Highways and byways in group-theoretic genome space Attila - - PowerPoint PPT Presentation

Highways and byways in group-theoretic genome space

Attila Egri-Nagy, joint work with Andrew Francis and Volker Gebhardt

Centre for Mathematics Research, School of Computing, Engineering and Mathematics University of Western Sydney

Phylomania 2013

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Questions

Is the distance a good enough measure? Can we use the number of shortest evolutionary paths? Maybe the ‘shape’ how these paths are put together...

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Biology→Math

Genome→ permutations 5 3 9 4 2 1 8 7 6 Genomic distance→ Length of geodesic words

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Groups, generator sets

Let G be a group with generators S = {s1, . . . , sn}. S∗ is the set of all finite sequences, words of the elements of S. The group element realized by the word w is denoted by w, thus w ∈ S∗ and w ∈ G.

Example

S =

• s1 = (1, 2), s2 = (2, 3)
• s1s2s1s2 = (1, 2)(2, 3)(1, 2)(2, 3) = (1, 2, 3)

So s1s2s1s2 = (1, 2, 3). sequences of generators ⇐ ⇒ sequences of events

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Cayley graph

The Cayley graph Γ(G, S) of G with respect to the generating set S is the directed graph with group elements as nodes and the labeled edges encoding the action of G on itself. Thus g

s

− → gs is an edge.

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Cayley graph of S3

Example

S =

• s1 = (1, 2), s2 = (2, 3)
• (2,3)

(1,2,3) 1 (1,3) 2 (1,3,2) 1 (1,2) 2 () 2 1

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Cayley graph of S3 – different generators

Example

S =

• s1 = (1, 2), s2 = (2, 3), s3 = (3, 1)
• (2,3)

(1,2,3) 1 (1,3,2) 3 (1,3) 2 1 (1,2) 3 2 () 2 3 1

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Geodesic distance, shortest path

The geodesic distance defined by dS(g1, g2) = |u|, where u is a minimal length word in S∗ with the property that g1u = g2 also denoted by g1

u

− → g2, and u is called a geodesic word. GeoS(g1, g2) is the set of all geodesic words from g1 to g2. What is GeoS(g1, g2)?

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A partial order defined by the geodesics

Due to a translation principle we can simpy write ℓ(g) instead of d(1, g). Similarly, we use Geo(g) instead of Geo(1, g).

Definition

For group elements g1, g2 ∈ G = S we write g1 ≤ g2 if ∃w = uv ∈ S∗ such that w = g2, u = g1, w ∈ Geo(g2), i.e. there is a geodesic from the identity to g2 and g1 is on it. Also called the prefix order, or weak order for Coxeter groups.

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Intervals

With the partial order closed intervals are defined in the obvious way [1, h] := {g ∈ G | 1 ≤ g ≤ h}

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Ranked poset

R0 R4 R1 R5 R2 R6 R3 R7 (0, 0) (4, 3) The rank-sets of the interval

• (0, 0), (3, 4)
• in Z × Z.

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Ranked poset

R0 R4 R1 R5 R2 R6 R3 R7 (0, 0) (4, 3)

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Length and size

In general there is no connection. (0, 0) (0, 4) y (0, 0) (2, 2) y x In Z2 two group elements with same length can have intervals of different size.

• [(0, 0), (0, 4)]
• = 5,
• [(0, 0), (2, 2)]
• = 9.

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Interval lattices in S3 =

• (1, 2, 3), (1, 2)
• AF,e-n@,VG (UWS CRM)

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S3 =

• (1, 2), (2, 3)
• AF,e-n@,VG (UWS CRM)

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S4 =

• (1, 2), (2, 3), (3, 4)
• AF,e-n@,VG (UWS CRM)

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S4 =

• (1, 2), (2, 3), (3, 4), (1, 4)
• AF,e-n@,VG (UWS CRM)

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Is it a lattice?

An obvious mathematical but biologically not so relevant question. A minimal counterexample would be:

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Trying with involutions

a b b c c a a b ab = bc = ca, ac = ba = cb. But since they are involutions, ba = cb = ⇒ c = bab

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Trying it with 2 generators

Minimal counterexamples a b a b b a a2 = b2, ab = ba For instance, a = (3, 4, 5), b = (1, 2)(3, 4, 5).

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C4 × C2 =

• (3, 4, 5, 6), (1, 2)(3, 4, 5, 6)
• (), (1, 2)
• AF,e-n@,VG (UWS CRM)

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Sperner property?

Sperner property: no antichain is bigger than the size of the maximal rank-set. Do these intervals have the Sperner property?

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Sperner property?

Sperner property: no antichain is bigger than the size of the maximal rank-set. Do these intervals have the Sperner property? NO. s4s3s1 = s4s1s3 = s3s1s2 = s1s3s2

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Anti-chains

Do anti-chains give the number of paths?

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Anti-chains

Do anti-chains give the number of paths? NO.

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Possible equivalence relations

The ultimate goal is to find equivalence classes of group elements.

1 Same length: ℓ(g1) = ℓ(g2). 2 Same ‘width’: | Geo(g1)| = | Geo(g2)|. Probably the most decisive

property for the biological application.

3 Same profile. 4 Same interval. AF,e-n@,VG (UWS CRM) Highways and Byways Phylomania 2013 24 / 32

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S4 =

• (1, 2), (2, 3), (3, 4), (1, 4)
• 1

3 2 3 1 4 1 3 3 1 (1,2,4,3) [ 0, 3 ] [ 3, 4 ] [ 4, 3 ] [ 3, 0 ] 1 2 4 2 4 4 4 2 2 3 (1,3,2,4) [ 0, 3 ] [ 3, 4 ] [ 4, 3 ] [ 3, 0 ] 1 2 3 4 4 2 4 4 2 2 (1,4,2,3) [ 0, 3 ] [ 3, 4 ] [ 4, 3 ] [ 3, 0 ] 1 3 4 3 1 2 1 3 1 3 (1,3,4,2) [ 0, 3 ] [ 3, 4 ] [ 4, 3 ] [ 3, 0 ]

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S4 =

• (1, 2), (2, 3), (3, 4), (1, 4)
• 1 2

3 4 4 1 4 1 2 2 3 3 (1,2,3,4) [ 0, 4 ] [ 4, 4 ] [ 4, 4 ] [ 4, 0 ] 1 4 2 3 2 1 1 2 3 3 4 4 (1,4,3,2) [ 0, 4 ] [ 4, 4 ] [ 4, 4 ] [ 4, 0 ] 1 2 3 4 2 1 2 1 4 3 3 4 (1,3) [ 0, 4 ] [ 4, 4 ] [ 4, 4 ] [ 4, 0 ]

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n = 5 all inversions circular linear length 4 7 11 [length,width] 7 14 30

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Number of paths

Assuming that we have an efficient algorithm for calculating the distance, we can also calculate the interval. For biological applications it is probably enough to estimate the interval by partially calculating it.

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Algorithm 1: Constructing the graded interval [g, h]. input : g, h ∈ G, S generator set, d distance function

• utput: [g, h] interval, Ri rank-sets

GradedInterval (g, h, S, d): n ← d(g, h); R0 ← {g}; foreach i ∈ {1, . . . , n} do Ri ← ∅; foreach g′ ∈ Ri−1 do foreach s ∈ S do if d(g′s, h) = n − i then Ri ← Ri ∪ g′s;

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TODO list

Study individual generating sets. (since no grand theory is available) Find the right interpretation in order to modify the distance function.

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Thank You!

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