  Theory of multi-dimensional representation of free energy surface of protein
folding is developed by adopting structural order parameters of multiple
regions in protein as multiple coordinates. Various scenarios of folding are
classified in terms of cooperativity within individual regions and interactions
among multiple regions and thus obtained classification is used to analyze the
folding process of several example proteins. Ribosomal protein S6, src-SH3
domain, CheY, barnase, and BBL domain are analyzed with the two-dimensional
representation by using a structure-based Hamiltonian model. Extension to the
higher dimensional representation leads to the finer description of the folding
process. Barnase, NtrC, and an ankyrin repeat protein are examined with the
three-dimensional representation. The multi-dimensional representation allows
us to directly address questions on folding pathways, intermediates, and
transition states.
