Microsecond dynamics in proteins by two-dimensional ESR: Predictions

Time-Frequency Analysis of Two-Dimensional Electron Spin Resonance Signals

Two-dimensional electron spin resonance (2D ESR) spectroscopy is a unique experimental technique for probing protein structure and dynamics, including processes that occur at the microsecond time scale. While it provides significant resolution enhancement over the one-dimensional experimental setup, spectral broadening and noise make extraction of spectral information highly challenging. Traditionally, two-dimensional Fourier transform (2D FT) is applied for the analysis of 2D ESR signals, although its efficiency is limited to stationary signals. In addition, it often fails to resolve overlapping peaks in 2D ESR. In this work, we propose a time-frequency analysis of 2D time-domain signals, which identifies all frequency peaks by decoupling a signal into its distinct constituent components via projection on the time-frequency plane. The method utilizes 2D undecimated discrete wavelet transform (2D UDWT) as an intermediate step in the analysis, followed by signal reconstruction and 2D FT. We have applied the method to a simulated 2D double quantum coherence (DQC) signal for validation and a set of experimental 2D ESR signals, demonstrating its efficiency in resolving overlapping peaks in the frequency domain, while displaying frequency evolution with time in case of non-stationary data.

Theory and Least Squares Fitting of CW ESR Saturation Spectra Using the MOMD Model

CW saturation experiments are widely used in ESR studies of relaxation processes in proteins and lipids. We develop the theory of saturation in ESR spectra in terms of its close relation with that of 2D-ELDOR. Our treatment of saturation is then based on the microscopic order macroscopic disorder (MOMD) model and can be used to fit the full CW saturation spectrum, rather than fitting just the peak-peak amplitude as a function of microwave field B ₁ as is commonly done. This requires fewer experiments to yield effects on T ₁, as well as provides a more extensive dynamic structural picture, for example, for scanning experiments on different protein sites. The code is released as a publicly available software package in Python that can be used to fit CW saturation spectra from biological samples of interest.

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 Exchange Processes Studied by Two-Dimensional ESR at 95 GHz

Exchange processes which include conformational change, protonation/deprotonation, and binding equilibria are routinely studied by 2D exchange NMR techniques, where information about the exchange of nuclei between environments with different NMR shifts is obtained from the development of cross-peaks. Whereas 2D NMR enables the real time study of millisecond and slower exchange processes, 2D ESR in the form of 2D-ELDOR (two-dimensional electron-electron double resonance) has the potential for such studies over the nanosecond to microsecond real time scales. Cross-peak development due to chemical exchange has been seen previously for semiquinones in ESR, but this is not possible for most common ESR probes, such as nitroxides, studied at typical ESR frequencies because, unlike NMR, the exchanging states yield ESR signals that are not resolved from each other within their respective line widths. But at 95 GHz, it becomes possible to resolve them in many cases because of the increased g-factor resolution. The 95 GHz instrumental developments occurring at ACERT now enable such studies. We demonstrate these new capabilities in two studies: (A) the protonation/deprotonation process for a pH-sensitive imidazoline spin label in aqueous solution where the exchange rate and the population ratio of the exchanging states are controlled by the concentration and pH of the buffer solution, respectively, and (B) a nitroxide radical partitioning between polar (aqueous) and nonpolar (phospholipid) environments in multilamellar lipid vesicles, where the cross-peak development arises from the exchange of the nitroxide between the two phases. This work represents the first example of the observation and analysis of cross-peaks arising from chemical exchange processes involving nitroxide spin labels.