Topoisomerases maintain the proper topological state of DNA. Human topoisomerase I removes DNA supercoils by clamping a duplex DNA segment, nicking one strand at a phosphodiester bond, covalently attaching to the 3' end of the nick, and allowing the DNA downstream of the cut to rotate around the intact strand. Using molecular dynamics simulations and umbrella sampling free energy calculations, we show that the rotation of downstream DNA in the grip of the enzyme that brings about release of positive or negative supercoils occurs by thermally assisted diffusion on ratchet energy profiles. The ratchetlike free-energy-versus-rotation profile that we compute provides a model for the function of topoisomerase in which the periodic maxima along the profile modulate the rate of supercoil relaxation, while the minima provide metastable conformational states for DNA religation. The results confirm previous experimental and computational work, and suggest that relaxation of the two types of supercoils involves distinct protein pathways. Additionally, simulations performed with the ternary complex of topoisomerase, DNA, and the chemotherapeutic drug topotecan show important differences in the mechanisms for supercoil relaxation when the drug is present, accounting for the relative values of relaxation rates measured in single-molecule experiments. Good agreement is found between rate constants from tweezer experiments and those calculated from simulations. Evidence is presented for the existence of semiopen states of the protein, which facilitate rotations after the initial one, as a result of biasing the protein into a conformation more favorable to strand rotation than the closed state required for nicking of the DNA.