Dynamics of protein folding: probing the kinetic network of folding-unfolding transitions with experiment and theory

Pushing Camera-Based Single-Molecule Kinetic Measurements to the Frame Acquisition Limit with Stroboscopic smFRET

Single-molecule fluorescence resonance energy transfer (smFRET) experiments permit detailed examination of microscopic dynamics. However, kinetic rate constants determined by smFRET are susceptible to systematic underestimation when the rate constants are comparable to the data acquisition rate. We demonstrate how such systematic errors in camera-based total internal reflection fluorescence microscopy experiments can be greatly reduced by using stroboscopic illumination/detection, allowing accurate rate constant determination up to the data sampling rate and yielding an order of magnitude increase in the dynamic range. Implementation of these stroboscopic smFRET ideas is straightforward, and the stroboscopically obtained data are compatible with multiple trajectory analysis methods, including dwell-time analysis and hidden Markov modeling. Such stroboscopic methods therefore offer a remarkably simple yet valuable addition to the smFRET toolkit, requiring only relatively modest modification to the normal data collection and analysis procedures.

Replica Exchange Molecular Dynamics of Diphenylalanine Amyloid Peptides in Electric Fields

The self-assembling propensity of amyloid peptides such as diphenylalanine (FF) allows them to form ordered, nanoscale structures, with biocompatible properties important for biomedical applications. Moreover, piezoelectric properties allow FF molecules and their aggregates (e.g., FF nanotubes) to be aligned in a controlled way by the application of external electric fields. However, while the behavior of FF nanostructures emerges from the biophysical properties of the monomers, the detailed responses of individual peptides to both temperature and electric fields are not fully understood. Here, we study the temperature-dependent conformational dynamics of FF peptides solvated in explicit water molecules, an environment relevant to biomedical applications, by using an enhanced sampling method, replica exchange molecular dynamics (REMD), in conjunction with applied electric fields. Our simulations highlight and overcome possible artifacts that may occur during the setup of REMD simulations of explicitly solvated peptides in the presence of external electric fields, a problem particularly important in the case of short peptides such as FF. The presence of the external fields could overstabilize certain conformational states in one or more REMD replicas, leading to distortions of the underlying potential energy distributions observed at each temperature. This can be overcome by correcting the REMD initial conditions to include the lower-energy conformations induced by the external field. We show that the converged REMD data can be analyzed using a Markovian description of conformational states and show that a rather complex, 3-state, temperature-dependent conformational dynamics in the absence of electric fields collapses to only one of these states in the presence of the electric fields. These details on the temperature- and electric-field-dependent thermodynamic and kinetic properties of small FF amyloid peptides can be useful in understanding and devising new methods to control their aggregation-prone biophysical properties and, possibly, the structural and biophysical properties of FF molecular nanostructures.

Long-time methods for molecular dynamics simulations: Markov State Models and Milestoning

Molecular dynamics (MD) studies of biomolecules require the ability to simulate complex biochemical systems with an increasingly larger number of particles and for longer time scales, a problem that cannot be overcome by computational hardware advances alone. A main problem springs from the intrinsically high-dimensional and complex nature of the underlying free energy landscape of most systems, and from the necessity to sample accurately such landscapes for identifying kinetic and thermodynamic states in the configurations space, and for accurate calculations of both free energy differences and of the corresponding transition rates between states. Here, we review and present applications of two increasingly popular methods that allow long-time MD simulations of biomolecular systems that can open a broad spectrum of new studies. A first approach, Markov State Models (MSMs), relies on identifying a set of configuration states in which the system resides sufficiently long to relax and loose the memory of previous transitions, and on using simulations for mapping the underlying complex energy landscape and for extracting accurate thermodynamic and kinetic information. The Markovian independence of the underlying transition probabilities creates the opportunity to increase the sampling efficiency by using sets of appropriately initialized short simulations rather than typically long MD trajectories, which also enhances sampling. This allows MSM-based studies to unveil bio-molecular mechanisms and to estimate free energy barriers with high accuracy, in a manner that is both systematic and relatively automatic, which accounts for their increasing popularity. The second approach presented, Milestoning, targets accurate studies of the ensemble of pathways connecting specific end-states (e.g., reactants and products) in a similarly systematic, accurate and highly automatic manner. Applications presented range from studies of conformational dynamics and binding of amyloid-forming peptides, cell-penetrating peptides and the DFG-flip dynamics in Abl kinase. As highlighted by the increasing number of studies using both methods, we anticipate that they will open new avenues for the investigation of systematic sampling of reactions pathways and mechanisms occurring on longer time scales than currently accessible by purely computational hardware developments.

