Malcolm H. Levitt

Theoretical studies of enzymic reactions: Dielectric, electrostatic and steric stabilization of the carbonium ion in the reaction of lysozyme

Abstract A general method for detailed study of enzymic reactions is presented. The method considers the complete enzyme-substrate complex together with the surrounding solvent and evaluates all the different quantum mechanical and classical energy factors that can affect the reaction pathway. These factors include the quantum mechanical energies associated with bond cleavage and charge redistribution of the substrate and the classical energies of steric and electrostatic interactions between the substrate and the enzyme. The electrostatic polarization of the enzyme atoms and the orientation of the dipoles of the surrounding water molecules is simulated by a microscopic dielectric model. The solvation energy resulting from this polarization is considerable and must be included in any realistic calculation of chemical reactions involving anything more than an isolated molecule in vacuo . Without it, acidic groups can never become ionized and the charge distribution on the substrate will not be reasonable. The same dielectric model can also be used to study the reaction of the substrate in solution. In this way the reaction in solution can be compared with the enzymic reaction. In this paper we study the stability of the carbonium ion intermediate formed in the cleavage of a glycosidic bond by lysozyme. It is found that electrostatic stabilization is an important factor in increasing the rate of the reaction step that leads to the formation of the carbonium ion intermediate. Steric factors, such as the strain of the substrate on binding to lysozyme, do not seem to contribute significantly.

Broadband dipolar recoupling in the nuclear magnetic resonance of rotating solids: A compensated C7 pulse sequence

We introduce an improved variant of the C7 pulse-sequence for efficient recoupling of spin-1/2 pair dipolar interactions in magic-angle spinning solid-state NMR spectroscopy. The tolerance of C7 toward isotropic as well as anisotropic chemical shift offsets and rf inhomogeneity is improved considerably by replacing the original basic element Cφ44=(2π)φ(2π)φ+π with the cyclically permuted element Cφ143=(π/2)φ(2π)φ+π(3π/2)φ. The improved performance of this permutationally offset stabilized variant of C7 is analyzed by average Hamiltonian theory to fifth order, numerical simulations, and demonstrated by experiments on powder samples of doubly 13C-labeled barium oxalate hemihydrate and diammonium fumarate.

Efficient dipolar recoupling in the NMR of rotating solids. A sevenfold symmetric radiofrequency pulse sequence

Abstract A new radiofrequency pulse sequence is introduced for the efficient reintroduction of magnetic dipolar couplings in the magic-angle-spinning NMR of solids. The sequence involves seven phase-shifted radiofrequency pulse cycles, timed to span two rotational periods of the sample. Double-quantum coherences are excited with high efficiency in a rotating powder sample of zinc acetate- 13 C 2 .

DOUBLE-QUANTUM HOMONUCLEAR ROTARY RESONANCE : EFFICIENT DIPOLAR RECOVERY IN MAGIC-ANGLE SPINNING NUCLEAR MAGNETIC RESONANCE

We describe an efficient method for the recovery of homonuclear dipole–dipole interactions in magic‐angle spinning NMR. Double‐quantum homonuclear rotary resonance (2Q‐HORROR) is established by fulfilling the condition ωr=2ω1, where ωr is the sample rotation frequency and ω1 is the nutation frequency around an applied resonant radio frequency (rf) field. This resonance can be used for double‐quantum filtering and measurement of homonuclear dipolar interactions in the presence of magic‐angle spinning. The spin dynamics depend only weakly on crystallite orientation allowing good performance for powder samples. Chemical shift effects are suppressed to zeroth order. The method is demonstrated for singly and doubly 13C labeled L‐alanine.

Assignment of carbon-13 NMR spectra via double-quantum coherence

The spin-spin coupling between carbon-13 nuclei is a particularly useful property on which to base a determination of the network of linkages within an organic molecule. One-bond carbon-carbon couplings are readily distinguished from long-range couplings on the basis of their magnitudes, identifying adjacent carbon atoms in an unambiguous manner. For materials with the natural isotopic abundance (l%), it is sufficient to consider only pairs of interacting carbon-13 nuclei in any given molecule, so the corresponding spin systems are very simple: AX, AB, or AZ. Recent experiments (Z-3) have exploited this principle through a technique which suppresses the strong resonances from isolated carbon-13 spins, revealing the weak satellite spectrum from molecules with two coupled carbon-13 spins. Good discrimination is achieved by excitation of double-quantum coherence (d-8), which exhibits a characteristic dependence on the phase of a radiofrequency pulse, different from that of single-quantum coherence or longitudinal magnetization. A phase-cycling procedure suppresses the unwanted strong signals leaving only signals derived from double-quantum coherence. Only the coupled carbon spins can generate such double-quantum signals. Each carbon site may be directly coupled to as many as four other sites, and since the coupling constants are often very similar in magnitude, assignment to specific pairs of carbon atoms cannot always be made on the basis of coupling constants alone. Fortunately the coupled spins can be identified by means of a different criterion-the frequency of the double-quantum coherence, which is equal to the sum of the chemical shifts of the two carbon sites, measured with respect to the transmitter frequency. Since double-quantum coherence may not be observed directly (5) it is converted into transverse nuclear magnetization and its frequency is determined by means of a two-dimensional Fourier transform experiment (5, 9). This method has recently been used to establish the connectivity of the carbon atoms in Sa-androstane (3). In its simple form the experiment uses the pulse sequence shown in Fig. la. The first three pulses serve to excite double-quantum coherence, which is allowed to evolve for a variable period tl, when the fourth pulse converts it back into detectable transverse nuclear magnetization. In this reconversion process, only the imaginary component of double-quantum coherence is recovered, leaving an ambiguity about the sense of precession of the doublequantum coherence during tl. In the resulting two-dimensional spectrum S(F,, F2), the signs of the double-quantum frequencies are not determined, and this could be a critical problem for spectra of any complexity. The purpose of the present communication is to describe an extension of the method which allows the sense of this precession to be determined by detecting the

