Protein Folding: A Perspective from Theory and Experiment

Effective diffusion in one-dimensional rough potential-energy landscapes

Diffusion in spatially rough, confining, one-dimensional continuous energy landscapes is treated using Zwanzig's proposal, which is based on the Smoluchowski equation. We show that Zwanzig's conjecture agrees with Brownian dynamics simulations only in the regime of small roughness. Our correction of Zwanzig's framework corroborates well with numerical results. A numerical simulation scheme based on our coarse-grained Langevin dynamics offers significant reductions in computational time. The mean first-passage time problem in the case of random roughness is treated. Finally, we address the validity of the separation of length scales assumption for the case of polynomial backgrounds and cosine-based roughness. Our results are applicable to hierarchical energy landscapes such as that of a protein's folding and transport processes in disordered media, where there is clear separation of length scale between smooth underlying potential and its rough perturbation.

Alternation of phases of regular and irregular dynamics in protein folding

The regularity of the dynamics in different phases of protein folding is investigated for a set of proteins which undergo a cooperative, two-state folding transition. To determine the degree of regularity of the dynamics, the fractal dimension of probability fluxes is calculated on the basis of simulated folding trajectories. It has been found that the phases of regular and irregular dynamics alternate as follows. In the initial (collapse) phase of folding, the dynamics are essentially regular. Then, as the protein comes to the basin of semicompact states that precedes the transition state, the dynamics become irregular. At the transition state, the dynamics are regularized again but become less regular when the nativelike states are explored. Depending on the specific conditions at which the protein folding was considered, some phases of the dynamics could not be well resolved, but no significant deviation from this general picture has been observed. The regularization of the dynamics at the transition state is discussed in relation to the recent studies of the Hamiltonian dynamics of small clusters, where both regular and chaotic dynamics were observed depending on the flatness of the energy surface at the transition state.

Protein unfolding from free-energy calculations: integration of the Gaussian network model with bond binding energies

Motivated by single molecule experiments, we study thermal unfolding pathways of four proteins, chymotrypsin inhibitor, barnase, ubiquitin, and adenylate kinase, using bond network models that combine bond energies and elasticity. The protein elasticity is described by the Gaussian network model (GNM), to which we add prescribed bond binding energies that are assigned to all (nonbackbone) connecting bonds in the GNM of native state and assumed identical for simplicity. Using exact calculation of the Helmholtz free energy for this model, we consider bond rupture single events. The bond designated for rupture is chosen by minimizing the free-energy difference for the process, over all (nonbackbone) bonds in the network. Plotting the free-energy profile along this pathway at different temperatures, we observe a few major partial unfolding, metastable or stable, states, that are separated by free-energy barriers and change role as the temperature is raised. In particular, for adenylate kinase we find three major partial unfolding states, which is consistent with single molecule FRET experiments [Pirchi et al., Nat. Commun. 2, 493 (2011)] for which hidden Markov analysis reveals between three and five such states. Such states can play a major role in enzymatic activity.

Chain length dependence of apomyoglobin folding: structural evolution from misfolded sheets to native helices

Very little is known about how protein structure evolves during the polypeptide chain elongation that accompanies cotranslational protein folding. This in vitro model study is aimed at probing how conformational space evolves for purified N-terminal polypeptides of increasing length. These peptides are derived from the sequence of an all-alpha-helical single domain protein, Sperm whale apomyoglobin (apoMb). Even at short chain lengths, ordered structure is found. The nature of this structure is strongly chain length dependent. At relatively short lengths, a predominantly non-native beta-sheet conformation is present, and self-associated amyloid-like species are generated. As chain length increases, alpha-helix progressively takes over, and it replaces the beta-strand. The observed trends correlate with the specific fraction of solvent-accessible nonpolar surface area present at different chain lengths. The C-terminal portion of the chain plays an important role by promoting a large and cooperative overall increase in helical content and by consolidating the monomeric association state of the full-length protein. Thus, a native-like energy landscape develops late during apoMb chain elongation. This effect may provide an important driving force for chain expulsion from the ribosome and promote nearly-posttranslational folding of single domain proteins in the cell. Nature has been able to overcome the above intrinsic misfolding trends by modulating the composition of the intracellular environment. An imbalance or improper functioning by the above modulating factors during translation may play a role in misfolding-driven intracellular disorders.