Originally published as Biophys J. BioFAST on October 7, 2005.
doi:10.1529/biophysj.105.068544
Biophysical Journal 90:340-356 (2006)
© 2006 The Biophysical Society
Dipolar Coupling between Nitroxide Spin Labels: The Development and Application of a Tether-in-a-Cone Model
Eric J. Hustedt *,
Richard A. Stein *,
Latsavongsakda Sethaphong *,
Suzanne Brandon *,
Zheng Zhou * and
Susan C. DeSensi 
* Department of Molecular Physiology and Biophysics, and Department of Chemistry, Vanderbilt University, Nashville, Tennessee 37232
Correspondence: Address reprint requests to Eric J. Hustedt, 735B Light Hall, Vanderbilt University, Nashville, TN 37232. Tel.: 615-322-3181; Fax: 615-322-7236; E-mail: eric.hustedt{at}vanderbilt.edu.
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ABSTRACT
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A tether-in-a-cone model is developed for the simulation of electron paramagnetic resonance spectra of dipolar coupled nitroxide spin labels attached to tethers statically disordered within cones of variable halfwidth. In this model, the nitroxides adopt a range of interprobe distances and orientations. The aim is to develop tools for determining both the distance distribution and the relative orientation of the labels from experimental spectra. Simulations demonstrate the sensitivity of electron paramagnetic resonance spectra to the orientation of the cones as a function of cone halfwidth and other parameters. For small cone halfwidths (<
40°), simulated spectra are strongly dependent on the relative orientation of the cones. For larger cone halfwidths, spectra become independent of cone orientation. Tether-in-a-cone model simulations are analyzed using a convolution approach based on Fourier transforms. Spectra obtained by the Fourier convolution method more closely fit the tether-in-a-cone simulations as the halfwidth of the cone increases. The Fourier convolution method gives a reasonable estimate of the correct average distance, though the distance distribution obtained can be significantly distorted. Finally, the tether-in-a-cone model is successfully used to analyze experimental spectra from T4 lysozyme. These results demonstrate the utility of the model and highlight directions for further development.
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INTRODUCTION
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Site-directed spin-labeling (SDSL) methods are being widely used to study both the structure and the functional dynamics of proteins (for reviews, see Hubbell and Altenbach and others (1
–6)). SDSL combines site-directed mutagenesis to introduce single cysteines at specific sites within a protein together with labeling by a cysteine-specific nitroxide probe, typically methanethiosulfonate spin label (MTSSL). Electron paramagnetic resonance (EPR) spectroscopy is then used to measure physical characteristics at that site including the mobility of the spin label and the accessibility of the side chain to relaxation agents such as O2 and Ni(EDDA). In double site-directed spin-labeling (DSDSL) studies, pairs of nitroxide spin-label probes are introduced at selected sites. EPR spectroscopy is then used to measure the distance or distance distribution between probes (for recent examples, see Klare et al. and others (7–14)). Distances obtained from DSDSL studies have been used to build models of protein structures and the structural transitions associated with protein function (e.g., 10,15,16). Reviews of various techniques for using EPR to measure interprobe distances have been recently collected in a single volume (17).
Continuous wave EPR (CW-EPR) spectroscopy can be used to measure distances up to 20–25 Å, whereas pulsed EPR methods can be used to measure distances up to 80 Å in favorable cases (18). Distances and distance distributions can be extracted from CW-EPR spectra of dipolar coupled nitroxides by a variety of approaches. In the case where the two nitroxides adopt a unique geometry with respect to each other, a single interelectron distance has been determined by computer lineshape simulation (19). In other cases, the use of different sets of assumptions allows distance determination by analysis of line height ratios (20), by measurement of the homogeneous linewidth (21), or by Fourier deconvolution/convolution methods (22–25). Various methods for measuring distances in the 7–24-Å range have been compared experimentally (26).
When two nitroxides adopt a specific, rigid geometry with respect to each other, both the relative distance and orientation between the two nitroxides can be determined by computer lineshape simulation (for reviews, see Hustedt and Beth (4,27)). In a study using a spin-labeled NAD+ ligand bound to polycrystalline glyceraldehyde-3-phosphate dehydrogenase (GAPDH), CW-EPR spectra were obtained at X-, Q-, and W-bands. Starting from the spin Hamiltonian and using a combination of simulated annealing and Marquardt-Levenberg algorithms, these multifrequency spectra were simultaneously analyzed by direct spectral simulation to obtain the distance between the unpaired electrons localized to the N-O bond of the nitroxides, the orientation of the interelectron vector in the frame of the nitroxide A- and g-tensors, and the three angles defining the relative orientation of the two nitroxide tensor frames (19).