Conformational analysis of replica exchange MD: Temperature-dependent Markov networks for FF amyloid peptides

Recent molecular modeling methods using Markovian descriptions of conformational states of biomolecular systems have led to powerful analysis frameworks that can accurately describe their complex dynamical behavior. In conjunction with enhanced sampling methods, such as replica exchange molecular dynamics (REMD), these frameworks allow the systematic and accurate extraction of transition probabilities between the corresponding states, in the case of Markov state models, and of statistically-optimized transition rates, in the case of the corresponding coarse master equations. However, applying automatically such methods to large molecular dynamics (MD) simulations, with explicit water molecules, remains limited both by the initial ability to identify good candidates for the underlying Markovian states and by the necessity to do so using good collective variables as reaction coordinates that allow the correct counting of inter-state transitions at various lag times. Here, we show that, in cases when representative molecular conformations can be identified for the corresponding Markovian states, and thus their corresponding collective evolution of atomic positions can be calculated along MD trajectories, one can use them to build a new type of simple collective variable, which can be particularly useful in both the correct state assignment and in the subsequent accurate counting of inter-state transition probabilities. In the case of the ubiquitously used root-mean-square deviation (RMSD) of atomic positions, we introduce the relative RMSD (RelRMSD) measure as a good reaction coordinate candidate. We apply this method to the analysis of REMD trajectories of amyloid-forming diphenylalanine (FF) peptides-a system with important nanotechnology and biomedical applications due to its self-assembling and piezoelectric properties-illustrating the use of RelRMSD in extracting its temperature-dependent intrinsic kinetics, without a priori assumptions on the functional form (e.g., Arrhenius or not) of the underlying conformational transition rates. The RelRMSD analysis enables as well a more objective assessment of the convergence of the REMD simulations. This type of collective variable may be generalized to other observables that could accurately capture conformational differences between the underlying Markov states (e.g., distance RMSD, the fraction of native contacts, etc.).

Peptide dimerization-dissociation rates from replica exchange molecular dynamics

We show how accurate rates of formation and dissociation of peptide dimers can be calculated using direct transition counting (DTC) from replica-exchange molecular dynamics (REMD) simulations. First, continuous trajectories corresponding to system replicas evolving at different temperatures are used to assign conformational states. Second, we analyze the entire REMD data to calculate the corresponding rates at each temperature directly from the number of transition counts. Finally, we compare the kinetics extracted directly, using the DTC method, with indirect estimations based on trajectory likelihood maximization using short-time propagators and on decay rates of state autocorrelation functions. For systems with relatively low-dimensional intrinsic conformational dynamics, the DTC method is simple to implement and leads to accurate temperature-dependent rates. We apply the DTC rate-extraction method to all-atom REMD simulations of dimerization of amyloid-forming NNQQ tetrapetides in explicit water. In an assessment of the REMD sampling efficiency with respect to standard MD, we find a gain of more than a factor of two at the lowest temperature.

One Peptide Reveals the Two Faces of α-Helix Unfolding-Folding Dynamics

The understanding of fast folding dynamics of single α-helices comes mostly from studies on rationally designed peptides displaying sequences with high helical propensity. The folding/unfolding dynamics and energetics of α-helix conformations in naturally occurring peptides remains largely unexplored. Here we report the study of a protein fragment analogue of the C-peptide from bovine pancreatic ribonuclease-A, RN80, a 13-amino acid residue peptide that adopts a highly populated helical conformation in aqueous solution. ¹H NMR and CD structural studies of RN80 showed that α-helix formation displays a pH-dependent bell-shaped curve, with a maximum near pH 5, and a large decrease in helical content in alkaline pH. The main forces stabilizing this short α-helix were identified as a salt bridge formed between Glu-2 and Arg-10 and the cation-π interaction involving Tyr-8 and His-12. Thus, deprotonation of Glu-2 or protonation of His-12 are essential for the RN80 α-helix stability. In the present study, RN80 folding and unfolding were triggered by laser-induced pH jumps and detected by time-resolved photoacoustic calorimetry (PAC). The photoacid proton release, amino acid residue protonation, and unfolding/folding events occur at different time scales and were clearly distinguished using time-resolved PAC. The partial unfolding of the RN80 α-helix, due to protonation of Glu-2 and consequent breaking of the stabilizing salt bridge between Glu-2 and Arg-10, is characterized by a concentration-independent volume expansion in the sub-microsecond time range (0.8 mL mol^-1, 369 ns). This small volume expansion reports the cost of peptide backbone rehydration upon disruption of a solvent-exposed salt bridge, as well as backbone intrinsic expansion. On the other hand, RN80 α-helix folding triggered by His-12 protonation and subsequent formation of a cation-π interaction leads to a microsecond volume contraction (-6.0 mL mol^-1, ∼1.7 μs). The essential role of two discrete side chain interactions, a salt bridge, and in particular a single cation-π interaction in the folding dynamics of a naturally occurring α-helix peptide is uniquely revealed by these data.

Optimized parameter selection reveals trends in Markov state models for protein folding

As molecular dynamics simulations access increasingly longer time scales, complementary advances in the analysis of biomolecular time-series data are necessary. Markov state models offer a powerful framework for this analysis by describing a system's states and the transitions between them. A recently established variational theorem for Markov state models now enables modelers to systematically determine the best way to describe a system's dynamics. In the context of the variational theorem, we analyze ultra-long folding simulations for a canonical set of twelve proteins [K. Lindorff-Larsen et al., Science 334, 517 (2011)] by creating and evaluating many types of Markov state models. We present a set of guidelines for constructing Markov state models of protein folding; namely, we recommend the use of cross-validation and a kinetically motivated dimensionality reduction step for improved descriptions of folding dynamics. We also warn that precise kinetics predictions rely on the features chosen to describe the system and pose the description of kinetic uncertainty across ensembles of models as an open issue.

