  Correlation patterns in multiple sequence alignments of homologous proteins
can be exploited to infer information on the three-dimensional structure of
their members. The typical pipeline to address this task, which we in this
paper refer to as the three dimensions of contact prediction, is to: (i) filter
and align the raw sequence data representing the evolutionarily related
proteins; (ii) choose a predictive model to describe a sequence alignment;
(iii) infer the model parameters and interpret them in terms of structural
properties, such as an accurate contact map. We show here that all three
dimensions are important for overall prediction success. In particular, we show
that it is possible to improve significantly along the second dimension by
going beyond the pair-wise Potts models from statistical physics, which have
hitherto been the focus of the field. These (simple) extensions are motivated
by multiple sequence alignments often containing long stretches of gaps which,
as a data feature, would be rather untypical for independent samples drawn from
a Potts model. Using a large test set of proteins we show that the combined
improvements along the three dimensions are as large as any reported to date.
