A master-equation approach to the description of proton-driven spin diffusion from crystal geometry using simulated zero-quantum lineshapes

Measurements of proton-driven carbon-13 spin diffusion (PDSD) by NMR spectroscopy are a central component of structural analyses of biomolecules in the solid-state. However, the quantitative link between experimental PDSD data and structural information is difficult to make. Here we observe that a master-equation approach can be used to model full PDSD dynamics accurately in polycrystalline (13)C-labelled L-histidine·HCl·H(2)O under magic-angle spinning. In the master-equation approach, PDSD rates and effective dipolar couplings are related by a function of the carbon-carbon zero-quantum lineshapes; we find that numerical simulations of the zero-quantum lineshapes are sufficiently accurate so as to allow the calculation of PDSD rates that are in good agreement with the measured rates, directly from crystal geometry and with no adjustable parameters. Finally, using carbon-carbon internuclear distances we illustrate the potential of the master-equation approach for structural studies. Generalisation of these results to proton-driven carbon-13 spin diffusion in more complex molecular systems is readily envisaged.     

Spin diffusion driven by R-symmetry sequences: applications to homonuclear correlation spectroscopy in MAS NMR of biological and organic solids

We present a family of homonuclear (13)C-(13)C magic angle spinning spin diffusion experiments, based on R2(n)(v) (n = 1 and 2, v = 1 and 2) symmetry sequences. These experiments are well suited for (13)C-(13)C correlation spectroscopy in biological and organic systems and are especially advantageous at very fast MAS conditions, where conventional PDSD and DARR experiments fail. At very fast MAS frequencies the R2(1)(1), R2(2)(1), and R2(2)(2) sequences result in excellent quality correlation spectra both in model compounds and in proteins. Under these conditions, individual R2(n)(v) display different polarization transfer efficiency dependencies on isotropic chemical shift differences: R2(2)(1) recouples efficiently both small and large chemical shift differences (in proteins these correspond to aliphatic-to-aliphatic and carbonyl-to-aliphatic correlations, respectively), while R2(1)(1) and R2(2)(2) exhibit the maximum recoupling efficiency for the aliphatic-to-aliphatic or carbonyl-to-aliphatic correlations, respectively. At moderate MAS frequencies (10-20 kHz), all R2(n)(v) sequences introduced in this work display similar transfer efficiencies, and their performance is very similar to that of PDSD and DARR. Polarization transfer dynamics and chemical shift dependencies of these R2-driven spin diffusion (RDSD) schemes are experimentally evaluated and investigated by numerical simulations for [U-(13)C,(15)N]-alanine and the [U-(13)C,(15)N] N-formyl-Met-Leu-Phe (MLF) tripeptide. Further applications of this approach are illustrated for several proteins: spherical assemblies of HIV-1 U-(13)C,(15)N CA protein, U-(13)C,(15)N-enriched dynein light chain DLC8, and sparsely (13)C/uniformly (15)N enriched CAP-Gly domain of dynactin. Due to the excellent performance and ease of implementation, the presented R2(n)(v) symmetry sequences are expected to be of wide applicability in studies of proteins and protein assemblies as well as other organic solids by MAS NMR spectroscopy.     

Theory for spin-lattice relaxation of spin probes on weakly deformable DNA

The weakly bending rod (WBR) model of double-stranded DNA (dsDNA) is adapted to analyze the internal dynamics of dsDNA as observed in electron paramagnetic resonance (EPR) measurements of the spin-lattice relaxation rate, R(1e), for spin probes rigidly attached to nucleic acid-bases. The WBR theory developed in this work models dsDNA base-pairs as diffusing rigid cylindrical discs connected by bending and twisting springs whose elastic force constants are kappa and alpha, respectively. Angular correlation functions for both rotational displacement and velocity are developed in detail so as to compute values for R(1e) due to four relaxation mechanisms: the chemical shift anisotropy (CSA), the electron-nuclear dipolar (END), the spin rotation (SR), and the generalized spin diffusion (GSD) relaxation processes. Measured spin-lattice relaxation rates in dsDNA under 50 bp in length are much faster than those calculated for the same DNAs modeled as rigid rods. The simplest way to account for this difference is by allowing for internal flexibility in models of DNA. Because of this discrepancy, we derive expressions for the spectral densities due to CSA, END, and SR mechanisms directly from a weakly bending rod model for DNA. Special emphasis in this development is given to the SR mechanism because of the lack of such detail in previous treatments. The theory developed in this paper provides a framework for computing relaxation rates from the WBR model to compare with magnetic resonance relaxation data and to ascertain the twisting and bending force constants that characterize DNA.     

