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March 2006Vineet Bafna CSE280b: Population Genetics Vineet Bafna/Pavel Pevzner www.cse.ucsd.edu/classes/sp05/cse291
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March 2006Vineet Bafna Population Genetics Individuals in a species (population) are phenotypically different. Often these differences are inherited (genetic). Studying these differences is important! Q:How predictive are these differences?
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March 2006Vineet Bafna EX:Population Structure 377 locations (loci) were sampled in 1000 people from 52 populations. 6 genetic clusters were obtained, which corresponded to 5 geographic regions (Rosenberg et al. Science 2003) Genetic differences can predict ethnicity. Africa EurasiaEast Asia America Oceania
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March 2006Vineet Bafna Scope of these lectures Basic terminology Key principles – Sources of variation – HW equilibrium – Linkage – Coalescent theory – Recombination/Ancestral Recombination Graph – Haplotypes/Haplotype phasing – Population sub-structure – Structural polymorphisms – Medical genetics basis: Association mapping/pedigree analysis
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March 2006Vineet Bafna Alleles Genotype: genetic makeup of an individual Allele: A specific variant at a location – The notion of alleles predates the concept of gene, and DNA. – Initially, alleles referred to variants that described a measurable phenotype (round/wrinkled seed) – Now, an allele might be a nucleotide on a chromosome, with no measurable phenotype. Humans are diploid, they have 2 copies of each chromosome. – They may have heterozygosity/homozygosity at a location – Other organisms (plants) have higher forms of ploidy. – Additionally, some sites might have 2 allelic forms, or even many allelic forms.
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March 2006Vineet Bafna What causes variation in a population? Mutations (may lead to SNPs) Recombinations Other genetic events (gene conversion) Structural Polymorphisms
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March 2006Vineet Bafna Single Nucleotide Polymorphisms 00000101011 10001101001 01000101010 01000000011 00011110000 00101100110 Infinite Sites Assumption: Each site mutates at most once
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March 2006Vineet Bafna Short Tandem Repeats GCTAGATCATCATCATCATTGCTAG GCTAGATCATCATCATTGCTAGTTA GCTAGATCATCATCATCATCATTGC GCTAGATCATCATCATTGCTAGTTA GCTAGATCATCATCATCATCATTGC 435335435335
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March 2006Vineet Bafna STR can be used as a DNA fingerprint Consider a collection of regions with variable length repeats. Variable length repeats will lead to variable length DNA Vector of lengths is a finger- print 4 2 3 5 1 3 2 3 1 5 3 loci individuals
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March 2006Vineet Bafna Recombination 00000000 11111111 00011111
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March 2006Vineet Bafna Gene Conversion Gene Conversion versus crossover – Hard to distinguish in a population
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March 2006Vineet Bafna Structural polymorphisms Large scale structural changes (deletions/insertions/inversions) may occur in a population.
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March 2006Vineet Bafna Topic 1: Basic Principles In a ‘stable’ population, the distribution of alleles obeys certain laws – Not really, and the deviations are interesting HW Equilibrium – (due to mixing in a population) Linkage (dis)-equilibrium – Due to recombination
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March 2006Vineet Bafna Hardy Weinberg equilibrium Consider a locus with 2 alleles, A, a p (respectively, q) is the frequency of A (resp. a) in the population 3 Genotypes: AA, Aa, aa Q: What is the frequency of each genotype If various assumptions are satisfied, (such as random mating, no natural selection), Then P AA =p 2 P Aa =2pq P aa =q 2
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March 2006Vineet Bafna Hardy Weinberg: why? Assumptions: – Diploid – Sexual reproduction – Random mating – Bi-allelic sites – Large population size, … Why? Each individual randomly picks his two chromosomes. Therefore, Prob. (Aa) = pq+qp = 2pq, and so on.
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March 2006Vineet Bafna Hardy Weinberg: Generalizations Multiple alleles with frequencies – By HW, Multiple loci?
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March 2006Vineet Bafna Hardy Weinberg: Implications The allele frequency does not change from generation to generation. Why? It is observed that 1 in 10,000 caucasians have the disease phenylketonuria. The disease mutation(s) are all recessive. What fraction of the population carries the disease? Males are 100 times more likely to have the “red’ type of color blindness than females. Why? Conclusion: While the HW assumptions are rarely satisfied, the principle is still important as a baseline assumption, and significant deviations are interesting.