Compensation for Pulse Imperfections in NMR Spin-Echo Experiments

Abstract The refocusing pulse R Y ( π ) of a Carr-Purcell spin-echo experiment may be replaced with a composite sequence of three pulses, R X ( π /2) R Y ( π ) R X ( π /2), which compensates the effects of pulse length errors due to spatial inhomogeneity of the radiofrequency field. An analysis in terms of rotation operators indicates that the overall effect of the sequence R X ( π /2 + Δ ) R Y ( π + 2 Δ ) R X ( π /2 + Δ ) is equivalent to an exact π pulse applied about an axis in the XY plane which is phase shifted by an angle Δ with respect to the Y axis. Good spin state inversion is therefore achieved, but there is a phase shift of 2Δ for odd-numbered echoes, which is canceled on even-numbered echoes. For spin systems with no homonuclear coupling, the degree of compensation exceeds that of the well-known Meiboom-Gill modificaation, while for coupled spin systems the composite pulse scheme provides effective compensation whereas the Meiboom-Gill method breaks down. The technique has been tested on the proton spectrum of 1,1,2-trichloroethane with deliberately misset refocusing pulses; gross fluctuations in intensity are observed unless the composite pulses are used. A slightly different composite pulse sequence is proposed for situations where radiofrequency offset effects are the dominant source of error.

Spin dynamics and thermodynamics in solid‐state NMR cross polarization

We investigate spin thermodynamic processes taking place during Hartmann–Hahn cross‐polarization experiments in solids, in which spin polarization is transferred between two nuclear spin species by the application of two strong rf fields. As is well known, optimum cross polarization is achieved for a particular ratio of the two rf field strengths called the Hartmann–Hahn condition; we find experimentally that this condition provides the optimum transfer not only on kinetic grounds, as is usually supposed, but also for thermodynamic reasons. By measurement of the evolution of the spin observables in a ferrocene single crystal we demonstrate the existence of a quasi‐equilibrium state with nonuniform spin temperature and less than maximum entropy. A modified spin thermodynamic theory is developed whose main features are the important role of the dipolar energy for the quasi‐equilibrium state and the existence of constants of the motion other than the total energy. A new cross‐polarization experiment is sugge...

Symmetrical composite pulse sequences for NMR population inversion. I. Compensation of radiofrequency field inhomogeneity

Abstract The interaction of a nuclear spin system with an inhomogenous or misset radio-frequency field is described by means of rotation operators. It is shown that it is possible to construct sequences of small numbers of pulses which rotate the magnetization vector in a manner analogous to a single pulse, but which have the advantage of being less sensitive to deviations in the radiofrequency field strength. Two new “composite pulses” are proposed which have this property and which convert longitudinal magnetization into transverse magnetization—a two-pulse sequence employing a 27π 3 radian phase shift, and a four-pulse sequence. The problem of inversion of spin populations by a composite pulse is then addressed. It is shown that composite pulse sequences can be constructed which take magnetization vectors from the z axis to the −z axis via trajectories possessing either approximate reflection symmetry, or approximate rotational symmetry. It is shown that this symmetry may cause additional error compensation, and two new pulse sequences are suggested for NMR population inversion—one of three pulses, and one of nine pulses. The sensitivity of these sequences to off-resonance effects is discussed.

Symmetry principles in the nuclear magnetic resonance of spinning solids: Heteronuclear recoupling by generalized Hartmann-Hahn sequences

General symmetry principles for rotor-synchronized pulse sequences in magic-angle spinning solid-state nuclear magnetic resonance are presented. The theory of symmetry-based pulse sequences using π pulse elements is presented for the first time. The symmetry theory is extended to the case of generalized Hartmann–Hahn sequences, in which rotor-synchronized rf irradiation is applied simultaneously to two isotopic spin species. The symmetry principles lead to heteronuclear selection rules. The symmetry theory is used to design pulse sequences which implement heteronuclear dipolar recoupling at the same time as decoupling homonuclear spin–spin interactions, and which also suppress chemical shift anisotropies. A number of specific pulse sequences based on these principles are listed. Experimental demonstrations are given of heteronuclear two-dimensional correlation spectroscopy, heteronuclear multiple-quantum spectroscopy, and the estimation of internuclear dipolar couplings.

Pulse sequence symmetries in the nuclear magnetic resonance of spinning solids: Application to heteronuclear decoupling

We develop the average Hamiltonian theory of a class of symmetrical radio-frequency pulse sequences in the NMR of rotating solids. Theorems are presented which allow one to predict the elimination of many average Hamiltonian terms, without detailed calculation. These results are applied to the problem of heteronuclear decoupling in the presence of rapid magic angle spinning. We present sequences which minimize the number of heteronuclear terms at the same time as recoupling the homonuclear interactions of the irradiated spins. The performance of the new sequences is tested on 13C labeled calcium formate. Experimental measurements of double-quantum 1H excitation indicate a relationship between good heteronuclear decoupling of the observed spin species and efficient recoupling of the irradiated spin species. The heteronuclear decoupling performance of the new sequences is significantly better than that obtained with an unmodulated radio-frequency field. The decoupling performance is improved further by brea...