Regularization of least squares problems in CHARMM parameter optimization by truncated singular value decompositions

We examine the use of the truncated singular value decomposition and Tikhonov regularization in standard form to address ill-posed least squares problems Ax = b that frequently arise in molecular mechanics force field parameter optimization. We illustrate these approaches by applying them to dihedral parameter optimization of genotoxic polycyclic aromatic hydrocarbon-DNA adducts that are of interest in the study of chemical carcinogenesis. Utilizing the discrete Picard condition and/or a well-defined gap in the singular value spectrum when A has a well-determined numerical rank, we are able to systematically determine truncation and in turn regularization parameters that are correspondingly used to produce truncated and regularized solutions to the ill-posed least squares problem at hand. These solutions in turn result in optimized force field dihedral terms that accurately parameterize the torsional energy landscape. As the solutions produced by this approach are unique, it has the advantage of avoiding the multiple iterations and guess and check work often required to optimize molecular mechanics force field parameters.