Exploring Phylogenetic Relationships in Drosophila withCiliate Operations

Jacob Herlin∗, Anna Nelson†, and Dr. Marion Scheepers†

∗Department of Mathematical Sciences, University of Northern Colorado,†Department of Mathematics, Boise State University

July 29, 2011

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 1 / 29

What is a phylogenetic relationship?

Phylogenetics is the study of evolutionary relationships between groups oforganisms. For this research, we focused on several organisms in the genusDrosophila. Using ciliate operations, we want to explore the possibility ofrelating via those operations the phylogenetic distance between twospecies to their known phylogeny.

Orthologs are genes that are common among various species

Relationships are illustrated using phylogenetic trees

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 2 / 29

What is a phylogenetic relationship?

Phylogenetics is the study of evolutionary relationships between groups oforganisms. For this research, we focused on several organisms in the genusDrosophila. Using ciliate operations, we want to explore the possibility ofrelating via those operations the phylogenetic distance between twospecies to their known phylogeny.

Orthologs are genes that are common among various species

Relationships are illustrated using phylogenetic trees

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 2 / 29

What is a phylogenetic relationship?

Phylogenetics is the study of evolutionary relationships between groups oforganisms. For this research, we focused on several organisms in the genusDrosophila. Using ciliate operations, we want to explore the possibility ofrelating via those operations the phylogenetic distance between twospecies to their known phylogeny.

Orthologs are genes that are common among various species

Relationships are illustrated using phylogenetic trees

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 2 / 29

Phylogenetic trees

Image courtesy of DroSpeGe: Drosophila Species Genomes. http://insects.eugenes.org/DroSpeGe

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 3 / 29

Using reversals to determine phylogeny

Genetic operations that cause DNA to be scrambled result from breakingand rejoining the chromosome

In 1938, Dobzhansky and Sturtevant discovered that the genearrangements that occurred in D. pseudoobscura were reversals. Sincethen, reversals have been used as the main genetic operation.

Using Drosophila melanogaster as the canonical reference species, onecan use number of reversals as a measure of evolutionary distance.

From Hannenhalli and Pevzner, it is known that the shortest reversalpath can be found in polynomial time.

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 4 / 29

Using reversals to determine phylogeny

Genetic operations that cause DNA to be scrambled result from breakingand rejoining the chromosome

In 1938, Dobzhansky and Sturtevant discovered that the genearrangements that occurred in D. pseudoobscura were reversals. Sincethen, reversals have been used as the main genetic operation.

Using Drosophila melanogaster as the canonical reference species, onecan use number of reversals as a measure of evolutionary distance.

From Hannenhalli and Pevzner, it is known that the shortest reversalpath can be found in polynomial time.

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 4 / 29

Using reversals to determine phylogeny

Genetic operations that cause DNA to be scrambled result from breakingand rejoining the chromosome

In 1938, Dobzhansky and Sturtevant discovered that the genearrangements that occurred in D. pseudoobscura were reversals. Sincethen, reversals have been used as the main genetic operation.

Using Drosophila melanogaster as the canonical reference species, onecan use number of reversals as a measure of evolutionary distance.

From Hannenhalli and Pevzner, it is known that the shortest reversalpath can be found in polynomial time.

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 4 / 29

Using reversals to determine phylogeny

Genetic operations that cause DNA to be scrambled result from breakingand rejoining the chromosome

In 1938, Dobzhansky and Sturtevant discovered that the genearrangements that occurred in D. pseudoobscura were reversals. Sincethen, reversals have been used as the main genetic operation.

Using Drosophila melanogaster as the canonical reference species, onecan use number of reversals as a measure of evolutionary distance.

From Hannenhalli and Pevzner, it is known that the shortest reversalpath can be found in polynomial time.

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 4 / 29

Micronucleus and macronucleus of ciliates

Ciliates are multinuclear protozoans found in aqueous environments.

The micronucleus contains very long strands of DNA that areencrypted versions of macronuclear DNA.

The macronucleus is larger than the micronucleus and contains shortstrands of DNA that have been multiplied.

The micronuclear DNA is decrypted to form macronuclear DNA usingthree ciliate operation (hi, ld, dlad).

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 5 / 29

Micronucleus and macronucleus of ciliates

Ciliates are multinuclear protozoans found in aqueous environments.

The micronucleus contains very long strands of DNA that areencrypted versions of macronuclear DNA.

The macronucleus is larger than the micronucleus and contains shortstrands of DNA that have been multiplied.

The micronuclear DNA is decrypted to form macronuclear DNA usingthree ciliate operation (hi, ld, dlad).

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 5 / 29

Micronucleus and macronucleus of ciliates

Ciliates are multinuclear protozoans found in aqueous environments.

The micronucleus contains very long strands of DNA that areencrypted versions of macronuclear DNA.

The macronucleus is larger than the micronucleus and contains shortstrands of DNA that have been multiplied.

The micronuclear DNA is decrypted to form macronuclear DNA usingthree ciliate operation (hi, ld, dlad).

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 5 / 29

Micronucleus and macronucleus of ciliates

Ciliates are multinuclear protozoans found in aqueous environments.

The micronucleus contains very long strands of DNA that areencrypted versions of macronuclear DNA.

The macronucleus is larger than the micronucleus and contains shortstrands of DNA that have been multiplied.

