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Terminology review

- Allele
- Locus
- Diploid
- SNP

Single Nucleotide Polymorphisms

Infinite Sites Assumption:

Each site mutates at most once

00000101011

10001101001

01000101010

01000000011

00011110000

00101100110

What causes variation in a population?

- Mutations (may lead to SNPs)
- Recombinations
- Other genetic events (gene conversion)
- Structural Polymorphisms

Gene Conversion

- Gene Conversion versus crossover
- Hard to distinguish in a population

Structural polymorphisms

- Large scale structural changes (deletions/insertions/inversions) may occur in a population.

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

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
- PAA=p2
- PAa=2pq
- Paa=q2

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.

Hardy Weinberg: Generalizations

- Multiple alleles with frequencies
- By HW,
- Multiple loci?

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 mutation?
- 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.

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.

The Infinite Sites Assumption

0 0 0 0 0 0 0 0

3

0 0 1 0 0 0 0 0

5

8

0 0 1 0 1 0 0 0

0 0 1 0 0 0 0 1

- 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

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

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.

The 4-gamete condition

- A column i partitions the set of species into two sets i0, and i1
- 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

i0

i1

4 Gamete Condition

- 4 Gamete Condition
- There exists a perfect phylogeny if and only if for all pair of columns (i,j), either j is not heterogenous w.r.t i0, or i1.
- 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)

4-gamete condition: proof

i

i0

i1

- Depending on which edge the mutation j occurs, either i0, or i1 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?

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.

Inclusion Property

- For any pair of columns i,j
- i < j if and only if i1 j1
- Note that if i<j then the edge containing i is an ancestor of the edge containing i

i

j

Example

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

Initially, there is a single clade r, and each node has r as its parent

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

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

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

u

B

D

A

C

E

Adding other columns

- Add other columns on edges using the ordering property

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

1

3

E

2

B

5

4

D

A

C

Unrooted case

- Switch the values in each column, so that 0 is the majority element.
- Apply the algorithm for the rooted case

Handling recombination

- A tree is not sufficient as a sequence may have 2 parents
- Recombination leads to loss of correlation between columns

Linkage (Dis)-equilibrium (LD)

- Consider sites A &B
- Case 1: No recombination
- Pr[A,B=0,1] = 0.25
- Linkage disequilibrium
- Case 2:Extensive recombination
- Pr[A,B=(0,1)=0.125
- Linkage equilibrium

A B

0 1

0 1

0 0

0 0

1 0

1 0

1 0

1 0

Handling recombination

- A tree is not sufficient as a sequence may have 2 parents
- Recombination leads to loss of correlation between columns

Recombination, and populations

- Think of a population of N individual chromosomes.
- The population remains stable from generation to generation.
- Without recombination, each individual has exactly one parent chromosome from the previous generation.
- With recombinations, each individual is derived from one or two parents.
- We will formalize this notion later in the context of coalescent theory.

Linkage (Dis)-equilibrium (LD)

- Consider sites A &B
- Case 1: No recombination
- Each new individual chromosome chooses a parent from the existing ‘haplotype’

A B

0 1

0 1

0 0

0 0

1 0

1 0

1 0

1 0

1 0

Linkage (Dis)-equilibrium (LD)

- Consider sites A &B
- Case 2: diploidy and recombination
- Each new individual chooses a parent from the existing alleles

A B

0 1

0 1

0 0

0 0

1 0

1 0

1 0

1 0

1 1

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

A B

0 1

0 1

0 0

0 0

1 0

1 0

1 0

1 0

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)

Measures of LD

- Consider two bi-allelic sites with alleles marked with 0 and 1
- Define
- P00 = Pr[Allele 0 in locus 1, and 0 in locus 2]
- P0* = Pr[Allele 0 in locus 1]
- Linkage equilibrium if P00 = P0* P*0
- D = abs(P00 - P0* P*0) = abs(P01 - P0* P*1) = …

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
- D(t) =(1-r) D(t-1) =(1-r)t D(0)

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.

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

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.

LD

D

N

N

D

D

N

0

1

1

0

0

1

LD and disease gene mapping problems

- Marker density?
- Complex diseases
- Population sub-structure

Human Samples

- We look at data from human samples
- Gabriel et al. Science 2002.
- 3 populations were sampled at multiple regions spanning the genome
- 54 regions (Average size 250Kb)
- SNP density 1 over 2Kb
- 90 Individuals from Nigeria (Yoruban)
- 93 Europeans
- 42 Asian
- 50 African American

Population specific recombination

- D’ was used as the measure between SNP pairs.
- SNP pairs were classified in one of the following
- Strong LD
- Strong evidence for recombination
- Others (13% of cases)
- This roughly favors out-of-africa. A Coalescent simulation can help give confidence values on this.

