Why Sequences?Objectives

Context Directed SwapsMathematical Results

Genome Remodeling in Developmental Time:Algorithms for Ciliates

Helen Wauck

Gustavus Adolphus College

AAAS Pacific Division Conference

June 24-27, 2012

Helen Wauck Genome Remodeling in Developmental Time: Algorithms for Ciliates

Why Sequences?Objectives

Context Directed SwapsMathematical Results

In collaboration with:Christopher Anderson, Marion Scheepers, and Marlena

Warnerrepresenting

Lewis and Clark College, Boise State University, andUniversity of Idaho

Helen Wauck Genome Remodeling in Developmental Time: Algorithms for Ciliates

Why Sequences?Objectives

Context Directed SwapsMathematical Results

Why Sequences?

Objectives

Context Directed Swaps

Mathematical Results

Helen Wauck Genome Remodeling in Developmental Time: Algorithms for Ciliates

Why Sequences?Objectives

Context Directed SwapsMathematical Results

Why Sequences?

I Sequences we are concerned with in this talk are permutationsof sequences of the form

[1, 2, 3, 4, 5, . . ., n], where n ∈ N

I Can be used to model a wide variety of natural mechanisms

I DNA sequences:I Canonical Ordering (Macronucleus): [1, 2, 3, 4, 5, 6, 7, 8]I Permutation (Micronucleus): [3, 6, 8, 5, 1, 4, 7, 2]

Helen Wauck Genome Remodeling in Developmental Time: Algorithms for Ciliates

Why Sequences?Objectives

Context Directed SwapsMathematical Results

Why Sequences?

I Sequences we are concerned with in this talk are permutationsof sequences of the form

[1, 2, 3, 4, 5, . . ., n], where n ∈ N

I Can be used to model a wide variety of natural mechanisms

I DNA sequences:I Canonical Ordering (Macronucleus): [1, 2, 3, 4, 5, 6, 7, 8]I Permutation (Micronucleus): [3, 6, 8, 5, 1, 4, 7, 2]

Helen Wauck Genome Remodeling in Developmental Time: Algorithms for Ciliates

Why Sequences?Objectives

Context Directed SwapsMathematical Results

Why Sequences?

I Sequences we are concerned with in this talk are permutationsof sequences of the form

[1, 2, 3, 4, 5, . . ., n], where n ∈ N

I Can be used to model a wide variety of natural mechanisms

I DNA sequences:

I Canonical Ordering (Macronucleus): [1, 2, 3, 4, 5, 6, 7, 8]I Permutation (Micronucleus): [3, 6, 8, 5, 1, 4, 7, 2]

Helen Wauck Genome Remodeling in Developmental Time: Algorithms for Ciliates

Why Sequences?Objectives

Context Directed SwapsMathematical Results

Why Sequences?

I Sequences we are concerned with in this talk are permutationsof sequences of the form

[1, 2, 3, 4, 5, . . ., n], where n ∈ N

I Can be used to model a wide variety of natural mechanisms

I DNA sequences:I Canonical Ordering (Macronucleus): [1, 2, 3, 4, 5, 6, 7, 8]

I Permutation (Micronucleus): [3, 6, 8, 5, 1, 4, 7, 2]

Helen Wauck Genome Remodeling in Developmental Time: Algorithms for Ciliates

Why Sequences?Objectives

Context Directed SwapsMathematical Results

Why Sequences?

I Sequences we are concerned with in this talk are permutationsof sequences of the form

[1, 2, 3, 4, 5, . . ., n], where n ∈ N

I Can be used to model a wide variety of natural mechanisms

I DNA sequences:I Canonical Ordering (Macronucleus): [1, 2, 3, 4, 5, 6, 7, 8]I Permutation (Micronucleus): [3, 6, 8, 5, 1, 4, 7, 2]

Helen Wauck Genome Remodeling in Developmental Time: Algorithms for Ciliates

Why Sequences?Objectives

Context Directed SwapsMathematical Results

Project Objectives

I Determine which permutations cannot occur in ciliatemicronuclei according to the mathematical model ofpointer-restricted rearrangements 1 2

I Determine if unsigned permutations that do occur in ciliatemicronuclei might occur because they are the most efficientpermutations for rearrangement using cds

