Dynamical effects in ab initio NMR calculations: classical force fields fitted to quantum forces

NMR crystallography of amino acids

The development of NMR crystallography methods requires a reliable database of chemical shifts measured for systems with known crystal structure. We measured and assigned carbon and hydrogen chemical shifts of twenty solid natural amino acids of known polymorphic structure, meticulously determined using powder X-ray diffraction. We then correlated the experimental data with DFT-calculated isotropic shieldings. The small size of the unit cell of most amino acids allowed for advanced computations using various families of DFT functionals, including generalized gradient approximation (GGA), meta-GGA and hybrid DFT functionals. We tested several combinations of functionals for geometry optimizations and NMR calculations. For carbon shieldings, the widely used GGA functional PBE performed very well, although an improvement could be achieved by adding shielding corrections calculated for isolated molecules using a hybrid functional. For hydrogen nuclei, we observed the best performance for NMR calculations carried out with structures optimized at the hybrid DFT level. The high fidelity of the calculations made it possible to assign additional signals that could not be assigned based on experiments alone, for example signals of two non-equivalent molecules in the unit cell of some of the amino acids.

Ligand 1 H NMR Chemical Shifts as Accurate Reporters for Protein-Ligand Binding Interfaces in Solution

The availability of high-resolution 3D structural information is crucial for investigating guest-host systems across a wide range of fields. In the context of drug discovery, the information is routinely used to establish and validate structure-activity relationships, grow initial hits from screening campaigns, and to guide molecular docking. For the generation of protein-ligand complex structural information, X-ray crystallography is the experimental method of choice, however, with limited information on protein flexibility. An experimentally verified structural model of the binding interface in the native solution-state would support medicinal chemists in their molecular design decisions. Here we demonstrate that protein-bound ligand 1 H NMR chemical shifts are highly sensitive and accurate probes for the immediate chemical environment of protein-ligand interfaces. By comparing the experimental ligand 1 H chemical shift values with those computed from the X-ray structure using quantum mechanics methodology, we identify significant disagreements for parts of the ligand between the two experimental techniques. We show that quantum mechanics/molecular mechanics (QM/MM) molecular dynamics (MD) ensembles can be used to refine initial X-ray co-crystal structures resulting in a better agreement with experimental 1 H ligand chemical shift values. Overall, our findings highlight the usefulness of ligand 1 H NMR chemical shift information in combination with a QM/MM MD workflow for generating protein-ligand ensembles that accurately reproduce solution structural data.

Importance of Nuclear Quantum Effects for NMR Crystallography

The resolving power of solid-state nuclear magnetic resonance (NMR) crystallography depends heavily on the accuracy of computational predictions of NMR chemical shieldings of candidate structures, which are usually taken to be local minima in the potential energy. To test the limits of this approximation, we systematically study the importance of finite-temperature and quantum nuclear fluctuations for ¹H, ¹³C, and ¹⁵N shieldings in polymorphs of three paradigmatic molecular crystals: benzene, glycine, and succinic acid. The effect of quantum fluctuations is comparable to the typical errors of shielding predictions for static nuclei with respect to experiments, and their inclusion improves the agreement with measurements, translating to more reliable assignment of the NMR spectra to the correct candidate structure. The use of integrated machine-learning models, trained on first-principles energies and shieldings, renders rigorous sampling of nuclear fluctuations affordable, setting a new standard for the calculations underlying NMR structure determinations.

NMR crystallography of molecular organics

Developments of NMR methodology to characterise the structures of molecular organic structures are reviewed, concentrating on the previous decade of research in which density functional theory-based calculations of NMR parameters in periodic solids have become widespread. With a focus on demonstrating the new structural insights provided, it is shown how "NMR crystallography" has been used in a spectrum of applications from resolving ambiguities in diffraction-derived structures (such as hydrogen atom positioning) to deriving complete structures in the absence of diffraction data. As well as comprehensively reviewing applications, the different aspects of the experimental and computational techniques used in NMR crystallography are surveyed. NMR crystallography is seen to be a rapidly maturing subject area that is increasingly appreciated by the wider crystallographic community.

