Vibrational subsystem analysis: A method for probing free energies and correlations in the harmonic limit

Assessing the Partial Hessian Approximation in QM/MM-Based Vibrational Analysis

The partial Hessian approximation is often used in vibrational analysis of quantum mechanics/molecular mechanics (QM/MM) systems because calculating the full Hessian matrix is computationally impractical. This approach aligns with the core concept of QM/MM, which focuses on the QM subsystem. Thus, using the partial Hessian approximation implies that the main interest is in the local vibrational modes of the QM subsystem. Here, we investigate the accuracy and applicability of the partial Hessian vibrational analysis (PHVA) approach as it is typically used within QM/MM, i.e., only the Hessian belonging to the QM subsystem is computed. We focus on solute-solvent systems with small, rigid solutes. To separate two of the major sources of errors, we perform two separate analyses. First, we study the effects of the partial Hessian approximation on local normal modes, harmonic frequencies, and harmonic IR and Raman intensities by comparing them to those obtained using full Hessians, where both partial and full Hessians are calculated at the QM level. Then, we quantify the errors introduced by QM/MM used with the PHVA by comparing normal modes, frequencies, and intensities obtained using partial Hessians calculated using a QM/MM-type embedding approach to those obtained using partial Hessians calculated at the QM level. Another aspect of the PHVA is the appearance of normal modes resembling the translation and rotation of the QM subsystem. These pseudotranslational and pseudorotational modes should be removed as they are collective vibrations of the atoms in the QM subsystem relative to a frozen MM subsystem and, thus, not well-described. We show that projecting out translation and rotation, usually done for systems in isolation, can adversely affect other normal modes. Instead, the pseudotranslational and pseudorotational modes can be identified and removed.

Scipion-EM-ProDy: A Graphical Interface for the ProDy Python Package within the Scipion Workflow Engine Enabling Integration of Databases, Simulations and Cryo-Electron Microscopy Image Processing

Macromolecular assemblies, such as protein complexes, undergo continuous structural dynamics, including global reconfigurations critical for their function. Two fast analytical methods are widely used to study these global dynamics, namely elastic network model normal mode analysis and principal component analysis of ensembles of structures. These approaches have found wide use in various computational studies, driving the development of complex pipelines in several software packages. One common theme has been conformational sampling through hybrid simulations incorporating all-atom molecular dynamics and global modes of motion. However, wide functionality is only available for experienced programmers with limited capabilities for other users. We have, therefore, integrated one popular and extensively developed software for such analyses, the ProDy Python application programming interface, into the Scipion workflow engine. This enables a wider range of users to access a complete range of macromolecular dynamics pipelines beyond the core functionalities available in its command-line applications and the normal mode wizard in VMD. The new protocols and pipelines can be further expanded and integrated into larger workflows, together with other software packages for cryo-electron microscopy image analysis and molecular simulations. We present the resulting plugin, Scipion-EM-ProDy, in detail, highlighting the rich functionality made available by its development.

Quantum harmonic free energies for biomolecules and nanomaterials

Obtaining the free energy of large molecules from quantum mechanical energy functions is a long-standing challenge. We describe a method that allows us to estimate, at the quantum mechanical level, the harmonic contributions to the thermodynamics of molecular systems of large size, with modest cost. Using this approach, we compute the vibrational thermodynamics of a series of diamond nanocrystals, and show that the error per atom decreases with system size in the limit of large systems. We further show that we can obtain the vibrational contributions to the binding free energies of prototypical protein-ligand complexes where exact computation is too expensive to be practical. Our work raises the possibility of routine quantum mechanical estimates of thermodynamic quantities in complex systems.

Surveying nonvisual arrestins reveals allosteric interactions between functional sites

Arrestins are important scaffolding proteins that are expressed in all vertebrate animals. They regulate cell-signaling events upon binding to active G-protein coupled receptors (GPCR) and trigger endocytosis of active GPCRs. While many of the functional sites on arrestins have been characterized, the question of how these sites interact is unanswered. We used anisotropic network modeling (ANM) together with our covariance compliment techniques to survey all the available structures of the nonvisual arrestins to map how structural changes and protein-binding affect their structural dynamics. We found that activation and clathrin binding have a marked effect on arrestin dynamics, and that these dynamics changes are localized to a small number of distant functional sites. These sites include α-helix 1, the lariat loop, nuclear localization domain, and the C-domain β-sheets on the C-loop side. Our techniques suggest that clathrin binding and/or GPCR activation of arrestin perturb the dynamics of these sites independent of structural changes.

Protein dynamics developments for the large scale and cryoEM: case study of ProDy 2.0

Cryo-electron microscopy (cryoEM) has become a well established technique with the potential to produce structures of large and dynamic supramolecular complexes that are not amenable to traditional approaches for studying structure and dynamics. The size and low resolution of such molecular systems often make structural modelling and molecular dynamics simulations challenging and computationally expensive. This, together with the growing wealth of structural data arising from cryoEM and other structural biology methods, has driven a trend in the computational biophysics community towards the development of new pipelines for analysing global dynamics using coarse-grained models and methods. At the centre of this trend has been a return to elastic network models, normal mode analysis (NMA) and ensemble analyses such as principal component analysis, and the growth of hybrid simulation methodologies that make use of them. Here, this field is reviewed with a focus on ProDy, the Python application programming interface for protein dynamics, which has been developed over the last decade. Two key developments in this area are highlighted: (i) ensemble NMA towards extracting and comparing the signature dynamics of homologous structures, aided by the recent SignDy pipeline, and (ii) pseudoatom fitting for more efficient global dynamics analyses of large and low-resolution supramolecular assemblies from cryoEM, revisited in the CryoDy pipeline. It is believed that such a renewal and extension of old models and methods in new pipelines will be critical for driving the field forward into the next cryoEM revolution.

