  The coding space of protein sequences is shaped by evolutionary constraints
set by requirements of function and stability. We show that the coding space of
a given protein family--the total number of sequences in that family--can be
estimated using models of maximum entropy trained on multiple sequence
alignments of naturally occuring amino acid sequences. We analyzed and
calculated the size of three abundant repeat proteins families, whose members
are large proteins made of many repetitions of conserved portions of ~ 30 amino
acids. While amino acid conservation at each position of the alignment explains
most of the reduction of diversity relative to completely random sequences, we
found that correlations between amino acid usage at different positions
significantly impact that diversity. We quantified the impact of different
types of correlations, functional and evolutionary, on sequence diversity.
Analysis of the detailed structure of the coding space of the families revealed
a rugged landscape, with many local energy minima of varying sizes with a
hierarchical structure, reminiscent of fustrated energy landscapes of spin
glass in physics. This clustered structure indicates a multiplicity of subtypes
within each family, and suggests new strategies for protein design.