The micronuclear DNA is decrypted to form macronuclear DNA usingthree ciliate operation (hi, ld, dlad).

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 5 / 29

Macronuclear vs. Micronuclear DNA

Micronuclear DNA has three elements:

1. Macronuclear destined sequences (MDSs)

2. Internal eliminated sequences (IESs)

3. Pointers occur on the flanks of the MDSs

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 6 / 29

Macronuclear vs. Micronuclear DNA

Micronuclear DNA has three elements:

1. Macronuclear destined sequences (MDSs)

2. Internal eliminated sequences (IESs)

3. Pointers occur on the flanks of the MDSs

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 6 / 29

Macronuclear vs. Micronuclear DNA

Micronuclear DNA has three elements:

1. Macronuclear destined sequences (MDSs)

2. Internal eliminated sequences (IESs)

3. Pointers occur on the flanks of the MDSs

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 6 / 29

ld operation

Step 1:

Step 2:

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 7 / 29

ld operation

Step 1:

Step 2:

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 7 / 29

ld operation

Step 3:

Step 4:

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 8 / 29

ld operation

Step 3:

Step 4:

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 8 / 29

ld operation

Step 3:

Step 4:

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 8 / 29

hi operation

Step 1:

Step 2:

Step 3a:

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 9 / 29

hi operation

Step 1:

Step 2:

Step 3a:

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 9 / 29

hi operation

Step 1:

Step 2:

Step 3a:

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 9 / 29

hi operation

Step 3b:

Step 4:

Original:

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 10 / 29

hi operation

Step 3b:

Step 4:

Original:

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 10 / 29

dlad operation

Step 1:

Step 2:

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 11 / 29

dlad operation

Step 1:

Step 2:

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 11 / 29

dlad operation

Step 3:

Step 4:

Original:

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 12 / 29

dlad operation

Step 3:

Step 4:

Original:

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 12 / 29

Data Collection

We collected data from Flybase.org, which is a database for Drosophilagenes and genomes. The data was in a precomputed text file with all thegenes in our reference species (D. melanogaster) and their orthologs onvarious species’ genome. We used the relative location and orientation ofgenes to produce signed permutations.

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 13 / 29

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 14 / 29

Data Analysis

We used the following species to create permutations based on theirgenomes.

1 Simulans

2 Sechellia

3 Yakuba

4 Erecta

5 Virilis

6 Grimshawi

7 Mojavensis

8 Melanogaster

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 15 / 29

Pointer Lists

We created an algorithm to find a path from a signed permutation back tothe canonical in terms of ciliate operations.

We start by mapping a signedpermutation where each elements represents a section of genome:

[1, 4,−3,−2, 6,−5]

onto a list of pairs of pointers:

[(1, 2), (4, 5), (4, 3), (3, 2), (6, 7), (6, 5)]

where each pair (a, b) represents a section of genome spanning from apointer a to a pointer b. In our algorithm, we give the pointers signs torepresent the orientation of each section, in stead of keeping them in pairs:

[1, 2, 4, 5,−4,−3,−3,−2, 6, 7,−6,−5]

We call this representation a pointer list.

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 16 / 29

Pointer Lists

We created an algorithm to find a path from a signed permutation back tothe canonical in terms of ciliate operations. We start by mapping a signedpermutation where each elements represents a section of genome:

[1, 4,−3,−2, 6,−5]

onto a list of pairs of pointers:

[(1, 2), (4, 5), (4, 3), (3, 2), (6, 7), (6, 5)]

where each pair (a, b) represents a section of genome spanning from apointer a to a pointer b. In our algorithm, we give the pointers signs torepresent the orientation of each section, in stead of keeping them in pairs:

[1, 2, 4, 5,−4,−3,−3,−2, 6, 7,−6,−5]

We call this representation a pointer list.

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 16 / 29

Pointer Lists

We created an algorithm to find a path from a signed permutation back tothe canonical in terms of ciliate operations. We start by mapping a signedpermutation where each elements represents a section of genome:

[1, 4,−3,−2, 6,−5]

onto a list of pairs of pointers:

[(1, 2), (4, 5), (4, 3), (3, 2), (6, 7), (6, 5)]

where each pair (a, b) represents a section of genome spanning from apointer a to a pointer b. In our algorithm, we give the pointers signs torepresent the orientation of each section, in stead of keeping them in pairs:

[1, 2, 4, 5,−4,−3,−3,−2, 6, 7,−6,−5]

We call this representation a pointer list.

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 16 / 29

Pointer Lists

We created an algorithm to find a path from a signed permutation back tothe canonical in terms of ciliate operations. We start by mapping a signedpermutation where each elements represents a section of genome:

[1, 4,−3,−2, 6,−5]

onto a list of pairs of pointers:

[(1, 2), (4, 5), (4, 3), (3, 2), (6, 7), (6, 5)]

where each pair (a, b) represents a section of genome spanning from apointer a to a pointer b. In our algorithm, we give the pointers signs torepresent the orientation of each section, in stead of keeping them in pairs:

[1, 2, 4, 5,−4,−3,−3,−2, 6, 7,−6,−5]

We call this representation a pointer list.

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 16 / 29

Pointer Lists

We define a pointer list formally as a list L = [x1, x2, . . . xn] that satisfiesthe following six conditions:

1 n is even.