1H‐19F REDOR‐filtered NMR spin diffusion measurements of domain size in heterogeneous polymers

Solid state NMR spectroscopy is inherently sensitive to chemical structure and composition and thus makes an ideal method to probe the heterogeneity of multicomponent polymers. Specifically, NMR spin diffusion experiments can be used to extract reliable information about spatial domain sizes on multiple length scales, provided that magnetization selection of one domain can be achieved. In this paper, we demonstrate the preferential filtering of protons in fluorinated domains during NMR spin diffusion experiments using ¹H‐¹⁹F heteronuclear dipolar dephasing based on rotational echo double resonance (REDOR) MAS NMR techniques. Three pulse sequence variations are demonstrated based on the different nuclei detected: direct ¹H detection, plus both ¹H?¹³C cross polarization and ¹H?¹⁹F cross polarization detection schemes. This ¹H‐¹⁹F REDOR‐filtered spin diffusion method was used to measure fluorinated domain sizes for a complex polymer blend. The efficacy of the REDOR‐based spin filter does not rely on spin relaxation behavior or chemical shift differences and thus is applicable for performing NMR spin diffusion experiments in samples where traditional magnetization filters may prove unsuccessful. This REDOR‐filtered NMR spin diffusion method can also be extended to other samples where a heteronuclear spin pair exists that is unique to the domain of interest.     

Exchange processes in nmr

The exchange of particles and the (noncorporeal) exchange of magnetization in systems are considered where nuclei exist in various regions which differ in their magnetic resonance properties. We start with the stochastic theory of nuclear magnetic resonance and relaxation in such systems. Two-region systems are treated in some detail; here, also, the modified Bloch equations are introduced. Experimental examples concerning adsorbent-adsorbate systems are given. Finally, the influence of cross-relaxation on the relaxation of molecules adsorbed on interfaces is taken into account, the corresponding theory being treated. NMR methods which allow to distinguish between particle and spin exchange are discussed.     

A suite of 2H NMR spin relaxation experiments for the measurement of RNA dynamics

A suite of (2)H-based spin relaxation NMR experiments is presented for the measurement of molecular dynamics in a site-specific manner in uniformly (13)C, randomly fractionally deuterated ( approximately 50%) RNA molecules. The experiments quantify (2)H R(1) and R(2) relaxation rates that can subsequently be analyzed to obtain information about dynamics on a pico- to nanosecond time scale. Sensitivity permitting, the consistency of the data can be evaluated by measuring all five rates that are accessible for a spin 1 particle and establishing that the rates obey relations that are predicted from theory. The utility of the methodology is demonstrated with studies of the dynamics of a 14-mer RNA containing the UUCG tetraloop at temperatures of 25 and 5 degrees C. The high quality of the data, even at 5 degrees C, suggests that the experiments will be of use for the study of RNA molecules that are as large as 30 nucleotides.     

Signal irreproducibility in high-field solution magnetic resonance experiments caused by spin turbulence

Turbulent spin dynamics arising from the joint action of radiation damping and the distant dipolar field are shown to generate irreproducible measurements in popular high-field, gradient-based magnetic resonance (MR) experiments, undermining the prevailing assumption of essentially predictable observables in MR. Sizeable fluctuations in echo amplitudes are reported and numerically simulated for pulsed gradient spin echo and stimulated echo diffusion measurements. The underlying microscopic dynamical instability is characterized by analysis of the finite-time Lyapunov exponents. Perturbations to the modulated magnetization are shown to render magic-angle gradients ineffective in suppressing signal fluctuations. Alternative approaches are suggested for cancelling out the feedback interactions leading to spin turbulence.     

