Relaxation mode analysis of a peptide system: comparison with principal component analysis

Intermediate scattering function for polymer molecules: An approach based on relaxation mode analysis

The theory of polymer dynamics describes the intermediate scattering function for a polymer molecule in terms of relaxation modes defined by normal coordinates for the corresponding coarse-grained model. However, due to the difficulty of defining the normal coordinates for arbitrary polymer molecules, it is generally challenging to express the intermediate scattering function for a polymer molecule in terms of relaxation modes. To overcome this challenge, we propose a general method to calculate the intermediate scattering function for a polymer molecule on the basis of a relaxation mode analysis approach [Takano and Miyashita, J. Phys. Soc. Jpn. 64, 3688 (1995)]. In the proposed method, relaxation modes defined by eigenfunctions in a Markov process are evaluated on the basis of the simulation results for a polymer molecule and used to calculate the intermediate scattering function for that molecule. To demonstrate the effectiveness of the present method, we simulate the dynamics of a linear polymer molecule in a dilute solution and apply it to the calculation of the intermediate scattering function for the polymer molecule. The evaluation results regarding the relaxation modes reasonably describe the intermediate scattering function on the length scale of the radius of gyration of the polymer molecule. Accordingly, we examine the contributions of the pure relaxation and oscillatory relaxation processes to the entire intermediate scattering function.

Structural and Computational Insights into Dynamics and Intermediate States of Orexin 2 Receptor Signaling

Orexin 2 receptor (OX2R) is a G protein-coupled receptor (GPCR) whose activation is crucial to regulation of the sleep-wake cycle. Recently, inactive and active state structures were determined from X-ray crystallography and cryo-electron microscopy single particle analysis, and the activation mechanisms have been discussed based on these static data. GPCRs have multiscale intermediate states during activation, and insights into these dynamics and intermediate states may aid the precise control of intracellular signaling by ligands in drug discovery. Molecular dynamics (MD) simulations are used to investigate dynamics induced in response to thermal perturbations, such as structural fluctuations of main and side chains. In this study, we proposed collective motions of the TM domain during activation by performing 30 independent microsecond-scale MD simulations for various OX2R systems and applying relaxation mode analysis. The analysis results suggested that TM3 had a vertical structural movement relative to the membrane surface during activation. In addition, we extracted three characteristic amino acid residues on TM3, i.e., Q1343.32, V1423.40, and R1523.50, which exhibited large conformational fluctuations. We quantitatively evaluated the changes in their equilibrium during activation in relation to the movement of TM3. We also discuss the regulation of ligand binding recognition and intracellular signal selectivity by changes in the equilibrium of Q1343.32 and R1523.50, respectively, according to MD simulations and GPCR database. Additionally, the OX2R-Gi signaling complex is stabilized in the conformation resembling a non-canonical (NC) state, which was previously proposed as an intermediate state during activation of neurotensin 1 receptor. Insights into the dynamics and intermediate states during activation gained from this study may be useful for developing biased agonists for OX2R.

Tutorial on how to build non-Markovian dynamic models from molecular dynamics simulations for studying protein conformational changes

Protein conformational changes play crucial roles in their biological functions. In recent years, the Markov State Model (MSM) constructed from extensive Molecular Dynamics (MD) simulations has emerged as a powerful tool for modeling complex protein conformational changes. In MSMs, dynamics are modeled as a sequence of Markovian transitions among metastable conformational states at discrete time intervals (called lag time). A major challenge for MSMs is that the lag time must be long enough to allow transitions among states to become memoryless (or Markovian). However, this lag time is constrained by the length of individual MD simulations available to track these transitions. To address this challenge, we have recently developed Generalized Master Equation (GME)-based approaches, encoding non-Markovian dynamics using a time-dependent memory kernel. In this Tutorial, we introduce the theory behind two recently developed GME-based non-Markovian dynamic models: the quasi-Markov State Model (qMSM) and the Integrative Generalized Master Equation (IGME). We subsequently outline the procedures for constructing these models and provide a step-by-step tutorial on applying qMSM and IGME to study two peptide systems: alanine dipeptide and villin headpiece. This Tutorial is available at https://github.com/xuhuihuang/GME_tutorials. The protocols detailed in this Tutorial aim to be accessible for non-experts interested in studying the biomolecular dynamics using these non-Markovian dynamic models.