A Bayesian approach to NMR crystal structure determination

Nuclear Magnetic Resonance (NMR) spectroscopy is particularly well suited to determine the structure of molecules and materials in powdered form. Structure determination usually proceeds by finding the best match between experimentally observed NMR chemical shifts and those of candidate structures. Chemical shifts for the candidate configurations have traditionally been computed by electronic-structure methods, and more recently predicted by machine learning. However, the reliability of the determination depends on the errors in the predicted shifts. Here we propose a Bayesian framework for determining the confidence in the identification of the experimental crystal structure, based on knowledge of the typical errors in the electronic structure methods. We demonstrate the approach on the determination of the structures of six organic molecular crystals. We critically assess the reliability of the structure determinations, facilitated by the introduction of a visualization of the similarity between candidate configurations in terms of their chemical shifts and their structures. We also show that the commonly used values for the errors in calculated ¹³C shifts are underestimated, and that more accurate, self-consistently determined uncertainties make it possible to use ¹³C shifts to improve the accuracy of structure determinations. Finally, we extend the recently-developed ShiftML model to render it more efficient, accurate, and, most importantly, to evaluate the uncertainties in its predictions. By quantifying the confidence in structure determinations based on ShiftML predictions we further substantiate that it provides a valid replacement for first-principles calculations in NMR crystallography.

Electron-phonon coupling from finite differences

The interaction between electrons and phonons underlies multiple phenomena in physics, chemistry, and materials science. Examples include superconductivity, electronic transport, and the temperature dependence of optical spectra. A first-principles description of electron-phonon coupling enables the study of the above phenomena with accuracy and material specificity, which can be used to understand experiments and to predict novel effects and functionality. In this topical review, we describe the first-principles calculation of electron-phonon coupling from finite differences. The finite differences approach provides several advantages compared to alternative methods, in particular (i) any underlying electronic structure method can be used, and (ii) terms beyond the lowest order in the electron-phonon interaction can be readily incorporated. But these advantages are associated with a large computational cost that has until recently prevented the widespread adoption of this method. We describe some recent advances, including nondiagonal supercells and thermal lines, that resolve these difficulties, and make the calculation of electron-phonon coupling from finite differences a powerful tool. We review multiple applications of the calculation of electron-phonon coupling from finite differences, including the temperature dependence of optical spectra, superconductivity, charge transport, and the role of defects in semiconductors. These examples illustrate the advantages of finite differences, with cases where semilocal density functional theory is not appropriate for the calculation of electron-phonon coupling and many-body methods such as the GW approximation are required, as well as examples in which higher-order terms in the electron-phonon interaction are essential for an accurate description of the relevant phenomena. We expect that the finite difference approach will play a central role in future studies of the electron-phonon interaction.

The application of tailor-made force fields and molecular dynamics for NMR crystallography: a case study of free base cocaine

Motional averaging has been proven to be significant in predicting the chemical shifts in ab initio solid-state NMR calculations, and the applicability of motional averaging with molecular dynamics has been shown to depend on the accuracy of the molecular mechanical force field. The performance of a fully automatically generated tailor-made force field (TMFF) for the dynamic aspects of NMR crystallography is evaluated and compared with existing benchmarks, including static dispersion-corrected density functional theory calculations and the COMPASS force field. The crystal structure of free base cocaine is used as an example. The results reveal that, even though the TMFF outperforms the COMPASS force field for representing the energies and conformations of predicted structures, it does not give significant improvement in the accuracy of NMR calculations. Further studies should direct more attention to anisotropic chemical shifts and development of the method of solid-state NMR calculations.

Temperature dependence of X-ray absorption and nuclear magnetic resonance spectra: probing quantum vibrations of light elements in oxides

Probing the quantum thermal fluctuations of nuclei in light-element oxides using XANES and NMR spectroscopies.

Enhanced NMR Discrimination of Pharmaceutically Relevant Molecular Crystal Forms through Fragment-Based Ab Initio Chemical Shift Predictions

Chemical shift prediction plays an important role in the determination or validation of crystal structures with solid-state nuclear magnetic resonance (NMR) spectroscopy. One of the fundamental theoretical challenges lies in discriminating variations in chemical shifts resulting from different crystallographic environments. Fragment-based electronic structure methods provide an alternative to the widely used plane wave gauge-including projector augmented wave (GIPAW) density functional technique for chemical shift prediction. Fragment methods allow hybrid density functionals to be employed routinely in chemical shift prediction, and we have recently demonstrated appreciable improvements in the accuracy of the predicted shifts when using the hybrid PBE0 functional instead of generalized gradient approximation (GGA) functionals like PBE. Here, we investigate the solid-state ¹³C and ¹⁵N NMR spectra for multiple crystal forms of acetaminophen, phenobarbital, and testosterone. We demonstrate that the use of the hybrid density functional instead of a GGA provides both higher accuracy in the chemical shifts and increased discrimination among the different crystallographic environments. Finally, these results also provide compelling evidence for the transferability of the linear regression parameters mapping predicted chemical shieldings to chemical shifts that were derived in an earlier study.

