data

• Easy to implement

Significance Thresholds

Suggestive Significant Mapping method P LOD P LOD Backcross 3.40E-03 1.9 1.00E-04 3.3 Intercross (2 df) 1.60E-03 2.8 5.20E-05 4.3

Lander, E. Kruglyak, L. Genetic dissection of complex traits: guidelines for interpreting and reporting linkage results Nature Genetics. 11, 241-7, 1995

Maximum likelihood methods

Marker genotype M Phenotypic value z Variance 2 Mean Qk

Maximum likelihood methods

L (z | MM) = (1-r)2 zQQ,2) + 2r ( 1-r) zQq,2) + r2 zqq,2) L (z | Mm) = r(1-r) zQQ,2) + (( 1-r)2+ r2)zQq,2) + r(1-r) zqq,2) L (z | mm) = r2 zQQ,2) + 2r ( 1-r) zQq,2) + (1-r)2 zqq,2)

Maximum likelihood methods

L (z | MM) = (1-r)2 zQQ,2) + 2r ( 1-r) zQq,2) + r2 zqq,2) L (z | Mm) = r(1-r) zQQ,2) + (( 1-r)2+ r2)zQq,2) + r(1-r) zqq,2) L (z | mm) = r2 zQQ,2) + 2r ( 1-r) zQq,2) + (1-r)2 zqq,2)

Interval mapping

M1

M1

M2

M2

Q

q

r1 r2

r12

Interval mapping

M1

M1

M2

M2

Q

q

r1 r2

r12

L (z | M1M1M2M2) = ((1-r1)2 (1-r2)

2 )/(1-r12)2zQQ,2) +

2r1r2 ( 1-r1) (1-r2)/(1-r12)

2 zQq,2) + (r1

2r22)/(1-r12)

2 zqq,2)

Maximum likelihoodTest statistic

LR = -2 ln (max Lr(z)/max L(z)) Lr(z) is maximum of the likelihood function under the null hypothesis of no segregating QTL (i.e. that the phenotypic distribution is a single normal)

Example

---------------------------------------| D2MIT21-D2MIT22 37.0 cM 0.0 0.294 -0.071 4.6% 5.069 | ************* 2.0 0.317 -0.074 5.3% 5.455 | ************** 4.0 0.341 -0.077 6.2% 5.861 | **************** 6.0 0.365 -0.077 7.0% 6.279 | ****************** 8.0 0.389 -0.076 8.0% 6.701 | ******************* 10.0 0.410 -0.073 8.9% 7.114 | ********************* 12.0 0.431 -0.068 9.7% 7.505 | *********************** 14.0 0.447 -0.061 10.5% 7.861 | ************************ 16.0 0.460 -0.054 11.0% 8.169 | ************************* 18.0 0.468 -0.046 11.4% 8.417 | ************************** 20.0 0.473 -0.039 11.7% 8.595 | *************************** 22.0 0.473 -0.033 11.6% 8.699 | *************************** 24.0 0.469 -0.026 11.4% 8.728 | *************************** 26.0 0.460 -0.020 11.0% 8.684 | *************************** 28.0 0.447 -0.015 10.4% 8.574 | *************************** 30.0 0.432 -0.009 9.6% 8.409 | ************************** 32.0 0.414 -0.004 8.8% 8.199 | ************************* 34.0 0.394 0.000 8.0% 7.956 | ************************ 36.0 0.373 0.004 7.1% 7.694 | *********************** ---------------------------------------| D2MIT22-D2MIT23 32.9 cM 0.0 0.363 0.006 6.7% 7.563 | *********************** 2.0 0.381 0.010 7.4% 7.705 | *********************** 4.0 0.399 0.013 8.2% 7.811 | ************************ 6.0 0.414 0.016 8.8% 7.867 | ************************ 8.0 0.425 0.019 9.3% 7.862 | ************************ 10.0 0.433 0.021 9.7% 7.786 | ************************ 12.0 0.438 0.021 9.9% 7.631 | *********************** 14.0 0.437 0.022 9.8% 7.394 | ********************** 16.0 0.431 0.023 9.6% 7.077 | ********************* 18.0 0.421 0.022 9.1% 6.684 | ******************* 20.0 0.405 0.019 8.4% 6.229 | ***************** 22.0 0.385 0.015 7.6% 5.726 | *************** 24.0 0.360 0.008 6.6% 5.196 | ************* 26.0 0.333 -0.002 5.6% 4.662 | *********** 28.0 0.303 -0.013 4.6% 4.146 | ********* 30.0 0.274 -0.026 3.8% 3.669 | ******* 32.0 0.246 -0.037 3.1% 3.244 | ***** ---------------------------------------| D2MIT23-D2MIT24 43.5 cM 0.0 0.235 -0.041 2.8% 3.080 | ***** 2.0 0.241 -0.052 3.0% 3.028 | ***** 4.0 0.247 -0.066 3.2% 2.966 | **** 6.0 0.251 -0.081 3.4% 2.894 | **** 8.0 0.255 -0.100 3.6% 2.812 | **** 10.0 0.256 -0.122 3.7% 2.721 | *** 12.0 0.255 -0.146 3.9% 2.620 | *** 14.0 0.251 -0.170 3.9% 2.511 | *** 16.0 0.245 -0.197 4.0% 2.396 | ** 18.0 0.236 -0.224 4.1% 2.275 | ** 20.0 0.225 -0.249 4.1% 2.149 | * 22.0 0.212 -0.267 4.1% 2.016 | * 24.0 0.197 -0.279 3.9% 1.876 | 26.0 0.181 -0.284 3.7% 1.728 | 28.0 0.163 -0.280 3.3% 1.574 | 30.0 0.145 -0.271 2.9% 1.416 | 32.0 0.127 -0.255 2.4% 1.261 | 34.0 0.109 -0.235 2.0% 1.113 | 36.0 0.091 -0.213 1.6% 0.978 | 38.0 0.074 -0.192 1.2% 0.860 | 40.0 0.059 -0.172 0.9% 0.759 | 42.0 0.046 -0.153 0.7% 0.676 | ---------------------------------------|

