Context Directed Reversals and the Ciliate Decryptome

Abstract

Prior studies of the efficiency of the block interchange (swap) and the reversal sorting operations on (signed) permutations identified specialized versions of the these operations. These specialized operations are here called context directed reversal, abbreviated cdr, and context directed swap, abbreviated cds. Prior works have also characterized which (signed) permutations are sortable by cdr or by cds.It is now known that when a permutation π is cds sortable in n steps, then any n consecutive applicable cds operations will sort π. Examples show that this is not the case for cdr. This phenomenon is the focus of this paper. It is proven that if a signed permutation is cdr sortable, then any cdr fixed point of it is cds sortable (the cds Rescue Theorem). The cds Rescue Theorem is discussed in the context of a mathematical model for ciliate micronuclear decryption.It is also known that if applications of cds to a permutation π reaches a cds fixed point in n steps, then any n consecutive applicable cds operations will terminate in a cds fixed point of π. This is not the case for cdr: It is proven that though for a given signed permutation the number of cdr operations leading to different cdr fixed points may be different from each other, the parity of the number of operations is the same (the cdr Parity Theorem). This result provides a solution to two previously formulated decision problems regarding certain combinatorial games.

Sorting by reversals and the theory of 4-regular graphs

“…For example, in both theories local complementation plays an important role -indeed, as we mentioned in Section 6 the Hannenhalli-Pevzner theorem was discovered independently in both theories. We remark that links between gene assembly in ciliates and sorting by reversals have also been appreciated in [18,29,30].…”

“…Since the theory of gene assembly in ciliates can also be fit into the theory of 4-regular multigraphs, we have also extended the links between the theories of gene assembly in ciliates and sorting by reversals as observed in [18,29,30].…”