Dynamic nuclear polarization assisted spin diffusion for the solid effect case

The dynamic nuclear polarization (DNP) process in solids depends on the magnitudes of hyperfine interactions between unpaired electrons and their neighboring (core) nuclei, and on the dipole-dipole interactions between all nuclei in the sample. The polarization enhancement of the bulk nuclei has been typically described in terms of a hyperfine-assisted polarization of a core nucleus by microwave irradiation followed by a dipolar-assisted spin diffusion process in the core-bulk nuclear system. This work presents a theoretical approach for the study of this combined process using a density matrix formalism. In particular, solid effect DNP on a single electron coupled to a nuclear spin system is considered, taking into account the interactions between the spins as well as the main relaxation mechanisms introduced via the electron, nuclear, and cross-relaxation rates. The basic principles of the DNP-assisted spin diffusion mechanism, polarizing the bulk nuclei, are presented, and it is shown that the polarization of the core nuclei and the spin diffusion process should not be treated separately. To emphasize this observation the coherent mechanism driving the pure spin diffusion process is also discussed. In order to demonstrate the effects of the interactions and relaxation mechanisms on the enhancement of the nuclear polarization, model systems of up to ten spins are considered and polarization buildup curves are simulated. A linear chain of spins consisting of a single electron coupled to a core nucleus, which in turn is dipolar coupled to a chain of bulk nuclei, is considered. The interaction and relaxation parameters of this model system were chosen in a way to enable a critical analysis of the polarization enhancement of all nuclei, and are not far from the values of (13)C nuclei in frozen (glassy) organic solutions containing radicals, typically used in DNP at high fields. Results from the simulations are shown, demonstrating the complex dependences of the DNP-assisted spin diffusion process on variations of the relevant parameters. In particular, the effect of the spin lattice relaxation times on the polarization buildup times and the resulting end polarization are discussed, and the quenching of the polarizations by the hyperfine interaction is demonstrated.     

Direct calculation of (1)H(2)O T(1) NMRD profiles and EPR lineshapes for the electron spin quantum numbers S = 1, 3/2, 2, 5/2, 3, 7/2, based on the stochastic Liouville equation combined with Brownian dynamics simulation

Direct calculation of electron spin relaxation and EPR lineshapes, based on Brownian dynamics simulation techniques and the stochastic Liouville equation approach (SLE-L) [Mol. Phys., 2004, 102, 1085-1093], is here generalized to high spin systems with spin quantum number S = 3/2, 2, 5/2, 3 and 7/2. A direct calculation method is demonstrated for electron spin-spin and spin-lattice relaxation, S-, X- and Q-band EPR-lineshapes and paramagnetic enhanced water proton T(1)- NMRD profiles. The main relaxation mechanism for the electron spin system is a stochastic second rank zero field splitting (ZFS). Brownian dynamics simulation techniques are used in describing a fluctuating ZFS interaction which comprises two parts namely the "permanent" part which is modulated by isotropic reorientation diffusion, and the transient part which is modulated by fast local distortion, which is also modelled by the isotropic rotation diffusion model. The SLE-L approach present is applicable both in the perturbation (Redfield) regime as well as outside the perturbation regime, in the so called slow motion regime.     

Broadband measurement of true transverse relaxation rates in systems with coupled protons: application to the study of conformational exchange

Accurate measurement of transverse relaxation rates in coupled spin systems is important in the study of molecular dynamics, but is severely complicated by the signal modulations caused by scalar couplings in spin echo experiments. The most widely used experiments for measuring transverse relaxation in coupled systems, CPMG and PROJECT, can suppress such modulations, but they also both suppress some relaxation contributions, and average relaxation rates between coupled spins. Here we introduce a new experiment which for the first time allows accurate broadband measurement of transverse relaxation rates of coupled protons, and hence the determination of exchange rate constants in slow exchange from relaxation measurements. The problems encountered with existing methods are illustrated, and the use of the new method is demonstrated for the classic case of hindered amide rotation and for the more challenging problem of exchange between helical enantiomers of a gold(i) complex.

