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Enumeration of ancestral configurations for matching gene trees and species trees
- Publication Year :
- 2016
-
Abstract
- Given a gene tree and a species tree, ancestral configurations represent the combinatorially distinct sets of gene lineages that can reach a given node of the species tree. They have been introduced as a data structure for use in the recursive computation of the conditional probability under the multispecies coalescent model of a gene tree topology given a species tree, the cost of this computation being affected by the number of ancestral configurations of the gene tree in the species tree. For matching gene trees and species trees, we obtain enumerative results on ancestral configurations. We study ancestral configurations in balanced and unbalanced families of trees determined by a given seed tree, showing that for seed trees with more than one taxon, the number of ancestral configurations increases for both families exponentially in the number of taxa n. For fixed n, the maximal number of ancestral configurations tabulated at the species tree root node and the largest number of labeled histories possible for a labeled topology occur for trees with precisely the same unlabeled shape. For ancestral configurations at the root, the maximum increases with \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$ k_0^n$$ \end{document}, where \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${{k_0}} \approx \bf1.5028$$ \end{document} is a quadratic recurrence constant. Under a uniform distribution over the set of labeled trees of given size, the mean number of root ancestral configurations grows with \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\sqrt {\bf 3 / 2} { {\bf ( 4 / 3 ) ^n}}$$ \end{document} and the variance with ∼\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${{\bf 1.4048 ( 1.8215 )} ^n}$$ \end{document}. The results provide a contribution to the combinatorial study of gene trees and species trees.
- Subjects :
- 0106 biological sciences
0301 basic medicine
K-ary tree
Genetic Speciation
Biology
010603 evolutionary biology
01 natural sciences
Coalescent theory
Evolution, Molecular
Combinatorics
03 medical and health sciences
Sequence Homology, Nucleic Acid
Genetics
Animals
Quantitative Biology::Populations and Evolution
Quantitative Biology - Populations and Evolution
Molecular Biology
Phylogeny
Research Articles
Seed tree
Models, Genetic
Phylogenetic tree
Populations and Evolution (q-bio.PE)
Conditional probability
Quantitative Biology::Genomics
Computational Mathematics
Tree (data structure)
030104 developmental biology
Taxon
Genes
Computational Theory and Mathematics
FOS: Biological sciences
Modeling and Simulation
Tree rearrangement
Algorithms
Subjects
Details
- Language :
- English
- Database :
- OpenAIRE
- Accession number :
- edsair.doi.dedup.....9f0231745ab13fa7f1a8c7455ad99d86