2 There is a unique i with µ = |xi | = min{|xj | : i ≤ j ≤ n}.3 There is a unique j with λ = |xi | = max{|xj | : i ≤ j ≤ n}.4 For each i ∈ {1, ..., n} with µ < |xi | < λ, there is a unique

j ∈ {1, ..., n}\{i} with |xi | = |xj |.5 For each odd i ∈ {1, ..., n}, xi ≤ xi+1 and xi · xi+1 > 0.

6 For each odd i , @ an odd j such that xi < xj < xi+1 < xj+1.

Example:[1, 2, 4, 5,−4,−3,−3,−2, 6, 7,−6,−5]

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 17 / 29

Pointer Lists

We define a pointer list formally as a list L = [x1, x2, . . . xn] that satisfiesthe following six conditions:

1 n is even.

2 There is a unique i with µ = |xi | = min{|xj | : i ≤ j ≤ n}.3 There is a unique j with λ = |xi | = max{|xj | : i ≤ j ≤ n}.4 For each i ∈ {1, ..., n} with µ < |xi | < λ, there is a unique

j ∈ {1, ..., n}\{i} with |xi | = |xj |.5 For each odd i ∈ {1, ..., n}, xi ≤ xi+1 and xi · xi+1 > 0.

6 For each odd i , @ an odd j such that xi < xj < xi+1 < xj+1.

Example:[1, 2, 4, 5,−4,−3,−3,−2, 6, 7,−6,−5]

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 17 / 29

Pointer Lists

We define a pointer list formally as a list L = [x1, x2, . . . xn] that satisfiesthe following six conditions:

1 n is even.

2 There is a unique i with µ = |xi | = min{|xj | : i ≤ j ≤ n}.

3 There is a unique j with λ = |xi | = max{|xj | : i ≤ j ≤ n}.4 For each i ∈ {1, ..., n} with µ < |xi | < λ, there is a unique

j ∈ {1, ..., n}\{i} with |xi | = |xj |.5 For each odd i ∈ {1, ..., n}, xi ≤ xi+1 and xi · xi+1 > 0.

6 For each odd i , @ an odd j such that xi < xj < xi+1 < xj+1.

Example:[1, 2, 4, 5,−4,−3,−3,−2, 6, 7,−6,−5]

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 17 / 29

Pointer Lists

We define a pointer list formally as a list L = [x1, x2, . . . xn] that satisfiesthe following six conditions:

1 n is even.

2 There is a unique i with µ = |xi | = min{|xj | : i ≤ j ≤ n}.3 There is a unique j with λ = |xi | = max{|xj | : i ≤ j ≤ n}.

4 For each i ∈ {1, ..., n} with µ < |xi | < λ, there is a uniquej ∈ {1, ..., n}\{i} with |xi | = |xj |.

5 For each odd i ∈ {1, ..., n}, xi ≤ xi+1 and xi · xi+1 > 0.

6 For each odd i , @ an odd j such that xi < xj < xi+1 < xj+1.

Example:[1, 2, 4, 5,−4,−3,−3,−2, 6, 7,−6,−5]

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 17 / 29

Pointer Lists

We define a pointer list formally as a list L = [x1, x2, . . . xn] that satisfiesthe following six conditions:

1 n is even.

2 There is a unique i with µ = |xi | = min{|xj | : i ≤ j ≤ n}.3 There is a unique j with λ = |xi | = max{|xj | : i ≤ j ≤ n}.4 For each i ∈ {1, ..., n} with µ < |xi | < λ, there is a unique

j ∈ {1, ..., n}\{i} with |xi | = |xj |.

5 For each odd i ∈ {1, ..., n}, xi ≤ xi+1 and xi · xi+1 > 0.

6 For each odd i , @ an odd j such that xi < xj < xi+1 < xj+1.

Example:[1, 2, 4, 5,−4,−3,−3,−2, 6, 7,−6,−5]

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 17 / 29

Pointer Lists

We define a pointer list formally as a list L = [x1, x2, . . . xn] that satisfiesthe following six conditions:

1 n is even.

2 There is a unique i with µ = |xi | = min{|xj | : i ≤ j ≤ n}.3 There is a unique j with λ = |xi | = max{|xj | : i ≤ j ≤ n}.4 For each i ∈ {1, ..., n} with µ < |xi | < λ, there is a unique

j ∈ {1, ..., n}\{i} with |xi | = |xj |.5 For each odd i ∈ {1, ..., n}, xi ≤ xi+1 and xi · xi+1 > 0.

6 For each odd i , @ an odd j such that xi < xj < xi+1 < xj+1.

Example:[1, 2, 4, 5,−4,−3,−3,−2, 6, 7,−6,−5]

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 17 / 29

Pointer Lists

We define a pointer list formally as a list L = [x1, x2, . . . xn] that satisfiesthe following six conditions:

1 n is even.

2 There is a unique i with µ = |xi | = min{|xj | : i ≤ j ≤ n}.3 There is a unique j with λ = |xi | = max{|xj | : i ≤ j ≤ n}.4 For each i ∈ {1, ..., n} with µ < |xi | < λ, there is a unique

j ∈ {1, ..., n}\{i} with |xi | = |xj |.5 For each odd i ∈ {1, ..., n}, xi ≤ xi+1 and xi · xi+1 > 0.

