The enhancement of the spin-lattice relaxation rate for nuclear spins in a ligand bound to a paramagnetic metal ion [known as the paramagnetic relaxation enhancement (PRE)] arises primarily through the dipole-dipole (DD) interaction between the nuclear spins and the electron spins. In solution, the DD interaction is modulated mostly by reorientation of the nuclear spin-electron spin axis and by electron spin relaxation. Calculations of the PRE are in general complicated, mainly because the electron spin interacts so strongly with the other degrees of freedom that its relaxation cannot be described by second-order perturbation theory or the Redfield theory. Three approaches to resolve this problem exist in the literature: The so-called slow-motion theory, originating from Swedish groups [Benetis et al., Mol. Phys. 48, 329 (1983); Kowalewski et al., Adv. Inorg. Chem. 57, (2005); Larsson et al., J. Chem. Phys. 101, 1116 (1994); T. Nilsson et al., J. Magn. Reson. 154, 269 (2002)] and two different methods based on simulations of the dynamics of electron spin in time domain, developed in Grenoble [Fries and Belorizky, J. Chem. Phys. 126, 204503 (2007); Rast et al., ibid. 115, 7554 (2001)] and Ann Arbor [Abernathy and Sharp, J. Chem. Phys. 106, 9032 (1997); Schaefle and Sharp, ibid. 121, 5387 (2004); Schaefle and Sharp, J. Magn. Reson. 176, 160 (2005)], respectively. In this paper, we report a numerical comparison of the three methods for a large variety of parameter sets, meant to correspond to large and small complexes of gadolinium(III) and of nickel(II). It is found that the agreement between the Swedish and the Grenoble approaches is very good for practically all parameter sets, while the predictions of the Ann Arbor model are similar in a number of the calculations but deviate significantly in others, reflecting in part differences in the treatment of electron spin relaxation. The origins of the discrepancies are discussed briefly.
A combined solid-state NMR and Molecular Dynamics simulation study of cellulose in urea aqueous solution and in pure water was conducted. It was found that the local concentration of urea is significantly enhanced at the cellulose/solution interface. There, urea molecules interact directly with the cellulose through both hydrogen bonds and favorable dispersion interactions, which seem to be the driving force behind the aggregation. The CP/MAS ^{13}C spectra was affected by the presence of urea at high concentrations, most notably the signal at 83.4 ppm, which has previously been assigned to C4 atoms in cellulose chains located at surfaces parallel to the (110) crystallographic plane of the cellulose Iβ crystal. Also dynamic properties of the cellulose surfaces, probed by spin-lattice relaxation time ^{13}CT _{1} measurements of C4 atoms, are affected by the addition of urea. Molecular Dynamics simulations reproduce the trends of the T _{1}measurements and lends new support to the assignment of signals from individual surfaces. That urea in solution is interacting directly with cellulose may have implications on our understanding of the mechanisms behind cellulose dissolution in alkali/urea aqueous solutions.
Proton/Fluoride spin-lattice ($T_1$) nuclear magnetic relaxation dispersion (NMRD) measurements of 1-butyl-3-methyl-$1H$-imidazolium hexa-fluorophosphate, [$C_4mim][PF_6]$, have been carried out using high field spectrometers and fast-field-cycling instrument at proton Larmor frequencies ranging from 10kHz to 40 MHz, at different temperatures. The NMRD profiles are interpreted by means of a simple relaxation model based on the inter- and intra-ionic dipole-dipole relaxation mechanism. Using an atomic molecular-ion dynamic simulation at 323 K the relevant spin dipole-dipole(DD) correlation functions are calculated. The results indicate the NMRD profiles can be rationalized using intra- and inter-ionic spin DD interactions, however, both are mainly modulated by ionic reorientation because of temporary correlations with cations, where modulation by translational diffusion plays a minor role. Reorientational dynamics of charge-neutral ion couples (i.e. $[C_4mim]^{...}[PF_6]$) and $[C_4mim]^{+}$ ions are in the nano-second (ns) time range whereas the reorientation of $[PF_6]{^-}$ is characterized by a reorientational correlation time in the pico-second (ps) regime. Based on the NMRD profiles we conclude the main relaxation mechanism for $[PF_6]{^-}$ is, due to fast internal reorientational motion, a partially averaged F-F intra and a F-H inter-ionic DD coupling as the anion resides in close proximity to its temporary oppositely charged cation partner. The F-$T_1$- NMRD data display a ns dispersions which is interpreted as being due to correlated reorientational modulations resultant from H-containing charge-neutral ion couple $[C_4mim]^{...}[PF_6]$. The analysis of ionicity is based on the free anion fraction, $f$ and it increase with temperature with $f$ $\rightarrow$ 1 at the highest temperatures investigated. The fraction is obtained from the H-F NMRD profiles as correlated-non-correlated dynamics of the ions. The analysis of $T_1$ relaxation rates of C, H, F and P at high fields cannot generally give the fraction of ion but are consistent with the interpretation based on the NMRD profiles with relaxation contributions due to DD-intra and -inter, CSA-intra (and -inter for C), including spin rotation for P. The investigation has led to a description of the mechanics governing ion transport in the title ionic liquid via identification of transient correlated/non-correlated ion dynamics.
Recently a model based on donor–donor energy migration (DDEM) was developed for examining structure–function properties of biomacromolecules such as proteins (J. Chem. Soc., Faraday Trans., 1996, 92, 1563). Unlike the extended Förster theory (EFT; J. Chem. Phys., 1996, 105, 10896) the DDEM model is straightforward to apply in the analyses of fluorescence depolarisation experiments, as obtained by the time-correlated single photon counting (TCSPC) technique. In order to test the validity of the DDEM model, the EFT was used to create synthetic depolarisation data. These mimic true TCSPC experiments and cover a wide range of physical conditions, which are difficult to arrange for in real experiments with model systems. In particular, the relative rate of DDEM () and the rotation correlation time () of the donor molecules was examined. The DDEM rates obtained from the analyses were compared to the true rates. From these results the relative error of the intramolecular distances were calculated. For values 1<12<25, the DDEM model is slightly overestimating the distances. Typically, the distances determined with the DDEM model are overestimated by 5–10%.
The performance of a magnetic resonance imaging contrast agent (CA) depends on several factors, including the relaxation times of the unpaired electrons in the CA. The electron spin relaxation time may be a key factor for the performance of new CAs, such as nanosized Gd_{2}O_{3} particles. The aim of this work is, therefore, to study changes in the magnetic susceptibility and the electron spin relaxation time of paramagnetic Gd_{2}O_{3} nanoparticles diluted with increasing amounts of diamagnetic Y_{2}O_{3}. Nanoparticles of (Gd_{x}Y_{1−x})_{2}O_{3} (0 ≤ x ≤ 1) were prepared by the combustion method and thoroughly characterized (by X-ray diffraction, transmission electron microscopy, thermogravimetry coupled with mass spectroscopy, photoelectron spectroscopy, Fourier transform infrared spectroscopy, and magnetic susceptibility measurements). Changes in the electron spin relaxation time were estimated by observations of the signal line width in electron paramagnetic resonance spectroscopy, and it was found that the line width was dependent on the concentration of yttrium, indicating that diamagnetic Y_{2}O_{3} may increase the electron spin relaxation time of Gd_{2}O_{3} nanoparticles.
