  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.
