Collective-variable selection and generative Hopfield-Potts models for protein-sequence families

Abstract

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 these protein families, i.e. the so-called residue conservation and covariation statistics. Pairwise Potts models are the maximum-entropy models achieving this. While being successful, these models depend on huge numbers of 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 collective-variable selection. It naturally leads to the formulation of statistical sequence models in terms of Hopfield-Potts models, with the Hopfield patterns being the collective variables. 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 generative models. The Hopfield patterns are interpretable in terms of sequence motifs and may be used to clusterize amino-acid sequences into functional subfamilies. 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.