The interaction between water protons and suitable quadrupolar nuclei (QN) can lead to quadrupole relaxation enhancement (QRE) of proton spins, provided the resonance condition between both spin transitions is fulfilled. This effect could be utilized as a frequency selective mechanism in novel, responsive T-1 shortening contrast agents (CAs) for magnetic resonance imaging (MRI). In particular, the proposed contrast mechanism depends on the applied external flux density-a property that can be exploited by special field-cycling MRI scanners. For the design of efficient CA molecules, exhibiting narrow and pronounced peaks in the proton T-1 relaxation dispersion, the nuclear quadrupole resonance (NQR) properties, as well as the spin dynamics of the system QN-H-1, have to be well understood and characterized for the compounds in question. In particular, the energy-level structure of the QN is a central determinant for the static flux densities at which the contrast enhancement appears. The energy levels depend both on the QN and the electronic environment, i.e., the chemical bonding structure in the CA molecule. In this work, the NQR properties of a family of promising organometallic compounds containing Bi-209 as QN have been characterized. Important factors like temperature, chemical structure, and chemical environment have been considered by NQR spectroscopy and ab initio quantum chemistry calculations. The investigated Bi-aryl compounds turned out to fulfill several crucial requirements: NQR transition frequency range applicable to clinical 1.5- and 3 T MRI systems, low temperature dependency, low toxicity, and tunability in frequency by chemical modification.
The hydration of the oxygen-evolving complex (OEC) was characterized in the dark stable S_{1} state of photosystem II using water R_{1}(ω) NMR dispersion (NMRD) profiles. The R_{1}(ω) NMRD profiles were recorded over a frequency range from 0.01 MHz to 40 MHz for both intact and Mn-depleted photosystem II core complexes from Thermosynechococcus vulcanus (T. vulcanus). The intact-minus-(Mn)-depleted difference NMRD profiles show a characteristic dispersion from approximately 0.03 MHz to 1 MHz, which is interpreted on the basis of the Solomon-Bloembergen-Morgan (SBM) and the slow motion theories as being due to a paramagnetic enhanced relaxation (PRE) of water protons. Both theories are qualitatively consistent with the S_{T} = 1, g = 4.9 paramagnetic state previously described for the S_{1} state of the OEC; however, an alternative explanation involving the loss of a separate class of long-lived internal waters due to the Mn-depletion procedure can presently not be ruled out. Using a point-dipole approximation the PRE-NMRD effect can be described as being caused by 1-2 water molecules that are located about 10 Å away from the spin center of the Mn_{4}CaO_{5 }cluster in the OEC. The application of the SBM theory to the dispersion observed for PSII in the S_{1} state is questionable, because the parameters extracted do not fulfil the presupposed perturbation criterion. In contrast, the slow motion theory gives a consistent picture indicating that the water molecules are in fast chemical exchange with the bulk (τ_{w} < 1 μs). The modulation of the zero-field splitting (ZFS) interaction suggests a (restricted) reorientation/structural equilibrium of the Mn_{4}CaO_{5} cluster with a characteristic time constant of τ_{ZFS} = 0.6-0.9 μs.
The hydration of a protein, peroxiredoxin 5, is obtained from a molecular dynamics simulation and compared with the picture of hydration which is obtained by analysing the water proton R_{1} NMRD profiles using a generally accepted relaxation model [K. Venu, V.P. Denisov and B. Halle, J. Am. Chem. Soc. 119,3122(1997)]. The discrepancy between the hydration pictures derived from the water R_{1}(ω_{ 0})-NMRD profiles and MD is relevant in a discussion of the factors behind the stretched NMRD profile, the distribution of orientationalorder parameters and residence times of buried water used in the NMRD model.
The water dynamics in the confined space of the zeolite ZSM-5 has bee ninvestigated by means of the field dependence of ^{1}H- and ^{2}H- spin-lattice relaxation rates using a 1T Stelar FFC2000 fast field-cycling instrument. The NMRD analysis of the experimental results indicates that the characteristic time dependence ( 50 ns to 1-2.4 μs) is due to water translational diffusion in narrow pores. The temperature dependence of the spin-lattice relaxation rates is weak.Zeolites with different counter ions( H^{+}, NH_{4}^{+} change the water hydration and the water translational diffusion in the pores drastically. The Zeolite-NH_{4}^{+} slow down the water motion with a factor of 2.The NMRD profiles show somewhat stretched character and is described by two Lorenzian which indicates that the distribution of pore sizes is broaden.The water^{ 1}H and ^{2}H spin lattice relaxation profiles give qualitatively information about water hydration in zeolites with different counter ions and is expected also to indicate structural changes of the zeolites.
Proton/Fluoride spin-lattice nuclear magnetic relaxation dispersion(NMRD) measurements of 1-Butyl-3-methylimidazolium-hexa fluorophosphate (BMIM[PF_{6}])have been carried out using a 1T Stelar FFC2000 fast-field-cycling instrument at proton Larmor frequencies ranging from 10 kHz to 40 MHz and at different temperatures. The NMRD profiles are interpreted by means of a simple relaxation modelbased on the inter- and intra-molecular dipole dipole relaxation mechanims. Using an atomic and a coarse-grained (CG)Molecular Dynamics (MD) simulations at temperature 323 K the relevant dipole-dipole correlation functions are calculated. The result indicate that the NMRD profiles can be rationalized using a combination of intra and inter molecular dipole-dipole couplings. However, both are mainly modulated by molecular reorientation whereas translation diffusion plays a minor role. The molecular reorientation dynamics of BMIM[PF_{6}] ,BMIM^{+} ion are in the nano secondtime regime whereas the reorientation of [PF6]^{-} is much faster and loses its correlation in the ps regime. The relaxation mechanism for [PF6]^{-} is H-F inter-molecular dipole-dipole coupling which is modulated by the reorientation of the H-containing molecule.
This work presents a new Brownian dynamics simulation method of translational diffusion on curved surfaces. This new method introduce any implicit defined surface into the stochastic differential equation describing Brownian motion on that surface. The surface curvature will thus enter the force term (A) in the stochastic differential equation dXt = A(Xt)dt + B(Xt)dWt describing an Itô process. We apply the method calculating time correlation functions relevant in nuclear magnetic resonance (NMR) relaxation and translational diffusion studies of cubic phases of lyotropic systems. In particularly we study some bicontinuous cubic liquid crystalline phases which can be described as triply periodic minimal surfaces. The curvature dependent spin relaxation of the Schwarz-P minimal surface is calculated. A comparison of relaxation is made with the more complex topology of the Neovius surface which is another minimal surface in the same space group, and with parallel displacement of the minimal surface which thus results in a nonminimal surface. The curvature dependent relaxation effects are determined by calculating the translational diffusion modulated time-correlation function which determine the relaxation rates of a quadrupole nuclei residing in the water–lipid interface. The results demonstrates that spin relaxation data can provide quantitative information about micro-structure of biocontinuous cubic phases and that it is sensitive to the topology of the surface and to parallel displacement of the model surface. Consequently, spin relaxation may be used as a complement to x-ray diffraction in order to discriminate between different microstructures. It is concluded that fast and accurate computer simulations experiments is needed to be able to interpret NMR relaxation experiments on curved surfaces. © 2002 American Institute of Physics.