Discovering Reaction Pathways, Slow Variables, and Committor Probabilities with Machine Learning

A significant challenge faced by atomistic simulations is the difficulty, and often impossibility, to sample the transitions between metastable states of the free-energy landscape associated with slow molecular processes. Importance-sampling schemes represent an appealing option to accelerate the underlying dynamics by smoothing out the relevant free-energy barriers, but require the definition of suitable reaction-coordinate (RC) models expressed in terms of compact low-dimensional sets of collective variables (CVs). While most computational studies of slow molecular processes have traditionally relied on educated guesses based on human intuition to reduce the dimensionality of the problem at hand, a variety of machine-learning (ML) algorithms have recently emerged as powerful alternatives to discover meaningful CVs capable of capturing the dynamics of the slowest degrees of freedom. Considering a simple paradigmatic situation in which the long-time dynamics is dominated by the transition between two known metastable states, we compare two variational data-driven ML methods based on Siamese neural networks aimed at discovering a meaningful RC model─the slowest decorrelating CV of the molecular process, and the committor probability to first reach one of the two metastable states. One method is the state-free reversible variational approach for Markov processes networks (VAMPnets), or SRVs─the other, inspired by the transition path theory framework, is the variational committor-based neural networks, or VCNs. The relationship and the ability of these methodologies to discover the relevant descriptors of the slow molecular process of interest are illustrated with a series of simple model systems. We also show that both strategies are amenable to importance-sampling schemes through an appropriate reweighting algorithm that approximates the kinetic properties of the transition.

Analysis of Protein Folding Simulation with Moving Root Mean Square Deviation

We apply moving root-mean-square deviation (mRMSD), which does not require a reference structure, as a method for analyzing protein dynamics. This method can be used to calculate the root-mean-square deviation (RMSD) of structure between two specified time points and to analyze protein dynamics behavior through time series analysis. We applied this method to the Trp-cage trajectory calculated by the Anton supercomputer and found that it shows regions of stable states as well as the conventional RMSD. In addition, we extracted a characteristic structure in which the side chains of Asp1 and Arg16 form hydrogen bonds near the most stable structure of the Trp-cage. We also determined that ≥20 ns is an appropriate time interval to investigate protein dynamics using mRMSD. Applying this method to NuG2 protein, we found that mRMSD can be used to detect regions of metastable states in addition to the stable state. This method can be applied to molecular dynamics simulations of proteins whose stable structures are unknown.

Chasing collective variables using temporal data-driven strategies

The convergence of free-energy calculations based on importance sampling depends heavily on the choice of collective variables (CVs), which in principle, should include the slow degrees of freedom of the biological processes to be investigated. Autoencoders (AEs), as emerging data-driven dimension reduction tools, have been utilised for discovering CVs. AEs, however, are often treated as black boxes, and what AEs actually encode during training, and whether the latent variables from encoders are suitable as CVs for further free-energy calculations remains unknown. In this contribution, we review AEs and their time-series-based variants, including time-lagged AEs (TAEs) and modified TAEs, as well as the closely related model variational approach for Markov processes networks (VAMPnets). We then show through numerical examples that AEs learn the high-variance modes instead of the slow modes. In stark contrast, time series-based models are able to capture the slow modes. Moreover, both modified TAEs with extensions from slow feature analysis and the state-free reversible VAMPnets (SRVs) can yield orthogonal multidimensional CVs. As an illustration, we employ SRVs to discover the CVs of the isomerizations of N-acetyl-N'-methylalanylamide and trialanine by iterative learning with trajectories from biased simulations. Last, through numerical experiments with anisotropic diffusion, we investigate the potential relationship of time-series-based models and committor probabilities.

Principal Component Analysis and Related Methods for Investigating the Dynamics of Biological Macromolecules

Principal component analysis (PCA) is used to reduce the dimensionalities of high-dimensional datasets in a variety of research areas. For example, biological macromolecules, such as proteins, exhibit many degrees of freedom, allowing them to adopt intricate structures and exhibit complex functions by undergoing large conformational changes. Therefore, molecular simulations of and experiments on proteins generate a large number of structure variations in high-dimensional space. PCA and many PCA-related methods have been developed to extract key features from such structural data, and these approaches have been widely applied for over 30 years to elucidate macromolecular dynamics. This review mainly focuses on the methodological aspects of PCA and related methods and their applications for investigating protein dynamics.

