Fokker-Planck formalism in magnetic resonance simulations

Spatially encoded pure-shift diffusion-ordered NMR spectroscopy yielded by chirp excitation

Spatially-encoded diffusion-ordered NMR spectroscopy (SPEN-DOSY) has emerged as a new time-efficient tool for the analysis of mixtures of small molecules in solution. Time efficiency is achieved using the concept of spatial parallelization of the effective gradient area, instead of the sequential incrementation used in conventional diffusion experiments. The data acquired with such sequences are then usually processed to extract diffusion coefficients, but cases when peak overlap in the 1H spectrum are difficult to address. Such limitation in conventional diffusion experiments is addressed via using the Pure Shift Yielded by CHirp Excitation (PSYCHE)-iDOSY sequence. Here we have adapted the PSYCHE-iDOSY sequence by using echo planar spectroscopic imaging (EPSI) to acquire SPEN-DOSY data. The pure shift mode of PSYCHE separates the overlapped components and a modified Stejskal-Tanner equation is used to extract the corresponding diffusion coefficient. The primary results obtained with the above-mentioned mixtures seem to open the possibility of separating complex mixtures in less time than PSYCHE-iDOSY.

Colloidal pathways of amorphous calcium carbonate formation lead to distinct water environments and conductivity

CaCO3 is the most abundant biomineral and a major constituent of incrustations arising from water hardness. Polycarboxylates play key roles in controlling mineralization. Herein, we present an analytical and spectroscopic study of polycarboxylate-stabilized amorphous CaCO3 (ACC) and its formation via a dense liquid precursor phase (DLP). Polycarboxylates facilitate pronounced, kinetic bicarbonate entrapment in the DLP. Since bicarbonate is destabilized in the solid state, DLP dehydration towards solid ACC necessitates the formation of locally calcium deficient sites, thereby inhibiting nucleation. Magic-angle spinning (MAS) nuclear magnetic resonance (NMR) spectroscopy of poly-aspartate-stabilized ACC reveals the presence of two distinct environments. The first contains immobile calcium and carbonate ions and structural water molecules, undergoing restricted, anisotropic motion. In the second environment, water molecules undergo slow, but isotropic motion. Indeed, conductive atomic force microscopy (C-AFM) reveals that ACC conducts electrical current, strongly suggesting that the mobile environment pervades the bulk of ACC, with dissolved hydroxide ions constituting the charge carriers. We propose that the distinct environments arise from colloidally stabilized interfaces of DLP nanodroplets, consistent with the pre-nucleation cluster (PNC) pathway.

Accounting for gradient non-uniformity in spatially-encoded diffusion-ordered NMR spectroscopy

Diffusion-ordered NMR spectroscopy (DOSY) is a powerful tool for the analysis of mixtures. Spatially-encoded (SPEN) DOSY makes it possible to collect a complete DOSY data set in a single scan, through spatial parallelisation of the gradient dimension. One limitation of current SPEN DOSY experiments is that the data is analysed assuming that the field gradient is uniform over the sample. This is usually not the case for high resolution NMR probes, and even less for triple-axis gradient probes. In this work, we have developed methods to account for gradient non-uniformity in the processing of SPEN DOSY experiment. We have first mapped the field gradient, using a stimulated echo (STE) NMR sequence with a weak readout gradient. We have then modified the calculation of the position-dependent effective gradient pulse area that is used in the analysis of SPEN DOSY data. The resulting model was validated through numerical simulations. A comparison of results obtained with and without inclusion of the effect of non-uniform gradients shows that the proposed approach increases the accuracy of SPEN DOSY experiments.