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March 2006Vineet Bafna Recombination 00000000 11111111 00011111
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March 2006Vineet Bafna What if there were no recombinations? Life would be simpler Each individual sequence would have a single parent (even for higher ploidy) The relationship is expressed as a tree.
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March 2006Vineet Bafna The Infinite Sites Assumption 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 1 0 0 0 3 8 5 The different sites are linked. A 1 in position 8 implies 0 in position 5, and vice versa. Some phenotypes could be linked to the polymorphisms Some of the linkage is “destroyed” by recombination
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March 2006Vineet Bafna Infinite sites assumption and Perfect Phylogeny Each site is mutated at most once in the history. All descendants must carry the mutated value, and all others must carry the ancestral value i 1 in position i 0 in position i
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March 2006Vineet Bafna Perfect Phylogeny Assume an evolutionary model in which no recombination takes place, only mutation. The evolutionary history is explained by a tree in which every mutation is on an edge of the tree. All the species in one sub-tree contain a 0, and all species in the other contain a 1. Such a tree is called a perfect phylogeny.
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March 2006Vineet Bafna The 4-gamete condition A column i partitions the set of species into two sets i 0, and i 1 A column is homogeneous w.r.t a set of species, if it has the same value for all species. Otherwise, it is heterogenous. EX: i is heterogenous w.r.t {A,D,E} i A 0 B 0 C 0 D 1 E 1 F 1 i0i0 i1i1
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March 2006Vineet Bafna 4 Gamete Condition – There exists a perfect phylogeny if and only if for all pair of columns (i,j), j is not heterogenous w.r.t i 0, or i 1. – Equivalent to – There exists a perfect phylogeny if and only if for all pairs of columns (i,j), the following 4 rows do not exist (0,0), (0,1), (1,0), (1,1)
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March 2006Vineet Bafna 4-gamete condition: proof (only if) Depending on which edge the mutation j occurs, either i 0, or i 1 should be homogenous. (only if) Every perfect phylogeny satisfies the 4- gamete condition (if) If the 4-gamete condition is satisfied, does a prefect phylogeny exist? i0i0 i1i1 i j
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March 2006Vineet Bafna Handling recombination A tree is not sufficient as a sequence may have 2 parents Recombination leads to loss of correlation between columns
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March 2006Vineet Bafna Linkage (Dis)-equilibrium (LD) Consider sites A &B Case 1: No recombination Each new individual chromosome chooses a parent from the existing ‘haplotype’ AB0101000010101010AB010100001010101000 1 0
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March 2006Vineet Bafna Linkage (Dis)-equilibrium (LD) Consider sites A &B Case 2: diploidy and recombination Each new individual chooses a parent from the existing alleles AB0101000010101010AB010100001010101000 1
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March 2006Vineet Bafna Linkage (Dis)-equilibrium (LD) Consider sites A &B Case 1: No recombination Each new individual chooses a parent from the existing ‘haplotype’ – Pr[A,B=0,1] = 0.25 Linkage disequilibrium Case 2: Extensive recombination Each new individual simply chooses and allele from either site – Pr[A,B=(0,1)=0.125 Linkage equilibrium AB0101000010101010AB010100001010101000
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March 2006Vineet Bafna LD In the absence of recombination, – Correlation between columns – The joint probability Pr[A=a,B=b] is different from P(a)P(b) With extensive recombination – Pr(a,b)=P(a)P(b)
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March 2006Vineet Bafna Measures of LD Consider two bi-allelic sites with alleles marked with 0 and 1 Define – P 00 = Pr[Allele 0 in locus 1, and 0 in locus 2] – P 0* = Pr[Allele 0 in locus 1] Linkage equilibrium if P 00 = P 0* P *0 D = abs(P 00 - P 0* P *0 ) = abs(P 01 - P 0* P *1 ) = …
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March 2006Vineet Bafna LD over time With random mating, and fixed recombination rate r between the sites, Linkage Disequilibrium will disappear – Let D (t) = LD at time t – P (t) 00 = (1-r) P (t-1) 00 + r P (t-1) 0* P (t-1) *0 – D (t) = P (t) 00 - P (t) 0* P (t) *0 = P (t) 00 - P (t-1) 0* P (t-1) *0 (HW) – D (t) =(1-r) D (t-1) =(1-r) t D (0)
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March 2006Vineet Bafna LD over distance Assumption – Recombination rate increases linearly with distance – LD decays exponentially with distance. The assumption is reasonable, but recombination rates vary from region to region, adding to complexity This simple fact is the basis of disease association mapping.