Existing methods for measuring transverse relaxation give incorrect results in coupled spin systems. Measuring true relaxation rates extends their utility.     

An NMR experiment for the accurate measurement of heteronuclear spin-lock relaxation rates

Rotating-frame relaxation rates, R(1)(rho), are often measured in NMR studies of protein dynamics. We show here that large systematic errors can be introduced into measured values of heteronuclear R(1)(rho) rates using schemes which are usually employed to suppress cross-correlation between dipole-dipole and CSA relaxation mechanisms. For example, in a scalar-coupled two-spin X-H spin system the use of (1)H WALTZ16 decoupling or (1)H pulses applied at regularly spaced intervals leads to a significant overestimation of heteronuclear R(1)(rho) values. The problem is studied experimentally and theoretically for (15)N-(1)H and (13)C-(1)H spin pairs, and simple schemes are described which eliminate the artifacts. The approaches suggested are essential replacements of existing methodology if accurate dynamics parameters are to be extracted from spin-lock relaxation data sets.     

Introduction to nuclear magnetic resonance

The basic principles of nuclear magnetic resonance (NMR) are presented in an elementary form using classical and elementary quantum mechanics and the experimental technique 1s explained. The motion of the magnetization by r.f. pulses, free induction decay and spectrum, transverse and longitudinal relaxation, local field and spin echo are described and the effects of molecular motion are discussed. The concepts of spin temperature and spin diffusion are presented and the advantage of using quadrupole nuclei is stressed. Finally, the specific problems of NMR in interface studies are considered and a typical example is given.     

Theoretical aspects of chemically induced magnetic polarization

The role of theory in guiding and simplifying interpretation of electron spin resonance experiments on photochemical and other reactions involving free radical intermediates is surveyed. Emphasis is on models which provide a physical picture as well as quantitative estimates for such phenomena as the radical pair mechanism of chemically induced electron spin polarization (CIDEP), the closely related process of spin exchange during radical-radical encounters, and spin lattice relaxation. Some specific topics discussed are: 1) an improved quantitative model of STo CIDEP combining an initial stage of polarization development followed partial loss of this polarization to spin exchange, 2) the relation between the spin exchange and recombination rate constants, and 3) simplification of spin-lattice relaxation in the common case of spin-rotation relaxation. The modification of the polarization processes in two-dimensional and closed three-dimensional systems is also discussed     

Measurement of NMR cross-polarization (CP) rate constants in the slow CP regime: relevance to structure determinations of zeolite-sorbate and other complexes by cp magic-angle spinning NMR

When analyzing I --> S variable contact time cross-polarization (CP) curves, the spin dynamics are usually assumed to be describable in the "fast CP regime" in which the growth of the S spin magnetization is governed by the rate of cross polarization while its decay is governed by the rate of I spin T1rho relaxation. However, in the investigation of the structures of zeolite-sorbate and other complexes by polarization transfer this will not necessarily be the case. We discuss the measurement of I --> S CP rate constants under the "slow CP regime" in which the rate of T1rho relaxation is fast compared to the rate of cross polarization, leading to a reversal of the usual assumptions such that the rate or growth is governed by the rate of I spin T1rho relaxation while the decay is governed by the rate of cross polarization (and the S spin T1rho relaxation). It is very important to recognize when a system is in the slow CP regime, as an analysis assuming the normal fast CP will lead to erroneous data. However, even when the slow CP regime is recognized, it is difficult to obtain absolute values for the CP rate constants from fits to standard CP curves, since the CP rate constant is correlated to the scaling factor, the contribution from 29Si T1rho relaxation is ignored, and it is difficult to obtain reliable data at very long contact times. The use of a 29Si{1H} CP "drain" or "depolarization" experiment, which measures absolute values of the CP rate constants, is therefore proposed as being most appropriate for theses situations. To illustrate the importance of these observations, measurements of the 1H-29Si CP rate constants in the p-dichlorobenzene/ZSM-5 sorbate-zeolite complex by 29Si{1H} CP and CP drain magic-angle spinning (MAS) NMR experiments are presented and compared and used to determine the location of the guest sorbate molecules in the cavities of the host zeolite framework.     