The solid effect of dynamic nuclear polarization in liquids

The solid-state effect of dynamic nuclear polarization (DNP) is operative also in viscous liquids where the dipolar interaction between the electronic and nuclear spins is partially averaged. The proper way to quantify the degree of averaging, and thus calculate the efficiency of the effect, should be based on the time-correlation function of the dipolar interaction. Here we use the stochastic Liouville equation formalism to develop a general theoretical description of the solid effect in liquids. The derived expressions can be used with different dipolar correlations functions depending on the assumed motional model. At high magnetic fields, the theory predicts DNP enhancements at small offsets, far from the classical solid-effect positions that are displaced by one nuclear Larmor frequency from the electronic resonance. The predictions are in quantitative agreement with such enhancement peaks observed at 9.4 T . These non-canonical peaks are not due to thermal mixing or the cross effect but exactly follow the dispersive component of the EPR line.

Theoretical analysis of flow effects in spatially encoded diffusion NMR

The measurement of translational diffusion coefficients by nuclear magnetic resonance (NMR) spectroscopy is essential in a broad range of fields, including organic, inorganic, polymer, and supramolecular chemistry. It is also a powerful method for mixture analysis. Spatially encoded diffusion NMR (SPEN DNMR)" is a time efficient technique to collect diffusion NMR data, which is particularly relevant for the analysis of samples that evolve in time. In many cases, motion other than diffusion is present in NMR samples. This is, for example, the case of flow NMR experiments, such as in online reaction monitoring and in the presence of sample convection. Such motion is deleterious for the accuracy of DNMR experiments in general and for SPEN DNMR in particular. Limited theoretical understanding of flow effects in SPEN DNMR experiments is an obstacle for their broader experimental implementation. Here, we present a detailed theoretical analysis of flow effects in SPEN DNMR and of their compensation, throughout the relevant pulse sequences. This analysis is validated by comparison with numerical simulation performed with the Fokker-Planck formalism. We then consider, through numerical simulation, the specific cases of constant, laminar, and convection flow and the accuracy of SPEN DNMR experiments in these contexts. This analysis will be useful for the design and implementation of fast diffusion NMR experiments and for their applications.

Field and magic angle spinning frequency dependence of proton resonances in rotating solids

Proton detection in solid state NMR is continuously developing and allows one to gain new insights in structural biology. Overall, this progress is a result of the synergy between hardware development, new NMR methodology and new isotope labeling strategies, to name a few factors. Even though current developments are rapid, it is worthwhile to summarize what can currently be achieved employing proton detection in biological solids. We illustrate this by analysing the signal-to-noise ratio (SNR) for spectra obtained for a microcrystalline α-spectrin SH3 domain protein sample by (i) employing different degrees of chemical dilution to replace protons by incorporating deuterons in different sites, by (ii) variation of the magic angle spinning (MAS) frequencies between 20 and 110 kHz, and by (iii) variation of the static magnetic field B₀. The experimental SNR values are validated with numerical simulations employing up to 9 proton spins. Although in reality a protein would contain far more than 9 protons, in a deuterated environment this is a sufficient number to achieve satisfactory simulations consistent with the experimental data. The key results of this analysis are (i) with current hardware, deuteration is still necessary to record spectra of optimum quality; (ii) 13CH3 isotopomers for methyl groups yield the best SNR when MAS frequencies above 100 kHz are available; and (iii) sensitivity increases with a factor beyond B0 3/2 with the static magnetic field due to a transition of proton-proton dipolar interactions from a strong to a weak coupling limit.

Theory and Least Squares Fitting of CW ESR Saturation Spectra Using the MOMD Model

CW saturation experiments are widely used in ESR studies of relaxation processes in proteins and lipids. We develop the theory of saturation in ESR spectra in terms of its close relation with that of 2D-ELDOR. Our treatment of saturation is then based on the microscopic order macroscopic disorder (MOMD) model and can be used to fit the full CW saturation spectrum, rather than fitting just the peak-peak amplitude as a function of microwave field B ₁ as is commonly done. This requires fewer experiments to yield effects on T ₁, as well as provides a more extensive dynamic structural picture, for example, for scanning experiments on different protein sites. The code is released as a publicly available software package in Python that can be used to fit CW saturation spectra from biological samples of interest.

