Statistical models for families of evolutionary related proteins have
recently gained interest: in particular pairwise Potts models, as those
inferred by the Direct-Coupling Analysis, have been able to extract information
about the three-dimensional structure of folded proteins, and about the effect
of amino-acid substitutions in proteins. These models are typically requested
to reproduce the one- and two-point statistics of the amino-acid usage in a
protein family, {\em i.e.}~to capture the so-called residue conservation and
covariation statistics of proteins of common evolutionary origin. Pairwise
Potts models are the maximum-entropy models achieving this. While being
successful, these models depend on huge numbers of {\em ad hoc} introduced
parameters, which have to be estimated from finite amount of data and whose
biophysical interpretation remains unclear. Here we propose an approach to
parameter reduction, which is based on selecting collective sequence motifs. It
naturally leads to the formulation of statistical sequence models in terms of
Hopfield-Potts models. These models can be accurately inferred using a mapping
to restricted Boltzmann machines and persistent contrastive divergence. We show
that, when applied to protein data, even 20-40 patterns are sufficient to
obtain statistically close-to-generative models. The Hopfield patterns form
interpretable sequence motifs and may be used to clusterize amino-acid
sequences into functional sub-families. However, the distributed collective
nature of these motifs intrinsically limits the ability of Hopfield-Potts
models in predicting contact maps, showing the necessity of developing models
going beyond the Hopfield-Potts models discussed here.Comment: 26 pages, 16 figures, to app. in PR