Lander-Green Algorithm. Biostatistics 666

Transcription

1 Lander-Green Algorith Biostatistics 666

2 Identity-by-Descent (IBD) A roerty o chroosoe stretches that descend ro the sae ancestor Two alleles are IBD i they descend ro the sae ounder chroosoe For a air o siblings, IBD generally deines the total nuber o IBD alleles between the two

3 P(Marker Genotye IBD State) IBD Sib CoSib 0 (a,b) (c,d) a b c d 0 0 (a,a) (b,c) a b c 0 0 (a,a) (b,b) a b 0 0 (a,b) (a,c) a b c a b c 0 (a,a) (a,b) 3 a b a b 0 (a,b) (a,b) a b a b + a b a b (a,a) (a,a) a 3 a a Prior Probability ¼ ½ ¼ Note: Assuing ordered genotyes, where (a,b)(c,d)

4 P(Marker Genotye IBD State) IBD Sib CoSib 0 (a,b) (c,d) a b c d 0 0 (a,a) (b,c) a b c 0 0 (a,a) (b,b) a b 0 0 (a,b) (a,c) a b c a b c 0 (a,a) (a,b) 3 a b a b 0 (a,b) (a,b) a b ( a b + a b ) a b 3 (a,a) (a,a) a a a Prior Probability ¼ ½ ¼ Note: Assuing unordered genotyes

5 Alternative likelihood Su over sets o IBD states at arker L P I P G I I P(I) Prior Probability o IBD state P(G I) Probability o genotyes given IBD state

6 Worked Exale 0.5 P( G IBD 0) P( G IBD) 3 P( G IBD ) 6 8 P( G) / /

7 Worked Exale II / / ) ( ) ( ), ( ) ( ) ( ) ( 6 /) /, ( /) /, ( /) /, ( /) /, ( q q q q q q q G G P G P G P G P G G G P G G G P G G G P G G G P G G G

8 Phenotye Inoration Probability o trait locus henotyes conditional on IBD state Deends on Trait locus allele requencies and q Penetrances or each genotye,,

9 For exale Phenotye robabilities or each IBD state in an ASP ASP Aected Sibling Pair P( ASP IBD is ) P( ASP IBD is) q q P( ASP IBD is 0) 3 q 3 q ( ) q q q( q q )

10 The Recobination Process The recobination raction is a easure o distance between two loci Probability that dierent alleles ro dierent grand-arents are inherited at soe locus It ilies the robability o change in IBD state or a air o chroosoes in siblings:

11 Transition Matrix or IBD States Allows calculation o IBD robabilities at arbitrary location conditional on linked arker Deends on recobination raction Conditional IBD Probabilities at distance 0 Known 0 (-) (-) IBD (-) (-) + (-) State (-) (-)

12 Genotye Likelihood or Two Loci Suation over 9 sets o IBD states: P(I) Prior Probability o IBD state P(G I) Probability o genotyes given IBD state ) ( ) ( ) ( ) ( I G P I G P I I P I P L I I

13 Worked Exale Two arkers Allele requencies A, q A = A B, q B = B / / / / Recobination raction = = 0.0

14 Alternative likelihood Su over sets o IBD states at each arker location P(I ) Prior Probability o IBD state at P(I i I i+ ) Transition robabilities or IBD states P(G i I i ) Probability o genotyes given IBD ) ( ) ( ) (... I i i i I i i i I G P I I P I P L

15 Moving along chroosoe Inut Vector v o IBD robabilities at location A Matrix T o transition robabilities AB Outut Vector v' o robabilities at location B Conditional on robabilities at location A For k IBD states, requires k oerations L( v' v) L( v ) T ( v v', i j j i j )

16 Bau Algorith Markov Model or IBD Vectors v l o robabilities at each location Transition atrix T between locations Key equations v l..l = v l-..l- T v l v l l.. = v l+ l+.. T v l v l.. = (v..l- T) v l (v l+.. T)

