Detecting coupled collective motions in protein by independent subspace analysis

Temporal Resolution of Activity-Related Solvation Dynamics in the TIM Barrel Enzyme Murine Adenosine Deaminase

Murine adenosine deaminase (mADA) is a prototypic system for studying the thermal activation of active site chemistry within the TIM barrel family of enzyme reactions. Previous temperature-dependent hydrogen deuterium exchange studies under various conditions have identified interconnected thermal networks for heat transfer from opposing protein-solvent interfaces to active site residues in mADA. One of these interfaces contains a solvent exposed helix-loop-helix moiety that presents the hydrophobic face of its long α-helix to the backside of bound substrate. Herein we pursue the time and temperature dependence of solvation dynamics at the surface of mADA, for comparison to established kinetic parameters that represent active site chemistry. We first created a modified protein devoid of native tryptophans with close to native kinetic behavior. Single site-specific tryptophan mutants were back inserted into each of the four positions where native tryptophans reside. Measurements of nanosecond fluorescence relaxation lifetimes and Stokes shift decays, that reflect time dependent environmental reoroganization around the photo-excited state of Trp*, display minimal temperature dependences. These regions serve as controls for the behavior of a new single tryptophan inserted into a solvent exposed region near the helix-loop-helix moiety located behind the bound substrate, Lys54Trp. This installed Trp displays a significantly elevated value for Ea ( k Stokes shift ) ; further, when Phe61 within the long helix positioned behind bound substrate is replaced by a series of aliphatic hydrophobic side chains, the trends in Ea ( k Stokes shift ) mirror the earlier reported impact of the same series of function-altering hydrophobic side chains on the activation energy of catalysis, Ea ( k cat ) .The reported experimental findings implicate a solvent initiated and rapid (>ns) protein restructuring that contributes to the enthalpic activation barrier to catalysis in mADA.

Realization of the structural fluctuation of biomolecules in solution: Generalized Langevin mode analysis

A new theoretical method, referred to as Generalized Langevin Mode Analysis (GLMA), is proposed to analyze the mode of structural fluctuations of a biomolecule in solution. The method combines the two theories in the statistical mechanics, or the Generalized Langevin theory and the RISM/3D-RISM theory, to calculate the second derivative, or the Hessian matrix, of the free energy surface of a biomolecule in aqueous solution, which consists of the intramolecular interaction among atoms in the biomolecule and the solvation free energy. The method is applied to calculate the wave-number spectrum of an alanine dipeptide in water for which the optical heterodyne-detected Raman-induced spectroscopy (RIKES) spectrum is available to compare with. The theoretical analysis reproduced the main features of the experimental spectrum with respect to the peak positions of the four bands around ~90 cm^-1 , ~240 cm^-1 , ~370 cm^-1 , and 400 cm^-1 , observed in the experimental spectrum, in spite that the physics involved in the two spectrum was not exactly the same: the experimental spectrum includes the contributions from the dipeptide and the water molecules interacting with the solute, while the theoretical one is just concerned with the solute molecule, influenced by solvation. Two major discrepancies between the theoretical and experimental spectra, one in the band intensity around ~100 cm^-1 , and the other in the peak positions around ~370 cm^-1 , are discussed in terms of the fluctuation mode of water molecules interacting with the dipeptide, which is not taken explicitly into account in the theoretical analysis.

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.

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.

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.

