Algebraic double cut and join

Abstract

Establishing a distance between genomes is a significant problem in computational genomics, because its solution can be used to establish evolutionary relationships including phylogeny. The “double cut and join” (DCJ) model of chromosomal rearrangement proposed by Yancopoulos et al. (Bioinformatics 21:3340–3346, 2005) has received attention as it can model inversions, translocations, fusion and fission on a multichromosomal genome that may contain both linear and circular chromosomes. In this paper, we realize the DCJ operator as a group action on the space of multichromosomal genomes. We study this group action, deriving some properties of the group and finding group-theoretic analogues for the key results in the DCJ theory.

Appendix: Some results about symmetric groups

This paper uses some standard results on symmetric groups that we collect here for ease of reference. More details on these results can be found in many undergraduate group theory textbooks, for example Fraleigh (2003).

A permutation is a bijection from a set $S$ to itself. $S$ is usually taken to be a set of natural numbers $\mathbf {n}=\{1,2,\ldots ,n\}$. A permutation can be written by specifying the value of the map on all the points.

Example

$$\begin{aligned} \pi = \Big (\begin{matrix} 1 &{}\quad 2 &{}\quad 3 &{}\quad 4 &{}\quad 5 &{}\quad 6 \\ 3 &{}\quad 4 &{}\quad 1 &{}\quad 6 &{}\quad 2 &{}\quad 5 \end{matrix}\Big ) \end{aligned}$$

is a permutation on the set $\{1,2,3,4,5,6\}$ that sends 1 to 3, 2 to 4, etc. The set of all permutations on the set ${\mathbf {n}}$ forms a group called the symmetric group and denoted by $S_{n}$.

1.1 Permutation multiplication

Since permutations are simply bijective functions, permutation multiplication is function composition. That is, to find the image of $i$ in the product $\pi _2\pi _1$, we do $\pi _2 \left( \pi _1(i)\right)$.

Example

Let $\pi _1 = \Big (\begin{matrix} 1 &{} 2 &{} 3 &{} 4 &{} 5 \\ 3 &{} 4 &{} 1 &{} 5 &{} 2 \end{matrix}\Big )$ and $\pi _2= \Big (\begin{matrix} 1 &{} 2 &{} 3 &{} 4 &{} 5 \\ 2 &{} 1 &{} 4 &{} 3 &{} 5 \end{matrix}\Big )$. The image of $1$ in the product $\pi _2\pi _1$ is $\pi _2 \left( \pi _1(1)\right) =\pi _2(3)=4$. So for each $i$, we have to “follow the string” – $\pi _1$ send $i$ to $j$, $\pi _2$ sends $j$ to $k$, so $i$ gets sent to $k$ by $\pi _2\pi _1$.

$$\begin{aligned} \Big (\begin{matrix} 1 &{} 2 &{} 3 &{} 4 &{} 5 \\ 2 &{} 1 &{} 4 &{} 3 &{} 5 \end{matrix}\Big ) \Big (\begin{matrix} 1 &{} 2 &{} 3 &{} 4 &{} 5 \\ 3 &{} 4 &{} 1 &{} 5 &{} 2 \end{matrix}\Big )= \Big (\begin{matrix} 1 &{} 2 &{} 3 &{} 4 &{} 5 \\ 4 &{} 3 &{} 2 &{} 5 &{} 1 \end{matrix}\Big ). \end{aligned}$$

1.2 Inverse of a permutation

Informally, a permutation $\pi \in S_{n}$ scrambles the elements of ${\mathbf {n}}$. The inverse of $\pi$ is the permutation that “undoes” the scrambling. Formally we define the identity permutation $\iota$ to be the permutation that maps $i$ to $i$ for all $i \in {\mathbf {n}}$.

