Coarse-grained biomolecular simulation with REACH: realistic extension algorithm via covariance Hessian

Coarse-graining of protein interactions provides a means of simulating large biological systems. Here, a coarse-graining method, REACH, is introduced, in which the force constants of a residue-scale elastic network model are calculated from the variance-covariance matrix obtained from atomistic molecular dynamics (MD) simulation. In test calculations, the C(alpha)-atoms variance-covariance matrices are calculated from the ensembles of 1-ns atomistic MD trajectories in monomeric and dimeric myoglobin, and used to derive coarse-grained force constants for the local and nonbonded interactions. Construction of analytical model functions of the distance-dependence of the interresidue force constants allows rapid calculation of the REACH normal modes. The model force constants from monomeric and dimeric myoglobin are found to be similar in magnitude to each other. The MD intra- and intermolecular mean-square fluctuations and the vibrational density of states are well reproduced by the residue-scale REACH normal modes without requiring rescaling of the force constant parameters. The temperature-dependence of the myoglobin REACH force constants reveals that the dynamical transition in protein internal fluctuations arises principally from softening of the elasticity in the nonlocal interactions. The REACH method is found to be a reliable way of determining spatiotemporal protein motion without the need for expensive computations of long atomistic MD simulations.     

Fluctuations and correlations in crystalline protein dynamics: a simulation analysis of staphylococcal nuclease

Understanding collective motions in protein crystals is likely to furnish insight into functional protein dynamics and will improve models for refinement against diffraction data. Here, four 10 ns molecular dynamics simulations of crystalline Staphylococcal nuclease are reported and analyzed in terms of fluctuations and correlations in atomic motion. The simulation-derived fluctuations strongly correlate with, but are slightly higher than, the values derived from the experimental B-factors. Approximately 70% of the atomic fluctuations are due to internal protein motion. For 65% of the protein atoms the internal fluctuations converge on the nanosecond timescale. Convergence is much slower for the elements of the interatomic displacement correlation matrix--of these, >80% converge within 1 ns for interatomic distances less, approximately <6 A, but only 10% for separations approximately =12 A. Those collective motions that converged on the nanosecond timescale involve mostly correlations within the beta-barrel or between alpha-helices of the protein. The R-factor with the experimental x-ray diffuse scattering for the crystal, which is determined by the displacement variance-covariance matrix, decreases to 8% after 10 ns simulation. Both the number of converged correlation matrix elements and the R-factor depend logarithmically on time, consistent with a model in which the number of energy minima sampled depends exponentially on the maximum energy barrier crossed. The logarithmic dependence is also extrapolated to predict a convergence time for the whole variance-covariance matrix of approximately 1 micros.     

Radial packing, order, and disorder in collagen fibrils.

Collagen fibrils resemble smectic, liquid crystals in being highly ordered axially but relatively disordered laterally. In some connective tissues, x-ray diffraction reveals three-dimensional crystallinity in the molecular packing within fibrils, although the continued presence of diffuse scatter indicates significant underlying disorder. In addition, several observations from electron microscopy suggest that the molecular packing is organized concentrically about the fibril core. In the present work, theoretical equatorial x-ray diffraction patterns for a number of models for collagen molecular packing are calculated and compared with the experimental data from tendon fibrils. None of the models suggested previously can account for both the crystalline Bragg peaks and the underlying diffuse scatter. In addition, models in which any of the nearest-neighbor, intermolecular vectors are perpendicular to the radial direction are inconsistent with the observed radial orientation of the principal approximately 4 nm Bragg spacing. Both multiple-start spiral and concentric ring models are devised in which one of the nearest-neighbor vectors is along the radial direction. These models are consistent with the radial orientation of the approximately 4 nm spacing, and energy minimization results in radially oriented crystalline domains separated by disordered grain boundaries. Theoretical x-ray diffraction patterns show a combination of sharp Bragg peaks and underlying diffuse scatter. Close agreement with the observed equatorial diffraction pattern is obtained. The concentric ring model is consistent with the observation that the diameters of collagen fibrils are restricted to discrete values.     

