  We review and further develop an analytical model that describes how
thermodynamic constraints on the stability of the native state influence
protein evolution in a site-specific manner. To this end, we represent both
protein sequences and protein structures as vectors: Structures are represented
by the principal eigenvector (PE) of the protein contact matrix, a quantity
that resembles closely the effective connectivity of each site; Sequences are
represented through the ``interactivity'' of each amino acid type, using novel
parameters that are correlated with hydropathy scales. These interactivity
parameters are more strongly correlated than the other hydropathy scales that
we examine with: (1) The change upon mutations of the unfolding free energy of
proteins with two-states thermodynamics; (2) Genomic properties as the
genome-size and the genome-wide GC content; (3) The main eigenvectors of the
substitution matrices. The evolutionary average of the interactivity vector
correlates very strongly with the PE of a protein structure. Using this result,
we derive an analytic expression for site-specific distributions of amino acids
across protein families in the form of Boltzmann distributions whose ``inverse
temperature'' is a function of the PE component. We show that our predictions
are in agreement with site-specific amino acid distributions obtained from the
Protein Data Bank, and we determine the mutational model that best fits the
observed site-specific amino acid distributions. Interestingly, the optimal
model almost minimizes the rate at which deleterious mutations are eliminated
by natural selection.
