Quantitative Genomics and Genetics BTRY 4830/6830; PBSB

Transcription

1 Quantitative Genomics Genetics BTRY 4830/6830; PBSB Lecture13: Introduction to genome-wide association studies (GWAS) II Jason Mezey March 16, 2017 (Th) 8:40-9:55

2 Announcements I will have fice hours today (!!) 3-5PM usual locations Homework #5 available tomorrow (Homework #4 due) Looking ahead: NO CLASS week March 20th (March 21 March 23) Midterm: Takehome March 29-March 31

3 Announcements II Quantitative Genomics Genetics - Spring 2017 BTRY 4830/6830; PBSB Midterm - available online 11:59PM, Tues., March 22 For midterm exam, due before 11:59PM, Fri., March 25 PLEASE NOTE THE FOLLOWING INSTRUCTIONS: 1. You are to complete this exam alone. The exam is open book, so you are allowed to use any books or information available online, your own notes your previously constructed code, etc. HOWEVER YOU ARE NOT ALLOWED TO COMMUNICATE OR IN ANY WAY ASK ANYONE FOR ASSISTANCE WITH THIS EXAM IN ANY FORM ( only exceptions are Afrah, Zijun, Dr. Mezey). As a non-exhaustive list this includes asking classmates or ANYONE else for advice or where to look for answers concerning problems, you are not allowed to ask anyone for access to ir notes or to even look at ir code wher constructed before exam or not, etc. You are refore only allowed to look at your own materials materials you can access on your own. In short, work on your own! Please note that you will be violating Cornell s honor code if you act orwise.

4 Announcements III 2. A complete answer to this exam will include two files: a SINGLE text file including all your R code, a SINGLE file including all your typed answers plots (where latter may be a scan as long as we can read it). Please note that for your R code, to get full credit for all problems, we must be able to run your code replicate all your results (with ease!). We will attempt to run your code if you do not do this but we will deduct points accordingly (note that no code no credit!). 3. Please pay attention to instructions complete ALL requirements for ALL questions, e.g. some questions ask for R code, plots, AND written answers. We will give partial credit so it is to you advantage to attempt every part every question. 4. The exam must be uploaded on CMS before 11:59PM Fri., March 25. It is your responsibility to make sure that it is in appropriate box before n no excuses will be accepted (power outages, computer problems, Cornell s internet slowed to a crawl, etc.). Remember: you are welcome to h this in early! We will deduct points for being late for exams received after this deadline (even if it is by minutes!!). Good Luck!!

5 Summary lecture 13 Last lecture, we introduced hyposis testing for quantitative genetic model Today we will provide a rigorous introduction to statistical foundation Genome-wide Association Study (GWAS) analysis We will also discuss concepts in population genetics as se relate to linkage disequilibrium, a concept which in turn is critical for understing analysis issues GWAS

6 Conceptual Overview Genetic System Does A1 -> A2 affect Y? Reject / DNR Measured individuals (genotype, phenotype) Regression model Sample or experimental pop Model params F-test Pr(Y X)

7 Review: genetic system Our goal in quantitative genetics / genomics is to identify loci (positions in genome) that contain causal mutations / polymorphisms / alleles X causal mutation or polymorphism - a position in genome where an experimental manipulation DNA produces an effect on phenotype on average or under specified conditions X Formally, we may represent this as follows: A 1! A 2 ) Y Z Our experiment will be a statistical experiment (sample inference!)

8 6 Review: genetic inference For our model focusing on one locus: Y µ + X a a + X d d + N(0, We have four possible parameters we could estimate: µ, a, d, However, for our purposes, we are only interested in genetic parameters testing following null 6 hyposis: [ 2 ) 2 H 0 : Cov(X a,y)0\ Cov(X d,y)0 H A : Cov(X a,y) 6 0[ Cov(X d,y) 6 0 \ OR H 0 : a 0\ d 0 H A : a 6 0[ d 60

9 Review: genetic estimation We will define a MLE for our parameters: [ µ, a, d] Recall that an MLE is simply a statistic (a function that takes a sample in outputs a number that is our estimate) In this case, our statistic will be a vector valued function that takes in vectors that represent our sample T (y, x a, x d ) ˆ 4 5 [ˆµ, ˆa, ˆd] Note that we calculate an MLE for this case just as we would any case (we use likelihood fixed sample where we identify parameter values that maximize this function) In linear regression case (just as with normal parameters) this has a closed form: 2 3 MLE( ˆ) (x T x) 1 x T y

10 Review: genetic hyposis testing We are going to test following hyposis: H 0 : a 0\ d 0 H A : a 6 0[ d 6 0 To do this, we need to construct following test statistic (for which we know distribution!): T (y, x a, x d H 0 : a 0\ d 0) Specifically, we are going to construct a likelihood ratio test (LRT) This is calculated using same structure that we have discussed (i.e. ratio likelihoods that take values parameters maximized under null alternative hyposis) In case a regression (not all cases!) we can write form LRT for our null in an alternative (but equivalent!) form In addition, our LRT has an exact distribution for all sample sizes n (!!)