Iron normal mode dynamics in (nitrosyl)iron(II)tetraphenylporphyrin from X-ray nuclear resonance data

The complete iron atom vibrational spectrum has been obtained by refinement of normal mode calculations to nuclear inelastic x-ray absorption data from (nitrosyl)iron(II)tetraphenylporphyrin, FeTPP(NO), a useful model for heme dynamics in myoglobin and other heme proteins. Nuclear resonance vibrational spectroscopy (NRVS) provides a direct measurement of the frequency and iron amplitude for all normal modes involving significant displacement of (57)Fe. The NRVS measurements on isotopically enriched single crystals permit determination of heme in-plane and out-of-plane modes. Excellent agreement between the calculated and experimental values of frequency and iron amplitude for each mode is achieved by a force-field refinement. Significantly, we find that the presence of the phenyl groups and the NO ligand leads to substantial mixing of the porphyrin core modes. This first picture of the entire iron vibrational density of states for a porphyrin compound provides an improved model for the role of iron atom dynamics in the biological functioning of heme proteins.     

Models with energy penalty on interresidue rotation address insufficiencies of conventional elastic network models

In this study, I present a new elastic network model, to our knowledge, that addresses insufficiencies of two conventional models—the Gaussian network model (GNM) and the anisotropic network model (ANM). It has been shown previously that the GNM is not rotation-invariant due to its energy, which penalizes rigid-body rotation (external rotation). As a result, GNM models are found contaminated with rigid-body rotation, especially in the most collective ones. A new model (EPIRM) is proposed to remove such external component in modes. The extracted internal motions result from a potential that penalizes interresidue stretching and rotation in a protein. The new model is shown to pertinently describe crystallographic temperature factors (B-factors) and protein open↔closed transitions. Also, the capability of separating internal and external motions in GNM slow modes permits reexamining important mechanochemical properties in enzyme active sites. The results suggest that catalytic residues stay closer to rigid-body rotation axes than their immediate backbone neighbors. I show that the cumulative density of states for EPIRM and ANM follow different power laws as functions of low-mode frequencies. When using a cutoff distance of 7.5 Å, The cumulative density of states of EPIRM scales faster than that of all-atom normal mode analysis and slower than that of simple lattices.     

Application of Elastic Network Models to Proteins in the Crystalline State

Normal mode analysis using elastic network models has grown popular for probing the low-frequency collective dynamics of proteins and other biomolecular assemblies. In most previous studies, these models were validated by comparing calculated atomic fluctuations for isolated proteins with experimental temperature factors determined in the crystalline state, although there were also hints that including crystal contacts in the calculations has an impact on the comparison. In this study, a set of 83 ultra-high resolution crystal structures with experimentally determined anisotropic displacement parameters is used to evaluate several C_α-based elastic network models that either ignore or treat the crystal environment in different ways; the latter include using periodic boundary conditions defined with respect to the asymmetric unit or the primitive unit cell as well as using the Born-von Kármán boundary condition that accounts for lattice vibrations. For all elastic network models, treating the crystal environment leads to better agreement with experimental anisotropic displacement parameters with the Born-von Kármán boundary condition giving the best agreement. Atomic correlations over the entire protein are clearly affected by the presence of the crystal contacts and fairly sensitive to the way that the crystal environment is treated. These observations highlight the importance of properly treating the protein system in an environment consistent with experiment when either evaluating approximate protein models or using approximate dynamic models in structural refinement application types. Finally, investigation of the scaling behaviors of the cumulative density of states and the heat capacity indicates that there are still gaps between simplified elastic models and all-atom models for proteins.     

HOMOGENEITY OF THE GENETIC VARIANCE-COVARIANCE MATRIX FOR ANTIPREDATOR TRAITS IN TWO NATURAL POPULATIONS OF THE GARTER SNAKE THAMNOPHIS ORDINOIDES

Quantitative genetic models of evolution rely on the genetic variance-covariance matrix to predict the phenotypic response to selection. Both prospective and retrospective studies of phenotypic evolution across generations rely on assumptions about the constancy of patterns of genetic covariance through time. In the absence of robust theoretical predictions about the stability of genetic covariances, this assumption must be tested with empirical comparisons of genetic parameters among populations and species. Genetic variance-covariance matrices were estimated for a suite of antipredator traits in two populations of the northwestern garter snake, Thamnophis ordinoides. The characters studied include color pattern and antipredator behaviors that interact to facilitate escape from predators. Significant heritabilities for all traits were detected in both populations. Genetic correlations and covariances were found among behaviors in both populations and between color pattern and behavior in one of the populations. Phenotypic means differed among populations, but pairwise comparisons revealed no heterogeneity of genetic parameters between the populations. The structure of the genetic variance-covariance matrix has apparently not changed significantly during the divergence of these two populations.     

