**2. The temperature and viscosity dependence of EPR spectra of spinlabeled macromolecules**

As it was noted in Introduction, an informational content of single EPR spectrum is rather limited. A case when both slow tumbling and fast label motions are indistinguishable from single lineshape is frequently encountered. The method of temperature and viscosity dependences (TVD) is found extremely helpful here, although other approaches also exist (e.g., mutifrequency EPR, where the relative sensitivity to motions on different time scales at various radiofrequency bands is utilized to resolve them). The TVD method developed throughout many years in a close connection with progress in computer hardware and software technologies. At present time, it can be utilized for unambiguous simulation of EPR spectra of spin-labeled macromolecules (Dudich et al., 1977; Timofeev, 1986, 1993, 1995) with easily available equipment (X-band spectrometer and personal computer) and minimum effort. The main idea this method is based on is pretty straightforward, once the TwIM model is assumed. In this model, the overall motion of spin label is composed of two independent components, one related to rapid motion of nitroxide-containing molecular fragment, and another to entire macromolecule tumbling. The former usually have correlation times in order of 100 ps or less, and, therefore, is *fast* on timescale of EPR method at X-Band. If this motion would be perfectly isotropic, its main effect is limited to some Lorentzian line broadening. In spin labeling applications, however, this is not usually the case, as the object (typically protein) spin label is attached to, significantly hinders nitroxide mobility. This is regarded as *motion anisotropy* in TwIM model, and it is quantified by order parameters. Its effect is much stronger: it changes the position of resonances, and this shift has severe impact on EPR spectrum. Effective spin Hamiltonian partial averaging is an efficient method to account for these changes. In fact, the only change in simple nitroxide spin Hamiltonian is different values of magnetic tensor *g* and hyperfine tensor *A*. This partial tensor averaging technique was utilized for studying anisotropic phases since early history of the spin labeling. Macromolecule tumbling correlation times lie typically in nanosecond range. The anisotropy of this motion is generally determined by the shape of macromolecule in liquid solution, and for most globular proteins the single isotropic "effective" correlation time (*τ*) is sufficient to describe the lineshape change induced by it. At X-Band, presumed here if no otherwise stated, and nitroxide spin labels, the *τ* values in range of 1-100 ns leads to significant changes in lineshape, which may be calculated by means of stochastic Liouville equation (SLE).

Henceforth we assume that the main EPR-observable effect of fast nitroxide fragment motion is changes in its *anisotropy*, quantified by order parameter *S*. For the slow tumbling, the corresponding quantity is its correlation time *τ*. This anisotropy-correlation time splitting presumed in TwIM model allows to eliminate much of the ambiguity from EPR spectra interpretation. But even in this case it may be very difficult to distinguishingly quantify the effects of both contributions from single lineshape. This is where temperatureviscosity dependence comes into play. Varying sample temperature, one can modulate both kinds of motion: *τ* decreases with temperature, and order parameter *S* typically do so due to increasing of molecular motion amplitudes. The viscosity of solution (which is also dependent on temperature), can be independently varied by addition of sucrose, glycerol or some polymer media. It mainly affects the correlation time *τ*, while interference with smallscale fast molecular motions is typically observed only at very high viscosity, where it is, strictly speaking, not "fast" anymore. In the TVD experiment, the set of EPR spectra is recorded, at different temperatures and solution viscosities. The processing of resulting dependence starts with analysis of separation between broad outer peaks in spectrum, according to the procedure described below.

### **2.1. The theory of temperature-viscosity dependences**

The method of temperature and viscosity dependence was originally proposed (Timofeev & Samarianov, 1995) to determine the correlation time *τ* of slow isotropic rotational diffusion of a macromolecule, and McConnell's order parameter *S* in axially symmetric case. This method was further developed and modified (Tkachev, 2010) to be suitable for smooth joining into spectra interpretation and simulation framework based on TwIM model. The primary quantity measured experimentally in this method is the separation between the broad outer peaks (BOPs) in X-Band conventional EPR spectra (absorption spectra derivative). Independence of slow Brownian diffusion of entire macromolecule treated as rigid entity, and fast anisotropic reorientations of spin label, as the basic concept of TwIM, is the fundamental proposition in TVD method. Both motion types narrow the lineshape, shrinking the separation between BOPs (referenced as 2A*'* here). If one considers fast motion by anisotropic (partial) averaging of effective spin Hamiltonian, the amount 2A*'* decreases will depend on motion ordering, or, speaking other way, its degree of anisotropy. The slow tumbling, which in many cases of globular proteins can be thought isotropic, lead to the same effect but now it is dependent on the correlation time. This means one can express an effective shift of BOPs as the sum of the shifts originating from these two motion components:

Uniform EPR Spectra Analysis of Spin-Labeled Macromolecules by Temperature and Viscosity Dependences 289

$$\mathbf{2A}\_{ZZ} - \mathbf{2A}^{\cdot} = \left(\mathbf{2A}\_{ZZ} - \mathbf{2\overline{A}}\right) + \left(\mathbf{2\overline{A}} - \mathbf{2A}^{\cdot}\right) = \boldsymbol{\Lambda}\_1 + \boldsymbol{\Lambda}\_2 \tag{1}$$

where 2A*ZZ* – the separation between BOPs in rigid-limit EPR ("powder") spectrum. It was originally found from experiment, that plotting the 2A*'* value versus the temperature and viscosity ratio .. , yields linear dependencies at constant temperature. The point where the extrapolated line crosses the ordinate axis yields the 2*A* value. In approximation of axial symmetry of the fast motion, it is possible to derive dependence of the first term *Δ*<sup>1</sup> , on McConell's order parameter *S*:

$$S = \frac{2\overline{A}\_{\parallel} - 2\overline{A}}{2\mathbf{A}\_{zz} - 2\mathbf{A}} - \frac{2\overline{A} - 2\mathbf{a}\_0}{2\mathbf{A}\_{zz} - 2\mathbf{a}\_0} \tag{2}$$

where <sup>0</sup> 3 *XX YY ZZ a = A +A +A /* is the isotropic hyperfine splitting constant of nitroxide radical. Therefore, the value of *Δ*1 is represented as:

$$\mathbf{A}\_{1} = \mathbf{2}\mathbf{A}\_{ZZ} - \mathbf{2}\overline{\mathbf{A}} = \left(\mathbf{2}\mathbf{A}\_{ZZ} - \mathbf{2}\mathbf{a}\_{0}\right)\left(\mathbf{1} - \mathbf{S}\right) \tag{3}$$

The second term *Δ*2 gives the shift of BOPs in relative to the 2*A* value. The value of <sup>2</sup> 2 2A*' <sup>Δ</sup> = A* can be evaluated in the following way. Spectral lineshape narrowing due to an exchange (Slichter, 1981), caused by slow rotation of a macromolecule with correlation time *τ*, is defined by the following expression (γ is a magnetogyric ratio):

$$
\Delta H^{'} \approx \Delta H \left[ 1 - \left( \frac{2}{\gamma \pi \Delta H} \right)^{2} \right] \tag{4}
$$

In the model of an exchange between two states we assume them to be parallel and perpendicular orientations relative to magnetic field (Z axis). Corresponding values of *ΔH* and *' ΔH* will be

$$
\Delta H = \mathfrak{2}\left(\overline{A}\_{\parallel} - \overline{A}\right) - \mathfrak{3}\left(\overline{A}\_{\parallel} - a\_0\right) = \mathfrak{3}\mathfrak{S}\left(A\_{ZZ} - a\_0\right) \tag{5}
$$

$$
\Delta H^{\dot{'}} = \mathbf{2}\left(\boldsymbol{A}\_{\parallel}^{\dot{'}} - \boldsymbol{A}^{\dot{'}}\right) = \mathbf{3}\left(\boldsymbol{A}\_{\parallel}^{\dot{'}} - \boldsymbol{a}\_{0}\right) \tag{6}
$$

Using expressions (2, 5, 6), we write *' ΔH* as follows:

$$
\Delta H^{'} = 2\left(\vec{A\_{\parallel}} - \vec{A^{\cdot}}\right) = \Delta H - \frac{3}{2}\Delta\_2 \tag{7}
$$

Now, combining (7) with the expressions (4, 5) we finally obtain

$$
\tau \mathbf{S} = \frac{1}{3\gamma \Delta\_0} \left(\frac{\Delta\_2}{\Delta\_0 \mathbf{S}}\right)^{-\frac{1}{2}} \mathbf{\hat{z}} \tag{8}
$$

where 0 0 2 *Δ ZZ =A a* .

288 Nitroxides – Theory, Experiment and Applications

means of stochastic Liouville equation (SLE).

according to the procedure described below.