Coarse Master Equations for Binding Kinetics of Amyloid Peptide Dimers

We characterize the kinetics of dimer formation of the short amyloid microcrystal-forming tetrapeptides NNQQ by constructing coarse master equations for the conformational dynamics of the system, using temperature replica-exchange molecular dynamics (REMD) simulations. We minimize the effects of Kramers-type recrossings by assigning conformational states based on their sequential time evolution. Transition rates are further estimated from short-time state propagators by maximizing the likelihood that the extracted rates agree with the observed atomistic trajectories without any a priori assumptions about their temperature dependence. Here, we evaluate the rates for both continuous replica trajectories that visit different temperatures and for discontinuous data corresponding to each REMD temperature. While the binding-unbinding kinetic process is clearly Markovian, the conformational dynamics of the bound NNQQ dimer has a complex character. Our kinetic analysis allows us to discriminate between short-lived encounter pairs and strongly bound conformational states. The conformational dynamics of NNQQ dimers supports a kinetically driven aggregation mechanism, in agreement with the polymorphic character reported for amyloid aggregates such as microcrystals and fibrils.

Early folding events during light harvesting complex II assembly in vitro monitored by pulsed electron paramagnetic resonance

Efficient energy transfer in the major light harvesting complex II (LHCII) of green plants is facilitated by the precise alignment of pigments due to the protein matrix they are bound to. Much is known about the import of the LHCII apoprotein into the chloroplast via the TOC/TIC system and its targeting to the thylakoid membrane but information is sparse about when and where the pigments are bound and how this is coordinated with protein folding. In vitro, the LHCII apoprotein spontaneously folds and binds its pigments if the detergent-solubilized protein is combined with a mixture of chlorophylls a and b and carotenoids. In the present work, we employed this approach to study apoprotein folding and pigment binding in a time-resolved manner by using pulsed electron paramagnetic resonance (EPR). Intra-molecular distances were measured before folding, after 255 ms and 40 s folding time in the absence of cryoprotectant, and in the fully folded and assembled LHCII. In accordance with earlier results, the most of the folding of the three membrane-spanning alpha helices precedes their apposition into the final tertiary structure. However, their formation follows different kinetics, partially extending into the final phase of LHCII formation during which much of the condensation of the pigment-protein structure occurs, presumably governed by the binding of chlorophyll b. A rough timetable is proposed to sort partial events into the LHCII formation process.

Inherent structure versus geometric metric for state space discretization

Inherent structure (IS) and geometry-based clustering methods are commonly used for analyzing molecular dynamics trajectories. ISs are obtained by minimizing the sampled conformations into local minima on potential/effective energy surface. The conformations that are minimized into the same energy basin belong to one cluster. We investigate the influence of the applications of these two methods of trajectory decomposition on our understanding of the thermodynamics and kinetics of alanine tetrapeptide. We find that at the microcluster level, the IS approach and root-mean-square deviation (RMSD)-based clustering method give totally different results. Depending on the local features of energy landscape, the conformations with close RMSDs can be minimized into different minima, while the conformations with large RMSDs could be minimized into the same basin. However, the relaxation timescales calculated based on the transition matrices built from the microclusters are similar. The discrepancy at the microcluster level leads to different macroclusters. Although the dynamic models established through both clustering methods are validated approximately Markovian, the IS approach seems to give a meaningful state space discretization at the macrocluster level in terms of conformational features and kinetics.

Complexity of the folding transition of the B domain of protein A revealed by the high-speed tracking of single-molecule fluorescence time series

The equilibrium unfolding transition of the B domain of protein A (BdpA) was investigated by using single-molecule fluorescence spectroscopy based on line-confocal detection of fast-flowing samples. The method achieved the time resolution of 120 μs and the observation time of a few milliseconds in the single-molecule time-series measurements of fluorescence resonance energy transfer (FRET). Two samples of BdpA doubly labeled with donor and acceptor fluorophores, the first possessing fluorophores at residues 22 and 55 (sample 1) and the second at residues 5 and 55 (sample 2), were prepared. The equilibrium unfolding transition induced by guanidium chloride (GdmCl) was monitored by bulk measurements and demonstrated that the both samples obey the apparent two-state unfolding. In the absence of GdmCl, the single-molecule FRET measurements for the both samples showed a single peak assignable to the native state (N). The FRET efficiency for N shifts to lower values as the increase of GdmCl concentration, suggesting the swelling of the native state structure. At the higher concentration of GdmCl, the both samples convert to the unfolded state (U). Near the unfolding midpoint for sample 1, the kinetic exchange between N and U causes the averaging of the two states and the higher values of the relative fluctuation. The time series for different molecules in U showed slightly different FRET efficiencies, suggesting the apparent heterogeneity. Thus, the high-speed tracking of fluorescence signals from single molecules revealed a complexity and heterogeneity hidden in the apparent two-state behavior of protein folding.