Quadratic spacing of the effective gradient area for spatially encoded diffusion NMR

Diffusion NMR experiments rely on the measurement of signal attenuation as a function of the area of diffusion-encoding pulsed magnetic-field gradients. In conventional experiments, arbitrary series of gradient values can be used, and different gradient spacing strategies have different advantages. Ultrafast diffusion NMR relies on the spatial parallelisation of effective gradient area values to collect full 2D diffusion data sets in a single scan. Until recently, only linear spacing was available. We have shown that quadratic spacing can be achieved using a tailored frequency swept pulse. Here we describe the design of the pulse and validate it with numerical spin simulations, that make it possible to check the effect of the quadratic spacing pulse at different stages of the pulse sequence. We also show that quadratic spacing makes it possible to use a recently reported analysis method for diffusion NMR, the Matrix Pencil Method. We describe the results obtained with the MPM and those obtained with the direct exponential curve resolution algorithm (DECRA), which also requires quadratic gradient spacing. Overall, these developments open new opportunities for applications of spatially encoded diffusion experiments, such as ultrafast DOSY NMR and ultrafast Laplace NMR.

High-field solution state DNP using cross-correlations

At the magnetic fields of common NMR instruments, electron Zeeman frequencies are too high for efficient electron-nuclear dipolar cross-relaxation to occur in solution. The rate of that process fades with the electron Zeeman frequency as ω^-2 - in the absence of isotropic hyperfine couplings, liquid state dynamic nuclear polarisation (DNP) in high-field magnets is therefore impractical. However, contact coupling and dipolar cross-relaxation are not the only mechanisms that can move electron magnetisation to nuclei in liquids: multiple cross-correlated (CC) relaxation processes also exist, involving various combinations of interaction tensor anisotropies. The rates of some of those processes have more favourable high-field behaviour than dipolar cross-relaxation, but due to the difficulty of their numerical - and particularly analytical - treatment, they remain largely uncharted. In this communication, we report analytical evaluation of every rotationally driven relaxation process in liquid state for 1e1n and 2e1n spin systems, as well as numerical optimisations of the steady-state DNP with respect to spin Hamiltonian parameters. A previously unreported cross-correlated DNP (CCDNP) mechanism was identified for the 2e1n system, involving multiple relaxation interference effects and inter-electron exchange coupling. Using simulations, we found realistic spin Hamiltonian parameters that yield stronger nuclear polarisation at high magnetic fields than dipolar cross-relaxation.

NMR Spin Relaxation Theory of Biomolecules Undergoing Highly Asymmetric Exchange with Large Interaction Partners

The transient interactions of proteins and other molecules with much larger structures, such as synthetic or biological nanoparticles, lead to certain types of enhanced nuclear magnetic resonance (NMR) spin relaxation effects, which can be accurately measured by multidimensional solution NMR techniques. These relaxation effects provide new information about the nanostructures and the protein, their interactions, internal dynamics, and associated kinetic and thermodynamic parameters, such as exchange rates and populations. Although theoretical treatments exist that cover either the fast or slow exchange limits, a theoretical treatment that applies to all practically relevant exchange processes is still missing. A unified theoretical framework is presented for this purpose based on a stochastic Liouville equation (SLE). It covers nuclear spin dynamics, overall rotational diffusion of both the protein and the nanostructure, the exchange process between a free state and a bound state, and internal protein dynamics. Although the numerical implementation of the SLE typically involves large matrices, it is shown here that it is computationally still tractable for situations relevant in practice. Application of the theory demonstrates how transverse relaxation is substantially impacted by the kinetics of binding on a wide range of exchange timescales. It is further shown that when exchange occurs on the appropriate timescale, transverse relaxation is able to report on internal dynamics far slower than observable by traditional transverse relaxation experiments. The SLE will allow the realistic and quantitative interpretation of experimental NMR data reporting about transient protein-nanoparticle interactions, thereby providing a powerful tool for the characterization of protein dynamics modes on a vast range of timescales including motions that may be functionally relevant.

