A lattice protein with an amyloidogenic latent state: stability and folding kinetics

Protein folding as a complex reaction: a two-component potential for the driving force of folding and its variation with folding scenario

The Helmholtz decomposition of the vector field of probability fluxes in a two-dimensional space of collective variables makes it possible to introduce a potential for the driving force of protein folding [Chekmarev, J. Chem. Phys. 139 (2013) 145103]. The potential has two components: one component (Φ) is responsible for the source and sink of the folding flow, which represent, respectively, the unfolded and native state of the protein, and the other (Ψ) accounts for the flow vorticity inherently generated at the periphery of the flow field and provides the canalization of the flow between the source and sink. Both components obey Poisson’s equations with the corresponding source/sink terms. In the present paper, we consider how the shape of the potential changes depending on the scenario of protein folding. To mimic protein folding dynamics projected onto a two-dimensional space of collective variables, the two-dimensional Müller and Brown potential is employed. Three characteristic scenarios are considered: a single pathway from the unfolded to the native state without intermediates, two parallel pathways without intermediates, and a single pathway with an off-pathway intermediate. To determine the probability fluxes, the hydrodynamic description of the folding reaction is used, in which the first-passage folding is viewed as a steady flow of the representative points of the protein from the unfolded to the native state. We show that despite the possible complexity of the folding process, the Φ-component is simple and universal in shape. The Ψ-component is more complex and reveals characteristic features of the process of folding. The present approach is potentially applicable to other complex reactions, for which the transition from the reactant to the product can be described in a space of two (collective) variables.

First passage analysis of the folding of a β-sheet miniprotein: is it more realistic than the standard equilibrium approach?

Simulations of first-passage folding of the antiparallel β-sheet miniprotein beta3s, which has been intensively studied under equilibrium conditions by A. Caflisch and co-workers, show that the kinetics and dynamics are significantly different from those for equilibrium folding. Because the folding of a protein in a living system generally corresponds to the former (i.e., the folded protein is stable and unfolding is a rare event), the difference is of interest. In contrast to equilibrium folding, the Ch-curl conformations become very rare because they contain unfavorable parallel β-strand arrangements, which are difficult to form dynamically due to the distant N- and C-terminal strands. At the same time, the formation of helical conformations becomes much easier (particularly in the early stage of folding) due to short-range contacts. The hydrodynamic descriptions of the folding reaction have also revealed that while the equilibrium flow field presented a collection of local vortices with closed ”streamlines”, the first-passage folding is characterized by a pronounced overall flow from the unfolded states to the native state. The flows through the locally stable structures Cs-or and Ns-or, which are conformationally close to the native state, are negligible due to detailed balance established between these structures and the native state. Although there are significant differences in the general picture of the folding process from the equilibrium and first-passage folding simulations, some aspects of the two are in agreement. The rate of transitions between the clusters of characteristic protein conformations in both cases decreases approximately exponentially with the distance between the clusters in the hydrogen bond distance space of collective variables, and the folding time distribution in the first-passage segments of the equilibrium trajectory is in good agreement with that for the first-passage folding simulations.

New insights into the folding of a β-sheet miniprotein in a reduced space of collective hydrogen bond variables: application to a hydrodynamic analysis of the folding flow

A new analysis of the 20 μs equilibrium folding/unfolding molecular dynamics simulations of the three-stranded antiparallel β-sheet miniprotein (beta3s) in implicit solvent is presented. The conformation space is reduced in dimensionality by introduction of linear combinations of hydrogen bond distances as the collective variables making use of a specially adapted principal component analysis (PCA); i.e., to make structured conformations more pronounced, only the formed bonds are included in determining the principal components. It is shown that a three-dimensional (3D) subspace gives a meaningful representation of the folding behavior. The first component, to which eight native hydrogen bonds make the major contribution (four in each beta hairpin), is found to play the role of the reaction coordinate for the overall folding process, while the second and third components distinguish the structured conformations. The representative points of the trajectory in the 3D space are grouped into conformational clusters that correspond to locally stable conformations of beta3s identified in earlier work. A simplified kinetic network based on the three components is constructed, and it is complemented by a hydrodynamic analysis. The latter, making use of "passive tracers" in 3D space, indicates that the folding flow is much more complex than suggested by the kinetic network. A 2D representation of streamlines shows there are vortices which correspond to repeated local rearrangement, not only around minima of the free energy surface but also in flat regions between minima. The vortices revealed by the hydrodynamic analysis are apparently not evident in folding pathways generated by transition-path sampling. Making use of the fact that the values of the collective hydrogen bond variables are linearly related to the Cartesian coordinate space, the RMSD between clusters is determined. Interestingly, the transition rates show an approximate exponential correlation with distance in the hydrogen bond subspace. Comparison with the many published studies shows good agreement with the present analysis for the parts that can be compared, supporting the robust character of our understanding of this "hydrogen atom" of protein folding.