Protein folding: understanding the role of water and the low Reynolds number environment as the peptide chain emerges from the ribosome and folds

The mechanism of protein folding during early stages of the process has three determinants. First, moving water molecules obey the rules of low Reynolds number physics without an inertial component. Molecular movement is instantaneous and size insensitive. Proteins emerging from the ribosome move and rotate without an external force if they change shape, forming and propagating helical structures that increases translocational efficiency. Forward motion ceases when the shape change or propelling force ceases. Second, application of quantum field theory to water structure predicts the spontaneous formation of low density coherent units of fixed size that expel dissolved atmospheric gases. Structured water layers with both coherent and non-coherent domains, form a sheath around the new protein. The surface of exposed hydrophobic amino acids is protected from water contact by small nanobubbles of dissolved atmospheric gases, 5 or 6 molecules on average, that vibrate, attracting even widely separated resonating nanobubbles. This force results from quantum effects, appearing only when the system is within and interacts with an oscillating electromagnetic field. The newly recognized quantum force sharply bends the peptide and is part of a dynamic field determining the pathway of protein folding. Third, the force initiating the tertiary folding of proteins arises from twists at the position of each hydrophobic amino acid, that minimizes surface exposure of the hydrophobic amino acids and propagates along the protein. When the total bend reaches 360°, the leading segment of water sheath intersects the trailing segment. This steric self-intersection expels water from overlapping segments of the sheath and by Newton׳s second law moves the polypeptide chain in an opposite direction. Consequently, with very few exceptions that we enumerate and discuss, tertiary structures are absent from proteins without hydrophobic amino acids, which control the early stages of protein folding and the overall shape of protein. Consequently, proteins only adopt a limited number of forms. The formation of quaternary structures is not necessarily prevented by the absence of hydrophobic amino acids.

Graph representation of protein free energy landscape

The thermodynamics and kinetics of protein folding and protein conformational changes are governed by the underlying free energy landscape. However, the multidimensional nature of the free energy landscape makes it difficult to describe. We propose to use a weighted-graph approach to depict the free energy landscape with the nodes on the graph representing the conformational states and the edge weights reflecting the free energy barriers between the states. Our graph is constructed from a molecular dynamics trajectory and does not involve projecting the multi-dimensional free energy landscape onto a low-dimensional space defined by a few order parameters. The calculation of free energy barriers was based on transition-path theory using the MSMBuilder2 package. We compare our graph with the widely used transition disconnectivity graph (TRDG) which is constructed from the same trajectory and show that our approach gives more accurate description of the free energy landscape than the TRDG approach even though the latter can be organized into a simple tree representation. The weighted-graph is a general approach and can be used on any complex system.

MD + QM correlations with tryptophan fluorescence spectral shifts and lifetimes

Principles behind quenching of tryptophan (Trp) fluorescence are updated and extended in light of recent 100-ns and 1-μs molecular dynamics (MD) trajectories augmented with quantum mechanical (QM) calculations that consider electrostatic contributions to wavelength shifts and quenching. Four studies are summarized, including (1) new insight into the single exponential decay of NATA, (2) a study revealing how unsuspected rotamer transitions affect quenching of Trp when used as a probe of protein folding, (3) advances in understanding the origin of nonexponential decay from 100-ns simulations on 19 Trps in 16 proteins, and (4) the correlation of wavelength with lifetime for decay-associated spectra (DAS). Each study strongly reinforces the concept that-for Trp-electron transfer-based quenching is controlled much more by environment electrostatic factors affecting the charge transfer (CT) state energy than by distance dependence of electronic coupling. In each case, water plays a large role in unexpected ways.

Spectral Rate Theory for Two-State Kinetics

Classical rate theories often fail in cases where the observable(s) or order parameter(s) used is a poor reaction coordinate or the observed signal is deteriorated by noise, such that no clear separation between reactants and products is possible. Here, we present a general spectral two-state rate theory for ergodic dynamical systems in thermal equilibrium that explicitly takes into account how the system is observed. The theory allows the systematic estimation errors made by standard rate theories to be understood and quantified. We also elucidate the connection of spectral rate theory with the popular Markov state modeling approach for molecular simulation studies. An optimal rate estimator is formulated that gives robust and unbiased results even for poor reaction coordinates and can be applied to both computer simulations and single-molecule experiments. No definition of a dividing surface is required. Another result of the theory is a model-free definition of the reaction coordinate quality. The reaction coordinate quality can be bounded from below by the directly computable observation quality, thus providing a measure allowing the reaction coordinate quality to be optimized by tuning the experimental setup. Additionally, the respective partial probability distributions can be obtained for the reactant and product states along the observed order parameter, even when these strongly overlap. The effects of both filtering (averaging) and uncorrelated noise are also examined. The approach is demonstrated on numerical examples and experimental single-molecule force-probe data of the p5ab RNA hairpin and the apo-myoglobin protein at low pH, focusing here on the case of two-state kinetics.

