Experiments and comprehensive simulations of the formation of a helical turn

Dissecting Multiple Pathways in the Relaxation Dynamics of Helix <==> Coil Transitions with Optimum Dimensionality Reduction

Fast kinetic experiments with dramatically improved time resolution have contributed significantly to understanding the fundamental processes in protein folding pathways involving the formation of a-helices and b-hairpin, contact formation, and overall collapse of the peptide chain. Interpretation of experimental results through application of a simple statistical mechanical model was key to this understanding. Atomistic description of all events observed in the experimental findings was challenging. Recent advancements in theory, more sophisticated algorithms, and a true long-term trajectory made way for an atomically detailed description of kinetics, examining folding pathways, validating experimental results, and reporting new findings for a wide range of molecular processes in biophysical chemistry. This review describes how optimum dimensionality reduction theory can construct a simplified coarse-grained model with low dimensionality involving a kinetic matrix that captures novel insights into folding pathways. A set of metastable states derived from molecular dynamics analysis generate an optimally reduced dimensionality rate matrix following transition pathway analysis. Analysis of the actual long-term simulation trajectory extracts a relaxation time directly comparable to the experimental results and confirms the validity of the combined approach. The application of the theory is discussed and illustrated using several examples of helix <==> coil transition pathways. This paper focuses primarily on a combined approach of time-resolved experiments and long-term molecular dynamics simulation from our ongoing work.

Kinetic pathway analysis of an α-helix in two protonation states: Direct observation and optimal dimensionality reduction

Thermodynamically stable conformers of secondary structural elements make a stable tertiary/quaternary structure that performs its proper biological function efficiently. Formation mechanisms of secondary and tertiary/quaternary structural elements from the primary structure are driven by the kinetic properties of the respective systems. Here we have carried out thermodynamic and kinetic characterization of an alpha helical heteropeptide in two protonation states, created with the addition and removal of a proton involving a single histidine residue in the primary structure. Applying far-UV circular dichroism spectroscopy, the alpha helix is observed to be significantly more stable in the deprotonated state. Nanosecond laser temperature jump spectroscopy monitoring time-resolved tryptophan fluorescence on the protonated conformer is carried out to measure the kinetics of this system. The measured relaxation rates at a final temperature between 296K and 314 K generated a faster component of 20 ns-11 ns and a slower component of 314 ns-198 ns. Atomically detailed characterization of the helix-coil kinetic pathways is performed based on all-atom molecular dynamics trajectories of the two conformers. Application of clustering and kinetic coarse-graining with optimum dimensionality reduction produced description of the trajectories in terms of kinetic models with two to five states. These models include aggregate states corresponding to helix, coil, and intermediates. The "coil" state involves the largest number of conformations, consistent with the expected high entropy of this structural ensemble. The "helix" aggregate states are found to be mixed with the full helix and partially folded forms. The experimentally observed higher helix stability in the deprotonated form of the alpha helical heteropeptide is reflected in the nature of the "helix" aggregate state arising from the kinetic model. In the protonated form, the "coil" state exhibits the lowest free energy and longest lifetime, while in the deprotonated form, it is the "helix" that is found to be most stable. Overall, the coarse grained models suggest that the protonation of a single histidine residue in the primary structure induces significant changes in the free energy landscape and kinetic network of the studied helix-forming heteropeptide.

Helix-Coil Transition Courses Through Multiple Pathways and Intermediates: Fast Kinetic Measurements and Dimensionality Reduction

Nanosecond laser temperature jumps with tryptophan fluorescence detection and molecular dynamics simulation with kinetic dimensionality reduction were used to study the helix-coil transition in a 21-residue α-helical heteropeptide. Analysis of the temperature- dependent relaxation dynamics of this heteropeptide identified a distinct faster component of 20-35 ns, besides a slower component of 300-400 ns at temperatures between 296 and 280 K. To understand the mechanism of progression from a non-structured coil state to a structured helical state, we carried out a 12 μs molecular dynamics simulation of this peptide system. Clustering and optimal dimensionality reduction were applied to the molecular dynamics trajectory to generate low-dimensional coarse-grained models of the underlying kinetic network in terms of 2-5 metastable states. In accord with the generally accepted understanding of the multiple conformations and high entropy of the unfolded ensemble of states, the "coil" metastable set contains the largest number of structures. Interestingly, the helix metastable state was also found to be structurally heterogeneous, consisting of the completely helical form and several partly folded conformers that interconvert at a time scale faster that global folding. The intermediate states contain the fewest structures, have lowest populations, and have the shortest lifetimes. As the number of considered metastable states increases, more intermediates and more folding paths appear in the coarse-grained models. One of these intermediates corresponds to the transition state for folding, which involves an "off-center" helical region over residues 11-16. The kinetic network model is consistent with a statistical picture of folding following a simple reaction coordinate counting the helical population of individual residues. On the basis of simulations, we propose that the fast relaxation time should be assigned to cooperative folding/unfolding of segments of 1-4 neighboring residues.

