15

Announcements: Finish chapter 10

Evolution

A fundamental random model in population genetics is the Moran model

We output n samples from a population of N evolving for time T

  Current = [0 0 ... 0]
  for t=1 to T
    for i=1 to N
       Next[i] = Mutate( Current[ Random(N) ] )
    Current = Next
  return Current[1:n]

At each generation, cells pick parents and copy their content with mutations

It is a haploid model. Can be extended to a diploid model

  Current = [0 0 ... 0]
  for t=1 to T
    for i=1 to N
       Next[i] = Mutate( SEX(Current[Random(N)],Current[Random(N)]) )
    Current = Next
  return Current[1:n]

Coallescent analysis

Give n samples which were evolving for an infinite time in a population of size T

A compact representation of the ancestral tree of the samples:

The coallescent is a stochastic process generating such trees

Population

Tree interpretations

• Phenetic (how similar)
• Cladistic (how closely related)

Molecular clock

Emiler Zuckerkandl and Linus Pauling

mitochondrial DNA (mtDNA) is used in many systematic studies

neutral mutations useful for closely related species

The Panda story

Human mtDNA tree

Trees in biology

Basic trees

• unrooted: each internal node has degree 3
• rooted: one internal node with degree 2, the rest 3

Nodes labeled by intermediate species

Edges labeled by differences (phenetic) or `time' (cladistic)

Basic input data

• distance: n x n matrix
• factors/characters: F x n matrix of discrete data

Distance-based tree reconstruction

computing distances in the tree

Additive tree reconstruction

• input: distance matrix d(i,j) from an unknown tree
• output: the tree compatable with d

Additive tree reconstruction

One insight

• if i,j are children of k: d(i,m)+d(j,m)-d(i,j) = 2 d(k,m)

Two significant insights

• d(i,k) + d(j,k) = d(i,j) --> k is right on the branch from i to j
• T' is T with x distance subtracted from all leaf edges --> d'(i,j) = d(i,j)-2x

How do we tell if d is tree-additive?

• try to build a tree and fail
• 4-point condition
  

Approximate Additive tree reconstruction

In practice, d(i,j) is not the tree distance, and must be regularized

Approximate additive tree reconstruction

• input: distance matrix d (maybe triangle inequality)
• output: d', additive matrix, with min ||d-d'||

Usually done with least squares but obvious variants with different d-metrics

Unweighted Pair Group Method with Arithmetic Mean

  UPGMA(d,n)
    Hierarchical clustering with
      d(C1,C2) = avg of d(x1,x2) for x1 in C1, x2 in C2
    Internal node has height = avg of d(x1,x2)
    Leaf have height = 0
    edge(i,j) = height of i - height of j

UPGMA problem case

NeighborJoining

Saitou and Nei - deceptively simple, yet works well

Key idea: attempt to reward both closeness and distinctness

  NJ(d,n)
    u(i) = (1/(n-2)) * SUM_k d(i,k)
    until there is 1 cluster
      pick i,j neighbors to merge minimizing d(i,j) - u(i) - u(j)
      create new node x with edge(x,i) and edge(x,j)
      d(x,k) = (d(i,k)+d(j,k)-d(i,j))/2
      edge(x,i) = (d(i,j) + u(i) - u(j))/2
      edge(x,j) = (d(i,j) + u(j) - u(i))/2
      remove i,j rows/columns from d

Character based methods

Each atom, i, has some discrete data associated with it

Examples

• morphology: wings? segmented body plan? ...
• alignment: `aat-ct-g', each position is a character

Character-based tree reconstruction

• input: atoms with character information
• output: a tree with characters on the internal nodes with minimum edge weight

Edge weight: d(i,j)=#(differences between i and j)

NOTE: character trees are not generally additive

• additive: perfect phylogeny

Advantages

• finer resolution
• infers something about the ancestor nodes
• less reliant on molecular clock hypothesis
• some characteristics are temporally ordered already

Parsimony problem

Biological motivation: each difference is an evolutionary event

Rarely should evolutionary events happen to two different ancestors

Most believable answer has fewest events (can be rephrased as a max likelihood problem)

Violations/confounding circumstances:

• convergent evolution
• non-evolutionary gene transfer
• mistake in character assessment

Two sorts of variables under consideration:

• Tree topology
• Internal characters

Small parsimony problem

This sub-problem isn't hard

• input: a tree topology on the atoms
• output: characters at each internal node

Large parsimony problem

This problem is hard

• input: the atoms
• output: the best scoring tree topology

Typical example

1. Form initial guess by NJ
2. Score the topology with small parsimony
3. For each internal edge, evaluate local change:
4. Take the best one and go to 2

Variations:

• more clever ways to get tree topology
• exhaustive search
• monte carlo, simulated annealing, etc