Proportion of Gaps and Fluctuations of the Optimal Score in Random Sequence Comparison

We study the asymptotic properties of optimal alignments when aligning two independent i.i.d. sequences over finite alphabet. Such kind of alignment is an important tool in many fields of applications including computational molecular biology. We are particularly interested in the (asymptotic) proportion of gaps of the optimal alignment. We show that when the limit of the average optimal score per letter (rescaled score) is considered as a function of the gap penalty, then given a gap penalty, the proportion of the gaps converges to the derivative of the limit score at that particular penalty. Such an approach, where the gap penalty is allowed to vary, has not been explored before. As an application, we solve the long open problem of the fluctuation of the optimal alignment in the case when the gap penalty is sufficiently large. In particular, we prove that for all scoring functions without a certain symmetry, as long as the gap penalty is large enough, the fluctuations of the optimal alignment score are of order square root of the length of the strings. This order was conjectured by Waterman [Phil. Trans. R. Soc. Lond. B 344(1):383–390, 1994] but disproves the conjecture of Chvatal and Sankoff in [J. Appl. Probab. 12:306–315, 1975].