and the upstream parts feel the same torque in opposite directions when DNA is nicked. Therefore, the upstream part should also be treated equally, and a restraining potential should be applied in some format. Therefore, we focused on the results of the simulation of these four systems, where both downstream and upstream duplexes are treated with external potentials. Normal mode analysis The structure at the end of the equilibration phase of molecular dynamics is minimized until a root-mean-square gradient of 10 5 kcal/mol/A˚ is reached in a series of 7500 cycles of steepest descent, followed by 8000 steps of the adopted basis Newton–Raphson procedure. This number of steps is needed such that, after discarding the six rigid-body modes, no negative eigenvalues are obtained. The lowest 1500 modes have been determined on this minimized structure by a iter- ative diagonalization scheme of the Hessian, where repetitive reduced-basis diagonalizations have been performed. These reduced basis is constructed partially from the not yet con- verged eigenvectors in Cartesian coordinates. This provides very efﬁcient large calculations on big systems (such as the one used here) by consuming less memory and will make possible repetitive calculations for a multitude of suggested mutants. The details of the procedure are given in (44). The projections of the lowest modes are done on the different reaction coordinates, chosen to be multidimensional vectors that correspond to the difference between the coordinates of the initial and the ﬁnal structures. Ordinary dot product of each mode with the reaction coordinates were performed. RESULTS Molecular dynamics simulations were run on a covalent com- plex with external constraining potentials on DNA (mimicking torque on either positively or negatively supercoiled DNA) that pushed it harmonically towards a target position obtained by applying, in 10 increments, full-circle (clockwise or counter-clockwise) rotations to the scissile strand, around an axis passing through the intact one. Six distinct DNA- topo-solvent systems (containing 86 694 atoms each, including water and counterions, and differing in how and on which region the constraints were applied, see Table 1) were simu- lated for 1.8 ns each. Out of the six runs, only four of them (system 1–4) were adept at relaxing supercoils. For the other two, the upstream DNA part was allowed to move, and it followed (together with the covalently attached protein) to a large extent the rotation of the downstream part, resulting in no effective relaxation. (For computational details, see the Materials and Methods section.) Although the ‘controlled-rotation’ mechanism suggests that the downstream duplex rotates around the intact strand (21), we have also tested an alternative model, in which DNA rotation is performed around the helical axis. This altern- ative model was however untenable due to the large under- lying free energy proﬁle relative to the strand rotation scheme (exceeding 70 kcal/mol for a mere sixth of a full rotation). Rotations around the helical axis were therefore ruled out. Nucleic Acids Research, 2005, Vol. 33, No. 20 6625 Distinct protein motions involved in positive versus negative supercoil relaxation Whether the protein, during strand rotation, stays in the closed form or opens its lips has been a focus of two recent studies with seemingly contradictory conclusions. In 2003, Carey et al. (45) reported the engineering of two cysteines into the oppos- ing lips that, after DNA-binding, formed a disulﬁde bond (a ‘distal clamp’, in reference to its relative distance from the active site, see Figure 1D) that sealed the two lips. They observed that the rate of DNA relaxation in the clamped form of the protein is comparable to the unclamped one, and con- cluded that relaxation occurs in a closed conformation. How- ever, later in the same year, Woo et al. (46) published the design of a disulﬁde bond that clamped the two lips at a position that was closer to both the active site Tyr723 and to the bound DNA duplex; we refer to it as the ‘proximal clamp’ (see Figure 1D). Contrary to what was found for the distal clamp, they observed that DNA rotation is inhibited within the locked protein. Because the above studies showed that the two subtle altera- tions in the ﬂexibility of the enzyme impacted differently on the enzyme’s activity, it is of fundamental interest to study the extent of protein breathing (or opening) during strand rotation. Does the wild-type protein stay or not in the closed clamp conformation during DNA rotation? If it does open up, then how and how much? Previously, any possible opening was thought to occur by parting the two lips, so it was these regions that have been sealed in the clamped mutants. We stress that, importantly, both the Carey et al. and the Woo et al. clamped mutants used positively supercoiled DNA substrates. We have set out to assess the opening of the protein by direct simulations of DNA rotation, in both the positive and negative direction, and have revealed unexpected structural dynamics (see the two movies in Supplementary Data). We have found that, while to relax positive supercoils the enzyme does tend to open up by a gradual separation of the two lips (see Figure 2A), the situation is quite different in the case of removal of neg- ative supercoils (see Figure 2B). For this case, our simulations indicate that the upper and lower cap open up by stretching the hinge region (from Leu429 to Lys436, positioned diametric- ally opposite the lips across the DNA helical cross-section), while the two lips do not signiﬁcantly change their relative positions. We ﬁnd thus that negative supercoil rotations induce the reverse opening motion relative to the opening of the lips induced by rotations of positive supercoils (see Figure 1C). Moreover, the opening amplitudes imparted by the two types of supercoiling are similar; the hinge stretching and the lip opening distances are both within 10–14 A˚ . This result is reproduced in all four of our relaxation-competent runs (systems 1–4, see Table 1). The relevant distances are plotted, as a function of DNA rotation, in the insets to Figure 2A and B. It is of interest that the hinge region hypothesized here to be important for negative supercoil relaxation, is also believed (21) to be involved in the hinge-bending motion needed to achieve an open conformation of the DNA-free enzyme that can part the lip regions wide enough to allow the entry of the duplex strand (opening step in Figure 1C). This might imply that this region has been designed to be ﬂexible both as a hinge and a stretch. What are the interaction details that bring about the two distinct mechanisms? The nose-cone helices (in the upper cap)