Simple, yet powerful methodologies for conformational sampling of proteins

Several biological functions, such as molecular recognition, enzyme catalysis, signal transduction, allosteric regulation, and protein folding, are strongly related to conformational transitions of proteins. These conformational transitions are generally induced as slow dynamics upon collective motions, including biologically relevant large-amplitude fluctuations of proteins. Although molecular dynamics (MD) simulation has become a powerful tool for extracting conformational transitions of proteins, it might still be difficult to reach time scales of the biological functions because the accessible time scales of MD simulations are far from biological time scales, even if straightforward conventional MD (CMD) simulations using massively parallel computers are employed. Thus, it is desirable to develop efficient methods to achieve canonical ensembles with low computational costs. From this perspective, we review several enhanced conformational sampling techniques of biomolecules developed by us. In our methods, multiple independent short-time MD simulations are employed instead of single straightforward long-time CMD simulations. Our basic strategy is as follows: (i) selection of initial seeds (initial structures) for the conformational sampling in restarting MD simulations. Here, the seeds should be selected as candidates with high potential to transit. (ii) Resampling from the selected seeds by initializing velocities in restarting short-time MD simulations. A cycle of these simple protocols might drastically promote the conformational transitions of biomolecules. (iii) Once reactive trajectories extracted from the cycles of short-time MD simulations are obtained, a free energy profile is evaluated by means of umbrella sampling (US) techniques with the weighted histogram analysis method (WHAM) as a post-processing technique. For the selection of the initial seeds, we proposed four different choices: (1) Parallel CaScade molecular dynamics (PaCS-MD), (2) Fluctuation Flooding Method (FFM), (3) Outlier FLOODing (OFLOOD) method, and (4) TaBoo SeArch (TBSA) method. We demonstrate applications of our methods to several biological systems, such as domain motions of proteins with large-amplitude fluctuations, conformational transitions upon ligand binding, and protein folding/refolding to native structures of proteins. Finally, we show the conformational sampling efficiencies of our methods compared with those by CMD simulations and other previously developed enhanced conformational sampling methods.

Nontargeted Parallel Cascade Selection Molecular Dynamics for Enhancing the Conformational Sampling of Proteins

Nontargeted parallel cascade selection molecular dynamics (nt-PaCS-MD) is proposed as an efficient conformational sampling method to enhance the conformational transitions of proteins, which is an extension of the original targeted PaCS-MD (t-PaCS-MD). The original PaCS-MD comprises cycles of (i) selection of initial structures for multiple independent MD simulations toward a predetermined target and (ii) conformational sampling by the independent MDs. In nt-PaCS-MD, structures that significantly deviate from an average are regarded as candidates that have high potential to address other metastable states and are chosen as the initial structures in the selection. To select significantly deviated structures, we examine the root-mean-square deviation (RMSD) of snapshots generated from the average structure based on Gram-Schmidt orthogonalization. nt-PaCS-MD was applied to the folding of the mini-protein chignolin in implicit solvent and to the open-closed conformational transitions of T4 lysozyme (T4L) and glutamine binding protein (QBP) in explicit solvent. We show that nt-PaCS-MD can reach chignolin's native state and can also cause the open-closed transition of T4L and QBP on a nanosecond time scale, which are very efficient in terms of conformational sampling and comparable to that with t-PaCS-MD.

Parallel Cascade Selection Molecular Dynamics (PaCS-MD) to generate conformational transition pathway

Parallel Cascade Selection Molecular Dynamics (PaCS-MD) is proposed as a molecular simulation method to generate conformational transition pathway under the condition that a set of "reactant" and "product" structures is known a priori. In PaCS-MD, the cycle of short multiple independent molecular dynamics simulations and selection of the structures close to the product structure for the next cycle are repeated until the simulated structures move sufficiently close to the product. Folding of 10-residue mini-protein chignolin from the extended to native structures and open-close conformational transition of T4 lysozyme were investigated by PaCS-MD. In both cases, tens of cycles of 100-ps MD were sufficient to reach the product structures, indicating the efficient generation of conformational transition pathway in PaCS-MD with a series of conventional MD without additional external biases. Using the snapshots along the pathway as the initial coordinates, free energy landscapes were calculated by the combination with multiple independent umbrella samplings to statistically elucidate the conformational transition pathways.