A MATHEMATICAL FRAMEWORK FOR EXACT MILESTONING

We give a mathematical framework for Exact Milestoning, a recently introduced algorithm for mapping a continuous time stochastic process into a Markov chain or semi-Markov process that can be efficiently simulated and analyzed. We generalize the setting of Exact Milestoning and give explicit error bounds for the error in the Milestoning equation for mean first passage times.

Exact milestoning

A new theory and an exact computer algorithm for calculating kinetics and thermodynamic properties of a particle system are described. The algorithm avoids trapping in metastable states, which are typical challenges for Molecular Dynamics (MD) simulations on rough energy landscapes. It is based on the division of the full space into Voronoi cells. Prior knowledge or coarse sampling of space points provides the centers of the Voronoi cells. Short time trajectories are computed between the boundaries of the cells that we call milestones and are used to determine fluxes at the milestones. The flux function, an essential component of the new theory, provides a complete description of the statistical mechanics of the system at the resolution of the milestones. We illustrate the accuracy and efficiency of the exact Milestoning approach by comparing numerical results obtained on a model system using exact Milestoning with the results of long trajectories and with a solution of the corresponding Fokker-Planck equation. The theory uses an equation that resembles the approximate Milestoning method that was introduced in 2004 [A. K. Faradjian and R. Elber, J. Chem. Phys. 120(23), 10880-10889 (2004)]. However, the current formulation is exact and is still significantly more efficient than straightforward MD simulations on the system studied.

Simulations of thermodynamics and kinetics on rough energy landscapes with milestoning

We investigated by computational means the kinetics and stationary behavior of stochastic dynamics on an ensemble of rough two-dimensional energy landscapes. There are no obvious separations of temporal scales in these systems, which constitute a simple model for the behavior of glasses and some biomaterials. Even though there are significant computational challenges present in these systems due to the large number of metastable states, the Milestoning method is able to compute their kinetic and thermodynamic properties exactly. We observe two clearly distinguished regimes in the overall kinetics: one in which diffusive behavior dominates and another that follows an Arrhenius law (despite the absence of a dominant barrier). We compare our results with those obtained with an exactly-solvable one-dimensional model, and with the results from the rough one-dimensional energy model introduced by Zwanzig. © 2015 Wiley Periodicals, Inc.

Modeling kinetics and equilibrium of membranes with fields: milestoning analysis and implication to permeation

Coarse graining of membrane simulations by translating atomistic dynamics to densities and fields with Milestoning is discussed. The space of the membrane system is divided into cells and the different cells are characterized by order parameters presenting the number densities. The dynamics of the order parameters are probed with Milestoning. The methodology is illustrated here for a phospholipid membrane system (a hydrated bilayer of DOPC (1,2-dioleoyl-sn-glycero-3-phosphocholine) lipid molecules). Significant inhomogeneity in membrane internal number density leads to complex free energy landscape and local maps of transition times. Dynamics and distributions of cavities within the membrane assist the permeation of nonpolar solutes such as xenon atoms. It is illustrated that quantitative and detailed dynamics of water transport through DOPC membrane can be analyzed using Milestoning with fields. The reaction space for water transport includes at least two slow variables: the normal to the membrane plane, and the water density.

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.

Molecular modeling of peptides

This article presents a review of the field of molecular modeling of peptides. The main focus is on atomistic modeling with molecular mechanics potentials. The description of peptide conformations and solvation through potentials is discussed. Several important computer simulation methods are briefly introduced, including molecular dynamics, accelerated sampling approaches such as replica-exchange and metadynamics, free energy simulations and kinetic network models like Milestoning. Examples of recent applications for predictions of structure, kinetics, and interactions of peptides with complex environments are described. The reliability of current simulation methods is analyzed by comparison of computational predictions obtained using different models with each other and with experimental data. A brief discussion of coarse-grained modeling and future directions is also presented.