Simple Elastic Network Models for Exhaustive Analysis of Long Double-Stranded DNA Dynamics with Sequence Geometry Dependence

Simple elastic network models of DNA were developed to reveal the structure-dynamics relationships for several nucleotide sequences. First, we propose a simple all-atom elastic network model of DNA that can explain the profiles of temperature factors for several crystal structures of DNA. Second, we propose a coarse-grained elastic network model of DNA, where each nucleotide is described only by one node. This model could effectively reproduce the detailed dynamics obtained with the all-atom elastic network model according to the sequence-dependent geometry. Through normal-mode analysis for the coarse-grained elastic network model, we exhaustively analyzed the dynamic features of a large number of long DNA sequences, approximately ∼150 bp in length. These analyses revealed positive correlations between the nucleosome-forming abilities and the inter-strand fluctuation strength of double-stranded DNA for several DNA sequences.     

Protein dynamics from X-ray crystallography: anisotropic, global motion in diffuse scattering patterns

Understanding X-ray crystallographic diffuse scattering is likely to improve our comprehension of equilibrium collective protein dynamics. Here, using molecular dynamics (MD) simulation, a detailed analysis is performed of the origins of diffuse scattering in crystalline Staphylococcal nuclease, for which the complete diffuse scattering pattern has been determined experimentally. The hydrogen-atom contribution and the scattering range over which the scattering can be considered to be a sum of solvent and protein scattering are determined. Two models of correlated protein motion are investigated by calculating the model-derived diffuse scattering and comparing with the scattering calculated directly from MD trajectories. In one model, previously used in diffuse scattering interpretation, the atomic displacement correlations decay isotropically with increasing separation. Model correlation lengths are obtained by refining the model scattering against the simulation-derived scattering pattern, and are found to be significantly different from those correlation lengths derived directly from the MD trajectories. Furthermore, the convergence between the model-derived and MD-derived scattering is poor. The second model, in which the displacement correlations are calculated from the principal components of the MD trajectories, is capable of fully reproducing the MD-derived diffuse scattering if the approximately 50% lowest-frequency modes are included. However, a small number ( approximately 10) of lowest-frequency and largest-amplitude modes dominates the diffuse scattering and thus the correlated protein motions. A detailed analysis of the principal components is performed. In particular, the effective free energy profile associated with each principle mode is analyzed and the eigenfrequency and damping coefficient computed using a model of Brownian dynamics. Those collective modes with effective frequencies below approximately 0.5 THz, including those that determine the diffuse scattering, are overdamped.     

The low-temperature heat capacity of solid proteins.

Several harmonic models of protein fluctuations are used to calculate the heat capacity. They get the spectral density of conformational modes from inelastic neutron scattering, normal mode calculations, or macroscopic elasticity (Debye model). It is assumed that the low-frequency spectral density depends only weakly on temperature and protein species. The Debye model predicts temperatures below which modes are primarily in their ground states: 10 and 80 K for the lattice and conformational modes, respectively. The models differ most below 100 K. The mode calculations yield the most accurate predictions, though all three models are within twofold of the data. The heat capacity has the power law form aTb for T less than 30 K. The experimental b's of proteins are 1.6-1.8, and the theoretical, 1.1-1.3. One possible explanation for the discrepancy is the occurrence of transitions between discrete conformations. All of the models approach the measured data in the range 100-200 K. They are very similar above 200 K, where the heat capacity includes significant contributions from bond stretching and bending. This masks the possible anharmonic behavior of the conformational modes. Hydration substantially increases the heat capacity above 200 K. This effect seems to be a consequence of conformational transitions that have higher energy than the ones seen with low hydration. The analysis also predicts that denaturation with constant hydration produces a negligible increase of heat capacity. The larger increment in solution arises from the different hydration of the folded and unfolded states, and is responsible for the existence of cold denaturation. This phenomenon is thus predicted not to occur when the hydration is constant.     