Dual effect of crowders on fibrillation kinetics of polypeptide chains revealed by lattice models

We have developed the lattice model for describing polypeptide chains in the presence of crowders. The influence of crowding confinement on the fibrillation kinetics of polypeptide chains is studied using this model. We observed the non-trivial behavior of the fibril formation time τfib that it decreases with the concentration of crowders if crowder sizes are large enough, but the growth is observed for crowders of small sizes. This allows us to explain the recent experimental observation on the dual effect of crowding particles on fibril growth of proteins that for a fixed crowder concentration the fibrillation kinetics is fastest at intermediate values of total surface of crowders. It becomes slow at either small or large coverages of cosolutes. It is shown that due to competition between the energetics and entropic effects, the dependence of τfib on the size of confined space is described by a parabolic function.

High-Resolution Free-Energy Landscape Analysis of α-Helical Protein Folding: HP35 and Its Double Mutant

The free-energy landscape can provide a quantitative description of folding dynamics, if determined as a function of an optimally chosen reaction coordinate. Here, we construct the optimal coordinate and the associated free-energy profile for all-helical proteins HP35 and its norleucine (Nle/Nle) double mutant, based on realistic equilibrium folding simulations [Piana et al. Proc. Natl. Acad. Sci. U.S.A.2012, 109, 17845]. From the obtained profiles, we directly determine such basic properties of folding dynamics as the configurations of the minima and transition states (TS), the formation of secondary structure and hydrophobic core during the folding process, the value of the pre-exponential factor and its relation to the transition path times, the relation between the autocorrelation times in TS and minima. We also present an investigation of the accuracy of the pre-exponential factor estimation based on the transition-path times. Four different estimations of the pre-exponential factor for both proteins give k₀^–1 values of approximately a few tens of nanoseconds. Our analysis gives detailed information about folding of the proteins and can serve as a rigorous common language for extensive comparison between experiment and simulation.

Reaction coordinates, one-dimensional Smoluchowski equations, and a test for dynamical self-consistency

We propose a method for identifying accurate reaction coordinates among a set of trial coordinates. The method applies to special cases where motion along the reaction coordinate follows a one-dimensional Smoluchowski equation. In these cases the reaction coordinate can predict its own short-time dynamical evolution, i.e., the dynamics projected from multiple dimensions onto the reaction coordinate depend only on the reaction coordinate itself. To test whether this property holds, we project an ensemble of short trajectory swarms onto trial coordinates and compare projections of individual swarms to projections of the ensemble of swarms. The comparison, quantified by the Kullback-Leibler divergence, is numerically performed for each isosurface of each trial coordinate. The ensemble of short dynamical trajectories is generated only once by sampling along an initial order parameter. The initial order parameter should separate the reactants and products with a free energy barrier, and distributions on isosurfaces of the initial parameter should be unimodal. The method is illustrated for three model free energy landscapes with anisotropic diffusion. Where exact coordinates can be obtained from Kramers-Langer-Berezhkovskii-Szabo theory, results from the new method agree with the exact results. We also examine characteristics of systems where the proposed method fails. We show how dynamical self-consistency is related (through the Chapman-Kolmogorov equation) to the earlier isocommittor criterion, which is based on longer paths.

TanCAR: A Novel Bispecific Chimeric Antigen Receptor for Cancer Immunotherapy

Targeted T cells are emerging as effective non-toxic therapies for cancer. Multiple elements, however, contribute to the overall pathogenesis of cancer through both distinct and redundant mechanisms. Hence, targeting multiple cancer-specific markers simultaneously could result in better therapeutic efficacy. We created a functional chimeric antigen receptor-the TanCAR, a novel artificial molecule that mediates bispecific activation and targeting of T cells. We demonstrate the feasibility of cumulative integration of structure and docking simulation data using computational tools to interrogate the design and predict the functionality of such a complex bispecific molecule. Our prototype TanCAR induced distinct T cell reactivity against each of two tumor restricted antigens, and produced synergistic enhancement of effector functions when both antigens were simultaneously encountered. Furthermore, the TanCAR preserved the cytolytic ability of T cells upon loss of one of the target molecules and better controlled established experimental tumors by recognition of both targets in an animal disease model. This proof-of-concept approach can be used to increase the specificity of effector cells for malignant versus normal target cells, to offset antigen escape or to allow for targeting the tumor and its microenvironment.Molecular Therapy-Nucleic Acids (2013) 2, e105; doi:10.1038/mtna.2013.32; published online 9 July 2013.

Thermodynamics of ideal proteinogenic homopolymer chains as a function of the energy spectrum E, helical propensity ω and enthalpic energy barrier

A reformulation and generalization of the Zwanzig model (ZW model) for ideal homopolymer chains poly-X, where X represents any of the twenty naturally occurring proteinogenic amino acid residues is presented. This reformulation and generalization provides a direct connection between coarse-grained parameters originally proposed in the ZW model with variables from the Lifson-Roig (LR) theory, such as the helical propensity per residue ω, and new variables introduced here, such as the energy gap Δ between unfolded and folded structures, as well as the ratio f of the energy scales involved. This enables us to discover the relevance of the energy spectrum E to the onset of configurational phase transitions. From the configurational partition function Q, thermodynamic properties such as the configurational entropy S, specific heat v and average energy <E> are calculated in terms of the number of residues K, temperature T, helical propensity ω and energy barrier ΔH for different poly-X chains in vacuo. Results obtained here provide substantial evidence that configurational phase transitions for ideal poly-X chains correspond to first-order phase transitions. An anomalous behavior of the thermodynamic functions <E>, Cv, S with respect to the number K of residues is also highlighted. On-going methods of solution are outlined.

