An introduction to maximum parsimony and compatibility

1
An introduction to maximum parsimony and
compatibility

• Trevor Bruen
• PhD Candidate
• McGill Centre for Bioinformatics

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Overview

• The point of this talk is to give a sense how
  discrete mathematics enters into phylogenetic and
  genetic inference.
• I will illustrate these ideas by describing two
  approaches in detail namely maximum compatibility
  and maximum parsimony.
• I will also show how ideas from these two
  criteria can be used to develop applications such
  as bounds and tests for recombination.
• My goal is to give the basis for further study in
  this type of area and to give greater insight
  into these methods.

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Outline

• Introduction to compatibility and parsimony
• Overview of basic notation/concepts
• Compatibility
• Compatibility as a graph theory problem
• Compatibility for pairs of characters
• Interpretation of compatibility
• Parsimony
• Parsimony score with connections to graph theory
• Connections between parsimony and compatibility
• Homoplasy
• Parsimony for pairs of characters
• Connections between SPRs/TBRs and parsimony
• Applications to recombination
• Parsimony as a consensus method

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Introduction

• Maximum parsimony and maximum compatibility that
  are used in phylogenetics, linguistics and
  population genetics
• Phylogenetics goal is to infer an evolutionary
  tree
• Linguistics often the same
• Population genetics uses compatibility for
  recombination
• For general phylogenetic inference with molecular
  data, likelihood (probability based) methods are
  generally preferred.
• BUT compatibility and parsimony are
  computationally tractable.
• ALSO the mathematics behind parsimony and
  compatibility is very well developed. We can
  show that parsimonylikelihood in certain
  circumstances (Tuffley and Steel 1997). This
  gives us insight in where to go in terms of
  research.

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Formalism

• A character is a mapping from a set of taxa to a
  set of states.
• In this case, XS1,S2,S3,S4
• Also, CA,C
• Informally, a character is a column in a
  multiple sequence alignment

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Binary Character / Splits

• If character has two states then it induces a
  split of the taxa set.
• Example Let X be the taxa set S1,S2,S3,S4.
  Let C be the state set A,C.
• Then S1,S2 S3,S4 is the split induced by
  the first character.
• In general a character induces a set of
  equivalence classes

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Tree and Labeling

• Informally we would like to be able to
  mathematically describe a tree and a labeling
  structure.
• In graph theory a tree T(V,E) consists of a
  graph with no cycles.
• Informally, we would also like to be able to add
  taxa (members of X) to our tree (actually the
  leaves).
• Define a labeling function (such that leaves of
  V(T) are labeled by members of X)

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X-Trees

• An X-tree consists of pair (T, phi) where phi is
  a labeling function that labels the leaves of T.
  
• Recall

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Extensions

• Informally, we have an X-tree consisting of the
  pair (T,phi). We also have a character chi. We
  need to relate the character to the tree.
• Define an extension of character as a function
  (which is consistent at the leaves with chi)
• Informally, an extension provides a description
  of how the internal vertices are labeled.

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Quick Summary

• Summary so far
• X-tree are trees along with functions labeling
  the leaves with members of X
• A character is a function from X into a state set
  C
• An extension is a labeling of the vertices of T
  with states of C

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Compatibility - Definition

• A character is compatible with a tree if and only
  if there exists an extension of the character to
  the tree so that the subgraphs induced by each of
  the states are connected.
• Example
• First tree character is compatible with tree
• Second tree character is incompatible since both
  As are disconnected

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Compatibility

• Problem definition Given a sequence of
  characters
• determine whether there exists a tree on
  which all character are compatible.
• Related problem Given a sequence of characters
• determine largest set of characters that are
  compatible with some tree

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Intersection Graph

• Suppose we have sequence of
• characters
• where
• Then each character induces a partition of X -
  I.e.
• Create a graph where the vertex set consists of
• There is an edge between two vertices iff only
  the intersection of the two subsets are non-empty

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Intersection Graph

• To figure out whether the sequence of characters
• are compatible, we will be able to determine
  this directly from the intersection graph.
• First we need to define two concepts a chordal
  graph and a restricted chordal completion of the
  intersection graph.

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Chordal Graphs

• A graph G(V,E) is chordal graph if every cycle
  with at least four vertices contains a chord (an
  edge connecting two non-consecutive vertices).
• A chordalization of graph is a graph G(V,E)
  where such that G is
  chordal

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Restricted Chordal Completions

• Imagine the vertices of our graph G(V,E) are
  colored. Then a restricted chordalization of G
  is a graph G(V,E), where G is chordal but all
  edges of G connect vertices of different colors.

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Restricted chordal completions

• A restricted chordal completion of the
  intersection graph is a chordalization where
  there is no edge between vertices that share the
  same character.
• In this case, the colors correspond to
  characters

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Main Theorem for Compatibility

• Let be a
  collection of characters. Then is
  compatible if and only if there is a restricted
  chordal completion of the intersection graph.

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Pairs of Characters

• A simple corollary of main theorem arises when we
  restrict our attention to two characters.
• Corollary Two characters
• are compatible if and only if the
  intersection graph, G for both characters is
  acyclic
• Proof (backwards direction) If graph is acyclic
  then it is chordal so the characters are
  compatible.
• (forward direction) OTOH Suppose G contains
  a cycle. Then any chordal completion of G must
  contain a three cycle. But no restricted
  completion of G can contain a three cycle! So G
  is acyclic.

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Interpretation

• Recall a set of characters are compatible with a
  X-tree if and only if there exists an extension
  of the character to the tree so that the
  subgraphs induced by each of the states are
  connected.
• Informally speaking this is a very strict
  condition. This corresponds to an all or
  nothing condition - either a character is
  compatible with a tree or it isnt. Relaxing
  this condition is the subject of the next
  section.

