Unfolded protein ensembles, folding trajectories, and refolding rate prediction.

Computer simulations can provide critical information on the unfolded ensemble of proteins under physiological conditions, by explicitly characterizing the geometrical properties of the diverse conformations that are sampled in the unfolded state. A general computational analysis across many proteins has not been implemented however. Here, we develop a method for generating a diverse conformational ensemble, to characterize properties of the unfolded states of intrinsically disordered or intrinsically folded proteins. The method allows unfolded proteins to retain disulfide bonds. We examined physical properties of the unfolded ensembles of several proteins, including chemical shifts, clustering properties, and scaling exponents for the radius of gyration with polymer length. A problem relating simulated and experimental residual dipolar couplings is discussed. We apply our generated ensembles to the problem of folding kinetics, by examining whether the ensembles of some proteins are closer geometrically to their folded structures than others. We find that for a randomly selected dataset of 15 non-homologous 2- and 3-state proteins, quantities such as the average root mean squared deviation between the folded structure and unfolded ensemble correlate with folding rates as strongly as absolute contact order. We introduce a new order parameter that measures the distance travelled per residue, which naturally partitions into a smooth "laminar" and subsequent "turbulent" part of the trajectory. This latter conceptually simple measure with no fitting parameters predicts folding rates in 0 M denaturant with remarkable accuracy (r = -0.95, p = 1 × 10(-7)). The high correlation between folding times and sterically modulated, reconfigurational motion supports the rapid collapse of proteins prior to the transition state as a generic feature in the folding of both two-state and multi-state proteins. This method for generating unfolded ensembles provides a powerful approach to address various questions in protein evolution, misfolding and aggregation, transient structures, and molten globule and disordered protein phases.

Determination of barrier heights and prefactors from protein folding rate data.

We determine both barrier heights and prefactors for protein folding by applying constraints determined from experimental rate measurements to a Kramers theory for folding rate. The theoretical values are required to match the experimental values at two conditions of temperature and denaturant that induce the same stability. Several expressions for the prefactor in the Kramers rate equation are examined: a random energy approximation, a correlated energy approximation, and an approximation using a single Arrhenius activation energy. Barriers and prefactors are generally found to be large as a result of implementing this recipe, i.e., the folding landscape is cooperative and smooth. Interestingly, a prefactor with a single Arrhenius activation energy admits no formal solution.

Protein folding rates correlate with heterogeneity of folding mechanism.

By observing trends in the folding kinetics of experimental 2-state proteins at their transition midpoints, and by observing trends in the barrier heights of numerous simulations of coarse-grained, C(alpha) model Go proteins, we show that folding rates correlate with the degree of heterogeneity in the formation of native contacts. Statistically significant correlations are observed between folding rates and measures of heterogeneity inherent in the native topology, as well as between rates and the variance in the distribution of either experimentally measured or simulated phi values.

Three-body interactions improve the prediction of rate and mechanism in protein folding models.

Here we study the effects of many-body interactions on rate and mechanism in protein folding by using the results of molecular dynamics simulations on numerous coarse-grained Calpha-model single-domain proteins. After adding three-body interactions explicitly as a perturbation to a Go-like Hamiltonian with native pairwise interactions only, we have found (i) a significantly increased correlation with experimental phi values and folding rates, (ii) a stronger correlation of folding rate with contact order, matching the experimental range in rates when the fraction of three-body energy in the native state is approximately 20%, and (iii) a considerably larger amount of three-body energy present in chymotripsin inhibitor than in the other proteins studied.

Speeding protein folding beyond the G(o) model: how a little frustration sometimes helps.

By perturbing a G(o) model toward a realistic protein Hamiltonian by adding non-native interactions, we find that the folding rate is in general enhanced as ruggedness is initially increased, as long as the protein is sufficiently large and flexible. Eventually, the rate drops rapidly toward zero when ruggedness significantly slows conformational transitions. Energy landscape arguments for thermodynamics and kinetics are coupled with a treatment of non-native collapse to elucidate this effect.

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 heterogeneously ordering, fast folding proteins, and we test our conclusions with lattice simulations. We find that both structural and energetic heterogeneity can lower the free energy barrier to folding. Correlating stronger contact energies with entropically likely contacts of a given native structure lowers the barrier, and anticorrelating the energies has the reverse effect. Designing in relatively mild energetic heterogeneity can eliminate the barrier completely at the transition temperature. Sequences with native energies tuned to fold uniformly, as well as sequences tuned to fold reliably by a single or a few routes, are rare. Sequences with weak native energetic 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 intrinsic biases in ordering arising from native topology. This theoretical framework allows us to investigate systematically the coupled effects of energy and topology in protein folding and to interpret recent experiments that investigate these effects.