Maximum likelihood (ML) is increasingly used as an optimality criterion for selecting evolutionary
trees, but finding the global optimum is a hard computational task. Because no
general analytic solution is known, numeric techniques such as hill climbing or expectation
maximization (EM), are used in order to find optimal parameters for a given tree. So far,
analytic solutions were derived only for the simplest model—three taxa, two state characters,
under a molecular clock. Four taxa rooted trees have two topologies—the fork (two subtrees
with two leaves each) and the comb (one subtree with three leaves, the other with a single
leaf). In a previous work, we devised a closed form analytic solution for the ML molecular
clock fork. In this work, we extend the state of the art in the area of analytic solutions
ML trees to the family of all four taxa trees under the molecular clock assumption. The
change from the fork topology to the comb incurs a major increase in the complexity of the
underlying algebraic system and requires novel techniques and approaches. We combine
the ultrametric properties of molecular clock trees with the Hadamard conjugation to derive
a number of topology dependent identities. Employing these identities, we substantially
simplify the system of polynomial equations. We finally use tools from algebraic geometry
(e.g., Gröbner bases, ideal saturation, resultants) and employ symbolic algebra software to
obtain analytic solutions for the comb. We show that in contrast to the fork, the comb has
no closed form solutions (expressed by radicals in the input data). In general, four taxa
trees can have multiple ML points. In contrast, we can now prove that under the molecular
clock assumption, the comb has a unique (local and global) ML point. (Such uniqueness
was previously shown for the fork.)
