Principal Component Analysis and Model Reduction for Dynamical Systems with Symmetry Constraints

Dimension reduction in molecular dynamics simulation is often realized through a principal component analysis based upon a singular value decomposition (SVD) of the trajectory. The left singular vectors of a truncated SVD provide the reduced basis. In many biological molecules, such as HIV1 protease, reflective or rotational symmetry should be present in the molecular configuration. Determining this symmetry allows one to provide SVD major modes of motion that best describe the symmetric movements of the protein. We present a method to compute the plane of reflective symmetry or the axis of rotational symmetry of a large set of points. Moreover, we develop an SVD that best approximates the given set while respecting the symmetry. Interesting subproblems arise in the presence of noisy data or in situations where most, but not all of the structure is symmetric. An important part of the determination of the axis of rotational symmetry or the plane of reflection symmetry is an iterative re-weighting scheme. This scheme is rapidly convergent in practice and seems to be very effective in ignoring outliers (points that do not respect the symmetry).