A detailed analysis of the previously developed (J. Chem. Phys. 1996, 105, 10896) extended Förster theory (EFT) is presented for analyzing electronic energy migration within pairs of donors (D). Synthetic data that mimics experimental time-correlated single photon counting data were generated and re-analyzed. To cover a wide dynamic range and various orientational restrictions, the rates of reorientation, as well as the orientational configurations of the interacting D-groups were varied. In general DD distances are recovered within an error limit of 5%, while the errors in orientational configurations are usually larger. The Maier−Saupe and cone potentials were used to generate an immense variety of orientational trajectories. The results obtained exhibit no significant dependence on the choice of potential function used for generating EFT data. Present work demonstrates how to overcome the classical “κ^{2}-problem” and the frequently applied approximation of κ^{2} = ^{2}/_{3} in the data analyses. This study also outlines the procedure for analyzing fluorescence depolarization data obtained for proteins, which are specifically labeled with D-groups. The EFT presented here brings the analyses of DDEM data to the same level of molecular detail as in ESR- and NMR-spectroscopy.
This paper discusses the process of energy migration transfer within reorientating chromophores using the stochastic master equation (SME) and the stochastic Liouville equation (SLE) of motion. We have found that the SME over-estimates the rate of the energy migration compared to the SLE solution for a case of weakly interacting chromophores. This discrepancy between SME and SLE is caused by a memory effect occurring when fluctuations in the dipole–dipole Hamiltonian (H(t)) are on the same timescale as the intrinsic fast transverse relaxation rate characterized by (1/T2). Thus the timescale critical for energy-transfer experiments is T2≈10−13 s. An extended SME is constructed, accounting for the memory effect of the dipole–dipole Hamiltonian dynamics. The influence of memory on the interpretation of experiments is discussed.
EPR line shapes can be calculated from the stochastic Liouville equation assuming a stochastic model for the reorientation of the spin probe. Here we use instead and for the first time a detailed molecular dynamics (MD) simulation to generate the stochastic input to the Langevin form of the Liouville equation. A 0.1 μs MD simulation at T = 50°C of a small lipid bilayer formed by 64 dipalmitoylphosphatidylcholine (DPPC) molecules at the water content of 23 water molecules per lipid was used. In addition, a 10 ns simulation of a 16 times larger system consisting of 32 DPPC molecules with a nitroxide spin moiety attached at the sixth position of the sn2 chain and 992 ordinary DPPC molecules, was used to investigate the extent of the perturbation caused by the spin probe. Order parameters, reorientational dynamics and the EPR FID curve were calculated for spin probe molecules and ordinary DPPC molecules. The timescale of the electron spin relaxation for a spin-moiety attached at the sixth carbon position of a DPPC lipid molecule is 11.9 × 10^{7} rad s^{−1} and for an unperturbed DPPC molecule it is 3.5 × 10^{7} rad s^{−1}.
An extended Förster theory (EFT) on electronic energy transfer is presented for the quantitative analysis of time-resolved fluorescence lifetime and depolarisation experiments. The EFT, which was derived from the stochastic Liouville equation, yields microscopic information concerning the reorientation correlation times, the order parameters, as well as inter chromophoric distances. Weakly interacting donor and acceptor groups, which reorient and interact in a pair wise fashion, are considered, under isotropic and anisotropic conditions. For the analysis of experiments it is shown that not only do we need to consider the orientational distributions of the transition dipoles, but the internal reorienting molecular dynamics within the pair which is of even greater importance. The latter determines the shape as well as the rate of the observed donor fluorescence and depolarisation decays, which are most often not mono-exponential functions. It is shown that the commonly used Förster theory is a special case of the EFT. Strategies are presented for applying the EFT, which makes use of Brownian dynamics simulation.
Monomethylmercury (MeHg) in fish from freshwater, estuarine and marine environments are a major global environmental issue. Mercury levels in biota are mainly controlled by the methylation of inorganic mercuric mercury (HgII) to MeHg in water, sediments and soils. There is, however, a knowledge gap concerning the mechanisms and rates of methylation of specific geochemical HgII species. Such information is crucial for a better understanding of variations in MeHg concentrations among ecosystems and, in particular, for predicting the outcome of currently proposed measures to mitigate mercury emissions and reduce MeHg concentrations in fish. To fill this knowledge gap we propose an experimental approach using HgII isotope tracers, with defined and geochemically important adsorbed and solid HgII forms in sediments, to study MeHg formation. We report HgII methylation rate constants, km, in estuarine sediments which span over two orders of magnitude depending on chemical form of added tracer: metacinnabar (β-201HgS(s)) < cinnabar (α-199HgS(s)) < HgII reacted with mackinawite (≡FeS-202HgII) < HgII bonded to natural organic matter (NOM-196HgII) < a typical aqueous tracer (198Hg(NO3)2(aq)). We conclude that a combination of thermodynamic and kinetic effects of HgII solid-phase dissolution and surface desorption control the HgII methylation rate in sediments and causes the large observed differences in km-values. The selection of relevant solid-phase and surface adsorbed HgII tracers will therefore be crucial to achieving biogeochemically accurate estimates of ambient HgII methylation rates.
Aqueous solutions of simple nickel(II) salts are a classical test case for theories of the paramagnetic relaxation enhancement (PRE) and its dependence on the magnetic field (nuclear magnetic relaxation dispersion, NMRD), going back to late fifties. We present here new experimental data, extending the NMRD range up to 21 T (900 MHz). In addition to salt solutions in (acidified) water, we have also measured on solutions containing glycerol. The aqueous solution data do not show any significant changes compared to the earlier experiments. The interpretation, based on the general ("slow-motion") theory is also similar to the earlier work from our laboratory. The NMRD-data in mixed solvents are qualitatively different, indicating that the glycerol not only changes the solution viscosity, but may also enter the first coordination sphere of the metal ion, resulting in lower symmetry complexes, characterized by non-vanishing averaged zero-field splitting. This hypothesis is corroborated by molecular dynamics simulations. A strategy appropriate for interpreting the NMRD-data for the chemically complicated systems of this type is proposed.