Computer simulation of quantum dynamics in a classical spin environment

In this paper, a formalism for studying the dynamics of quantum systems coupled to classical spin environments is reviewed. The theory is based on generalized antisymmetric brackets and naturally predicts open-path off-diagonal geometric phases in the evolution of the density matrix. It is shown that such geometric phases must also be considered in the quantum–classical Liouville equation for a classical bath with canonical phase space coordinates; this occurs whenever the adiabatics basis is complex (as in the case of a magnetic field coupled to the quantum subsystem). When the quantum subsystem is weakly coupled to the spin environment, non-adiabatic transitions can be neglected and one can construct an effective non-Markovian computer simulation scheme for open quantum system dynamics in classical spin environments. In order to tackle this case, integration algorithms based on the symmetric Trotter factorization of the classical-like spin propagator are derived. Such algorithms are applied to a model comprising a quantum two-level system coupled to a single classical spin in an external magnetic field. Starting from an excited state, the population difference and the coherences of this two-state model are simulated in time while the dynamics of the classical spin is monitored in detail. It is the author’s opinion that the numerical evidence provided in this paper is a first step toward developing the simulation of quantum dynamics in classical spin environments into an effective tool. In turn, the ability to simulate such a dynamics can have a positive impact on various fields, among which, for example, nanoscience.     

Nuclear spin relaxation driven by intermolecular dipolar interactions: the role of solute-solvent pair correlations in the modeling of spectral density functions

Nuclear spin relaxation provides useful information related to the dynamics of molecular systems. When relaxation is driven by intermolecular dipolar interactions, the relevant spectral density functions (SDFs) also have significant contributions, in principle, from distant spins all over the dynamic range typically probed by NMR experiments such as NOESY. In this work, we investigate the intermolecular dipolar spin relaxation as driven by the relative diffusion of solvent and solute molecules taking place under a central force field, and we examine the relevant implications for (preferential) solvation studies. For this purpose, we evaluate the SDFs by employing a numerical approach based on spatial discretization of the time-propagation equation, and we supply an analytical solution for the simplest case of a steplike mean-field potential. Several situations related to different solute-solvent pair correlation functions are examined in terms of static/dynamic effects and relaxation modes, and some conclusions are drawn about the interpretation of NOE measurements. While we confirm previous results concerning the spoiling effect of long-range spins (Halle, B. J. Chem. Phys. 2003, 119, 12372), we also show that SDFs are sufficiently sensitive to pair correlation functions that useful, yet rather complicated, inferences can be made on the nature of the solvation shell.     

Pulsed field gradient NMR study of phenol binding and exchange in dispersions of hollow polyelectrolyte capsules

The distribution and exchange dynamics of phenol molecules in colloidal dispersions of submicron hollow polymeric capsules is investigated by pulsed field gradient NMR (PFG-NMR). The capsules are prepared by layer-by-layer assembly of polyelectrolyte multilayers on silica particles, followed by dissolution of the silica core. In capsule dispersion, (1)H PFG echo decays of phenol are single exponentials, implying fast exchange of phenol between a free site and a capsule-bound site. However, apparent diffusion coefficients extracted from the echo decays depend on the diffusion time, which is typically not the case for the fast exchange limit. We attribute this to a particular regime, where apparent diffusion coefficients are observed, which arise from the signal of free phenol only but are influenced by exchange with molecules bound to the capsule, which exhibit a very fast spin relaxation. Indeed, relaxation rates of phenol are strongly enhanced in the presence of capsules, indicating binding to the capsule wall rather than encapsulation in the interior. We present a quantitative analysis in terms of a combined diffusion-relaxation model, where exchange times can be determined from diffusion and spin relaxation experiments even in this particular regime, where the bound site acts as a relaxation sink. The result of the analysis yields exchange times between free phenol and phenol bound to the capsule wall, which are on the order of 30 ms and thus slower than the diffusion controlled limit. From bound and free fractions an adsorption isotherm of phenol to the capsule wall is extracted. The binding mechanism and the exchange mechanism are discussed. The introduction of the global analysis of diffusion as well as relaxation echo decays presented here is of large relevance for adsorption dynamics in colloidal systems or other systems, where the standard diffusion echo decay analysis is complicated by rapidly relaxing boundary conditions.     