6 For each odd i , @ an odd j such that xi < xj < xi+1 < xj+1.

Example:[1, 2, 4, 5,−4,−3,−3,−2, 6, 7,−6,−5]

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 17 / 29

ld

The ld operation is represented by the removal of pairs of the same pointerthat are adjacent. This is equivalent to joining two sections that arecorrectly adjacent.

[1, 2, 4, 5,−4,−3,−3,−2, 6, 7,−6, 5]

[1, 2, 4, 5,−4,−2, 6, 7,−6,−5]

Formally, ld is a function that maps a pointer list of length n to a pointerlist of length n-2 as such:

[x1, x2, . . . xi , xi+1, . . . xn]⇒ [x1, x2, . . . xi−1, xi+2, . . . xn]

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 18 / 29

ld

The ld operation is represented by the removal of pairs of the same pointerthat are adjacent. This is equivalent to joining two sections that arecorrectly adjacent.

[1, 2, 4, 5,−4,−3,−3,−2, 6, 7,−6, 5]

[1, 2, 4, 5,−4,−2, 6, 7,−6,−5]

Formally, ld is a function that maps a pointer list of length n to a pointerlist of length n-2 as such:

[x1, x2, . . . xi , xi+1, . . . xn]⇒ [x1, x2, . . . xi−1, xi+2, . . . xn]

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 18 / 29

ld

The ld operation is represented by the removal of pairs of the same pointerthat are adjacent. This is equivalent to joining two sections that arecorrectly adjacent.

[1, 2, 4, 5,−4,−3,−3,−2, 6, 7,−6, 5]

[1, 2, 4, 5,−4,−2, 6, 7,−6,−5]

Formally, ld is a function that maps a pointer list of length n to a pointerlist of length n-2 as such:

[x1, x2, . . . xi , xi+1, . . . xn]⇒ [x1, x2, . . . xi−1, xi+2, . . . xn]

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 18 / 29

ld

The ld operation is represented by the removal of pairs of the same pointerthat are adjacent. This is equivalent to joining two sections that arecorrectly adjacent.

[1, 2, 4, 5,−4,−3,−3,−2, 6, 7,−6, 5]

[1, 2, 4, 5,−4,−2, 6, 7,−6,−5]

Formally, ld is a function that maps a pointer list of length n to a pointerlist of length n-2 as such:

[x1, x2, . . . xi , xi+1, . . . xn]⇒ [x1, x2, . . . xi−1, xi+2, . . . xn]

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 18 / 29

hi

The hi operation is represented by moving together two pointers ofopposite orientation with a reversal, setting up an ld move. Note thatreversing a section changes the signs of each element.

[1, 2, 4, 5,−4,−2, 6, 7,−6,−5]

[1, 2, 4, 4,−5,−2, 6, 7,−6,−5]

Formally, hi is a function that maps a pointer list of length n to a pointerlist of the same length as such:

[x1, x2, ...,xi , xi+1, ..., xj , xj+1, ..., xn]

⇒ [x1, x2, ..., xi ,−xj , ...,−xi+1, xj+1, ..., xn]

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 19 / 29

hi

The hi operation is represented by moving together two pointers ofopposite orientation with a reversal, setting up an ld move. Note thatreversing a section changes the signs of each element.

[1, 2, 4, 5,−4,−2, 6, 7,−6,−5]

[1, 2, 4, 4,−5,−2, 6, 7,−6,−5]

Formally, hi is a function that maps a pointer list of length n to a pointerlist of the same length as such:

[x1, x2, ...,xi , xi+1, ..., xj , xj+1, ..., xn]

⇒ [x1, x2, ..., xi ,−xj , ...,−xi+1, xj+1, ..., xn]

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 19 / 29

hi

The hi operation is represented by moving together two pointers ofopposite orientation with a reversal, setting up an ld move. Note thatreversing a section changes the signs of each element.

[1, 2, 4, 5,−4,−2, 6, 7,−6,−5]

[1, 2, 4, 4,−5,−2, 6, 7,−6,−5]

Formally, hi is a function that maps a pointer list of length n to a pointerlist of the same length as such:

[x1, x2, ...,xi , xi+1, ..., xj , xj+1, ..., xn]

⇒ [x1, x2, ..., xi ,−xj , ...,−xi+1, xj+1, ..., xn]

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 19 / 29

hi

The hi operation is represented by moving together two pointers ofopposite orientation with a reversal, setting up an ld move. Note thatreversing a section changes the signs of each element.

[1, 2, 4, 5,−4,−2, 6, 7,−6,−5]

[1, 2, 4, 4,−5,−2, 6, 7,−6,−5]

Formally, hi is a function that maps a pointer list of length n to a pointerlist of the same length as such:

[x1, x2, ...,xi , xi+1, ..., xj , xj+1, ..., xn]

⇒ [x1, x2, ..., xi ,−xj , ...,−xi+1, xj+1, ..., xn]

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 19 / 29

dlad

The dlad operation is represented by finding a 4-tuple of pointers(xi , xj , xk , xl) where xi = xj , and xk = xl and i < k < j < l . Then, youtake the section xjxl , including the pointers, and the section in-between,but not including, the pointers xi and xk , and swapping them, setting uptwo ld-moves.

