Variational Theory for Site Resolved Protein Folding Free Energy Surfaces

Abstract

We present a microscopic variational theory for the free energy surface of a fast folding protein that allows folding kinetics to be resolved to the residue level using Debye-Waller factors as local order parameters. We apply the method to the l-repressor protein and compare with site directed mutagenesis experiments. The formation of native structure and the free energy profile along the folding route are shown to be well described by the capillarity approximation but with some fine structure due to local folding topology. [S0031-9007(98)07855-7] PACS numbers: 87.15. -v Proteins fold on a configurational energy landscape that has the shape of a funnel [1]. As the protein moves down the funnel towards the native state, incomplete cancellation of the entropy and energy losses may result in free energy barriers. So far, proteins that fold fast exhibit single exponential kinetics [2], consistent with a free energy profile that has a single highest barrier along the progress coordinate. Central issues are the origin of the free energy barrier for fast folding proteins and how the ensemble of structures which represent the bottleneck is to be characterized. We address these questions using a variational approximation that describes ensembles of partially folded proteins at the highest level of resolution, i.e., the specific role of individual residues in guiding the protein to the native state is quantified. In the laboratory, Fersht has developed a probe of the transition state or bottleneck ensemble through protein engineering kinetic studies in which the sequence of the protein is altered by replacing residues one at a time [3]. The experiment yields the fraction of the time that the mutated site is in the native conformation in the bottleneck ensemble by comparing folding rates of the mutant to the wild type. Since this can be done for any residue in the sequence, these studies are inherently "site resolved." Resolving the transition state ensemble to this level is one way to monitor the average of the many routes taken as proteins fold.Previous analytic mean field theories and simulations have produced energy landscapes in one or two global dimensions characterizing the folding ensemble [4]. We develop here a free energy profile for proteins with a funneled landscape that is completely site resolved, i.e., one dimension per residue, by extending the mean field variational calculations presented in [5]. The underlying Hamiltonian explicitly incorporates chain stiffness and connectivity while the approximation employs a variational density that monitors local order parameters for folding akin to the Debye-Waller factors (also called temperature factors) for individual residues seen in x-ray crystallography.The basic Hamiltonian for an interacting polymer chain is H H chain 1 H int , where H chain is backbone potential and H int are the interactions between distant monomers along the chain. H chain is an effective harmonic potential bH chain 1͞2 P r i ? G ij ? r j 1 B P r 2 i , where ͕r i ͖ are the positions of t...

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