  Recognition of pathogens relies on families of proteins showing great
diversity. Here we construct maximum entropy models of the sequence repertoire,
building on recent experiments that provide a nearly exhaustive sampling of the
IgM sequences in zebrafish. These models are based solely on pairwise
correlations between residue positions, but correctly capture the higher order
statistical properties of the repertoire. Exploiting the interpretation of
these models as statistical physics problems, we make several predictions for
the collective properties of the sequence ensemble: the distribution of
sequences obeys Zipf's law, the repertoire decomposes into several clusters,
and there is a massive restriction of diversity due to the correlations. These
predictions are completely inconsistent with models in which amino acid
substitutions are made independently at each site, and are in good agreement
with the data. Our results suggest that antibody diversity is not limited by
the sequences encoded in the genome, and may reflect rapid adaptation to
antigenic challenges. This approach should be applicable to the study of the
global properties of other protein families.