Frequency response of a protein to local conformational perturbations

Signals created by local perturbations are known to propagate long distances through proteins via backbone connectivity and nonbonded interactions. In the current study, signal propagation from the flexible ligand binding loop to the rest of Protein Tyrosine Phosphatase 1B (PTP1B) was investigated using frequency response techniques. Using restrained Targeted Molecular Dynamics (TMD) potential on WPD and R loops, PTP1B was driven between its crystal structure conformations at different frequencies. Propagation of the local perturbation signal was manifested via peaks at the fundamental frequency and upper harmonics of 1/f distributed spectral density of atomic variables, such as Cα atoms, dihedral angles, or polar interaction distances. Frequency of perturbation was adjusted high enough (simulation length >∼10×period of a perturbation cycle) not to be clouded by random diffusional fluctuations, and low enough (<∼0.8 ns(-1)) not to attenuate the propagating signal and enhance the contribution of the side-chains to the dissipation of the signals. Employing Discrete Fourier Transform (DFT) to TMD simulation trajectories of 16 cycles of conformational transitions at periods of 1.2 to 5 ns yielded Cα displacements consistent with those obtained from crystal structures. Identification of the perturbed atomic variables by statistical t-tests on log-log scale spectral densities revealed the extent of signal propagation in PTP1B, while phase angles of the filtered trajectories at the fundamental frequency were used to cluster collectively fluctuating elements. Hydrophobic interactions were found to have a higher contribution to signal transduction between side-chains compared to the role of polar interactions. Most of in-phase fluctuating residues on the signaling pathway were found to have high identity among PTP domains, and located over a wide region of PTP1B including the allosteric site. Due to its simplicity and efficiency, the suggested technique may find wide applications in identification of signaling pathways of different proteins.

Molecular mechanisms in the activation of abscisic acid receptor PYR1

The pyrabactin resistance 1 (PYR1)/PYR1-like (PYL)/regulatory component of abscisic acid (ABA) response (RCAR) proteins comprise a well characterized family of ABA receptors. Recent investigations have revealed two subsets of these receptors that, in the absence of ABA, either form inactive homodimers (PYR1 and PYLs 1–3) or mediate basal inhibition of downstream target type 2C protein phosphatases (PP2Cs; PYLs 4–10) respectively in vitro. Addition of ABA has been shown to release the apo-homodimers yielding ABA-bound monomeric holo-receptors that can interact with PP2Cs; highlighting a competitive-interaction process. Interaction selectivity has been shown to be mediated by subtle structural variations of primary sequence and ligand binding effects. Now, the dynamical contributions of ligand binding on interaction selectivity are investigated through extensive molecular dynamics (MD) simulations of apo and holo-PYR1 in monomeric and dimeric form as well as in complex with a PP2C, homology to ABA insensitive 1 (HAB1). Robust comparative interpretations were enabled by a novel essential collective dynamics approach. In agreement with recent experimental findings, our analysis indicates that ABA-bound PYR1 should efficiently bind to HAB1. However, both ABA-bound and ABA-extracted PYR1-HAB1 constructs have demonstrated notable similarities in their dynamics, suggesting that apo-PYR1 should also be able to make a substantial interaction with PP2Cs, albeit likely with slower complex formation kinetics. Further analysis indicates that both ABA-bound and ABA-free PYR1 in complex with HAB1 exhibit a higher intra-molecular structural stability and stronger inter-molecular dynamic correlations, in comparison with either holo- or apo-PYR1 dimers, supporting a model that includes apo-PYR1 in complex with HAB1. This possibility of a conditional functional apo-PYR1-PP2C complex was validated in vitro. These findings are generally consistent with the competitive-interaction model for PYR1 but highlight dynamical contributions of the PYR1 structure in mediating interaction selectivity suggesting added degrees of complexity in the regulation of the competitive-inhibition.