1D. M. Prescott, A. Ehrenfeucht, G. Rozenberg, Template-guided recombination for IES elimination and

unscrambling of genes in stichotrichous ciliates, Journal of Theoretical Biology, Volume 222, Issue 3, 7 June 2003,Pages 323-330

2A. Angeleska, N. Jonoska, M. Saito, L. F. Landweber, RNA-guided DNA assembly, Journal of Theoretical

Biology, Volume 248, Issue 4, 21 October 2007, Pages 706-720

Helen Wauck Genome Remodeling in Developmental Time: Algorithms for Ciliates

Why Sequences?Objectives

Context Directed SwapsMathematical Results

Project Objectives

I Determine which permutations cannot occur in ciliatemicronuclei according to the mathematical model ofpointer-restricted rearrangements 1 2

I Determine if unsigned permutations that do occur in ciliatemicronuclei might occur because they are the most efficientpermutations for rearrangement using cds

1D. M. Prescott, A. Ehrenfeucht, G. Rozenberg, Template-guided recombination for IES elimination and

unscrambling of genes in stichotrichous ciliates, Journal of Theoretical Biology, Volume 222, Issue 3, 7 June 2003,Pages 323-330

2A. Angeleska, N. Jonoska, M. Saito, L. F. Landweber, RNA-guided DNA assembly, Journal of Theoretical

Biology, Volume 248, Issue 4, 21 October 2007, Pages 706-720

Helen Wauck Genome Remodeling in Developmental Time: Algorithms for Ciliates

Why Sequences?Objectives

Context Directed SwapsMathematical Results

Project Objectives

I Determine which permutations cannot occur in ciliatemicronuclei according to the mathematical model ofpointer-restricted rearrangements 1 2

I Determine if unsigned permutations that do occur in ciliatemicronuclei might occur because they are the most efficientpermutations for rearrangement using cds

1D. M. Prescott, A. Ehrenfeucht, G. Rozenberg, Template-guided recombination for IES elimination and

unscrambling of genes in stichotrichous ciliates, Journal of Theoretical Biology, Volume 222, Issue 3, 7 June 2003,Pages 323-330

2A. Angeleska, N. Jonoska, M. Saito, L. F. Landweber, RNA-guided DNA assembly, Journal of Theoretical

Biology, Volume 248, Issue 4, 21 October 2007, Pages 706-720

Helen Wauck Genome Remodeling in Developmental Time: Algorithms for Ciliates

Why Sequences?Objectives

Context Directed SwapsMathematical Results

Context Directed Swaps vs. Arbitrary Swaps

Arbitrary swaps:

I can swap any two blocks of a sequence

I always reduce a permutation to the identity

I Ex:

Helen Wauck Genome Remodeling in Developmental Time: Algorithms for Ciliates

Why Sequences?Objectives

Context Directed SwapsMathematical Results

Context Directed Swaps vs. Arbitrary Swaps

Arbitrary swaps:

I can swap any two blocks of a sequence

I always reduce a permutation to the identity

I Ex:

Helen Wauck Genome Remodeling in Developmental Time: Algorithms for Ciliates

Why Sequences?Objectives

Context Directed SwapsMathematical Results

Context Directed Swaps vs. Arbitrary Swaps

Arbitrary swaps:

I can swap any two blocks of a sequence

I always reduce a permutation to the identity

I Ex:

Helen Wauck Genome Remodeling in Developmental Time: Algorithms for Ciliates

Why Sequences?Objectives

Context Directed SwapsMathematical Results

Context Directed Swaps vs. Arbitrary Swaps

Arbitrary swaps:

I can swap any two blocks of a sequence

I always reduce a permutation to the identity

I Ex:

Helen Wauck Genome Remodeling in Developmental Time: Algorithms for Ciliates

Why Sequences?Objectives

Context Directed SwapsMathematical Results

Context Directed Swaps vs. Arbitrary Swaps

Arbitrary swaps:

I can swap any two blocks of a sequence

I always reduce a permutation to the identity

I Ex:

Helen Wauck Genome Remodeling in Developmental Time: Algorithms for Ciliates

Why Sequences?Objectives

Context Directed SwapsMathematical Results

Context Directed Swaps vs. Arbitrary Swaps

Arbitrary swaps:

I can swap any two blocks of a sequence

I always reduce a permutation to the identity

I Ex:

Helen Wauck Genome Remodeling in Developmental Time: Algorithms for Ciliates

Why Sequences?Objectives

Context Directed SwapsMathematical Results

Context Directed Swaps vs. Arbitrary Swaps

Arbitrary swaps:

I can swap any two blocks of a sequence

I always reduce a permutation to the identity

I Ex:

Helen Wauck Genome Remodeling in Developmental Time: Algorithms for Ciliates

Why Sequences?Objectives

Context Directed SwapsMathematical Results

Context Directed Swaps vs. Arbitrary Swaps

Arbitrary swaps:

I can swap any two blocks of a sequence

I always reduce a permutation to the identity

I Ex:

Helen Wauck Genome Remodeling in Developmental Time: Algorithms for Ciliates

Why Sequences?Objectives

Context Directed SwapsMathematical Results

Context Directed Swaps vs. Arbitrary Swaps

Context Directed Swaps (cds):I can only swap blocks whose pointer patterns match

I For a sequence of length n, every term has one pointerassociated with it if the term is 1 or n and 2 pointers otherwise

I In order to swap two blocks in a sequence, the followingpointer pattern must occur, where p and q are distinctpointers: p, q, p, q.

I Ex:

Helen Wauck Genome Remodeling in Developmental Time: Algorithms for Ciliates

Why Sequences?Objectives

Context Directed SwapsMathematical Results

Context Directed Swaps vs. Arbitrary Swaps

Context Directed Swaps (cds):

I can only swap blocks whose pointer patterns matchI For a sequence of length n, every term has one pointer

associated with it if the term is 1 or n and 2 pointers otherwiseI In order to swap two blocks in a sequence, the following

pointer pattern must occur, where p and q are distinctpointers: p, q, p, q.

I Ex:

Helen Wauck Genome Remodeling in Developmental Time: Algorithms for Ciliates

Why Sequences?Objectives

Context Directed SwapsMathematical Results

Context Directed Swaps vs. Arbitrary Swaps

Context Directed Swaps (cds):I can only swap blocks whose pointer patterns match

I For a sequence of length n, every term has one pointerassociated with it if the term is 1 or n and 2 pointers otherwise

I In order to swap two blocks in a sequence, the followingpointer pattern must occur, where p and q are distinctpointers: p, q, p, q.

I Ex:

Helen Wauck Genome Remodeling in Developmental Time: Algorithms for Ciliates

Why Sequences?Objectives

Context Directed SwapsMathematical Results

Context Directed Swaps vs. Arbitrary Swaps

Context Directed Swaps (cds):I can only swap blocks whose pointer patterns match

I For a sequence of length n, every term has one pointerassociated with it if the term is 1 or n and 2 pointers otherwise

I In order to swap two blocks in a sequence, the followingpointer pattern must occur, where p and q are distinctpointers: p, q, p, q.

I Ex:

Helen Wauck Genome Remodeling in Developmental Time: Algorithms for Ciliates

Why Sequences?Objectives

Context Directed SwapsMathematical Results

Context Directed Swaps vs. Arbitrary Swaps

Context Directed Swaps (cds):I can only swap blocks whose pointer patterns match

I For a sequence of length n, every term has one pointerassociated with it if the term is 1 or n and 2 pointers otherwise

I In order to swap two blocks in a sequence, the followingpointer pattern must occur, where p and q are distinctpointers: p, q, p, q.

I Ex:

Helen Wauck Genome Remodeling in Developmental Time: Algorithms for Ciliates

Why Sequences?Objectives

Context Directed SwapsMathematical Results

Context Directed Swaps vs. Arbitrary Swaps

Context Directed Swaps (cds):I can only swap blocks whose pointer patterns match

I For a sequence of length n, every term has one pointerassociated with it if the term is 1 or n and 2 pointers otherwise

I In order to swap two blocks in a sequence, the followingpointer pattern must occur, where p and q are distinctpointers: p, q, p, q.

