Two-dimensional Fourier-transform electron spin resonance in complex fluids

Microsecond dynamics in proteins by two-dimensional ESR. II. Addressing computational challenges

Two-dimensional electron-electron double resonance (2D-ELDOR) provides extensive insight into molecular motions. Recent developments permitting experiments at higher frequencies (95 GHz) provide molecular orientational resolution, enabling a clearer description of the nature of the motions. In previous work, we provided simulations for the case of domain motions within proteins that are themselves slowly tumbling in a solution. In order to perform these simulations, it was found that the standard approach of solving the relevant stochastic Liouville equation using the efficient Lanczos algorithm for this case breaks down, so algorithms were employed that rely on the Arnoldi iteration. While they lead to accurate simulations, they are very time-consuming. In this work, we focus on a variant known as the rational Arnoldi algorithm. We show that this can achieve a significant reduction in computation time. The stochastic Liouville matrix, which is of very large dimension, N, is first reduced to a much smaller dimension, m, e.g., from N ∼ O(10⁴) to m ∼ 60, that spans the relevant Krylov subspace from which the spectrum is predicted. This requires the selection of the m frequency shifts to be utilized. A method of adaptive shift choice is introduced to optimize this selection. We also find that these procedures help in optimizing the pruning procedure that greatly reduces the dimension of the initial N dimensional stochastic Liouville matrix in such subsequent computations.

Microsecond dynamics in proteins by two-dimensional ESR: Predictions

Two-dimensional electron-electron double resonance (2D-ELDOR) provides extensive insight into molecular motions. Recent developments permitting experiments at higher frequencies (95 GHz) provide molecular orientational resolution, enabling a clearer description of the nature of the motions. In this work, simulations are provided for the example of domain motions within proteins that are themselves slowly tumbling in solution. These show the nature of the exchange cross-peaks that are predicted to develop in real time from such domain motions. However, we find that the existing theoretical methods for computing 2D-ELDOR experiments over a wide motional range begin to fail seriously when applied to very slow motions characteristic of proteins in solution. One reason is the failure to obtain accurate eigenvectors and eigenvalues of the complex symmetric stochastic Liouville matrices describing the experiment when computed by the efficient Lanczos algorithm in the range of very slow motion. Another, perhaps more serious, issue is that these matrices are "non-normal," such that for the very slow motional range even rigorous diagonalization algorithms do not yield the correct eigenvalues and eigenvectors. We have employed algorithms that overcome both these issues and lead to valid 2D-ELDOR predictions even for motions approaching the rigid limit. They are utilized to describe the development of cross-peaks in 2D-ELDOR at 95 GHz for a particular case of domain motion.

Focus: Two-dimensional electron-electron double resonance and molecular motions: The challenge of higher frequencies

The development, applications, and current challenges of the pulsed ESR technique of two-dimensional Electron-Electron Double Resonance (2D ELDOR) are described. This is a three-pulse technique akin to 2D Exchange Nuclear Magnetic Resonance, but involving electron spins, usually in the form of spin-probes or spin-labels. As a result, it required the extension to much higher frequencies, i.e., microwaves, and much faster time scales, with π/2 pulses in the 2-3 ns range. It has proven very useful for studying molecular dynamics in complex fluids, and spectral results can be explained by fitting theoretical models (also described) that provide a detailed analysis of the molecular dynamics and structure. We discuss concepts that also appear in other forms of 2D spectroscopy but emphasize the unique advantages and difficulties that are intrinsic to ESR. Advantages include the ability to tune the resonance frequency, in order to probe different motional ranges, while challenges include the high ratio of the detection dead time vs. the relaxation times. We review several important 2D ELDOR studies of molecular dynamics. (1) The results from a spin probe dissolved in a liquid crystal are followed throughout the isotropic → nematic → liquid-like smectic → solid-like smectic → crystalline phases as the temperature is reduced and are interpreted in terms of the slowly relaxing local structure model. Here, the labeled molecule is undergoing overall motion in the macroscopically aligned sample, as well as responding to local site fluctuations. (2) Several examples involving model phospholipid membranes are provided, including the dynamic structural characterization of the boundary lipid that coats a transmembrane peptide dimer. Additionally, subtle differences can be elicited for the phospholipid membrane phases: liquid disordered, liquid ordered, and gel, and the subtle effects upon the membrane, of antigen cross-linking of receptors on the surface of plasma membrane, vesicles can be observed. These 2D ELDOR experiments are performed as a function of mixing time, Tm, i.e., the time between the second and third π/2 pulses, which provides a third dimension. In fact, a fourth dimension may be added by varying the ESR frequency/magnetic field combination. Therefore, (3) it is shown how continuous-wave multifrequency ESR studies enable the decomposition of complex dynamics of, e.g., proteins by virtue of their respective time scales. These studies motivate our current efforts that are directed to extend 2D ELDOR to higher frequencies, 95 GHz in particular (from 9 and 17 GHz), in order to enable multi-frequency 2D ELDOR. This required the development of quasi-optical methods for performing the mm-wave experiments, which are summarized. We demonstrate state-of-the-art 95 GHz 2D ELDOR spectroscopy through its ability to resolve the two signals from a spin probe dissolved in both the lipid phase and the coexisting aqueous phase. As current 95 GHz experiments are restricted by limited spectral coverage of the π/2 pulse, as well as the very short T2 relaxation times of the electron spins, we discuss how these limitations are being addressed.

