Geometry Dynamics of α-Helices in Different Class I Major Histocompatibility Complexes

Figure 6

Capturing helix motions with polynomials of different degrees. Panel (a) is an illustration of a helix bending movement. We created a model of an ideal linear helix (blue) comprising only atoms whose axis is gradually bent by a mathematically well-defined function. From this function we can easily derive the curvature () and compare it to the curvature we measure (detect) from the polynomial fitted to the helical axis () that was calculated by the fragment-fitting method. From this comparison we derive the relative error; see (11). Panel (b) shows the relative errors of bending motion for polynomial degrees 1 to 8. Panel (c) is an illustration of a helix hinge movement. We created a model of an ideal linear helix (blue) comprising only atoms and split it into two parts. One part was 10-atom long and the other part was 20-atom long. Then one part was rotated around a pivotal point as to simulate a hinge movement. The curvature integral of the helical axis was compared to the hinge angle by calculating the relative error. Panel (d) shows the relative errors of hinge motion for polynomials degrees 1 to 8. Polynomials of second degree were found to reproduce the bending and hinge angles with minimal relative errors.