Introduction to Bioinformatics

1
Introduction to Bioinformatics

• Phylogenetics
• Part II
• Distance-Based Methods

2
Distance Matrix

• (Evolutionary) Distance
• Many possible measures
• Fraction of sites that differ between two
  sequences
• of changes needed to convert one sequence to
  another (count mismatches, substitution models,
  )
• Distance Matrix
• Matrix of pairwise distances between all
  sequences
• Used to generate tree
• Varies with construction method, distance metric

3
Distance in Phylogenetic Tree

• Distances are ultrametric if
• Same rate of change on all branches in tree (rare
  in practice)
• All leaves equidistant from root
• Also known as a molecular clock
• Distance matrix must satisfy the following
  3-point condition
• For any three leaves i, j, k, distances dij, dik,
  djk
• two of three distances are equal and third

4
Distance in Phylogenetic Tree

• Distances are additive if
• Distance between any two leaves i j on tree
  sum of lengths of edges connecting i j
• Distance matrix must satisfy the following
  4-point condition
• For any four leaves i, j, k, m, two of the
  distances dijdkm, dikdjm, dimdjk are equal and
  greater than the third
• In fact, the difference is 2 x the length of the
  bridge edge(s)

5
UPGMA

• UPGMA (Unweighted Pair Group Method using
  Arithmetic Averages) Sokal Michener 1958
• Algorithm
• 1. Find pair of sequences (or clusters) A, B with
  smallest distance dAB
• 2. Insert join for A, B at tree height ½ dAB.
  A and B thus form a new cluster.
• 3. Update distance of any other sequence/cluster
  X to new cluster as ½ (dAX dBX)
• 4. Repeat until all sequences / clusters joined
• 5. Produces rooted tree
• Assumptions
• Distances for tree are ultrametric
• Branch lengths for 2 leaves same after join
• Distances for tree are additive

similar algorithms vary at this step
6
UPGMA Example

• Given sequences
• Build distance matrix

7
UPGMA Example

• Form clusters
• Next step?

8
Transformed Distance Method

• Weakness of UPGMA
• Assume constant evolution rate across lineage
• Example Consider sequences A, B, C, and D is
  Figure 4.5. UPGMA cluster A and C first.
• Transformed Distance Method J. Farris, 1977
• Take advantage of the power of an outgroup
• Similar to UPGMA except for the distance matrix
• Algorithm
• Select an outgroup D
• Transformed distance between i and j
• dij (dij diD djD)/2 (?dkD)/n
• where n is ingroups
• Run UPGMA with matrix of dij

9
Transformed Distance Method

• Example
• Select D as the outgroup
• Calculate transformed distance
• (?dkD)/n (dAD dBD dCD)/3
• (12 15 10)/3 37/3
• dAB (dAB dAD dBD)/2 37/3
• (9 12 15)/2 37/3 10/3
• dAC (dAC dAD dCD)/2 37/3
• (8 12 10)/2 37/3 16/3
• dBC (dBC dBD dCD)/2 37/3
• (11 15 10)/2 37/3 16/3
• Construct new distance matrix
• Run UPGMA

10
Transformed Distance Method

• Example (contd)
• How do you compute the length of a lineage?

11
Neighbor-Joining Method

• Goal
• Join closest neighbors (nodes w / same parent) in
  tree
• Avoids problem with UPGMA when rates of change
  differ
• Example
• Closest leaves not neighbors in correct tree, but
  joined first by UPGMA (see previous example)
• Assumptions
• Rate of change can differ
• Branch lengths may differ after join
• Branch lengths for tree are additive

12
Neighbor-Joining Method

• Approach
• To find closest pair of neighbors
• Reduce branch length for a node by
  (approximately) the average distance of the node
  from all other nodes
• Find smallest distance between nodes (after
  reduction)
• Definitions
• For all pairs of nodes A B in set of all nodes
  L, let
• dA,B distance between A,B
• RX ? dX,N where N ? L (total distance from X to
  all N)
• rX RX / (n 2),where n of nodes
• (normalized divergence from X to all other nodes)
• QA,B (n 2) dA,B (RA RB) (rate-corrected
  distance)
• Key property - 2 nodes w/ minimum Q are always
  neighbors!

13
Neighbor-Joining Method

• Algorithm Saitou Nei 1987, Studier Keppler
  1988
• 1. Begin with star tree all sequences as nodes
  in L
• 2. Find pair of nodes A B ? L with minimum QA,B
• 3. Create insert new join (node K) w/ branch
  lengths
• dA,K ½ (dA,B rA rB)
• dB,K ½ (dA,B rB rA)
• 4. For remaining nodes C ? L, update distance to
  K as
• dK,C ½ (dA,C dB,C dA,B)
• 5. Insert K and remove A, B from L
• 6. Repeat steps 2-5 until only two nodes left

K
A
B
14
Neighbor-Joining Method

• Example