Dipolar pathways in multi-spin and multi-dimensional dipolar EPR spectroscopy

Abstract

Dipolar electron paramagnetic resonance (EPR) experiments, such as double electron-electron resonance (DEER), measure distributions of nanometer-scale distances between unpaired electrons, which provide valuable information for structural characterization of proteins and other macromolecular systems. We present an extension to our previously published general model based on dipolar pathways valid for multi-dimensional dipolar EPR experiments with more than two spin-1/2 labels. We examine the 4-pulse DEER and TRIER experiments in terms of dipolar pathways and show experimental results confirming the theoretical predictions. This extension to the dipolar pathways model allows the analysis of previously challenging datasets and the extraction of multivariate distance distributions.

Fig. 1. General multi-dimensional pulse sequence. The sequence consists of N pulses (black boxes) separated by the time intervals T_n. The spin echo is detected after the last time interval T_N. Several time intervals can be incremented/decremented (as indicated by the vectors Δ_d = (Δ_d,1,…,Δ_d,N)) according to the different experimental time axes t_d. The number of different time axes t_d defines the dimensionality D of the experiment.

Fig. 2. Schematic representation of dipolar interactions in different multi-spin systems. The spins are shown as black arrows and all possible interspin distances r_q are shown as dashed lines for a two-spin system (M = 2, Q = 1), a three-spin system (M = 3, Q = 3), and a four-spin system (M = 4, Q = 6).

Fig. 3. Schematic representation of a multi-spin dipolar pathway, exemplified in a three-spin system. During a sequence of pulses (shown as black boxes) and time intervals T_n, a multi-spin dipolar pathway with multi-spin dipolar phase accumulation factors is composed of Q pair dipolar phase accumulation factors s_k,q (shown as white lines) that describe whether the dipolar phase corresponding to the q-th dipolar interaction frequency ω_dip,q increases (+1), decreases (−1), or remains constant (0). Each pair dipolar pathway is colored as its corresponding dipolar interaction.

Fig. 4. Schematic representation of the sequential simplification of the multi-spin dipolar pathways models. An example subset of five different multi-spin dipolar pathways with multi-spin accumulation factors for the one-dimensional three-pulse DEER experiment on a three-spin system is shown as boxes containing the individual modulated (|δ_p| > 0, colored) and unmodulated (|δ_p| = 0, grey) pair dipolar pathways s_k,q, which can be collected into a unique multi-spin dipolar pathway 1 with multi-spin accumulation factors where all unmodulated pair pathways are represented by a generic pair dipolar pathway. Other sets of multi-spin dipolar pathways can be collected into unique multi-spin dipolar pathways 2 to 8 with accumulation factors . The set of eight different unique multi-spin dipolar pathways can be described by one unmodulated pathway (grey) and two unique pair dipolar pathways with pair accumulation factors s₁ (turquoise) and s₂ (magenta). The amplitudes of the different pathways are given on top of each box.

Fig. 5. Pair dipolar pathways of the multi-spin 4-pulse DEER experiment. The 4-pulse DEER pulse sequence is shown on top, with every probe pulse represented as a black box and the pump pulse as a green box. The table contains a subset of the most commonly encountered pair dipolar pathways s_p along with their harmonics δ_p and refocusing times t_ref,p. The pathways are ordered in decreasing estimated amplitude for commonly reported experimental conditions. The refocusing times are illustrated as dashed turquoise lines on top of a schematic 4-pulse DEER dipolar signal shown as a black line.

Fig. 6. Pair dipolar pathways of the multi-spin TRIER experiment. The TRIER pulse sequence is shown on top, every probe pulse represented as a black box, the t₁-shifted pump pulse as a green box, and the t₂-shifted pump pulse as a red box. The table lists all possible pair dipolar pathways s_p along with their harmonics δ_p and refocusing times t_ref,p modulated along t₁ (green), t₂ (red), and both (black). The pathways are ordered in decreasing estimated amplitude for commonly reported experimental conditions. In the left panel, the refocusing times of the one-dimensional modulated contributions are illustrated as dashed green and red lines, and the two-dimensional modulated contributions as black circles on top of a schematic TRIER dipolar signal shown as greyscale contours.

Fig. 7. Global analysis with DeerLab of a series of X-band 4-pulse DEER of oligoPPE triradical T111 (top right) acquired with τ₁ = 0.4 μs and τ₂ = 6 μs, and different levels of microwave power attenuation. The experimental datasets are shown in the left panel as grey dots, along with the model fits and unmodulated contributions shown as solid and dashed blue lines, respectively. The contributions from two-spin interactions are shown as turquoise lines, and the contributions arising from three-spin interactions are shown as red lines. For clarity, only the first 3 μs out of the 6.4 μs of the recorded trace are shown (the full traces are shown in Fig. S1 in the ESI†). The right panel shows a globally fitted trivariate distance distribution. The univariate marginal distributions are shown as filled areas, and the bivariate marginal distributions are shown as colored contours. The fitted distribution is shown in blue, and the HSC simulation is shown in grey.

