6630 Nucleic Acids Research, 2005, Vol. 33, No. 20 Figure 4. Average RMS deviations for all four systems; (A) for system 1, (B) for system 2, (C) for system 3, and (D) for system 4. RMSDs were calculated on heavy atoms of protein’s domains, and plotted as a function of the rotation angle. Legends in inset are the same for all panels. RMSDs were calculated by superimposing the coordinates of a domain (at every 100 MD steps) on the initial coordinates at the beginning of the rotation. directions of internal motions; see Materials and Methods for technical details). Such analyses have provided considerable insight into the nature of collective motions in many proteins (58–62) and have showed that, in most systems where an initial and a ﬁnal structure are available, the ﬁrst few low frequency modes are sufﬁcient to describe the large-scale con- formational changes involved in going from one structure to the other (63,64). This strategy has worked well, including for protein–DNA complexes (65), as well as for systems as large as the ribosome (66). The difference vectors between the initial and the ﬁnal (after a 360 DNA rotation) of the protein for the 4 relaxation- competent systems we have studied have been regarded as reaction coordinates, and the lowest 100 modes have been projected on the these reaction coordinates. The percent con- tributions to the reaction coordinate are plotted in Figure 5A (system 1 and 2), and the corresponding plot is presented in Figure 5B (for system 3 and 4). Signiﬁcant contributions appear especially from the lowest 30 non-zero modes. As the reaction coordinate describes the breathing of the protein (whether this involves opening of the lips or stretching of the hinge), the signiﬁcant contributions document the protein’s large inherent ﬂexibility which, perhaps by design, allows for ﬂexing during DNA rotation. These implications are particularly attractive in the light of recently reported involve- ment of concerted protein motions in the activity of enzymes [see (67) and discussion below]. Interestingly, it is observed that the contributions to relaxa- tion of positive supercoils are almost a mirror image of the contributions to relaxation of negative supercoils with respect to the mode number. This vividly demonstrates that the same modes are responsible for both positive and negative supercoil relaxations. The crystal structure ‘ground-state’ of the protein therefore seems to encode (and to almost equally favor) both lip opening and hinge stretching in a see-saw motion of the upper cap relative to the lower one. Normal mode analysis is a linear approximation of local dynamics, in distinction from molecular dynamics. It, non- etheless, is in good agreement with the MD results. Even if the normal modes were calculated in the absence of torque on DNA, torque was implicitly considered because it was applied to generate the reaction coordinates in the two directions. It therefore affected how the supercoiling sign of the DNA sub- strate directed protein motion. This is important because, for topo IV (a type II enzyme), studies of positive versus negative relaxation showed that for preferential cleavage of positively