17 Pictorial Reresentation Single Marker Let Conditional Right Conditional Full Likelihood

18 Lander-Green Algorith Factorize likelihood by arker Each ste assigns hase For one arker For all individuals in the edigree Colexity e n Large nuber o arkers Assues no intererence Relatively sall edigrees

19 Lander-Green Recie. List ossible outcoes or all eioses Total o n- ossibilities Two outcoes or each o n eiosis Fix outcoe o one eiosis or each o ounder. For each arker calculate vector v Stores P(Genotyes Inheritance) or each ossible eiotic outcoe Calculated by drawing ounders allele grah and then using roducts o allele requencies

20 Lander-Green Recie 3. Calculate transition atrices T,+ Eleent ij is the robability o outcoe j at location + given outcoe i at location Deends on how any eiotic outcoes change or reain the sae change in eiotic outcoe = no change = - Next, run Markov-Chain See next age

21 Lander-Green Recie. Markov chain A) Start at irst arker, = Let conditional robability vector L =v B) Calculate L T,+ Vector*Matrix ultilication C) Next robability vector L + =L T,+ v + Coonent-wise vector ultilication D) Increent, reeat B) and C) until done

22 Lander-Green Recie 5. Overall likelihood is L last arker Su over all eleents o last vector Proortional to likelihood, issing rior All eiotic outcoes are equally likely, so divide this su by the nuber o eleents in vector Presto!

23 Enuerate Possibilities Enuerate genelow atterns Gene-low attern: Sets transitted allele or each eiosis Ilies ounder allele or each individual Meiosis Meiosis Meiosis V V V3 V

24 Lander-Green inheritance vector At each arker location l Deine inheritance vector v l n eleents Meiotic outcoes seciied in index bit Likelihood or each gene low attern Conditional on observed genotyes at location l L L L L L L L L L L L L L L L L

25 Founder Allele Sets Calculated or each arker individually Founder allele labels coatible with: Observed genotyes or all individuals The current gene low attern v Likelihood o soe ounder allele set is just the roduct o likelihoods o individual alleles

26 Observed Genotyes For each aily For each arker Soe attern o observed genotyes????

27 Gene low attern In turn, seciy gene low throughout the edigree For each individual, we know recisely what ounder allele they carry

28 Cobine the two Conditional on gene low (aka, inheritance vector) In this case one ounder allele set, {,,,?}? Conditional likelihood P(allele ) 3 P(any allele)

29 Founder Allele Grah General set o rules or inding ounder allele sets Grou ounder alleles transitted to the sae genotyed individuals I an allele asses through an hoozygote or two dierent heterozygotes its state is known Ilies state o all other alleles in grou

30 Founder Allele Grah II Possible grous o ounder alleles Any Allele Singleton Passes only through untyed individuals One Possible State For Each Allele At least one allele asses through hoozygotes or dierent heterozygotes Two Possible States For Each Allele All alleles ass through identical heterozygotes Arbitrarily ixing one ounder allele sets others

31 A ore colex exale 3

32 A ore colex exale C D E F A B G H {,} {,} {,} {,} {,3} {,}

33 Possible ounder allele states Founder Alleles in Corresonding Grou Allele States Probability (B) (any allele) (A,C,E) (,,) or (,,) P()²P()+P()P()² (D,E,F,G) (,,3,) P()P()P(3)P()

34 So ar For each arker Suarize inoration or all individuals In a vector o conditional likelihoods Next ste Cobine inoration along the genoe

35 Next Ste Assue recobination events are indeendent Distance between events exonential No. o events in an interval is Poisson Moving ro let-to-right along a, cuulate inoration ro each arker

36 Lander-Green Markov Model Transition atrix T n T v l..l = v l-..l- T n v l v l l.. = v l+ l+.. T n v l v l.. = (v..l- T n ) v l (v l+.. T n )

37 Inoration at single arkers At each akrer Vector o conditional likelihoods I or soe arker, grand-aternal allele were transitted Say, v = [L 0] And the next arker were coletely uninorative Say, v + = [L + L + ]

38 Cuulating inoration data at data at data at, data at ) ( ) ( ) ( 0 a L L L L L L L L v v v L L L v