Differences in ligand-induced protein dynamics extracted from an unsupervised deep learning approach correlate with protein-ligand binding affinities

Prediction of protein–ligand binding affinity is a major goal in drug discovery. Generally, free energy gap is calculated between two states (e.g., ligand binding and unbinding). The energy gap implicitly includes the effects of changes in protein dynamics induced by ligand binding. However, the relationship between protein dynamics and binding affinity remains unclear. Here, we propose a method that represents ligand-binding-induced protein behavioral change with a simple feature that can be used to predict protein–ligand affinity. From unbiased molecular simulation data, an unsupervised deep learning method measures the differences in protein dynamics at a ligand-binding site depending on the bound ligands. A dimension reduction method extracts a dynamic feature that strongly correlates to the binding affinities. Moreover, the residues that play important roles in protein–ligand interactions are specified based on their contribution to the differences. These results indicate the potential for binding dynamics-based drug discovery.

Differences in ligand-induced protein dynamics extracted as a single feature from a deep learning-based analysis of MD simulations correlate with ligand binding affinity.

Identifying the leading dynamics of ubiquitin: A comparison between the tICA and the LE4PD slow fluctuations in amino acids' position

Molecular Dynamics (MD) simulations of proteins implicitly contain the information connecting the atomistic molecular structure and proteins' biologically relevant motion, where large-scale fluctuations are deemed to guide folding and function. In the complex multiscale processes described by MD trajectories, it is difficult to identify, separate, and study those large-scale fluctuations. This problem can be formulated as the need to identify a small number of collective variables that guide the slow kinetic processes. The most promising method among the ones used to study the slow leading processes in proteins' dynamics is the time-structure based on time-lagged independent component analysis (tICA), which identifies the dominant components in a noisy signal. Recently, we developed an anisotropic Langevin approach for the dynamics of proteins, called the anisotropic Langevin Equation for Protein Dynamics or LE4PD-XYZ. This approach partitions the protein's MD dynamics into mostly uncorrelated, wavelength-dependent, diffusive modes. It associates with each mode a free-energy map, where one measures the spatial extension and the time evolution of the mode-dependent, slow dynamical fluctuations. Here, we compare the tICA modes' predictions with the collective LE4PD-XYZ modes. We observe that the two methods consistently identify the nature and extension of the slowest fluctuation processes. The tICA separates the leading processes in a smaller number of slow modes than the LE4PD does. The LE4PD provides time-dependent information at short times and a formal connection to the physics of the kinetic processes that are missing in the pure statistical analysis of tICA.

Conformational Fluctuations in GTP-Bound K-Ras: A Metadynamics Perspective with Harmonic Linear Discriminant Analysis

Biomacromolecules often undergo significant conformational rearrangements during function. In proteins, these motions typically consist in nontrivial, concerted rearrangement of multiple flexible regions. Mechanistic, thermodynamics, and kinetic predictions can be obtained via molecular dynamics simulations, provided that the simulation time is at least comparable to the relevant time scale of the process of interest. Because of the substantial computational cost, however, plain MD simulations often have difficulty in obtaining sufficient statistics for converged estimates, requiring the use of more-advanced techniques. Central in many enhanced sampling methods is the definition of a small set of relevant degrees of freedom (collective variables) that are able to describe the transitions between different metastable states of the system. The harmonic linear discriminant analysis (HLDA) has been shown to be useful for constructing low-dimensional collective variables in various complex systems. Here, we apply HLDA to study the free-energy landscape of a monomeric protein around its native state. More precisely, we study the K-Ras protein bound to GTP, focusing on two flexible loops and on the region associated with oncogenic mutations. We perform microsecond-long biased simulations on the wild type and on G12C, G12D, G12 V mutants, describe the resulting free-energy landscapes, and compare our predictions with previous experimental and computational studies. The fast interconversion between open and closed macroscopic states and their similar thermodynamic stabilities are observed. The mutation-induced effects include the alternations of the relative stabilities of different conformational states and the introduction of many microscopic metastable states. Together, our results demonstrate the applicability of the HLDA-based protocol for the conformational sampling of multiple flexible regions in folded proteins.

