ETNA: Equilibrium Transitions Network and Arrhenius Equation for Extracting Folding Kinetics from REMD Simulations

Antibody binding modulates the dynamics of the membrane-bound prion protein

Misfolding of the cellular prion protein (PrP^C) is associated with lethal neurodegeneration. PrP^C consists of a flexible tail (residues 23-123) and a globular domain (residues 124-231) whose C-terminal end is anchored to the cell membrane. The neurotoxic antibody POM1 and the innocuous antibody POM6 recognize the globular domain. Experimental evidence indicates that POM1 binding to PrP^C emulates the influence on PrP^C of the misfolded prion protein (PrP^Sc) while the binding of POM6 has the opposite biological response. Little is known about the potential interactions between flexible tail, globular domain, and the membrane. Here, we used atomistic simulations to investigate how these interactions are modulated by the binding of the Fab fragments of POM1 and POM6 to PrP^C and by interstitial sequence truncations to the flexible tail. The simulations show that the binding of the antibodies restricts the range of orientations of the globular domain with respect to the membrane and decreases the distance between tail and membrane. Five of the six sequence truncations influence only marginally this distance and the contact patterns between tail and globular domain. The only exception is a truncation coupled to a charge inversion mutation of four N-terminal residues, which increases the distance of the flexible tail from the membrane. The interactions of the flexible tail and globular domain are modulated differently by the two antibodies.

Unsupervised Learning Methods for Molecular Simulation Data

Unsupervised learning is becoming an essential tool to analyze the increasingly large amounts of data produced by atomistic and molecular simulations, in material science, solid state physics, biophysics, and biochemistry. In this Review, we provide a comprehensive overview of the methods of unsupervised learning that have been most commonly used to investigate simulation data and indicate likely directions for further developments in the field. In particular, we discuss feature representation of molecular systems and present state-of-the-art algorithms of dimensionality reduction, density estimation, and clustering, and kinetic models. We divide our discussion into self-contained sections, each discussing a specific method. In each section, we briefly touch upon the mathematical and algorithmic foundations of the method, highlight its strengths and limitations, and describe the specific ways in which it has been used-or can be used-to analyze molecular simulation data.

Equilibration of Protein States: A Time Dependent Free-Energy Disconnectivity Graph

The process of equilibration of protein states in a three-stranded antiparallel β-sheet miniprotein is studied using a time-dependent free energy disconnectivity graph. To determine the rates of transitions, the molecular dynamics simulation results of a recent work (Kalgin, I. V.; J. Phys. Chem. B 2013, 117, 6092) are employed. The vertices of the graph are the free energies of characteristic states of the protein, and the edges are the transition state free energies. To determine the latter, the "complete" partition function (Eyring, 1935) is used, which includes the translational partition function corresponding to the ballistic motion of the system along the reaction coordinate. The distance along the reaction coordinate that enters the translational partition function is taken to be proportional to the observation time and thus measures the number of representative points that cross the transition state surface during given time. As the time increases, the free energy barriers between the clusters of characteristic conformations (native-like, helical, and β-sheet conformations of different degree of organization) decrease and (local) equilibrium between the clusters is established. With time, these clusters are grouped into larger clusters, extending the equilibrium to a larger portion of protein states.

Folding of a β-sheet miniprotein: probability fluxes, streamlines, and the potential for the driving force

In this work we continue the study of the first-passage folding of an antiparallel β-sheet miniprotein (beta3s) that was initiated in the previous work [Kalgin et al. J. Phys. Chem. B, 2014, 118, 4287]. We consider a larger ensemble of folding trajectories, which allows us to gain a closer insight into the folding dynamics. In particular, we calculate the potential for the driving force of folding in a reduced space of collective variables. The potential has two components. One component (Φ) is responsible for the source and sink of the folding flow, which are formed, respectively, in the regions of the unfolded and native states of the protein, and the other (Ψ) accounts for the flow vorticity inherently generated at the sides of the reaction channel and provides the canalization of the folding flow between the source and sink. We show that both components obey Poisson's equations with the corresponding source/sink terms. The resulting components have a very simple form: the Φ-surface consists of two well-defined peaks of different signs, which correspond, respectively, to the source and sink of the folding flow, and the Ψ-surface consists of two ridges of different signs that connect the source and sink of the flow.