Fourier methods assume the lineshape to be the convolution of a dipolar broadening function with the sum of the spectra of the isolated probes at the two sites (22–25). These methods inherently assume that there is an isotropic averaging of the relative orientation between the nitroxides and are used to obtain estimates of the average interspin distance and the width of the distance distribution.
Fourier convolution methods and the spectral simulation approach used to analyze data for spin-labeled NAD+ bound to GAPDH represent two extremes. Fourier convolution methods have been successfully used to give interspin distances in a number of studies. On the other hand, the highly ordered nature of the spin-labeled NAD+/GAPDH complex required the detailed consideration of the orientation of the nitroxides. These results raise questions about the analysis of data from DSDSL studies where there is often partial ordering between the two probes. The apparent mobility of the MTSSL side chain depends strongly on the label site (28). Labels at buried sites give EPR spectra indicative of immobilized labels, as can labels at sites that have some degree of tertiary contact. Steric interactions at these sites restrict the local dynamics of the MTSSL side chain. The mobility of the MTSSL side chain, as reflected in the inverse of the central linewidth, is a measure of both the amplitude and the rate of constrained anisotropic rotational diffusion (5). Simulations of the X-band EPR lineshapes from MTSSL and other methanethiosulfonate spin labels at helix surface sites in T4 lysozyme (T4L) have been used to determine rates and the associated order parameters of rotational dynamics (29). Similar results have been obtained from a combined analysis of spectra collected at 9 and 250 GHz (30). These results indicate that the local motion of the MTSSL side chain is highly constrained, the overall motion of the probe is coupled to the backbone, and the lineshape of MTSSL retains sensitivity to backbone dynamics even at helix surface sites (5). Similarly it is important to consider whether because of the site-specific constraints on the local order of the MTSSL side chains, the EPR spectrum of a pair of dipolar coupled labels retains sensitivity to their relative orientation. Careful consideration of the effects of the relative orientation of the probes may lead to increased accuracy in the determination of the relative distance distribution between probes. In addition, the relative orientation may itself be used as a constraint for building structural models.
The vast majority of SDSL studies of protein structure utilize the MTSSL label, which has five chemical bonds between the 
-carbon of the cysteine and the five-membered nitroxide ring. Although detailed consideration of the preferred orientation and flexibility about each of these bonds is possible (29,31), in this work a simple model is developed in an initial attempt to treat the inherent flexibility of the tether linking the nitroxide to the protein in the context of simulation of the EPR spectra of dipolar coupled spin labels. The unpaired electron is placed at the end of a tether that adopts all possible angles within a cone of variable halfwidth. Hence, both the interelectron distance and the relative orientation of the two nitroxides are determined by the position of the tethers within their cones. This model then, inherently, produces a distribution of interelectron distances. At the same time, the degree of orientational disorder between the probes is also a function of the various parameters of the model. The tether-in-a-cone model can thus be used to test the sensitivity of CW-EPR spectra of dipolar coupled nitroxides to the degree of disorder in both the interelectron distance and orientation.
Simulations are presented below that demonstrate the sensitivity of EPR spectra to the various parameters of the tether-in-a-cone model. In particular, these simulations demonstrate the sensitivity of CW-EPR spectra to the relative orientation of the two cones as a function of the cone halfwidth. These simulations are used to determine how much disorder is required to obscure the effects of the relative orientation of the spin labels. These simulations are then used as artificial data for analysis by a Fourier convolution method. The distance distributions obtained by Fourier convolution analysis are compared to the known distance distributions from the tether-in-a-cone model. From these results, it can be determined how well the Fourier convolution method does as a function of the degree of disorder. Finally, the tether-in-a-cone model can itself be used as a tool for analyzing data from DSDSL studies as demonstrated by analysis of selected data from DSDSL studies of T4L.
Portions of this work have been previously published as an abstract (32).
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METHODS
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Tether-in-a-cone model
The spin Hamiltonian in the high-field approximation for a pair of dipolar coupled nitroxides is given by
 | (1) |
where ße is the Bohr magneton, g1 and g2 are the tensors defining the interaction of the electron spin of nitroxide 1 (
) and nitroxide 2 (
) with the DC magnetic field (
), 
n is the Larmor frequency of the nitrogen nucleus, 
e is the gyromagnetic ratio for the electron, A1 and A2 are the hyperfine tensors defining the interaction of the nitrogen nuclear spins (
or 
) with or 
D is the unique element of the dipolar coupling tensor, and J is the scalar exchange interaction. Treatment of this spin Hamiltonian has been described in detail (4,19). The g-tensor and the hyperfine (A-) tensor elements for the two nitroxides, as well as the dipolar coupling tensor, D, are all determined by a set of Euler angle transformations (see Fig. 1). In the tether-in-a-cone model, the nitroxides are placed at the end of tethers of length, q, which are separated by a distance, p, at their bases. The tethers adopt all possible angles within cones of halfwidth µmax resulting in a distribution of internitroxide distances and orientations.