I Ex:

Helen Wauck Genome Remodeling in Developmental Time: Algorithms for Ciliates

Why Sequences?Objectives

Context Directed SwapsMathematical Results

Context Directed Swaps vs. Arbitrary Swaps

I Do not always reduce a permutation to the identity:

Helen Wauck Genome Remodeling in Developmental Time: Algorithms for Ciliates

Why Sequences?Objectives

Context Directed SwapsMathematical Results

Context Directed Swaps vs. Arbitrary Swaps

I Do not always reduce a permutation to the identity:

Helen Wauck Genome Remodeling in Developmental Time: Algorithms for Ciliates

Why Sequences?Objectives

Context Directed SwapsMathematical Results

Context Directed Swaps vs. Arbitrary Swaps

I Do not always reduce a permutation to the identity:

Helen Wauck Genome Remodeling in Developmental Time: Algorithms for Ciliates

Why Sequences?Objectives

Context Directed SwapsMathematical Results

Context Directed Swaps vs. Arbitrary Swaps

I Do not always reduce a permutation to the identity:

Helen Wauck Genome Remodeling in Developmental Time: Algorithms for Ciliates

Why Sequences?Objectives

Context Directed SwapsMathematical Results

Context Directed Swaps vs. Arbitrary Swaps

I Do not always reduce a permutation to the identity:

Helen Wauck Genome Remodeling in Developmental Time: Algorithms for Ciliates

Why Sequences?Objectives

Context Directed SwapsMathematical Results

Context Directed Swaps vs. Arbitrary Swaps

Recall previous example for arbitrary swaps:

Helen Wauck Genome Remodeling in Developmental Time: Algorithms for Ciliates

Why Sequences?Objectives

Context Directed SwapsMathematical Results

Context Directed Swaps vs. Arbitrary Swaps

Recall previous example for arbitrary swaps:

Helen Wauck Genome Remodeling in Developmental Time: Algorithms for Ciliates

Why Sequences?Objectives

Context Directed SwapsMathematical Results

Context Directed Swaps vs. Arbitrary Swaps

Recall previous example for arbitrary swaps:

Helen Wauck Genome Remodeling in Developmental Time: Algorithms for Ciliates

Why Sequences?Objectives

Context Directed SwapsMathematical Results

Context Directed Swaps vs. Arbitrary Swaps

Recall previous example for arbitrary swaps:

Helen Wauck Genome Remodeling in Developmental Time: Algorithms for Ciliates

Why Sequences?Objectives

Context Directed SwapsMathematical Results

Context Directed Swaps vs. Arbitrary Swaps

Recall previous example for arbitrary swaps:

Helen Wauck Genome Remodeling in Developmental Time: Algorithms for Ciliates

Why Sequences?Objectives

Context Directed SwapsMathematical Results

Context Directed Swaps vs. Arbitrary Swaps

The pointer constraints in cds correspond to one of three DNAoperations conjectured to occur in ciliate micronuclear decryption.

Modeling these operations for ciliate micronuclear decryptionrequires also considering signed sequences.

Work so far has focused on cds for unsigned sequences only.

Helen Wauck Genome Remodeling in Developmental Time: Algorithms for Ciliates

Why Sequences?Objectives

Context Directed SwapsMathematical Results

Context Directed Swaps vs. Arbitrary Swaps

The pointer constraints in cds correspond to one of three DNAoperations conjectured to occur in ciliate micronuclear decryption.

Modeling these operations for ciliate micronuclear decryptionrequires also considering signed sequences.

Work so far has focused on cds for unsigned sequences only.

Helen Wauck Genome Remodeling in Developmental Time: Algorithms for Ciliates

Why Sequences?Objectives

Context Directed SwapsMathematical Results

Context Directed Swaps vs. Arbitrary Swaps

The pointer constraints in cds correspond to one of three DNAoperations conjectured to occur in ciliate micronuclear decryption.

Modeling these operations for ciliate micronuclear decryptionrequires also considering signed sequences.

Work so far has focused on cds for unsigned sequences only.

Helen Wauck Genome Remodeling in Developmental Time: Algorithms for Ciliates

Why Sequences?Objectives

Context Directed SwapsMathematical Results

Context Directed Swaps vs. Arbitrary Swaps

The pointer constraints in cds correspond to one of three DNAoperations conjectured to occur in ciliate micronuclear decryption.

Modeling these operations for ciliate micronuclear decryptionrequires also considering signed sequences.

Work so far has focused on cds for unsigned sequences only.

