  With the aim to study the relationship between protein sequences and their
native structures, we adopt vectorial representations for both sequence and
structure. The structural representation is based on the Principal Eigenvector
of the fold's contact matrix (PE). As recently shown, the latter encodes
sufficient information for reconstructing the whole contact matrix. The
sequence is represented through a Hydrophobicity Profile (HP), using a
generalized hydrophobicity scale that we obtain from the principal eigenvector
of a residue-residue interaction matrix and denote it as interactivity scale.
Using this novel scale, we define the optimal HP of a protein fold, and
predict, by means of stability arguments, that it is strongly correlated with
the PE of the fold's contact matrix. This prediction is confirmed through an
evolutionary analysis, which shows that the PE correlates with the HP of each
individual sequence adopting the same fold and, even more strongly, with the
average HP of this set of sequences. Thus, protein sequences evolve in such a
way that their average HP is close to the optimal one, implying that neutral
evolution can be viewed as a kind of motion in sequence space around the
optimal HP. Our results indicate that the correlation coefficient between
N-dimensional vectors constitutes a natural metric in the vectorial space in
which we represent both protein sequences and protein structures, which we call
Vectorial Protein Space. In this way, we define a unified framework for
sequence to sequence, sequence to structure, and structure to structure
alignments. We show that the interactivity scale is nearly optimal both for the
comparison of sequences with sequences and sequences with structures.
