1. (1) Schedule Mar 15Linkage disequilibrium (LD) mapping Mar 17LD mapping Mar 22Guest speaker, Dr Yang Mar 24Overview Attend ENAR Biometrical meeting in Austin from Mar 20 to 23 (2) Projects - Work on a problem learnt in the class -Select a problem from your own projects

2. What I have learnt from my trip to Seattle -Fred Hutchinson Cancer Research Center -University of Washington Statistical Genetics of Complex Traits Single Nucleotide Polymorphisms (SNPs) Haplotype blocks HIV/AIDS dynamics Cancer progression

3. Statistical Genetics of Complex Traits Rongling Wu, Chang-Xing Ma and George Casella  Springer-Verlag New York Linkage, Disequilibrium and QTL

4. Linkage Disequilibrium Linkage analysis – controlled crosses (backcross or F2) and structured pedigrees (grandparent- parent-children generation) Linkage disequilibrium analysis – Natural population Linkage mapping is used in plant and animal genetics, as well as human genetics of diseases like cancers. LD mapping is used for human genetics of diseases like HIV/AIDS and SARS.

5. Mixture model-based likelihood without marker information L(y|  ) =  i=1 n [½f 1 (y i ) + ½f 0 (y i )] Height QTL genotype Sample (cm, y) Qqqq 1184½ ½ 2185 ½ ½ 3180 ½ ½ 4182 ½ ½ 5167 ½ ½ 6169 ½ ½ 7165 ½ ½ 8166 ½ ½ Linkage mapping - backcross

6. Mixture model-based likelihood with marker information L(y,M|  ) =  i=1 n [  1|i f 1 (y i ) +  0|i f 0 (y i )] Sam- Height Marker genotype QTL ple(cm, y) M1M2Qqqq 1184Mm (1)Nn (1)1 0 2185Mm (1)Nn (1) 1 0 3180Mm (1)Nn (1) 1 0 4182Mm (1)nn (0)1-   5167mm (0)nn (1)  1-  6169mm (0)nn (0)0 1 7165mm (0)nn (0)0 1 8166mm (0)Nn (0) 0 1 Prior prob. Linkage mapping - backcross

7. Conditional probabilities of the QTL genotypes (missing) based on marker genotypes (observed) L(y,M|  ) =  i=1 n [  1|i f 1 (y i ) +  0|i f 0 (y i )] =  i=1 n1 [1 f 1 (y i ) + 0 f 0 (y i )]Conditional on 11 (n 1 )   i=1 n2 [(1-  ) f 1 (y i ) +  f 0 (y i )]Conditional on 10 (n 2 )   i=1 n3 [  f 1 (y i ) + (1-  ) f 0 (y i )]Conditional on 01 (n 3 )   i=1 n4 [0 f 1 (y i ) + 1 f 0 (y i )]Conditional on 00 (n 4 ) Linkage mapping - backcross

8. Normal distributions of phenotypic values for each QTL genotype group f 1 (y i ) = 1/(2  2 ) 1/2 exp[-(y i -  1 ) 2 /(2  2 )],  1 =  + a* f 0 (y i ) = 1/(2  2 ) 1/2 exp[-(y i -  0 ) 2 /(2  2 )],  0 =  Linkage mapping - backcross

9. Differentiating L with respect to each unknown parameter, setting derivatives equal zero and solving the log-likelihood equations L(y,M|  ) =  i=1 n [  1|i f 1 (y i ) +  0|i f 0 (y i )] log L(y,M|  ) =  i=1 n log[  1|i f 1 (y i ) +  0|i f 0 (y i )] Define  1|i =  1|i f 1 (y i )/[  1|i f 1 (y i ) +  0|i f 0 (y i )](1)  0|i =  0|i f 1 (y i )/[  1|i f 1 (y i ) +  0|i f 0 (y i )](2)  1 =  i=1 n (  1|i y i )/  i=1 n  1|i (3)  0 =  i=1 n (  0|i y i )/  i=1 n  0|i (4)  2 = 1/n  i=1 n [  1|i (y i -  1 ) 2 +  0 |i (y i -  0 ) 2 ](5)  = (  i=1 n2  1|i +  i=1 n3  0 |i )/(n 2 +n 3 )(6) Linkage mapping - backcross

