# MAGIC design. and other topics. Karl Broman. Biostatistics & Medical Informatics University of Wisconsin Madison

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1 MAGIC design and other topics Karl Broman Biostatistics & Medical Informatics University of Wisconsin Madison biostat.wisc.edu/ kbroman github.com/kbroman

2 CC founders compgen.unc.edu 2

3 MAGIC lines Valdar et al., Genetics 172:1783,

4 MAGIC lines combine mix fix Valdar et al., Genetics 172:1783,

5 MAGIC lines combine mix fix How many? Valdar et al., Genetics 172:1783,

6 MAGIC lines combine mix fix How many? Which? Valdar et al., Genetics 172:1783,

7 MAGIC lines combine mix fix How many? How long? Which? Valdar et al., Genetics 172:1783,

8 MAGIC lines combine mix fix How many? Which? How long? How? Valdar et al., Genetics 172:1783,

9 But first... G 0 A B C D E F G H G 1 A B C D E F G H G 2 ABCD EFGH G 2 F 1 G 2 F 2 G 2 F 4

10 Simulation results Pr ( recombination in RIL ) 7/ recombination fraction 5

12 Result for selfing Haldane & Waddington, Genetics 16:357,

13 Result for sib-mating Haldane & Waddington, Genetics 16:357,

14 Simulation results Pr ( recombination in RIL ) 7/ recombination fraction 9

15 Simulation results Pr ( recombination in RIL ) 7/ R = a r / ( 1 + b r )? recombination fraction 9

16 Non-linear regression out <- nls( R ~ a*r/(1 + b*r), data = data.frame(r=r, R=R), start = list(a=4, b=6)) summary(out) 10

17 Non-linear regression out <- nls( R ~ a*r/(1 + b*r), data = data.frame(r=r, R=R), start = list(a=4, b=6)) summary(out) Estimate Std. Error a b

18 Non-linear regression out <- nls( R ~ a*r/(1 + b*r), data = data.frame(r=r, R=R), start = list(a=4, b=6)) summary(out) More data Estimate Std. Error Estimate Std. Error a a b b

19 Simulation results Pr ( recombination in RIL ) 7/ R = 7 r / ( r ) recombination fraction 11

20 3-point coincidence r ij = recombination fraction for interval (i, j) Assume r 12 = r 23 = r. Coincidence = c = Pr(double recombinant)/r 2 = Pr(rec n in 23 rec n in 12)/Pr(rec n in 23) No interference = 1 Positive interference < 1 Negative interference > 1 Generally c is a function of r 12

21 Coincidence function Coincidence Meiosis 2 way RILs by selfing 2 way RILs by sib mating 8 way RILs by sib mating No interference Mouse interference recombination fraction Broman, Genetics 169:1133,

22 non-markov property { } Pr(M3 = A M 2 = E, M 1 = x) log 2 Pr(M 3 = A M 2 = E) 2 log 2 probability ratio recombination fraction x = A x = B x = C x = E/F x = G Broman, Genetics 169:1133,

23 Coincidence formula C = (1 + 6r)[ r 848r2 + 5c(7 28r 368r r 3 ) 2c 2 (49 324r + 452r 2 )r 2 16c 3 (1 2r)r 4 ] 49(1 + 12r 12cr 2 )[5 + 10r 4(2 + c)r 2 + 8cr 3 ] Teuscher & Broman, Genetics 175:1267,

24 The CC again G 0 A B C D E F G H G 1 A B C D E F G H G 2 ABCD EFGH G 2 F 1 G 2 F 2 G 2 F 16

25 Crazy table Table 4 Two-locus haplotype probabilities at generation F k in the formation of four-way RIL by sibling mating Chr. Individual Prototype No. states Probability of each A Random AA 4 AB 4 1 6r 2 4ð116rÞ 2 27r23rs 122r1s k 6r 2 27r13rs 122r2s 1 4ð116rÞs 4 4ð116rÞs 4 r 10r 2 2ð116rÞ 1 2r2rs 122r1s 4ð116rÞs 4 k 2 10r 2 2r1rs 4ð116rÞs 122r2s 4 k k Pre-CC Genotype Probabilities 409 AC 8 X Female AA 2 AB 2 AC 4 CC 1 X Male AA 2 AB 2 AC 4 CC 1 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s 5 4r 2 212r15 and t 5 r 2 210r15; the autosomal haplotype probabilities are valid for r, 1 2. r 2r 2 2ð116rÞ 2 13r1rs 122r1s 4ð116rÞs 4 k 1 2r 2 13r2rs 4ð116rÞs 122r2s 4 1 3ð114rÞ k 4r 2 6ð11rÞ 2 3 2ð4r 2 13rÞt13r 2 k 25r 12r1t 4r 3 1ð4r 2 13rÞt13r 2 25r 12r2t 4ð4r r11Þt 4 4ð4r 2 15r11Þt 4 2r 3ð114rÞ 1 r 2 1 k 2r 1 3ð11rÞ r 2 2ð2r 2 1rÞt 12r1t k 2r 3 16r 2 1ð2r 2 1rÞt 12r2t 2ð4r r11Þt 4 2ð4r 2 15r11Þt 4 2r 3ð114rÞ 2 r 2 1 k 9r 2 6ð11rÞ r1rt 12r1t k 9r 2 15r2rt 12r2t 4ð4r r11Þt 4 4ð4r 2 15r11Þt 4 1 3ð114rÞ k 9r 1 3ð11rÞ r1rt 12r1t 2ð4r 2 15r11Þt 4 k 2 k 9r 2 15r2rt 2ð4r 2 15r11Þt 12r2t 4 1 3ð114rÞ k r 1 3ð11rÞ 2 3 2ð8r 3 1r 2 23rÞt210r 2 15r 12r1t k r 3 1ð8r 3 1r 2 23rÞt210r 2 15r 12r2t 2ð4r 4 235r 3 229r r15Þ 4 2ð4r 4 235r 3 229r 2 115r15Þ 4 2r 3ð114rÞ 2 2r 2 1 k r 1 3ð11rÞ 2 4 1ð5r 3 2rÞt210r 3 15r 2 k 12r1t r 1 4 2ð5r 3 2rÞt210r 3 15r 2 12r2t 4r 4 235r 3 229r 2 115r15 4 4r 4 235r 3 229r 2 115r15 4 2r 3ð114rÞ 1 r 2 1 k 2r 2 3ð11rÞ 2 4 1ð2r 3 2r 2 1rÞt219r 3 k 15r 12r1t 2r 4 2ð2r 3 2r 2 1rÞt219r 3 15r 12r2t 2ð4r 4 235r 3 229r r15Þ 4 2ð4r 4 235r 3 229r 2 115r15Þ 4 1 3ð114rÞ k 2r 1 3ð11rÞ 2 4 1ð2r 3 2r 2 1rÞt219r 3 15r 12r1t k 2r 4 2ð2r 3 2r 2 1rÞt219r 3 15r 12r2t 4r 4 235r 3 229r r15 4 4r 4 235r 3 229r 2 115r15 4 k k k k k k k k Broman, Genetics 190:403,