A model of the paramagnetic relaxation enhancement is developed in terms of electron-spin relaxation caused by the zero-field splitting (ZFS) fluctuating in time due to a coupling between the electron-spin variables and quantum vibrations. The ZFS interaction provides a coupling between the electron-spin variables and vibrational degrees of freedom, and is represented as a Taylor series expansion in a set of vibrational modes (normal coordinates). A two-level harmonic oscillator subsystem is assumed, and the electron-spin relaxation associated with T2V and T1V vibrational relaxation is considered. The description of vibrationally induced electron-spin dynamics is incorporated into the calculations of the paramagnetic relaxation enhancement by the Solomon–Bloembergen–Morgan approach as well as in the framework of the general slow-motion theory. The theoretical predictions are compared with the experimental paramagnetic relaxation enhancement values for the Ni(H2O) complex in aqueous solution. The parameters required by the model are obtained from quantum chemical and molecular dynamics studies. Comparison is made between the current model and its recently published classical counterpart.
By combining H-2-NMRD and CP/MAS C-13-NMR measurements of water-based cellulose gels and of water swollen pulps it was possible to estimate the nature of the interior structure of cellulose fibril aggregates. A set of samples with high cellulose purity and low charge was used. The interpretation of data was based on a relaxation model describing the exchange dynamics for deuterium exchange between water molecules and cellulose hydroxyl groups. The theoretical model used made it possible to calculate cellulose surface-to-volume ratios (q-values) from both H-2-NMRD and CP/MAS C-13-NMR data. Good consistency between H-2-NMRD and CP/MAS C-13-NMR data was found. In all investigated samples the cellulose fibril aggregates showed a different degree of "openness" interpreted as the presence of interstitial water inside fibril aggregates. One result also showed that an increased degree of fibril aggregate openness results from the TEMPO-oxidation. Common to all samples was that in the water swollen state water molecules could access part of the fibril aggregate interior.
The CP/MAS 13C NMR line shape of cellulose I has been qualitatively analyzed by direct simulations using the Ornstein–Uhlenbeck stochastic process and the Kubo model. Both approaches describe a anhydroglucose C4 carbon as a oscillator with fluctuating Larmor frequency. The NMR resonance frequency is written , where the fluctuating part with zero mean was modelled as a stationary Markov diffusion process.
The simulation results both motivates the use of multiple line shapes when fitting CP/MAS 13C NMR spectra recorded on cellulose I and gives some insights into why signals from crystalline cellulose I give rise to Lorentzian line shapes.
A theoretical analysis of the paramagnetically enhanced water proton spin–lattice relaxation of a hydrated Gd^{3+} ion is combined with Molecular Dynamics (MD) simulations. The electron–proton dipole–dipole correlation function, CDDp(), as well as the pseudo-rotation (PR) model of the transient zero-field splitting (ZFS) are evaluated with the help of the data from MD simulations. The fast local water motion in the first hydration shell, i.e. the wagging and rocking motions, is found not to change the mono exponential character of the dipole correlation function CDDp(), but is important in the time dependence of the transient ZFS interaction.The dynamics of the transient ZFS interaction is modeled as the water-induced electric field gradient tensor at the site of the metal ion. This approach follows the ideas of the pseudo-rotation model, describing the fluctuating zero-field interaction as a constant amplitude in the principal frame but reorienting according to a rotational diffusion equation of motion. The MD results indicate that the pseudo-rotation model gives a multi-exponential correlation function which oscillates at short times and is described by three exponential terms. The time scale is shorter than previously assumed but contain an intermediate time constant (1–2 ps). The electron spin resonance (ESR) spectral width at half height at frequencies of X-band, Q-band, 75 MHz, 150 MHz and 225 MHz can be reproduced at 320 K without any contributions from 4th or 6th rank ZFS interactions. Consequently, there are two mutually inconsistent dynamic models of the ZFS interaction which can describe the water proton T_{1}-NMRD (nuclear magnetic resonance dispersion) profile and the field dependent ESR spectra of the hydrated Gd(III) complex equally well.
Urea in the lysozyme solvation shell has been studied by utilizing a combination of urea , water NMR relaxation experiments and a molecular dynamics simulation of the urea–lysozyme system. Samples with lysozyme in the native fold in water as well as in 3 M urea have been studied, as well as denatured lysozyme in a 8.5 M urea solvent. The spin relaxation rates of the samples with folded protein show a clear field dependence, which is consistent with a few urea molecules having long residence times on the protein surface and preferentially located in pockets and grooves on the protein. By combining the 3 M urea NMR relaxation data and data from the MD simulation, a full parameter set of the relaxation model is found which can successfully predict the experimental relaxation rates of the 3 M urea sample. However, in the parameter fitting it is evident that the rotational dynamics of urea in the MD simulation is slightly too fast to be consistent with the NMR relaxation rates, perhaps a result of the fast dynamics of the TIP3P water model. The relaxation rates of urea in the proximity of the unfolded lysozyme lack field dependence, which can be interpreted as a loss of pockets and grooves on the denatured protein. The extracted model parameters from the 3 M sample are adjusted and tested on a simple model of the unfolded protein sample and are seen to be in agreement with the relaxation rates. It is shown that the combination of NMR relaxation and MD simulations can be used to create a microscopic picture of the solvent at the protein interface, which can be used for example in the study of chemical denaturation.
Molecular dynamics simulations of chymotrypsin inhibitor 2 in both water and in 10 M urea have been compared with respect to the energies of interaction between protein and solvent. The analysis yield clear and detailed information regarding the enthalpic driving force of urea-induced protein denaturation. The protein is kept in the folded structure by applying positional restraints on the alpha-carbons, thereby creating an equilibrium system from which appropriate driving forces for denaturation can be obtained. All protein atoms are classified as belonging to the backbone, the polar side chains or to the hydrophobic side chains. The interaction energies are extracted for each class separately. The commonly proposed mechanisms of urea denaturation, i.e. that urea interacts mainly with the backbone or with the hydrophobic side chains, can then be tested. The results show that urea decreases the Lennard-Jones interaction energies between protein and solvent by a large amount. The electrostatic energies are almost unaffected by the switch of solvent. The energetically favorable interaction between CI2 and the urea solvent will function as a driving force for the protein to increase its solvent accessible surface area as compared to the folded protein in water. The magnitude of the decrease in the Lennard-Jones energies for the hydrophobic and the hydrophilic side chains and for the backbone were similar. We therefore conclude that urea interacts favorably with the whole protein surface and that all parts of the protein are important in urea-induced denaturation.
The dynamics of chymotrypsin inhibitor 2 (CI2) in water, as well as in 10M urea, have been studied by Molecular Dynamics simulations. The analysis aims at investigating how local protein processes are affected by urea and how the perturbation by urea on the local level manifests itself in the kinetics of the global unfolding. The results show that the effect of urea on local processes depends upon the type of process at hand. An isolated two-residue contact on the surface of CI2 has a decreased frequency of rupture in the urea solvent. This is in contrast to the increased frequency of rupture of the hydrogen bonds in secondary structure elements in the urea solvent. It is proposed that the increase in the unfolding rates of complex protein processes is based upon the retardation of the refolding rate of small scale, isolated processes.