A molecular simulation protocol to avoid sampling redundancy and discover new states

BACKGROUND

For biomacromolecules or their assemblies, experimental knowledge is often restricted to specific states. Ambiguity pervades simulations of these complex systems because there is no prior knowledge of relevant phase space domains, and sampling recurrence is difficult to achieve. In molecular dynamics methods, ruggedness of the free energy surface exacerbates this problem by slowing down the unbiased exploration of phase space. Sampling is inefficient if dwell times in metastable states are large.

METHODS

We suggest a heuristic algorithm to terminate and reseed trajectories run in multiple copies in parallel. It uses a recent method to order snapshots, which provides notions of "interesting" and "unique" for individual simulations. We define criteria to guide the reseeding of runs from more "interesting" points if they sample overlapping regions of phase space.

RESULTS

Using a pedagogical example and an α-helical peptide, the approach is demonstrated to amplify the rate of exploration of phase space and to discover metastable states not found by conventional sampling schemes. Evidence is provided that accurate kinetics and pathways can be extracted from the simulations.

CONCLUSIONS

The method, termed PIGS for Progress Index Guided Sampling, proceeds in unsupervised fashion, is scalable, and benefits synergistically from larger numbers of replicas. Results confirm that the underlying ideas are appropriate and sufficient to enhance sampling.

GENERAL SIGNIFICANCE

In molecular simulations, errors caused by not exploring relevant domains in phase space are always unquantifiable and can be arbitrarily large. Our protocol adds to the toolkit available to researchers in reducing these types of errors. This article is part of a Special Issue entitled "Recent developments of molecular dynamics".

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.

The mechanism of the converter domain rotation in the recovery stroke of myosin motor protein

Upon ATP binding, myosin motor protein is found in two alternative conformations, prerecovery state M* and postrecovery state M**. The transition from one state to the other, known as the recovery stroke, plays a key role in the myosin functional cycle. Despite much recent research, the microscopic details of this transition remain elusive. A critical step in the recovery stroke is the rotation of the converter domain from "up" position in prerecovery state to "down" position in postrecovery state that leads to the swing of the lever arm attached to it. In this work, we demonstrate that the two rotational states of the converter domain are determined by the interactions within a small structural motif in the force-generating region of the protein that can be accurately modeled on computers using atomic representation and explicit solvent. Our simulations show that the transition between the two states is controlled by a small helix (SH1) located next to the relay helix and relay loop. A small translation in the position of SH1 away from the relay helix is seen to trigger the transition from "up" state to "down" state. The transition is driven by a cluster of hydrophobic residues I687, F487, and F506 that make significant contributions to the stability of both states. The proposed mechanism agrees well with the available structural and mutational studies.

Distribution of Reciprocal of Interatomic Distances: A Fast Structural Metric

We present a structural metric based on the Distribution of Reciprocal of Interatomic Distances (DRID) for evaluating geometrical similarity between two conformations of a molecule. A molecular conformation is described by a vector of 3N orientation-independent DRID descriptors where N is the number of molecular centroids, for example, the non-hydrogen atoms in all nonsymmetric groups of a peptide. For two real-world applications (pairwise comparison of snapshots from an explicit solvent simulation of a protease/peptide substrate complex and implicit solvent simulations of reversible folding of a 20-residue β-sheet peptide), the DRID-based metric is shown to be about 5 and 15 times faster than coordinate root-mean-square deviation (cRMSD) and distance root-mean-square deviation (dRMSD), respectively. A single core of a mainstream processor can perform about 10(8) DRID comparisons in one-half a minute. Importantly, the DRID metric has closer similarity to kinetic distance than does either cRMSD or dRMSD. Therefore, DRID is suitable for clustering molecular dynamics trajectories for kinetic analysis, for example, by Markov state models. Moreover, conformational space networks and free energy profiles derived by DRID-based clustering preserve the kinetic information.

