Optimal alignments of longest common subsequences and their path properties

TL;DR

This paper studies the properties of optimal alignment paths for sequences, showing that related sequences have smaller differences in extremal alignments and that these differences grow logarithmically, which can indicate sequence relatedness.

Contribution

The paper introduces the existence of extremal optimal alignments and demonstrates their differences as a way to characterize sequence relatedness, supported by theoretical proofs and simulations.

Findings

Related sequences have smaller extremal alignment differences.

Differences between extremal alignments grow logarithmically for homologous sequences.

Simulations support theoretical results on alignment path properties.

Abstract

We investigate the behavior of optimal alignment paths for homologous (related) and independent random sequences. An alignment between two finite sequences is optimal if it corresponds to the longest common subsequence (LCS). We prove the existence of lowest and highest optimal alignments and study their differences. High differences between the extremal alignments imply the high variety of all optimal alignments. We present several simulations indicating that the homologous (having the same common ancestor) sequences have typically the distance between the extremal alignments of much smaller size than independent sequences. In particular, the simulations suggest that for the homologous sequences, the growth of the distance between the extremal alignments is logarithmical. The main theoretical results of the paper prove that (under some assumptions) this is the case, indeed. The paper suggests that the properties of the optimal alignment paths characterize the relatedness of the sequences.