An Upper Bound on the Convergence Rate of a Second Functional in Optimal Sequence Alignment

TL;DR

This paper establishes a probabilistic upper bound of order n^{0.75} on the deviation of optimal sequence alignment scores relative to a second scoring function, advancing understanding of alignment microstructure.

Contribution

It provides the first probabilistic upper bound on the deviation of optimal alignment scores relative to a second scoring function, extending existing alignment theory.

Findings

Upper bound of order n^{0.75} on score deviation

Advances understanding of alignment microstructure

Extends previous theoretical work

Abstract

Consider finite sequences $X_{[1,n]}=X_1\dots X_n$ and $Y_{[1,n]}=Y_1\dots Y_n$ of length $n$, consisting of i.i.d.\ samples of random letters from a finite alphabet, and let $S$ and $T$ be chosen i.i.d.\ randomly from the unit ball in the space of symmetric scoring functions over this alphabet augmented by a gap symbol. We prove a probabilistic upper bound of linear order in $n^{0.75}$ for the deviation of the score relative to $T$ of optimal alignments with gaps of $X_{[1,n]}$ and $Y_{[1,n]}$ relative to $S$. It remains an open problem to prove a lower bound. Our result contributes to the understanding of the microstructure of optimal alignments relative to one given scoring function, extending a theory begun by the first two authors.