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FIGURE 1 Diagram defining parameters for the tether-in-a-cone model. The axes X, Y, and Z are the axes of the two nitroxides with X along the N-O bond and Z perpendicular to the plane of the nitroxide and with all subscripts referring to nitroxide 1 or nitroxide 2. The two cones are separated by  at their bases and their relative orientation is determined by the angles  1,  2, and 2 (2 not shown). The orientation of the two cones with respect to the magnetic field,  is determined by the angles  and  ( not shown). The orientations of the tethers,  and  within their respective cones are determined by the angles  , µ, and  ( and not shown). The orientation of the tethers with respect to the nitroxide axis systems are determined by the angles , ß, and ( and not shown). Those angles that are not shown for the sake of simplicity correspond to Euler angle rotations about the appropriate Z axis as defined in Eqs. 2–8.
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A set of seven rotation operators define the various transformations that are needed to determine the elements of the nitroxide g- and A-tensors and the dipolar coupling tensor. The orientations of the tether axes with respect to the two nitroxide axis frames are determined by the transformations 
and 
for nitroxides 1 and 2, respectively. The orientations of the tether axes within the cones are determined by the transformations 
and 
for nitroxides 1 and 2, respectively. The orientation of the two cones with respect to each other is determined by 
for nitroxide 1 and 
for nitroxide 2. Finally, the orientation of the nitroxide pair with respect to the magnetic field is determined by the transformation, 
Using this set of Euler angle transformations, the g- and A-tensor elements in the laboratory frame for nitroxides 1 and 2 are given by
 | (2) |
where 
and 
are the diagonal g-tensors and 
and 
are the diagonal A-tensors for the two nitroxides. All of the Euler angle rotation matrices are defined as follows,
 | (3) |
In this work it is assumed that the two nitroxides have the same principal values for their g- and A-tensors (gxx = 2.0082, gyy = 2.0060, gzz = 2.0023, Axx = 6.5 Gauss, Ayy = 5.5 Gauss, Azz = 36.0 Gauss) except as noted below for the analyses of data from T4L.
The dipolar coupling depends on the angle, 
between the interelectron vector, 
and the magnetic field vector,
 | (4) |
where
 | (5) |
The vector is determined by
 | (6) |
and 
written as
 | (7) |
with
 | (8) |
where and 
are vectors representing the two tethers and is the vector of length p connecting their respective origins (see Fig. 1).
A spectrum is calculated by averaging over a set of angles to account for an isotropic distribution of nitroxide pairs in the lab frame (as determined by and ) and to account for a range of tether positions within the two cones assuming square-well potentials of equal halfwidth (µmax).
 | (9) |
For a pair of 14N nitroxide spin labels, there are nine possible combinations of nitrogen nuclear spin states. For each nuclear spin state there are four allowed electron spin transitions. Therefore, S(H0;1,ß1,1,2,ß2,2,1,2,2,1,µ1,1,2,µ2,2,,) is the sum of 36 first-derivative Lorentzian lineshapes. The calculation of the center position and transition probabilities from the spin Hamiltonian (Eq. 1) for each of these Lorentzian lines has been described in detail elsewhere (19). In all of the calculations presented here a Lorentzian linewidth of 2.0 Gauss is used except as noted below for the analyses of data from T4L.
A number of different spherical codes for efficient averaging over the surface of a sphere as required by Eq. 9 have been used for simulation of EPR spectra (33–35). In this work a variation of the SPIRAL code (36,37) has been used. The SPIRAL code defined below is equivalent to that previously used by Wong and Roos (37) in defining a set of values but differs slightly in how the corresponding values of are generated. Let
 | (10) |
then
 | (11) |
and
 | (12) |
The value of Nloop determines the tightness of the spiral, i.e., the number of turns from = 0 to 2
as goes from (or /2) to 0. A total of 
spirals are used with each spiral starting at equally spaced values of from 0 to 2.