Proteins are known to denature in high concentrations of compounds such as urea or guanidinium chloride. However, the mechanism by which urea and guanidinium chloride destabilizes proteins is not yet known, despite many decades of reasearch. Attempts have been made to understand protein denaturation on a thermodynamic level as well as on a molecular level. The long term goal in the field is to merge the results of these two types of studies into one mechanism that covers both the microscopic and the macroscopic level. In this text we firstly review thermodynamic studies as well as spectroscopic and computer simulation studies of chemical denaturation. The results of the different types of studies is then merged together in order to find a consistent view on chemical denaturation. In contrast to common belief in the field, a high degree of consensus is found between the different studies and a molecular mechanism of urea-induced protein denaturation can therefore be proposed.
An extended Förster theory (EFT) is derived and outlined for electronic energy migration between two fluorescent molecules which are chemically identical, but photophysically non-identical. These molecules exhibit identical absorption and fluorescence spectra, while their fluorescence lifetimes differ. The latter means that the excitation probability becomes irreversible. Unlike the case of equal lifetimes, which is often referred to as, donor–donor energy migration (DDEM), the observed fluorescence relaxation is then no longer invariant to the energy migration process. To distinguish, the present case is therefore referred to as partial donor–donor energy migration (PDDEM). The EFT of PPDEM is described by a stochastic master equation (SME), which has been derived from the stochastic Liouville equation (SLE) of motion. The SME accounts for the reorienting as well as the translational motions of the interacting chromophores. Synthetic fluorescence lifetime and depolarisation data that mimics time-correlated single photon counting experiments have been generated and re-analysed. The rates of reorientation, as well as the orientational configurations of the interacting D-groups were examined. Moreover the EFT of PPDEM overcomes the classical 2-problem and the frequently applied approximation of 2 = 2/3 in the data analyses. An outline for the analyses of fluorescence lifetime and depolarisation data is also given, which might prove applicable to structural studies of D-labelled macromolecules, e.g. proteins. The EFT presented here brings the analyses of PDDEM data to the same level of molecular detail as that used in ESR- and NMR-spectroscopy.
The extended Förster theory (EFT) of electronic energy transport accounts for translational and rotational dynamics, which are neglected by the classical Förster theory (FT). EFT has been developed for electronic energy transfer within donor-acceptor pairs [Isaksson, et al, Phys. 16 Chem. Chem. Phys., 9, 1941(2007)] and donor-donor pairs [Johansson, et al, J. Chem. Phys., 105, 10896 (1996); Norlin, et al, Phys. Chem. Chem. Phys., 10, 6962(2008)]. For donors that exhibit different or identical non-exponential fluorescence relaxation within a donor-donor pair, the process of reverberating energy migration is reversible to a higher or lower degree. Here the impact of the EFT has been studied with respect to its influence on fluorescence quantum yields, fluorescence lifetimes as well as depolarisation experiments. The FT predicts relative fluorescence quantum yields which usually agree with the EFT within experimental accuracy, however, substantial deviations occurs in the steady-state and in particular the time-resolved depolarisation data.
A mathematical framework for translational Brownian motion on hypersurfaces is presented, using an imbedding of the surface and Ito diffusions in the ambient space. This includes a survey of Ito calculus and differential geometry. Computational methods for time correlation functions relevant to spin relaxation studies on curved interfaces are given, and explicit calculations of time correlation functions and order parameters for a “Rippled” surface are presented.
A four-site exchange model is developed in order to explain deuterium -nuclear magnetic relaxation dispersion (NMRD) profiles of acetonitrile in silica pore systems. The four-site exchange model comprises a bulk, surface and two types of burried or cavity sites. It is found that the residence time of acetonitrile- at a flat Si-surface is less than 100 ps. No bilayer-like ordering of acetonitrile is formed at the Si-surface because no quadrupole splitting was observed. The dispersion in the deuterium T1-NMRD profiles are due to relatively few so-called beta-sites with molecular residence time in the range 0.2-2 micro seconds. This deuterium T-NMR dispersion experiment suggest that the retention time of different analysts can be studied in terms of their residence time in beta sites.
Cellular uptake of dissolved methylmercury (MeHg) by phytoplankton is the most important point of entry for MeHg into aquatic food webs. However, the process is not fully understood. In this study we investigated the influence of chemical speciation on rate constants for MeHg accumulation by the freshwater green microalga Selenastrum capricornutum. We used six MeHg–thiol complexes with moderate but important structural differences commonly found in the environment. Rate constants for MeHg interactions with cells were determined for the MeHg–thiol treatments and a control assay containing the thermodynamically less stable MeHgOH complex. We found both elevated amounts of MeHg associated with whole cells and higher MeHg association rate constants in the control compared to the thiol treatments. Furthermore, the association rate constants were lower when algae were exposed to MeHg complexes with thiols of larger size and more “branched” chemical structure compared to complexes with simpler structure. The results further demonstrated that the thermodynamic stability and chemical structure of MeHg complexes in the medium is an important controlling factor for the rate of MeHg interactions with the cell surface, but not for the MeHg exchange rate across the membrane. Our results are in line with uptake mechanisms involving formation of MeHg complexes with cell surface ligands prior to internalization.
The 2H2O NMR powder line shapes and relaxation times, T1 and T2, of the liquid crystal L, the intermediate P and the gel L phases of dipalmitoylphosphatidylcholine (DPPC)/2H2O-system are analysed. The water structure and dynamics of the lipid/water interfaces of DPPC in the hydration regime, where all water molecules are associated to the interface, are described in terms of orientational order parameters and correlation times. The line shape of the ripple phase (P) is analysed assuming model parameters of the gel or liquid crystalline phase. The narrow line shape of the ripple phase is partly due to an extra average of the quadrupole interaction because of lateral diffusion along the curved surface, reducing the splitting with a factor 0.5–0.2 depending on the nature of the curved ripple surface. However, more importantly, an extra reduction of the quadrupole splitting may be due to the same reorganization of water, among bound sites with different signs of the order parameter, which also explains the increase in the quadrupole splitting with temperature observed in the liquid crystalline phase. The linewidths in 14N MAS NMR spectra clearly indicate slow dynamics of the polar headgroup in the ripple phase. The results indicate that the headgroup hydrations of the ripple and liquid crystalline phases are similar, while the acyl chains are still in their gel state in the ripple phase. The increased headgroup area introduces a stress, as confirmed by the slow headgroup dynamics, which causes the bilayer to curve in the ripple phase.
Present study combines NMR relaxation and line shape analysis for heavy water in the lamellar liquid crystalline phase of DDAO/(2)-H2O, NMR spin-lattice, spin-spin relaxation times and the quadrupole splitting are measured at two temperatures and three different water contents in the hydration regime. A molecular picture of water hydration of the DDAO/(2)-H2O interface is extracted, which indicates a much more rapid water translational diffusion along the detergent interface, as compared to phospholipid interfaces. The local order and dynamics of the bound water are, however, not changing much between the two interfaces. This indicates, that local interactions of water with the headgroup are not much dependent on the actual phase or detergent system. This work also presents clear experimental evidence for a dip at the magic angle of the H-2 powder spectrum, as theoretically predicted. Raising the temperature removes this observed dip at the isotropic frequency. This corresponds to an increase in the correlation time tau (c) from 8.5 ns at 25 degreesC to 20 ns at 55 degreesC, where tau (c) is related to translational dynamics of water along the detergent/water interface. However, this counter-intuitive increase in tau (c) with temperature may be interpreted as a reorganization of water at the interface, as is further supported by increasing quadrupolar splittings with increasing temperature.
