CSE280b: Population Genetics

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CSE280b Population Genetics

• Vineet Bafna/Pavel Pevzner

www.cse.ucsd.edu/classes/sp05/cse291
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Population Genetics

• Individuals in a species (population) are
  phenotypically different.
• Often these differences are inherited (genetic).
• Studying these differences is important!
• QHow predictive are these differences?

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EXPopulation 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.

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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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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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What causes variation in a population?

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

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Single Nucleotide Polymorphisms
Infinite Sites Assumption Each site mutates at
most once
00000101011 10001101001 01000101010 01000000011 00
011110000 00101100110
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Short Tandem Repeats
GCTAGATCATCATCATCATTGCTAG GCTAGATCATCATCATTGCTAGTT
A GCTAGATCATCATCATCATCATTGC GCTAGATCATCATCATTGCTAG
TTA GCTAGATCATCATCATTGCTAGTTA GCTAGATCATCATCATCATC
ATTGC
4 3 5 3 3 5
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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 3 5 1 3 2 3 1 5 3
individuals
loci
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Recombination
00000000 11111111 00011111
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Gene Conversion

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

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Structural polymorphisms

• Large scale structural changes (deletions/insertio
  ns/inversions) may occur in a population.

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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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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
• PAAp2
• PAa2pq
• Paaq2

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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) pqqp 2pq,
  and so on.

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Hardy Weinberg Generalizations

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

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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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Recombination
00000000 11111111 00011111
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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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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

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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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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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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
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4 Gamete Condition

• 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 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)

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4-gamete condition proof (only if)

• 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?

i
j
i0
i1
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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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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
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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
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Linkage (Dis)-equilibrium (LD)

• Consider sites A B
• Case 1 No recombination
• Each new individual chooses a parent from the
  existing haplotype
• PrA,B0,1 0.25
• Linkage disequilibrium
• Case 2 Extensive recombination
• Each new individual simply chooses and allele
  from either site
• PrA,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
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LD

• In the absence of recombination,
• Correlation between columns
• The joint probability PrAa,Bb is different
  from P(a)P(b)
• With extensive recombination
• Pr(a,b)P(a)P(b)

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Measures of LD

• Consider two bi-allelic sites with alleles marked
  with 0 and 1
• Define
• P00 PrAllele 0 in locus 1, and 0 in locus 2
• P0 PrAllele 0 in locus 1
• Linkage equilibrium if P00 P0 P0
• D abs(P00 - P0 P0) abs(P01 - P0 P1)

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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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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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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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LD can be used to map disease genes
LD
0 1 1 0 0 1
D N N D D N

• LD decays with distance from the disease allele.
• By plotting LD, one can short list the region
  containing the disease gene.

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LD and disease gene mapping problems

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

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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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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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Wright Fisher Model of Evolution

• Fixed population size from generation to
  generation
• Random mating

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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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Coalescent theory

• Insight 2
• Much of the genealogy is irrelevant, because it
  disappears.
• Better to go backwards

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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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Coalescent theory

• Consider k individuals.
• Probability that no pair coalesces after 1
  generation
• Probability that no pair coalesces after t
  generations

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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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Coalescent approximation

• At any step, there are 1 lt k lt 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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Typical coalescents

• 4 random examples with n6 (Note that we do not
  need to specify N. Why?)
• Expected time to coalesce?

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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(Ttot)

1
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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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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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Coalescent with Recombination

• An individual may have one parent, or 2 parents

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ARG Coalescent with recombination

• Given mutation rate ?, recombination rate ?,
  population size 2N (diploid), sample size n.
• How can you generate the ARG (topologybranch
  lengths) efficiently?
• How will you generate sequences for n
  individuals?
• Given sequence data, can you reconstruct the ARG
  (topology)

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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
• An individual arises because of a recombination
  event between two individuals (It will have 2
  parents).
• Two individuals coalesce
• Neither (Each individual has a distinct parent)
• Multiple events (low probability)

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Recombination

• We ignore the case of multiple (gt 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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ARG

• 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

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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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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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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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Inclusion Property

• For any pair of columns i,j
• i lt j if and only if i1 ? j1
• Note that if iltj then the edge containing i is an
  ancestor of the edge containing i

i
j
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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
Initially, there is a single clade r, and each
node has r as its parent
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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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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
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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
B
2
5
4
D
A
C
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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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