A comparative study of structures and structural transitions of secondary transporters with the LeuT fold

Secondary active transporters from several protein families share a core of two five-helix inverted repeats that has become known as the LeuT fold. The known high-resolution protein structures with this fold were analyzed by structural superposition of the core transmembrane domains (TMDs). Three angle parameters derived from the mean TMD axes correlate with accessibility of the central binding site from the outside or inside. Structural transitions between distinct conformations were analyzed for four proteins in terms of changes in relative TMD arrangement and in internal conformation of TMDs. Collectively moving groups of TMDs were found to be correlated in the covariance matrix of elastic network models. The main features of the structural transitions can be reproduced with the 5 % slowest normal modes of anisotropic elastic network models. These results support the rocking bundle model for the major conformational change between the outward- and inward-facing states of the protein and point to an important role for the independently moving last TMDs of each repeat in occluding access to the central binding site. Occlusion is also supported by flexing of some individual TMDs in the collectively moving bundle and hash motifs.     

Predicting the order in which contacts are broken during single molecule protein stretching experiments

We combine two methods to enable the prediction of the order in which contacts are broken under external stretching forces in single molecule experiments. These two methods are Gō-like models and elastic network models. The Gō-like models have shown remarkable success in representing many aspects of protein behavior, including the reproduction of experimental data obtained from atomic force microscopy. The simple elastic network models are often used successfully to predict the fluctuations of residues around their mean positions, comparing favorably with the experimentally measured crystallographic B-factors. The behavior of biomolecules under external forces has been demonstrated to depend principally on their elastic properties and the overall shape of their structure. We have studied in detail the muscle protein titin and green fluorescent protein and tested for ten other proteins. First, we stretch the proteins computationally by performing stochastic dynamics simulations with the Gō-like model. We obtain the force-displacement curves and unfolding scenarios of possible mechanical unfolding. We then use the elastic network model to calculate temperature factors (B-factors) and compare the slowest modes of motion for the stretched proteins and compare them with the predicted order of breaking contacts between residues in the Gō-like model. Our results show that a simple Gaussian network model is able to predict contacts that break in the next time stage of stretching. Additionally, we have found that the contact disruption is strictly correlated with the highest force exerted by the backbone on these residues. Our prediction of bond-breaking agrees well with the unfolding scenario obtained with the Gō-like model. We anticipate that this method will be a useful new tool for interpreting stretching experiments.     

Conventional NMA as a better standard for evaluating elastic network models

Normal mode analysis (NMA) is an important tool for studying protein dynamics. Because of the complexity of conventional NMA that uses an all‐atom model and a semi‐empirical force field, many simplified NMA models have been developed, some of which are known as elastic network models. The quality of these simplified NMA models was assessed mostly by evaluating their predictions against experimental B‐factors, and rarely by comparing them with the original NMA. In this work, we take the effort to create a publicly accessible dataset of proteins with their minimized structures, NMA modes, and mean‐square fluctuations. Then, for the first time, we evaluate the quality of individual normal modes of several widely used elastic network models by comparing them with the conventional NMA. Our results demonstrate that the conventional NMA presents a better and more complete evaluation measure of the quality of elastic network models. This realization should be very helpful in improving current or designing new, higher quality elastic network models. Moreover, using the conventional NMA as the standard of evaluation, a number of interesting and significant insights into the elastic network models are gained. Proteins 2015; 83:259–267. © 2014 Wiley Periodicals, Inc.     

Theoretical Formulation of Principal Components Analysis to Detect and Correct for Population Stratification

The Eigenstrat method, based on principal components analysis (PCA), is commonly used both to quantify population relationships in population genetics and to correct for population stratification in genome-wide association studies. However, it can be difficult to make appropriate inference about population relationships from the principal component (PC) scatter plot. Here, to better understand the working mechanism of the Eigenstrat method, we consider its theoretical or “population” formulation. The eigen-equation for samples from an arbitrary number () of populations is reduced to that of a matrix of dimension , the elements of which are determined by the variance-covariance matrix for the random vector of the allele frequencies. Solving the reduced eigen-equation is numerically trivial and yields eigenvectors that are the axes of variation required for differentiating the populations. Using the reduced eigen-equation, we investigate the within-population fluctuations around the axes of variation on the PC scatter plot for simulated datasets. Specifically, we show that there exists an asymptotically stable pattern of the PC plot for large sample size. Our results provide theoretical guidance for interpreting the pattern of PC plot in terms of population relationships. For applications in genetic association tests, we demonstrate that, as a method of correcting for population stratification, regressing out the theoretical PCs corresponding to the axes of variation is equivalent to simply removing the population mean of allele counts and works as well as or better than the Eigenstrat method.     

