Kinetics of protein folding: The dynamics of globally connected rough energy landscapes with biases

Abstract

We study relaxation within a biased rough energy landscape as a means of investigating protein folding kinetics. The energies of the conformational states of the protein are taken to be random variables. The degree of similarity of each state with the native is described by an order parameter ρ. The energy is, on average, a decreasing function of ρ, in line with the principle of minimal frustration. The limiting case model described here has global connectivity. There are no restrictions on the change in order parameter in the elementary moves through conformational space. The folding kinetics vary as a function of both the energy gap between the folded and unfolded states and the roughness of the energy landscape. This model yields curved Arrhenius plots that are qualitatively similar to those seen in experiments and in simulations of protein folding. We show that in globally connected models, the Laplace transformed concentrations of species obey a frequency dependent generalization of the law of mass action. We show that when states with intermediate degrees of ordering are included in a globally connected model, one obtains much the same qualitative behavior as some experiments that are often used to justify the existence of specific intermediates along a sequential pathway even though there is no such path in the model. Also we show that for short times, populations of ensembles of configurations are dominated by entropic considerations: an observation in harmony with studies on initial events in folding based on more detailed models. Thus, we believe that this model sheds light on the interpretation of experiments that probe protein folding kinetics.