Influence of subdiffusive motion on spin relaxation and spin effects in radical pairs

Specific features of spin relaxation and the kinetics of spin effect generation in radical pairs (RPs) undergoing subdiffusive relative motion are studied in detail. Two types of processes are analyzed: (1) spin relaxation in biradicals, resulting from anomalously slow subdiffuisive reorientation (with the correlation function P(t) approximately (wt)(-alpha), where 0 < alpha < 1) and (2) spin effect generation in subdiffusion-assisted RP recombination. Analysis is made with the use of the non-Markovian stochastic Liouville equation (SLE) derived within the continuous time random walk approach. The SLE predicts anomalous (very slow and nonexponential) spin relaxation in biradicals which results in some peculiarities of the spectrum of the system. In RP recombination, the subdiffusive relative motion shows itself in slow dependence of the reaction yield Y(r)() on reactivity and parameters of the RP spin Hamiltonian and anomalous electron spin polarization of escaped radicals. The spectrum of the reaction yield detected magnetic resonance, that is, the Y(r)() dependence on the frequency omega of microwave field, is found to be strongly non-Lorenzian with the width determined by the field strength omega(1) and very broad wings depending on alpha. Analysis shows that the majority of interesting, specific features of the observables in both systems are controlled only by the parameter alpha.     

Nuclear magnetic resonance proton dipolar order relaxation in thermotropic liquid crystals: a quantum theoretical approach

By means of the Jeener-Broekaert nuclear magnetic resonance pulse sequence, the proton spin system of a liquid crystal can be prepared in quasiequilibrium states of high dipolar order, which relax to thermal equilibrium with the molecular environment with a characteristic time (T1D). Previous studies of the Larmor frequency and temperature dependence of T1D in thermotropic liquid crystals, that included field cycling and conventional high-field experiments, showed that the slow hydrodynamic modes dominate the behavior of T1D, even at high Larmor frequencies. This noticeable predominance of the cooperative fluctuations (known as order fluctuations of the director, OFD) could not be explained by standard models based on the spin-lattice relaxation theory in the limit of high temperature (weak order). This fact points out the necessity of investigating the role of the quantum terms neglected in the usual high temperature theory of dipolar order relaxation. In this work, we present a generalization of the proton dipolar order relaxation theory for highly correlated systems, which considers all the spins belonging to correlated domains as an open quantum system interacting with quantum bath. As starting point, we deduce a formulation of the Markovian master equation of relaxation for the statistical spin operator, valid for all temperatures, which is suitable for introducing a dipolar spin temperature in the quantum regime, without further assumptions about the form of the spin-lattice Hamiltonian. In order to reflect the slow dynamics occurring in correlated systems, we lift the usual short-correlation-time assumption by including the average over the motion of the dipolar Hamiltonian together with the Zeeman Hamiltonian into the time evolution operator. In this way, we calculate the time dependence of the spin operators in the interaction picture in a closed form, valid for high magnetic fields, bringing into play the spin-spin interactions within the microscopic time scale. Then, by adopting the spin-temperature density operator to represent the collective state of the spin system, and removing the traditional hypothesis of high temperature, we deduce an expression for the first order quantum contribution to T1D (-1), in terms of spectral densities, with coefficients in form of spin traces. The properties that distinguish our result from the high-temperature T1D (-1) are as follows. (a) It is exclusively associated to cooperative fluctuations. (b) Because of its quantum character, it relies on both considering the lattice degrees of freedom quantum mechanically and including the spin-spin interactions in the microscopic time scale. With regard to the average dipolar Hamiltonian, only the nonsecular part plays a relevant role. (c) Associated with the structure of the spin operator involved in the quantum contribution, a term arises which is proportional to the number of spins in the correlated molecular domains, showing that the quantum contribution may be of macroscopic size in highly correlated systems. When applied to nematic liquid crystals, the new term exhibits the typical nu(-1/2) Larmor frequency dependence through the spectral density of the OFD, in consistence with the experimental results.