  We derive an analytic expression for site-specific stationary distributions
of amino acids from the Structurally Constrained Neutral (SCN) model of protein
evolution with conservation of folding stability. The stationary distributions
that we obtain have a Boltzmann-like shape, and their effective temperature
parameter, measuring the limit of divergent evolutionary changes at a given
site, can be predicted from a site-specific topological property, the principal
eigenvector of the contact matrix of the native conformation of the protein.
These analytic results, obtained without free parameters, are compared with
simulations of the SCN model and with the site-specific amino acid
distributions obtained from the Protein Data Bank. These results also provide
new insights into how the topology of a protein fold influences its
designability, i.e. the number of sequences compatible with that fold. The
dependence of the effective temperature on the principal eigenvector decreases
for longer proteins, a possible consequence of the fact that selection for
thermodynamic stability becomes weaker in this case.