[2, 8, 10, 11, 9,−2,−1, 10, 8, 9,−12,−11]

[2, 8, 10, 10, 8, 9, 9,−2,−1, 11,−12,−11]

Formally, this maps a list of length n to a list of the lame length:

[x1, . . . xi , . . . xk , . . . xj , . . . xl , . . . xn]

⇒ [x1, . . . xi , xj , . . . xl , xk , . . . xj−1, xi+1, . . . xk−1, xl+1, . . . xn]

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 20 / 29

dlad

The dlad operation is represented by finding a 4-tuple of pointers(xi , xj , xk , xl) where xi = xj , and xk = xl and i < k < j < l . Then, youtake the section xjxl , including the pointers, and the section in-between,but not including, the pointers xi and xk , and swapping them, setting uptwo ld-moves.

[2, 8, 10, 11, 9,−2,−1, 10, 8, 9,−12,−11]

[2, 8, 10, 10, 8, 9, 9,−2,−1, 11,−12,−11]

Formally, this maps a list of length n to a list of the lame length:

[x1, . . . xi , . . . xk , . . . xj , . . . xl , . . . xn]

⇒ [x1, . . . xi , xj , . . . xl , xk , . . . xj−1, xi+1, . . . xk−1, xl+1, . . . xn]

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 20 / 29

dlad

The dlad operation is represented by finding a 4-tuple of pointers(xi , xj , xk , xl) where xi = xj , and xk = xl and i < k < j < l . Then, youtake the section xjxl , including the pointers, and the section in-between,but not including, the pointers xi and xk , and swapping them, setting uptwo ld-moves.

[2, 8, 10, 11, 9,−2,−1, 10, 8, 9,−12,−11]

[2, 8, 10, 10, 8, 9, 9,−2,−1, 11,−12,−11]

Formally, this maps a list of length n to a list of the lame length:

[x1, . . . xi , . . . xk , . . . xj , . . . xl , . . . xn]

⇒ [x1, . . . xi , xj , . . . xl , xk , . . . xj−1, xi+1, . . . xk−1, xl+1, . . . xn]

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 20 / 29

dlad

The dlad operation is represented by finding a 4-tuple of pointers(xi , xj , xk , xl) where xi = xj , and xk = xl and i < k < j < l . Then, youtake the section xjxl , including the pointers, and the section in-between,but not including, the pointers xi and xk , and swapping them, setting uptwo ld-moves.

[2, 8, 10, 11, 9,−2,−1, 10, 8, 9,−12,−11]

[2, 8, 10, 10, 8, 9, 9,−2,−1, 11,−12,−11]

Formally, this maps a list of length n to a list of the lame length:

[x1, . . . xi , . . . xk , . . . xj , . . . xl , . . . xn]

⇒ [x1, . . . xi , xj , . . . xl , xk , . . . xj−1, xi+1, . . . xk−1, xl+1, . . . xn]

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 20 / 29

boundary-ld

The boundary ld move (b-ld) maps lists of length 4 onto lists of length 2.

It only operates on lists of the following form:

[x ,m,m′, x ]

where m,m′ ∈ {±µ,±λ} and x /∈ {±µ,±λ} is some pointer. It maps assuch:

[x ,m,m′, x ]⇒ [m′,m]

For example,[−2,−1,−3,−2]⇒ [−3,−1]

.A list is considered sorted if it is in the form [µ, λ] or [−λ,−µ] Thus, if aboundary-ld move is done, it will always be the final move in the sorting.

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 21 / 29

boundary-ld

The boundary ld move (b-ld) maps lists of length 4 onto lists of length 2.It only operates on lists of the following form:

[x ,m,m′, x ]

where m,m′ ∈ {±µ,±λ} and x /∈ {±µ,±λ} is some pointer.

It maps assuch:

[x ,m,m′, x ]⇒ [m′,m]

For example,[−2,−1,−3,−2]⇒ [−3,−1]

.A list is considered sorted if it is in the form [µ, λ] or [−λ,−µ] Thus, if aboundary-ld move is done, it will always be the final move in the sorting.

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 21 / 29

boundary-ld

The boundary ld move (b-ld) maps lists of length 4 onto lists of length 2.It only operates on lists of the following form:

[x ,m,m′, x ]

where m,m′ ∈ {±µ,±λ} and x /∈ {±µ,±λ} is some pointer. It maps assuch:

[x ,m,m′, x ]⇒ [m′,m]

For example,[−2,−1,−3,−2]⇒ [−3,−1]

.A list is considered sorted if it is in the form [µ, λ] or [−λ,−µ] Thus, if aboundary-ld move is done, it will always be the final move in the sorting.

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 21 / 29

boundary-ld

The boundary ld move (b-ld) maps lists of length 4 onto lists of length 2.It only operates on lists of the following form:

[x ,m,m′, x ]

where m,m′ ∈ {±µ,±λ} and x /∈ {±µ,±λ} is some pointer. It maps assuch:

[x ,m,m′, x ]⇒ [m′,m]

For example,[−2,−1,−3,−2]⇒ [−3,−1]

.A list is considered sorted if it is in the form [µ, λ] or [−λ,−µ] Thus, if aboundary-ld move is done, it will always be the final move in the sorting.

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 21 / 29

The Algorithm

(1) Map the signed permutation onto a list of signed pointers.

