Elastic energy of proteins and the stages of protein folding

We propose a universal elastic energy for proteins, which depends only on the radius of gyration Rg and the residue number N . It is constructed using physical arguments based on the hydrophobic effect and hydrogen bonding. Adjustable parameters are fitted to data from the computer simulation of the folding of a set of proteins using the CSAW (conditioned self-avoiding walk) model. The elastic energy gives rise to scaling relations of the form Rg ∼ N ν in different regions. It shows three folding stages characterized by the progression with exponents ν = 3/5, 3/7, 2/5, which we identify as the unfolded stage, pre-globule, and molten globule, respectively. The pre-globule goes over to the molten globule via a break in behavior akin to a first-order phase transition, which is initiated by a sudden acceleration of hydrogen bonding.