Detailed microscopic unfolding pathways of an α-helix and a β-hairpin: direct observation and molecular dynamics

We present a combined experimental and computational study of unfolding pathways of a model 21-residue α-helical heteropeptide (W1H5-21) and a 16-residue β-hairpin (GB41-56). Experimentally, we measured fluorescence energy transfer efficiency as a function of temperature, employing natural tryptophans as donors and dansylated lysines as acceptors. Secondary structural analysis was performed with circular dichroism and Fourier transform infrared spectroscopy. Our studies present markedly different unfolding pathways of the two elementary secondary structural elements. During thermal denaturation, the helical peptide exhibits an initial decrease in length, followed by an increase, while the hairpin undergoes a systematic increase in length. In the complementary computational part of the project, we performed microsecond length replica-exchange molecular dynamics simulations of the peptides in explicit solvent, yielding a detailed microscopic picture of the unfolding processes. For the α-helical peptide, we found a large heterogeneous population of intermediates that are primarily frayed single helices or helix-turn-helix motifs. Unfolding starts at the termini and proceeds through a stable helical region in the interior of the peptide but shifted off-center toward the C-terminus. The simulations explain the experimentally observed non-monotonic variation of helix length with temperature as due primarily to the presence of frayed-end single-helix intermediate structures. For the β-hairpin peptide, our simulations indicate that folding is initiated at the turn, followed by formation of the hairpin in zipper-like fashion, with Cα···Cα contacts propagating from the turn to termini and hairpin hydrogen bonds forming in parallel with these contacts. In the early stages of hairpin formation, the hydrophobic side-chain contacts are only partly populated. Intermediate structures with low numbers of β-hairpin hydrogen bonds have very low populations. This is in accord with the "broken zipper" model of Scheraga. The monotonic increase in length with temperature may be explained by the zipper-like breaking of the hairpin hydrogen bonds and backbone contacts.

Non-exponential kinetics and a complete folding pathway of an α-helical heteropeptide: direct observation and comprehensive molecular dynamics

We have performed a combined experimental and computational study of the folding of a 21-residue α-helix-forming heteropeptide (WH21). Temperature jump kinetics with improved dynamic range at several temperatures revealed non-exponential relaxation that could be well described with two time constants of 20 and 300 ns at 298 K. In the computational part, we performed multi-microsecond molecular dynamics simulations of WH21 in explicit water, using the AMBER03 and OPLS/AA potentials. The simulations were in good agreement with available experimental data on helix content and relaxation times. On the basis of 70 individual transitions, we identified the main pathways of helix unfolding. Three paths were found in both force fields, with unfolding progressing through (1) N-terminus, C-terminus, and center; (2) C-terminus, N-terminus, and center; and (3) C-terminus, center, and N-terminus. An additional fourth path starting in the central region and expanding to the termini was detected only with AMBER03. Intermediate structures sampled along the pathway included a central helix with frayed termini, an off-center helix, and a helical hairpin. The simulations suggest that the short relaxation should be assigned to partly cooperative fluctuations of several neighboring hydrogen bonds. Overall, by a combination of ultrafast kinetic measurements and detailed microscopic description through comprehensive molecular dynamics, we have obtained important new insights into the helix folding process.

Computational study of peptide permeation through membrane: Searching for hidden slow variables

Atomically detailed molecular dynamics trajectories in conjunction with Milestoning are used to analyze the different contributions of coarse variables to the permeation process of a small peptide (N-acetyl-L-tryptophanamide, NATA) through a 1,2-dioleoyl-sn-glycero-3-phosphocholine (DOPC) membrane. The peptide reverses its overall orientation as it permeates through the biological bilayer. The large change in orientation is investigated explicitly but is shown to impact the free energy landscape and permeation time only moderately. Nevertheless, a significant difference in permeation properties of the two halves of the membrane suggests the presence of other hidden slow variables. We speculate, based on calculation of the potential of mean force, that a conformational transition of NATA makes significant contribution to these differences. Other candidates for hidden slow variables may include water permeation and collective motions of phospholipids.

Analyzing milestoning networks for molecular kinetics: definitions, algorithms, and examples

Network representations are becoming increasingly popular for analyzing kinetic data from techniques like Milestoning, Markov State Models, and Transition Path Theory. Mapping continuous phase space trajectories into a relatively small number of discrete states helps in visualization of the data and in dissecting complex dynamics to concrete mechanisms. However, not only are molecular networks derived from molecular dynamics simulations growing in number, they are also getting increasingly complex, owing partly to the growth in computer power that allows us to generate longer and better converged trajectories. The increased complexity of the networks makes simple interpretation and qualitative insight of the molecular systems more difficult to achieve. In this paper, we focus on various network representations of kinetic data and algorithms to identify important edges and pathways in these networks. The kinetic data can be local and partial (such as the value of rate coefficients between states) or an exact solution to kinetic equations for the entire system (such as the stationary flux between vertices). In particular, we focus on the Milestoning method that provides fluxes as the main output. We proposed Global Maximum Weight Pathways as a useful tool for analyzing molecular mechanism in Milestoning networks. A closely related definition was made in the context of Transition Path Theory. We consider three algorithms to find Global Maximum Weight Pathways: Recursive Dijkstra's, Edge-Elimination, and Edge-List Bisection. The asymptotic efficiency of the algorithms is analyzed and numerical tests on finite networks show that Edge-List Bisection and Recursive Dijkstra's algorithms are most efficient for sparse and dense networks, respectively. Pathways are illustrated for two examples: helix unfolding and membrane permeation. Finally, we illustrate that networks based on local kinetic information can lead to incorrect interpretation of molecular mechanisms.