Helen Wauck Genome Remodeling in Developmental Time: Algorithms for Ciliates

Why Sequences?Objectives

Context Directed SwapsMathematical Results

Context Directed Swaps vs. Arbitrary Swaps

Theorem (Christie)

There is an algorithm that performs arbitrary swaps and achievesthe minimal number of swaps needed. This algorithm hascomplexity O(n2) with n the length of the permutation. 3

For cds there are partial results on this complexity.

3D.A. Christie, Sorting permutations by block interchanges, InformationProcessing Letters 60 (1996), 165 - 169

Helen Wauck Genome Remodeling in Developmental Time: Algorithms for Ciliates

Why Sequences?Objectives

Context Directed SwapsMathematical Results

Context Directed Swaps vs. Arbitrary Swaps

Theorem (Christie)

There is an algorithm that performs arbitrary swaps and achievesthe minimal number of swaps needed. This algorithm hascomplexity O(n2) with n the length of the permutation. 3

For cds there are partial results on this complexity.

3D.A. Christie, Sorting permutations by block interchanges, InformationProcessing Letters 60 (1996), 165 - 169

Helen Wauck Genome Remodeling in Developmental Time: Algorithms for Ciliates

Why Sequences?Objectives

Context Directed SwapsMathematical Results

Context Directed Swaps vs. Arbitrary Swaps

Theorem (Christie)

There is an algorithm that performs arbitrary swaps and achievesthe minimal number of swaps needed. This algorithm hascomplexity O(n2) with n the length of the permutation. 3

For cds there are partial results on this complexity.

3D.A. Christie, Sorting permutations by block interchanges, InformationProcessing Letters 60 (1996), 165 - 169

Helen Wauck Genome Remodeling in Developmental Time: Algorithms for Ciliates

Why Sequences?Objectives

Context Directed SwapsMathematical Results

The Special Sequence

TheoremOf all unsigned sequences of length 2n, the only sequence forwhich any pair of pointers is cds-competent and which can be putinto the correct order using only cds is of the form

[1, 3, . . . , 2n − 1, 2, 4, . . . , 2n].

There are many different cds paths to a correct ordering of thissequence:

TheoremThere are exactly K (n) = (2n−1)!

2n−1 ways to arrange the sequence[1, 3, . . . , 2n − 1, 2, 4, . . . , 2n], n ≥ 2 into the correct order usingcontext directed swaps.

Helen Wauck Genome Remodeling in Developmental Time: Algorithms for Ciliates

Why Sequences?Objectives

Context Directed SwapsMathematical Results

The Special Sequence

TheoremOf all unsigned sequences of length 2n, the only sequence forwhich any pair of pointers is cds-competent and which can be putinto the correct order using only cds is of the form

[1, 3, . . . , 2n − 1, 2, 4, . . . , 2n].

There are many different cds paths to a correct ordering of thissequence:

TheoremThere are exactly K (n) = (2n−1)!

2n−1 ways to arrange the sequence[1, 3, . . . , 2n − 1, 2, 4, . . . , 2n], n ≥ 2 into the correct order usingcontext directed swaps.

Helen Wauck Genome Remodeling in Developmental Time: Algorithms for Ciliates

Why Sequences?Objectives

Context Directed SwapsMathematical Results

The Special Sequence

TheoremOf all unsigned sequences of length 2n, the only sequence forwhich any pair of pointers is cds-competent and which can be putinto the correct order using only cds is of the form

[1, 3, . . . , 2n − 1, 2, 4, . . . , 2n].

There are many different cds paths to a correct ordering of thissequence:

TheoremThere are exactly K (n) = (2n−1)!

2n−1 ways to arrange the sequence[1, 3, . . . , 2n − 1, 2, 4, . . . , 2n], n ≥ 2 into the correct order usingcontext directed swaps.

Helen Wauck Genome Remodeling in Developmental Time: Algorithms for Ciliates

Why Sequences?Objectives

Context Directed SwapsMathematical Results

The Special Sequence

TheoremOf all unsigned sequences of length 2n, the only sequence forwhich any pair of pointers is cds-competent and which can be putinto the correct order using only cds is of the form

[1, 3, . . . , 2n − 1, 2, 4, . . . , 2n].

There are many different cds paths to a correct ordering of thissequence:

TheoremThere are exactly K (n) = (2n−1)!

2n−1 ways to arrange the sequence[1, 3, . . . , 2n − 1, 2, 4, . . . , 2n], n ≥ 2 into the correct order usingcontext directed swaps.

