Formulation of Zeeman modulation as a signal filter

CW-EPR Spectral Simulations: Solid State

This chapter summarizes the core concepts underlying the simulation of EPR spectra from biological samples in the solid state, from a user perspective. The key choices and decisions that have to be made by a user when simulating an experimental EPR spectrum are outlined. These include: the choice of the simulation model (the network of spins and the associated spin Hamiltonian), the dynamic regime (solid, liquid, slow motion), the level of theory used in the simulation (matrix diagonalization, perturbation theory, etc.), the treatment of orientational order and disorder (powder, crystal, partial ordering), the inclusion of the effects of structural disorder (strains), the effects of other line broadening mechanisms (unresolved hyperfine couplings, relaxation), and the inclusion of experimental distortions (field modulation, power saturation, filtering). Additionally, the salient aspects of utilizing least-squares fitting algorithms to aid the analysis of experimental spectra with the help of simulations are outlined. Although drawing from the experience gained from implementing EasySpin and from interacting with EasySpin's user base, this chapter applies to any EPR simulation software.

Statistical reconstruction algorithms for continuous wave electron spin resonance imaging

Electron spin resonance imaging (ESRI) is an important branch of ESR that deals with heterogeneous samples ranging from semiconductor materials to small live animals and even humans. ESRI can produce either spatial images (providing information about the spatially dependent radical concentration) or spectral-spatial images, where an extra dimension is added to describe the absorption spectrum of the sample (which can also be spatially dependent). The mapping of oxygen in biological samples, often referred to as oximetry, is a prime example of an ESRI application. ESRI suffers frequently from a low signal-to-noise ratio (SNR), which results in long acquisition times and poor image quality. A broader use of ESRI is hampered by this slow acquisition, which can also be an obstacle for many biological applications where conditions may change relatively quickly over time. The objective of this work is to develop an image reconstruction scheme for continuous wave (CW) ESRI that would make it possible to reduce the data acquisition time without degrading the reconstruction quality. This is achieved by adapting the so-called "statistical reconstruction" method, recently developed for other medical imaging modalities, to the specific case of CW ESRI. Our new algorithm accounts for unique ESRI aspects such as field modulation, spectral-spatial imaging, and possible limitation on the gradient magnitude (the so-called "limited angle" problem). The reconstruction method shows improved SNR and contrast recovery vs. commonly used back-projection-based methods, for a variety of simulated synthetic samples as well as in actual CW ESRI experiments.

Direct simulation of magnetic resonance relaxation rates and line shapes from molecular trajectories

We simulate spin relaxation processes, which may be measured by either continuous wave or pulsed magnetic resonance techniques, using trajectory-based simulation methodologies. The spin-lattice relaxation rates are extracted numerically from the relaxation simulations. The rates obtained from the numerical fitting of the relaxation curves are compared to those obtained by direct simulation from the relaxation Bloch-Wangsness-Abragam-Redfield theory (BWART). We have restricted our study to anisotropic rigid-body rotational processes, and to the chemical shift anisotropy (CSA) and a single spin-spin dipolar (END) coupling mechanisms. Examples using electron paramagnetic resonance (EPR) nitroxide and nuclear magnetic resonance (NMR) deuterium quadrupolar systems are provided. The objective is to compare those rates obtained by numerical simulations with the rates obtained by BWART. There is excellent agreement between the simulated and BWART rates for a Hamiltonian describing a single spin (an electron) interacting with the bath through the chemical shift anisotropy (CSA) mechanism undergoing anisotropic rotational diffusion. In contrast, when the Hamiltonian contains both the chemical shift anisotropy (CSA) and the spin-spin dipolar (END) mechanisms, the decay rate of a single exponential fit of the simulated spin-lattice relaxation rate is up to a factor of 0.2 smaller than that predicted by BWART. When the relaxation curves are fit to a double exponential, the slow and fast rates extracted from the decay curves bound the BWART prediction. An extended BWART theory, in the literature, includes the need for multiple relaxation rates and indicates that the multiexponential decay is due to the combined effects of direct and cross-relaxation mechanisms.

Multisite EPR oximetry from multiple quadrature harmonics

Multisite continuous wave (CW) electron paramagnetic resonance (EPR) oximetry using multiple quadrature field modulation harmonics is presented. First, a recently developed digital receiver is used to extract multiple harmonics of field modulated projection data. Second, a forward model is presented that relates the projection data to unknown parameters, including linewidth at each site. Third, a maximum likelihood estimator of unknown parameters is reported using an iterative algorithm capable of jointly processing multiple quadrature harmonics. The data modeling and processing are applicable for parametric lineshapes under nonsaturating conditions. Joint processing of multiple harmonics leads to 2-3-fold acceleration of EPR data acquisition. For demonstration in two spatial dimensions, both simulations and phantom studies on an L-band system are reported.