Modeling Electronic Response Properties with an Explicit-Electron Machine Learning Potential

Explicit-electron force fields introduce electrons or electron pairs as semiclassical particles in force fields or empirical potentials, which are suitable for molecular dynamics simulations. Even though semiclassical electrons are a drastic simplification compared to a quantum-mechanical electronic wave function, they still retain a relatively detailed electronic model compared to conventional polarizable and reactive force fields. The ability of explicit-electron models to describe chemical reactions and electronic response properties has already been demonstrated, yet the description of short-range interactions for a broad range of chemical systems remains challenging. In this work, we present the electron machine learning potential (eMLP), a new explicit electron force field in which the short-range interactions are modeled with machine learning. The electron pair particles will be located at well-defined positions, derived from localized molecular orbitals or Wannier centers, naturally imposing the correct dielectric and piezoelectric behavior of the system. The eMLP is benchmarked on two newly constructed data sets: eQM7, an extension of the QM7 data set for small molecules, and a data set for the crystalline β-glycine. It is shown that the eMLP can predict dipole moments, polarizabilities, and IR-spectra of unseen molecules with high precision. Furthermore, a variety of response properties, for example, stiffness or piezoelectric constants, can be accurately reproduced.

Anharmonic Vibrational Calculations Based on Group-Localized Coordinates: Applications to Internal Water Molecules in Bacteriorhodopsin

An efficient anharmonic vibrational method is developed exploiting the locality of molecular vibration. Vibrational coordinates localized to a group of atoms are employed to divide the potential energy surface (PES) of a system into intra- and inter-group contributions. Then, the vibrational Schrödinger equation is solved based on a PES, in which the inter-group coupling is truncated at the harmonic level while accounting for the intra-group anharmonicity. The method is applied to a pentagonal hydrogen bond network (HBN) composed of internal water molecules and charged residues in a membrane protein, bacteriorhodopsin. The PES is calculated by the quantum mechanics/molecular mechanics (QM/MM) calculation at the level of B3LYP-D3/aug-cc-pVDZ. The infrared (IR) spectrum is computed using a set of coordinates localized to each water molecule and amino acid residue by second-order vibrational quasi-degenerate perturbation theory (VQDPT2). Benchmark calculations show that the proposed method yields the N-D/O-D stretching frequencies with an error of 7 cm-1 at the cost reduced by more than five times. In contrast, the harmonic approximation results in a severe error of 150 cm-1. Furthermore, the size of QM regions is carefully assessed to find that the QM regions should include not only the pentagonal HBN itself but also its HB partners. VQDPT2 calculations starting from transient structures obtained by molecular dynamics simulations have shown that the structural sampling has a significant impact on the calculated IR spectrum. The incorporation of anharmonicity, sufficiently large QM regions, and structural samplings are of essential importance to reproduce the experimental IR spectrum. The computational spectrum paves the way for decoding the IR signal of strong HBNs and helps elucidate their functional roles in biomolecules.

Insights into the substrate binding mechanism of SULT1A1 through molecular dynamics with excited normal modes simulations

Sulfotransferases (SULTs) are phase II drug-metabolizing enzymes catalyzing the sulfoconjugation from the co-factor 3′-phosphoadenosine 5′-phosphosulfate (PAPS) to a substrate. It has been previously suggested that a considerable shift of SULT structure caused by PAPS binding could control the capability of SULT to bind large substrates. We employed molecular dynamics (MD) simulations and the recently developed approach of MD with excited normal modes (MDeNM) to elucidate molecular mechanisms guiding the recognition of diverse substrates and inhibitors by SULT1A1. MDeNM allowed exploring an extended conformational space of PAPS-bound SULT1A1, which has not been achieved up to now by using classical MD. The generated ensembles combined with docking of 132 SULT1A1 ligands shed new light on substrate and inhibitor binding mechanisms. Unexpectedly, our simulations and analyses on binding of the substrates estradiol and fulvestrant demonstrated that large conformational changes of the PAPS-bound SULT1A1 could occur independently of the co-factor movements that could be sufficient to accommodate large substrates as fulvestrant. Such structural displacements detected by the MDeNM simulations in the presence of the co-factor suggest that a wider range of drugs could be recognized by PAPS-bound SULT1A1 and highlight the utility of including MDeNM in protein–ligand interactions studies where major rearrangements are expected.

Normal mode analysis of membrane protein dynamics using the vibrational subsystem analysis

The vibrational subsystem analysis is a useful approach that allows for evaluating the spectrum of modes of a given system by integrating out the degrees of freedom accessible to the environment. The approach could be utilized for exploring the collective dynamics of a membrane protein (system) coupled to the lipid bilayer (environment). However, the application to membrane proteins is limited due to high computational costs of modeling a sufficiently large membrane environment unbiased by end effects, which drastically increases the size of the investigated system. We derived a recursive formula for calculating the reduced Hessian of a membrane protein embedded in a lipid bilayer by decomposing the membrane into concentric cylindrical domains with the protein located at the center. The approach allows for the design of a time- and memory-efficient algorithm and a mathematical understanding of the convergence of the reduced Hessian with respect to increasing membrane sizes. The application to the archaeal aspartate transporter GltPh illustrates its utility and efficiency in capturing the transporter's elevator-like movement during its transition between outward-facing and inward-facing states.

A revised formulation of the generalized subsystem vibrational analysis (GSVA)

In this work, a simplified formulation of our recently developed generalized subsystem vibrational analysis (GSVA) for obtaining intrinsic fragmental vibrations (J Chem Theory Comput 14:2558, 2018) is presented. In contrast to the earlier implementation, which requires the explicit definition of a non-redundant set of internal coordinate parameters to be constructed for the subsystem, the new implementation circumvents this process by employing massless Eckart conditions to the subsystem fragment paired with a Gram-Schmidt orthogonalization to span the same internal vibration space indirectly. This revised version of GSVA (rev-GSVA) can be applied to equilibrium structure as well as transition state structure, and it has been incorporated into the open-source package UniMoVib (https://github.com/zorkzou/UniMoVib).

Supplementary Information

The online version contains supplementary material available at 10.1007/s00214-021-02727-y.

