Towards a systematic classification of protein folds

A lattice model Hamiltonian is suggested for protein structures that can explain the division into structural fold classes during the folding process. Proteins are described by chains of secondary structure elements, with the hinges in between being the important degrees of freedom. The protein structures are given a unique name, which simultaneously represent a linear string of physical coupling constants describing hinge spin interactions. We have defined a metric and a precise distance measure between the fold classes. An automated procedure is constructed in which any protein structure in the usual protein data base coordinate format can be transformed into the proposed chain representation. Taking into account hydrophobic forces we have found a mechanism for the formation of domains with a unique fold containing predicted magic numbers {4,6,9,12,16,18,…} of secondary structures and multiples of these domains. It is shown that the same magic numbers are robust and occur as well for packing on other nonclosed packed lattices. We have performed a statistical analysis of available protein structures and found agreement with the predicted preferred abundances of proteins with a predicted magic number of secondary structures. Thermodynamic arguments for the increased abundance and a phase diagram for the folding scenario are given. This includes an intermediate high symmetry phase, the parent structures, between the molten globule and the native states. We have made an exhaustive enumeration of dense lattice animals on a cubic lattice for acceptance number $Z=4\mathrm{}$ and $Z=5\mathrm{}$ up to 36 vertices.