Microsecond dynamics in proteins by two-dimensional ESR. II. Addressing computational challenges

Two-dimensional electron-electron double resonance (2D-ELDOR) provides extensive insight into molecular motions. Recent developments permitting experiments at higher frequencies (95 GHz) provide molecular orientational resolution, enabling a clearer description of the nature of the motions. In previous work, we provided simulations for the case of domain motions within proteins that are themselves slowly tumbling in a solution. In order to perform these simulations, it was found that the standard approach of solving the relevant stochastic Liouville equation using the efficient Lanczos algorithm for this case breaks down, so algorithms were employed that rely on the Arnoldi iteration. While they lead to accurate simulations, they are very time-consuming. In this work, we focus on a variant known as the rational Arnoldi algorithm. We show that this can achieve a significant reduction in computation time. The stochastic Liouville matrix, which is of very large dimension, N, is first reduced to a much smaller dimension, m, e.g., from N ∼ O(10⁴) to m ∼ 60, that spans the relevant Krylov subspace from which the spectrum is predicted. This requires the selection of the m frequency shifts to be utilized. A method of adaptive shift choice is introduced to optimize this selection. We also find that these procedures help in optimizing the pruning procedure that greatly reduces the dimension of the initial N dimensional stochastic Liouville matrix in such subsequent computations.

-Frequency-swept pulses for ultrafast spatially encoded NMR

Ultrafast NMR based on spatial encoding yields arbitrary multidimensional spectra in a single scan. The dramatic acceleration afforded by spatial parallelisation makes it possible to capture transient species and processes, and has notably been applied to the monitoring of reactions and the analysis of hyperpolarised species. At the heart of ultrafast NMR lies the spatially sequential manipulation of nuclear spins. This is virtually always achieved by combining a swept radio-frequency pulse with a magnetic field gradient pulse. The dynamics of nuclear spins during these pulse sequence elements is key to understand and design ultrafast NMR experiments, and can often be described by surprisingly simple models. This article describes the spatial encoding of relaxation, chemical shift and diffusion in a common framework and discusses directions for future developments.

Markovian exchange phenomena in magnetic resonance and the Lindblad equation

Nuclear Magnetic Resonance (NMR) experiments involve coherent and incoherent dynamics. Incoherent dynamics give rise to nuclear spin relaxation typically accounted for by a differential equation known as a quantum master equation. Out of the existing master equations the Lindblad equation is particularly important. A central theorem of open quantum theory states that a relaxation process represents a quantum Markovian process if and only if it can be described by a Lindblad master equation. This in turn raises serious questions regarding the compatibility of conventional NMR relaxation theory and one of the most fundamental results of open quantum theory. The NMR treatment of important Markovian relaxation mechanisms such as Brownian motion or chemical exchange events do in general not result in a master equation in Lindblad form and would therefore be at odds with the mathematically rigorous theory of open quantum systems. We explore this apparent conflict for two important chemical exchange phenomena and show that their conventional treatment may be recast into a suitable Lindblad equation. Similarities between spatial diffusion and multi-site exchange events allow us to extend our results to nuclear spin relaxation due to translational diffusion. As a result we are able to show that the NMR treatment of intramolecular exchange and translational diffusion represent valid quantum Markovian processes compatible with the rigorous definitions of open quantum theory.

Spin dynamics during chirped pulses: applications to homonuclear decoupling and broadband excitation

Swept-frequency pulses have found applications in a wide range of areas including spectroscopic techniques where efficient control of spins is required. For many of these applications, a good understanding of the evolution of spin systems during these pulses plays a vital role, not only in describing the mechanism of techniques, but also in enabling new methodologies. In magnetic resonance spectroscopy, broadband inversion, refocusing, and excitation using these pulses are among the most used applications in NMR, ESR, MRI, and in vivo MRS. In the present survey, a general expression for chirped pulses will be introduced, and some numerical approaches to calculate the spin dynamics during chirped pulses via solutions of the well-known Liouville-von Neumann equation and the lesser-explored Wei-Norman Lie algebra along with comprehensive examples are presented. In both cases, spin state trajectories are calculated using the solution of differential equations. Additionally, applications of the proposed methods to study the spin dynamics during the PSYCHE pulse element for broadband homonuclear decoupling and the CHORUS sequence for broadband excitation will be presented.