Mega-Dalton biomolecular motion captured from electron microscopy reconstructions

The vibrational analysis of elastic models suggests that the essential motions of large biomolecular assemblies can be captured efficiently at an intermediate scale without requiring knowledge of the atomic structure. While prior work has established a theoretical foundation for this analysis, we demonstrate here on experimental electron microscopy maps that vibrational modes indeed describe functionally relevant movements of macromolecular machines. The clamp closure in bacterial RNA polymerase, the ratcheting of 30S and 50S subunits of the ribosome, and the dynamic flexibility of chaperonin CCT are extracted directly from single electron microscopy structures at 15-27 A resolution. The striking agreement of the presented results with experimentally observed motions suggests that the motion of the large scale machinery in the cell is surprisingly independent of detailed atomic interactions and can be quite reasonably described as a motion of elastic bodies.     

Spectral broadening of the Soret band in myoglobin: an interpretation by the full spectrum of low-frequency modes from a normal modes analysis

In this work the temperature dependence of the Soret band line shape in carbon-monoxy myoglobin is re-analyzed by using both the full correlator approach in the time domain and the frequency domain approach. The new analyses exploit the full density of vibrational states of carbon-monoxy myoglobin available from normal modes analysis, and avoid the artificial division of the entire set of vibrational modes coupled to the Soret transition into "high-frequency" and "low-frequency" subsets; the frequency domain analysis, however, makes use of the so-called short-times approximation, while the time domain one avoids it. Time domain and frequency domain analyses give very similar results, thus supporting the applicability of the short-times approximation to the analysis of hemeprotein spectra; in particular, they clearly indicate the presence of spectral heterogeneity in the Soret band of carbon-monoxy myoglobin. The analyses also show that a temperature dependence of the Gaussian width parameter steeper than the hyperbolic cotangent law predicted by the Einstein harmonic oscillator and/or a temperature dependence of inhomogeneous broadening are not sufficient to obtain quantitative information on the magnitude of an-harmonic contributions to the iron-heme plane motion. However, the dependence of the previous two quantities may be used to obtain semiquantitative information on the overall coupling of the Soret transition to the low-frequency modes and therefore on the dynamic properties of the heme pocket in different states of the protein.     

Sexual dimorphism in the quantitative-genetic architecture of floral, leaf, and allocation traits in Silene latifolia

Many species exhibit sexual dimorphism in a variety of characters, and the underlying genetic architecture of dimorphism potentially involves sex-specific differences in the additive-genetic variance-covariance matrix (G) of dimorphic traits. We investigated the quantitative-genetic structure of dimorphic traits in the dioecious plant Silene latifolia by estimating G (including within-sex matrices, G(m), G(f), and the between-sex variance-covariance matrix, B), and the phenotypic variance-covariance matrix (P) for seven traits. Flower number was the most sexually dimorphic trait, and was significantly genetically correlated with all traits within each sex. Negative genetic correlations between flower size and number suggested a genetic trade-off in investment, but positive environmental correlations between the same traits resulted in no physical evidence for a trade-off in the phenotype. Between-sex genetic covariances for homologous traits were always greater than 0 but smaller than 1, showing that some, but not all, of the variation in traits is caused by genes or alleles with sex-limited expression. Using common principal-components analysis (CPCA), a maximum-likelihood (ML) estimation approach, and element-by-element comparison to compare matrices, we found that G(m) and G(f) differed significantly in eigenstructure because of dissimilarity in covariances involving leaf traits, suggesting the presence of variation in sex-limited genes with pleiotropic effects and/or linkage between sex-limited loci. The sex-specific structure of G is expected to cause differences in the correlated responses to selection within each sex, promoting the further evolution and maintenance of dimorphism.