Equations 10–12 define a SPIRAL code for averaging over the angles and . The SPIRAL code for averaging over the cone angles µ and is defined using a similar set of equations with µ restricted between 0 
µ µmax. A third angle, , which corresponds to rotation about the tether axis, is also included. Note that rotations by the angle do not change the interelectron distance.
 | (13) |
In all of the simulations presented in this work, it is assumed that the lengths of the tethers q = |Q1|= |Q2| and the angles defining the orientations of the tethers with respect to the nitroxide, = 1= 2 and ß = ß1 = ß2, are the same for both labels. For all the calculations presented here, N = 1 and is fixed to be 0°. Using the SPIRAL codes as defined in Eqs. 10–13, the integral in Eq. 9 becomes
 | (14) |
In the tether-in-a-cone model, the interelectron distance, R, is a function of the values of p and q, the relative orientation of the two cones (as determined by the angles 1, 2, and 2), and the cone angles µ and for both nitroxides. Averaging over the angles µ1, 1, µ2, and 2 results in a distribution of interelectron distances, P(R). The values of R obtained by averaging over these angles are binned and P(R) is represented as a histogram at 99 values of R between R = 0 and R = p + 3q. The distance distribution, P(R), is characterized by its average value, 
R
,
 | (15) |
and its standard deviation, 
R,
 | (16) |
Convolution analysis
Convolution-based methods have been widely used to analyze EPR spectra of dipolar coupled nitroxides (22–25). It is assumed that the spectrum of the dipolar coupled nitroxides, C(H0), is given by the convolution of the spectrum of the spatially isolated nitroxides, M(H0), with a dipolar broadening function, 
(H0).
 | (17) |
The function (H0) is assumed to be a sum of Pake patterns corresponding to a distribution of interspin distances. Rabenstein and Shin (23) obtained (H0) by Fourier deconvolution of experimentally determined C(H0) and M(H0) spectra and then determined an average distance and width of the distance distribution from (H0). Noise in the Fourier transform of (H0), which is an inherent result of deconvolution, can be dealt with by fitting to a sum of two Gaussian functions (25,38). Alternatively, Steinhoff and co-workers (24) avoided the deconvolution step by assuming a Gaussian distribution of interspin distances to calculate (H0), convolving (H0) and M(H0), comparing the result to the experimental data, and finding the Gaussian distance distribution that gave the best fit. Hubbell and co-workers (22) have developed an interactive approach using a combination of deconvolution and convolution steps to obtain a histogram of interspin distances.
In this work, the approach of Steinhoff and co-workers is used (24). The distribution of interspin distances, PG(R), is taken to be a Gaussian centered at R0 of width 
R.
 | (18) |
The broadening function, (H0), is calculated as the sum of a large number (100,000 or more) of Pake patterns. A lower limit distance cutoff, determined by the sweep width (SW, in Gauss) of the experimental spectra, Rmin(Å) = 30.3 x SW(–1/3), is used so as not to include Pake patterns whose singularities lie outside of the scan range of the data being analyzed. A dipolar broadened spectrum is calculated by convolving (H0) and M(H0). The fit between the calculated spectrum and the spectrum being analyzed is then optimized by adjusting R0 and R using Marquardt-Levenberg methods to minimize 
2, the sum of the squares of the differences between the data and the fit (see Hustedt et al. (39)). In some cases, the distance distribution, PG(R), was taken to be the sum of two Gaussians. The fitting parameters are then the centers and widths of the two Gaussians and their relative amplitudes. When analyzing simulated spectra of dipolar coupled nitroxides, the spectrum of the spatially isolated nitroxides, M(H0), was calculated by setting p = 100 Å, q = 0 Å, and µmax = 0°. When analyzing data from DSDSL studies of T4L, M(H0) was taken to be the equally weighted sum of the normalized EPR spectra of the two spin-labeled single cysteine mutants at the corresponding sites.
The quality of the fits to the data is expressed by the correlation coefficient
 | (19) |
where Yi are the values of the data, Fi are the values of the fit to the data, and N is the total number of points in each. To account for incomplete labeling when analyzing data from DSDSL experiments on T4L, the best fit to the data was obtained as a linear combination of: 1), the spectrum of the spatially isolated nitroxides, M(H0), 2), the calculated spectrum of dipolar coupled nitroxides, C(H0), and 3), a constant baseline correction as described previously (19).
 | (20) |
Analysis of data using the tether-in-a-cone model
The tether-in-a-cone model has been incorporated into a computer software package designed for the nonlinear least-squares analysis of EPR spectra (19,39). The program uses a combination of simulated annealing and Marquardt-Levenberg algorithms to find the best fit parameters to a given spectrum. The principal values of the g-tensors, and 
and A-tensors, and 
were obtained from fitting the EPR spectra of the corresponding spin-labeled single cysteine mutants as previously described (39). The Lorentzian linewidth was taken to be the average value obtained from the analysis of the two singly labeled spectra. Values of ß, , and q were obtained from the X4X5 model (29) as described below. Values of p have been estimated from the C to C distance measured from the crystal structure of wild-type T4L (Protein Data Bank (PDB), 3LZM (40)). Data from DSDSL studies of T4L were analyzed using a combination of multiple independent runs of simulated annealing followed by further minimization by the Marquardt-Levenberg algorithm and rigorous determination of confidence intervals (19,39) in an attempt to find the values of 1, 2, 2, and µmax, which give the global 2 minimum. To account for incomplete labeling, the best fit to the data was obtained as a linear combination of: 1), the equally weighted sum of the spectra of the spin-labeled single cysteine mustants, M(H0), 2), the calculated spectrum of dipolar coupled nitroxides, S(H0), and 3), a constant baseline correction as described previously (19).
 | (21) |
Computer software
All computer programs used to generate the tether-in-a-cone-model simulations and to perform the analyses of data were written in FORTRAN. Anyone interested in obtaining the FORTRAN code developed in this work should contact the authors.