Helen Wauck Genome Remodeling in Developmental Time: Algorithms for Ciliates

Why Sequences?Objectives

Context Directed SwapsMathematical Results

The Special Sequence

TheoremThe minimum number of cds’s needed to arrange a sequence of theformat [1, 3, . . . , 2n− 1, 2, 4, . . . , 2n] into the correct order is n + 1.

This result is also true of arbitrary swaps applied to this sequence.

More generally:

TheoremIf a sequence of length ` starts with the term 1 and ends with theterm `, then the minimal number of cds’s needed to get it into thecorrect order is equal to the minimal number of arbitrary blockinterchanges needed to get it into the correct order.

Helen Wauck Genome Remodeling in Developmental Time: Algorithms for Ciliates

Why Sequences?Objectives

Context Directed SwapsMathematical Results

The Special Sequence

TheoremThe minimum number of cds’s needed to arrange a sequence of theformat [1, 3, . . . , 2n− 1, 2, 4, . . . , 2n] into the correct order is n + 1.

This result is also true of arbitrary swaps applied to this sequence.

More generally:

TheoremIf a sequence of length ` starts with the term 1 and ends with theterm `, then the minimal number of cds’s needed to get it into thecorrect order is equal to the minimal number of arbitrary blockinterchanges needed to get it into the correct order.

Helen Wauck Genome Remodeling in Developmental Time: Algorithms for Ciliates

Why Sequences?Objectives

Context Directed SwapsMathematical Results

The Special Sequence

TheoremThe minimum number of cds’s needed to arrange a sequence of theformat [1, 3, . . . , 2n− 1, 2, 4, . . . , 2n] into the correct order is n + 1.

This result is also true of arbitrary swaps applied to this sequence.

More generally:

TheoremIf a sequence of length ` starts with the term 1 and ends with theterm `, then the minimal number of cds’s needed to get it into thecorrect order is equal to the minimal number of arbitrary blockinterchanges needed to get it into the correct order.

Helen Wauck Genome Remodeling in Developmental Time: Algorithms for Ciliates

Why Sequences?Objectives

Context Directed SwapsMathematical Results

The Special Sequence

TheoremThe minimum number of cds’s needed to arrange a sequence of theformat [1, 3, . . . , 2n− 1, 2, 4, . . . , 2n] into the correct order is n + 1.

This result is also true of arbitrary swaps applied to this sequence.

More generally:

TheoremIf a sequence of length ` starts with the term 1 and ends with theterm `, then the minimal number of cds’s needed to get it into thecorrect order is equal to the minimal number of arbitrary blockinterchanges needed to get it into the correct order.

Helen Wauck Genome Remodeling in Developmental Time: Algorithms for Ciliates

Why Sequences?Objectives

Context Directed SwapsMathematical Results

General Properties

TheoremIterated application of cds to an unsigned sequence of length nresults in at least 1 of the n fixed points of cds: 4

[1, 2, 3, 4, 5, . . . , n][2, 3, 4, 5, . . . , n, 1][3, 4, 5, . . . , n, 1, 2]

...[n, 1, 2, . . . , n − 1]

4J. Herlin, A. Nelson and M. Scheepers, Can a ciliate compute evolutionarydistance?, in progress

Helen Wauck Genome Remodeling in Developmental Time: Algorithms for Ciliates

Why Sequences?Objectives

Context Directed SwapsMathematical Results

General Properties

TheoremIterated application of cds to an unsigned sequence of length nresults in at least 1 of the n fixed points of cds: 4

[1, 2, 3, 4, 5, . . . , n][2, 3, 4, 5, . . . , n, 1][3, 4, 5, . . . , n, 1, 2]

...[n, 1, 2, . . . , n − 1]

4J. Herlin, A. Nelson and M. Scheepers, Can a ciliate compute evolutionarydistance?, in progress

Helen Wauck Genome Remodeling in Developmental Time: Algorithms for Ciliates

Why Sequences?Objectives

Context Directed SwapsMathematical Results

General Properties

TheoremIterated application of cds to an unsigned sequence of length nresults in at least 1 of the n fixed points of cds: 4

[1, 2, 3, 4, 5, . . . , n]

[2, 3, 4, 5, . . . , n, 1][3, 4, 5, . . . , n, 1, 2]

...[n, 1, 2, . . . , n − 1]

