The maximum penalty criterion for ridge regression: application to the calibration of the force constant in elastic network models

Multivariate regression is a widespread computational technique that may give meaningless results if the explanatory variables are too numerous or highly collinear. Tikhonov regularization, or ridge regression, is a popular approach to address this issue. We reveal here a formal analogy between ridge regression and statistical mechanics, where the objective function is comparable to a free energy and the ridge parameter plays the role of temperature. This analogy suggests two new criteria to select a suitable ridge parameter: the specific-heat (Cv) and the maximum penalty (MP) fits. We apply these methods to the calibration of the force constant in elastic network models (ENM). This key parameter determines the amplitude of the predicted atomic fluctuations, and is commonly obtained by fitting crystallographic B-factors. However, rigid-body motions are often partially neglected in such fits, even though their importance has been repeatedly stressed. Considering the full set of rigid-body and internal degrees of freedom bears significant risks of overfitting, due to strong correlations between explanatory variables, and requires thus careful regularization. Using simulated data, we show that ridge regression with the Cv or MP criterion markedly reduces the error of the estimated force constant, its across-protein variation, and the number of proteins with unphysical values of the fit parameters, in comparison with popular regularization schemes such as generalized cross-validation. When applied to protein crystals, the new methods provide a more robust calibration of ENM force constants, even though rigid-body motions account on average for more than 80% of the amplitude of B-factors. While MP emerges as the optimal choice for fitting crystallographic B-factors, the Cv fit is more robust to the nature of the data, and is thus an interesting candidate for other applications.