Genome rearrangements as double coset Markov chains

Area of Science:

• Computational Biology
• Mathematical Biology
• Genomics

Background:

• Permutation groups are essential for modeling genome rearrangements and evolutionary relationships.
• Double cosets offer a framework for analyzing symmetries in biological systems, particularly circular genomes with repeated genes.

Purpose of the Study:

• To establish a correspondence between circular genomes with repeated genes and double coset spaces.
• To derive formulas for the sizes and number of double cosets in specific cases.
• To utilize this representation for defining Markov chain models of genome evolution.

Main Methods:

• Modeling circular genomes with repeated genes using the double coset space S_λ \ B_n / D_n.
• Deriving formulas for the sizes of double cosets.
• Developing Markov chains based on the derived representation to simulate genome rearrangement processes.

Main Results:

• Formulas for the sizes of double cosets were derived, providing insights into genome structures.
• The size of double cosets corresponds to the probability distribution of genomes under uniform sampling.
• A representation was established for defining Markov chains that model key genome rearrangement events.

Conclusions:

• The developed framework provides a robust mathematical model for studying genome evolution.
• The derived formulas and Markov chains offer new tools for computational genomics and evolutionary biology research.
• This work bridges abstract algebra and biological applications, enhancing our understanding of genomic diversity and evolutionary dynamics.