Large-scale magnetic resonance simulations: A tutorial

Computational modeling is becoming an essential tool in magnetic resonance to design and optimize experiments, test the performance of theoretical models, and interpret experimental data. Recent theoretical research and software development made possible simulations of large spin systems, for example, proteins with thousands of spins, in reasonable time. In the last few years, the Fokker-Planck formalism also re-emerged due to its ability to handle spatial dynamics. The purpose of this tutorial is to describe advantages and disadvantages of the most common formalisms, the latest developments and strategies to improve the computational efficiency, and to guide users in the setting up of a simulation using the Spinach software.

Microsecond dynamics in proteins by two-dimensional ESR: Predictions

Two-dimensional electron-electron double resonance (2D-ELDOR) provides extensive insight into molecular motions. Recent developments permitting experiments at higher frequencies (95 GHz) provide molecular orientational resolution, enabling a clearer description of the nature of the motions. In this work, simulations are provided for the example of domain motions within proteins that are themselves slowly tumbling in solution. These show the nature of the exchange cross-peaks that are predicted to develop in real time from such domain motions. However, we find that the existing theoretical methods for computing 2D-ELDOR experiments over a wide motional range begin to fail seriously when applied to very slow motions characteristic of proteins in solution. One reason is the failure to obtain accurate eigenvectors and eigenvalues of the complex symmetric stochastic Liouville matrices describing the experiment when computed by the efficient Lanczos algorithm in the range of very slow motion. Another, perhaps more serious, issue is that these matrices are "non-normal," such that for the very slow motional range even rigorous diagonalization algorithms do not yield the correct eigenvalues and eigenvectors. We have employed algorithms that overcome both these issues and lead to valid 2D-ELDOR predictions even for motions approaching the rigid limit. They are utilized to describe the development of cross-peaks in 2D-ELDOR at 95 GHz for a particular case of domain motion.

A master equation for spin systems far from equilibrium

The quantum dynamics of spin systems is often treated by a differential equation known as the master equation, which describes the trajectories of spin observables such as magnetization components, spin state populations, and coherences between spin states. The master equation describes how a perturbed spin system returns to a state of thermal equilibrium with a finite-temperature environment. The conventional master equation, which has the form of an inhomogeneous differential equation, applies to cases where the spin system remains close to thermal equilibrium, which is well satisfied for a wide variety of magnetic resonance experiments conducted on thermally polarized spin systems at ordinary temperatures. However, the conventional inhomogeneous master equation may fail in the case of hyperpolarized spin systems, when the spin state populations deviate strongly from thermal equilibrium, and in general where there is a high degree of nuclear spin order. We highlight a simple case in which the inhomogeneous master equation clearly fails, and propose an alternative master equation based on Lindblad superoperators which avoids most of the deficiencies of previous proposals. We discuss the strengths and limitations of the various formulations of the master equation, in the context of spin systems which are far from thermal equilibrium. The method is applied to several problems in nuclear magnetic resonance and to spin-isomer conversion.

Quo vadis EPR?

Complexity of paramagnetic catalysts and materials increases, and the same applies to systems targeted by integrative structural biology. Hence, EPR spectroscopists must find ways to enhance information content of their data. I argue that a third major wave of method development in EPR spectroscopy, which is triggered by recent advances in digital electronics and computing, can achieve this. Transfer of NMR methods to EPR will go on, but part of the new EPR methodology will depend on completely new concepts.