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March 2006Vineet Bafna LD and disease mapping Consider a mutation that is causal for a disease. The goal of disease gene mapping is to discover which gene (locus) carries the mutation. Consider every polymorphism, and check: – There might be too many polymorphisms – Multiple mutations (even at a single locus) that lead to the same disease Instead, consider a dense sample of polymorphisms that span the genome
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March 2006Vineet Bafna LD can be used to map disease genes LD decays with distance from the disease allele. By plotting LD, one can short list the region containing the disease gene. 011001011001 DNNDDNDNNDDN LD
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March 2006Vineet Bafna LD and disease gene mapping problems Marker density? Complex diseases Population sub-structure
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March 2006Vineet Bafna Population Genetics Often we look at these equilibria (Linkage/HW) and their deviations in specific populations These deviations offer insight into evolution. However, what is Normal? A combination of empirical (simulation) and theoretical insight helps distinguish between expected and unexpected.
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March 2006Vineet Bafna Topic 2: Simulating population data We described various population genetic concepts (HW, LD), and their applicability The values of these parameters depend critically upon the population assumptions. – What if we do not have infinite populations – No random mating (Ex: geographic isolation) – Sudden growth – Bottlenecks – Ad-mixture It would be nice to have a simulation of such a population to test various ideas. How would you do this simulation?
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March 2006Vineet Bafna Wright Fisher Model of Evolution Fixed population size from generation to generation Random mating
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March 2006Vineet Bafna Coalescent model Insight 1: – Separate the genealogy from allelic states (mutations) – First generate the genealogy (who begat whom) – Assign an allelic state (0) to the ancestor. Drop mutations on the branches.
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March 2006Vineet Bafna Coalescent theory Insight 2: – Much of the genealogy is irrelevant, because it disappears. – Better to go backwards
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March 2006Vineet Bafna Coalescent theory (Kingman) Input – (Fixed population (N individuals), random mating) Consider 2 individuals. – Probability that they coalesce in the previous generation (have the same parent)= Probability that they do not coalesce after t generations=
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March 2006Vineet Bafna Coalescent theory Consider k individuals. – Probability that no pair coalesces after 1 generation – Probability that no pair coalesces after t generations is time in units of N generations
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March 2006Vineet Bafna Coalescent approximation Insight 3: – Topology is independent of coalescent times – If you have n individuals, generate a random binary topology Iterate (until one individual) – Pick a pair at random, and coalesce Insight 4: – To generate coalescent times, there is no need to go back generation by generation
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March 2006Vineet Bafna Coalescent approximation At any step, there are 1 <= k <= n individuals To generate time to coalesce (k to k-1 individuals) – Pick a number from exponential distribution with rate k(k-1)/2 – Mean time to coalescence = 2/(k(k-1))
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March 2006Vineet Bafna Typical coalescents 4 random examples with n=6 (Note that we do not need to specify N. Why?) Expected time to coalesce?
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March 2006Vineet Bafna Coalescent properties Expected time for the last step The last step is half of the total time to coalesce Studying larger number of individuals does not change numbers tremendously EX: Number of mutations in a population is proportional to the total branch length of the tree – E(T tot ) =1
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March 2006Vineet Bafna Variants (exponentially growing populations) If the population is growing exponentially, the branch lengths become similar, or even star-like. Why? With appropriate scaling of time, the same process can be extended to various scenarios: male- female, hermaphrodite, segregation, migration, etc.