Time-dependent principal component analysis: A unified approach to high-dimensional data reduction using adiabatic dynamics

Systematic reduction of the dimensionality is highly demanded in making a comprehensive interpretation of experimental and simulation data. Principal component analysis (PCA) is a widely used technique for reducing the dimensionality of molecular dynamics (MD) trajectories, which assists our understanding of MD simulation data. Here, we propose an approach that incorporates time dependence in the PCA algorithm. In the standard PCA, the eigenvectors obtained by diagonalizing the covariance matrix are time independent. In contrast, they are functions of time in our new approach, and their time evolution is implemented in the framework of Car-Parrinello or Born-Oppenheimer type adiabatic dynamics. Thanks to the time dependence, each of the step-by-step structural changes or intermittent collective fluctuations is clearly identified, which are often keys to provoking a drastic structural transformation but are easily masked in the standard PCA. The time dependence also allows for reoptimization of the principal components (PCs) according to the structural development, which can be exploited for enhanced sampling in MD simulations. The present approach is applied to phase transitions of a water model and conformational changes of a coarse-grained protein model. In the former, collective dynamics associated with the dihedral-motion in the tetrahedral network structure is found to play a key role in crystallization. In the latter, various conformations of the protein model were successfully sampled by enhancing structural fluctuation along the periodically optimized PC. Both applications clearly demonstrate the virtue of the new approach, which we refer to as time-dependent PCA.

Dimeric allostery mechanism of the plant circadian clock photoreceptor ZEITLUPE

In Arabidopsis thaliana, the Light-Oxygen-Voltage (LOV) domain containing protein ZEITLUPE (ZTL) integrates light quality, intensity, and duration into regulation of the circadian clock. Recent structural and biochemical studies of ZTL indicate that the protein diverges from other members of the LOV superfamily in its allosteric mechanism, and that the divergent allosteric mechanism hinges upon conservation of two signaling residues G46 and V48 that alter dynamic motions of a Gln residue implicated in signal transduction in all LOV proteins. Here, we delineate the allosteric mechanism of ZTL via an integrated computational approach that employs atomistic simulations of wild type and allosteric variants of ZTL in the functional dark and light states, together with Markov state and supervised machine learning classification models. This approach has unveiled key factors of the ZTL allosteric mechanisms, and identified specific interactions and residues implicated in functional allosteric changes. The final results reveal atomic level insights into allosteric mechanisms of ZTL function that operate via a non-trivial combination of population-shift and dynamics-driven allosteric pathways.

Molecular Dynamics Simulations for the Determination of the Characteristic Structural Differences between Inactive and Active States of Wild Type and Mutants of the Orexin2 Receptor

The orexin2 receptor (OX2R), which is classified as a class A G protein-coupled receptor (GPCR), is the target of our study. We performed over 20 several-microsecond-scale molecular dynamics simulations of the wild type and mutants of OX2R to extract the characteristics of the structural changes taking place in the active state. We introduced mutations that exhibited the stable inactive state and the constitutively active state in class A GPCRs. In these simulations, significant characteristic structural changes were observed in the V309^6.40Y mutant, which corresponded to a constitutively active mutant. These conformational changes include the outward movement of the transmembrane helix 6 (TM6) and the inward movement of TM7, which are common structural changes in the activation of GPCRs. In addition, we extracted a suitable index for the quantitative evaluation of the active and inactive states of GPCRs, namely, the inter-atomic distance of Cα atoms between x(3.46) and Y(7.53). The structures of the inactive and active states solved by X-ray crystallography and cryo-electron microscopy can be classified using the inter-atomic distance. Furthermore, we clarified that the inward movement of TM7 requires the swapping of M305^6.36 on TM6 and L367^7.56 on TM7. Finally, we discussed the structural advantages of TM7 inward movement for GPCR activation.

Universal Relation between Instantaneous Diffusivity and Radius of Gyration of Proteins in Aqueous Solution

Protein conformational fluctuations are highly complex and exhibit long-term correlations. Here, molecular dynamics simulations of small proteins demonstrate that these conformational fluctuations directly affect the protein's instantaneous diffusivity D_{I}. We find that the radius of gyration R_{g} of the proteins exhibits 1/f fluctuations that are synchronous with the fluctuations of D_{I}. Our analysis demonstrates the validity of the local Stokes-Einstein-type relation D_{I}∝1/(R_{g}+R_{0}), where R_{0}∼0.3  nm is assumed to be a hydration layer around the protein. From the analysis of different protein types with both strong and weak conformational fluctuations, the validity of the Stokes-Einstein-type relation appears to be a general property.