Evaluating Simple Ab Initio Models of the Hydrated Electron: The Role of Dynamical Fluctuations

Despite its importance in electron transfer reactions and radiation chemistry, there has been disagreement over the fundamental nature of the hydrated electron, such as whether or not it resides in a cavity. Mixed quantum/classical simulations of the hydrated electron give different structures depending on the pseudopotential employed, and ab initio models of computational necessity use small numbers of water molecules and/or provide insufficient statistics to compare to experimental observables. A few years ago, Kumar et al. (J. Phys. Chem. A 2015, 119, 9148) proposed a minimalist ab initio model of the hydrated electron with only a small number of explicitly treated water molecules plus a polarizable continuum model (PCM). They found that the optimized geometry had four waters arranged tetrahedrally around a central cavity, and that the calculated vertical detachment energy and radius of gyration agreed well with experiment, results that were largely independent of the level of theory employed. The model, however, is based on a fixed structure at 0 K and does not explicitly incorporate entropic contributions or the thermal fluctuations that should be associated with the room-temperature hydrated electron. Thus, in this paper, we extend the model of Kumar et al. by running Born-Oppenheimer molecular dynamics (BOMD) of a small number of water molecules with an excess electron plus PCM at room temperature. We find that when thermal fluctuations are introduced, the level of theory chosen becomes critical enough when only four waters are used that one of the waters dissociates from the cluster with certain density functionals. Moreover, even with an optimally tuned range-separated hybrid functional, at room temperature the tetrahedral orientation of the 0 K first-shell waters is entirely lost and the central cavity collapses, a process driven by the fact that the explicit water molecules prefer to make H-bonds with each other more than with the excess electron. The resulting average structure is quite similar to that produced by a noncavity mixed quantum/classical model, so that the minimalist 4-water BOMD models suffer from problems similar to those of noncavity models, such as predicting the wrong sign of the hydrated electron's molar solvation volume. We also performed BOMD with 16 explicit water molecules plus an extra electron and PCM. We find that the inclusion of an entire second solvation shell of explicit water leads to little change in the outcome from when only four waters were used. In fact, the 16-water simulations behave much like those of water cluster anions, in which the electron localizes at the cluster surface, showing that PCM is not acceptable for use in minimalist models to describe the behavior of the bulk hydrated electron. For both the 4- and 16-water models, we investigate how the introduction of thermal motions alters the predicted absorption spectrum, vertical detachment energy, and resonance Raman spectrum of the simulated hydrated electron. We also present a set of structural criteria that can be used to numerically determine how cavity-like (or not) a particular hydrated electron model is. All of the results emphasize that the hydrated electron is a statistical object whose properties are inadequately captured using only a small number of explicit waters, and that a proper treatment of thermal fluctuations is critical to understanding the hydrated electron's chemical and physical behavior.

Multi-level free energy simulation with a staged transformation approach

Combining multiple levels of theory in free energy simulations to balance computational accuracy and efficiency is a promising approach for studying processes in the condensed phase. While the basic idea has been proposed and explored for quite some time, it remains challenging to achieve convergence for such multi-level free energy simulations as it requires a favorable distribution overlap between different levels of theory. Previous efforts focused on improving the distribution overlap by either altering the low-level of theory for the specific system of interest or ignoring certain degrees of freedom. Here, we propose an alternative strategy that first identifies the degrees of freedom that lead to gaps in the distributions of different levels of theory and then treats them separately with either constraints or restraints or by introducing an intermediate model that better connects the low and high levels of theory. As a result, the conversion from the low level to the high level model is done in a staged fashion that ensures a favorable distribution overlap along the way. Free energy components associated with different steps are mostly evaluated explicitly, and thus, the final result can be meaningfully compared to the rigorous free energy difference between the two levels of theory with limited and well-defined approximations. The additional free energy component calculations involve simulations at the low level of theory and therefore do not incur high computational costs. The approach is illustrated with two simple but non-trivial solution examples, and factors that dictate the reliability of the result are discussed.

Use of Density Functional Based Tight Binding Methods in Vibrational Circular Dichroism

Vibrational circular dichroism (VCD) is a spectroscopic technique used to resolve the absolute configuration of chiral systems. Obtaining a theoretical VCD spectrum requires computing atomic polar and axial tensors on top of the computationally demanding construction of the force constant matrix. In this study we evaluated a VCD model in which all necessary quantities are obtained with density functional based tight binding (DFTB) theory. The analyzed DFTB parametrizations fail at providing accurate vibrational frequencies and electric dipole gradients but yield reasonable normal modes at a fraction of the computational cost of density functional theory (DFT). Thus, by applying DFTB in composite methods along with DFT, we show that it is possible to obtain accurate VCD spectra at a much lower computational demand.

Frequency Range Selection Method for Vibrational Spectra

Theoretical calculations of vibrational properties are widely used to explain and predict experimental spectra. However, with standard quantum chemical methods all molecular motions are considered, which is rather time-consuming for large molecules. Because typically only a specific spectral region is of experimental interest, we propose here an efficient method that allows calculation of only a selected frequency interval. After a computationally cheap low-level estimate of the molecular motions, the computational time is proportional to the number of normal modes needed to describe this frequency range. Results for a medium-sized molecule show a reduction in computational time of up to 1 order of magnitude with negligible loss in accuracy. We also show that still larger computational savings are possible by using an additional intensity-selection procedure.

Dissecting conformational changes in APP's transmembrane domain linked to ε-efficiency in familial Alzheimer's disease

The mechanism by which familial Alzheimer's disease (FAD) mutations within the transmembrane domain (TMD) of the Amyloid Precursor Protein (APP) affect ε-endoproteolysis is only poorly understood. Thereby, mutations in the cleavage domain reduce ε-efficiency of γ-secretase cleavage and some even shift entry into production lines. Since cleavage occurs within the TMD, a relationship between processing and TMD structure and dynamics seems obvious. Using molecular dynamic simulations, we dissect the dynamic features of wild-type and seven FAD-mutants into local and global components. Mutations consistently enhance hydrogen-bond fluctuations upstream of the ε-cleavage sites but maintain strong helicity there. Dynamic perturbation-response scanning reveals that FAD-mutants target backbone motions utilized in the bound state. Those motions, obscured by large-scale motions in the pre-bound state, provide (i) a dynamic mechanism underlying the proposed coupling between binding and ε-cleavage, (ii) key sites consistent with experimentally determined docking sites, and (iii) the distinction between mutants and wild-type.