The direct calculation method of slow-motion EPR spectra is presented based on the stochastic Liouville equation in the Langevin form, using the trajectories of Brownian dynamics (BD) simulations. We have developed a model of EPR slow-motion lineshapes describing a probe molecule residing in a curved bilayer system. Two dynamic processes are active: one is local reorientation motion of the lipid chain, and the second is the lateral diffusion of the spin probe along the curved lipid bilayer surface. The trajectories of two independent BD simulations are combined in order to describe the stochastic fluctuation of the electron spin-lattice Hamiltonian. The method of obtaining the Langevin equation describing lateral diffusion from the diffusion equation is discussed in detail. We solve the stochastic Liouville-von Neumann equation in the semiclassical approximation. EPR slow-motion lineshapes are obtained together with electron spin correlation functions. The synthesis of classical simulation methods and the lineshape model is illustrated by calculating a number of curvature dependent EPR lineshapes. Two curved model surfaces are considered namely the rippled z=a sin(bx) surface and the so called "Baltic Sea" z=a[sin(bx)+sin(by)].
A detailed analysis of the classic Stern–Gerlach experiment is presented. An analytical simple solution is presented for the quantum description of the translational and spin dynamics of a silver atom in a magnetic field with a gradient along a single z-direction. This description is then used to obtain an approximate quantum description of the more realistic case with a magnetic field gradient also in a second y-direction. An explicit relation is derived for how an initial off center deviation in the y-direction affects the final result observed at the detector. This shows that the “mouth shape” pattern at the detector observed in the original Stern–Gerlach experiment is a generic consequence of the gradient in the y-direction. This is followed by a discussion of the spin dynamics during the entry of the silver atom into the magnet. An analytical relation is derived for a simplified case of a field only along the z-direction. A central question for the conceptual understanding of the Stern–Gerlach experiment has been how an initially unpolarized spin ends up in a polarized state at the detector. It is argued that this can be understood with the use of the adiabatic approximation. When the atoms first experience the magnetic field outside the magnet, there is in general a change in the spin state, which transforms from a degenerate eigenstate in the absence of a field into one of two possible non-degenerate states in the field. If the direction of the field changes during the passage through the device, there is a corresponding adiabatic change of the spin state. It is shown that an application of the adiabatic approximation in this way is consistent with the previously derived exact relations.
We analyze a coincidence Stern-Gerlach measurement often discussed in connection with the derivation and illustration of Bell's theorem. The treatment is based on our recent analysis of the original Stern-Gerlach experiment (PCCP, 14, 1677‐1684 (2012)), where it is concluded that it is necessary to include a spin relaxation process to account for the experimental observations. We consider two limiting cases of a coincidence measurement using both an analytical and a numerical description. In on limit relaxation effects are neglected. In this case the correlation between the two spins present in the initial state is conserved during the passage through the magnets. However, at exit the z coordinate along the magnetic field gradient is randomly distributed between the two extreme values. In the other limit T_{2} relaxation is assumed to be fast relative to the time of flight through the magnet. In this case the z coordinate takes one of two possible values as observed in the original Stern‐Gerlach experiment. Due to the presence of a relaxation process involving transfer of angular momentum between particle and magnet the initially entangled spin state changes character leading to a loss of correlation between the two spins. In the original derivations of Bell's theorem based on a coincidence Stern‐Gerlach setup one assumes both a perfect correlation between the spins and only two possible values for the z‐coordinate on exit. According to the present calculations one can satisfy either of these conditions but not both simultaneously.
The classical Stern-Gerlach experiment is analyzed with an emphasis on the spin dynamics. The central question asked is whether there occurs a relaxation of the spin angular momentum during the time the particle passes through the Stern-Gerlach magnet. We examine in particular the transverse relaxation, involving angular momentum exchange between the spin of the particles and the spins of the magnet. A method is presented describing relaxation effects at an individual particle level. This leads to a stochastic equation of motion for the spins. This is coupled to a classical equation of motion for the particle translation. The experimental situation is then modeled through simulations of individual trajectories using two sets of parameter choices and three different sets of initial conditions. The two main conclusions are: (A) if the coupling between the magnet and the spin is solely described by the Zeeman interaction with the average magnetic field the simulations show a clear disagreement with the experimental observation of Stern and Gerlach. (B) If one, on the other hand, also allows for a T(2) relaxation time shorter than the passage time one can obtain a practically quantitative agreement with the experimental observations. These conclusions are at variance with the standard textbook explanation of the Stern-Gerlach experiment.
Snowflakes and ordinary hexagonal ice were studied measuring water proton spin–lattice relaxation rate R1(ωI)-nuclear magnetic resonance dispersion (NMRD) profiles at proton Larmor frequencies ranging from 1 to 30 MHz and at different temperatures ranging from −2◦C to −10◦C. The spin–spin relaxation rate 1/ 1/T2(ωI) was determined at a single Larmor frequency of 16.3 MHz. The high-field wing of the proton R1(ωI)-NMRD profile was characterised by two parameters: a correlation time τc which described the dipole–dipole spectral density, and the relaxation rate at low fields R ^{max} _{real} (0) which was determined from T _{2} . The correlation time τc depended on the dynamic model used. A rotation diffusion model yield approximatively 3μs at −3◦C to about 5μs at 10◦C, whereas for a more realistic six-site discrete exchange model, the correlation times decreased slightly to about 80% for the same temperature interval. Proton dipole–dipole interactions were divided into intramolecular and intermolecular contributions where the intermolecular contribution was about 0.4–0.8 × the intramolecular contribution. It was not possible to discriminate between the dynamic models or to detect ice/water interface effects by comparing the NMRD data from snowflakes with ordinary hexagonal ice data.
Ganglioside GM1 (GM1) micelles have been studied by means of water proton T1 NMRD experiment. The ﬁeld dependent spin-lattice relaxation rates were measured for Larmor frequencies ranging from 0.1 to 40 MHz and for two micelle concentrations at three temperatures (T=10,15,20oC). The proton T1 NMRD-proﬁles are well described by assuming two proton pools are responsible for the dispersion curves. The proton pools are characterized by an effective correlation time and a proton fraction. The largest correlation time, τc,1 ≈ 130−160 ns, is determined by the low ﬁeld part of the NMRD proﬁle. The second correlation time, τc,2 ≈ 12 ns, is determined by the high ﬁeldpartoftheNMRDproﬁle. Theradiusoftheganglioside micelles has previously been determined as about 54 using ﬂuorescence experiments and with Stoke-Einstein relation the reorientation correlation time becomes τR= 120-165 ns depending on the temperature dependence of the water viscosity. It is thus plausible to identify one pool of waterprotons, characterized by the largest effective correlation time, as corresponding to waters residing in the headgroup withanorderparameterS6=0andτc,1 ≈ τR orcorresponding to labile protons with a τc,1as the mean life time. The proton NMRD proﬁle reveal a second Lorenzian which also can eitherbelabileandexchangingGangliosideprotonsorwater moleculesresidingintheheadgroupwithameanlifetimeas approximately 12 ns. The proton NMRD experiment cannot discriminate between these two cases.