The Theory of Ultra-Coarse-Graining. 1. General Principles

Coarse-grained (CG) models provide a computationally efficient means to study biomolecular and other soft matter processes involving large numbers of atoms correlated over distance scales of many covalent bond lengths and long time scales. Variational methods based on information from simulations of finer-grained (e.g., all-atom) models, for example the multiscale coarse-graining (MS-CG) and relative entropy minimization methods, provide attractive tools for the systematic development of CG models. However, these methods have important drawbacks when used in the "ultra-coarse-grained" (UCG) regime, e.g., at a resolution level coarser or much coarser than one amino acid residue per effective CG particle in proteins. This is due to the possible existence of multiple metastable states "within" the CG sites for a given UCG model configuration. In this work, systematic variational UCG methods are presented that are specifically designed to CG entire protein domains and subdomains into single effective CG particles. This is accomplished by augmenting existing effective particle CG schemes to allow for discrete state transitions and configuration-dependent resolution. Additionally, certain conclusions of this work connect back to single-state force matching and open up new avenues for method development in that area. These results provide a formal statistical mechanical basis for UCG methods related to force matching and relative entropy CG methods and suggest practical algorithms for constructing optimal approximate UCG models from fine-grained simulation data.

Thermally induced protein unfolding probed by isotope-edited IR spectroscopy

Infrared (IR) spectroscopy has been widely utilized for the study of protein folding, unfolding, and misfolding processes. We have previously developed a theoretical method for calculating IR spectra of proteins in the amide I region. In this work, we apply this method, in combination with replica-exchange molecular dynamics simulations, to study the equilibrium thermal unfolding transition of the villin headpiece subdomain (HP36). Temperature-dependent IR spectra and spectral densities are calculated. The spectral densities correctly reflect the unfolding conformational changes in the simulation. With the help of isotope labeling, we are able to capture the feature that helix 2 of HP36 loses its secondary structure before global unfolding occurs, in agreement with experiment.

Alzheimer Aβ peptide interactions with lipid membranes: fibrils, oligomers and polymorphic amyloid channels

Fibrillar aggregates of misfolded amyloid proteins are involved in a variety of diseases such as Alzheimer disease (AD), type 2 diabetes, Parkinson, Huntington and prion-related diseases. In the case of AD amyloid β (Aβ) peptides, the toxicity of amyloid oligomers and larger fibrillar aggregates is related to perturbing the biological function of the adjacent cellular membrane. We used atomistic molecular dynamics (MD) simulations of Aβ_9–40 fibrillar oligomers modeled as protofilament segments, including lipid bilayers and explicit water molecules, to probe the first steps in the mechanism of Aβ-membrane interactions. Our study identified the electrostatic interaction between charged peptide residues and the lipid headgroups as the principal driving force that can modulate the further penetration of the C-termini of amyloid fibrils or fibrillar oligomers into the hydrophobic region of lipid membranes. These findings advance our understanding of the detailed molecular mechanisms and the effects related to Aβ-membrane interactions, and suggest a polymorphic structural character of amyloid ion channels embedded in lipid bilayers. While inter-peptide hydrogen bonds leading to the formation of β-strands may still play a stabilizing role in amyloid channel structures, these may also present a significant helical content in peptide regions (e.g., termini) that are subject to direct interactions with lipids rather than with neighboring Aβ peptides.

Simple few-state models reveal hidden complexity in protein folding

Markov state models constructed from molecular dynamics simulations have recently shown success at modeling protein folding kinetics. Here we introduce two methods, flux PCCA+ (FPCCA+) and sliding constraint rate estimation (SCRE), that allow accurate rate models from protein folding simulations. We apply these techniques to fourteen massive simulation datasets generated by Anton and Folding@home. Our protocol quantitatively identifies the suitability of describing each system using two-state kinetics and predicts experimentally detectable deviations from two-state behavior. An analysis of the villin headpiece and FiP35 WW domain detects multiple native substates that are consistent with experimental data. Applying the same protocol to GTT, NTL9, and protein G suggests that some beta containing proteins can form long-lived native-like states with small register shifts. Even the simplest protein systems show folding and functional dynamics involving three or more states.