Analysis of molecular dynamics simulations of 10-residue peptide, chignolin, using statistical mechanics: Relaxation mode analysis and three-dimensional reference interaction site model theory

Molecular dynamics simulation is a fruitful tool for investigating the structural stability, dynamics, and functions of biopolymers at an atomic level. In recent years, simulations can be performed on time scales of the order of milliseconds using special purpose systems. Since the most stable structure, as well as meta-stable structures and intermediate structures, is included in trajectories in long simulations, it is necessary to develop analysis methods for extracting them from trajectories of simulations. For these structures, methods for evaluating the stabilities, including the solvent effect, are also needed. We have developed relaxation mode analysis to investigate dynamics and kinetics of simulations based on statistical mechanics. We have also applied the three-dimensional reference interaction site model theory to investigate stabilities with solvent effects. In this paper, we review the results for designing amino-acid substitution of the 10-residue peptide, chignolin, to stabilize the misfolded structure using these developed analysis methods.

Normal mode analysis and beyond

Normal mode analysis provides a powerful tool in biophysical computations. Particularly, we shed light on its application to protein properties because they directly lead to biological functions. As a result of normal mode analysis, the protein motion is represented as a linear combination of mutually independent normal mode vectors. It has been widely accepted that the large amplitude motions throughout the entire protein molecule can be well described with a few low-frequency normal modes. Furthermore, it is possible to represent the effect of external perturbations, e.g., ligand binding, hydrostatic pressure, as the shifts of normal mode variables. Making use of this advantage, we are able to explore mechanical properties of proteins such as Young's modulus and compressibility. Within thermally fluctuating protein molecules under physiological conditions, tightly packed amino acid residues interact with each other through heat and energy exchanges. Since the structure and dynamics of protein molecules are highly anisotropic, the flow of energy and heat should also be anisotropic. Based on the harmonic approximation of the heat current operator, it is possible to analyze the communication map of a protein molecule. By using this method, the energy transfer pathways of photoactive yellow protein were calculated. It turned out that these pathways are similar to those obtained via the Green-Kubo formalism with equilibrium molecular dynamics simulations, indicating that normal mode analysis captures the intrinsic nature of the transport properties of proteins.

Folding pathways of NuG2-a designed mutant of protein G-using relaxation mode analysis

Dynamic analysis methods are important for analyzing long simulations such as folding simulations. Relaxation mode analysis, which approximately extracts slow modes and rates, has been applied in molecular dynamics (MD) simulations of protein systems. Previously, we showed that slow modes are suitable for analyzing simulations in which large conformational changes occur. Here, we applied relaxation mode analysis to folding simulations of a designed mutant of protein G, NuG2, to investigate its folding pathways. The folding simulations of NuG2 were previously performed for this mutant with Anton. In the present study, the free energy surfaces were calculated by projecting the coordinates on the axis of the slow relaxation modes obtained from relaxation mode analysis. We classified various characteristic states such as native, nativelike, intermediate, and random states and clarified two main folding pathways. In the early folding process, the first and second β strands formed an N-terminal β-sheet. After the early folding process, the fourth β strand formed along the first β strand in the same or opposite direction as the native structure; two characteristic intermediate states were identified. Finally, the intermediate structures folded to the native structure in the folding process. Relaxation mode analysis can be applied to folding simulations of complex proteins to investigate their folding processes.

Improving collective variables: The case of crystallization

Several enhanced sampling methods, such as umbrella sampling or metadynamics, rely on the identification of an appropriate set of collective variables. Recently two methods have been proposed to alleviate the task of determining efficient collective variables. One is based on linear discriminant analysis; the other is based on a variational approach to conformational dynamics and uses time-lagged independent component analysis. In this paper, we compare the performance of these two approaches in the study of the homogeneous crystallization of two simple metals. We focus on Na and Al and search for the most efficient collective variables that can be expressed as a linear combination of X-ray diffraction peak intensities. We find that the performances of the two methods are very similar. Wherever the different metastable states are well-separated, the method based on linear discriminant analysis, based on its harmonic version, is to be preferred because simpler to implement and less computationally demanding. The variational approach, however, has the potential to discover the existence of different metastable states.

t-Distributed Stochastic Neighbor Embedding Method with the Least Information Loss for Macromolecular Simulations