26 Time to fixation Pr( fixed by F k ) selfing way, sib mating, X chr 2 way, sib mating, autosome 4 way, sib mating, X chr 4 way, sib mating, autosome Generation F k Broman, Genetics 190:403,

27 Map expansion 7 6 Eight way Four way 5 Map expansion 4 3 Two way 2 1 Sib mating Selfing Generation F k Broman, Genetics 190:403,

28 MAGIC lines combine mix fix How many? Which? How long? How? Valdar et al., Genetics 172:1783,

29 MAGIC is magic Genetic diversity High-precision mapping Predictable linkage disequilibrium Phenotype replicates to reduce individual variation Pool phenotypes from multiple labs, environments, treatments Genotype once Cool name 21

30 MAGIC is magic Genetic diversity High-precision mapping Predictable linkage disequilibrium Phenotype replicates to reduce individual variation Pool phenotypes from multiple labs, environments, treatments Genotype once Cool name 21

31 The goal Identify QTL Power Mapping precision 22

32 The goal Identify QTG Power Mapping precision 22

33 The goal Identify QTG Power Mapping precision Estimate QTL allele frequencies 22

34 Principles Avoid population structure Tradeoff between power for de novo discovery and mapping precision More QTL to find more QTL getting in the way? More QTL alleles less information about each Are QTL alleles common or rare? 23

35 How many founders? More More general use More QTL Greater precision Estimate allele frequencies Fewer Lower residual variance Greater power for a particular QTL? Better power for epistasis Rare alleles are less rare Haplotype analysis in founders 24

36 Which founders? Diverse Interesting No breeding problems Balanced: star phylogeny 25

37 How much mixing? More mixing Greater mapping precision...but lower power for de novo mapping Potential for population structure, missing alleles Random mating or curated mating? Start with many random cross directions? 26

38 Selfing or DH? Inbreeding gives added recombination But not so much as at the mixing stage If doubled haploids are feasible, use them 27

39 Key analysis issues How to deal with the multiple alleles? Full model (an effect for each allele) Diallelic QTL model Random effects model (like BLUP) How to account for multiple QTL? Stepwise selection Bayesian model averaging Random effect for polygenes 28

40 Sharing is also key The greatest power of MAGIC comes from sharing Pooling data, exploring multiple environments/treatments Common software needs Analysis software, database infrastructure Our students need to learn the same stuff Joint training opportunities 29

41 R/qtl Lines of code R C idea man svn git Year 30

42 R/qtl: Good things Hidden Markov model code Many methods Extensible Open 31

43 R/qtl: Not-so-good Some really bad code Scantwo is 4% of R code and 20% of C code, with a 1354-line R function The stupidest R code ever: for(i in 1:n) { temp[i] <- all(data[2,1:i]=="") if(!temp[i]) break } The central data structure is too restrictive Can t handle multiple individuals per genotype Memory mis-management Lack of connections to genome databases Largely one developer (who is also the support staff) 32

44 qtlhd Re-implementation of R/qtl Aimed at high-throughput computing High-dimensional data Dense markers, high-dim phenotypes, modern cross designs Separate from R But accessible from R (and ruby and python) Interactive graphics Connections to genome DBs/browsers github.com/qtlhd/qtlhd (Still at an exploratory stage) 33

45 How many founders? Summary Tradeoff between diversity and information about particular alleles Which founders? Diverse, interesting, no breeding problems, star phylogeny How long to mix? Tradeoff between power and precision How to fix? Doubled haploids are great if feasible Let s share! Lines, data, software, training 34

46 How many founders? Summary Tradeoff between diversity and information about particular alleles Which founders? Diverse, interesting, no breeding problems, star phylogeny How long to mix? Tradeoff between power and precision How to fix? Doubled haploids are great if feasible Let s collaborate! Lines, data, software, training 34

47 Acknowledgments RIL calculations R/qtl James Crow Bret Larget David Levin Dan Naiman Timo Seppäläinen Friedrich Teuscher Danny Arends Gary Churchill Ritsert Jansen Pjotr Prins Śaunak Sen Hao Wu Brian Yandell Funding: NIH R01 GM

Recombination and Linkage Karl W Broman Biostatistics & Medical Informatics University of Wisconsin Madison http://www.biostat.wisc.edu/~kbroman The genetic approach Start with the phenotype; find genes

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