Uncovering evolutionary history: new methods for inferring phylogenies

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Uncovering evolutionary history new methods for
inferring phylogenies
Tom Nye, Wally Gilks (MRC BSU)
Pietro Liò (Computer Laboratory, Cambridge)

• RSS Manchester
• 12 October 2005

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Winding Back the Evolutionary Clock

• Biologists want to reconstruct the evolutionary
  history of genes, genomes, and species
• Evolutionary history helps us to understand the
  genomes we see today
• Phylogenetic trees represent evolutionary
  relationships between species and are a vital
  ingredient in many biological analyses

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Todays Talk

• An introduction to phylogeny
• Maximum likelihood and distance-matrix frameworks
• A new distance-matrix approach

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Evolutionary Trees

• Evolutionary relationships can be represented by
  trees, called phylogenies
• Leaf nodes are extant species
• Internal nodes are speciation events
• Branch lengths show evolutionary distance

Moa
Cassowary
Emu
Kiwi
Ostrich
Rhea
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Biological Data
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Sequence Evolution
AAGCTGATC
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Models of Nucleotide Substitution

• Model sites along the DNA string as evolving
  independently
• Continuous time Markov chain with states A,C,G,T
  
• Define
• Pij (t) Prob (in state j at time t given in
  state i at t 0)
• So that
• P(t) exp ?tQ
• where
• Q is the instantaneous rate matrix
• ? is the rate of mutation events, ?t represents
  branch length
• Various models available for Q

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Molecular Clocks

• Branch lengths represent evolutionary distance
  (typically number of nucleotide substitutions)
• Rates of change may vary between branches
• Molecular clock no rate variation

t
No Clock
With Clock
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Tree Likelihood

• Given a tree topology and branch lengths,
  evaluate the likelihood of the tree under the
  substitution model

AAGCGCATG AACCTCATC TAGCCGTTC TAGCCCTGC G
AGCAGTTC
Likelihood Prob(G,T,C,C,A Tree)
???? Prob(w) Prob(zw) Prob(Az) Prob(Cz)
? Prob(yw) Prob(Cy)
? Prob(xy) Prob(Gx) Prob(Tx)
x y z w

• Full likelihood is obtained by taking product
  over nucleotide sites

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Likelihood Maximization

• We can search for the maximum likelihood tree
• Pick an initial topology
• Find the optimal set of branch lengths
• Is this the highest likelihood we have seen?
• Pick a new topology

• Tree space is huge the search is computationally
  intensive

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Distance-Matrix Approaches

• Given a matrix of evolutionary distances,
  estimate the tree that gave rise to those
  distances

Tree unknown
Distance matrix known

• Even if we knew the underlying tree, in practice
  any observed distance matrix will not match it
  perfectly
• branch length gives the expected number of
  substitutions at one site
• the distance matrix is unlikely to match any tree

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Comments

• Distance matrices
• The distance matrix summarizes the information in
  the full sequence data set
• Data loss problematic for widely diverged
  sequences
• Distance matrix is obtained from sequence data
  using a substitution model many ways to do this
• Comparison with likelihood
• Distance matrix methods are less sophisticated...
• ... but they are much faster!

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Least Squares Fitting

• Suppose we are given a tree topology and a
  distance matrix how would we find branch lengths
  on the tree?
• For two leaves i,j denote
• true distances on tree tij
• observed distances dij
• Assume that observed distances are unbiased
  estimates of the true distances
• Use branch lengths tij that minimize the error
  term

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Neighbour Joining

• Neighbour Joining (NJ) is defined by an
  agglomerative algorithm

i
i
u
u
j
j
Start with star-like topology
Consider joining every pair of nodes i,j in turn
Replace nodes i,j by a single node u
Calculate branch length estimates via least
squares
Continue...
Join the pair i,j that minimizes the sum of
branch lengths
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Comments

• NJ is hard to justify statistically...
• ... but it works surprisingly well!
• Recent improvements to the algorithm have not
  introduced a thorough statistical framework

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Our Methodology

• New distance-matrix method for constructing
  phylogenies
• Motivated by the example of gene families but
  also applies to species trees
• Essential ingredients
• Distribution free, moment-based approach
• Handles variance/covariance of distances more
  thoroughly than existing distance-matrix methods

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Motivation Families of Paralogs

• Certain genes have many copies within
• the same genome
• Examples olfactory receptors, proteases, kinases
• Appear to have evolved through duplications of
  individual genes, clusters of genes, and
  rearrangements within gene clusters
• Phylogenetic tree for these genes ? history of
  gene duplication
• Could we construct a more sophisticated history?
• Block duplications of more than one gene
• A history of linear arrangement along the genome

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Assumptions (1)

• Molecular clock setting necessary in order to
  consider events in which more than one gene is
  duplicated
• In a block duplication, two or more genes are
  copied at the same time
• The observed distances dij are the result of a
  random process perturbing the underlying true
  tree T

T
D
i
j
k
l
i
j
k
l
dij denotes distance between i,j in D
tij denotes path between i,j in T
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Assumptions (2)

• The observed distances dij are the result of a
  random process perturbing the underlying true
  tree T, that satisfies

T
i
j
k
l

• Note that we do not need a complete description
  of the perturbation process just consider
  moments

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Building trees

• Adopt an agglomerative approach winding back
  the clock

- Try joining every pair of nodes
- Can also consider block joins

• Pick the best join according to some statistical
  score

- Update the set of distances between nodes
Etc...
- Continue recursively

• We need to specify
• How to score join events
• How to fix the time for a join
• How to update the distances at each iteration

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Scoring Joins (1)

• Suppose we have constructed T as far back as some
  time t. What is the covariance matrix for
  distances between nodes?

i
t
Tt
i
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Scoring Joins (2)
j
i
t
Time ti
Tt
i
j
Next consider the expectation and variance of
distances between the black nodes
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Scoring Joins (3)

• Score tree using the goodness-of-fit of the
  calculated distances dt to expected distances
• Under suitable asymptotic assumptions this is a
  ?2 statistic
• The distance vector dt is n x n so the covariance
  matrix is n2 x n2 and inverting it is potentially
  O(n6)
• However, it can be inverted algebraically in
  O(n2) steps, and score evaluated in O(n) steps

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Results

• Construction of large trees
• Comparison with other methods (NJ) in progress...
• Issues
• As a purely phylogenetic method it is held back
  by the molecular clock assumption
• It is not a complete approach to inferring
  historical arrangements of paralogous genes,
  although it can incorporate duplication of more
  than one gene at a time

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Conclusions

• Full probabilistic models for constructing
  phylogenies are unsuitable if there are many
  leaves
• Existing distance-matrix methods could be
  improved upon
• We have a new distance-matrix approach that
  improves upon standard approaches to variance /
  covariance
• Future Work
• Could we build our approach to covariance into a
  setting with no molecular clock?
• Can we develop approaches that combine
  phylogenetic and arrangement information to build
  evolutionary histories?

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