X4X5 model
The X4X5 model proposed by Hubbell and co-workers (29,31) has been used to obtain estimated values for the two angles that determine the orientation of the tether axis with respect to the nitroxide frame, and ß, and the tether length, q. In the X4X5 model the torsion angles of the first three bonds between the C of the labeled cysteine and the nitroxide ring are fixed, whereas the torsion angles for the fourth and fifth bonds are allowed to adopt all possible angles between 0 and 360°. From the x-ray crystal structure of spin-labeled mutants of T4L, the MTSSL at a solvent-exposed site is expected to have X1 
–60°, X2 –60°, and X3 –90° or X3 +90° (29,31). Using the known structure of T4L (PDB, 3LZM (40)), the appropriate residue was mutated to cysteine and labeled with MTSSL in the conformation X1 = –60°, X2 = –60°, and X3 = –90°. The torsion angles were set using the University of California San Francisco Chimera package (Computer Graphics Laboratory, UCSF; supported by National Institutes of Health (NIH) P41 RR-01081). Using the PyMOL molecular graphics system (Delano Scientific LLC, San Carlos, CA), the nitroxide ring was rotated in steps of 30° about X4 and X5 and the appropriate coordinates saved at each step. From the set of 144 orientations, average values of various parameters relating the nitroxide to the C of the labeled cysteine residue were calculated. For a spin label at residue 65 of T4L, the values calculated are q = 7.08 Å, ß = 78.9°, and = 330.5°. Very similar results have been obtained for spin labels at the other T4L sites (61, 68, and 69) that have been labeled in this study and using an alternate conformation (X1 = 180°, X2 = +60°, and X3 = +90°).
SDSL of T4 lysozyme
Plasmids containing the cysteine-less T4L gene and certain mutants (single cysteine mutants at sites 65 and 69, double cysteine mutants at 65/68 and 65/69) were generously provided by Dr. Hassane Mchaourab (Vanderbilt University, Nashville, TN). Other cysteine mutations (single cysteine mutants at sites 61 and 68, double cysteine mutant at 61/65) were engineered into the cysteine-less T4L gene by the overlap extension method (41). For all the T4L mutants, the entire coding region was confirmed by DNA sequencing. T4L mutants were expressed and purified as previously described (28). Briefly, plasmids were transformed into competent Escherichia coli K38 cells; 1 mM isopropyl ß-thiogalactoside was added to log phase cultures to induce protein expression for 90 min. The cell pellet was resuspended in a buffer containing 25 mM Tris, 25 mM MOPS, and 0.2 mM EDTA (pH 7.6). The cells were then disrupted by sonication. After 30 min centrifugation at 10,000 rpm, the supernatant was passed through a 0.22-µm filter. The flowthrough was then loaded on a Resource S cation-exchange column (Amersham Bioscience, Piscataway, NJ) and eluted with a NaCl gradient from 0 to 1 M. Protein concentration was determined by ultraviolet absorption at 280 nm using an extinction coefficient of 1.228 cm2mg–1. The purity of all T4L mutant proteins was at least 95%, as determined by SDS-PAGE (42). Single or double cysteine mutants were spin labeled with a 10-fold or 20-fold molar excess of 1-oxyl-2,2,5,5-tetramethyl-3-pyrroline-3-methyl methanethiosulfonate spin label (Toronto Research Chemicals, North York, Ontario, Canada) at room temperature for 10 min and then at 4°C overnight. Unreacted label was removed from all samples using a HiTrap desalting column (Amersham Bioscience, Piscataway, NJ) with desalting buffer containing 100 mM NaCl, 20 mM MOPS, 0.1 mM EDTA, and 0.02% azide (pH 7.0). Proteins were then concentrated in an Amicon Ultra-4 centrifugal filter device (5 kDa nominal molecular weight limit; Millipore, Bedford, MA).