Fig. 8. Global analysis with DeerLab of a series of X-band 4-pulse DEER of oligoPPE triradical T011 (top right) acquired with τ₁ = 0.4 μs and τ₂ = 12 μs, and different levels of microwave power attenuation. The experimental datasets are shown in the left panel as grey dots, along with the model fits and unmodulated contributions shown as solid and dashed blue lines, respectively. The contributions from two-spin interactions are shown as turquoise lines, and the contributions arising from three-spin interactions are shown as red lines. For clarity, only the first 3 μs out of the 12.1 μs of the recorded trace are shown (the full traces are shown in Fig. S2 in the ESI†). The right panel shows a globally fitted trivariate distance distribution. The univariate marginal distributions are shown as filled areas, and the bivariate marginal distributions are shown as colored contours. The fitted distribution is shown in blue, and the HSC simulation is shown in grey.

Fig. 9. Global analysis with DeerLab of a series of Q-band 4-pulse DEER of triply MTSL-labeled Rpo47 protein (ribbon model top right, PDB : 1GO3) acquired with τ₁ = 0.4 μs and τ₂ = 9 μs, and different levels of microwave power attenuation. The experimental datasets are shown in the left panel as grey dots, along the model fits and unmodulated contributions shown as solid and dashed blue lines, respectively. The contributions from two-spin interactions are shown as turquoise lines, and the contributions arising from three-spin interactions are shown as red lines. The right panel shows a globally fitted trivariate distance distribution. The univariate marginal distributions are shown as filled areas, and the bivariate marginal distributions are shown as colored contours. The fitted distribution is shown in blue, and the MMMx simulation is shown in grey.

Fig. 10. Global analysis with DeerLab of a series of X-band 4-pulse DEER of a oligoPPE tetraradical (top right), acquired with τ₁ = 0.4 μs and τ₂ = 8 μs, and different levels of pump pulse power. The experimental datasets are shown in the left panel as grey dots and the model fits and unmodulated contributions shown as solid and dashed blue lines, respectively. The contributions from two-spin interactions are shown as turquoise lines and the contributions arising from three-spin interactions are shown as red lines. For clarity, only the first 3 μs out of the 8.1 μs of the recorded trace are shown (the full traces are shown in Fig. S3 in the ESI†). The right panel shows the globally fitted hexavariate distance distribution. The univariate marginal distributions are shown as filled areas and the bivariate marginal distributions are shown as colored contours. The fitted distribution is shown in blue and the HSC simulation is shown in grey.

Fig. 11. Analysis with DeerLab of the Q-band TRIER data on oligoPPE triradical T111 (top right), acquired with τ₁ = 0.4 μs, τ₂ = 4.6 μs, and τ₃ = 0.9 μs. The two-dimensional experimental dataset is shown in the left panel as filled colored contours along with the model fit shown as greyscale contour lines. The signal integrals along each dimension are shown in the insets as grey dots for the experimental data and as a solid blue line for the model fit and a dashed blue line for the unmodulated contribution. The right panel shows the fitted trivariate distance distribution. The univariate marginal distributions are shown as filled areas and the bivariate marginal distributions are shown as colored contours. The fitted distribution is shown in blue and the HSC simulation is shown in grey for reference.

Fig. 12. Analysis with DeerLab of the Q-band TRIER data on oligoPPE triradical T011, (top right) acquired with τ₁ = 0.4 μs, τ₂ = 2.8 μs, and τ₃ = 0.9 μs. The two-dimensional experimental dataset is shown in the left panel as filled colored contours along the model fit shown as greyscale contour lines. The signal integrals along each dimension are shown in the insets as grey dots for the experimental data and as a solid blue line for the model fit and a dashed blue line for the unmodulated contribution. The right panel shows the fitted trivariate distance distribution. The univariate marginal distributions are shown as filled areas and the bivariate marginal distributions are shown as colored contours. The fitted distribution is shown in blue and the HSC simulation is shown in grey.

Fig. 13. Analysis with DeerLab of the Q-band TRIER data of triply MTSL-labeled Rpo47 protein (ribbon model top right, PDB : 1GO3), acquired with τ₁ = 0.4 μs, τ₂ = 2.8 μs, and τ₃ = 0.4 μs. The two-dimensional experimental dataset is shown in the left panel as filled colored contours along the model fit shown as greyscale contour lines. The signal integrals along each dimension are shown in the insets as grey dots for the experimental data and as a solid blue line for the model fit and a dashed blue line for the unmodulated contribution. The right panel shows the fitted trivariate distance distribution. The univariate marginal distributions are shown as filled areas and the bivariate marginal distributions are shown as colored contours. The fitted distribution is shown in blue and the MMMx simulation is shown in grey.