Alex D. Bain

Coherence levels and coherence pathways in NMR. A simple way to design phase cycling procedures

Abstract The paper presents a simple way of calculating what a phase cycling procedure in a pulse NMR experiment does. This is done by exploiting the fact that if a pulse is phase shifted by angle φ and if it changes the coherence level by Δm, then the effect on the coherence is a multiplication by eiΔmφ The coherence level is a fundamental quantum number of the system-the z component of the Liouville space angular momentum—and is a generalization of the number of quanta in a multiple-quantum transition. The sequence of coherence levels that a signal passes through between the first pulse and the time it reaches the receiver is the coherence pathway. This pathway, combined with the above formula, provides a powerful method for analyzing pulse NMR experiments.

ApoA-I mimetic peptides with differing ability to inhibit atherosclerosis also exhibit differences in their interactions with membrane bilayers.

Two homologous apoA-I mimetic peptides, 3F-2 and 3F14, differ in their in vitro antiatherogenic properties (Epand, R. M., Epand, R. F., Sayer, B. G., Datta, G., Chaddha, M., and Anantharamaiah, G. M. (2004) J. Biol. Chem. 279, 51404-51414). In the present work, we demonstrate that the peptide 3F-2, which has more potent anti-inflammatory activity in vitro when administered intraperitoneally to female apoE null mice (20 μg/mouse/day) for 6 weeks, inhibits atherosclerosis (lesion area 15,800 ± 1000 μm2, n = 29), whereas 3F14 does not (lesion area 20,400 ± 1000 μm2, n = 26) compared with control saline administered (19,900 ± 1400 μm2, n = 22). Plasma distribution of the peptides differs in that 3F-2 preferentially associates with high density lipoprotein, whereas 3F14 preferentially associates with apoB-containing particles. After intraperitoneal injection of 14C-labeled peptides, 3F14 reaches a higher maximal concentration and has a longer half-time of elimination than 3F-2. A study of the effect of these peptides on the motional and organizational properties of phospholipid bilayers, using several NMR methods, demonstrates that the two peptides insert to different extents into membranes. 3F-2 with aromatic residues at the center of the nonpolar face partitions closer to the phospholipid head group compared with 3F14. In contrast, only 3F14 affects the terminal methyl group of the acyl chain, decreasing the 2H order parameter and at the same time also decreasing the molecular motion of this methyl group. This dual effect of 3F14 can be explained in terms of the cross-sectional shape of the amphipathic helix. These results support the proposal that the molecular basis for the difference in the biological activities of the two peptides lies with their different interactions with membranes.

Properties of Mixtures of Cholesterol with Phosphatidylcholine or with Phosphatidylserine Studied by 13 C Magic Angle Spinning Nuclear Magnetic Resonance

The behavior of cholesterol is different in mixtures with phosphatidylcholine as compared with phosphatidylserine. In (13)C cross polarization/magic angle spinning nuclear magnetic resonance spectra, resonance peaks of the vinylic carbons of cholesterol are a doublet in samples containing 0.3 or 0.5 mol fraction cholesterol with 1-palmitoyl-2-oleoyl phosphatidylserine (POPS) or in cholesterol monohydrate crystals, but a singlet with mixtures of cholesterol and 1-palmitoyl-2-oleoyl phosphatidylcholine (POPC). At these molar fractions of cholesterol with POPS, resonances of the C-18 of cholesterol appear at the same chemical shifts as in pure cholesterol monohydrate crystals. These resonances do not appear in samples of POPS with 0.2 mol fraction cholesterol or with POPC up to 0.5 mol fraction cholesterol. In addition, there is another resonance from the cholesterol C18 that appears in all of the mixtures of phospholipid and cholesterol but not in pure cholesterol monohydrate crystals. Using direct polarization, the fraction of cholesterol present as crystallites in POPS with 0.5 mol fraction cholesterol is found to be 80%, whereas with the same mol fraction of cholesterol and POPC none of the cholesterol is crystalline. After many hours of incubation, cholesterol monohydrate crystals in POPS undergo a change that results in an increase in the intensity of certain resonances of cholesterol monohydrate in (13)C cross polarization/magic angle spinning nuclear magnetic resonance, indicating a rigidification of the C and D rings of cholesterol but not other regions of the molecule.

Superspin in NMR: Application to the ABX system

Abstract Superspin is defined to be the angular momentum associated with operators on a nuclear spin system. These operators correspond to magnetizations and the effect of a pulse on these magnetizations is most easily calculated using this formalism. The derivation of the superspin characteristics of the magnetizations in an ABX system is given in detail.

