Abstract
This paper introduces a new way to define a genome rearrangement distance, using the concept of mean first passage time from probability theory. Crucially, this distance provides a genuine metric on genome space. We develop the theory and introduce a link to a graph-based zeta function. The approach is very general and can be applied to a wide variety of group-theoretic models of genome evolution.
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ARF acknowledges the support of the Australian Research Council via Discovery Project DP180102215.
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Appendices
Appendix A. Evolutionary models
Table 2 shows a range of group-based models that this approach can be applied to. Each corresponds to a particular group and generating set.
Appendix B. Cayley graphs for \(S_4\)
The Cayley graphs of \(S_4\) with standard and with circular generators are shown for reference in Figs. 4 and 5 respectively.
Cayley graph of \(S_4\) with standard Coxeter generators. Edges are coded dashed blue for multiplication on the right by \((1\ 2)\), dotted red for \((2\ 3)\), and black for \((3\ 4)\) (group action is also on the right) (color figure online)
Cayley graph of \(S_4\) with circular generators. Edges are coded dashed blue for multiplication on the right by \((1\ 2)\), dotted red for \((2\ 3)\), black for \((3\ 4)\), and dash-dotted green for \((1\ 4)\) (group action is also on the right) (color figure online)
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Francis, A.R., Wynn, H.P. A mean first passage time genome rearrangement distance. J. Math. Biol. 80, 1971–1992 (2020). https://doi.org/10.1007/s00285-020-01487-w
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DOI: https://doi.org/10.1007/s00285-020-01487-w