A NMR line shape/spin relaxation model is developed for (H2O)-H-2 studies of structure and dynamics of the lipid-water interfaces of phosphatidylcholine bilayers. A line shape function describing the orientational dependence of (H2O)-H-2 is derived. In addition, also expressions of the observed quadrupole splitting and the spin-lattice relaxation rate are derived within the same dynamic model. The model comprises two chemically interchanging fractions of water namely, "free", and "bound". There are four molecular parameters characterizing the "bound" water of the lipid water interface, namely, (1) the fraction water molecules bound to lipid molecules, (2) the local water order parameter S-0(Pd), (3) the order parameter S-0(dD) which describes the averaged "bound" water, and an effective correlation time (4) tau(c), characterizing water translational diffusion at the interface. This model allows for analyzing quadrupole splittings, spin-lattice relaxation rates, and water powder line shapes. Thus, dynamics as well as structural information about the water molecules residing in the water lipid interface may be extracted. In the reinterpretation of (H2O)-H-2 powder spectra obtained for lamellar phases of dipalmitoylphosphatidylcholine (DPPC), the results clearly indicate that S-0(dD)(L-alpha) > S-0(dD)(L-beta') when comparing the liquid crystalline phase with 10-11 water molecules per lipid molecule and the gel phase with 3.5-4.2 water molecules per lipid molecule. Whereas the order of the perturbed water is similar in both phases, S-0(Pd)(L-alpha) approximate to S-0(Pd)(L-beta'). The lateral diffusion is characterized by a correlation time tau(c) > 60 ns but cannot be determined without measuring the spin-lattice relaxation measurements.
The field dependence of the proton (I) spin-lattice relaxation rate is calculated for a dipole-dipole coupled spin pair, (I = 1/2) - (S = 1), where the quadrupole nucleus (S) is ^{2}H or ^{14}N with asymmetry parameter η = 0. The observed relaxation profile shows a marked enhancement for equal proton Larmor and quadrupole spin frequencies (i.e. ω_{I} = ω_{Q}). This phenomenon is referred to as the quadrupole dip, and has been observed, for instance, in ^{14}N-^{1}H amide groups of immobilized proteins. In this work, an analysis of the observed relaxation enhancement is presented when the dipole-dipole coupling and the quadrupole interaction are modulated by the overall re-orientational motion. A characteristic low field dispersion is observed when (3/2) τ_{R}ω_{I} ≥ 1, where τ_{R} is the rotational correlation time and ω_{I} is the proton Larmor frequency. At higher fields, the relaxation peak exhibits a Lorentzian-like line shape, , which is centred at the quadrupole frequency. The quadrupole spin system shows a spin-lattice T_{1Q} and a spin-spin relaxation time T_{2Q} that become equal in the zero field limit. In the slow tumbling limit, the quadrupole spin relaxation times, T_{1Q}, T_{2Q}, are equal to (3/2)τ_{R}.
An analysis, based on the stochastic Liouville approach, is presented of the R(1)-NMRD or field dependent spin-lattice relaxation rate of amide protons. The proton relaxivity, displayed as R(1)-NMRD profiles, is calculated for a reorienting (1)H-(14)N spin group, where the inter spin coupling is due to spin dipole-dipole coupling or the scalar coupling. The quadrupole nucleus (14)N has an asymmetry parameter eta = 0.4 and a quadrupole interaction which is modulated by the overall reorientational motion of the protein. In the very slow reorientational regime, omega(Q)tau(R)>> 1 and tau(R)>/= 2.0 mus, both the dipole-dipole coupling and the scalar coupling yield a T(1)-NMRD profile with three marked peaks of proton spin relaxation enhancement. These peaks appear when the proton Larmor frequency, omega(I), matches the nuclear quadrupole spin transition frequencies: omega(1) = omega(Q)2eta/3, omega(2) = omega(Q)(1 -eta/3) and omega(3) = omega(Q)(1 + eta/3), and the quadrupole spin system thus acts as a relaxation sink. The relative relaxation enhancements of the peaks are different for the dipole-dipole and the scalar coupling. Considering the dipole-dipole coupling, the low frequency peak, omega(1), is small compared to the high field peaks whereas for the scalar coupling the situation is changed. For slow tumbling proteins with a correlation time of tau(R) = 400 ns, omega(2) and omega(3) are not resolved but become one relatively broad peak. At even faster reorientation, tau(R) < 60 ns, the marked peaks disappear. In this motional regime, the main effect of the cross relaxation phenomenon is a subtle perturbation of the main amide proton T(1) NMRD dispersion. The low field part of it can be approximately described by a Lorentzian function: R(DD,SC)(0.01)/(1 + (omega(I)tau(R)3/2)(2)) whereas the high field part coincides with R(DD,SC)(0.01)/(1 + (omega(I)tau(R))(2)).
For immobilized protein the water proton T _{1}-NMRD profile displays three enhanced relaxation peaks (QP). For slow tumbling proteins these relaxation peaks are not experimentally observed. However, the theoretically determined QP effect on the amide proton T _{1}-NMRD profile displays a distorted Lorentzian dispersion profile. The question arises as to whether there is also a distortion of the water-proton T _{1}-NMRD profile due to QP. The model of Sunde and Halle [J. Magn. Reson. 203, 257 (2010)] predicts a decreasing QP relaxation contribution and, with the aid of a model for tumbling proteins [P.-O. Westlund, Phys. Chem. Chem. Phys, 12, 3136 (2010)], it is shown that the QP effect is absent in water-proton T _{1}-NMRD profiles for slow tumbling proteins with τ_{R} < 1 µs, τ_{I}.
It is demonstrated, using a Liouville formalism, that the relative motion of two atoms can result in the emission of photons and conversely that photons can be absorbed to excite the relative translational motion. The mechanism responsible for the energy transfer between the radiation field and the translational motion of the atoms is a dynamic version of the long-range Casimir-Polder interaction between two fixed atoms. The phenomenon is analogous to the dynamic Casimir effect discussed for moving macro- (or meso)scopic objects and we term it the dynamic Casimir-Polder effect. The absorption or emission is a two-photon process and we find that the transition probability is proportional to the spectral density of a correlation function involving the relative translational motion of two atoms. An energy transfer only occurs for photons with energies smaller than or of the same magnitude as the thermal energy. The effect provides a microscopic mechanism for establishing thermal equilibrium between the radiation field and a gas. A sufficiently large volume of gas would be perceived as a black-body radiator. Applications of the dynamic Casimir-Polder effect might be found in the microscopic description of the cosmic low-temperature black-body radiation.
