Self-Similarity of solvent-accessible surfaces of biological and synthetical macromolecules

The quantification of surface roughness of globular proteins and synthetical macromolecules in the globular state is discussed using the concept of fractality. The Hausdorff dimension as a measure for local and global fractality of surfaces is applied. To calculate the Hausdorff dimension of any surface at a high level of accuracy, a new algorithm is presented that is based on a triangulated solvent-accessible molecular surface. It can be demonstrated that protein surfaces (as calculated on the basis of experimentally determined structures) as well as surfaces of globular polyethylene (PE) conformers (calculated on the basis of structural information basing on extensive Monte Carlo and molecular dynamics simulations) in fact show self-similarity within a reasonable yardstick range, at least in a global statistical sense. The same is true for parts of a protein surface provided that these regions are not too small. The concept of self-similarity breaks down when individual surface points are considered. The results obtained for the fractal dimension of PE surfaces (average fractal dimension D = 2.23) lead to the conclusion that protein surfaces probably do not exhibit a unique and specific degree of geometrical complexity (or surface roughness) characterized by a fractal dimension of approximately D = 2.2 as was argued in the past. It is clear that the concept of self-similarity is helpful for the classification of surface roughness of large molecules, but it seems questionable whether this concept is useful for the identification of active sites or other questions related to the field of molecular recognition. © John Wiley & Sons, Inc.