The Pfaffian Structure of CFN Phylogenetic Networks.

Area of Science:

• Computational Biology and Bioinformatics
• Algebraic Statistics
• Phylogenetics

Background:

• Algebraic techniques have historically advanced phylogenetic identifiability and reconstruction.
• Phylogenetic invariants are crucial for understanding evolutionary models.
• The Cavender-Farris-Neyman (CFN) model is a key model in phylogenetics.

Purpose of the Study:

• To characterize phylogenetic invariants for the CFN model on phylogenetic networks.
• To develop a description of invariants useful for network inference.
• To simplify the representation of invariants for computational efficiency.

Main Methods:

• Studied the ideal of phylogenetic invariants for the CFN model on sunlet networks.
• Utilized Pfaffians to parameterize an affine open patch of the CFN sunlet model.
• Established an isomorphism to a determinantal variety and derived an explicit Gröbner basis.

Main Results:

• The parameterization of the CFN sunlet model factors through skew-symmetric matrices via Pfaffians.
• The affine patch is isomorphic to a determinantal variety with a simplified Gröbner basis.
• This basis uses $\left(\begin{array}{c}n\\ 2\end{array}\right)$ coordinates instead of $2^n$ .

Conclusions:

• Sunlet networks with at least 6 leaves are identifiable using the derived polynomials.
• Extensive simulations demonstrate the polynomials' accuracy in inferring correct networks from DNA data.
• This algebraic approach offers a computationally efficient method for phylogenetic network inference.