In this paper, we use the term gene to represent an atomic evolutionary unit that has never been broken due to breakpoints caused by any operations (duplication or rearrangement). If two genes are derived from a common ancestral gene, then they belong to the same gene family. We use g[x] to represent the gene x in genome g. Also, if two genes from the same family x are in the same genome g, then we denote these genes as g[x.i] and g[x.j] (i = j). A chromosome of a modern or ancestral genome consists of a list of genes where each gene has a sign (orientation) that is either positive (+) or negative ( —). The reverse complement of a chromosome is obtained by reversing the list and flipping the sign of each gene. A genome is a set of chromosomes.
If genome g contains gene x, then the predecessor pg (x) is defined as the gene that immediately precedes x on the same chromosome. Predecessor has a sign. In the opposite orientation, pg (—x) immediately precedes —x in the reverse complement of the same chromosome. We set pg(x) = <&a if x appears first on a chromosome. The successor sg (x) of x is defined analogously. And we also set sg (x) = if x appears last on a chromosome. For instance, let g have the chromosome (1 —4.1 —3 4.2 5 2). Then pg(1) = $A, pg(2) = 5, pg(—3) = —4.1, Sg(—4.1) = —3, pg( —1) = 4.1, Sg(—5) = —4.2, etc.
In addition to speciation events, the original ancestral genes evolve through large-scale evolutionary operations which include insertion/deletion, rearrangements (inversion, translocation, fusion/fission), and tandem and segmental duplications. Consequently, we have a different number of genes and different gene orders in present day genomes. Our goal is to reconstruct the order and orientation of genes in the target ancestral genome. We call each reconstructed chromosome a contiguous ancestral region (CAR).
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