Quantifying proton NMR coherent linewidth in proteins under fast MAS conditions: a second moment approach

Proton detected solid-state NMR under fast magic-angle-spinning (MAS) conditions is currently redefining the applications of solid-state NMR, in particular in structural biology. Understanding the contributions to the spectral linewidth is thereby of paramount importance. When disregarding the sample-dependent inhomogeneous contributions, the NMR proton linewidth is defined by homogeneous broadening, which has incoherent and coherent contributions. Understanding and disentangling these different contributions in multi-spin systems like proteins is still an open issue. The coherent contribution is mainly caused by the dipolar interaction under MAS and is determined by the molecular structure and the proton chemical shifts. Numerical simulation approaches based on numerically exact direct integration of the Liouville-von Neumann equation can give valuable information about the lineshape, but are limited to small spin systems (<12 spins). We present an alternative simulation method for the coherent contributions based on the rapid and partially analytic calculation of the second moments of large spin systems. We first validate the method on a simple system by predicting the ¹⁹F linewidth in CaF₂ under MAS. We compare simulation results to experimental data for microcrystalline ubiquitin (deuterated 100% back-exchanged at 110 kHz and fully-protonated at 125 kHz). Our results quantitatively explain the observed linewidth per-residue basis for the vast majority of residues.

Quantum mechanical MRI simulations: Solving the matrix dimension problem

We propose a solution to the matrix dimension problem in quantum mechanical simulations of MRI (magnetic resonance imaging) experiments on complex molecules. This problem is very old; it arises when Kronecker products of spin operators and spatial dynamics generators are taken-the resulting matrices are far too large for any current or future computer. However, spin and spatial operators individually have manageable dimensions, and we note here that the action by their Kronecker products on any vector may be computed without opening those products. This eliminates large matrices from the simulation process. MRI simulations for coupled spin systems of complex metabolites in three dimensions with diffusion, flow, chemical kinetics, and quantum mechanical treatment of spin relaxation are now possible. The methods described in this paper are implemented in versions 2.4 and later of the Spinach library.

Diffusional attenuation during soft pulses: A Zangger-Sterk pure shift iDOSY experiment

Diffusion-ordered spectroscopy experiments in which existing delays in a parent pulse sequence are used for diffusion encoding - iDOSY experiments - are potentially attractive because of their simplicity and sensitivity. However the calculation of diffusional attenuation in Zangger-Sterk pure shift iDOSY experiments is a very difficult problem to attack analytically, and is more easily approached numerically. Numerical simulations show that for typical experimental conditions, the dependence of diffusional attenuation on diffusion-encoding gradient amplitude is well represented by a shifted Gaussian function. The shift in gradient can be calculated analytically for the limiting case where the selective pulse is replaced by a hard 180° pulse at its midpoint; numerical simulations show that the effect of using different shapes of selective pulse is to scale down this limiting gradient shift by a constant factor that depends on the pulse shape used. The practical consequence is that under the experimental conditions appropriate for small molecules, the pure shift iDOSY method should allow good diffusion coefficient measurements to be made if appropriate allowance is made for the change in effective diffusion-encoding gradient. Parallel sets of numerical simulations and experiments are presented, and a practical application of a Zangger-Sterk pure shift iDOSY experiment to a simple test mixture is illustrated.

Spin-Scenario: A flexible scripting environment for realistic MR simulations

In this paper we present a new open source package, Spin-Scenario, aimed at developing an intuitive, flexible and unique scripting framework able to cover many aspects of simulations in both MR imaging and MR spectroscopy. For this purpose, we adopted the Liouville space model as the standard computing engine and let the consequent computational burden be afforded by parallel computing techniques. Benefitting from the powerful Lua scripting language, the pulse sequence programming syntax was specially designed to offer an extremely concise way of scripting. Moreover, the built-in dataflow graph based optimal control scheme enables an efficient optimization of shaped pulses or multiple cooperative pulses for real-life experiment evaluations. As the name states, the users are expected to be able to realize their creative ideas like a scenarist that creates a scenario script and looks at the spin actors acting accordingly. The validation of the framework was demonstrated with several examples within MR imaging and MR spectroscopy. Spin-Scenario is available for download at https://github.com/spin-scenario.