Vibrational intensities in the mobile block Hessian approximation

Herein we present a proof of concept for the recovery of vibrational intensities from density functional theory vibrational calculations performed in the Mobile Block Hessian (MBH) approximation, which constrains the internal degrees of freedom of designated subsets of a molecule. We compare and contrast the behaviour of this methodology with respect to conventional vibrational calculations, and characterise the performance and accuracy of our method with respect to the size of MBH constrained regions within a variety of species. We demonstrate the viability of this method as a means by which to obtain vibrational intensities for regions of interest within a molecule whilst potentially dramatically reducing computational expense with respect to conventional all-atom vibrational calculations, and discuss caveats for application.

Methods for Efficiently and Accurately Computing Quantum Mechanical Free Energies for Enzyme Catalysis

Enzyme activity is inherently linked to free energies of transition states, ligand binding, protonation/deprotonation, etc.; these free energies, and thus enzyme function, can be affected by residue mutations, allosterically induced conformational changes, and much more. Therefore, being able to predict free energies associated with enzymatic processes is critical to understanding and predicting their function. Free energy simulation (FES) has historically been a computational challenge as it requires both the accurate description of inter- and intramolecular interactions and adequate sampling of all relevant conformational degrees of freedom. The hybrid quantum mechanical molecular mechanical (QM/MM) framework is the current tool of choice when accurate computations of macromolecular systems are essential. Unfortunately, robust and efficient approaches that employ the high levels of computational theory needed to accurately describe many reactive processes (ie, ab initio, DFT), while also including explicit solvation effects and accounting for extensive conformational sampling are essentially nonexistent. In this chapter, we will give a brief overview of two recently developed methods that mitigate several major challenges associated with QM/MM FES: the QM non-Boltzmann Bennett's acceptance ratio method and the QM nonequilibrium work method. We will also describe usage of these methods to calculate free energies associated with (1) relative properties and (2) along reaction paths, using simple test cases with relevance to enzymes examples.

Bridging between NMA and Elastic Network Models: Preserving All-Atom Accuracy in Coarse-Grained Models

Dynamics can provide deep insights into the functional mechanisms of proteins and protein complexes. For large protein complexes such as GroEL/GroES with more than 8,000 residues, obtaining a fine-grained all-atom description of its normal mode motions can be computationally prohibitive and is often unnecessary. For this reason, coarse-grained models have been used successfully. However, most existing coarse-grained models use extremely simple potentials to represent the interactions within the coarse-grained structures and as a result, the dynamics obtained for the coarse-grained structures may not always be fully realistic. There is a gap between the quality of the dynamics of the coarse-grained structures given by all-atom models and that by coarse-grained models. In this work, we resolve an important question in protein dynamics computations—how can we efficiently construct coarse-grained models whose description of the dynamics of the coarse-grained structures remains as accurate as that given by all-atom models? Our method takes advantage of the sparseness of the Hessian matrix and achieves a high efficiency with a novel iterative matrix projection approach. The result is highly significant since it can provide descriptions of normal mode motions at an all-atom level of accuracy even for the largest biomolecular complexes. The application of our method to GroEL/GroES offers new insights into the mechanism of this biologically important chaperonin, such as that the conformational transitions of this protein complex in its functional cycle are even more strongly connected to the first few lowest frequency modes than with other coarse-grained models.

Proteins and other biomolecules are not static but are constantly in motion. Moreover, they possess intrinsic collective motion patterns that are tightly linked to their functions. Thus, an accurate and detailed description of their motions can provide deep insights into their functional mechanisms. For large protein complexes with hundreds of thousands of atoms or more, an atomic level description of the motions can be computationally prohibitive, and so coarse-grained models with fewer structural details are often used instead. However, there can be a big gap between the quality of motions derived from atomic models and those from coarse-grained models. In this work, we solve an important problem in protein dynamics studies: how to preserve the atomic-level accuracy in describing molecular motions while using coarse-grained models? We accomplish this by developing a novel iterative matrix projection method that dramatically speeds up the computations. This method is significant since it promises accurate descriptions of protein motions approaching an all-atom level even for the largest biomolecular complexes. Results shown here for a large molecular chaperonin demonstrate how this can provide new insights into its functional process.

Lightweight object oriented structure analysis: tools for building tools to analyze molecular dynamics simulations

LOOS (Lightweight Object Oriented Structure-analysis) is a C++ library designed to facilitate making novel tools for analyzing molecular dynamics simulations by abstracting out the repetitive tasks, allowing developers to focus on the scientifically relevant part of the problem. LOOS supports input using the native file formats of most common biomolecular simulation packages, including CHARMM, NAMD, Amber, Tinker, and Gromacs. A dynamic atom selection language based on the C expression syntax is included and is easily accessible to the tool-writer. In addition, LOOS is bundled with over 140 prebuilt tools, including suites of tools for analyzing simulation convergence, three-dimensional histograms, and elastic network models. Through modern C++ design, LOOS is both simple to develop with (requiring knowledge of only four core classes and a few utility functions) and is easily extensible. A python interface to the core classes is also provided, further facilitating tool development.

Multiple environment single system quantum mechanical/molecular mechanical (MESS-QM/MM) calculations. 1. Estimation of polarization energies

In combined quantum mechanical/molecular mechanical (QM/MM) free energy calculations, it is often advantageous to have a frozen geometry for the quantum mechanical (QM) region. For such multiple-environment single-system (MESS) cases, two schemes are proposed here for estimating the polarization energy: the first scheme, termed MESS-E, involves a Roothaan step extrapolation of the self-consistent field (SCF) energy; whereas the other scheme, termed MESS-H, employs a Newton–Raphson correction using an approximate inverse electronic Hessian of the QM region (which is constructed only once). Both schemes are extremely efficient, because the expensive Fock updates and SCF iterations in standard QM/MM calculations are completely avoided at each configuration. They produce reasonably accurate QM/MM polarization energies: MESS-E can predict the polarization energy within 0.25 kcal/mol in terms of the mean signed error for two of our test cases, solvated methanol and solvated β-alanine, using the M06-2X or ωB97X-D functionals; MESS-H can reproduce the polarization energy within 0.2 kcal/mol for these two cases and for the oxyluciferin–luciferase complex, if the approximate inverse electronic Hessians are constructed with sufficient accuracy.