Microsecond folding dynamics of apomyoglobin at acidic pH

Apomyolgobin (apoMb) is an important model for understanding the folding mechanism of helical proteins. This study focuses on a partially structured state of sperm whale apoMb populated at pH 4.2 (M-state), which structurally resembles a late kinetic intermediate in the formation of the native state (N) at higher pH. The thermodynamics and cooperativity of apoMb folding at pH 4.2 and 6.2 were studied by global analysis of the urea-induced unfolding transitions monitored by tryptophan fluorescence and circular dichroism. The kinetics of folding and unfolding of apoMb at pH 4.2 was measured over a time window from 40 to 850 μs, using fluorescence-detected continuous-flow measurements. Our observation of biphasic kinetics provides clear evidence for rapid (<100 μs) accumulation of previously unresolved intermediate states in both refolding and unfolding experiments. Quantitative kinetic modeling of the results, using a four-state mechanism with two intermediates on a direct route between the unfolded and folded states (U↔I↔L↔M), gave new insight into the conformational states and barriers that precede the rate-limiting step in the formation of the N-state of apoMb.

Conformational dynamics of human IAPP monomers

We study the conformational dynamics of the human Islet Amyloid Polypeptide (hIAPP) molecule - a 37 residue-long peptide associated to type 2 diabetes - using molecular dynamics (MD) simulations. We identify partially structured conformational states of the hIAPP monomer, categorized by both end-to-end distance and secondary structure, as suggested by previous experimental and computational studies. The MD trajectories of hIAPP are analyzed using data-driven methods, in particular principal component analysis, in order to identify preferred conformational states of the amylin monomer and to discuss their relative stability as compared to corresponding states in the amylin dimer. These potential hIAPP conformational states could be further tested and described experimentally, or in conjunction with modern computational analysis tools such as Markov state-based methods for extracting kinetics and thermodynamics from atomistic MD trajectories.

Simulations of tryptophan fluorescence dynamics during folding of the villin headpiece

Protein folding kinetics is commonly monitored by changes in tryptophan (Trp) fluorescence intensity. Considerable recent discussion has centered on whether the fluorescence of the single Trp in the much-studied, fast-folding villin headpiece C-terminal domain (HP35) accurately reflects folding kinetics, given the general view that quenching is by a histidine cation (His(+)) one turn away in an α-helix (helix III) that forms early in the folding process, according to published MD simulations. To help answer this question, we ran 1.0 μs MD simulations on HP35 (N27H) and a faster-folding variant in its folded form at 300 K and used the coordinates and force field charges with quantum calculations to simulate fluorescence quenching caused by electron transfer to the local amide and to the His(+). The simulations demonstrate that quenching by His(+) in the fully formed helix III is possible only during certain Trp and His(+) rotamer and solvent conformations, the propensity of which is a variable that can allow Trp fluorescence to report the global folding rate, as recent experiments imply.

Spectroscopic studies of protein folding: linear and nonlinear methods

Although protein folding is a simple outcome of the underlying thermodynamics, arriving at a quantitative and predictive understanding of how proteins fold nevertheless poses huge challenges. Therefore, both advanced experimental and computational methods are continuously being developed and refined to probe and reveal the atomistic details of protein folding dynamics and mechanisms. Herein, we provide a concise review of recent developments in spectroscopic studies of protein folding, with a focus on new triggering and probing methods. In particular, we describe several laser-based techniques for triggering protein folding/unfolding on the picosecond and/or nanosecond timescales and various linear and nonlinear spectroscopic techniques for interrogating protein conformations, conformational transitions, and dynamics.

UV resonance Raman studies of the NaClO4 dependence of poly-L-lysine conformation and hydrogen exchange kinetics

We used 204 nm excitation UV Resonance Raman (UVRR) spectroscopy to examine the effects of NaClO(4) on the conformation of poly-L-lysine (PLL). The presence of NaClO(4) induces the formation of α-helix, π-helix/bulge, and turn conformations. The dependence of the AmIII(3) frequency on the peptide Ψ Ramachandran angle allows us to experimentally determine the conformational population distributions and the energy landscape of PLL along the Ramachandran Ψ angle. We also used UVRR to measure the NaClO(4) concentration dependence of PLL amide hydrogen exchange kinetics. Exchange rates were determined by fitting the D(2)O exchanging PLL UVRR AmII' band time evolution. Hydrogen exchange is slowed at high NaClO(4) concentrations. The PLL AmII' band exchange kinetics at 0.0, 0.2, and 0.35 M NaClO(4) can be fit by single exponentials, but the AmII' band kinetics of PLL at 0.8 M NaClO(4) requires a double exponential fit. The exchange rates for the extended conformations were monitored by measuring the C(α)-H band kinetics. These kinetics are identical to those of the AmII' band until 0.8 M NaClO(4) whereupon the extended conformation exchange becomes clearly faster than that of the α-helix-like conformations. Our results indicate that ClO(4)(-) binds to the PLL backbone to protect it from OH(-) exchange catalysis. In addition, ClO(4)(-) binding also slows the conformational exchange between the extended and α-helix-like conformations, probably by increasing the activation barriers for conformational interchanges.