4J. Herlin, A. Nelson and M. Scheepers, Can a ciliate compute evolutionarydistance?, in progress

Helen Wauck Genome Remodeling in Developmental Time: Algorithms for Ciliates

Why Sequences?Objectives

Context Directed SwapsMathematical Results

General Properties

TheoremIterated application of cds to an unsigned sequence of length nresults in at least 1 of the n fixed points of cds: 4

[1, 2, 3, 4, 5, . . . , n][2, 3, 4, 5, . . . , n, 1][3, 4, 5, . . . , n, 1, 2]

...[n, 1, 2, . . . , n − 1]

4J. Herlin, A. Nelson and M. Scheepers, Can a ciliate compute evolutionarydistance?, in progress

Helen Wauck Genome Remodeling in Developmental Time: Algorithms for Ciliates

Why Sequences?Objectives

Context Directed SwapsMathematical Results

General Properties

TheoremLet S be an unsigned sequence of length n, and let k be an integerwhere 1 ≤ k ≤ n− 1. If S begins with k + 1 and ends with k, thenS is irresolvable.

TheoremIf an unsigned sequence of length n contains the pattern. . . , n, 1, . . ., then the sequence is irresolvable.

Helen Wauck Genome Remodeling in Developmental Time: Algorithms for Ciliates

Why Sequences?Objectives

Context Directed SwapsMathematical Results

General Properties

TheoremLet S be an unsigned sequence of length n, and let k be an integerwhere 1 ≤ k ≤ n− 1. If S begins with k + 1 and ends with k, thenS is irresolvable.

TheoremIf an unsigned sequence of length n contains the pattern. . . , n, 1, . . ., then the sequence is irresolvable.

Helen Wauck Genome Remodeling in Developmental Time: Algorithms for Ciliates

Why Sequences?Objectives

Context Directed SwapsMathematical Results

General Properties

TheoremLet S be an unsigned sequence of length n, and let k be an integerwhere 1 ≤ k ≤ n− 1. If S begins with k + 1 and ends with k, thenS is irresolvable.

TheoremIf an unsigned sequence of length n contains the pattern. . . , n, 1, . . ., then the sequence is irresolvable.

Helen Wauck Genome Remodeling in Developmental Time: Algorithms for Ciliates

Why Sequences?Objectives

Context Directed SwapsMathematical Results

General Properties

TheoremAn unsigned sequence of length n can be rearranged into canonicalorder if it begins with the term 1 or ends with the term n.

Theorem

Fix n ∈ N. Define cn =

(1 2 · · · nn n − 1 · · · 1

). Then for each

sequence b of length n, the following are equivalent:

1. b is cds invertible

2. c−1n ◦ b ◦ cn is cds invertible

Helen Wauck Genome Remodeling in Developmental Time: Algorithms for Ciliates

Why Sequences?Objectives

Context Directed SwapsMathematical Results

General Properties

TheoremAn unsigned sequence of length n can be rearranged into canonicalorder if it begins with the term 1 or ends with the term n.

Theorem

Fix n ∈ N. Define cn =

(1 2 · · · nn n − 1 · · · 1

). Then for each

sequence b of length n, the following are equivalent:

1. b is cds invertible

2. c−1n ◦ b ◦ cn is cds invertible

Helen Wauck Genome Remodeling in Developmental Time: Algorithms for Ciliates

Why Sequences?Objectives

Context Directed SwapsMathematical Results

General Properties

TheoremAn unsigned sequence of length n can be rearranged into canonicalorder if it begins with the term 1 or ends with the term n.

Theorem

Fix n ∈ N. Define cn =

(1 2 · · · nn n − 1 · · · 1

). Then for each

sequence b of length n, the following are equivalent:

1. b is cds invertible

2. c−1n ◦ b ◦ cn is cds invertible

Helen Wauck Genome Remodeling in Developmental Time: Algorithms for Ciliates

Why Sequences?Objectives

Context Directed SwapsMathematical Results

General Properties

TheoremAn unsigned sequence of length n can be rearranged into canonicalorder if it begins with the term 1 or ends with the term n.

