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A comprehensive theory for interpreting two-dimensional Fourier transform (2D-FT) electron spin resonance (ESR) experiments that is based on the stochastic Liouville equation is presented. It encompasses the full range of motional rates from fast through very slow motions, and it also provides for microscopic as well as macroscopic molecular ordering. In these respects it is as sophisticated in its treatment of molecular dynamics as the theory currently employed for analyzing cw ESR spectra. The general properties of the pulse propagator superoperator, which describes the microwave pulses in Liouville space, are analyzed in terms of the coherence transfer pathways appropriate for COSY (correlation spectroscopy), SECSY (spin–echo correlation spectroscopy), and 2D-ELDOR (electron–electron double resonance) sequences wherein either the free-induction decay (FID) or echo decay is sampled. Important distinctions are made among the sources of inhomogeneous broadening, which include (a) incomplete spectral averaging in the slow-motional regime, (b) unresolved superhyperfine structure and related sources, and (c) microscopic molecular ordering but macroscopic disorder (MOMD). The differing effects these sources of inhomogeneous broadening have on the two mirror image coherence pathways observed in the dual quadrature 2D experiments, as well as on the auto vs crosspeaks of 2D-ELDOR, is described. The theory is applied to simulate experiments of nitroxide spin labels in complex fluids such as membrane vesicles, where the MOMD model applies and these distinctions are particularly relevant, in order to extract dynamic and ordering parameters. The recovery of homogeneous linewidths from FID-based COSY experiments on complex fluids with significant inhomogeneous broadening is also described. The theory is applied to the ultraslow motional regime, and a simple method is developed to determine rotational rates from the broadening of the autopeaks of the 2D-ELDOR spectra as a function of the mixing time, which is due to the development of ‘‘motional crosspeaks.’’ The application of this method to recent experiments with nitroxide probes illustrates that rotational correlation times as slow as milliseconds may be measured. It is shown how 2D-ELDOR can be useful to distinguish between the cases of very slow motional (SM) rates with little or no ordering and of very high ordering (HO) but substantial motional rates even though the cw ESR spectra are virtually the same. The effects of motion and of microscopic ordering on the nuclear modulation patterns in 2D-FT-ESR are compared, and it is suggested that these effects could be utilized to further distinguish between SM and HO cases. Key aspects of the challenging computational problems are discussed, and algorithms are described which lead to significant reductions in computation time as needed to permit nonlinear least-squares fitting of the theory to experiments.

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Rigid-limit 250-GHz electron spin resonance (FIR-ESR) spectra have been studied for a series of phosphatidylcholine spin labels (n-PC, where n = 5, 7, 10, 12, 16) in pure lipid dispersions of dipalmitoylphosphatidylcholine (DPPC) and 1-palmitoyl-2-oleoylphosphatidylcholine (POPC), as well as dispersions of DPPC containing the peptide gramicidin A (GA) in a 1:1 molar ratio. The enhanced g-tensor resolution of 250-GHz ESR for these spin labels permitted a careful study of the nitroxide g-tensor as a function of spin probe location and membrane composition. In particular, as the spin label is displaced from the polar head group, Azz decreases and gxx increases as they assume values typical of a nonpolar environment, appropriate for the hydrophobic alkyl chains in the case of pure lipid dispersions. The field shifts of spectral features due to changes in gxx are an order of magnitude larger than those from changes in Azz. The magnetic tensor parameters measured in the presence of GA were characteristic of a polar environment and showed only a very weak dependence of Azz and gxx on label position. These results demonstrate the significant influence of GA on the local polarity along the lipid molecule, and may reflect increased penetration of water into the alkyl chain region of the lipid in the presence of GA. The spectra from the pure lipid dispersions also exhibit a broad background signal that is most significant for 7-, 10-, and 12-PC, and is more pronounced in DPPC than in POPC. It is attributed to spin probe aggregation yielding spin exchange narrowing. The addition of GA to DPPC essentially suppressed the broad background signal observed in pure DPPC dispersions.

Theoretical Microscopic Titration Curves (THEMATICS) may be used to identify chemically important residues in active sites of enzymes by characteristic deviations from the normal, sigmoidal Henderson-Hasselbalch titration behavior. Clusters of such deviant residues in physical proximity constitute reliable predictors of the location of the active site. Originally the residues with deviant predicted behavior were identified by human observation of the computed titration curves. However, it is preferable to select the unusual residues by mathematically well-defined criteria, in order to reduce the chance of error, eliminate any possible biases, and substantially speed up the selection process. Here we present some simple statistical tests that constitute such selection criteria. The first derivatives of the predicted titration curves resemble distribution functions and are normalized. The moments of these first derivative functions are computed. It is shown that the third and fourth moments, measures of asymmetry and kurtosis, respectively, are good measures of the deviations from normal behavior. Results are presented for 44 different enzymes. Detailed results are given for 4 enzymes with 4 different types of chemistry: arginine kinase from Limulus polyphemus (horseshoe crab); beta-lactamase from Escherichia coli; glutamate racemase from Aquifex pyrophilus; and 3-isopropylmalate dehydrogenase from Thiobacillus ferrooxidans. The relationship between the statistical measures of nonsigmoidal behavior in the predicted titration curves and the catalytic activity of the residue is discussed.