Free Energy Guided Sampling

A free energy-guided sampling (FEGS) method is proposed for accelerating exploration of conformational space in unbiased molecular dynamics. Using the cut-based free energy profile and Markov state models, FEGS speeds up sampling of the canonical ensemble by iteratively restarting multiple short simulations in parallel from regions of the free energy surface visited rarely. This exploration stage is followed by a refinement stage in which multiple independent runs are initiated from Boltzmann distributed conformations. Notably, FEGS does not require either collective variables or reaction coordinates and can control the kinetic distance from the starting conformation. We applied FEGS to the alanine dipeptide, which has a human-comprehensible two-dimensional free energy landscape, and a three-stranded antiparallel β-sheet peptide of 20 residues whose folding/unfolding process is governed by a delicate interplay of enthalpy and entropy. For these two systems, FEGS speeds up the exploration of conformational space by 1 to 2 orders of magnitude with respect to conventional sampling and preserves the basins and barriers on the free energy profile.

Optimal use of data in parallel tempering simulations for the construction of discrete-state Markov models of biomolecular dynamics

Parallel tempering (PT) molecular dynamics simulations have been extensively investigated as a means of efficient sampling of the configurations of biomolecular systems. Recent work has demonstrated how the short physical trajectories generated in PT simulations of biomolecules can be used to construct the Markov models describing biomolecular dynamics at each simulated temperature. While this approach describes the temperature-dependent kinetics, it does not make optimal use of all available PT data, instead estimating the rates at a given temperature using only data from that temperature. This can be problematic, as some relevant transitions or states may not be sufficiently sampled at the temperature of interest, but might be readily sampled at nearby temperatures. Further, the comparison of temperature-dependent properties can suffer from the false assumption that data collected from different temperatures are uncorrelated. We propose here a strategy in which, by a simple modification of the PT protocol, the harvested trajectories can be reweighted, permitting data from all temperatures to contribute to the estimated kinetic model. The method reduces the statistical uncertainty in the kinetic model relative to the single temperature approach and provides estimates of transition probabilities even for transitions not observed at the temperature of interest. Further, the method allows the kinetics to be estimated at temperatures other than those at which simulations were run. We illustrate this method by applying it to the generation of a Markov model of the conformational dynamics of the solvated terminally blocked alanine peptide.

Delineation of folding pathways of a β-sheet miniprotein

Several methods have been developed in the past few years for the analysis of molecular dynamics simulations of biological (macro)molecules whose complexity is difficult to capture by simple projections of the free-energy surface onto one or two geometric variables. The locally scaled diffusion map (LSDMap) method is a nonlinear dimensionality reduction technique for describing the dynamics of complex systems in terms of a few collective coordinates. Here, we compare LSDMap to two previously developed approaches for the characterization of the configurational landscape associated with the folding dynamics of a three-stranded antiparallel β-sheet peptide, termed Beta3s. The analysis is aided by an improved procedure for extracting pathways from the equilibrium transition network, which enables calculation of pathway-specific cut-based free energy profiles. We find that the results from LSDMap are consistent with analysis based on transition networks and allow a coherent interpretation of metastable states and folding pathways in terms of different time scales of transitions between minima on the free energy projections.

Extracting physically intuitive reaction coordinates from transition networks of a beta-sheet miniprotein

Simulations are important for understanding complex reactions, but their interpretation is challenging owing to the large number of degrees of freedom typically involved. To address this issue, various means for relating the dynamics of a stochastic system to its structural and energetic features have been introduced. Here, we show how two leading approaches can be combined to advantage. We use the network of transitions observed in a reversible folding/unfolding simulation of a 20-residue three-stranded antiparallel beta-sheet peptide (beta3s) to estimate the probabilities of committing to stable states (the native state and major nonnative states), and these then serve as the basis for an efficient statistical procedure for identifying physical variables that describe the dynamics. We find that a single coordinate that jointly characterizes the formation of the two native turns of beta3s can adequately describe the overall folding process, despite its complex nature. Additional features associated with major pathways leading from individual nonnative states are resolved; indeed, a key result is an improved understanding of the unfolded state. Connections to other methods for analyzing complex reactions are discussed.

Network models for molecular kinetics and their initial applications to human health

Molecular kinetics underlies all biological phenomena and, like many other biological processes, may best be understood in terms of networks. These networks, called Markov state models (MSMs), are typically built from physical simulations. Thus, they are capable of quantitative prediction of experiments and can also provide an intuition for complex conformational changes. Their primary application has been to protein folding; however, these technologies and the insights they yield are transferable. For example, MSMs have already proved useful in understanding human diseases, such as protein misfolding and aggregation in Alzheimer's disease.