Investigation of routes and funnels in protein folding by free energy functional methods

We use a free energy functional theory to elucidate general properties of h eterogeneously ordering, fast folding proteins, and we test our conclusions with lattice simulations. We find that both structural and energetic heter ogeneity can lower the free energy barrier to folding. Correlating stronger contact energies with entropically likely contacts of a given native struc ture lowers the barrier, and anticorrelating the energies has the reverse e ffect. Designing in relatively mild energetic heterogeneity can eliminate t he barrier completely at the transition temperature. Sequences with native energies tuned to fold uniformly, as well as sequences tuned to fold reliab ly by a single or a few routes, are rare. Sequences with weak native energe tic heterogeneity are more common; their folding kinetics is more strongly determined by properties of the native structure. Sequences with different distributions of stability throughout the protein may still be good folders to the same structure. A measure of folding route narrowness is introduced that correlates with rate and that can give information about the intrinsi c biases in ordering arising from native topology. This theoretical framewo rk allows us to investigate systematically the coupled effects of energy an d topology in protein folding and to interpret recent experiments that inve stigate these effects.