Molecular phylogeny, the art of reconstructing the history
of a sample of sequences from the leaves of a tree, is often performed
within the framework of additive metric trees [1]. If distances between
leaves are tree-additive, the tree topology as well as the branch lengths
may be reconstructed. A very general distance concept is the so-called
`paralinear distance' [2,3], which is a function of pairwise joint
probabilities of the letters at the leaves. It is tree-additive if the
underlying substitution process is a Markov process on a tree, where the
transition matrices are allowed to vary from branch to branch.
We address two questions: 1) How are the branch lengths belonging to
the
paralinear distance related to the substitution process on the tree?
2) Can information about the substitution process (as opposed to just the
tree geometry) be inferred from pairwise joint distributions of letters
at the leaves?
The answers are: 1) The paralinear branch length may be interpreted
as the sum of relative probability fluxes, integrated over the branch,
and averaged over the forward and reverse directions of the Markov process,
where the averaging reflects our ignorance of the position of the root
[4]. 2) The Markov transition matrices across branches may not be inferred
separately, but return-trip matrices (forward and back again across a branch)
may be inferred, up to conjugacy in the case of internal branches [5].
[1] M.S. Waterman et al., J. Theor. Biol. 64 (1977),
199--217.
[2] M.A. Steel, Appl. Math. Lett. 7 (1994), 19--25.
[3] J. A. Lake, PNAS 91 (1994), 1455--1459.
[4] E. Baake, A. v. Haeseler, Theor. Pop. Biol. 55
(1999), 166--175.
[5] E. Baake, Math. Biosci. 154 (1998), 1--21.