Sampling rare events: Statistics of local sequence alignments

A method to calculate probability distributions in regions where the events are very unlikely (e.g., $p\sim {10}^{-40})$ is presented. The basic idea is to map the underlying model on a physical system. The system is simulated at a low temperature, such that preferably configurations with originally low probabilities are generated. Since the distribution of such a physical system is known, the original unbiased distribution can be obtained. As an application, local alignment of protein sequences is studied. The deviation of the distribution $p\left(S\right)\mathrm{}$ of optimum scores from the extreme-value distribution is quantified. This deviation decreases with growing sequence length.