High-frequency (250 GHz) electron paramagnetic resonance (EPR) spectra in the limit of motional narrowing have been studied for a nitroxide spin probe diffusing in a low-viscosity isotropic solvent. The enhanced sensitivity of the 250-GHz spectrum to rotational modulation of the g tensor reveals new details of the microscopic motion of the spin probe. Specifically, it is shown how independent linear constraints imposed by line-width measurements a t 250 G H z and a lower frequency may be used to fully determine the diffusion tensor R in the general case R, # Ry # R,. This procedure is demonstrated experimentally for the perdeuterated Tempone (PDT) probe diffusing in toluene-& where analysis of the line widths obtained at 250 and 9.5 G H z gives values for the anisotropy parameters px = R,/R, and py = R,/R, of 1.8 f 0.2 and 1.5 f 0.3, respectively. Thus, this molecule is found to exhibit small deviations from spherically symmetric reorientation. The enhanced accuracy of determining the g tensor from rigid-limit spectra obtained a t 250 G H z is an important feature of such highfrequency studies. Existing theoretical expressions for motionally narrowed nitroxide line widths have been modified appropriately for the high-field case by including a completely anisotropic diffusion tensor and the I4N nuclear Zeeman interaction, which becomes important a t fields above about 5 T.

A 250-GHz electron paramagnetic resonance (EPR) study of the slow rotational diffusion of two spin probes in toluene, viz., perdeuterated 2,2,6,6-tetramethyl-4-piperidone (PDT) and 3-doxylcholestane (CSL) is presented.EPR spectra were obtained in the slow-motional and near-rigid limit regions, which corresponds to rotational correlation times 1&l0 > T R > 10-6 s. These two probes differ significantly in size and shape, permitting a detailed exploration of the sensitivity of 250-GHz EPR to different aspects of the molecular dynamics such as rotational anisotropy and nowBrownian diffusion. Nonlinear least-squares fitting based on full stochastic Liouville calculations provides a sensitive means for discriminating amongst motional models. PDT in toluened8 is found to be well described by an approximate free diffusion model, whereas the larger spin probe, CSL, is best described by Brownian diffusion. The slow-motional spectra at 250 GHz are most sensitive to the diffusional model, the (geometric) mean diffusional rate, and axial diffusional anisotropy but less sensitive to rhombic deviations from an axially symmetric diffusion tensor (Le., to the general case R, # Ry # Rz), The slow-motional spectra of PDT were fit using anisotropic diffusion parameters determined from fast-motional spectra but are not very sensitive to such small anisotropies. For the case of Brownian diffusion, CSL was best fit with IVY Ry/(RZRx)lI2 = 9.0 (where they axis is the long axis of the molecule and x and z are perpendicular axes), which differs appreciably from the fast-motional value of IVY = 4.3 * 0.2 (and px 5 R,/R, = 0.5).However, a mixed model of free-diffusional motion about t h e y axis with Brownian motion of this axis yields an IVY close to the fast-motional value with comparable overall quality in fit compared to full Brownian motion.An important feature of the 250-GHz studies is the ability to measure very accurately the magnetic tensors needed for the motional studies. The theoretical modifications needed for inclusion of a fully anisotropic rotational diffusion tensor in the slow-motional EPR simulations are also given.

A number of groups have utilized molecular dynamics (MD) to calculate slow-motional electron paramagnetic resonance (EPR) spectra of spin labels attached to biomolecules. Nearly all such calculations have been based on some variant of the trajectory method introduced by Robinson, Slutsky and Auteri (J. Chem. Phys. 1992,96, 2609-2616). Here we present an alternative approach that is specifically adapted to the diffusion operator-based stochastic Liouville equation (SLE) formalism that is also widely used to calculate slow-motional EPR line shapes. Specifically, the method utilizes MD trajectories to derive diffusion parameters such as the rotational diffusion tensor, diffusion tilt angles, and expansion coefficients of the orienting potential, which are then used as direct inputs to the SLE line shape program. This approach leads to a considerable improvement in computational efficiency over trajectory-based methods, particularly for high frequency, high field EPR. It also provides a basis for deconvoluting the effects of local spin label motion and overall motion of the labeled molecule or domain: once the local motion has been characterized by this approach, the label diffusion parameters may be used in conjunction with line shape analysis at lower EPR frequencies to characterize global motions. The method is validated by comparison of the MD predicted line shapes to experimental high frequency (250 GHz) EPR spectra.