2D-ELDOR study of heterogeneity and domain structure changes in plasma membrane vesicles upon cross-linking of receptors

2D electron-electron double resonance (2D-ELDOR) with the "full Sc-" method of analysis is applied to the study of plasma membrane vesicles. Membrane structural changes upon antigen cross-linking of IgE receptors (IgE-FcεRI) in plasma membrane vesicles (PMVs) isolated from RBL-2H3 mast cells are investigated, for the first time, by means of these 2D-ELDOR techniques. Spectra of 1-palmitoyl-2-(16-doxyl stearoyl) phosphatidylcholine (16-PC) from PMVs before and after this stimulation at several temperatures are reported. The results demonstrate a coexistence of liquid-ordered (L(o)) and liquid-disordered (L(d)) components. We find that upon cross-linking, the membrane environment is remodeled to become more disordered, as shown by a moderate increase in the population of the L(d) component. This change in the relative amount of the L(o) versus L(d) components upon cross-linking is consistent with a model wherein the IgE receptors, which when clustered by antigen to cause cell stimulation, lead to more disordered lipids, and their dynamic and structural properties are slightly altered. This study demonstrates that 2D-ELDOR, analyzed by the full Sc- method, is a powerful approach for capturing the molecular dynamics in biological membranes. This is a particular case showing how 2D-ELDOR can be applied to study physical processes in complex systems that yield subtle changes.

A new Lanczos-based algorithm for simulating high-frequency two-dimensional electron spin resonance spectra

The Lanczos algorithm (LA) is a useful iterative method for the reduction of a large matrix to tridiagonal form. It is a storage efficient procedure requiring only the preceding two Lanczos vectors to compute the next. The quasi-minimal residual (QMR) method is a powerful method for the solution of linear equation systems, Ax = b. In this report we provide another application of the QMR method: we incorporate QMR into the LA to monitor the convergence of the Lanczos projections in the reduction of large sparse matrices. We demonstrate that the combined approach of the LA and QMR can be utilized efficiently for the orthogonal transformation of large, but sparse, complex, symmetric matrices, such as are encountered in the simulation of slow-motional 1D- and 2D-electron spin resonance (ESR) spectra. Especially in the 2D-ESR simulations, it is essential that we store all of the Lanczos vectors obtained in the course of the LA recursions and maintain their orthogonality. In the LA-QMR application, the QMR weight matrix mitigates the problem that the Lanczos vectors lose orthogonality after many LA projections. This enables substantially more Lanczos projections, as required to achieve convergence for the more challenging ESR simulations. It, therefore, provides better accuracy for the eigenvectors and the eigenvalues of the large sparse matrices originating in 2D-ESR simulations than does the previously employed method, which is a combined approach of the LA and the conjugate-gradient (CG) methods, as evidenced by the quality and convergence of the 2D-ESR simulations. Our results show that very slow-motional 2D-ESR spectra at W-band (95 GHz) can be reliably simulated using the LA-QMR method, whereas the LA-CG consistently fails. The improvements due to the LA-QMR are of critical importance in enabling the simulation of high-frequency 2D-ESR spectra, which are characterized by their very high resolution to molecular orientation.