Dimensionality reduction methods are usually applied on molecular dynamics simulations of macromolecules for analysis and visualization purposes. It is normally desired that suitable dimensionality reduction methods could clearly distinguish functionally important states with different conformations for the systems of interest. However, common dimensionality reduction methods for macromolecules simulations, including predefined order parameters and collective variables (CVs), principal component analysis (PCA), and time-structure based independent component analysis (t-ICA), only have limited success due to significant key structural information loss. Here, we introduced the t-distributed stochastic neighbor embedding (t-SNE) method as a dimensionality reduction method with minimum structural information loss widely used in bioinformatics for analyses of macromolecules, especially biomacromolecules simulations. It is demonstrated that both one-dimensional (1D) and two-dimensional (2D) models of the t-SNE method are superior to distinguish important functional states of a model allosteric protein system for free energy and mechanistic analysis. Projections of the model protein simulations onto 1D and 2D t-SNE surfaces provide both clear visual cues and quantitative information, which is not readily available using other methods, regarding the transition mechanism between two important functional states of this protein.

Conformational change of a biomolecule studied by the weighted ensemble method: Use of the diffusion map method to extract reaction coordinates

We simulate the nonequilibrium ensemble dynamics of a biomolecule using the weighted ensemble method, which was introduced in molecular dynamics simulations by Huber and Kim and further developed by Zuckerman and co-workers. As the order parameters to characterize its conformational change, we here use the coordinates derived from the diffusion map (DM) method, one of the manifold learning techniques. As a concrete example, we study the kinetic properties of a small peptide, chignolin in explicit water, and calculate the conformational change between the folded and misfolded states in a nonequilibrium way. We find that the transition time scales thus obtained are comparable to those using previously employed hydrogen-bond distances as the order parameters. Since the DM method only uses the 3D Cartesian coordinates of a peptide, this shows that the DM method can extract the important distance information of the peptide without relying on chemical intuition. The time scales are compared well with the previous results using different techniques, non-Markovian analysis and core-set milestoning for a single long trajectory. We also find that the most significant DM coordinate turns out to extract a dihedral angle of glycine, and the previously studied relaxation modes are well correlated with the most significant DM coordinates.

Multipoint segmental repulsive potential for entangled polymer simulations with dissipative particle dynamics

A model is developed for simulating entangled polymers by dissipative particle dynamics (DPD) using the segmental repulsive potential (SRP). In contrast to previous SRP models that define a single-point interaction on each bond, the proposed SRP model applies a dynamically adjustable multipoint on the bond. Previous SRP models could not reproduce the equilibrium properties of Groot and Warren's original DPD model [R. D. Groot and P. B. Warren, J. Chem. Phys. 107, 4423 (1997)] because the introduction of a single SRP induces a large excluded volume, whereas, the proposed multipoint SRP (MP-SRP) introduces a cylindrical effective excluded bond volume. We demonstrate that our MP-SRP model exhibits equilibrium properties similar to those of the original DPD polymers. The MP-SRP model parameters are determined by monitoring the number of topology violations, thermodynamic properties, and the polymer internal structure. We examine two typical DPD polymers with different bond-length distributions; one of them was used in the modified SRP model by Sirk et al. [J. Chem. Phys. 136, 134903 (2012)], whereas the other was used in the original DPD model. We demonstrate that for both polymers, the proposed MP-SRP model captures the entangled behaviors of a polymer melt naturally, by calculating the slowest relaxation time of a chain in the melt and the shear relaxation modulus. The results indicate that the proposed MP-SRP model can be applied to a variety of DPD polymers.

Analysis of Structural Stability of Chignolin

We discuss the stability of an entire protein and the influence of main chains and side chains of individual amino acids to investigate the protein-folding mechanism. For this purpose, we calculated the solvation free-energy contribution of individual atoms using the three-dimensional reference interaction site model with the atomic decomposition method. We generated structures of chignolin miniprotein by a molecular dynamics simulation and classified them into six types: native 1, native 2, misfolded 1, misfolded 2, intermediate, and unfolded states. The total energies of the native (-171.1 kcal/mol) and misfolded (-171.2 kcal/mol) states were almost the same and lower than those of the intermediate (-158.5 kcal/mol) and unfolded (-148.1 kcal/mol) states; however, their components were different. In the native state, the side-chain interaction between Thr6 and Thr8 is important for the formation of π-turn. On the other hand, the hydrogen bonds between the atoms of the main chains in the misfolded state become stronger than those in the intermediate state.