Comparing the intrinsic dynamics of multiple protein structures using elastic network models

BACKGROUND

Elastic network models (ENMs) are based on the simple idea that a protein can be described as a set of particles connected by springs, which can then be used to describe its intrinsic flexibility using, for example, normal mode analysis. Since the introduction of the first ENM by Monique Tirion in 1996, several variants using coarser protein models have been proposed and their reliability for the description of protein intrinsic dynamics has been widely demonstrated. Lately an increasing number of studies have focused on the meaning of slow dynamics for protein function and its potential conservation through evolution. This leads naturally to comparisons of the intrinsic dynamics of multiple protein structures with varying levels of similarity.

SCOPE OF REVIEW

We describe computational strategies for calculating and comparing intrinsic dynamics of multiple proteins using elastic network models, as well as a selection of examples from the recent literature.

MAJOR CONCLUSIONS

The increasing interest for comparing dynamics across protein structures with various levels of similarity, has led to the establishment and validation of reliable computational strategies using ENMs. Comparing dynamics has been shown to be a viable way for gaining greater understanding for the mechanisms employed by proteins for their function. Choices of ENM parameters, structure alignment or similarity measures will likely influence the interpretation of the comparative analysis of protein motion.

GENERAL SIGNIFICANCE

Understanding the relation between protein function and dynamics is relevant to the fundamental understanding of protein structure-dynamics-function relationship. This article is part of a Special Issue entitled Recent developments of molecular dynamics.

Efficiently computing pathway free energies: New approaches based on chain-of-replica and Non-Boltzmann Bennett reweighting schemes

BACKGROUND

Accurately modeling condensed phase processes is one of computation's most difficult challenges. Include the possibility that conformational dynamics may be coupled to chemical reactions, where multiscale (i.e., QM/MM) methods are needed, and this task becomes even more daunting.

METHODS

Free energy simulations (i.e., molecular dynamics), multiscale modeling, and reweighting schemes.

RESULTS

Herein, we present two new approaches for mitigating the aforementioned challenges. The first is a new chain-of-replica method (off-path simulations, OPS) for computing potentials of mean force (PMFs) along an easily defined reaction coordinate. This development is coupled with a new distributed, highly-parallel replica framework (REPDstr) within the CHARMM package. Validation of these new schemes is carried out on two processes that undergo conformational changes. First is the simple torsional rotation of butane, while a much more challenging glycosidic rotation (in vacuo and solvated) is the second. Additionally, a new approach that greatly improves (i.e., possibly an order of magnitude) the efficiency of computing QM/MM PMFs is introduced and compared to standard schemes. Our efforts are grounded in the recently developed method for efficiently computing QM-based free energies (i.e., QM-Non-Boltzmann Bennett, QM-NBB). Again, we validate this new technique by computing the QM/MM PMF of butane's torsional rotation.

CONCLUSIONS

The OPS-REPDstr method is a promising new approach that overcomes many limitations of standard pathway simulations in CHARMM. The combination of QM-NBB with pathway techniques is very promising as it offers significant advantages over current procedures.

GENERAL SIGNIFICANCE

Efficiently computing potentials of mean force is a major, unresolved, area of interest. This article is part of a Special Issue entitled Recent developments of molecular dynamics.

Correcting for the free energy costs of bond or angle constraints in molecular dynamics simulations

BACKGROUND

Free energy simulations are an important tool in the arsenal of computational biophysics, allowing the calculation of thermodynamic properties of binding or enzymatic reactions. This paper introduces methods to increase the accuracy and precision of free energy calculations by calculating the free energy costs of constraints during post-processing. The primary purpose of employing constraints for these free energy methods is to increase the phase space overlap between ensembles, which is required for accuracy and convergence.

METHODS

The free energy costs of applying or removing constraints are calculated as additional explicit steps in the free energy cycle. The new techniques focus on hard degrees of freedom and use both gradients and Hessian estimation. Enthalpy, vibrational entropy, and Jacobian free energy terms are considered.

RESULTS

We demonstrate the utility of this method with simple classical systems involving harmonic and anharmonic oscillators, four-atomic benchmark systems, an alchemical mutation of ethane to methanol, and free energy simulations between alanine and serine. The errors for the analytical test cases are all below 0.0007kcal/mol, and the accuracy of the free energy results of ethane to methanol is improved from 0.15 to 0.04kcal/mol. For the alanine to serine case, the phase space overlaps of the unconstrained simulations range between 0.15 and 0.9%. The introduction of constraints increases the overlap up to 2.05%. On average, the overlap increases by 94% relative to the unconstrained value and precision is doubled.

CONCLUSIONS

The approach reduces errors arising from constraints by about an order of magnitude. Free energy simulations benefit from the use of constraints through enhanced convergence and higher precision.

GENERAL SIGNIFICANCE

The primary utility of this approach is to calculate free energies for systems with disparate energy surfaces and bonded terms, especially in multi-scale molecular mechanics/quantum mechanics simulations. This article is part of a Special Issue entitled Recent developments of molecular dynamics.

Computing the Free Energy along a Reaction Coordinate Using Rigid Body Dynamics

The calculations of potential of mean force along complex chemical reactions or rare events pathways are of great interest because of their importance for many areas in chemistry, molecular biology, and material science. The major difficulty for free energy calculations comes from the great computational cost for adequate sampling of the system in high-energy regions, especially close to the reaction transition state. Here, we present a method, called FEG-RBD, in which the free energy gradients were obtained from rigid body dynamics simulations. Then the free energy gradients were integrated along a reference reaction pathway to calculate free energy profiles. In a given system, the reaction coordinates defining a subset of atoms (e.g., a solute, or the quantum mechanics (QM) region of a quantum mechanics/molecular mechanics simulation) are selected to form a rigid body during the simulation. The first-order derivatives (gradients) of the free energy with respect to the reaction coordinates are obtained through the integration of constraint forces within the rigid body. Each structure along the reference reaction path is separately subjected to such a rigid body simulation. The individual free energy gradients are integrated along the reference pathway to obtain the free energy profile. Test cases provided demonstrate both the strengths and weaknesses of the FEG-RBD method. The most significant benefit of this method comes from the fast convergence rate of the free energy gradient using rigid-body constraints instead of restraints. A correction to the free energy due to approximate relaxation of the rigid-body constraint is estimated and discussed. A comparison with umbrella sampling using a simple test case revealed the improved sampling efficiency of FEG-RBD by a factor of 4 on average. The enhanced efficiency makes this method effective for calculating the free energy of complex chemical reactions when the reaction coordinate can be unambiguously defined by a small subset of atoms within the system.

