Finding the consensus shape for a protein family

We define and prove properties of the consensus shape for a protein family, a protein-like structure that provides a compact summary of the significant structural information for a protein family. If all members of the protein family exhibit a geometric relationship between corresponding α-carbons, then that relationship is preserved in the consensus shape. In particular, distances and angles that are consistent across family members are preserved. For the consensus shape, the spacing between successive α-carbons is variable, with small distances in regions where the members of the protein family exhibit significant variation and large distances (up to the standard spacing of about 3.8 Å) in regions where the family members agree. Despite this non-protein-like characteristic, the consensus shape preserves and highlights important structural information. We describe an iterative algorithm for computing the consensus shape and prove that the algorithm converges. As a by-product we produce a multiple structure alignment for the family members. We present the results of experiments in which we build consensus shapes for several known protein families.