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The gapped local alignment score of two random sequences follows a Gumbeldistribution. If computers could estimate the parameters of the Gumbeldistribution within one second, the use of arbitrary alignment scoring schemescould increase the sensitivity of searching biological sequence databases overthe web. Accordingly, this article gives a novel equation for the scaleparameter of the relevant Gumbel distribution. We speculate that the equationis exact, although present numerical evidence is limited. The equation involvesascending ladder variates in the global alignment of random sequences. Inglobal alignment simulations, the ladder variates yield stopping timesspecifying random sequence lengths. Because of the random lengths, and becauseour trial distribution for importance sampling occurs on a different samplespace from our target distribution, our study led to a mapping theorem, whichled naturally in turn to an efficient dynamic programming algorithm for theimportance sampling weights. Numerical studies using several popular alignmentscoring schemes then examined the efficiency and accuracy of the resultingsimulations.

Estimating the Gumbel scale parameter for local alignment of random sequences by importance sampling with stopping times