Effect of nucleosome distortion on the linking deficiency in relaxed minichromosomes

The wrapping of closed circular DNA on a protein surface, followed by relaxation with a topoisomerase and removal of proteins, produces a characteristic DNA linking deficiency, delta Lk. We show that the magnitude of delta Lk depends upon the surface shape, and we calculate changes in delta Lk caused by particular distortions of the protein wrapping surface. If the DNA remains attached to the surface during distortion, the DNA winding number, phi, is not altered. The change in delta Lk is then equal to the change in the surface linking number, SLk, which is a straightforward measure of the wrapping of the DNA around the surface. For left-handed wrapping, as in a nucleosome, SLk = -n, the number of times that the DNA axis winds around the axis of the protein complex. We calculate values of SLk for the helical wrapping of a constant length of DNA on protein surfaces having the shapes of cylinders and of ellipsoids and hyperboloids of revolution. If the equatorial radius of the protein is fixed, change in shape from a cylinder to a hyperboloid increases SLk, while the corresponding change to an ellipsoid reduces SLk. We apply the general results to the interpretation of experiments in which minichromosomes are relaxed with topoisomerase at various temperatures and delta Lk is determined. The result is that a distortion of the nucleosome core by at most 5% (the change in the radius at the axial extremity relative to the equator) is sufficient to explain the observed delta Lk changes.