1. QTL mapping in animals

2. It works

3. QTL mapping in animals It works It’s cheap

4. QTL mapping in animals It works It’s cheap It’s relevant to human studies

5. Genomic resource Nature December 5 2002

6. No more crosses?

7. In silico mapping

8. Method

10. Recombinant Inbreds F 0 Parental Generation F 1 Generation F 2 Generation Interbreeding for approximately 20 generations to produce recombinant inbreds

11. RI strain phenotypes

12. RI strain genotypes

13. QTL for airway responsiveness

15. 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.

17. Experimentally verified QTL for airway responsiveness 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).

18. Inbred Strain Cross

19. Quantitative Trait Locus Detection

20. Marker QTL M m Q q r

21. M m Q q r MM QQ Qq qq Mm QQ Qq qq mm QQ Qq qq

22. Marker QTL MM QQ Mm QQ P (QQ | MM) = (1-r) 2 P (Qq | MM) = 2r(1-r) P (qq | MM) = r 2 (1-r) 2 + 2r(1-r) + r 2

23. QTL Genotypic values Alleles at the QTL: q and Q Additive value: a Degree of dominance: d  QQ =  + 2a  Qq =  + a(1+d)  qq = 

24. Mean values for marker genotypes

26. Two things follow Contrasts of single marker means can be used to detect QTL

27. QTLeffects.xls Example

30. REAL_DATA/Real data.xls

31. Two things follow Contrasts of single marker means can be used to detect QTL Estimates of position and effect are confounded

32. Additive and dominance estimates

33. Flanking markers M1M1 m1m1 M2M2 m2m2

34. M1M1 m1m1 M2M2 m2m2 M1M1 M2M2 M1M1 M2m2 M1M1 m2m2 M1m1 M2M2 M1m1 M2m2 M1m1 m2m2 m1m1 M2M2 m1m1 M2m2 m1m1 m2m2

35. Interval mapping M1M1 m1m1 M2M2 m2m2 Q q r1r1 r2r2 r 12

36. Interval mapping M1M1 m1m1 M2M2 m2m2 Q q r1r1 r2r2 r 12 r 2 =( r 12 – r 1 )/(1-2r 1 ) No interference r 2 = r 12 - r 1 Complete interference

37. Interval mapping M 1 M 1 M 2 M 2 M1M1 m1m1 M2M2 m2m2 Q q r1r1 r2r2 r 12 p(M 1 QM 2 | M 1 QM 2 ) = ((1-r 1 ) (1-r 2 )/2) 2

38. Interval mapping M1M1 m1m1 M2M2 m2m2 Q q r1r1 r2r2 r 12 p(QQ|M 1 M 1 M 2 M 2 ) = ((1-r 1 ) 2 (1-r 2 ) 2 )/(1-r 12 ) 2 p(Qq|M 1 M 1 M 2 M 2 ) = (2r 1 r 2 (1-r 1 ) (1-r 2 ) )/(1-r 12 ) 2 p(qq|M 1 M 1 M 2 M 2 ) = (r 1 2 r 2 2 )/(1-r 12 ) 2

39. Significance thresholds

40. Permutation tests to establish thresholds Empirical threshold values for quantitative trait mapping GA 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.

41. Permutation tests Trait values are randomly reassigned to genotypes 10,000 re-samplings for 1% value

42. Permutation tests Robust to departures from normality Robust to missing or erroneous data Easy to implement

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

44. Maximum likelihood methods

47. Interval mapping M1M1 M1M1 M2M2 M2M2 Q q r1r1 r2r2 r 12

48. Interval mapping M1M1 M1M1 M2M2 M2M2 Q q r1r1 r2r2 r 12

49. Maximum likelihood Test statistic

50. Example SIMULATED_DATA WinQTL

51. Linear models

52.  QQ =  + a  Qq =  + d  qq =  - a

53. Linear models  QQ =  + a  Qq =  + d  qq =  - a z j =  + a. x (M j ) + d. y (M j ) + e j

54. Linear models  QQ =  + a  Qq =  + d  qq =  - a z j =  + a. x (M j ) + d. y (M j ) + e j x (Mj) = p(QQ | Mj) – p (qq| Mj) y (Mj) = p(Qq | Mj)

55. Linear models x (Mj) = p(QQ | Mj) – p (qq| Mj) x(M 1 M 1 M 2 M 2 ) (1-r 1 ) 2 (1-r 2 ) 2 -(r 1 2 r 2 2 ) (1-r 12 ) 2 =

56. Linear models x (Mj) = p(QQ | Mj) – p (qq| Mj) x(M 1 M 1 M 2 M 2 ) (1-r 1 ) 2 (1-r 2 ) 2 -(r 1 2 r 2 2 ) (1-r 12 ) 2 = 2r 1 r 2 (1-r 1 ) (1-r 2 ) y(M 1 M 1 M 2 M 2 ) (1-r 12 ) 2 = y (Mj) = p(Qq | Mj)

57. Significance test LR = n ln (SS T /SS E ) = -n ln (1-r 2 ) Degrees of freedom are the number of estimated QTL parameters, plus one for the map position

58. Matrix statement of Haley Knott regression  r1 = (X T r1 X r1 ) -1 X T r1 z ith row of matrix X r1 : (1,x(M i,r 1 ), y(M i,r 1 ))

59. Example Regression example.xls

60. Problems of QTL detection Linked QTLs corrupt the position estimates Unlinked QTLs decreases the power of QTL detection

61. Extensions to linear regression Composite interval mapping Multiple interval mapping

62. Composite interval mapping ZB Zeng Precision mapping of quantitative trait loci Genetics, Vol 136, 1457-1468, 1994 http://statgen.ncsu.edu/qtlcart/cartographer.html

63. Composite interval mapping

64. M1M1 M2M2 M1M1 M2M2 QQQ

65. M -1 M1M1 M2M2 M3M3 M1M1 M2M2 M3M3 QQQ

66. Composite interval mapping M -1 M1M1 M2M2 M3M3 M1M1 M2M2 M3M3 QQQ z j =  + a. x (M j ) + d. y (M j ) +  k=i, i+1 b k. x kj + e j

67. Example SIMULATED_DATA WinQTL

68. Multiple Interval Mapping

71. Example?