Statistical analyses of protein folding rates from the view of quantum transition

Understanding protein folding rate is the primary key to unlock the fundamental physics underlying protein structure and its folding mechanism.Especially,the temperature dependence of the folding rate remains unsolved in the literature.Starting from the assumption that protein folding is an event of quantum transition between molecular conformations,we calculated the folding rate for all two-state proteins in a database and studied their temperature dependencies.The non-Arrhenius temperature relation for 16 proteins,whose experimental data had previously been available,was successfully interpreted by comparing the Arrhenius plot with the first-principle calculation.A statistical formula for the prediction of two-state protein folding rate was proposed based on quantum folding theory.The statistical comparisons of the folding rates for 65 two-state proteins were carried out,and the theoretical vs.experimental correlation coefficient was 0.73.Moreover,the maximum and the minimum folding rates given by the theory were consistent with the experimental results.     

Folding/Unfolding Kinetics of Apomyoglobin

The equilibrium and kinetic folding/unfolding of apomyoglobin (ApoMb) were studied at pH 6.2, 11 °C by recording tryptophan fluorescence. The equilibrium unfolding of ApoMb in the presence of urea was shown to involve accumulation of an intermediate state, which had a higher fluorescence intensity as compared with the native and unfolded states. The folding proceeded through two kinetic phases, a rapid transition from the unfolded to the intermediate state and a slow transition from the intermediate to the native state. The accumulation of the kinetic intermediate state was observed in a wide range of urea concentrations. The intermediate was detected even in the region corresponding to the unfolding limb of the chevron plot. Urea concentration dependence was obtained for the observed folding/unfolding rate. The shape of the dependence was compared with that of two-state proteins characterized by a direct transition from the unfolded to the native state.     

Transition state contact orders correlate with protein folding rates

We have used molecular dynamics simulations restrained by experimental phi values derived from protein engineering experiments to determine the structures of the transition state ensembles of ten proteins that fold with two-state kinetics. For each of these proteins we then calculated the average contact order in the transition state ensemble and compared it with the corresponding experimental folding rate. The resulting correlation coefficient is similar to that computed for the contact orders of the native structures, supporting the use of native state contact orders for predicting folding rates. The native contacts in the transition state also correlate with those of the native state but are found to be about 30% lower. These results show that, despite the high levels of heterogeneity in the transition state ensemble, the large majority of contributing structures have native-like topologies and that the native state contact order captures this phenomenon.     

Can contact potentials reliably predict stability of proteins?

The simplest approximation of interaction potential between amino acid residues in proteins is the contact potential, which defines the effective free energy of a protein conformation by a set of amino acid contacts formed in this conformation. Finding a contact potential capable of predicting free energies of protein states across a variety of protein families will aid protein folding and engineering in silico on a computationally tractable time-scale. We test the ability of contact potentials to accurately and transferably (across various protein families) predict stability changes of proteins upon mutations. We develop a new methodology to determine the contact potentials in proteins from experimental measurements of changes in protein's thermodynamic stabilities (DeltaDeltaG) upon mutations. We apply our methodology to derive sets of contact interaction parameters for a hierarchy of interaction models including solvation and multi-body contact parameters. We test how well our models reproduce experimental measurements by statistical tests. We evaluate the maximum accuracy of predictions obtained by using contact potentials and the correlation between parameters derived from different data-sets of experimental (DeltaDeltaG) values. We argue that it is impossible to reach experimental accuracy and derive fully transferable contact parameters using the contact models of potentials. However, contact parameters may yield reliable predictions of DeltaDeltaG for datasets of mutations confined to the same amino acid positions in the sequence of a single protein.     

Determination of ultrafast protein folding rates from loop formation dynamics

Quenching of the triplet state of tryptophan by contact with cysteine can be used to measure the kinetics of loop formation in unfolded proteins. Here we show that cysteine quenching dynamics also provide a novel method for measuring folding rates when the exchange between folded and unfolded states is faster than the unquenched triplet lifetime (approximately 100 micros). We use this technique to investigate folding/unfolding kinetics of the 35 residue headpiece subdomain of the protein villin, which contains a single tryptophan residue and was engineered to contain a cysteine residue at the N terminus. At intermediate concentrations of denaturant the time-course of the triplet decay consists of two relaxations, the rates and amplitudes of which reveal the fast kinetics for folding and unfolding of this protein. The folding rates extracted using a simple kinetic model are close to those reported previously from laser-induced temperature-jump experiments that employ the change in tryptophan fluorescence as a probe. However, the results differ significantly from those reported from dynamic NMR line shape analysis on a variant with methionine at the N terminus, an issue that remains to be resolved. The analysis of the triplet quenching kinetics also shows that the quenching rates in the unfolded state increase with decreasing denaturant concentration, indicating a compaction of the unfolded protein.     

Relationships between the folding rate constant and the topological parameters of small two-state proteins based on general random walk model

In this paper, we propose an analytically tractable model of protein folding based on one-dimensional general random walk. A second-order differential equation for the mean folding time of a single protein is constructed which can be used to derive the observed relationship between the folding rate constant and the number of native contacts. The parameters appearing in the model can be determined by fitting the theoretical prediction to the experimental result. In addition, taking into account the fact that the number of native contacts is almost proportional to the relative contact order, we can also explain the observed relationship between the folding rate constant and the relative contact order.     

Sensitivity of the Folding Pathway to the Details of Amino Acid Sequence

Experimental data on the structure of the transition state demonstrate that proteins with the same topology as a rule have similar folding nuclei (the structured formed part of the transition state). In this review discussed are the experimental works showing that the position of folding nuclei is different among proteins with the same topology. These facts emphasize that the folding pathway is sensitive to the details of amino acid sequence.     

On the role of some conserved and nonconserved amino acid residues in the transitional state and intermediate of apomyoglobin folding

The contributions of some amino acid residues in the A, B, G, and H helices to the formation of the folding nucleus and folding intermediate of apomyoglobin were estimated. The effects of point substitutions of Ala for hydrophobic amino acid residues on the structural stability of the native (N) protein and its folding intermediate (I), as well as on the folding/unfolding rates for four mutant apomyoglobin forms, were studied. The equilibrium and kinetic studies of the folding/unfolding rates of these mutant proteins in a wide range of urea concentrations demonstrated that their native state was considerably destabilized as compared with the wild-type protein, whereas the stability of the intermediate state changed moderately. It was shown that the amino acid residues in the A, G, and H helices contributed insignificantly to the stabilization of the apomyoglobin folding nucleus in the rate-limiting I ? N transition, taking place after the formation of the intermediate, whereas the residue of the B helix was of great importance in the formation of the folding nucleus in this transition.     

Optimal potentials for predicting inter-helical packing in transmembrane proteins

A set of pairwise contact potentials between amino acid residues in transmembrane helices was determined from the known native structure of the transmembrane protein (TMP) bacteriorhodopsin by the method of perceptron learning, using Monte Carlo dynamics to generate suitable "decoy" structures. The procedure of finding these decoys is simpler than for globular proteins, since it is reasonable to assume that helices behave as independent, stable objects and, therefore, the search in the conformational space is greatly reduced. With the learnt potentials, the association of the helices in bacteriorhodopsin was successfully simulated. The folding of a second TMP (the helix-dimer glycophorin A) was then accomplished with only a refinement of the potentials from a small number of decoys.