2D-ELDOR using full S(c-) fitting and absorption lineshapes

Recent progress in developing 2D-ELDOR (2D electron-electron double resonance) techniques to better capture molecular dynamics in complex fluids, particularly in model and biological membranes, is reported. The new "full S(c-) method", which corrects the spectral analysis for the phase distortion effects present in the experiments, is demonstrated to enhance the sensitivity of 2D-ELDOR in reporting on molecular dynamics in complex membrane environments. That is, instead of performing spectral fitting in the magnitude mode, our new method enables simultaneous fitting of both the real and imaginary components of the S(c-) signal. The full S(c-) fitting not only corrects the phase distortions in the experimental data but also more accurately determines instrumental dead times. The phase corrections applied to the S(c-) spectrum enable the extraction of the pure absorption-mode spectrum, which is characterized by much better resolution than the magnitude-mode spectrum. In the absorption mode, the variation of homogeneous broadening, which reports on the dynamics of the spin probe, can even be observed by visual inspection. This new method is illustrated with results from model membranes of dipalmitoyl-sn-glycero-phosphatidylcholine (DPPC)-cholesterol binary mixtures, as well as with results from plasma membrane vesicles of mast cells. In addition to the dynamic parameters, which provide quantitative descriptions for membranes at the molecular level, the high-resolution absorption spectra themselves may be used as a "fingerprint" to characterize membrane phases and distinguish coexisting components in biomembranes. Thus we find that 2D-ELDOR is greatly improved with the new "full S(c-) method" especially for exploring the complexity of model and biological membranes.

Collective Fluctuations in Ordered Fluids Investigated by Two-Dimensional Electron−Electron Double Resonance Spectroscopy

Two-dimensional electron-electron double resonance (2D-ELDOR) is a technique that is sensitive to the dynamical processes affecting spin labels in complex fluid environments. In ordered fluids, such as membrane vesicles, the 2D-ELDOR experiment is affected by the molecular tumbling in the locally ordered environment. This motion occurs on two different time scales, the faster molecular motion relative to the local director, and the slower collective fluctuations of the director field. In the experimental study of Patyal, Crepeau, and Freed (Biophys. J. 1997, 73, 2201), it was found that the widths of the autopeaks of the 2D-ELDOR spectrum increased as a function of the mixing time. In the present work, a theory is developed for the effects of director fluctuations on the autopeaks in the 2D-ELDOR experiment by employing an analytical solution of the stochastic Liouville equation for which the director field is treated as a multidimensional Gaussian process, as previously developed by Frezzato, Kothe, and Moro (J. Phys. Chem. B 2001, 105, 1281 and J. Phys. Chem. B 2004, 108, 9505). Good agreement is found between theory and experiment, notably the only adjustable parameter is k, the bending elastic modulus of the membrane. The values of k = 11 x 10(-20) J for 1,2-dipalmitoyl-sn-glycero-phosphatidylcholine (DPPC) vesicles and k = 15 x 10(-20) J for DPPC/gramicidin A (5:1) vesicles, both at 45 degrees C, were found from the analysis and agree well with previous related measurements by other physical techniques. This establishes 2D-ELDOR as a useful technique to study the elastic properties of biological membranes.

Effects of finite pulse width on two-dimensional Fourier transform electron spin resonance

Two-dimensional (2D) Fourier transform ESR techniques, such as 2D-ELDOR, have considerably improved the resolution of ESR in studies of molecular dynamics in complex fluids such as liquid crystals and membrane vesicles and in spin labeled polymers and peptides. A well-developed theory based on the stochastic Liouville equation (SLE) has been successfully employed to analyze these experiments. However, one fundamental assumption has been utilized to simplify the complex analysis, viz. the pulses have been treated as ideal non-selective ones, which therefore provide uniform irradiation of the whole spectrum. In actual experiments, the pulses are of finite width causing deviations from the theoretical predictions, a problem that is exacerbated by experiments performed at higher frequencies. In the present paper we provide a method to deal with the full SLE including the explicit role of the molecular dynamics, the spin Hamiltonian and the radiation field during the pulse. The computations are rendered more manageable by utilizing the Trotter formula, which is adapted to handle this SLE in what we call a "Split Super-Operator" method. Examples are given for different motional regimes, which show how 2D-ELDOR spectra are affected by the finite pulse widths. The theory shows good agreement with 2D-ELDOR experiments performed as a function of pulse width.