Catch bond-like kinetics of helix cracking: network analysis by molecular dynamics and milestoning

The first events of unfolding of secondary structure under load are considered with Molecular Dynamics simulations and Milestoning analysis of a long helix (126 amino acids). The Mean First Passage Time is a non-monotonic function of the applied load with a maximum of 3.6 ns at about 20 pN. Network analysis of the reaction space illustrates the opening and closing of an off-pathway trap that slows unfolding at intermediate load levels. It is illustrated that the nature of the reaction networks changes as a function of load, demonstrating that the process is far from one-dimensional.

Automated Optimization of Potential Parameters

An algorithm and software to refine parameters of empirical energy functions according to condensed phase experimental measurements are discussed. The algorithm is based on sensitivity analysis and local minimization of the differences between experiment and simulation as a function of potential parameters. It is illustrated for a toy problem of alanine dipeptide and is applied to folding of the peptide WAAAH. The helix fraction is highly sensitive to the potential parameters while the slope of the melting curve is not. The sensitivity variations make it difficult to satisfy both observations simultaneously. We conjecture that there is no set of parameters that reproduces experimental melting curves of short peptides that are modeled with the usual functional form of a force field.

On the foldability of tryptophan-containing tetra- and pentapeptides: an exhaustive molecular dynamics study

Short peptides serve as minimal model systems to decipher the determinants of foldability due to their simplicity arising from their smaller size, their ability to echo protein-like structural characteristics, and their direct implication in force field validation. Here, we describe an effort to identify small peptides that can still form stable structures in aqueous solutions. We followed the in silico folding of a selected set of 8640 tryptophan-containing tetra- and pentapeptides through 15 210 molecular dynamics simulations amounting to a total of 272.46 μs using explicit representation of the solute and full treatment of the electrostatics. The evaluation and sorting of peptides is achieved through scoring functions, which include terms based on interatomic vector distances, atomic fluctuations, and rmsd matrices between successive frames of a trajectory. Highly scored peptides are studied further via successive simulation rounds of increasing simulation length and using different empirical force fields. Our method suggested only a handful of peptides with strong foldability prognosis. The discrepancies between the predictions of the various force fields for such short sequences are also extensively discussed. We conclude that the vast majority of such short peptides do not adopt significantly stable structures in water solutions, at least based on our computational predictions. The present work can be utilized in the rational design and engineering of bioactive peptides with desired molecular properties.

Early events in helix unfolding under external forces: a milestoning analysis

Initial events of helix breakage as a function of load are considered using molecular dynamics simulations and milestoning analysis. A helix length of ∼100 amino acids is considered as a model for typical helices found in molecular machines and as a model that minimizes end effects for early events of unfolding. Transitions of individual amino acids (averaged over the helix's interior residues) are examined and its surrounding hydrogen bonds are considered. Dense kinetic networks are constructed that, with milestoning analysis, provide the overall kinetics of early breakage events. Network analysis and selection of MaxFlux pathways illustrate that load impacts unfolding mechanisms in addition to time scales. At relatively high (100 pN) load levels, the principal intermediate is the 3(10)-helix, while at relatively low (10 pN) levels the π-helix is significantly populated, albeit not as an unfolding intermediate. Coarse variables are examined at different levels of resolution; the rate of unfolding illustrates remarkable stability under changes in the coarsening. Consistent prediction of about ∼5 ns for the time of a single amino-acid unfolding event are obtained. Hydrogen bonds are much faster coarse variables (by about 2 orders of magnitude) compared to backbone torsional transition, which gates unfolding and thereby provides the appropriate coarse variable for the initiation of unfolding. Results provide an atomic description of "catch-bond" behavior, based on alternative pathways, in which unfolding of a simple protein structural element occurs over longer timescales for intermediate (10 pN) loads than for zero (0 pN) or large (100 pN) loads.