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Adaptive evolution is, to a large extent, a complex combinatorial optimization process. Such processes can be characterized as “uphill walks on rugged fitness landscapes”. Concrete examples of fitness landscapes include the distribution of any specific functional property such as the capacity to catalyze a specific reaction, or bind a specific ligand, in “protein space”. In particular, the property might be the affinity of all possible antibody molecules for a specific antigenic determinant. That affinity landscape presumably plays a critical role in maturation of the immune response. In this process, hypermutation and clonal selection act to select antibody V region mutant variants with successively higher affinity for the immunizing antigen. The actual statistical structure of affinity landscapes, although knowable, is currently unknown. Here, we analyze a class of mathematical models we call NK models. We show that these …