A Distribution Function Arising in Computational Biology

Karlin and Altschul in their statistical analysis for multiple highscoring segments in molecular sequences introduced a distribution function which gives the probability there are at least r distinct and consistently ordered segment pairs all with score at least x. For long sequences this distribution can be expressed in terms of the distribution of the length of the longest increasing subsequence in a random permutation. Within the past few years, this last quantity has been extensively studied in the mathematics literature. The purpose of this note is to summarize these new mathematical developments in a form suitable for use in computational biology. Dedicated to Barry McCoy on the occasion of his sixtieth birthday. 1 The Distribution Function Karlin and Altschul [8] in their statistical analysis for multiple high-scoring segments in molecular sequences, introduced the following distribution function: Let F (r; y) denote the probability that there are at least r distinct and consistently ordered segment pairs all with score at least x. They further introduced a parameter y = KNe−λx where K and λ are parameters related to the scoring system, see [8] for details. We use the parameter y without further reference to x. For long sequences (N → ∞) this distribution function is well approximated by [8] F (r; y) = e−y ∞