14N overtone NMR under MAS: Signal enhancement using cross-polarization methods

Polarization transfer methods are widely adopted for the purpose of correlating different nuclear species as well as to achieve signal enhancement. Polarization transfer from ¹H to the ¹⁴N overtone transition (Δm = 2) can be achieved using cross polarization methods under magic-angle spinning conditions, where spin locks of the order of several milliseconds can be obtained on common bio-solids (α-glycine and N-acetylvaline). Signal enhancement factors up to 4.4 per scan, can be achieved under favorable conditions, despite MHz-sized quadrupolar interaction. Moreover, we present a detailed theoretical treatment and accurate numerical simulations which are in excellent agreement the unusual experimental matching conditions observed for cross-polarization to ¹⁴N overtone.

Refocusing CSA during magic angle spinning rotating-frame relaxation experiments

We examine coherent evolution of spin-locked magnetization during magic-angle spinning (MAS), in the context of relaxation experiments designed to probe chemical exchange (rotating-frame relaxation (R_1ρ)). Coherent evolution is expected in MAS based rotating-frame relaxation decay experiments if matching conditions are met (such as, ω₁ = nω_r) and if the chemical shielding anisotropy (CSA) is substantial. We show here using numerical simulations and experiments that even when such matching requirements are avoided (e.g., ω₁ < 0.5ω_r, ∼1.5ω_r, >2.5ω_r), coherent evolution of spin-locked magnetization with large CSA is still considerable. The coherent evolution has important consequences on the analysis of relaxation decay and the ability to extract accurate information of interest about dynamics. We present a pulse sequence that employs rotary echoes and refocuses CSA contributions, allowing for more sensitive measurement of rotating-frame relaxation with less interference from coherent evolution. In practice, the proposed pulse sequence, REfocused CSA Rotating-frame Relaxation (RECRR) is robust to carrier frequency offset, B₁-field inhomogeneity, and slight miscalibrations of the refocusing pulses.

Ultrafast Maximum-Quantum NMR Spectroscopy for the Analysis of Aromatic Mixtures

Maximum-quantum (MaxQ) NMR experiments have been introduced to overcome issues related to peak overlap and high spectral density in the NMR spectra of aromatic mixtures. In MaxQ NMR, spin systems are separated on the basis of the highest-quantum coherence that they can form. MaxQ experiments are however time consuming and methods have been introduced to accelerate them. In this article, we demonstrate the ultrafast, single-scan acquisition of MaxQ NMR spectra using spatial encoding of the multiple-quantum dimension. So far, the spatial encoding methodology has been applied only for the encoding of up to double-quantum coherences, and here we show that it can be extended to higher coherence orders, to yield a massive reduction of the acquisition time of multi-quantum spectra of aromatic mixtures, and also to monitor chemical reactions.

Deep neural network processing of DEER data

The established model-free methods for the processing of two-electron dipolar spectroscopy data [DEER (double electron-electron resonance), PELDOR (pulsed electron double resonance), DQ-EPR (double-quantum electron paramagnetic resonance), RIDME (relaxation-induced dipolar modulation enhancement), etc.] use regularized fitting. In this communication, we describe an attempt to process DEER data using artificial neural networks trained on large databases of simulated data. Accuracy and reliability of neural network outputs from real experimental data were found to be unexpectedly high. The networks are also able to reject exchange interactions and to return a measure of uncertainty in the resulting distance distributions. This paper describes the design of the training databases, discusses the training process, and rationalizes the observed performance. Neural networks produced in this work are incorporated as options into Spinach and DeerAnalysis packages.

1H line width dependence on MAS speed in solid state NMR - Comparison of experiment and simulation

Recent developments in magic angle spinning (MAS) technology permit spinning frequencies of ≥100 kHz. We examine the effect of such fast MAS rates upon nuclear magnetic resonance proton line widths in the multi-spin system of β-Asp-Ala crystal. We perform powder pattern simulations employing Fokker-Plank approach with periodic boundary conditions and ¹H-chemical shift tensors calculated using the bond polarization theory. The theoretical predictions mirror well the experimental results. Both approaches demonstrate that homogeneous broadening has a linear-quadratic dependency on the inverse of the MAS spinning frequency and that, at the faster end of the spinning frequencies, the residual spectral line broadening becomes dominated by chemical shift distributions and susceptibility effects even for crystalline systems.