Combining solid-state NMR spectroscopy with first-principles calculations - a guide to NMR crystallography

Recent advances in the application of first-principles calculations of NMR parameters to periodic systems have resulted in widespread interest in their use to support experimental measurement. Such calculations often play an important role in the emerging field of "NMR crystallography", where NMR spectroscopy is combined with techniques such as diffraction, to aid structure determination. Here, we discuss the current state-of-the-art for combining experiment and calculation in NMR spectroscopy, considering the basic theory behind the computational approaches and their practical application. We consider the issues associated with geometry optimisation and how the effects of temperature may be included in the calculation. The automated prediction of structural candidates and the treatment of disordered and dynamic solids are discussed. Finally, we consider the areas where further development is needed in this field and its potential future impact.

Modeling Polymorphic Molecular Crystals with Electronic Structure Theory

Interest in molecular crystals has grown thanks to their relevance to pharmaceuticals, organic semiconductor materials, foods, and many other applications. Electronic structure methods have become an increasingly important tool for modeling molecular crystals and polymorphism. This article reviews electronic structure techniques used to model molecular crystals, including periodic density functional theory, periodic second-order Møller-Plesset perturbation theory, fragment-based electronic structure methods, and diffusion Monte Carlo. It also discusses the use of these models for predicting a variety of crystal properties that are relevant to the study of polymorphism, including lattice energies, structures, crystal structure prediction, polymorphism, phase diagrams, vibrational spectroscopies, and nuclear magnetic resonance spectroscopy. Finally, tools for analyzing crystal structures and intermolecular interactions are briefly discussed.

Temperature Dependence of NMR Parameters Calculated from Path Integral Molecular Dynamics Simulations

The influence of temperature on NMR chemical shifts and quadrupolar couplings in model molecular organic solids is explored using path integral molecular dynamics (PIMD) and density functional theory (DFT) calculations of shielding and electric field gradient (EFG) tensors. An approach based on convoluting calculated shielding or EFG tensor components with probability distributions of selected bond distances and valence angles obtained from DFT-PIMD simulations at several temperatures is used to calculate the temperature effects. The probability distributions obtained from the quantum PIMD simulations, which includes nuclear quantum effects, are significantly broader and less temperature dependent than those obtained with conventional DFT molecular dynamics or with 1D scans through the potential energy surface. Predicted NMR observables for the model systems were in excellent agreement with experimental data.

Fine refinement of solid state structure of racemic form of phospho-tyrosine employing NMR Crystallography approach

We present step by step facets important in NMR Crystallography strategy employing O-phospho-dl-tyrosine as model sample. The significance of three major techniques being components of this approach: solid state NMR (SS NMR), X-ray diffraction of powdered sample (PXRD) and theoretical calculations (Gauge Invariant Projector Augmented Wave; GIPAW) is discussed. Each experimental technique provides different set of structural constraints. From the PXRD measurement the size of the unit cell, space group and roughly refined molecular structure are established. SS NMR provides information about content of crystallographic asymmetric unit, local geometry, molecular motion in the crystal lattice and hydrogen bonding pattern. GIPAW calculations are employed for validation of quality of elucidation and fine refinement of structure. Crystal and molecular structure of O-phospho-dl-tyrosine solved by NMR Crystallography is deposited at Cambridge Crystallographic Data Center under number CCDC 1005924.

Fine refinement of solid-state molecular structures of Leu- and Met-enkephalins by NMR crystallography

This paper presents a methodology that allows the fine refinement of the crystal and molecular structure for compounds for which the data deposited in the crystallographic bases are of poor quality. Such species belong to the group of samples with molecular disorder. In the Cambridge Crystallographic Data Center (CCDC), there are approximately 22,000 deposited structures with an R-factor over 10. The powerful methodology we present employs crystal data for Leu-enkephalin (two crystallographic forms) with R-factor values of 14.0 and 8.9 and for Met-enkephalin (one form) with an R-factor of 10.5. NMR crystallography was employed in testing the X-ray data and the quality of the structure refinement. The GIPAW (gauge invariant projector augmented wave) method was used to optimize the coordinates of the enkephalins and to compute NMR parameters. As we reveal, this complementary approach makes it possible to generate a reasonable set of new coordinates that better correlate to real samples. This methodology is general and can be employed in the study of each compound possessing magnetically active nuclei.