10. Mixture model-based likelihood without marker information Suppose there is natural population with a segregating QTL of two alternative alleles, Q and q, Prob(Q)=q, Prob(q)=1-q → Prob(QQ)=q 2, Prob(Qq)=2q(1-q), Prob(qq)=(1-q) 2 L(y|  ) =  i=1 n [[q 2 f 2 (y i ) + 2q(1-q)f 1 (y i ) + (1-q) 2 f 0 (y i )] Height QTL genotype Sample (cm, y) QQQqqq 1184q 2 2q(1-q)(1-q) 2 2185 q 2 2q(1-q)(1-q) 2 3180 q 2 2q(1-q)(1-q) 2 4182 q 2 2q(1-q)(1-q) 2 5167 q 2 2q(1-q)(1-q) 2 6169 q 2 2q(1-q)(1-q) 2 7165 q 2 2q(1-q)(1-q) 2 8166 q 2 2q(1-q)(1-q) 2 Linkage disequilibrium mapping – natural population

11. Association between marker and QTL -Marker, Prob(M)=p, Prob(m)=1-p -QTL, Prob(Q)=q, Prob(q)=1-q Four haplotypes: Prob(MQ)=p 11 =pq+D p=(p 11 +p 10 )/2 Prob(Mq)=p 10 =p(1-q)-Dq=(p 11 +p 01 )/2 Prob(mQ)=p 01 =(1-p)q-DD=p 11 p 00 -p 10 p 01 Prob(mq)=p 00 =(1-p)(1-q)+D Linkage disequilibrium mapping – natural population

12. QQQqqqObs MMp 11 2 2p 11 p 10 p 10 2 n 2 Mm2p 11 p 01 2(p 11 p 00 +p 10 p 01 )2p 10 p 00 n 1 mmp 01 2 2p 01 p 00 p 00 2 n 0 MMp 11 2 2p 11 p 10 p 10 2 n 2 p 2 p 2 p 2 Mm2p 11 p 01 2(p 11 p 00 +p 10 p 01 )2p 10 p 00 n 1 2p(1-p)2p(1-p)2p(1-p) mmp 01 2 2p 01 p 00 p 00 2 n 0 (1-p) 2 (1-p) 2 (1-p) 2 Joint and conditional (  j|i ) genotype prob. between marker and QTL

13. Mixture model-based likelihood with marker information L(y,M|  )=  i=1 n [  2|i f 2 (y i ) +  1|i f 1 (y i ) +  0|i f 0 (y i )] Sam- Height Marker genotype QTL genotype ple(cm, y) M QQQqqq 1184MM (2)  2|i  1|i  0|i 2185MM (2)  2|i  1|i  0|i 3180Mm (1)  2|i  1|i  0|i 4182Mm (1)  2|i  1|i  0|i 5167Mm (1)  2|i  1|i  0|i 6169Mm (1)  2|i  1|i  0|i 7165mm (0)  2|i  1|i  0|i 8166mm (0)  2|i  1|i  0|i Prior prob. Linkage disequilibrium mapping – natural population

14. Conditional probabilities of the QTL genotypes (missing) based on marker genotypes (observed) L(y,M|  ) =  i=1 n [  2|i f 2 (y i ) +  1|i f 1 (y i ) +  0|i f 0 (y i )] =  i=1 n2 [  2|2i f 2 (y i ) +  1|2i f 1 (y i ) +  0|2i f 0 (y i )] Conditional on 2 (n 2 )   i=1 n1 [  2|1i f 2 (y i ) +  1|1i f 1 (y i ) +  0|1i f 0 (y i )] Conditional on 1 (n 1 )   i=1 n0 [  2|0i f 2 (y i ) +  1|0i f 1 (y i ) +  0|0i f 0 (y i )] Conditional on 0 (n 0 ) Linkage disequilibrium mapping – natural population

