Analysis of internal loops within the RNA secondary structure in almost quadratic time

Motivation: Evaluating all possible internal loops is one of the key steps in predicting the optimal secondary structure of an RNA molecule. The best algorithm available runs in time O(L-3), L is the length of the RNA. Results: We propose a new algorithm for evaluating internal loops, its run-time is O(M*log(2)L), M < L-2 is a number of possible nucleotide pairings. We created a software tool Afold which predicts the optimal secondary structure of RNA molecules of lengths up to 28 000 nt, using a computer with 2 Gb RAM. We also propose algorithms constructing sets of conditionally optimal multi-branch loop free (MLF) structures, e.g. the set that for every possible pairing (x, y) contains an optimal MLF structure in which nucleotides x and y form a pair. All the algorithms have run-time O(M*log(2)L).