Molecular interactions of Alzheimer's Aβ protofilaments with lipid membranes

Amyloid fibrils and peptide oligomers play central roles in the pathology of Alzheimer's disease, type 2 diabetes, Parkinson's disease, Huntington's disease, and prion-related disease. Here, we investigate the molecular interactions between preformed amyloid β (Aβ) molecular protofilaments and lipid bilayer membranes, in the presence of explicit water molecules, using computational models and all-atom molecular dynamics. These interactions play an important role in the stability and function of both Aβ fibrils and the adjacent cellular membrane. Taking advantage of the symmetry-related and directional properties of the protofilaments, we build models that cover several relative protofilament-membrane orientations. Our molecular dynamics simulations reveal the relative contributions of different structural elements to the dynamics and stability of Aβ protofilament segments near membranes, and the first steps in the mechanism of fibril-membrane interactions. During this process, we observe a significant alteration of the side-chain contact pattern in protofilaments, although a fraction of the characteristic β-sheet content is preserved. As a major driving force, we identify the electrostatic interactions between Aβ charged side chains, including E22, D23, and K28, and lipid headgroups. Together with hydrogen bonding with atoms from lipid headgroups, these interactions can facilitate the penetration of hydrophobic C-terminal amino acids through the lipid headgroup region, which can finally lead both to further loss of the initial fibril structure and to local membrane-thinning effects. Our results may guide new experiments that could test the extent to which the structural features of water-formed amyloid fibrils are preserved, lost, or reshaped by membrane-mediated interactions.

Effects of solvents on the intrinsic propensity of peptide backbone conformations

We investigated the effects of solvents on the intrinsic propensity of peptide backbone conformations based on molecular dynamics simulations. The results show that compared with pure water, aqueous urea decreases the helix propensity. In comparison, methanol decreases the polyproline II (PPII) propensity. Such a solvent dependence of the intrinsic propensity of the backbone conformation is correlated with the solvent dependence of the hydration of the backbone groups and the formation probability of the local intrapeptide hydrogen bonds. Aqueous urea which has low ability to stabilize the local intrapeptide hydrogen bonds disfavors the helical conformation. Whereas, methanol which has low ability to hydrate the backbone groups disfavors the polyproline II conformation. In addition, the solvent effects can be further modulated by the side chains of the peptides. The solvent effects of the intrinsic propensity of peptide backbone conformations observed in this work suggest that changing the intrinsic propensity of the protein backbone conformations can partly contribute to the solvent-induced protein structure and dynamics variations. These results will be useful in understanding the solvent dependence of the conformational distributions of the unfolded proteins or peptides (or intrinsically disordered proteins) in which the global tertiary interactions are less important than that in the well-folded proteins.

MSMBuilder2: Modeling Conformational Dynamics at the Picosecond to Millisecond Scale

Markov State Models provide a framework for understanding the fundamental states and rates in the conformational dynamics of biomolecules. We describe an improved protocol for constructing Markov State Models from molecular dynamics simulations. The new protocol includes advances in clustering, data preparation, and model estimation; these improvements lead to significant increases in model accuracy, as assessed by the ability to recapitulate equilibrium and kinetic properties of reference systems. A high-performance implementation of this protocol, provided in MSMBuilder2, is validated on dynamics ranging from picoseconds to milliseconds.

Are Peptides Good Two-State Folders?

The folding kinetics of proteins is frequently single-exponential, as basins of folded and unfolded conformations are well separated by a high barrier. However, for relatively short peptides, a two-state character of folding is rather the exception than the rule. In this work, we use a Zwanzig-type model of protein conformational dynamics to study the dependence of folding kinetics on the protein chain length, M. The analysis is focused on the gap in the eigenvalue spectrum of the rate matrix that describes the protein's conformational dynamics. When there is a large gap between the two smallest in magnitude nonzero eigenvalues, the corresponding relaxation times have qualitatively different physical interpretations. The longest of these two times characterizes the interbasin equilibration (i.e., folding), whereas the second time characterizes the intrabasin relaxation. We derive approximate analytical solutions for the two eigenvalues that show how they depend on M. From these solutions, we infer that there is a large gap between the two, and thus, the kinetics is essentially single-exponential when M is large enough such that 2(M+1) is much larger than M(2).

The protein folding network indicates that the ultrafast folding mutant of villin headpiece subdomain has a deeper folding funnel

Protein folding is a dynamic process with continuous transitions among different conformations. In this work, the dynamics in the protein folding network of villin headpiece subdomain (HP35) has been investigated based on multiple reversible folding trajectories of HP35 and its ultrafast folding mutant where sub-angstrom folding was achieved. The four folding states were clearly separated on the network, validating the classification of the states. Examination of the eight conformers with different formation of the individual helices revealed high plasticity of the three helices in all the four states. A consistent feature between the wild type and mutant protein is the dominant conformer 111 (all three helices formed) in the folded state and conformers 111 and 011 (helices II and III formed) in the major intermediate state, indicating the critical role of helices II and III in the folding mechanism. When compared to the wild type, the folding landscape of the ultrafast folding mutant exhibited a deeper folding funnel towards the folded state. The very beginning of the folding (0-10 ns) was very similar for both protein variants but it soon diverged and displayed different folding pathways. Although going through the major intermediate state is the dominant pathway for both, it was also observed that some folding went through the minor intermediate state for the mutant. The intriguing difference resulting from the mutation at two residues in helix III has been carefully analyzed and discussed in details.