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March 2006Vineet Bafna Simulating population data Generate a coalescent (Topology + Branch lengths) For each branch length, drop mutations with rate Generate sequence data Note that the resulting sequence is a perfect phylogeny. Given such sequence data, can you reconstruct the coalescent tree? (Only the topology, not the branch lengths) Also, note that all pairs of positions are correlated (should have high LD).
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March 2006Vineet Bafna Coalescent with Recombination An individual may have one parent, or 2 parents
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March 2006Vineet Bafna ARG: Coalescent with recombination Given: mutation rate , recombination rate , population size 2N (diploid), sample size n. How can you generate the ARG (topology+branch lengths) efficiently? How will you generate sequences for n individuals? Given sequence data, can you reconstruct the ARG (topology)
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March 2006Vineet Bafna Recombination Define r as the probability of recombining. – Note that the parameter is a caled value which will be defined later Assume k individuals in a generation. The following might happen: 1. An individual arises because of a recombination event between two individuals (It will have 2 parents). 2. Two individuals coalesce 3. Neither (Each individual has a distinct parent) 4. Multiple events (low probability)
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March 2006Vineet Bafna Recombination We ignore the case of multiple (> 1) events in one generation Pr (No recombination) = 1-kr Pr (No coalescence) Consider scaled time in units of 2N generations. Thus the number of individuals increase with rate kr2N, and decrease with rate The value 2rN is usually small, and therefore, the process will ultimately coalesce to a single individual (MRCA)
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March 2006Vineet Bafna Let k = n, Define Iterate until k= 1 – Choose time from an exponential distribution with rate – Pick event as recombination with probability – If event is recombination, choose an individual to recombine, and a position, else choose a pair to coalesce. – Update k, and continue ARG What is the flaw in this procedure?
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March 2006Vineet Bafna Simulating sequences on the ARG Generate topology and branch lengths as before For each recombination, generate a position. Next generate mutations at random on branch lengths – For a mutation, select a position as well.
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March 2006Vineet Bafna Recombination events and Given , n, can you compute the expected number of recombination events? It can be shown that E(n, ) = log (n) The question that people are really interested in Given a set of sequences from a population, compute the recombination rate Given a population reconstruct the most likely history (as an ancestral recombination graph) We will address this question in subsequent lectures
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March 2006Vineet Bafna An algorithm for constructing a perfect phylogeny We will consider the case where 0 is the ancestral state, and 1 is the mutated state. This will be fixed later. In any tree, each node (except the root) has a single parent. – It is sufficient to construct a parent for every node. In each step, we add a column and refine some of the nodes containing multiple children. Stop if all columns have been considered.
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March 2006Vineet Bafna Inclusion Property For any pair of columns i,j – i < j if and only if i 1 j 1 Note that if i<j then the edge containing i is an ancestor of the edge containing i i j
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March 2006Vineet Bafna Example 1 2 3 4 5 A 1 1 0 0 0 B 0 0 1 0 0 C 1 1 0 1 0 D 0 0 1 0 1 E 1 0 0 0 0 r A BCDE Initially, there is a single clade r, and each node has r as its parent
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March 2006Vineet Bafna Sort columns Sort columns according to the inclusion property (note that the columns are already sorted here). This can be achieved by considering the columns as binary representations of numbers (most significant bit in row 1) and sorting in decreasing order 1 2 3 4 5 A 1 1 0 0 0 B 0 0 1 0 0 C 1 1 0 1 0 D 0 0 1 0 1 E 1 0 0 0 0
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March 2006Vineet Bafna Add first column In adding column i – Check each edge and decide which side you belong. – Finally add a node if you can resolve a clade r A B C D E 1 2 3 4 5 A 1 1 0 0 0 B 0 0 1 0 0 C 1 1 0 1 0 D 0 0 1 0 1 E 1 0 0 0 0 u
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March 2006Vineet Bafna Adding other columns Add other columns on edges using the ordering property r E B C D A 1 2 3 4 5 A 1 1 0 0 0 B 0 0 1 0 0 C 1 1 0 1 0 D 0 0 1 0 1 E 1 0 0 0 0 1 2 4 3 5
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March 2006Vineet Bafna Unrooted case Switch the values in each column, so that 0 is the majority element. Apply the algorithm for the rooted case
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March 2006Vineet Bafna
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March 2006Vineet Bafna
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