Definition 8.1

(Inverse) Let $\pi \in S_{n}$. Then the inverse of $\pi$ is the permutation $\pi ^{-1}$ such that

$$\begin{aligned} \pi \pi ^{-1}=\iota \quad \text {and}\quad \pi \pi ^{-1}=\iota . \end{aligned}$$

If $\pi ^{-1}$ is the inverse of $\pi$ then $\pi$ is the inverse of $\pi ^{-1}$. That is, $(\pi ^{-1})^{-1}=\pi$. In general, $(\pi _1\pi _2)^{-1}=\pi _2^{-1}\pi _1^{-1}.$

1.3 Cycles and cycle decomposition

For a permutation $\pi \in S_{n}$, if we repeatedly apply $\pi$ to any $i \in {\mathbf {n}}$,

$$\begin{aligned} i \mathop {\rightarrow }\limits ^{\pi } \pi (i) \mathop {\rightarrow }\limits ^{\pi } \pi ^2(i) \ldots , \end{aligned}$$

we must eventually (say after $k$ steps) reach $i$ again since ${\mathbf {n}}$ is a finite set. If there is some $j \in {\mathbf {n}}$ which does not occur in this sequence, then we can form a similar sequence for $j$, and keep doing this until every element of ${\mathbf {n}}$ occurs in some sequence.

Definition 8.2

(Cycle) Let $i_1,i_2,\ldots i_k$ be $k$ distinct integers in ${\mathbf {n}}$. A cycle $\pi _c$ written as $(i_1,i_2,\ldots ,i_k)$ is a permutation in $S_n$ defined as

$$\begin{aligned} \pi _c(i_s):= {\left\{ \begin{array}{ll} i_{s+1} &{} \text { if } i_s \in \{i_1,i_2,\ldots i_{k-1}\}\\ i_1 &{} \text { if } i_s=i_k \\ i_s &{} \text { otherwise.} \end{array}\right. } \end{aligned}$$

A 2-cycle is a cycle of length 2. That is, $\pi =(i,j)$ means that $\pi (i)=j,\pi (j)=i$ and $\pi (k)=k$ if $k \ne i,j$. A cycle of length 2 is also called a transposition.

Two cycles are said to be disjoint if they have no elements in common.

Theorem 8.3

Any permutation $\pi \in S_{n}$ can be written as a product of disjoint cycles.

Example

Let $\pi = \Big (\begin{matrix} 1 &{} 2 &{} 3 &{} 4 &{} 5 &{} 6 \\ 3 &{} 4 &{} 1 &{} 6 &{} 2 &{} 5 \end{matrix}\Big )$. $\pi$ can be written as

$$\begin{aligned} \pi =(1,3)(2,4,6,5). \end{aligned}$$

This way of writing a permutation is referred to as cycle notation. There is a unique way of writing a permutation as a product of disjoint cycles, up to the ordering of the cycles (they commute) and cyclic equivalence of each cycle (e.g. $(1,2,3)=(2,3,1)=(3,1,2)$). Since the sizes of the disjoint cycles will always add to $n$ (including if necessary some 1-cycles), we can define the cycle type as follows.

Definition 8.4

(Cycle type) The cycle type of a permutation $\pi$ is the partition $\lambda \vdash n$ whose components are the sizes of the cycles in the disjoint cycle decomposition of $\pi$.

Example

The cycle type of $\pi =(1,3)(2,4,6,5)$ is (4, 2) since it has one cycle of length 2 and one cycle of length 4.

1.4 Conjugation

Definition 8.5

Let $\pi ,g \in S_{n}$. The conjugate of $\pi$ by $g$ is defined to be the permutation $g \pi g^{-1}$, and we say that $\pi$ and $g\pi g^{-1}$ are conjugate permutations.

Theorem 8.6

Let $\pi _1$ and $\pi _2$ be permutations on the set $\mathbf {n}$, then $\pi _1$ and $\pi _2$ are conjugate in $S_{n}$ if and only if they have the same cycle type.

1.5 Permutation as product of transpositions

Theorem 8.7

Any permutation $\pi \in S_{n}$ can be written as a product of transpositions.

Example

The permutation $\pi =(1,3)(2,4,6,5)$ can be written as

$$\begin{aligned} \pi =(1,3)(2,4,6,5)=(1,3)(2,5)(2,6)(2,4). \end{aligned}$$

While the decomposition of a permutation into a product of disjoint cycles is unique, the decomposition of a permutation into a product of transpositions is not unique. However the number of transpositions used must be either always be even, or always be odd.

Theorem 8.8

A permutation $\pi \in S_n$ can be expressed as a product of either an even number of transpositions or an odd number of transpositions, but not both.

Definition 8.9

A permutation is said to be even if it can be written as a product of an even number of transpositions. Otherwise it is said to be an odd permutation.