Protein pyrabactin resistance 1 (PYR1) belongs to a group of PYR1-like (PYL) proteins that regulate plant development and responses to conditions of drought and salinity. Recent studies have reported characterization of their molecular structures as well as elucidation of important aspects of their function; highlighting their roles as receptors for the stress responsive phytohormone, abscisic acid (ABA). However details of the molecular mechanisms regulating their receptor signalling remain enigmatic. In this work, we use molecular dynamics simulations complemented by a sophisticated statistical-mechanical analysis to investigate structural and dynamical properties of PYR1 protein and how its interaction with ABA modifies receptor-protein complex formation. Our results provide detailed insight into how the PYR1-mediated inactivation of its downstream phosphatase target is regulated by homodimer formation and yield new hypotheses, supported by in vitro experiments, for further investigation. Ultimately, this knowledge provides insight into how plants respond to stress, with potential applications in the development of crops with improved growth characteristics and higher stress tolerance.

Exploring the essential collective dynamics of interacting proteins: application to prion protein dimers

Essential collective dynamics is a promising and robust approach for analysing the slow motions of macromolecules from short molecular dynamics trajectories. In this study, an extension of the method to treat a collection of interacting protein molecules is presented. The extension is applied to investigate the effects of dimerization on the collective dynamics of ovine prion protein molecules in two different arrangements. Examination of the structural plasticity shows that aggregation has a restricting effect on the local mobility of the prion protein molecules in the interfacial regions. Domain motions of the two dimeric ovine prion protein conformations are distinctly different and can be related to interatomic correlations at the interface. Correlated motions are among the slow collective modes extensively analysed by considering both main-chain and side-chain atoms. Correlation maps reveal the existence of a vast network of dynamically correlated side groups, which extends beyond individual subunits via interfacial interconnections. The network is formed by a core of hydrophobic side chains surrounded by hydrophilic groups at the periphery. The relevance of these findings are discussed in the context of mutations associated with prion diseases. The binding free energy of the dimeric conformations is evaluated to probe their thermodynamic stability. The descriptions afforded by the analysis of the essential collective dynamics of the prion dimers are consistent with their binding free energies. The agreement validates the extension of the methodology and provides a means of interpreting the collective dynamics in terms of the thermodynamic stability of ovine prion proteins.

Comparing Conformational Ensembles Using the Kullback-Leibler Divergence Expansion

We present a thermodynamical approach to identify changes in macromolecular structure and dynamics in response to perturbations such as mutations or ligand binding, using an expansion of the Kullback-Leibler Divergence that connects local population shifts in torsion angles to changes in the free energy landscape of the protein. While the Kullback-Leibler Divergence is a known formula from information theory, the novelty and power of our implementation lies in its formal developments, connection to thermodynamics, statistical filtering, ease of visualization of results, and extendability by adding higher-order terms. We present a formal derivation of the Kullback-Leibler Divergence expansion and then apply our method at a first-order approximation to molecular dynamics simulations of four protein systems where ligand binding or pH titration is known to cause an effect at a distant site. Our results qualitatively agree with experimental measurements of local changes in structure or dynamics, such as NMR chemical shift perturbations and hydrogen-deuterium exchange mass spectrometry. The approach produces easy-to-analyze results with low background, and as such has the potential to become a routine analysis when molecular dynamics simulations in two or more conditions are available. Our method is implemented in the MutInf code package and is available on the SimTK website at https://simtk.org/home/mutinf.

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

This article reports the first attempt to apply the relaxation mode analysis method to a simulation of a biomolecular system. In biomolecular systems, the principal component analysis is a well-known method for analyzing the static properties of fluctuations of structures obtained by a simulation and classifying the structures into some groups. On the other hand, the relaxation mode analysis has been used to analyze the dynamic properties of homopolymer systems. In this article, a long Monte Carlo simulation of Met-enkephalin in gas phase has been performed. The results are analyzed by the principal component analysis and relaxation mode analysis methods. We compare the results of both methods and show the effectiveness of the relaxation mode analysis.