EPR spectroscopy
All EPR spectra were collected at X-band using a Bruker EMX spectrometer (BrukerBiospin, Billerica, MA) equipped with a TM110 cavity with a Dewar insert and a Bruker variable temperature control unit. All spectra were recorded at –30°C using 100 kHz Zeeman modulation of 1 Gauss amplitude and a microwave power of 5 mW. Samples of spin-labeled T4L in desalting buffer plus 70% (w/w) glycerol were placed in 50-µl glass capillaries (VWR International, West Chester, PA) and flame sealed. The 50-µl capillaries were placed in a 5-mm quartz tube containing silicone fluid (Thomas Scientific, Swedesboro, NJ) held within the Dewar insert.
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RESULTS
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Model calculations
Fig. 2 shows simulations calculated for the tether-in-a-cone model as a function of the distance between the bases of the two cones, p, with all other parameters fixed. For comparison, the upper simulation (Fig. 2 A) is a spectrum of spatially isolated nitroxides where p = 100 Å, q = 0 Å, and µmax = 0° so that R = 100 Å, R = 0 Å, and the dipolar coupling is effectively zero. The other panels show simulations for p = 16 Å (Fig. 2 B), p = 12 Å (Fig. 2 C), and p = 8 Å (Fig. 2 D) with q = 6 Å and µmax = 30°. The insets at the upper right of these panels (and in subsequent figures) show histograms representing the interspin distance distribution, P(R). As p decreases, the center of P(R) moves to shorter distances and the corresponding simulated EPR spectrum becomes broader, which is indicative of increased dipolar coupling.
Fig. 3 shows the effect of increasing the tether length, q, with all other parameters fixed. As the tether length is increased from q = 2 Å (Fig. 3 A) through q = 6 Å (Fig. 3 B) to q = 10 Å (Fig. 3 C), splittings that are clearly resolved at the high and low field z-turning points in the simulated spectra for the smallest tether lengths are broadened out as the width of the distance distribution, P(R), increases accordingly. Fig. 4 shows similar results as a function of the cone halfwidth for µmax = 10° (Fig. 4 A), µmax = 30° (Fig. 4 B), and µmax = 60° (Fig. 4 C). As the cone halfwidth increases, resolved dipolar splittings in the spectra are broadened out and the width of P(R) increases. The calculations shown in Figs. 3 B and 4 B have been performed for the same values of q = 6 Å and µmax = 30°. Given the orientation of the cones as determined by the angles 1 = 90°, 2 = 0°, and 2 = 90°, this results in a distance distribution centered at R = 12.05 Å with a width of R = 1.88 Å. Reducing the tether length to q = 2 Å (Fig. 3 A; R = 12.02 Å and R = 0.71 Å) or reducing the cone halfwidth to µmax = 10° (Fig. 4 A; R = 12.00 Å and R = 0.62 Å) have similar effects on the resulting distance distribution and EPR spectra. Increasing the tether length to q = 10 Å (Fig. 3 C; R = 12.56 Å and R = 3.49 Å) or increasing the cone halfwidth to µmax = 60° (Fig. 4 C; R = 12.05 Å and R = 3.34 Å) likewise have similar effects on the resulting distance distribution and EPR spectra.
Fig. 5 shows simulations for the tether-in-a-cone model calculated for various values of four angles (ß, 1, 2, 2) for fixed values of p = 8 Å, q = 6 Å, and µmax = 30°. Comparing the spectra in Fig. 5, A (ß = 0°) and B (ß = 90°), shows that the simulated EPR spectra depend strongly on the orientation of the tether axis in the frame of the nitroxide, whereas the corresponding distance distributions are independent of the angles , ß, and (results for and not shown). The spectra in Fig. 5, A and C–F, show the effect of changes in the relative orientation of the two cones as determined by the angles 1, 2, and 2. The relative orientation of the cones is represented by the diagrams at the left of each panel in Fig. 5 and in subsequent figures. Both the average and the width of the distance distribution depend strongly on the relative orientation of the cones. Therefore, the lineshape changes observed in Fig. 5 as a function of 1, 2, and 2 do not reflect changes in cone orientation alone but also reflect significant changes in P(R). Fig. 6 shows the corresponding simulations for a larger value of the distance between the bases of the two cones (p = 12 Å, q = 6 Å, and µmax = 30°). As in Fig. 5, the results of Fig. 6 show that the angles 1, 2, and 2 influence both the EPR lineshape and the distance distribution, P(R).
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FIGURE 5 EPR spectra calculated for the tether-in-a-cone model as a function of various angles. For all calculations, p = 8 Å, q = 6 Å, and µmax = 30°. (A) = ß = = 0° and 1 = 2 = 2 = 0°. (B) = 0°, ß = 90°, = 0°, and 1 = 2 = 2 = 0°. (C) = ß = = 0°, 1 = 0°, 2 = 0°, and 2 = 90°. (D) = ß = = 0°, 1 = 90°, 2 = 0°, and 2 = 90°. (E) = ß = = 0°, 1 = 90°, 2 = 90°, and 2 = 90°. (F) = ß = = 0°, 1 = 90°, 2 = 180°, and 2 = 90°. The insets (top right) show a histogram representing P(R). The diagrams to the left of each panel represent the relative orientation of the cones as determined by 1, 2, and 2.