(2) Search for and do the first possible ld. If one is done, go to (2). If nold is found, go to (3).

(3) Check if a boundary-ld can be done. If it can, do it.

(4) Check if the list is sorted. If it is, end the program. Otherwise, go to(5)

(5) Search through the list, and keep memory of pairs ofoppositely-oriented pointers and of pairs of equally-oriented pointers.

(6) Search through the list of equally-oriented pointers for a possible dladmove. If one is found, do it and go to (2). If none is found, go to (7).

(7) Do the hi represented by the first element of the list ofequally-oriented pairs, then go to (2).

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 22 / 29

The Algorithm

(1) Map the signed permutation onto a list of signed pointers.

(2) Search for and do the first possible ld. If one is done, go to (2). If nold is found, go to (3).

(3) Check if a boundary-ld can be done. If it can, do it.

(4) Check if the list is sorted. If it is, end the program. Otherwise, go to(5)

(5) Search through the list, and keep memory of pairs ofoppositely-oriented pointers and of pairs of equally-oriented pointers.

(6) Search through the list of equally-oriented pointers for a possible dladmove. If one is found, do it and go to (2). If none is found, go to (7).

(7) Do the hi represented by the first element of the list ofequally-oriented pairs, then go to (2).

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 22 / 29

The Algorithm

(1) Map the signed permutation onto a list of signed pointers.

(2) Search for and do the first possible ld. If one is done, go to (2). If nold is found, go to (3).

(3) Check if a boundary-ld can be done. If it can, do it.

(4) Check if the list is sorted. If it is, end the program. Otherwise, go to(5)

(5) Search through the list, and keep memory of pairs ofoppositely-oriented pointers and of pairs of equally-oriented pointers.

(6) Search through the list of equally-oriented pointers for a possible dladmove. If one is found, do it and go to (2). If none is found, go to (7).

(7) Do the hi represented by the first element of the list ofequally-oriented pairs, then go to (2).

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 22 / 29

The Algorithm

(1) Map the signed permutation onto a list of signed pointers.

(2) Search for and do the first possible ld. If one is done, go to (2). If nold is found, go to (3).

(3) Check if a boundary-ld can be done. If it can, do it.

(4) Check if the list is sorted. If it is, end the program. Otherwise, go to(5)

(5) Search through the list, and keep memory of pairs ofoppositely-oriented pointers and of pairs of equally-oriented pointers.

(6) Search through the list of equally-oriented pointers for a possible dladmove. If one is found, do it and go to (2). If none is found, go to (7).

(7) Do the hi represented by the first element of the list ofequally-oriented pairs, then go to (2).

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 22 / 29

The Algorithm

(1) Map the signed permutation onto a list of signed pointers.

(2) Search for and do the first possible ld. If one is done, go to (2). If nold is found, go to (3).

(3) Check if a boundary-ld can be done. If it can, do it.

(4) Check if the list is sorted. If it is, end the program. Otherwise, go to(5)

(5) Search through the list, and keep memory of pairs ofoppositely-oriented pointers and of pairs of equally-oriented pointers.

(6) Search through the list of equally-oriented pointers for a possible dladmove. If one is found, do it and go to (2). If none is found, go to (7).

(7) Do the hi represented by the first element of the list ofequally-oriented pairs, then go to (2).

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 22 / 29

The Algorithm

(1) Map the signed permutation onto a list of signed pointers.

(2) Search for and do the first possible ld. If one is done, go to (2). If nold is found, go to (3).

(3) Check if a boundary-ld can be done. If it can, do it.

(4) Check if the list is sorted. If it is, end the program. Otherwise, go to(5)

(5) Search through the list, and keep memory of pairs ofoppositely-oriented pointers and of pairs of equally-oriented pointers.

(6) Search through the list of equally-oriented pointers for a possible dladmove. If one is found, do it and go to (2). If none is found, go to (7).

(7) Do the hi represented by the first element of the list ofequally-oriented pairs, then go to (2).

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 22 / 29

The Algorithm

(1) Map the signed permutation onto a list of signed pointers.

(2) Search for and do the first possible ld. If one is done, go to (2). If nold is found, go to (3).

(3) Check if a boundary-ld can be done. If it can, do it.

(4) Check if the list is sorted. If it is, end the program. Otherwise, go to(5)

(5) Search through the list, and keep memory of pairs ofoppositely-oriented pointers and of pairs of equally-oriented pointers.

(6) Search through the list of equally-oriented pointers for a possible dladmove. If one is found, do it and go to (2). If none is found, go to (7).

(7) Do the hi represented by the first element of the list ofequally-oriented pairs, then go to (2).

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 22 / 29

The Algorithm

(1) Map the signed permutation onto a list of signed pointers.

(2) Search for and do the first possible ld. If one is done, go to (2). If nold is found, go to (3).

(3) Check if a boundary-ld can be done. If it can, do it.

(4) Check if the list is sorted. If it is, end the program. Otherwise, go to(5)

(5) Search through the list, and keep memory of pairs ofoppositely-oriented pointers and of pairs of equally-oriented pointers.

(6) Search through the list of equally-oriented pointers for a possible dladmove. If one is found, do it and go to (2). If none is found, go to (7).

(7) Do the hi represented by the first element of the list ofequally-oriented pairs, then go to (2).

