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Population Genetics Basics. 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?.

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terminology review
Terminology review
  • Allele
  • Locus
  • Diploid
  • SNP
single nucleotide polymorphisms
Single Nucleotide Polymorphisms

Infinite Sites Assumption:

Each site mutates at most once

00000101011

10001101001

01000101010

01000000011

00011110000

00101100110

what causes variation in a population
What causes variation in a population?
  • Mutations (may lead to SNPs)
  • Recombinations
  • Other genetic events (gene conversion)
  • Structural Polymorphisms
recombination
Recombination

00000000

11111111

00011111

gene conversion
Gene Conversion
  • Gene Conversion versus crossover
    • Hard to distinguish in a population
structural polymorphisms
Structural polymorphisms
  • Large scale structural changes (deletions/insertions/inversions) may occur in a population.
topic 1 basic principles
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
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
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
Hardy Weinberg: Generalizations
  • Multiple alleles with frequencies
    • By HW,
  • Multiple loci?
hardy weinberg implications
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.
recombination13
Recombination

00000000

11111111

00011111

what if there were no recombinations
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
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
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
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
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
  • 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
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
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
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
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
  • 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
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
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
Unrooted case
  • Switch the values in each column, so that 0 is the majority element.
  • Apply the algorithm for the rooted case
handling recombination
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
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 recombination30
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
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.