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Embeddability of Kimura 3ST Markov matrices
Jordi Roca-Lacostena1, Jesús Fernández-Sánchez1
1Departament de Matemátiques, Universitat Politécnica de Catalunya, Spain.
Journal of Theoretical Biology
|February 21, 2018
Summary
This study analyzes Kimura 3ST Markov matrices, revealing how eigenvalues determine embeddability and calculating their volume. It demonstrates mutation rates are not always identifiable from substitution probabilities.
Area of Science:
- Mathematical Biology
- Evolutionary Genetics
- Matrix Theory
Background:
- Markov matrices are fundamental in modeling evolutionary processes.
- The Kimura 3ST model is a standard for DNA sequence evolution.
- Understanding matrix embeddability is crucial for evolutionary modeling.
Purpose of the Study:
- To characterize the embeddability of generic Kimura 3ST Markov matrices using eigenvalues.
- To compute the relative volume of these matrices within the broader space of Markov matrices.
- To investigate the identifiability of mutation rates from substitution probabilities.
Main Methods:
- Eigenvalue analysis of Kimura 3ST Markov matrices.
- Volume computation within the space of Markov matrices.
- Illustrative examples for identifiability and symmetry analysis.
Main Results:
- Embeddability of Kimura 3ST Markov matrices is characterized by their eigenvalues.
- The relative volume of these matrices can be computed.
- Mutation rates are generally not identifiable from substitution probabilities.
- Symmetries in mutation probabilities do not imply symmetries in mutation rates.
Conclusions:
- Eigenvalue-based characterization provides insights into Kimura 3ST Markov matrix structure.
- The study quantifies the space occupied by these matrices.
- Identifiability issues highlight limitations in inferring evolutionary parameters from sequence data.
