Pairwise Compatibility

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Pairwise Compatibility

The first step in working with compatibility, is parallel to the small parsimony problem (see 8.2.1): Given a tree T with labeled leaves, find the best compatibility score that can be achieved for that tree, i.e., the maximum number, over all possible labelings of internal nodes, of characters compatible with the fully-labeled tree. This can be done easily using Fitch's algorithm (see 8.2.1).

The more interesting problem here is of course that of ``large compatibility'' - finding the best phylogeny given only the data matrix M. We shall tackle this problem through the notions of pairwise compatibility and joint compatibility.

$\begin{definition}Two characters $c_1$\space and $c_2$\space are said to be {\em... ...that both $c_1$\space and $c_2$\space are compatible with $T$ . \end{definition}$

$\begin{definition}Characters $c_1, \ldots, c_k$\space are said to be {\em jointl... ...space such that $\forall i: c_i$\space is compatible with $T$ . \end{definition}$

We will present two theorems. The first, by Wilson [16], identifies pairwise compatible characters:

$\begin{theorem}% latex2html id marker 241 {Pairwise Compatibility Test:}\\ For ... ...malPhylogeny}; then $PC(c,c')$\space iff $S_{cc'} \ne \{0,1\}^2$ . \end{theorem}$

$\begin{proof}% latex2html id marker 250 Assume $S_{cc'} \ne \{0,1\}^2$ . Then th... ... direction is simple, and is left as an exercise to the reader. \qed \end{proof}$

**Figure 8.6:** A schematic description of a tree that is compatible with two characters, having 3 combined values (see proof of theorem 8.1).
$\includegraphics{lec08_figs/paircomptree.ps}$

The next, somewhat surprising, theorem by Estabrook et al. [7] identifies jointly compatible sets of binary characters:
$\begin{theorem}{Pairwise Compatibility Theorem:}\\ All binary characters in a s... ...\space are jointly compatible iff $\forall c,c' \in S, PC(c,c')$ . \end{theorem}$
Note that the theorem does not hold for characters that are not binary. We will not present a proof for this theorem.

So the problem of ``large compatibility'' is reduced to the problem of finding the largest jointly compatible set of characters, which amounts to finding a maximum clique in the pairwise-compatibility graph, defined as:

$\begin{displaymath}G=(V,E);\ V=\left\{v_1,\ldots,v_m\right\};\ E=\left\{(v_i,v_j): PC(c_i,c_j)\right\}\end{displaymath}$

This seems to be of no great help, because as we know, finding a maximum clique in a graph is an NP-hard problem. However, there are algorithms, such as Bron and Kerbosch's [1] Branch-and-Bound clique-finding algorithm, which seem to work very well with biological data. All in all, compatibility methods usually run faster than parsimony methods for the same data.
Perfect Phylogeny

The problem of perfect phylogeny (also called full compatibility) is to decide if all the characters are jointly compatible. For binary characters the problem is easy: it is solved by checking if the pairwise-compatibility graph is a complete graph. For non-binary characters the problem is NP-hard (it relates to some properties of chordal graphs).

Next: Finding the Tree Up: Compatibility Previous: Compatibility and Parsimony

Peer Itsik
2001-01-01