11 Review: genetic inference We now have everything we need to construct a hyposis test for: H 0 : a 0\ d 0 H A : a 6 0[ d 6 0 This is equivalent to testing following: H 0 : Cov(X, Y ) 0 For a linear regression, we use F-statistic for our sample: F [2,n 3] (y, x a, x d ) MSM MSE We n determine a p-value using distribution F- statistic under null: pval(f [2,n 3] (y, x a, x d ))

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13 Review: Quantitative genomic analysis I We now know how to assess null hyposis as to wher a polymorphism has a causal effect on our phenotype Occasionally we will assess this hyposis for a single genotype In quantitative genomics, we generally do not know location causal polymorphisms in genome We refore perform a hyposis test many genotypes throughout genome This is a genome-wide association study (GWAS)

14 Review: Quantitative genomic analysis II Analysis in a GWAS raises (at least) two issues we have not yet encountered: An analysis will consist many hyposis tests (not just one) We ten do not test causal polymorphism (usually) Note that this latter issue is a bit strange (!?) - how do we assess causal polymorphisms if we have not measured causal polymorphism? Also note that causal genotypes will begin to be measured in our GWAS with next-generation sequencing data (but issue will still be present!)

15 Review: correlation among genotypes If we test a (non-causal) genotype that is correlated with causal genotype AND if correlated genotypes are in same position in genome THEN we can identify genomic position casual genotype (!!) This is case in genetic systems (why!?) Do we know which genotype is causal in this scenario? Copyright: Journal Diabetes its Complications; Science Direct; Vendramini et al

16 The Manhattan plot I We will consider a number visualization tools for analyzing GWAS data For moment, we will introduce Manhattan plot This is a plot genotypes on x-axis on y-axis -log p-values (base 10) (why?) resulting from each hyposis test each genotype Each point on plot is refore a single p-value corresponding to a single measured genotype We are looking for sets points with high -log p-value position a causal polymorphism

17 The Manhattan plot II: examples MTRR Log P Chromosome

18 Linkage Disequilibrium (LD) Mapping position a causal polymorphism in a GWAS requires re to be LD for genotypes that are both physically linked close to each or AND that markers that are eir far apart or on different chromosomes to be in equilibrium Note that disequilibrium includes both linkage disequilibrium AND or types disequilibrium (!!), e.g. gametic phase disequilibrium Chr. 1 LD equilibrium, linkage A B C D equilibrium, no linkage Chr. 2

19 6 Rigorous formulation GWAS analysis I For a GWAS, we assume that re could be causal polymorphisms X(Xa, Xd) that are BOTH in same physical position genome AND are correlated (in linkage disequilibrium) with polymorphisms that we have measured X (Xa,Xd ): Corr(X, X 0 ) >> 0 Note we are using Corr(X,X ) not specifying Xa, Xd because eir or both Xa, Xd could be correlated with XXa, Xd For analysis a GWAS with N measured genotypes (2 alleles each) a normal (error) 6 [ phenotype 6 we perform N hyposis tests: H 0 : H A : 0 a 0\ 0 d 0 0 a 6 0[ 0 d 6 0 Y X 0 a 0 a + X 0 0 d + For genotypes / sets genotypes (which sets?) for which we reject null, we assume that this indicates a position causal polymorphism (we have mapped position)

20 Rigorous formulation GWAS analysis II Polymorphisms for which we reject null are tags (in linkage disequilibrium with) causal polymorphisms The true parameter values regression model for tags: 0 a 0 The more correlated tag with causal polymorphism, closer tag parameter estimates to parameter causal polymorphism (on average): where y will be Corr(X, X 0 ) 0 Corr(X, X 0 ) >> 0 Corr(X, X 0 ) 1 0 d 0 ˆa 0, ˆd 0 ˆa a, ˆd d This means that for tag markers, we are getting a really bad estimate true tag parameters but a good estimate causal polymorphism parameters, which is what we depend on in a GWAS (!!)