First principle chemical kinetics in zeolites: the methanol-to-olefin process as a case study

To optimally design next generation catalysts a thorough understanding of the chemical phenomena at the molecular scale is a prerequisite. Apart from qualitative knowledge on the reaction mechanism, it is also essential to be able to predict accurate rate constants. Molecular modeling has become a ubiquitous tool within the field of heterogeneous catalysis. Herein, we review current computational procedures to determine chemical kinetics from first principles, thus by using no experimental input and by modeling the catalyst and reacting species at the molecular level. Therefore, we use the methanol-to-olefin (MTO) process as a case study to illustrate the various theoretical concepts. This process is a showcase example where rational design of the catalyst was for a long time performed on the basis of trial and error, due to insufficient knowledge of the mechanism. For theoreticians the MTO process is particularly challenging as the catalyst has an inherent supramolecular nature, for which not only the Brønsted acidic site is important but also organic species, trapped in the zeolite pores, must be essentially present during active catalyst operation. All these aspects give rise to specific challenges for theoretical modeling. It is shown that present computational techniques have matured to a level where accurate enthalpy barriers and rate constants can be predicted for reactions occurring at a single active site. The comparison with experimental data such as apparent kinetic data for well-defined elementary reactions has become feasible as current computational techniques also allow predicting adsorption enthalpies with reasonable accuracy. Real catalysts are truly heterogeneous in a space- and time-like manner. Future theory developments should focus on extending our view towards phenomena occurring at longer length and time scales and integrating information from various scales towards a unified understanding of the catalyst. Within this respect molecular dynamics methods complemented with additional techniques to simulate rare events are now gradually making their entrance within zeolite catalysis. Recent applications have already given a flavor of the benefit of such techniques to simulate chemical reactions in complex molecular environments.

Elastic Network Models of Nucleic Acids Flexibility

Elastic network models (ENMs) are a useful tool for describing large scale motions in protein systems. While they are well validated in the context of proteins, relatively little is known about their applicability to nucleic acids, whose different architecture does not necessarily warrant comparable performance. In this study we thoroughly evaluate and optimize the efficiency of popular ENMs for capturing RNA and DNA flexibility. We also introduce two alternative models in which the strength of elastic connections at a coarse-grained level is governed by distance distribution at atomic resolution. For each of the considered ENMs we report the optimal length of spring connections as well as the scaling of elastic force constants that provides the best agreement of vibrational frequencies with normal modes based on atomic force field. In order to determine the absolute values of force constants we introduce a novel method based on the overlap of pseudoinverse of Hessian matrices.

Protein engineering and the use of molecular modeling and simulation: the case of heterodimeric Fc engineering

Computational and structure guided methods can make significant contributions to the development of solutions for difficult protein engineering problems, including the optimization of next generation of engineered antibodies. In this paper, we describe a contemporary industrial antibody engineering program, based on hypothesis-driven in silico protein optimization method. The foundational concepts and methods of computational protein engineering are discussed, and an example of a computational modeling and structure-guided protein engineering workflow is provided for the design of best-in-class heterodimeric Fc with high purity and favorable biophysical properties. We present the engineering rationale as well as structural and functional characterization data on these engineered designs.

The interplay of structure and dynamics: insights from a survey of HIV-1 reverse transcriptase crystal structures

HIV-1 reverse transcriptase (RT) is a critical drug target for HIV treatment, and understanding the exact mechanisms of its function and inhibition would significantly accelerate the development of new anti-HIV drugs. It is well known that structure plays a critical role in protein function, but for RT, structural information has proven to be insufficient-despite enormous effort-to explain the mechanism of inhibition and drug resistance of non-nucleoside RT inhibitors. We hypothesize that the missing link is dynamics, information about the motions of the system. However, many of the techniques that give the best information about dynamics, such as solution nuclear magnetic resonance and molecular dynamics simulations, cannot be easily applied to a protein as large as RT. As an alternative, we combine elastic network modeling with simultaneous hierarchical clustering of structural and dynamic data. We present an extensive survey of the dynamics of RT bound to a variety of ligands and with a number of mutations, revealing a novel mechanism for drug resistance to non-nucleoside RT inhibitors. Hydrophobic core mutations restore active-state motion to multiple functionally significant regions of HIV-1 RT. This model arises out of a combination of structural and dynamic information, rather than exclusively from one or the other.

PIM: phase integrated method for normal mode analysis of biomolecules in a crystalline environment

In this study, a normal mode analysis, named phase integrated method (PIM), is developed for computing modes of biomolecules in a crystalline environment. PIM can calculate low-frequency modes on one or a few asymmetric units (AUs) and generate exact modes of a whole unit cell according to space group symmetry, while the translational symmetry between unit cells is maintained via the periodic boundary condition. Therefore, the method can dramatically reduce computational cost in mode calculation in the presence of crystal symmetry. PIM also has an option to map modes onto a single AU to form an orthonormalized mode set, which can be directly applied to normal-mode-based thermal parameter refinement in X-ray crystallography. The performance of PIM was tested on all 65 space groups available in protein crystals (one protein for each space group) and on another set of 83 ultra-high-resolution X-ray structures. The results showed that considering space group symmetry in mode calculation is crucial for accurately describing vibrational motion in a crystalline environment. Moreover, the optimal inter-AU packing stiffness was found to be about 60% of that of intra-AU interactions (non-bonded interaction only).

