SCNSMAA F03 Phylogenetic Distance and Coxeter Groups 1

Phylogenetic Distance

andCoxeter Groups

David J. Hunter

Department of Mathematics, Westmont College

SCNSMAA F03 Phylogenetic Distance and Coxeter Groups 2

The Phylogeny Problem

Given genetic data from a set of related organisms,

reconstruct the evolutionary tree.

SCNSMAA F03 Phylogenetic Distance and Coxeter Groups 3

Distance Based Methods

Definition: The phylogenetic distance between two

genomes is the number of evolutionary events (e.g.,

mutations) that occurred in the transition from one

genome to the other.

Given reasonably accurate estimates for the pairwise

phylogenetic distances among a set of genomes, methods

exist that will reconstruct the true topology of the

evolutionary tree. (See Saitou and Nei, 1987.)

Many estimators exist: breakpoint distance, inversion

distance, EDE, IEBP, etc.

SCNSMAA F03 Phylogenetic Distance and Coxeter Groups 4

Inversions

An inversion is a mutation of a chromosome where a

sequence of genes is broken off and glued back in reverse

order. The directionality of the inverted sequence is

toggled.

1 2 3 4 5 6 7 8 9 10 11 1 2 −5 −4 −3 6 7 8 9 10 11

SCNSMAA F03 Phylogenetic Distance and Coxeter Groups 5

Algebraic Model

Let n be the number of genes. The set of possible

genomes can be modeled as the group Bn of signed

permutations. Secretly,

Bn∼= Z/2 o Sn

but it is more convenient to view Bn as a subgroup of

O(n).

SCNSMAA F03 Phylogenetic Distance and Coxeter Groups 6

Generators of Bn:

1. . .

0 1. . .

1 0. . .

1

,

1. . .

−1. . .

1

Note: These are reflections in Rn.

SCNSMAA F03 Phylogenetic Distance and Coxeter Groups 7

Finite groups of reflections are Coxeter groups, so they

have the following presentation:

Generators: s1, s2, . . . , sn

Relations: (sisj)o(sisj) = 1

Many results in Coxeter group theory deal with this

presentation.

Key Fact: In the case of Bn the set {s1, s2, . . . , sn}contains all the inversions of length 2.

Therefore, word length l(α) can be used as a phylogenetic

distance estimator (related to inversion distance).

SCNSMAA F03 Phylogenetic Distance and Coxeter Groups 8

Since Bn is a finite reflection group, we have

Theorem: [Humphreys] The Poincare polynomial

Wn(t) of Bn has the following factorization, where l(α) is

word length:

Wn(t) =∑

α∈Bn

tl(α) =n∏

i=1

t2i − 1

t− 1

i.e., Wn(t) is a generating function for P (n, k), the

number of words of length k. For example, when n = 4,

W4(t) = 1 + 4t + 9t2 + 16t3 + 24t4 + 32t5 + 39t6

+44t7 + 46t8 + 44t9 + 39t10 + 32t11

+24t12 + 16t13 + 9t14 + 4t15 + t16

SCNSMAA F03 Phylogenetic Distance and Coxeter Groups 9

Distribution of word length for elements of B10

SCNSMAA F03 Phylogenetic Distance and Coxeter Groups 10

It is possible to solve for the coefficients of Wn(t) for

k ≤ 2n:

P (n, k) =

n + k + 1

k

+

∑i≥1

(−1)i

n + k − 1− i(3i− 1)

k − i(3i− 1)

+

n + k − 1− i(3i + 1)

k − i(3i + 1)

where(

ij

)= 0 if j < 0.

SCNSMAA F03 Phylogenetic Distance and Coxeter Groups 11

Computer simulations indicate that inversion distance

tends to underestimate the true phylogenetic distance.

(See, for example, Moret, Tang, Wang, Warnow, 2001.)

Actual Number of Inversions (Simulated)

Inve

rsio

n D

ista

nce

SCNSMAA F03 Phylogenetic Distance and Coxeter Groups 12

Questions

• Is it possible to quantify how bad the estimate is

(and correct it)?

• Are there other phylogenetic distance estimators

lurking in Bn?

• What about the unsigned case?

• ???

SCNSMAA F03 Phylogenetic Distance and Coxeter Groups 13

References

• James E. Humphreys, Reflection Groups and Coxeter

Groups, Cambridge Study #29, 1990.

• B. M. E. Moret et. al., Steps toward accurate

reconstructions of phylogenies from gene-order data,

LNCS 2149, 2001.

• N. Saitou and M. Nei, The Neighbor-joining method:

A new method for reconstructing phylogenetic trees,

Mol. Biol. Evol., 1987.