A number of equally optimal alignments inherently exist in the sequence and structure comparisons among proteins. To represent the sub-optimal alignments systematically, we have developed a method of generating probabilistic alignments for sequences and structures, by which the correspondence between pairs of residues is evaluated in a probabilistic manner. Our method uses the periodic boundary condition to avoid the entropy artifact favoring full-length matches. In the structure comparison, the environmental effects are incorporated by the mean-field approximation. We applied this method in comparisons of two pairs of proteins with internal symmetry; the first set were proteins of TIM-barrel fold and the second were β-trefoil fold. These pairs are expected to have distinct sub-optimal alignments suitable for probabilistic description with the periodic boundary. It was shown that the sequence and structure alignments are consistent with each other and that the alignments with the highest probability represent circular permutation.
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