Elastic Network Models are Robust to Variations in Formalism

Understanding the functions of biomolecules requires insight not only from structures, but from dynamics as well. Often, the most interesting processes occur on time scales too slow for exploration by conventional molecular dynamics (MD) simulations. For this reason, alternative computational methods such as elastic network models (ENMs) have become increasingly popular. These simple, coarse-grained models represent molecules as beads connected by harmonic springs; the system's motions are solved analytically by normal mode analysis. In the past few years, many different formalisms for performing ENM calculations have emerged, and several have been optimized using all-atom MD simulations. In contrast to other studies, we have compared the various formalisms in a systematic, quantitative way. In this study, we optimize many ENM functional forms using a uniform dataset containing only long (> 1 μs) all-atom MD simulations. Our results show that all models once optimized produce spring constants for immediate neighboring residues that are orders of magnitude stiffer than more distal contacts. In addition, the statistical significance of ENM performance varied with model resolution. We also show that fitting long trajectories does not improve ENM performance due to a problem inherent in all network models tested: they underestimate the relative importance of the most concerted motions. Finally, we characterize ENMs' resilience by tessellating the parameter space to show that broad ranges of parameters produce similar quality predictions. Taken together our data reveals that choice of spring function and parameters are not vital to performance of a network model and that simple parameters can by derived "by hand" when no data is available for fitting, thus illustrating the robustness of these models.

Density-cluster NMA: A new protein decomposition technique for coarse-grained normal mode analysis

Normal mode analysis has emerged as a useful technique for investigating protein motions on long time scales. This is largely due to the advent of coarse-graining techniques, particularly Hooke's Law-based potentials and the rotational-translational blocking (RTB) method for reducing the size of the force-constant matrix, the Hessian. Here we present a new method for domain decomposition for use in RTB that is based on hierarchical clustering of atomic density gradients, which we call Density-Cluster RTB (DCRTB). The method reduces the number of degrees of freedom by 85-90% compared with the standard blocking approaches. We compared the normal modes from DCRTB against standard RTB using 1-4 residues in sequence in a single block, with good agreement between the two methods. We also show that Density-Cluster RTB and standard RTB perform well in capturing the experimentally determined direction of conformational change. Significantly, we report superior correlation of DCRTB with B-factors compared with 1-4 residue per block RTB. Finally, we show significant reduction in computational cost for Density-Cluster RTB that is nearly 100-fold for many examples.

Separating the role of protein restraints and local metal-site interaction chemistry in the thermodynamics of a zinc finger protein

We express the effective Hamiltonian of an ion-binding site in a protein as a combination of the Hamiltonian of the ion-bound site in vacuum and the restraints of the protein on the site. The protein restraints are described by the quadratic elastic network model. The Hamiltonian of the ion-bound site in vacuum is approximated as a generalized Hessian around the minimum energy configuration. The resultant of the two quadratic Hamiltonians is cast into a pure quadratic form. In the canonical ensemble, the quadratic nature of the resultant Hamiltonian allows us to express analytically the excess free energy, enthalpy, and entropy of ion binding to the protein. The analytical expressions allow us to separate the roles of the dynamic restraints imposed by the protein on the binding site and the temperature-independent chemical effects in metal-ligand coordination. For the consensus zinc-finger peptide, relative to the aqueous phase, the calculated free energy of exchanging Zn(2+) with Fe(2+), Co(2+), Ni(2+), and Cd(2+) are in agreement with experiments. The predicted excess enthalpy of ion exchange between Zn(2+) and Co(2+) also agrees with the available experimental estimate. The free energy of applying the protein restraints reveals that relative to Zn(2+), the Co(2+), and Cd(2+)-site clusters are more destabilized by the protein restraints. This leads to an experimentally testable hypothesis that a tetrahedral metal binding site with minimal protein restraints will be less selective for Zn(2+) over Co(2+) and Cd(2+) compared to a zinc finger peptide. No appreciable change is expected for Fe(2+) and Ni(2+). The framework presented here may prove useful in protein engineering to tune metal selectivity.

MSCALE: A General Utility for Multiscale Modeling

The combination of theoretical models of macromolecules that exist at different spatial and temporal scales has become increasingly important for addressing complex biochemical problems. This work describes the extension of concurrent multiscale approaches, introduces a general framework for carrying out calculations, and describes its implementation into the CHARMM macromolecular modeling package. This functionality, termed MSCALE, generalizes both the additive and subtractive multiscale scheme (e.g. QM/MM ONIOM-type), and extends its support to classical force fields, coarse grained modeling (e.g. ENM, GNM, etc.), and a mixture of them all. The MSCALE scheme is completely parallelized with each subsystem running as an independent, but connected calculation. One of the most attractive features of MSCALE is the relative ease of implementation using the standard MPI communication protocol. This allows external access to the framework and facilitates the combination of functionality previously isolated in separate programs. This new facility is fully integrated with free energy perturbation methods, Hessian based methods, and the use of periodicity and symmetry, which allows the calculation of accurate pressures. We demonstrate the utility of this new technique with four examples; (1) subtractive QM/MM and QM/QM calculations; (2) multi-force field alchemical free energy perturbation; (3) integration with the SANDER module of AMBER and the TINKER package to gain access to potentials not available in CHARMM; and (4) mixed resolution (i.e. coarse grain / all-atom) normal mode analysis. The potential of this new tool is clearly established and in conclusion an interesting mathematical problem is highlighted and future improvements are proposed.

Normal mode analysis with molecular geometry restraints: bridging molecular mechanics and elastic models

A new method for normal mode analysis is reported for all-atom structures using molecular geometry restraints (MGR). Similar to common molecular mechanics force fields, the MGR potential contains short- and long-range terms. The short-range terms are defined by molecular geometry, i.e., bond lengths, angles and dihedrals; the long-range term is similar to that in elastic network models. Each interaction term uses a single force constant parameter, and is determined by fitting against a set of known structures. Tests on proteins/non-proteins show that MGR can produce low frequency eigenvectors closer to all-atom force-field-based methods than conventional elastic network models. Moreover, the "tip effect", found in low frequency eigenvectors in elastic network models, is reduced in MGR to the same level of the modes produced by force-field-based methods. The results suggest that molecular geometry plays an important role, in addition to molecular shape, in determining low frequency deformational motions. MGR does not require initial energy minimization, and is applicable to almost any structure, including the one with missing atoms, bad contacts, or bad geometries, frequently observed in low-resolution structure determination and refinement. The method bridges the two major representations in normal mode analyses, i.e., the molecular mechanics models and elastic network models.