Gabriel et al., Science 2002

Haplotype Blocks

- A haplotype block is a region of low recombination.
- Define a region as a block if less than 5% of the pairs show strong recombination
- Much of the genome is in blocks.
- Distribution of block sizes vary across populations.

Testing Out-of-Africa

- Generate simulations with and without migration.
- Check size of haplotype blocks.
- Does it vary when migrations are allowed?
- When the ‘new’ population has a bottleneck?
- If there was a bottleneck that created European and Asian populations, can we say anything about frequency of alleles that are ‘African specific’?
- Should they be high frequency, or low frequency in African populations?

Haplotype Block: implications

- The genome is mostly partitioned into haplotype blocks.
- Within a block, there is extensive LD.
- Is this good, or bad, for association mapping?

Coalescent reconstruction

- Reconstructing likely coalescents

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.

Inclusion Property

- For any pair of columns i,j
- i < j if and only if i1 j1
- Note that if i<j then the edge containing i is an ancestor of the edge containing i

i

j

Example

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

Initially, there is a single clade r, and each node has r as its parent

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

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

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

u

B

D

A

C

E

Adding other columns

- Add other columns on edges using the ordering property

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

1

3

E

2

B

5

4

D

A

C

Unrooted case

- Important point is that the perfect phylogeny condition does not change when you interchange 1s and 0s at a column.
- Switch the values in each column, so that 0 is the majority element.
- Apply the algorithm for the rooted case.
- Homework: show that this is a correct algorithm

Population sub-structure can increase LD

Pop. A

0 .. 1

0 .. 1

0 .. 0

1 .. 1

0 .. 1

0 .. 1

0 .. 1

0 .. 1

0 .. 1

p1=0.1

q1=0.9

P11=0.1

D=0.01

Pop. B

1 .. 0

1 .. 0

0 .. 0

1 .. 1

1 .. 0

1 .. 0

1 .. 0

1 .. 0

1 .. 0

p1=0.9

q1=0.1

P11=0.1

D=0.01

- Consider two populations that were isolated and evolving independently.
- They might have different allele frequencies in some regions.
- Pick two regions that are far apart (LD is very low, close to 0)

Recent ad-mixing of population

- If the populations came together recently (Ex: African and European population), artificial LD might be created.
- D = 0.15 (instead of 0.01), increases 10-fold
- This spurious LD might lead to false associations
- Other genetic events can cause LD to arise, and one needs to be careful

0 .. 1

0 .. 1

0 .. 0

1 .. 1

0 .. 1

0 .. 1

0 .. 1

0 .. 1

0 .. 1

Pop. A+B

p1=0.5

q1=0.5

P11=0.1

D=0.1-0.25=0.15

1 .. 0

1 .. 0

0 .. 0

1 .. 1

1 .. 0

1 .. 0

1 .. 0

1 .. 0

1 .. 0

Determining population sub-structure

- Given a mix of people, can you sub-divide them into ethnic populations.
- Turn the ‘problem’ of spurious LD into a clue.
- Find markers that are too far apart to show LD
- If they do show LD (correlation), that shows the existence of multiple populations.
- Sub-divide them into populations so that LD disappears.

Determining Population sub-structure

0 .. 1

0 .. 1

0 .. 0

1 .. 1

0 .. 1

0 .. 1

0 .. 1

0 .. 1

0 .. 1

1 .. 0

1 .. 0

0 .. 0

1 .. 1

1 .. 0

1 .. 0

1 .. 0

1 .. 0

1 .. 0

- Same example as before:
- The two markers are too similar to show any LD, yet they do show LD.
- However, if you split them so that all 0..1 are in one population and all 1..0 are in another, LD disappears

Iterative algorithm for population sub-structure

- Define
- N = number of individuals (each has a single chromosome)
- k = number of sub-populations.
- Z {1..k}N is a vector giving the sub-population.
- Zi=k’ => individual i is assigned to population k’
- Xi,j = allelic value for individual i in position j
- Pk,j,l = frequency of allele l at position j in population k

Example

- Ex: consider the following assignment
- P1,1,0 = 0.9
- P2,1,0 = 0.1

1

1

1

1

1

1

1

1

1

1

2

2

2

2

2

2

2

2

2

2

0 .. 1

0 .. 1

0 .. 0

1 .. 1

0 .. 1

0 .. 1

0 .. 1

0 .. 1

0 .. 1

0 .. 1

1 .. 0

1 .. 0

0 .. 0

1 .. 1

1 .. 0

1 .. 0

1 .. 0

1 .. 0

1 .. 0

1 .. 0

Goal

- X is known.
- P, Z are unknown.
- The goal is to estimate Pr(P,Z|X)
- Various learning techniques can be employed.
- maxP,Z Pr(X|P,Z) (Max likelihood estimate)
- maxP,Z Pr(X|P,Z) Pr(P,Z) (MAP)
- Sample P,Z from Pr(P,Z|X)
- Here a Bayesian (MCMC) scheme is employed to sample from Pr(P,Z|X). We will only consider a simplified version

Algorithm:Structure

- Iteratively estimate
- (Z(0),P(0)), (Z(1),P(1)),.., (Z(m),P(m))
- After ‘convergence’, Z(m) is the answer.
- Iteration
- Guess Z(0)
- For m = 1,2,..
- Sample P(m) from Pr(P | X, Z(m-1))
- Sample Z(m) from Pr(Z | X, P(m))
- How is this sampling done?

