Dr. Jonathan Burns
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## Biological Background

Some genera of ciliates, such as Oxytricha and Stylonychia, undergo massive genome rearrangements and provide model organisms to study DNA recombination. A common feature of all ciliates is the presence of two types of nuclei: a germline micronucleus (MIC) and a transcritionally-active somatic macronucleus (MAC). During sexual reproduction, the old parental macronucleus disintegrates and a new macronucleus forms from a copy of the micronucleus. During this process, macronuclear chromosomes assemble through DNA processing events that delete 90-98% of the DNA content of the micronucleus. The precursor DNA regions in the micronucleus are frequently interrupted by non-coding DNA segments, and each macronuclear locus may be present in the micronucleus as several nonconsecutive, permuted, and possibly inverted DNA segments.

The precursor micronuclear segments which are retained in the product macronuclear genome are called macronuclear-destined sequences (MDSs). The last few nucleotides of an MDS are repeated in the beginning of the next consecutive MDS. This mutual sequence is called a pointer sequence for the two MDSs, and possibly helps to guide the recombination process.

Figure 1: Ciliates can be found in freshwater ponds all over the world.

Figure 2: During conjugation, the precursor MIC segments (MDSs) rearrange to form the new product MAC genome. The regions where the MDSs overlap in the product MAC are the pointers.

## Rearrangement Models

The reordering of precursor sequences into product sequences can be represented abstractly in several forms:

Rearrangment Maps and Patterns.

The ordering and orientation of the precursor MIC MDSs, relative to the product MAC sequence, can be recorded as a rearrangement map. The rearrangement map $$\alpha$$ for the scrambled loci in Figure 2 is $$\alpha = \overline{M}_1 M_2 M_3 M_5 M_4,$$ where the bar over $$\overline{M}_1$$ denotes that MDS 1 is inverted.

The mutually oriented, consecutive MDSs in a rearrangement map — i.e., $$M_i M_{i+1}$$ and $$\overline{M}_{i+1} \overline{M}_i$$ — do not increase its scrambling complexity, so one may form the reduced map of a rearrangement map by merging these MDSs and relabeling its order indices. For example, the reduced map $$\alpha^R$$ of $$\alpha$$ is $$\alpha^R = \overline{M}_1 M_2 M_4 M_3.$$ Note that the non-scrambled rearrangement maps $$M_1 M_2 \dots M_n$$ and $$\overline{M}_n \overline{M}_{n-1} \dots \overline{M}_1$$ simplify to the reduced maps $$M_1$$ and $$\overline{M}_1$$, respectively.

Each rearrangement map is labeled according to the implied orientations of the precursor and product sequences. However these orientations are somewhat arbitrary, in the absence a further biological criterium, since both the precusor and product sequences represent double stranded DNA.

Figure 3: A rearrangement pattern correponding to the rearrangement maps $$\alpha = M_2 \overline{M}_3 M_1 M_4$$, $$\alpha^I = \overline{M}_4 \overline{M}_1 M_3 \overline{M}_2$$, $$\alpha^A = \overline{M}_3 M_2 \overline{M}_4 \overline{M}_1$$, and $$\alpha^{AI} = M_1 M_4 \overline{M}_2 M_3$$ which correspond to the choice of precursor and product strands 1 & 3, 2 & 3, 1 & 4, and 2 & 4, respectively.

Assembly Graphs.

The precursor and product sequences of a rearrangement pattern can be represented as a path and a transversal, respectively, of a spatial graph with 4-valent rigid vertices, called an assembly graph. The vertices of the assembly graph represent the pointers, and the edges correpsond with the rest of the precursor regions.

Orienting the assembly graph corresponds to choosing an orientation of the precursor DNA, and orienting the Hamitonian path corresponds to choosing an orientation the for the product DNA.

Figure 4: The assembly graph representing the rearrangement map in Figure 3.

Double Occurrence Words.

Chord Diagrams.