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 22 / 29

Some Theorems

Theorem

The algorithm runs in polynomial time. Specifically, the worst-casecomplexity is O(n3).

Theorem

A correctly-formed pointer list of length n > 4 is always in the domain ofof an hi, dlad, ld or boundary ld move.

Theorem

The algorithm will always find a path to either [µ, λ] or [−λ,−µ], an thuswill always terminate.

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 23 / 29

Some Theorems

Theorem

The algorithm runs in polynomial time. Specifically, the worst-casecomplexity is O(n3).

Theorem

A correctly-formed pointer list of length n > 4 is always in the domain ofof an hi, dlad, ld or boundary ld move.

Theorem

The algorithm will always find a path to either [µ, λ] or [−λ,−µ], an thuswill always terminate.

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 23 / 29

Some Theorems

Theorem

The algorithm runs in polynomial time. Specifically, the worst-casecomplexity is O(n3).

Theorem

A correctly-formed pointer list of length n > 4 is always in the domain ofof an hi, dlad, ld or boundary ld move.

Theorem

The algorithm will always find a path to either [µ, λ] or [−λ,−µ], an thuswill always terminate.

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 23 / 29

The Algorithm

We proved that a pointer list is always in the domain of a ciliate operationby contradiction.

We assumed the following conditions:

(1) For every i where 1 ≤ i < n, xi 6= xi+1. (ld moves can’t happen)

(2) For every |xi | = |xj |, xi = xj . (hi moves can’t happen)

(3) For every xi = xj and xk = xl where i < j and k < l the intervals(xi , xk) and (xj , xl) are either disjoint, or one is a proper subset of theother. (dlad moves can’t happen)

We then showed that it is impossible for a list to fit these conditions andto still be a pointer list.This, and the fact that the ciliate operations all produce pointer lists,prove that the algorithm halts.

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 24 / 29

The Algorithm

We proved that a pointer list is always in the domain of a ciliate operationby contradiction. We assumed the following conditions:

(1) For every i where 1 ≤ i < n, xi 6= xi+1. (ld moves can’t happen)

(2) For every |xi | = |xj |, xi = xj . (hi moves can’t happen)

(3) For every xi = xj and xk = xl where i < j and k < l the intervals(xi , xk) and (xj , xl) are either disjoint, or one is a proper subset of theother. (dlad moves can’t happen)

We then showed that it is impossible for a list to fit these conditions andto still be a pointer list.This, and the fact that the ciliate operations all produce pointer lists,prove that the algorithm halts.

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 24 / 29

The Algorithm

We proved that a pointer list is always in the domain of a ciliate operationby contradiction. We assumed the following conditions:

(1) For every i where 1 ≤ i < n, xi 6= xi+1. (ld moves can’t happen)

(2) For every |xi | = |xj |, xi = xj . (hi moves can’t happen)

(3) For every xi = xj and xk = xl where i < j and k < l the intervals(xi , xk) and (xj , xl) are either disjoint, or one is a proper subset of theother. (dlad moves can’t happen)

We then showed that it is impossible for a list to fit these conditions andto still be a pointer list.This, and the fact that the ciliate operations all produce pointer lists,prove that the algorithm halts.

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 24 / 29

The Algorithm

We proved that a pointer list is always in the domain of a ciliate operationby contradiction. We assumed the following conditions:

(1) For every i where 1 ≤ i < n, xi 6= xi+1. (ld moves can’t happen)

(2) For every |xi | = |xj |, xi = xj . (hi moves can’t happen)

(3) For every xi = xj and xk = xl where i < j and k < l the intervals(xi , xk) and (xj , xl) are either disjoint, or one is a proper subset of theother. (dlad moves can’t happen)

We then showed that it is impossible for a list to fit these conditions andto still be a pointer list.This, and the fact that the ciliate operations all produce pointer lists,prove that the algorithm halts.

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 24 / 29

The Algorithm

We proved that a pointer list is always in the domain of a ciliate operationby contradiction. We assumed the following conditions:

(1) For every i where 1 ≤ i < n, xi 6= xi+1. (ld moves can’t happen)

(2) For every |xi | = |xj |, xi = xj . (hi moves can’t happen)

(3) For every xi = xj and xk = xl where i < j and k < l the intervals(xi , xk) and (xj , xl) are either disjoint, or one is a proper subset of theother. (dlad moves can’t happen)

We then showed that it is impossible for a list to fit these conditions andto still be a pointer list.

This, and the fact that the ciliate operations all produce pointer lists,prove that the algorithm halts.

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 24 / 29

The Algorithm

We proved that a pointer list is always in the domain of a ciliate operationby contradiction. We assumed the following conditions:

(1) For every i where 1 ≤ i < n, xi 6= xi+1. (ld moves can’t happen)

(2) For every |xi | = |xj |, xi = xj . (hi moves can’t happen)

(3) For every xi = xj and xk = xl where i < j and k < l the intervals(xi , xk) and (xj , xl) are either disjoint, or one is a proper subset of theother. (dlad moves can’t happen)

We then showed that it is impossible for a list to fit these conditions andto still be a pointer list.This, and the fact that the ciliate operations all produce pointer lists,prove that the algorithm halts.