New technologies in electron spin resonance

New electron spin resonance (ESR) technologies have been developed, which have led to new and improved applications. (a) The development of two-dimensional Fourier transform (FT) ESR required spectrometers providing intense pi/2 microwave pulses of very short (3-5 ns) duration, wide bandwidths, and very short dead times. It has enabled studies that resolve sophisticated details of molecular dynamics in complex fluids. (b) Methods that produce multiple quantum coherences by pulsed ESR now enable accurate measurements of large distances (>12A). (c) One of the most important advances has been the extension of ESR to high magnetic fields and high frequencies. This has benefited from the utilization of quasi-optical methods, especially above 150 GHz. The greatly improved orientational resolution and the faster "snapshot" of motions that are provided by ESR at high frequencies enhance studies of molecular dynamics. The use of both high and lower frequencies enables one to unravel faster and slower modes from the complex dynamics of fluids and macromolecules. (d) The development of FT-ESR imaging required substantial pulsed field gradients lasting only 50-100 ns. ESR imaging is effective in studying diffusion in fluids. Areas for further development are also described.

Electron-spin resonance study of aggregation of gramicidin in dipalmitoylphosphatidylcholine bilayers and hydrophobic mismatch

The effect of aggregation of gramicidin A' (GA) on the phase structure of dipalmitoylphosphatidylcholine (DPPC) multilamellar vesicles was studied by cw-ESR using a chain-labeled lipid (16PC) at temperatures between 30 degrees and 45 degreesC that span the main phase transition of DPPC. Boundary lipids were observed only in dispersions with GA/DPPC molar ratios >1:15, where GA aggregates. Detailed fits by nonlinear least squares (NLLS) methods are consistent with the boundary lipid being characterized by a large negative order parameter ( approximately -0.4), indicative of a dynamic bending of the end of the acyl chain, and a substantially reduced motion, about an order of magnitude slower than that of the bulk lipid. The NLLS analysis compares favorably with a recent two-dimensional Fourier transform ESR study on DPPC/GA vesicles, which accurately discerned the bulk lipid. The detailed ESR observables are discussed in terms of the ordering effect of GA at low concentration of GA, the dissociation of the GA channel and the dynamic bending of the end chain segment of boundary lipid at high concentration of GA, and of HII phase formation induced by GA. It is suggested that these phenomena can be interpreted in terms of the combined effects of partial dehydration of the lipid headgroup by the GA and of the hydrophobic mismatch between GA and DPPC molecules. Substantial hysteresis is observed for heating versus cooling cycles, but only for a GA/DPPC molar ratio >1:15. This is consistent with the aggregation of GA molecules at high concentrations.

Absorption lineshapes in two-dimensional electron spin resonance and the effects of slow motions in complex fluids

A methodology for obtaining pure absorption two-dimensional electron spin resonance spectra is presented for the case of large inhomogeneous broadening and/or slow motions. For slow motions, the spectra consist of "complex Lorentzians" superimposed with complex weighting factors, presenting a challenge to obtaining absorption spectra. It is shown how absorption-type spectra can be recovered for the two-pulse COSY and SECSY experiments in such cases. For three-pulse 2D ELDOR experiments, absorption lineshapes can be obtained for the autopeaks, whereas the cross peaks would be of mixed-mode character, in general. However, for practical cases the dispersive components in the cross peaks will be relatively small. Theoretical and experimental absorption spectra are provided to illustrate the method and to show the improved resolution obtained from absorption lineshapes. In particular, the variation in linewidths across a SECSY spectrum, which is a key component in elucidating motional dynamics, is clearly rendered in the pure absorption mode. A convenient method for introducing the necessary phase corrections for the slow-motional spectra is also provided.