We analyze the spin coincidence experiment considered by Bell in the derivation of Bells theorem. We solve the equation of motion for the spin system with a spin Hamiltonian, H_{z}, where the magnetic field is only in the z-direction. For the specific case of the coincidence experiment where the two magnets have the same orientation the Hamiltonian H_{z} commutes with the total spin I_{z}, which thus emerges as a constant of the motion. Bells argument is then that an observation of spin up at one magnet A necessarily implies spin down at the other B. For an isolated spin system A-B with classical translational degrees of freedom and an initial spin singlet state there is no force on the spin particles A and B. The spins are fully entangled but none of the spin particles A or B are deflected by the Stern-Gerlach magnets. This result is not compatible with Bells assumption that spin 1/2 particles are deected in a Stern-Gerlach device. Assuming a more realistic Hamiltonian H_{z} + H_{x} including a gradient in x direction the total I_{z} is not conserved and fully entanglement is not expected in this case. The conclusion is that Bells theorem is not applicable to spin coincidence measurement originally discussed by Bell.
The low field ESR lineshape and the electron spin-lattice relaxation correlation function are calculated using the stochastic Liouville theory for an effective electron spin quantum number S = 1. When an axially symmetric permanent zero field splitting provides the dominant relaxation mechanism, and when it is much larger than the rotational diffusion constant, it is shown that both electron spin correlation functions S(0)S(t) (n = 0,1) are characterized by the same relaxation time tau(S) = (4D(R))(-1). This confirms the conjectures made by Schaefle and Sharp, J. Chem. Phys., 2004, 121, 5287 and by Fries and Belorizky, J. Chem. Phys., 2005, 123, 124510, based on numerical results using a different formalism. The stochastic Liouville approach also gives the paramagnetically enhanced nuclear spin relaxation time constants, T(1) and T(2), and the ESR lineshape function I(omega). In particular, the L-band (B(0) = 0.035 T) ESR spectrum of a low symmetry Ni(ii)-complex with a cylindrical ZFS tensor is shown to be detectable at sufficiently slowly reorientation of the complex. The analysis shows that the L-band spectrum becomes similar to the zero-field spectrum with a electron spin relaxation time tau(S) = (4D(R))(-1).
A generalization of the modified SBM theory is developed in closed analytical form. The theory is applied to describe the paramagnetically enhanced water proton spin–lattice relaxation rates of the aqueous-systems containing a gadolinium(S=7/2) complex(MS-325) in the presence or absence of human serum albumin (HSA). MS-325 binds to HSA: in the absence of the protein the reorientational time, τR, is short, but when HSA is added τR becomes much longer. In this way, the effect of reorientational motion, static (Δs), and transient (Δt) zero-field splitting (ZFS) interactions on both the water proton relaxivity and the Gd ESR lineshapes are investigated.
Two dynamic models of electron spin relaxation are presented, characterized by transient and static ZFS-interactions. X-, Q-, and W-bands ESR spectra of MS-325+HSA are analyzed in order to describe the effect on the electron spin system upon binding to a macromolecule. A computer program based on this theory is developed which calculates solvent water proton T1 NMRD profiles and the corresponding X-, Q-, U-, and W-bands ESR lineshapes.
A complete description of the T1-NMRD profiles and the ESR lineshape of Gd(III) complexes (S = 7/2) was presented using second-order perturbation theory (GSBM) by Zhou et al. [J. Magn. Reson. 167 (2004) 147]. This report compares the GSBM with the stochastic Liouville approach (SLA) to determine the validity of the closed analytical expressions of NMRD and the ESR lineshape functions. Both approaches give the same results at high fields while a very small divergence is observed for X- and W-band ESR lineshapes when the magnitude of the perturbation term times the correlation time approaches the limit of the perturbation regime, ΔZFSτf ≈ 0.1. There was a clear discrepancy between the theoretical GSBM X-band spectrum and the recorded ESR spectrum of the Gd(III) MS-325 + HSA complex. This is probably due to a slow-motion effect caused by a slow modulation of the ZFS interaction. The characteristic correlation time of this slow modulation is in the range of 150 ps, which therefore cannot be due to the reorientational motion of the whole MS-325 + HSA complex.
Water proton T1-NMRD profiles of the Gd(H2O)83+ complex have been recorded at three temperatures and at four concentrations of glycerol. The analysis is performed using both the generalized Solomon–Bloembergen–Morgan (GSBM) theory [J. Magn. Reson. 167(2004), 147–160], and the stochastic Liouville approach (SLA). The GSBM approach uses a two processes dynamic model of the zero-field splitting (ZFS) correlation function whereas SLA uses a single process model. Both models reproduce the proton T1-NMRD profiles well. However, the model parameters extracted from the two analyses, yield different ESR X-band spectra which moreover do not reproduce the experimental ESR spectra. It is shown that the analyses of the proton T1-NMRD profiles recorded for a solution Gd(H2O)83+ ions are relatively insensitive to the slow modulation part of dynamic model of the ZFS interaction correlation function. The description of the electron spin system results in a very small static ZFS, while recent ESR lineshape analysis indicates that the contribution from the static ZFS is important. Analysis of proton T1-NMRD profiles of Gd(H2O)83+ complex do result in a description of the electron spin system but these microscopic parameters are uncertain unless they also are tested in a ESR-lineshape analysis.
X-band ESR spectra of Gd-aqua complex in various weight concentration of glycerol have been recorded at four temperatures. The interpretation of the ESR linewidth is preformed using both the stochastic Liouville approach (SLA) and a perturbation theory. The SLA uses a one dynamic model of the zero-field splitting whereas the perturbation approach uses a two dynamic model. Both models can reproduce the variation of the linewidth with respect to viscosity. In the SLA model, both the zero-field splitting (ZFS) interaction and the correlation time vary with the glycerol content. In the two dynamic perturbation model, only the correlation times are viscosity dependent. The two models give different NMRD profiles.
A new computational method is developed for calculating H-2 NMR lineshapes of H2O in microheterogeneous systems, such as lyotropic liquid crystals that exhibit curved lipid/water interfaces. The method presented is based on the stochastic Liouville equation (SLE) in its Langevin form. This means that the Liouville equation of motion is combined with Brownian dynamics simulations to describe the stochastic spin - lattice Liouvillian. The NMR relaxation is caused by translational diffusion of the heavy water molecules, along the curved (H2O)-H-2/lipid interface. The model used is a nodal surface approximation of the cubic symmetric gyroid minimal surface. This unit cell is then isotropically expanded or distorted in two dimension. The changes in (H2O)-H-2 NMR lineshapes have been calculated for the enlarged or the distorted cubic unit cell. The timescale of the residual quadrupole interaction, which determines the NMR lineshape, ranges from the Redfield regime to the slow-motional regime depending on the curvature of the interface. The distortion of the cubic phase illustrates the possibility to explore the intermediate interfaces of a phase transition, by means of (H2O)-H-2 lineshape analysis.