Exact calculation, using angular momentum, of combined Zeeman and quadrupolar interactions in NMR

The NMR of nuclei with spins greater than ½ is often strongly influenced by the quadrupole interaction. This combination of Zeeman and quadrupole terms can usually be treated using perturbation theory, but an exact calculation is also needed. We explain an exact approach that eliminates the evaluation of commutators of complicated operators. Instead, the calculation is based on matrix elements of the Liouvillian, the commutator with the Hamiltonian. The spectrum can then be calculated directly from the eigenvectors and eigenvalues of the Liouvillian. With the aid of angular momentum methods, it can be shown that the quadupole interaction for spin I is fully determined by only (2I −1) reduced matrix elements—for spin 3/2, this means only two quantities. The exact nature of the various basis operators is not needed, since the calculation only needs the angular momentum quantum numbers. The full Liouvillian matrix can be calculated from selection rules and the Wigner-Eckart theorem. Furthermore, we present an expression for these reduced matrix elements which is valid for any spin. This theory covers the whole range from quadrupole-perturbed NMR spectra to Zeeman-perturbed nuclear quadrupole resonance.

Simulation of many-spin system dynamics via sparse matrix methodology

A sparse-matrix-based numerical method is constructed to simulate NMR spectra of many-spin systems, including the effects of chemical exchange and/or relaxation. The associated computational demands are predicted to scale like O(22n), as the number of spins n increases. This is vastly superior to the inevitable O(26n) scaling of conventional Householder-based methodology. The improved scaling is verified via numerical computations of simple isomerization systems with four to nine spins. The new method is based on splitting the propagator and use of Chebyshev polynomial expansion of the exponential function.

Liouvillians in NMR: the direct method revisited.

1. Notation and basic principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 1.1. Preamble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 1.2. Vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 1.3. Angular momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 1.4. Operators and superoperators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 1.5. Useful definitions and equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 2. The direct method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 2.1. Spin-1/2 and the Bloch Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 2.2. Calculation of matrix elements of the Liouvillian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 3. The rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 3.1. Basis sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 3.2. Formulae for specific NMR interactions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 4. Existence of formulae for any commutator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 5. Angular momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 5.1. Wigner–Eckart theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 5.2. Selection rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 6. Details of rules for Liouvillians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 6.1. Zeeman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 6.2. Radiofrequency pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 6.3. Quadrupole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 6.4. Scalar coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 6.5. Dipolar coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 7. Derivation of reduced matrix elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 8. Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 8.1. The Single Spin-1/2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

Ground state gas and solution phase conformational dynamics of polar processes: Furfural systems

The conductorlike continuum solvation model, modified for ab initio in the quantum chemistry program GAMESS, implemented at the Moller–Plesset Order 2 (MP2) level of theory has been applied to a group of push–pull pyrrole systems to illustrate the effects of donor/acceptor and solvation on the stability and energetics of such systems. The most accurate theoretical gas and solution phase data to date has been presented for the parent furan-2-carbaldehyde (furfural) system, and predictions made for three additional analogues, thiophene-2-carbaldehyde, pyrrole2-carbaldehyde, and, cyclopentadiene-1-carbaldehyde. Solvent effects on internal rotational barriers in all systems were evaluated over six different values of dielectric, using the new method. Calculated electrostatic energies are shown to be highly sensitive to level of theory incorporated.

Definitive Solid-State 185/187Re NMR Spectral Evidence for and Analysis of the Origin of High-Order Quadrupole-Induced Effects for I = 5/2

Rhenium-185/187 solid-state nuclear magnetic resonance (SSNMR) experiments using NaReO(4) and NH(4)ReO(4) powders provide unambiguous evidence for the existence of high-order quadrupole-induced effects (HOQIE) in SSNMR spectra. Fine structure, not predicted by second-order perturbation theory, has been observed in the (185/187)Re SSNMR spectrum of NaReO(4) at 11.75 T, where the ratio of the Larmor frequency (ν(0)) to the quadrupole frequency (ν(Q)) is ∼2.6. This is the first experimental observation that under static conditions, HOQIE can directly manifest in SSNMR powder patterns as additional fine structure. Using NMR simulation software which includes the quadrupole interaction (QI) exactly, extremely large (185/187)Re nuclear quadrupole coupling constants (C(Q)) are accurately determined. QI parameters are confirmed independently using solid-state (185/187)Re nuclear quadrupole resonance (NQR). We explain the spectral origin of the HOQIE and provide general guidelines that may be used to assess when HOQIE may impact the interpretation of the SSNMR powder pattern of any spin-5/2 nucleus in a large, axially symmetric electric field gradient (EFG). We also quantify the errors incurred when modeling SSNMR spectra for any spin-5/2 nucleus within an axial EFG using second-order perturbation theory. Lastly, we measure rhenium chemical shifts in the solid state for the first time.

Chapter 2 - Chemical Exchange

A general overview of chemical exchange and dynamic NMR is presented. This includes a brief overview of the theory, some comments on new techniques and methodology and some personal comments from the author on trends and opinions. Following that are some recent examples of applications (2005 and later). These are mainly within the chemical field (including solid-state NMR), but there are some references to studies of exchange in biological macromolecules. The list of applications is by no means complete, but it is hoped that most of the major fields have been mentioned, and that the selection is representative. The conclusion is that dynamic effects are very widespread in the NMR of many molecules, and that there is a wide range of experiments that are becoming accessible to almost all NMR spectroscopists.