21 Linkage Disequilibrium (LD) Mapping position a causal polymorphism in a GWAS requires re to be LD for genotypes that are both physically linked close to each or AND that markers that are eir far apart or on different chromosomes to be in equilibrium Note that disequilibrium includes both linkage disequilibrium AND or types disequilibrium (!!), e.g. gametic phase disequilibrium Chr. 1 LD equilibrium, linkage A B C D equilibrium, no linkage Chr. 2

22 Population versus quantitative genetics The oretical explanations statistical modeling used to underst linkage disequilibrium (LD) belong to field population genetics Population genetics (genomics) - a field concerned with modeling processes that lead to observed genomic variation in a population (i.e. field is concerned with explaining patterns DNA in a population) Quantitative genetics (genomics) - a field concerned with understing inferring relationship between genotypes an phenotypes Since LD describes pattern genomes in a population, understing LD is province population genetics We will discuss just most basic population genetic concepts critical for our purposes but I encourage you to take a class in pop gen

23 Hardy-Weinberg equilibrium I LD has two components: linkage (dis)equilibrium Linkage refers to physical linkage alleles on a chromosome, e.g. if alleles A1 at one locus B1 at a second locus are on same chromosome (y are on same molecule) y are linked Disequilibrium refers to any set alleles at two or more loci that are not in Hardy-Weinberg (H-W) equilibrium H-W equilibrium to a statistical description pattern alleles in a population If a population is in H-W equilibrium: 1. alleles an individual polymorphism are in equilibrium 2. sets alleles at two or more polymorphic sites are in equilibrium (where later concerns LD)

24 Hardy-Weinberg equilibrium II Theoretical conditions for that lead to H-W equilibrium are defined within population genetics, i.e. infinite population size, rom mating, no selection, no mutation, no gene flow, meiotic drive, or linkage (do se ever apply in real populations!?) 1. Under H-W equilibrium, if we represent probability A1 allele in a population or a sample as Pr(A1)p! ) probability A2 allele as Pr(A2)q1-p, n probability genotypes in population are: Pr(A 1,A 1 )Pr(A 1 )Pr(A 1 )p 2 Pr(A 1,A 2 )2Pr(A 1 )Pr(A 2 )2pq Pr(A 2,A 2 )Pr(A 2 )Pr(A 2 )q 2 2. Under H-W equilibrium, for two polymorphic sites A B, for all i, j, k, l we have: Pr(A i A j B k B l )Pr(A i A j )Pr(B k B l )Pr(A i )Pr(A j )Pr(B k )Pr(B l ) (Corr(X a,a,x a,b ) 0) (Corr(X a,a,x d,b ) 0) (Corr(X d,a,x a,b ) 0) (Corr(X d,a,x d,b ) 0)

25 Hardy-Weinberg equilibrium III Note that H-W refers to independence alleles occurring toger both WITHIN a polymorphic site BETWEEN polymorphic sites: Pr(A i A j B k B l )Pr(A i A j )Pr(B k B l )Pr(A i )Pr(A j )Pr(B k )Pr(B l ) Notation note 1: Instead probabilities alleles genotypes, we ten refer to se as frequencies (for our purposes, y are equivalent!) Notation note 2: We have some notation gridlock in literature, e.g. p for Pr(A1) (also a parameter binomial!!) xij is ten used to refer to probability / frequency AiAj (instead as a r.v.!!) For a sample n individuals in a sample or population, less frequent allele ( allele with lower probability) is minor allele this frequency (probability) is minor allele frequency (MAF) As an example, consider a sample n5 individuals with following observed genotypes: A1A1, A1A2, A1A2, A2A2, A2A2, n frequency allele A1 4/10, frequency allele A2 6/10 allele A1 is minor allele MAF 0.4

26 Linkage Disequilibrium (LD) I Two polymorphic sites in genome are in Disequilibrium if: Pr(A i B j,a k B l ) Pr(A i A k )Pr(B j B l ) (Corr(X a,a,x a,b ) 0) (Corr(X a,a,x d,b ) 0) (Corr(X d,a,x a,b ) 0) (Corr(X d,a,x d,b ) 0) Two polymorphic sites in genome in Linkage Disequilibrium (LD) if y are in Disequilibrium AND y physically linked on a chromosome (!!) Note that Disequilibrium ( LD) do not refer to H-W within a polymorphic site ONLY between sites, e.g. two sites may be in LD but genotype frequencies within two sites may be in H-W equilibrium We now know what correlation between tag causal polymorphism is referring to (!!), simply set one to A (X ) or to B (X)!