SIMULATED_DATA

WinQTL

Linear modelszik = + bi + eik

kth individual of marker genotype i

Linear models

QQ = + a Qq = + d qq = - a

Linear models

QQ = + a Qq = + d qq = - a

zj = + a . x (Mj) + d . y (Mj) + ej

Linear models

QQ = + a Qq = + d qq = - a

zj = + a . x (Mj) + d . y (Mj) + ej

x (Mj) = p(QQ | Mj) – p (qq| Mj)

y (Mj) = p(Qq | Mj)

Linear modelsx (Mj) = p(QQ | Mj) – p (qq| Mj)

x(M1M1M2M2)(1-r1) 2(1-r2)2 -(r1

2r22)

(1-r12)2=

Linear modelsx (Mj) = p(QQ | Mj) – p (qq| Mj)

x(M1M1M2M2)(1-r1) 2(1-r2)2 -(r1

2r22)

(1-r12)2=

2r1r2(1-r1) (1-r2)y(M1M1M2M2) (1-r12)2=

y (Mj) = p(Qq | Mj)

Significance test

LR = n ln (SST/SSE) = -n ln (1-r2)

Degrees of freedom are the number of estimated QTL parameters, plus one for the map position

Matrix statement of Haley Knott regression

r1 = (XTr1 Xr1) -1 XT

r1 z

ith row of matrix Xr1: (1,x(Mi,r1), y(Mi,r1))

Example

Regression example.xls

0 0 0.15 0.15 0.3 0.3Marker Genotypes Phenotypes x y x y x yM1M1M2M2 5.6 1 0 0.91 0.9 1 0M1M1M2M2 5.4 1 0 0.91 0.9 1 0M1M1M2m2 5.3 1 0 0.56 0.4 0 1M1m1M2m2 3.9 0 1 0 0.85 0 1M1m1M2m2 3.3 0 1 0 0.85 0 1M1m1M2M2 3.6 0 1 0.35 0.6 1 0M1m1M2M2 3.7 0 1 0.35 0.6 1 0m1m1M2m2 3.9 -1 0 -0.56 0.4 0 1m1m1M2m2 3.5 -1 0 -0.56 0.4 0 1m1m1m2m2 1.1 -1 0 -0.91 0.9 -1 0m1m1m2m2 0.8 -1 0 -0.91 0.9 -1 0

Problems of QTL detection

• Linked QTLs corrupt the position estimates

• Unlinked QTLs decreases the power of QTL detection

Extensions to linear regression

• Composite interval mapping

• Multiple interval mapping

Composite interval mapping

ZB Zeng Precision mapping of quantitative trait lociGenetics, Vol 136, 1457-1468, 1994

http://statgen.ncsu.edu/qtlcart/cartographer.html

Composite interval mapping

Composite interval mapping

M1 M2

M1 M2QQ Q

Composite interval mapping

M-1 M1 M2 M3

M-1 M1 M2 M3QQ Q

Composite interval mapping

M-1 M1 M2 M3

M-1 M1 M2 M3QQ Q

zj = + a . x (Mj) + d . y (Mj)

+ k=i, i+1

bk . xkj + ej

Example

SIMULATED_DATA

WinQTL

Multiple Interval Mapping

Multiple Interval Mapping

Multiple Interval Mapping

Example?