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Parsimony

• Informally given an leaf labeled tree and a
  character, how can we define the fit of the
  character to the tree?
• Consider a character, along with an
  extension to a leaf labeled tree. Then
  the length of the extension is the number edges
  where
• Define the parsimony score of a character on a
  tree as the length of a minimal extension of the
  character to the tree. Denote this value by

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Parsimony

• Then the maximum parsimony score for a set of
  characters
• on a tree is defined as
• The tree that minimizes this score is referred to
  as the maximum parsimony tree.

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Parsimony and graph theory

• A minimal cut-set for a leaf-labeled tree T(V,E)
  and a character is a minimal set of edges
  whose removal ensure that if
  that x and y are in different components.
• Claim There is a bijection between the set of
  minimal cut sets and minimal extensions. So the
  cardinality of the minimal cut set is equal to
  the parsimony score.

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Parsimony and Graph Theory

• Recall Mengers Theorem (1927) Let G(V,E) be a
  graph with V1 and V2 as two disjoint subsets of
  V. Then the minimum number of edges whose
  removal from G leaves vertices of V1 and V2 in
  different components is equal to the maximum
  number of edge disjoint paths between V1 and V2.
• Corollary For a binary character, the maximal
  number of edge disjoint paths corresponds to the
  parsimony score.

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Compatibility and parsimony

• Recall let
  be a collection of characters. Then
  is compatible if and only if there is a
  restricted chordal completion of the intersection
  graph.
• Question How can characterize parsimony with
  respect to an intersection graph?

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Compatibility Graph

• Recall Each character induces a partition of X -
  I.e.
• A block for a character
• is a subset taxa on which is constant.
• Thus we may identify the blocks of
• with the vertices of the intersection
  graph.

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Character Refinement

• A character refines another character
  if
• implies
• Thus characters that refine other characters
  correspond to refinements of the partition

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Compatibility and Parsimony

• Recall Let
  be a collection of characters. Then
  is compatible if and only if there is a
  restricted chordal completion of the intersection
  graph.
• Main

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Special Case Two characters

• Recall Two characters are compatible if and only
  if the intersection graph, G for both characters
  is acyclic
• Using the previous theorem we can show that the
  parsimony score for two
• characters corresponds to
• where k is the number of components in the graph.
• Note This score corresponds to the maximum
  parsimony score over all trees.

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Homoplasy

• Recall The parsimony score of a character on a
  tree, corresponds to minimum number of
  changes of a character on a tree.
• Informally What is an intuitive way to think
  about the parsimony score?
• Define the homoplasy of character on a tree as

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Homoplasy

• Note that with equality
  if and only if is convex on T
• Informally Homoplasy corresponds to the number
  of extra mutations of the character on the
  tree. These extra mutations correspond to
  recurrent mutations
• Informally Thus a character is not compatible
  on a tree iff it cannot be placed on a tree
  without extra mutations.

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Homoplasy For Two Characters

• Recall The parsimony score for a pair of
  characters can be found directly from the
  bipartite intersection graph.
• Recall This score corresponds to an optimum
  over all trees.
• Thus for two characters, we can define a pairwise
  homoplasy score as
• Recall Up to now homoplasy refers to extra
  mutations on a tree.

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A second look at homoplasy

• Example Two characters with a pairwise
  homoplasy score equal to one.
• Informally We have seen that the homoplasy
  corresponds to the number of extra mutations on
  a tree.
• But in certain situations, this is biologically
  implausible. The state 1 may correspond to a
  mutation that has only arisen once. In this
  case, the fact that the pairs of characters are
  incompatible can be explained by a recombination
  event.
• This will be defined more precisely later.

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A quick aside - tree distances.

• Differences between leaf labeled trees can be
  defined using various metrics - e.g. Subtree
  Prune and Regrafts
• A subtree prune and regraft corresponds to a
  specific re-arrangement of a tree.
• For two leaf-labeled trees, dSPR(T1, T2) is
  minimum SPRs between T1 and T2

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Homoplasy for two characters

• Theorem If and are two
  characters then corresponds
  to the minimum number of SPRs from any
  leaf-labled tree on which is compatible
  to any leaf labeled tree on which is
  compatible!
• Informally Thus we have a whole new
  interpretation of homoplasy.

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Application - Testing for Recombination

• If recombination has occurred sites will have
  different histories
• Nearby sites will tend to have greater
  genealogical correlation than distant sites
• Idea If recombination has occurred,
  genealogical correlation will be partially
  reflected by a tendency for pairs of closely
  linked sites to have than less homoplasy than
  distant sites

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Test for Recombination

• Idea We would like to distinguish between two
  possibilities - recurrent mutation and
  recombination.
• Idea Use previous observations to develop test
  for recombination.
• H0 Single history describe all sites.
• H0 Nearby sites share no more compatibility
  than arbitrary pairs of sites
• Use statistic to capture information and
  solve analytically for p-values

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Application Parsimony and supertrees

• Supertree MRP - parsimony with characters that
  represent trees.
• What does homoplasy mean in this context?

Courtesy of TREE 12315-322
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Parsimony as a consensus tree

• Recall If and are two
  characters then corresponds
  to the minimum number of SPRs from any
  leaf-labeled tree on which is compatible
  to any leaf labeled tree on which is
  compatible.
• Informally This can be generalized to show that
  the maximum parsimony tree for a set of charaters
  
• minimizes the SPR distance to each of the
  set of tree on which each character is compatible

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Acknowledgements

• Thanks for listening!
• Background and further reading
• Phylogenetics, Semple and Steel (book 2003)
• Some results I presented are not on this book -
  they are from work I have worked on. Please talk
  to me if you are interested.
• I have many other references- please see me if
  interested.