Distance and Parsimony methods (Part 2)

Parsimony-based methods

• Character-based data: 4 nucleotides ACGT or 20 aminoacids => matrix of aligned characters
• It does not rely on models of evolution
• One seeks the tree that minimizes the amount of evolutionary change required to explain the data
• Justification:
  • Ockham’s razor: when two hypothesis provide equally valid explanations for a phenomenon, the simpler one should always be preferred
  • More character-state changes imply a more complex hypothesis because homoplasy (sharing identical character states that cannot be explained by inheritance from a common ancestor) is an ad hoc hypothesis
• Parsimony represents a useful fall back method when model-based methods cannot be used due to computational limitations

Assumptions

• Parsimony methods are most effective when rate of evolution is slow, but this is not a necessary assumption
• Parsimony methods can perform well under high rates of evolution as long as there are no pathological inequalities (long-branch attraction: Felsenstein zone)
• The only real assumption of parsimony is independence among characters

Methodology

1. Determine the amount of character change required to explain the data by a given tree
2. Search over all possible tree topologies

• We need to be able to calculate the length of a proposed tree which is defined as the amount of character change implied by a most parsimonious reconstruction of internal nodes
• Just as in MSA, we need to have costs for substitutions (equal costs or unequal costs)

Example: Evaluate the length of the ((W,Y),(X,Z)); tree given the site:

W:G
X:C
Y:A
Z:C

• Full solution: see this YouTube video
• Note that this is only one site! We need to repeat this process for every site and add up the lengths
• More on Newick (parenthetical) format here

Just as in MSA, we cannot do this by hand and there are dynamic programming algorithms that help us (what was dynamic programming?):

• Fitch algorithm (HB Box 8.2) for equal costs
• Sankoff algorithm (HB Box 8.1) for unequal costs

Fitch algorithm

1) Root the tree in a random place (parsimony score is not affected by the root)

2) Calculate the state-set $X_i$ for each internal node $i$ corresponding the set of states that can be assigned to each node so that the minimum possible length of the subtree can be achieved. Let $L(i)$ and $R(i)$ be the left and right child descendant nodes of $i$ respectively.

2.1) Form the intersection of the two child state sets: $X_{L(i)} \cap X_{R(i)}$

2.2) If the intersection is non-empty, set $X_i$ equal to this intersection and the accumulated length for this node as the sum of the accumulated lengths for the two child nodes: $s_i=s_{L(i)}+s_{R(i)}$

2.3) If the intersection is empty, let $X_i$ be equal to the union of the two child sets: $X_{L(i)} \cup X_{R(i)}$ and set the accumulated length for this node as the sum of the accumulated lengths for the two child nodes plus one: $s_i=s_{L(i)}+s_{R(i)}+1$

Example: Evaluate the length of the ((W,Y),(X,Z)); tree given the site:

W:G
X:C
Y:A
Z:C

using the Fitch algorithm.

• Full solution: see this YouTube video
• Homework: Redo the algorithm with different root positions to verify that you get the same length

Phylogenetic inference: Maximum Parsimony (MP) tree: Step 1) Evaluate the parsimony score of a given tree (length) with Fitch algorithm. Step 2) Search the space of trees until you find the optimum.

Some downsides

• Parsimony methods have been shown to produce inconsistent trees
• Read more in Felsenstein 1978