Example

- Choose Z at random, so each individual is assigned to be in one of 2 populations. See example.
- Now, we need to sample P(1) from Pr(P | X, Z(0))
- Simply count
- Nk,j,l = number of people in pouplation k which have allele l in position j
- pk,j,l = Nk,j,l / N

1

2

2

1

1

2

1

2

1

2

1

2

2

1

1

2

1

2

2

1

0 .. 1

0 .. 1

0 .. 0

1 .. 1

0 .. 1

0 .. 1

0 .. 1

0 .. 1

0 .. 1

0 .. 1

1 .. 0

1 .. 0

0 .. 0

1 .. 1

1 .. 0

1 .. 0

1 .. 0

1 .. 0

1 .. 0

1 .. 0

Example

- Nk,j,l = number of people in population k which have allele l in position j
- pk,j,l = Nk,j,l / Nk,j,*
- N1,1,0 = 4
- N1,1,1 = 6
- p1,1,0 = 4/10
- p1,2,0 = 4/10
- Thus, we can sample P(m)

1

2

2

1

1

2

1

2

1

2

1

2

2

1

1

2

1

2

2

1

0 .. 1

0 .. 1

0 .. 0

1 .. 1

0 .. 1

0 .. 1

0 .. 1

0 .. 1

0 .. 1

0 .. 1

1 .. 0

1 .. 0

0 .. 0

1 .. 1

1 .. 0

1 .. 0

1 .. 0

1 .. 0

1 .. 0

1 .. 0

Sampling Z

- Pr[Z1 = 1] = Pr[”01” belongs to population 1]?
- We know that each position should be in linkage equilibrium and independent.
- Pr[”01” |Population 1] = p1,1,0 * p1,2,1 =(4/10)*(6/10)=(0.24)
- Pr[”01” |Population 2] = p2,1,0 * p2,2,1 = (6/10)*(4/10)=0.24
- Pr [Z1 = 1] = 0.24/(0.24+0.24) = 0.5

Assuming, HWE, and LE

Sampling

- Suppose, during the iteration, there is a bias.
- Then, in the next step of sampling Z, we will do the right thing
- Pr[“01”| pop. 1] = p1,1,0 * p1,2,1 = 0.7*0.7 = 0.49
- Pr[“01”| pop. 2] = p2,1,0 * p2,2,1 =0.3*0.3 = 0.09
- Pr[Z1 = 1] = 0.49/(0.49+0.09) = 0.85
- Pr[Z6 = 1] = 0.49/(0.49+0.09) = 0.85
- Eventually all “01” will become 1 population, and all “10” will become a second population

1

1

1

2

1

2

1

2

1

1

2

2

2

1

2

2

1

2

2

1

0 .. 1

0 .. 1

0 .. 0

1 .. 1

0 .. 1

0 .. 1

0 .. 1

0 .. 1

0 .. 1

0 .. 1

1 .. 0

1 .. 0

0 .. 0

1 .. 1

1 .. 0

1 .. 0

1 .. 0

1 .. 0

1 .. 0

1 .. 0

Allowing for admixture

- Define qi,k as the fraction of individual i that originated from population k.
- Iteration
- Guess Z(0)
- For m = 1,2,..
- Sample P(m),Q(m) from Pr(P,Q | X, Z(m-1))
- Sample Z(m) from Pr(Z | X, P(m),Q(m))

Estimating Z (admixture case)

- Instead of estimating Pr(Z(i)=k|X,P,Q), (origin of individual i is k), we estimate Pr(Z(i,j,l)=k|X,P,Q)

i,1

i,2

j

Results: Thrush data

- For each individual, q(i) is plotted as the distance to the opposite side of the triangle.
- The assignment is reliable, and there is evidence of admixture.

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)

Oceania

Eurasia

East Asia

America

Africa

Population sub-structure:research problem

- Systematically explore the effect of admixture. Can admixture be predicted for a locus, or for an individual
- The sampling approach may or may not be appropriate. Formulate as an optimization/learning problem:
- (w/out admixture). Assign individuals to sub-populations so as to maximize linkage equilibrium, and hardy weinberg equilibrium in each of the sub-populations
- (w/ admixture) Assign (individuals, loci) to sub-populations

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