Chapter 2: Electron Paramagnetic Resonance
2.9 EPR Data Analysis
EPR spectra can be analyzed in several different ways, depending on the type of experiment and conclusions desired. The most qualitative analysis involves noting general changes in EPR lineshape, indicative of some structural transition taking place in the sample. While this is useful during acquisition to verify sample preparation and compare previous hypotheses, more rigorous analysis is certainly possible (Fig. 22).
X-Band
W-Band
3350 3400 11900 11950 12000 12050 121003450 34850349003495035000350503510035150Q-Band
X-axis Y-axis Z-axis Powder (random) 100G wide 200G wide 300G wideFig. 21. Multifrequency EPR. Simulated spectra of a typical nitroxide label with magnetic field
oriented along the label’s x (red), y (blue), and z (green) axis, along with isotropic orientation (black) at X, Q, and W microwave frequency bands. As the microwave frequency is increased, the oriented components overlap less and less. This is also observed by the increasing complexity (number of lines) in the powder spectrum (bottom). Therefore, higher frequency bands possess greater orientation sensitivity.
Measuring the distance in magnetic field units between the low and high resonance lines (splitting) is a way to calculate spin label mobility, as an increase in the rate of rotational diffusion will cause this splitting to decrease:
[ ]
Eq. 23
where T||′ is the splitting for the sample of interest, and T|| is the splitting of an immobilized version of the sample. Eq. 23 is an empirical formula valid for spin-labeled proteins exhibiting slow, restricted motion [109].
Also of interest is the width of the resonance lines (linewidth), although unlike the splitting, the linewidth is affected by multiple phenomena, including the rate of rotational
Magnetic Field Strength
Deriv
ativ
e
A
bso
rptio
n
2T||′
2L′
Fig. 22. EPR spectrum lineshape measurements. A conventional EPR spectrum is typically
characterized by the splitting between the low and high field resonances (2T||’) and the linewidth of the low field resonance (2L’). These measurements can be used to determine rotational mobility and spin-spin interactions.
diffusion and presence of neighboring spin labels via spin-spin interaction. In the case of dilute spin label, an equation similar to Eq. 23 can be used in conjunction with the linewidth (as measured from the low field resonance line as half width at half maximum L′):
[ ] Eq. 24 where again, the primed quantity refers to the sample as perturbed by rotational diffusion, and the unprimed quantity refers to the immobilized sample. Eq. 24 is valid for slow, restricted motion as well, although Eq. 23 is typically used instead of Eq. 24, due to possible effects from spin-spin interactions.
Determining the order parameter S from EPR spectra is commonly accomplished using Eq. 21, assuming the protein undergoes rapid restricted motion [110]. The spectral splitting corresponding to the protein sample of interest (2T||′), a fully immobilized protein sample (2T||), and a rapid isotropic protein sample (2T0) are required, so typically multiple samples of the same protein must be prepared. Freezing or lyophilizing the protein can be used to obtain T||′, while heating or truncating the protein in solution can be used to obtain T0.
Saturation transfer EPR spectra are analyzed based on peak height ratios and integrated intensity. These have been measured previously for spin-labeled hemoglobin, a well-characterized globular protein with a known rotational correlation time based on the Stokes-Einstein formula (Eq. 17) [111]. This spin-labeled hemoglobin was mixed into solutions containing a specific amount of glycerol to produce a desired viscosity, and thus a variety of rotational correlation times. The peak height ratios were measured from the
saturation transfer EPR spectra, and plotted against the calculated rotational correlation times. Using these plots, a sample with an unknown rotational correlation time can be determined from the peak height ratios or the integrated intensity measured from the saturation transfer EPR spectrum. The integrated intensity parameter is commonly used in STEPR analysis, due to its minimal sensitivity to orientation anisotropy and well- defined value, though C′/C is also common due to even less sensitivity to orientation anisotropy [112]. 1.0 0.5 0.0 -0.5 -1.0 2.0 1.5 1.0 0.5 0.0 2.0 1.5 1.0 0.5 0.0 L′′ /L 1.00 0.75 0.50 0.25 0.00 C ′/C H ′′/ H IST = ∫V 2 ′
L
L′′
C′
C
H′′
H
-8 -7 -6 -5 -4 -3 ∞ -8 -7 -6 -5 -4 -3 ∞ -8 -7 -6 -5 -4 -3 ∞ LOG(R/ s) LOG(R/ s) LOG(R/ s)-8 -7 -6 -5 -4 -3 ∞ LOG(R/ s)
Fig. 23. Measuring rotational diffusion from saturation transfer EPR. STEPR spectra are analyzed
by measuring intensities at specific locations in the spectrum (red, top left) or the integral of the spectrum (integrated intensity, IST). These values are then compared to correlation time plots obtained with standard samples of maleimide spin labeled hemoglobin in glycerol to determine the rotational correlation time for the sample of interest. Figures adapted from [111].
Precisely determining the rate and amplitude of rotational diffusion, as well as orientation distributions, requires analysis of EPR spectra beyond lineshape measurements. In these cases, we can fit EPR spectra through least-squares minimization and spectral simulation [113]. For a rigid limit (R > 100 ns) sample containing one or several orientations, the simulation is relatively straightforward, as there are static and well-defined magnetic parameters corresponding to non-exchanging states, at least on the EPR timescale defined by T2 relaxation. Additionally, if the diffusion is much faster than T2, then we recover an isotropic case with complete averaging and a single state. The simulation becomes more complicated when rotational diffusion on the order of T2 is present, due to multiple states and incomplete rotational averaging generating additional
Fig. 24. Fitting of EPR spectra by simulation. Conventional EPR spectra of nitroxide spin labels
can be simulated and fit by least squares minimization (right). Key parameters of the EPR spectrum, including electron g, hyperfine T, rotational diffusion R, order parameter S, and probe
states. However, several dedicated EPR simulation programs have been developed by Jack Freed, David Schneider, and David Budil [110, 113, 114]. These programs utilize Sturm-Lioville theory along with parallelized matrix calculations.to quickly simulate these slow, but not immobilized, EPR spectra. As a result, we can simulate conventional EPR spectra of nitroxide spin labels in the fast, intermediate, and slow limits.
Fitting EPR spectra by simulation can be challenging due to the number of variables in the fit. A typical EPR simulation model will contain ~10 parameters unique to the sample, including but not limited to electron g tensor, hyperfine T tensor, rotational diffusion tensor, and order parameter. Assumptions are often employed to constrain these fits, including previous measurements on similar samples (frozen, randomly oriented, etc), and physical considerations like axial or isotropic symmetries. Additionally, only a few parameters are varied at one time, and correlation matrices produced by the fitting software can be consulted to control for overparameterization. Global fitting of several related sample spectra (parallel/perpendicular bicelles, X/Q-band, etc) is also a common tactic. Several models are often compared, as the concept of a ‘best’ model is somewhat elusive. Despite these shortcomings, this is perhaps the most quantitative method of EPR data analysis.
Chapter 3. Protein-Protein Interactions in Calcium Transport Regulation