27 Linkage Disequilibrium (LD) II Note that value LD for identifying causal polymorphisms depends on following: With a tag depends on re not being disequilibrium among polymorphisms that are not physically linked (!!) Polymorphisms that are close to each tend to be in higher LD than those that are furr apart on a chromosome Let s next consider biological explanation as to why this is case, starting with polymorphisms on different chromosomes n with polymorphisms on same chromosome

28 Linkage Disequilibrium (LD) Mapping position a causal polymorphism in a GWAS requires re to be LD for genotypes that are both physically linked close to each or AND that markers that are eir far apart or on different chromosomes to be in equilibrium Note that disequilibrium includes both linkage disequilibrium AND or types disequilibrium (!!), e.g. gametic phase disequilibrium Chr. 1 LD equilibrium, linkage A B C D equilibrium, no linkage Chr. 2

29 Different chromosomes I Polymorphisms on different chromosomes tend to be in equilibrium because independent assortment rom mating, i.e. rom matching gametes to form zygotes Copyright:

30 Different chromosomes II Polymorphisms on different chromosomes tend to be in equilibrium because independent assortment rom mating, i.e. rom matching gametes to form zygotes

31 Different chromosomes III More formally, we represent independent assortment as: Pr(A i B k )Pr(A i )Pr(B k ) For rom pairing gametes to produce zygotes: Pr(A i B k,a j B l )Pr(A i B k )Pr(A j B l ) Putting this toger for rom pairing gametes to produce zygotes we get conditions for equilibrium: Pr(A i B k,a j B l )Pr(A i B k )Pr(A j B l ) Pr(A i )Pr(A j )Pr(B k )Pr(B l )Pr(A i A j )Pr(B k B l ) (Corr(X a,a,x a,b ) 0) (Corr(X a,a,x d,b ) 0) (Corr(X d,a,x a,b ) 0) (Corr(X d,a,x d,b ) 0)

32 Same chromosome I For polymorphisms on same chromosome, y are linked so if y are in disequilibrium, y are in LD In general, polymorphisms that are closer toger on a chromosome are in greater LD than polymorphisms that are furr apart (exactly what we need for GWAS!) This is because recombination, biological process by which chromosomes exchange sections during meiosis Since recombination events occur at rom throughout a chromosome (approximately!), furr apart two polymorphisms are, greater probability a recombination event between m Since more recombination events that occur between polymorphisms, closer y get to equilibrium, this means markers closer toger tend to be in greater LD

33 Same chromosome II In diploids, recombination occurs between pairs chromosomes during meiosis ( formation gametes) Note that this results in taking alleles that were physically linked on different chromosomes physically linking m on same chromosome

34 Same chromosome III To see how recombination events tend to increase equilibrium, consider an extreme example where alleles A1 B1 always occur toger on a chromosome A2 B2 always occur toger on a chromosome: Pr(A 1 B 2 ) 0, Pr(A 2 B 1 )0 Corr(X a,a,x a,b ) 1 AND Corr(X d,a,x d,b )1 If re is a recombination event, most chromosomes are A1-B1 A2-B2 but now re is an A1-B2 A2-B1 chromosome such that: Pr(A 1 B 2 ) 6 0, Pr(A 2 B 1 ) 6 0 Corr(X a,a,x a,b ) 1 AND Corr(X d,a,x d,b ) 1 Note recombination events disproportionally lower probabilities more frequent pairs! This means over time, polymorphisms will tend to increase equilibrium (decrease LD) Since more recombination events, greater equilibrium, polymorphisms that are furr apart will tend to be in greater equilibrium, those closer toger in greater LD

35 Linkage Disequilibrium (LD) Mapping position a causal polymorphism in a GWAS requires re to be LD for genotypes that are both physically linked close to each or AND that markers that are eir far apart or on different chromosomes to be in equilibrium Note that disequilibrium includes both linkage disequilibrium AND or types disequilibrium (!!), e.g. gametic phase disequilibrium Chr. 1 LD equilibrium, linkage A B C D equilibrium, no linkage Chr. 2

36 That s it for today Next lecture: continued discussion GWAS statistical, analysis, interpretation issues!