Validating and improving elastic network models with molecular dynamics simulations

Elastic network models (ENMs) are a class of simple models intended to represent the collective motions of proteins. In contrast to all-atom molecular dynamics simulations, the low computational investment required to use an ENM makes them ideal for speculative hypothesis-testing situations. Historically, ENMs have been validated via comparison to crystallographic B-factors, but this comparison is relatively low-resolution and only tests the predictions of relative flexibility. In this work, we systematically validate and optimize a number of ENM-type models by quantitatively comparing their predictions to microsecond-scale all-atom simulations of three different G protein coupled receptors. We show that, despite their apparent simplicity, well-optimized ENMs perform remarkably well, reproducing the protein fluctuations with an accuracy comparable to what one would expect from all-atom simulations run for several hundred nanoseconds.

Comparative study of various normal mode analysis techniques based on partial Hessians

Standard normal mode analysis becomes problematic for complex molecular systems, as a result of both the high computational cost and the excessive amount of information when the full Hessian matrix is used. Several partial Hessian methods have been proposed in the literature, yielding approximate normal modes. These methods aim at reducing the computational load and/or calculating only the relevant normal modes of interest in a specific application. Each method has its own (dis)advantages and application field but guidelines for the most suitable choice are lacking. We have investigated several partial Hessian methods, including the Partial Hessian Vibrational Analysis (PHVA), the Mobile Block Hessian (MBH), and the Vibrational Subsystem Analysis (VSA). In this article, we focus on the benefits and drawbacks of these methods, in terms of the reproduction of localized modes, collective modes, and the performance in partially optimized structures. We find that the PHVA is suitable for describing localized modes, that the MBH not only reproduces localized and global modes but also serves as an analysis tool of the spectrum, and that the VSA is mostly useful for the reproduction of the low frequency spectrum. These guidelines are illustrated with the reproduction of the localized amine-stretch, the spectrum of quinine and a bis-cinchona derivative, and the low frequency modes of the LAO binding protein.

Optimal modeling of atomic fluctuations in protein crystal structures for weak crystal contact interactions

The accurate modeling of protein dynamics in crystalline states holds keys to the understanding of protein dynamics relevant to functions. In this study, we used coarse-grained elastic network models (ENMs) to explore the atomic fluctuations of a protein structure that interacts with its crystalline environment, and evaluated the modeling results using the anisotropic displacement parameters (ADPs) obtained from x-ray crystallography. To ensure the robustness of modeling results, we used three ENM schemes for assigning force constant combined with three boundary conditions for treating the crystalline environment. To explore the role of crystal contact interactions in the modeling of ADPs, we varied the strength of interactions between a protein structure and its environment. For a list of 83 high-resolution crystal structures, we found that the optimal modeling of ADPs, as assessed by a variety of metrics, is achieved for weak protein-environment interactions (compared to the interactions within a protein structure). As a result, the ADPs are dominated by contributions from rigid-body motions of the entire protein structure, and the internal protein dynamics is only weakly perturbed by crystal packing. Our finding of weak crystal contact interactions is also corroborated by the calculations of residue-residue contact energy within a protein structure and between protein molecules using a statistical potential.

CHARMM: the biomolecular simulation program

CHARMM (Chemistry at HARvard Molecular Mechanics) is a highly versatile and widely used molecular simulation program. It has been developed over the last three decades with a primary focus on molecules of biological interest, including proteins, peptides, lipids, nucleic acids, carbohydrates, and small molecule ligands, as they occur in solution, crystals, and membrane environments. For the study of such systems, the program provides a large suite of computational tools that include numerous conformational and path sampling methods, free energy estimators, molecular minimization, dynamics, and analysis techniques, and model-building capabilities. The CHARMM program is applicable to problems involving a much broader class of many-particle systems. Calculations with CHARMM can be performed using a number of different energy functions and models, from mixed quantum mechanical-molecular mechanical force fields, to all-atom classical potential energy functions with explicit solvent and various boundary conditions, to implicit solvent and membrane models. The program has been ported to numerous platforms in both serial and parallel architectures. This article provides an overview of the program as it exists today with an emphasis on developments since the publication of the original CHARMM article in 1983.

Approximate normal mode analysis based on vibrational subsystem analysis with high accuracy and efficiency

Normal mode analysis (NMA) has been proven valuable in modeling slow conformational dynamics of biomolecular structures beyond the reach of direct molecular simulations. However, it remains computationally expensive to directly solve normal modes for large biomolecular systems. In this study, we have evaluated the accuracy and efficiency of two approximate NMA protocols-one based on our recently proposed vibrational subsystem analysis (VSA), the other based on the rotation translation block (RTB), in comparison with standard NMA that directly solves a full Hessian matrix. By properly accounting for flexibility within blocks of residues or atoms based on a subsystem-environment partition, VSA-based NMA has attained a much higher accuracy than RTB and much lower computing cost than standard NMA. Therefore, VSA enables accurate and efficient calculations of normal modes from all-atom or coarse-grained potential functions, which promise to improve conformational sampling driven by low-frequency normal modes.

Normal modes for large molecules with arbitrary link constraints in the mobile block Hessian approach

In a previous paper [Ghysels et al., J. Chem. Phys. 126, 224102 (2007)] the mobile block Hessian (MBH) approach was presented. The method was designed to accurately compute vibrational modes of partially optimized molecular structures. The key concept was the introduction of several blocks of atoms, which can move as rigid bodies with respect to a local, fully optimized subsystem. The choice of the blocks was restricted in the sense that none of them could be connected, and also linear blocks were not taken into consideration. In this paper an extended version of the MBH method is presented that is generally applicable and allows blocks to be adjoined by one or two common atoms. This extension to all possible block partitions of the molecule provides a structural flexibility varying from very rigid to extremely relaxed. The general MBH method is very well suited to study selected normal modes of large macromolecules (such as proteins and polymers) because the number of degrees of freedom can be greatly reduced while still keeping the essential motions of the molecular system. The reduction in the number of degrees of freedom due to the block linkages is imposed here directly using a constraint method, in contrast to restraint methods where stiff harmonic couplings are introduced to restrain the relative motion of the blocks. The computational cost of this constraint method is less than that of an implementation using a restraint method. This is illustrated for the alpha-helix conformation of an alanine-20-polypeptide.