An analytic study of the phase transition line in local sequence alignment with gaps

A detailed analytic study of the log-linear phase transition of the Smith–Waterman local alignment algorithm is presented. A rectangular alignment lattice is introduced to facilitate the statistical analysis for alignment with gaps. With a few simplifying assumptions, we obtain an analytic expression for the loci of the phase transition line. Our result reproduces the exact and conjectured values for the very large and very small gap costs; the latter corresponds to the related problem of the longest common subsequence. For intermediate values of gap costs, our result is not exact, although a comparison to numerical results yielded a difference of no more than several percent.