Theorem

Fix n ∈ N. Define cn =

(1 2 · · · nn n − 1 · · · 1

). Then for each

sequence b of length n, the following are equivalent:

1. b is cds invertible

2. c−1n ◦ b ◦ cn is cds invertible

Helen Wauck Genome Remodeling in Developmental Time: Algorithms for Ciliates

Why Sequences?Objectives

Context Directed SwapsMathematical Results

General Properties

TheoremAn unsigned sequence of length n can be rearranged into canonicalorder if it begins with the term 1 or ends with the term n.

Theorem

Fix n ∈ N. Define cn =

(1 2 · · · nn n − 1 · · · 1

). Then for each

sequence b of length n, the following are equivalent:

1. b is cds invertible

2. c−1n ◦ b ◦ cn is cds invertible

Helen Wauck Genome Remodeling in Developmental Time: Algorithms for Ciliates

Why Sequences?Objectives

Context Directed SwapsMathematical Results

Acknowledgments

I Prof. Jim Smith, Boise State University, for consultation anduse of his laboratory

I Prof. Hans Lipps and Dr. Franziska Jonnson of the Universityof Witten/Herdecke, Germany, for a gift of DNA of the ciliateStylonychia lemnae

I Boise State University Math REU 2012

I NSF REU funding via grant DMS 1062857

I NIH INBRE funding

Helen Wauck Genome Remodeling in Developmental Time: Algorithms for Ciliates

Why Sequences?Objectives

Context Directed SwapsMathematical Results

Acknowledgments

I Prof. Jim Smith, Boise State University, for consultation anduse of his laboratory

I Prof. Hans Lipps and Dr. Franziska Jonnson of the Universityof Witten/Herdecke, Germany, for a gift of DNA of the ciliateStylonychia lemnae

I Boise State University Math REU 2012

I NSF REU funding via grant DMS 1062857

I NIH INBRE funding

Helen Wauck Genome Remodeling in Developmental Time: Algorithms for Ciliates

Why Sequences?Objectives

Context Directed SwapsMathematical Results

Acknowledgments

I Prof. Jim Smith, Boise State University, for consultation anduse of his laboratory

I Prof. Hans Lipps and Dr. Franziska Jonnson of the Universityof Witten/Herdecke, Germany, for a gift of DNA of the ciliateStylonychia lemnae

I Boise State University Math REU 2012

I NSF REU funding via grant DMS 1062857

I NIH INBRE funding

Helen Wauck Genome Remodeling in Developmental Time: Algorithms for Ciliates

Why Sequences?Objectives

Context Directed SwapsMathematical Results

Acknowledgments

I Prof. Jim Smith, Boise State University, for consultation anduse of his laboratory

I Prof. Hans Lipps and Dr. Franziska Jonnson of the Universityof Witten/Herdecke, Germany, for a gift of DNA of the ciliateStylonychia lemnae

I Boise State University Math REU 2012

I NSF REU funding via grant DMS 1062857

I NIH INBRE funding

Helen Wauck Genome Remodeling in Developmental Time: Algorithms for Ciliates

Why Sequences?Objectives

Context Directed SwapsMathematical Results

Acknowledgments

I Prof. Jim Smith, Boise State University, for consultation anduse of his laboratory

I Prof. Hans Lipps and Dr. Franziska Jonnson of the Universityof Witten/Herdecke, Germany, for a gift of DNA of the ciliateStylonychia lemnae

I Boise State University Math REU 2012

I NSF REU funding via grant DMS 1062857

I NIH INBRE funding

Helen Wauck Genome Remodeling in Developmental Time: Algorithms for Ciliates

Why Sequences?Objectives

Context Directed SwapsMathematical Results

Acknowledgments

I Prof. Jim Smith, Boise State University, for consultation anduse of his laboratory

I Prof. Hans Lipps and Dr. Franziska Jonnson of the Universityof Witten/Herdecke, Germany, for a gift of DNA of the ciliateStylonychia lemnae

I Boise State University Math REU 2012

I NSF REU funding via grant DMS 1062857

I NIH INBRE funding

Helen Wauck Genome Remodeling in Developmental Time: Algorithms for Ciliates

Why Sequences?Objectives

Context Directed SwapsMathematical Results

Thank you!

Helen Wauck Genome Remodeling in Developmental Time: Algorithms for Ciliates

Why Sequences?Objectives

Context Directed SwapsMathematical Results

Thank you!

Helen Wauck Genome Remodeling in Developmental Time: Algorithms for Ciliates