Distributions on bicoloured evolutionary trees : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University

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DISTRIBUTIONS ON BICOLOURED EVOLUTIONARY TREES

A thesis p resented in partial fulfilment of the requirements for the degree of

Doctor of Philosophy in Mathematics at

Massey University

M ichael Anthony Steel February, 1 989

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Massey University Library

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ABSTRACT

A central and challenging problem in contemporary biology is how to accurately reconstruct evol utionary trees from DNA sequence data.

This thesis addresses three themes fro m this endeavour -- comparison , consistency and confidence i ntervals -- by analysing distributions arising from phylogenetic trees.

Toward the first theme, the d istribution of the sym metric d ifference

metric o n pairs of binary and phylogenetic trees is studied , and a n u mber of n ew resu lts o btained. These theorems, as well as a result o n another tree metric answer previous conjectures i n this area. Also u nder the theme of comparison , we analyse distributions on bicoloured trees arising from the p rinciple of parsimony. A streaml ined proof is given of an

eleg ant theorem which allows an efficient com parison of how m uch better a m axim u m parsi mony tree fits given data than a randomly-chosen tree. A dual distri bution , where the tree is fixed and the data varies is also

analysed, answering a recent u nsolved problem.

We then consider the theoretical accuracy of tree-bu i lding methods, concentrating on the statistical property of consistency. Under a simple stochastic model on bicoloured trees, conditio ns for the consistency of frequently-used methods based o n parsimony and compatibility are examined. lt is shown that even i n "best possible" conditions both methods can be inconsistent, though a strong sufficient condition for compatibility is give n . The analysis is extended for a molecular clock.

Finally, p rocedu res are described for placing confidence i ntervals aro u nd phylogenies, and limitations on the sort of confidence intervals possible are given . Ways to efficiently implement these procedu res are the n considered -- in particular, approximate methods, applications t o sets of taxa of size fou r, and simplifications u nder a molecu lar clock.

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The rate that sequence data must grow as a fu nction of the n u mber of taxa for co nfide nce inte rvals to converge to a single tree is also considered.

The arg u ments i n this thesis are primarily combinatorial and stochastic.

I n the hope that their i mpl ications wil l also i nterest biologists, som e

space h as been given t o motivating and explaining t h e biological relevance of the resu lts presented.

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Comb i nat o ri al

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Stochas t i c

Chart i l l u strati n g t h e f l ow of re s u l ts f rom on e s e cti on to anoth e r

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Introduction

Notation/Table of Symbols

--trees

--resolved subtrees --forests

--bicoloured trees Section Two:

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--distribution generating functions --distribution on pairs of trees

--absolute (non-asymptotic) inequalities --asymptotic range of the distribution --mon oton icity

--description of the metric from below --distribution on PT(n)

--co mparison with other metrics

--induced subtrees and minimal similarity --span n i ng sets

--consensus trees --efficiency

--constraints with two colours --lower bou nds on i nformation loss

--vector spaces of edge sets

--(weakly-) con necting trees and forests --distributions arising fro m parsimony --path/edge dual ity

3

5 I I 1 5 1 8

29 34 38 43 46 48 5 1 54

56 62 66 68

73 76

82 84 90 1 00

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--stochastic p reliminaries --the model

--central o bservations --invariants

--partitio n probabi lities

--convergence and consistency

--consistent recovery of trees from dissimilarities --consistency of parsimony and compatibility --sufficient conditions

--consistency u nder a molecular clock

--confidence inte rvals --approximate methods --exact solutions

--efficiency (I) --efficiency (11)

--a X2 test (molecu lar clock) Appen d ix

References

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