  Several methods are available for the detection of covarying positions from a
multiple sequence alignment (MSA). If the MSA contains a large number of
sequences, information about the proximities between residues derived from
covariation maps can be sufficient to predict a protein fold. If the structure
is already known, information on the covarying positions can be valuable to
understand the protein mechanism.
  In this study we have sought to determine whether a multivariate extension of
traditional mutual information (MI) can be an additional tool to study
covariation. The performance of two multidimensional MI (mdMI) methods,
designed to remove the effect of ternary/quaternary interdependencies, was
tested with a set of 9 MSAs each containing <400 sequences, and was shown to be
comparable to that of methods based on maximum entropy/pseudolikelyhood
statistical models of protein sequences. However, while all the methods tested
detected a similar number of covarying pairs among the residues separated by <
8 {\AA} in the reference X-ray structures, there was on average less than 65%
overlap between the top scoring pairs detected by methods that are based on
different principles.
  We have also attempted to identify whether the difference in performance
among methods is due to different efficiency in removing covariation
originating from chains of structural contacts. We found that the reason why
methods that derive partial correlation between the columns of a MSA provide a
better recognition of close contacts is not because they remove chaining
effects, but because they filter out the correlation between distant residues
that originates from general fitness constraints. In contrast we found that
true chaining effects are expression of real physical perturbations that
propagate inside proteins, and therefore are not removed by the derivation of
partial correlation between variables.