Relaxation mode analysis for molecular dynamics simulations of proteins

Molecular dynamics simulation is a powerful method for investigating the structural stability, dynamics, and function of biopolymers at the atomic level. In recent years, it has become possible to perform simulations on time scales of the order of milliseconds using special hardware. However, it is necessary to derive the important factors contributing to structural change or function from the complicated movements of biopolymers obtained from long simulations. Although some analysis methods for protein systems have been developed using increasing simulation times, many of these methods are static in nature (i.e., no information on time). In recent years, dynamic analysis methods have been developed, such as the Markov state model and relaxation mode analysis (RMA), which was introduced based on spin and homopolymer systems. The RMA method approximately extracts slow relaxation modes and rates from trajectories and decomposes the structural fluctuations into slow relaxation modes, which characterize the slow relaxation dynamics of the system. Recently, this method has been applied to biomolecular systems. In this article, we review RMA and its improved versions for protein systems.

Two-step relaxation mode analysis with multiple evolution times applied to all-atom molecular dynamics protein simulation

Proteins implement their functionalities when folded into specific three-dimensional structures, and their functions are related to the protein structures and dynamics. Previously, we applied a relaxation mode analysis (RMA) method to protein systems; this method approximately estimates the slow relaxation modes and times via simulation and enables investigation of the dynamic properties underlying the protein structural fluctuations. Recently, two-step RMA with multiple evolution times has been proposed and applied to a slightly complex homopolymer system, i.e., a single [n]polycatenane. This method can be applied to more complex heteropolymer systems, i.e., protein systems, to estimate the relaxation modes and times more accurately. In two-step RMA, we first perform RMA and obtain rough estimates of the relaxation modes and times. Then, we apply RMA with multiple evolution times to a small number of the slowest relaxation modes obtained in the previous calculation. Herein, we apply this method to the results of principal component analysis (PCA). First, PCA is applied to a 2-μs molecular dynamics simulation of hen egg-white lysozyme in aqueous solution. Then, the two-step RMA method with multiple evolution times is applied to the obtained principal components. The slow relaxation modes and corresponding relaxation times for the principal components are much improved by the second RMA.

Spotting the difference in molecular dynamics simulations of biomolecules

Comparing two trajectories from molecular simulations conducted under different conditions is not a trivial task. In this study, we apply a method called Linear Discriminant Analysis with ITERative procedure (LDA-ITER) to compare two molecular simulation results by finding the appropriate projection vectors. Because LDA-ITER attempts to determine a projection such that the projections of the two trajectories do not overlap, the comparison does not suffer from a strong anisotropy, which is an issue in protein dynamics. LDA-ITER is applied to two test cases: the T4 lysozyme protein simulation with or without a point mutation and the allosteric protein PDZ2 domain of hPTP1E with or without a ligand. The projection determined by the method agrees with the experimental data and previous simulations. The proposed procedure, which complements existing methods, is a versatile analytical method that is specialized to find the "difference" between two trajectories.

Efficient Conformational Search Based on Structural Dissimilarity Sampling: Applications for Reproducing Structural Transitions of Proteins

Structural Dissimilarity Sampling (SDS) is proposed as an efficient conformational search method to promote structural transitions essential for the biological functions of proteins. In SDS, initial structures are selected based on structural dissimilarity, and conformational resampling is repeated. Conformational resampling is performed as follows: (I) arrangement of initial structures for a diverse distribution at the edge of a conformational subspace and (II) promotion of the structural transitions with multiple short-time molecular dynamics (MD) simulations restarting from the diversely distributed initial structures. Cycles of (I) and (II) are repeated to intensively promote structural transitions because conformational resampling from the initial structures would quickly expand conformational distributions toward unvisited conformational subspaces. As a demonstration, SDS was first applied to maltodextrin binding protein (MBP) in explicit water to reproduce structural transitions between the open and closed states of MBP. Structural transitions of MBP were successfully reproduced with SDS in nanosecond-order simulation times. Starting from both the open and closed forms, SDS successfully reproduced the structural transitions within 25 cycles (a total of 250 ns of simulation time). For reference, a conventional long-time (500 ns) MD simulation under NPT (300 K and 1 bar) starting from the open form failed to reproduce the structural transition. In addition to the open-closed motions of MBP, SDS was applied to folding processes of the fast-folding proteins (chignolin, Trp-cage, and villin) and successfully sampled their native states. To confirm how the selections of initial structures affected conformational sampling efficiency, numbers of base sets for characterizing structural dissimilarity of initial structures were addressed in distinct trials of SDS. The parameter searches showed that the conformational sampling efficiency was relatively insensitive with respect to the numbers of base sets, indicating the robustness of SDS for actual applications.

