Four-Taxon Problem

The "Four-Taxon Problem" and the "Three-Taxon Statement"

      Among four taxa A, B, C, & D, there are exactly three hypotheses of relationship:
            either A is most closely related to B, or to C, or to D
            (This statement covers all possible relationships between the four taxa)
      We want to evaluate the hypotheses in the form of the Three-Taxon Statement:
       "X and Y are more closely related to each other than either is to Z"
            The alternative hypotheses can be shown as networks with four branches and an internode

If (for example), A is most closely related to B
       A & B will share characters inherited from their common ancestor
       These changes will occur on internode between the pairs

In the four-taxon problem, seven types of SNP distributions can be identified
    (for a detailed analysis, see Notes on Parsimony Analysis)

Types 1 - 4 are uninformative:
           They give no information about relationships, because
   all hypotheses require the same number of changes [see HOMEWORK],
                   so none is more parsimonious than the others.
            Position 1 is invariant, and is the most common type.
            Position 2 requires a single change in one taxon, in all hypotheses

Types 5, 6 & 7 are informative:
       Two taxa share one state, other two share another
                They give information about relationships,
                because one hypothesis requires fewer changes than the other two
                  & is therefore more parsimonious than the others

   Position 5 indicates that A & B are most closely related:
      The first hypothesis explains the distribution of SNPs with a single change,
         the latter two require two changes each
     The first hypothesis is a more parsimonious explanation of the data than the others.

By the same logic:
   Position 6 indicates that A & C are most closely related.
   Position 7 indicates that A & D are most closely related.

Homework: 1) Explain why Positions 2, 3, & 4 are uninformative.
(Hint: how many changes are required by each, for the three hypotheses
2) For the three networks above, sketch the changes required by sites of Positions 6 & 7