THE SHAPE OF THE ONE-DIMENSIONAL PHYLOGENETIC LIKELIHOOD FUNCTION.

Area of Science:

• Computational Biology
• Phylogenetics
• Mathematical Biology

Background:

• Phylogenetic likelihood models are fundamental to inferring evolutionary relationships.
• One-dimensional likelihood functions, derived by optimizing all but one parameter, are key to understanding model behavior.
• The shape of these functions impacts the reliability of phylogenetic inference.

Purpose of the Study:

• To introduce a mathematical framework for characterizing one-dimensional phylogenetic likelihood functions.
• To identify conditions guaranteeing at most one stationary point (maximum likelihood) for these functions.
• To investigate the complexity of these functions, particularly under the Kimura 2-parameter model.

Main Methods:

• Developed an algebraic framework analyzing frequency patterns and polynomial representations of likelihood functions.
• Derived conditions for unimodal one-dimensional likelihood functions.
• Constructed examples for the Kimura 2-parameter model demonstrating multimodal likelihood functions.

Main Results:

• Established conditions for simple phylogenetic models (e.g., Jukes-Cantor, Felsenstein 1981) that ensure at most one stationary point.
• Demonstrated that the Kimura 2-parameter model can yield one-dimensional likelihood functions with multiple stationary points.
• Proved that these functions under the Kimura 2-parameter model are dense in a space of continuous functions.

Conclusions:

• One-dimensional likelihood functions can be more complex than assumed in standard phylogenetic algorithms.
• The Kimura 2-parameter model exhibits complexities that require careful consideration in phylogenetic inference.
• The developed mathematical framework provides tools to analyze and understand these complexities.