Reconstruction of the first-derivative EPR spectrum from multiple harmonics of the field-modulated continuous wave signal

Selection of the amplitude of magnetic field modulation for continuous wave electron paramagnetic resonance (EPR) often is a trade-off between sensitivity and resolution. Increasing the modulation amplitude improves the signal-to-noise ratio, S/N, at the expense of broadening the signal. Combining information from multiple harmonics of the field-modulated signal is proposed as a method to obtain the first derivative spectrum with minimal broadening and improved signal-to-noise. The harmonics are obtained by digital phase-sensitive detection of the signal at the modulation frequency and its integer multiples. Reconstruction of the first-derivative EPR line is done in the Fourier conjugate domain where each harmonic can be represented as the product of the Fourier transform of the 1st derivative signal with an analytical function. The analytical function for each harmonic can be viewed as a filter. The Fourier transform of the 1st derivative spectrum can be calculated from all available harmonics by solving an optimization problem with the goal of maximizing the S/N. Inverse Fourier transformation of the result produces the 1st derivative EPR line in the magnetic field domain. The use of modulation amplitude greater than linewidth improves the S/N, but does not broaden the reconstructed spectrum. The method works for an arbitrary EPR line shape, but is limited to the case when magnetization instantaneously follows the modulation field, which is known as the adiabatic approximation.

Simulation of nitroxide electron paramagnetic resonance spectra from brownian trajectories and molecular dynamics simulations

A simulated continuous wave electron paramagnetic resonance spectrum of a nitroxide spin label can be obtained from the Fourier transform of a free induction decay. It has been previously shown that the free induction decay can be calculated by solving the time-dependent stochastic Liouville equation for a set of Brownian trajectories defining the rotational dynamics of the label. In this work, a quaternion-based Monte Carlo algorithm has been developed to generate Brownian trajectories describing the global rotational diffusion of a spin-labeled protein. Also, molecular dynamics simulations of two spin-labeled mutants of T4 lysozyme, T4L F153R1, and T4L K65R1 have been used to generate trajectories describing the internal dynamics of the protein and the local dynamics of the spin-label side chain. Trajectories from the molecular dynamics simulations combined with trajectories describing the global rotational diffusion of the protein are used to account for all of the dynamics of a spin-labeled protein. Spectra calculated from these combined trajectories correspond well to the experimental spectra for the buried site T4L F153R1 and the helix surface site T4L K65R1. This work provides a framework to further explore the modeling of the dynamics of the spin-label side chain in the wide variety of labeling environments encountered in site-directed spin labeling studies.

Post-processing of EPR spectra by convolution filtering: calculation of a harmonics' series and automatic separation of fast-motion components from spin-label EPR spectra

This communication reports on post-processing of continuous wave EPR spectra by a digital convolution with filter functions that are subjected to differentiation or the Kramers-Krönig transform analytically. In case of differentiation, such a procedure improves spectral resolution in the higher harmonics enhancing the relative amplitude of sharp spectral features over the broad lines. At the same time high-frequency noise is suppressed through filtering. These features are illustrated on an example of a Lorentzian filter function that has a principal advantage of adding a known magnitude of homogeneous broadening to the spectral shapes. Such spectral distortion could be easily and accurately accounted for in the consequent least-squares data modeling. Application examples include calculation of higher harmonics from pure absorption echo-detected EPR spectra and resolving small hyperfine coupling that are unnoticeable in conventional first derivative EPR spectra. Another example involves speedy and automatic separation of fast and broad slow-motion components from spin-label EPR spectra without explicit simulation of the slow motion spectrum. The method is illustrated on examples of X-band EPR spectra of partially aggregated membrane peptides.

Application of magnetic field over-modulation for improved EPR linewidth measurements using probes with Lorentzian lineshape

Magnetic field modulation in CW electron paramagnetic resonance (EPR) is used for signal detection. However, it can also distort signal lineshape. In experiments where the linewidth information is of particular importance, small modulation amplitude is usually used to limit the lineshape distortion. The use of small modulation amplitude, however, results in low signal-to-noise ratio and therefore affects the precision of linewidth measurements. Recently, a new spectral simulation model has been developed enabling accurate fitting of modulation-broadened EPR spectra in liquids. Since the use of large modulation amplitude (over-modulation) can significantly enhance the EPR signal, the precision of linewidth measurements is therefore greatly improved. We investigated the over-modulation technique in EPR oximetry experiments using the oxygen-sensing probe lithium octa-n-butoxy-substitued naphthalocyanine (LiNc-BuO). Modulation amplitudes 2-18 times the intrinsic linewidth of the probe were applied to increase the spectral signal-to-noise ratio. The intrinsic linewidth of the probe at different oxygen concentrations was accurately extracted through curve fitting from the enhanced spectra. Thus, we demonstrated that the over-modulation model is also applicable to particulate oxygen-sensing probes such as LiNc-BuO and that the lineshape broadening induced by oxygen is separable from that induced by over-modulation. Therefore, the over-modulation technique can be used to enhance sensitivity and improve linewidth measurements for EPR oximetry with particulate oxygen-sensing probes with Lorentzian lineshape. It should be particularly useful for in vivo oxygen measurements, in which direct linewidth measurements may not be feasible due to inadequate signal-to-noise ratio.