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FIGURE 6 EPR spectra calculated for the tether-in-a-cone model as a function of various angles. For all calculations, p = 12 Å, q = 6 Å, and µmax = 30°. (A) = ß = = 0° and 1 = 2 = 2 = 0°. (B) = 0°, ß = 90°, = 0°, and 1 = 2 = 2 = 0°. (C) = ß = = 0°, 1 = 0°, 2 = 0°, and 2 = 90°. (D) = ß = = 0°, 1 = 90°, 2 = 0°, and 2 = 90°. (E) = ß = = 0°, 1 = 90°, 2 = 90°, and 2 = 90°. (F) = ß = = 0°, 1 = 90°, 2 = 180°, and 2 = 90°. The insets (top right) show a histogram representing P(R). The diagrams to the left of each panel represent the relative orientation of the cones as determined by 1, 2, and 2.
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The results in Figs. 5 and 6 demonstrate that the average value and width of P(R) are determined, in part, by the angles 1, 2, and 2. Both R and R are also determined by p, q, and µmax. To isolate the effects of the cone orientation on the calculated EPR spectrum, the five simulations shown in Fig. 7 have been performed for different values of 1, 2, and 2 with p, q, and µmax adjusted so that P(R) is approximately equal for the different cone orientations. Using the distance distribution calculated for 1 = 0°, 2 = 0°, and 2 = 90°, p = 10.66 Å (adjusted so that R = 8 Å), q = 6 Å, and µmax = 30° as a template, values of p, q, and µmax were obtained by nonlinear least-squares analysis, which best fit this P(R) for alternate values of 1, 2, and 2. From the parameter sets obtained in this way, the EPR spectra and P(R) distributions in Fig. 7 were then calculated. As shown in Table 1, the parameters used in Fig. 7 all give a P(R) with R 8 Å and R 1.5 Å. The calculations in Figs. 8–10 have been performed in a similar way so that within each figure, all simulations give distance distributions that have approximately the same values of R and R. The results in Fig. 8 were calculated starting with 1 = 2 = 2 = 90°, p = 8.51 Å, q = 6 Å, and µmax = 10° giving a P(R) with R 12 Å and R 0.7 Å. Likewise for Fig. 9, all of the calculations gave a distance distribution with R 12 Å and R 1.9 Å; and for Fig. 10, R 12 Å and R 3.4 Å. It is apparent from these results that for small values of R, there are large differences in the calculated EPR spectrum as a function of 1, 2, and 2 (see Fig. 8). These differences are reduced at intermediate values of R (see Fig. 9) and are small or negligible at larger values of R (see Fig. 10). Each one of the simulations in Figs. 7–10 (solid gray lines) was then analyzed using a Fourier convolution method and the results are overlaid (dashed black lines) as described below.
Convolution analysis
When using convolution-based methods to analyze the spectrum of a pair of dipolar coupled nitroxides, the assumption is made that the orientation effects that are explicitly modeled in the tether-in-a-cone model can be neglected. The results presented in Figs. 7–10
represent an opportunity to test the robustness of the convolution approach under situations resulting in a wide range of EPR lineshapes for dipolar coupled nitroxides.
The spectra simulated for the tether-in-a-cone model shown in Figs. 7–10 have been fit using a convolution approach (Eq. 17). The spectrum of the spatially isolated nitroxide, M(H0), is shown in Fig. 2 A. The overlaid fits, C(H0), are shown as dashed black lines in Figs. 7–10. The Gaussian distance distributions, PG(R), are shown as black lines overlaid on the histogram of distances obtained from the tether-in-a-cone model. The results of analysis by the Fourier convolution method are summarized in Table 1, which gives the values of p, q, and µmax along with the angles 1, 2, and 2 used in the simulations. These six parameters determine the distance distribution, P(R), from the tether-in-a-cone model. The distance distribution is described by an average distance, R, and variance, R. The average distance, R0, and width, R, obtained by convolution analysis assuming a Gaussian distance distribution are also given in Table 1 as are the correlation coefficients (Eq. 19) for the fits obtained by the Fourier convolution method to the spectra simulated for the tether-in-a-cone model. The values of the correlation coefficients for the fits using the single Gaussian distance distributions are plotted in Fig. 11 versus q, µmax, and R. There is no apparent relation between the correlation coefficient and the tether length, q (Fig. 11 A). On the other hand, the correlation coefficient steadily increases with µmax up to 40° (Fig. 11 B).