An extrapolation method for the efficient calculation of molecular response properties within Born-Oppenheimer molecular dynamics

The calculation of molecular response properties in dynamic molecular systems is a major challenge that requires sampling over many steps of, e.g., Born-Oppenheimer molecular dynamics (BO-MD) simulations. We present an extrapolation scheme to accelerate such calculations for multiple steps within BO-MD trajectories or equivalently within other sampling methods of conformational space. The extrapolation scheme is related to the one introduced by Pulay and Fogarasi [Chem. Phys. Lett., 2004, 386, 272] for self-consistent field (SCF) energy calculations. We extend the extrapolation to the quantities within our density matrix-based Laplace-transformed coupled perturbed SCF (DL-CPSCF) method that allows for linear-scaling calculations of response properties for large molecular systems. Here, we focus on the example of calculating NMR chemical shifts for which the number of required DL-CPSCF iterations reduces by roughly 40-70%.

Contribution of high-energy conformations to NMR chemical shifts, a DFT-BOMD study

This paper highlights the relevance of including the high-energy conformational states sampled by Born-Oppenheimer molecular dynamics (BOMD) in the calculation of time-averaged NMR chemical shifts. Our case study is the very flexible glycerol molecule that undergoes interconversion between conformers in a nonrandom way. Along the sequence of structures from one backbone conformer to another, transition states have been identified. The three (13)C NMR chemical shifts of the molecule were estimated by averaging their calculated values over a large set of BOMD snapshots. The simulation time needed to obtain a good agreement with the two signals present in the experimental spectrum is shown to be dependent on the atomic orbital basis set used for the dynamics, with a necessary longer trajectory for the most extended basis sets. The large structural deformations with respect to the optimized conformer geometries that occur along the dynamics are related to a kinetically driven conformer distribution. Calculated conformer type populations are in good agreement with experimental gas phase microwave results.

Errors in the calculation of (27)Al nuclear magnetic resonance chemical shifts

Computational chemistry is an important tool for signal assignment of 27Al nuclear magnetic resonance spectra in order to elucidate the species of aluminum(III) in aqueous solutions. The accuracy of the popular theoretical models for computing the 27Al chemical shifts was evaluated by comparing the calculated and experimental chemical shifts in more than one hundred aluminum(III) complexes. In order to differentiate the error due to the chemical shielding tensor calculation from that due to the inadequacy of the molecular geometry prediction, single-crystal X-ray diffraction determined structures were used to build the isolated molecule models for calculating the chemical shifts. The results were compared with those obtained using the calculated geometries at the B3LYP/6-31G(d) level. The isotropic chemical shielding constants computed at different levels have strong linear correlations even though the absolute values differ in tens of ppm. The root-mean-square difference between the experimental chemical shifts and the calculated values is approximately 5 ppm for the calculations based on the X-ray structures, but more than 10 ppm for the calculations based on the computed geometries. The result indicates that the popular theoretical models are adequate in calculating the chemical shifts while an accurate molecular geometry is more critical.

Interrogation of the protein-protein interactions between human BRCA2 BRC repeats and RAD51 reveals atomistic determinants of affinity