Large-scale NMR simulations in liquid state: A tutorial

Liquid state nuclear magnetic resonance is the only class of magnetic resonance experiments for which the simulation problem is solved comprehensively for spin systems of any size. This paper contains a practical walkthrough for one of the many available simulation packages - Spinach. Its unique feature is polynomial complexity scaling: the ability to simulate large spin systems quantum mechanically and with accurate account of relaxation, diffusion, chemical processes, and hydrodynamics. This paper is a gentle introduction written with a PhD student in mind.

Dynamic nuclear polarization for sensitivity enhancement in modern solid-state NMR

The field of dynamic nuclear polarization has undergone tremendous developments and diversification since its inception more than 6 decades ago. In this review we provide an in-depth overview of the relevant topics involved in DNP-enhanced MAS NMR spectroscopy. This includes the theoretical description of DNP mechanisms as well as of the polarization transfer pathways that can lead to a uniform or selective spreading of polarization between nuclear spins. Furthermore, we cover historical and state-of-the art aspects of dedicated instrumentation, polarizing agents, and optimization techniques for efficient MAS DNP. Finally, we present an extensive overview on applications in the fields of structural biology and materials science, which underlines that MAS DNP has moved far beyond the proof-of-concept stage and has become an important tool for research in these fields.

Microsecond Time Scale Proton Rotating-Frame Relaxation under Magic Angle Spinning

This paper deals with the theoretical foundation of proton magic angle spinning rotating-frame relaxation (R_1ρ) and establishes the range of validity and accuracy of the presented approach to describe low-amplitude microsecond time scale motion in the solid state. Beside heteronuclear dipolar and chemical shift anisotropy interactions, a major source of relaxation for protons is the homonuclear dipolar interaction. For this latter relaxation process, no general analytical equation has been published until now, which would describe the R_1ρ relaxation at any spinning speed, spin-lock field, or tilt angle. To validate the derived equations, we compared the analytical relaxation rates, obtained by solving the master equation within the framework of Redfield theory, with numerically simulated relaxation rates. We found that for small opening angles (∼10°), the relaxation rates obtained with stochastic Liouville simulations agree well with the analytical Redfield relaxation rates for a large range of motional correlation times. However, deviations around the rotary-resonance conditions highlight the fact that Redfield treatment of the solid-state relaxation rates can only provide qualitative insights into the microsecond time scale motion.

Microscopic interpretation and generalization of the Bloch-Torrey equation for diffusion magnetic resonance

In order to bridge microscopic molecular motion with macroscopic diffusion MR signal in complex structures, we propose a general stochastic model for molecular motion in a magnetic field. The Fokker-Planck equation of this model governs the probability density function describing the diffusion-magnetization propagator. From the propagator we derive a generalized version of the Bloch-Torrey equation and the relation to the random phase approach. This derivation does not require assumptions such as a spatially constant diffusion coefficient, or ad hoc selection of a propagator. In particular, the boundary conditions that implicitly incorporate the microstructure into the diffusion MR signal can now be included explicitly through a spatially varying diffusion coefficient. While our generalization is reduced to the conventional Bloch-Torrey equation for piecewise constant diffusion coefficients, it also predicts scenarios in which an additional term to the equation is required to fully describe the MR signal.

Spatially encoded 2D and 3D diffusion-ordered NMR spectroscopy

The separation of 2D spectra of components in mixtures is accelerated with spatial encoding.

Efficient simulation of ultrafast magnetic resonance experiments

We present a convenient and powerful simulation formalism for ultrafast NMR spectroscopy. The formalism is based on the Fokker–Planck equation that supports systems with complicated combinations of classical spatial dynamics and quantum mechanical spin dynamics.