Toward a molecular theory of early and late events in monomer to amyloid fibril formation

Quantitative understanding of the kinetics of fibril formation and the molecular mechanism of transition from monomers to fibrils is needed to obtain insights into the growth of amyloid fibrils and more generally self-assembly multisubunit protein complexes. Significant advances using computations of protein aggregation in a number of systems have established generic and sequence-specific aspects of the early steps in oligomer formation. Theoretical considerations, which view oligomer and fibril growth as diffusion in a complex energy landscape, and computational studies, involving minimal lattice and coarse-grained models, have revealed general principles governing the transition from monomeric protein to ordered fibrillar aggregates. Detailed atomistic calculations have explored the early stages of the protein aggregation pathway for a number of amyloidogenic proteins, most notably amyloid β- (Aβ-) protein and fragments from proteins linked to various diseases. These computational studies have provided insights into the role of sequence, role of water, and specific interatomic interactions underlying the thermodynamics and dynamics of elementary kinetic steps in the aggregation pathway. Novel methods are beginning to illustrate the structural basis for the production of Aβ-peptides through interactions with secretases in the presence of membranes. We show that a variety of theoretical approaches, ranging from scaling arguments to minimal models to atomistic simulations, are needed as a complement to experimental studies probing the principles governing protein aggregation.

Information entropy as a measure of nonexponentiality of waiting-time distributions

It is shown that the information entropy based on waiting-time distributions (WTDs) offers a natural and robust measure of nonexponentiality of the distributions in the form of the Schrödinger-Brillouin negentropy, or equivalently the Kullback-Leibler divergence, and has a straightforward interpretation in terms of transition state theory. Other measures of nonexponentiality of WTDs, based on comparison of the standard deviation and the median with the mean waiting time, are also discussed. The theoretical analysis is illustrated with results from protein folding studies.

Diffusive reaction dynamics on invariant free energy profiles

A fundamental problem in the analysis of protein folding and other complex reactions in which the entropy plays an important role is the determination of the activation free energy from experimental measurements or computer simulations. This article shows how to combine minimum-cut-based free energy profiles (F(C)), obtained from equilibrium molecular dynamics simulations, with conventional histogram-based free energy profiles (F(H)) to extract the coordinate-dependent diffusion coefficient on the F(C) (i.e., the method determines free energies and a diffusive preexponential factor along an appropriate reaction coordinate). The F(C), in contrast to the F(H), is shown to be invariant with respect to arbitrary transformations of the reaction coordinate, which makes possible partition of configuration space into basins in an invariant way. A "natural coordinate," for which F(H) and F(C) differ by a multiplicative constant (constant diffusion coefficient), is introduced. The approach is illustrated by a model one-dimensional system, the alanine dipeptide, and the folding reaction of a double beta-hairpin miniprotein. It is shown how the results can be used to test whether the putative reaction coordinate is a good reaction coordinate.

All-atom computer simulations of amyloid fibrils disaggregation

Amyloidlike fibrils are found in many fatal diseases, including Alzheimer's disease, type II diabetes mellitus, transmissible spongiform encephalopathies, and prion diseases. These diseases are linked to proteins that have partially unfolded, misfolded, and aggregated into amyloidlike fibrils. The kinetics of amyloidlike fibrils aggregation is still hotly debated and remains an important open question. We have utilized the GNNQQNY crystal structure and high-temperature molecular dynamics simulation in explicit solvent to study the disaggregation mechanism of the GNNQQNY fibrils and to infer its likely aggregation pathways. A hexamer model and a 12-mer model both with two parallel beta-sheets separated by a dry side-chain interface were adopted in our computational analysis. A cumulative time of 1 micros was simulated for the hexamer model at five different temperatures (298 K, 348 K, 398 K, 448 K, and 498 K), and a cumulative time of 2.1 micros was simulated for the 12-mer model at four temperatures (298 K, 398 K, 448 K, and 498 K). Our disaggregation landscape and kinetics analyses indicate that tetramers probably act as the transition state in both the hexamer and the 12-mer simulations. In addition, the 12-mer simulations show that the initial aggregation nucleus is with eight peptides. Furthermore, the landscape is rather flat from 8-mers to 12-mers, indicating the absence of major barriers once the initial aggregation nucleus forms. Thus, the likely aggregation pathway is from monomers to the initial nucleus of 8-mers with tetramers as the transition state. Transition state structure analysis shows that the two dominant transition state conformations are tetramers in the 3-1 and 2-2 arrangements. The predominant nucleus conformations are in peptide arrangements maximizing dry side-chain contacts. Landscape and kinetics analyses also indicate that the parallel beta-sheets form earlier than the dry side-chain contacts during aggregation. These results provide further insights in understanding the early fibrils aggregation.

One-dimensional barrier-preserving free-energy projections of a beta-sheet miniprotein: new insights into the folding process

The conformational space of a 20-residue three-stranded antiparallel beta-sheet peptide (double hairpin) was sampled by equilibrium folding/unfolding molecular dynamics simulations for a total of 20 micros. The resulting one-dimensional free-energy profiles (FEPs) provide a detailed description of the free-energy basins and barriers for the folding reaction. The similarity of the FEPs obtained using the probability of folding before unfolding (pfold) or the mean first passage time supports the robustness of the procedure. The folded state and the most populated free-energy basins in the denatured state are described by the one-dimensional FEPs, which avoid the overlap of states present in the usual one- or two-dimensional projections. Within the denatured state, a basin with fluctuating helical conformations and a heterogeneous entropic state are populated near the melting temperature at about 11% and 33%, respectively. Folding pathways from the helical basin or enthalpic traps (with only one of the two hairpins formed) reach the native state through the entropic state, which is on-pathway and is separated by a low barrier from the folded state. A simplified equilibrium kinetic network based on the FEPs shows the complexity of the folding reaction and indicates, as augmented by additional analyses, that the basins in the denatured state are connected primarily by the native state. The overall folding kinetics shows single-exponential behavior because barriers between the non-native basins and the folded state have similar heights.