The breast cancer suppressor BRCA2 controls the recombinase RAD51 in the reactions that mediate homologous DNA recombination, an essential cellular process required for the error-free repair of DNA double-stranded breaks. The primary mode of interaction between BRCA2 and RAD51 is through the BRC repeats, which are ∼35 residue peptide motifs that interact directly with RAD51 in vitro. Human BRCA2, like its mammalian orthologues, contains 8 BRC repeats whose sequence and spacing are evolutionarily conserved. Despite their sequence conservation, there is evidence that the different human BRC repeats have distinct capacities to bind RAD51. A previously published crystal structure reports the structural basis of the interaction between human BRC4 and the catalytic core domain of RAD51. However, no structural information is available regarding the binding of the remaining seven BRC repeats to RAD51, nor is it known why the BRC repeats show marked variation in binding affinity to RAD51 despite only subtle sequence variation. To address these issues, we have performed fluorescence polarisation assays to indirectly measure relative binding affinity, and applied computational simulations to interrogate the behaviour of the eight human BRC-RAD51 complexes, as well as a suite of BRC cancer-associated mutations. Our computational approaches encompass a range of techniques designed to link sequence variation with binding free energy. They include MM-PBSA and thermodynamic integration, which are based on classical force fields, and a recently developed approach to computing binding free energies from large-scale quantum mechanical first principles calculations with the linear-scaling density functional code onetep. Our findings not only reveal how sequence variation in the BRC repeats directly affects affinity with RAD51 and provide significant new insights into the control of RAD51 by human BRCA2, but also exemplify a palette of computational and experimental tools for the analysis of protein-protein interactions for chemical biology and molecular therapeutics.

The atomic scale interactions that occur at the interfaces between proteins are fundamental to all biological processes. One such critical interface is formed between the proteins, human BRCA2 and RAD51. BRCA2 binds to and delivers RAD51 to sites of DNA damage, where RAD51 mediates the error-free repair of double-stranded DNA breaks. Mutations in BRCA2 have been linked to breast cancer predisposition. Therefore, an accurate picture of the interactions between these two proteins is of great importance. BRCA2 interacts with RAD51 via eight “BRC repeats” that are similar, but not identical, in sequence. Due to lack of experimental structural information regarding the binding of seven of the eight BRC repeats to RAD51, it is unknown how subtle sequence variations in the repeats translate to measurable variations in their binding affinity. We have used a range of computational methods, firstly based on classical force fields, and secondly based on first principles quantum mechanical techniques whose computational cost scales linearly with the number of atoms, allowing us to perform calculations on the entire protein complex. This is the first study comparing all eight BRC repeats at the atomic scale and our results provide critical insights into the control of RAD51 by human BRCA2.

The PAW/GIPAW approach for computing NMR parameters: a new dimension added to NMR study of solids

In 2001, Mauri and Pickard introduced the gauge including projected augmented wave (GIPAW) method that enabled for the first time the calculation of all-electron NMR parameters in solids, i.e. accounting for periodic boundary conditions. The GIPAW method roots in the plane wave pseudopotential formalism of the density functional theory (DFT), and avoids the use of the cluster approximation. This method has undoubtedly revitalized the interest in quantum chemical calculations in the solid-state NMR community. It has quickly evolved and improved so that the calculation of the key components of NMR interactions, namely the shielding and electric field gradient tensors, has now become a routine for most of the common nuclei studied in NMR. Availability of reliable implementations in several software packages (CASTEP, Quantum Espresso, PARATEC) make its usage more and more increasingly popular, maybe indispensable in near future for all material NMR studies. The majority of nuclei of the periodic table have already been investigated by GIPAW, and because of its high accuracy it is quickly becoming an essential tool for interpreting and understanding experimental NMR spectra, providing reliable assignments of the observed resonances to crystallographic sites or enabling a priori prediction of NMR data. The continuous increase of computing power makes ever larger (and thus more realistic) systems amenable to first-principles analysis. In the near future perspectives, as the incorporation of dynamical effects and/or disorder are still at their early developments, these areas will certainly be the prime target.

First Principles-Based Calculations of Free Energy of Binding: Application to Ligand Binding in a Self-Assembling Superstructure

The accurate prediction of ligand binding affinities to a protein remains a desirable goal of computational biochemistry. Many available methods use molecular mechanics (MM) to describe the system, however, MM force fields cannot fully describe the complex interactions involved in binding, specifically electron transfer and polarization. First principles approaches can fully account for these interactions, and with the development of linear-scaling first principles programs, it is now viable to apply first principles calculations to systems containing tens of thousands of atoms. In this paper, a quantum mechanical Poisson-Boltzmann surface area approach is applied to a model of a protein-ligand binding cavity, the "tennis ball" dimer. Results obtained from this approach demonstrate considerable improvement over conventional molecular mechanics Poisson-Boltzmann surface area due to the more accurate description of the interactions in the system. For the first principles calculations in this study, the linear-scaling density functional theory program ONETEP is used, allowing the approach to be applied to receptor-ligand complexes of pharmaceutical interest that typically include thousands of atoms.