Relaxation mode analysis and Markov state relaxation mode analysis for chignolin in aqueous solution near a transition temperature

It is important to extract reaction coordinates or order parameters from protein simulations in order to investigate the local minimum-energy states and the transitions between them. The most popular method to obtain such data is principal component analysis, which extracts modes of large conformational fluctuations around an average structure. We recently applied relaxation mode analysis for protein systems, which approximately estimates the slow relaxation modes and times from a simulation and enables investigations of the dynamic properties underlying the structural fluctuations of proteins. In this study, we apply this relaxation mode analysis to extract reaction coordinates for a system in which there are large conformational changes such as those commonly observed in protein folding/unfolding. We performed a 750-ns simulation of chignolin protein near its folding transition temperature and observed many transitions between the most stable, misfolded, intermediate, and unfolded states. We then applied principal component analysis and relaxation mode analysis to the system. In the relaxation mode analysis, we could automatically extract good reaction coordinates. The free-energy surfaces provide a clearer understanding of the transitions not only between local minimum-energy states but also between the folded and unfolded states, even though the simulation involved large conformational changes. Moreover, we propose a new analysis method called Markov state relaxation mode analysis. We applied the new method to states with slow relaxation, which are defined by the free-energy surface obtained in the relaxation mode analysis. Finally, the relaxation times of the states obtained with a simple Markov state model and the proposed Markov state relaxation mode analysis are compared and discussed.

Dimensionality of Collective Variables for Describing Conformational Changes of a Multi-Domain Protein

Collective variables (CVs) are often used in molecular dynamics simulations based on enhanced sampling algorithms to investigate large conformational changes of a protein. The choice of CVs in these simulations is essential because it affects simulation results and impacts the free-energy profile, the minimum free-energy pathway (MFEP), and the transition-state structure. Here we examine how many CVs are required to capture the correct transition-state structure during the open-to-close motion of adenylate kinase using a coarse-grained model in the mean forces string method to search the MFEP. Various numbers of large amplitude principal components are tested as CVs in the simulations. The incorporation of local coordinates into CVs, which is possible in higher dimensional CV spaces, is important for capturing a reliable MFEP. The Bayesian measure proposed by Best and Hummer is sensitive to the choice of CVs, showing sharp peaks when the transition-state structure is captured. We thus evaluate the required number of CVs needed in enhanced sampling simulations for describing protein conformational changes.

Diffusion Monte Carlo study on temporal evolution of entropy and free energy in nonequilibrium processes

A computational scheme to describe the temporal evolution of thermodynamic functions in stochastic nonequilibrium processes of isothermal classical systems is proposed on the basis of overdamped Langevin equation under given potential and temperature. In this scheme the associated Fokker-Planck-Smoluchowski equation for the probability density function is transformed into the imaginary-time Schrödinger equation with an effective Hamiltonian. The propagator for the time-dependent wave function is expressed in the framework of the path integral formalism, which can thus represent the dynamical behaviors of nonequilibrium molecular systems such as those conformational changes observed in protein folding and ligand docking. The present study then employs the diffusion Monte Carlo method to efficiently simulate the relaxation dynamics of wave function in terms of random walker distribution, which in the long-time limit reduces to the ground-state eigenfunction corresponding to the equilibrium Boltzmann distribution. Utilizing this classical-quantum correspondence, we can describe the relaxation processes of thermodynamic functions as an approach to the equilibrium state with the lowest free energy. Performing illustrative calculations for some prototypical model potentials, the temporal evolutions of enthalpy, entropy, and free energy of the classical systems are explicitly demonstrated. When the walkers initially start from a localized configuration in one- or two-dimensional harmonic or double well potential, the increase of entropy usually dominates the relaxation dynamics toward the equilibrium state. However, when they start from a broadened initial distribution or go into a steep valley of potential, the dynamics are driven by the decrease of enthalpy, thus causing the decrease of entropy associated with the spatial localization. In the cases of one- and two-dimensional asymmetric double well potentials with two minimal points and an energy barrier between them, we observe a nonequilibrium behavior that the system entropy first increases with the broadening of the initially localized walker distribution and then it begins to decrease along with the trapping at the global minimum of the potential, thus leading to the minimization of the free energy.