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Decoding Natural Behavior from Neuroethological Embedding

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The model-specific Markov embedding problem for symmetric group-based models.

Muhammad Ardiyansyah¹, Dimitra Kosta², Kaie Kubjas³

¹Department of Mathematics and Systems Analysis, Aalto University, Espoo, Finland.

Journal of Mathematical Biology

|September 9, 2021

Summary

We introduce model embeddability for phylogenetic models, defining conditions for Markov matrices derived from rate matrices within specific model structures. This research provides criteria for model embeddable matrices, with applications to synthetic DNA models.

Keywords:

Embedding problem Evolutionary models Group-based models Markov generator Markov matrix

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Area of Science:

Computational Biology
Probability Theory
Phylogenetics

Background:

The embedding problem in probability theory involves Markov matrices as matrix exponentials of rate matrices.
Phylogenetic models often assume specific structures for these rate matrices.

Purpose of the Study:

To define and characterize 'model embeddability' for Markov matrices within structured phylogenetic models.
To establish necessary and sufficient conditions for model embeddability in symmetric group-based models.

Main Methods:

Investigating the relationship between Markov matrices and their rate matrices under model constraints.
Analyzing eigenvalues of symmetric group-based matrices to derive embeddability conditions.
Applying the developed criteria to specific models, such as hachimoji models.

Main Results:

A characterization of model embeddable Markov matrices for symmetric group-based phylogenetic models is provided.
Necessary and sufficient conditions for model embeddability are derived using matrix eigenvalues.
The study demonstrates an application to eight-state hachimoji models for synthetic DNA.

Conclusions:

The developed criteria for model embeddability offer a precise way to classify matrices within structured phylogenetic models.
This work facilitates the computation of volumes for sets of model embeddable Markov matrices.
The findings are particularly relevant for synthetic DNA modeling and evolutionary studies.