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 24 / 29

Data Analysis

The algorithm produced these numbers of each move for the Muller Aelement, shown with the known time since their divergence from D.melanogaster.

species hi dlad b-ld known divergence time

D. sechellia 7 488 0 5.4 myaD. simulans 8 268 0 5.4 myaD. erecta 3 170 0 12.6 myaD. yakuba 11 105 1 12.6 myaD. mojavensis 41 407 0 62 myaD. virilis 33 403 0 62 myaD. grimshawi 36 381 1 62 mya

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 25 / 29

Data Analysis

The algorithm produced these total numbers for every Muller elementadded together:

species hi dlad b-ld known divergence time

D. sechellia 15 1460 3 5.4 myaD. simulans 24 508 0 5.4 myaD. erecta 30 658 0 12.6 myaD. yakuba 47 438 1 12.6 myaD. mojavensis 189 986 2 62 myaD. virilis 194 1278 2 62 myaD. grimshawi 197 1459 3 62 mya

While this algorithm does not necessarily produce the shortest overall pathin the number of combined hi, dlad and ld moves, we conjecture that thisis the shortest possible number of hi moves.

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 26 / 29

Data Analysis

The algorithm produced these total numbers for every Muller elementadded together:

species hi dlad b-ld known divergence time

D. sechellia 15 1460 3 5.4 myaD. simulans 24 508 0 5.4 myaD. erecta 30 658 0 12.6 myaD. yakuba 47 438 1 12.6 myaD. mojavensis 189 986 2 62 myaD. virilis 194 1278 2 62 myaD. grimshawi 197 1459 3 62 mya

While this algorithm does not necessarily produce the shortest overall pathin the number of combined hi, dlad and ld moves, we conjecture that thisis the shortest possible number of hi moves.

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 26 / 29

The Future

Here are some further areas we have to explore:

1. We conjecture that our algorithm minimizes the number of hi movesin its reversal path. We have yet to prove this.

2. Develop an algorithm to find the shortest possible ciliate operationpath.

3. Look at more species in the Drosophila genus and see if the correlationbetween ciliate operation path length and divergence time holds.

4. Explore the possibility of using ciliate operations to solve mathematicalproblems, such as the word and conjugacy problems in groups.

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 27 / 29

The Future

Here are some further areas we have to explore:

1. We conjecture that our algorithm minimizes the number of hi movesin its reversal path. We have yet to prove this.

2. Develop an algorithm to find the shortest possible ciliate operationpath.

3. Look at more species in the Drosophila genus and see if the correlationbetween ciliate operation path length and divergence time holds.

4. Explore the possibility of using ciliate operations to solve mathematicalproblems, such as the word and conjugacy problems in groups.

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 27 / 29

The Future

Here are some further areas we have to explore:

1. We conjecture that our algorithm minimizes the number of hi movesin its reversal path. We have yet to prove this.

2. Develop an algorithm to find the shortest possible ciliate operationpath.

3. Look at more species in the Drosophila genus and see if the correlationbetween ciliate operation path length and divergence time holds.

4. Explore the possibility of using ciliate operations to solve mathematicalproblems, such as the word and conjugacy problems in groups.

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 27 / 29

The Future

Here are some further areas we have to explore:

1. We conjecture that our algorithm minimizes the number of hi movesin its reversal path. We have yet to prove this.

2. Develop an algorithm to find the shortest possible ciliate operationpath.

3. Look at more species in the Drosophila genus and see if the correlationbetween ciliate operation path length and divergence time holds.

4. Explore the possibility of using ciliate operations to solve mathematicalproblems, such as the word and conjugacy problems in groups.

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 27 / 29

Bibliography

Sridhar Hannenhalli, Pavel A. Pevzner Transforming Cabbage intoTurnip: Polynomial Algorithm for Sorting Signed Permutations byReversals. Journal of the ACM, Vol. 46, No. 1, 1999.

Pavel Pevzner, Glenn Tesler Genome Rearrangements in MammalianEvolution: Lessons from Human and House Henomes. GenomeResearch, Vol. 13, 2003.

Arjun Bhutkar, Stephen W. Schaeffer, Susan M. Russo, Mu Xu,Temple F. Smith, William M. Gelbart Chromosomal RearrangementInferred From Comparisons of 12 Drosophila Genomes. Genetics, Vol197, 2008.

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 28 / 29

Bibliography (cont.)

Jose M. Ranz, Damien Maurin, Yuk S. Chan, Marchin Von Grotthuss,LeDeana W. Hillier, John Roote, Michael Ashburner, Casey M.Bergman Principles of Genome Evolution in the Drosophilamelanogaster Species Group. PLoS Biology, Vol. 5, Issue 6, 2007.

Andrzej Ehrenfeucht, Tero Harju, Ion Petre, David M. Prescott,Grzegorz Rozenberg Computation in Living Cells. Springer-VerlagBerlin Heidelberg, 2004.

S. Tweedie, M. Ashburner, K. Falls, P. Leyland, P. McQuilton, S.Marygold, G. Millburn, D. Osumi-Sutherland, A. Schroeder, R. Seal,H. Zhang and The FlyBase Consortium FlyBase: enhancing DrosophilaGene Ontology annotations. Nucleic Acids Research, Vol. 37, 2009.

J. Herlin, A. Nelson and M. Scheepers () Phylogenetic Relationships July 29, 2011 29 / 29