15. Normal distributions of phenotypic values for each QTL genotype group f 2 (y i ) = 1/(2  2 ) 1/2 exp[-(y i -  2 ) 2 /(2  2 )],  2 =  + a f 1 (y i ) = 1/(2  2 ) 1/2 exp[-(y i -  1 ) 2 /(2  2 )],  1 =  + d f 0 (y i ) = 1/(2  2 ) 1/2 exp[-(y i -  0 ) 2 /(2  2 )],  0 =  - a Linkage disequilibrium mapping – natural population

16. Differentiating L with respect to each unknown parameter, setting derivatives equal zero and solving the log-likelihood equations L(y,M|  ) =  i=1 n [  2|i f 2 (y i ) +  1|i f 1 (y i ) +  0|i f 0 (y i )] log L(y,M|  ) =  i=1 n log[  2|i f 2 (y i ) +  1|i f 1 (y i ) +  0|i f 0 (y i )] Define  2|i =  2|i f 1 (y i )/[  2|i f 2 (y i ) +  1|i f 1 (y i ) +  0|i f 0 (y i )](1)  1|i =  1|i f 1 (y i )/[  2|i f 2 (y i ) +  1|i f 1 (y i ) +  0|i f 0 (y i )](2)  0|i =  0|i f 1 (y i )/[  2|i f 2 (y i ) +  1|i f 1 (y i ) +  0|i f 0 (y i )](3)  1 =  i=1 n (  1|i y i )/  i=1 n  1|i (4)  0 =  i=1 n (  0|i y i )/  i=1 n  0|i (5)  2 = 1/n  i=1 n [  1|i (y i -  1 ) 2 +  0|i (y i -  0 ) 2 ](6) Linkage disequilibrium mapping – natural population

17. Complete dataPrior prob QQQqqqObs MMp 11 2 2p 11 p 10 p 10 2 n 2 Mm2p 11 p 01 2(p 11 p 00 +p 10 p 01 )2p 10 p 00 n 1 mmp 01 2 2p 01 p 00 p 00 2 n 0 QQQqqqObs MMn 22 n 21 n 20 n 2 Mmn 12 n 11 n 10 n 1 mmn 02 n 01 n 00 n 0 p 11 =[2n 22 + (n 21 +n 12 ) +  n 22 ]/2n, p 10 =[2n 20 + (n 21 +n 10 ) + (1-  )n 22 ]/2n, p 01 =[2n 02 + (n 12 +n 01 ) + (1-  )n 22 ]/2n, p 11 =[2n 00 + (n 10 +n 01 ) +  n 22 ]/2n,  =p 11 p 00 /(p 11 p 00 +p 10 p 01 )

18. Incomplete (observed) data Posterior prob QQQqqqObs MM  2|2i  1|2i  0|2i n 2 Mm  2|1i  1|1i  0|1i n 1 mm  2|0i  1|0i  0|0i n 0 p 11 =1/2n{  i=1 n2 [2  2|2i +  1|2i ]+  i=1 n1 [  2|1i +  1|1i ],(7) p 10 =1/2n{  i=1 n2 [2  0|2i +  1|2i ]+  i=1 n1 [  0|1i +(1-  )  1|1i ],(8) p 01 =1/2n{  i=1 n0 [2  2|0i +  1|0i ]+  i=1 n1 [  2|1i +(1-  )  1|1i ],(9) p 00 =1/2n{  i=1 n2 [2  0|0i +  1|0i ]+  i=1 n1 [  0|1i +  1|1i ] (10)

19. EM algorithm (1) Give initiate values  (0) =(  2,  1,  0,  2,p 11,p 10,p 01,p 00 ) (0) (2) Calculate  2|i (1),  1|i (1) and  0|i (1) using Eqs. 1-3, (3) Calculate  (1) using  2|i (1),  1|i (1) and  0|i (1) based on Eqs. 4-10, (4) Repeat (2) and (3) until convergence.

20. Example: Human Obesity