The tether-in-a-cone model simulations in Figs. 7–10 have also been analyzed by the Fourier convolution method assuming a bimodal Gaussian distribution of distances. This approach does, in general, lead to modest improvements in the fit of the convolved spectrum to the tether-in-a-cone model simulation. Nevertheless, in all of the examples considered here, the correlation coefficient between the distance distribution, P(R), from the tether-in-a-cone model and the distance distribution, PG(R), from the Fourier convolution method decreases when a bimodal Gaussian distance distribution is used. Fig. 12 A shows an example of a tether-in-a-cone model simulation fit by the Fourier convolution method assuming a bimodal Gaussian distance distribution. The fit obtained by the Fourier convolution method to the tether-in-a-cone model simulation is improved even as the use of a bimodal PG(R) results in a less accurate representation of the true P(R) (cf. Fig. 7 B). As the correlation coefficient, 
corr, for the EPR spectra increases from 0.985 to 0.993 on going from a single Gaussian to a bimodal Gaussian PG(R), the correlation coefficient between P(R) and PG(R) decreases from 0.936 to 0.848.
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FIGURE 12 Simulated EPR spectra (gray lines) analyzed using a Fourier convolution method assuming a bimodal Gaussian distribution. The fits obtained by the Fourier convolution method are shown as dashed lines. The insets show the actual distance distribution for the simulated EPR spectrum (gray histogram) and the distance distribution obtained by the Fourier convolution method (dashed line). (A) Same simulation as in Fig. 7 B. (B) Simulation is the sum of those in Figs. 7 B and 9 B. (C) Simulation is the sum of those in Figs. 7 B and 10 B. In the insets to panels B and C the dotted lines were obtained by fitting the histograms to a bimodal Gaussian distribution. The dotted spectra were generated by the Fourier convolution method using these bimodal Gaussian distributions.
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Simulated EPR spectra that correspond to a truly bimodal distance distribution have also been analyzed by the Fourier convolution method assuming a bimodal Gaussian distribution of distances. The simulated spectrum (solid gray line) in Fig. 12 B is the equally weighted sum of those in Fig. 7 B (R = 8 Å) and Fig. 9 B (R = 12 Å). The simulated spectrum in Fig. 12 C is the equally weighted sum of those in Fig. 7 B (R = 8 Å) and Fig. 10 B (R = 12 Å). The best fits obtained by the Fourier convolution method assuming a bimodal Gaussian distribution of distances are shown as dashed lines. In both cases, the Fourier convolution method gives a good estimate (10.2 Å, Fig. 12 B; and 10.0 Å, Fig. 12 C) of the actual average distance between the labels (R = 10.0 Å). Nevertheless, as shown in the insets at the top right of each panel, the bimodal PG(R) recovered from Fourier convolution analysis (dashed lines) does not represent the true bimodal shape of the distance distribution (gray histogram). The dotted lines overlaid on the histograms in Fig. 12, B and C, were obtained by fitting the P(R) directly to a bimodal Gaussian distribution. The bimodal PG(R) distributions so obtained were then used to generate the dotted lines overlaid on the spectra in Fig. 12, B and C. The spectral fits obtained by first fitting the distance distributions (dotted lines; corr = 0.972 for Fig. 12 B and corr = 0.982 for Fig. 12 C) are statistically worse than those obtained by fitting the spectra themselves (dashed lines; corr = 0.989 for Fig. 12 B and corr = 0.994 for Fig. 12 C).
Analysis of data
The EPR spectra of four spin-labeled single cysteine T4L mutants (61, 65, 68, and 69) and three spin-labeled double cysteine T4L mutants (61/65, 65/68, and 65/69) were collected at –30°C in 70% glycerol. Under these conditions the effect of the global tumbling of the protein and all local dynamics on the linear CW-EPR spectra are negligible and the spectra of the four single cysteine mutants are nearly identical. The four spectra of the singly labeled mutants were analyzed to give g- and A-tensor values and Lorentzian linewidths (results not shown). The three sum of singles EPR spectra (61 + 65, 65 + 68, and 65 + 69), given by the equally weighted sum of the normalized EPR spectra of the spin-labeled single cysteine mutants, are shown on the left side of Fig. 13 (green lines) with the EPR spectra of the three spin-labeled double cysteine mutants (61/65, 65/68, and 65/69) overlaid on an expanded scale (x5; black lines). The EPR spectra of the doubly labeled mutants are also shown on the right side of Fig. 13 (black circles) with the results of analysis by the convolution method (red lines) and the results of fitting to the tether-in-a-cone model (blue lines) overlaid.
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ABSTRACT
INTRODUCTION