**2.1. The theory of temperature-viscosity dependences** 

history of the spin labeling. Macromolecule tumbling correlation times lie typically in nanosecond range. The anisotropy of this motion is generally determined by the shape of macromolecule in liquid solution, and for most globular proteins the single isotropic "effective" correlation time (*τ*) is sufficient to describe the lineshape change induced by it. At X-Band, presumed here if no otherwise stated, and nitroxide spin labels, the *τ* values in range of 1-100 ns leads to significant changes in lineshape, which may be calculated by

Henceforth we assume that the main EPR-observable effect of fast nitroxide fragment motion is changes in its *anisotropy*, quantified by order parameter *S*. For the slow tumbling, the corresponding quantity is its correlation time *τ*. This anisotropy-correlation time splitting presumed in TwIM model allows to eliminate much of the ambiguity from EPR spectra interpretation. But even in this case it may be very difficult to distinguishingly quantify the effects of both contributions from single lineshape. This is where temperatureviscosity dependence comes into play. Varying sample temperature, one can modulate both kinds of motion: *τ* decreases with temperature, and order parameter *S* typically do so due to increasing of molecular motion amplitudes. The viscosity of solution (which is also dependent on temperature), can be independently varied by addition of sucrose, glycerol or some polymer media. It mainly affects the correlation time *τ*, while interference with smallscale fast molecular motions is typically observed only at very high viscosity, where it is, strictly speaking, not "fast" anymore. In the TVD experiment, the set of EPR spectra is recorded, at different temperatures and solution viscosities. The processing of resulting dependence starts with analysis of separation between broad outer peaks in spectrum,

The method of temperature and viscosity dependence was originally proposed (Timofeev & Samarianov, 1995) to determine the correlation time *τ* of slow isotropic rotational diffusion of a macromolecule, and McConnell's order parameter *S* in axially symmetric case. This method was further developed and modified (Tkachev, 2010) to be suitable for smooth joining into spectra interpretation and simulation framework based on TwIM model. The primary quantity measured experimentally in this method is the separation between the broad outer peaks (BOPs) in X-Band conventional EPR spectra (absorption spectra derivative). Independence of slow Brownian diffusion of entire macromolecule treated as rigid entity, and fast anisotropic reorientations of spin label, as the basic concept of TwIM, is the fundamental proposition in TVD method. Both motion types narrow the lineshape, shrinking the separation between BOPs (referenced as 2A*'* here). If one considers fast motion by anisotropic (partial) averaging of effective spin Hamiltonian, the amount 2A*'* decreases will depend on motion ordering, or, speaking other way, its degree of anisotropy. The slow tumbling, which in many cases of globular proteins can be thought isotropic, lead to the same effect but now it is dependent on the correlation time. This means one can express an effective

shift of BOPs as the sum of the shifts originating from these two motion components:

Hence, the dependence between the BOP shift, correlation time *τ* of a macromolecule tumbling, and order parameter *S* of fast spin label motion, should be found in the following form:

$$
\sigma \mathbf{S} = a \left(\frac{\Delta\_2}{\Delta\_0 \mathbf{S}}\right)^{-b} \tag{9}
$$

Empirical parameters *a* and *b* can be found from a set of simulated EPR spectra at various values of *S* and *τ* . The formula (9) is deduced in the dimensionless form, therefore parameter *a* has the dimension of nanoseconds. Parameters *a* and *b* are dependent on changes of initial magnetic tensors and individual line width. But their mutual dependence (they always derived simultaneously, and are not independent) makes splitting (1) weakly affected by these changes. For the most of experiments with spin-labeled samples the value of *a* is close to 1.1 ns, and parameter b ≈ 1.3.

To establish a link with experimentally measurable values – temperature *Т* and viscosity of a solution *η* , the straightforward way is to call for Stokes-Einstein relation (describing rotational motion of a sphere in a viscous media):

$$
\tau = \frac{V\eta}{kT} \tag{10}
$$

In this case expression (9) takes a form:

$$
\Delta\_2 = \Delta\_0 \mathcal{S} \left( \frac{akT}{SV\eta} \right)^{\frac{1}{b}} \Leftrightarrow \mathcal{2}\mathcal{A}^\prime = \mathcal{2}\overline{\mathcal{A}} - \Delta\_0 \mathcal{S} \left( \frac{akT}{SV\eta} \right)^{\frac{1}{b}} \tag{11}
$$

The overall result here is that the separation between BOPs is a linear function of *T / η b* . Experimental measurement of this dependence by varying the temperature and the viscosity, it is possible to find 2*A* , and corresponding order parameter *S*, and, according to the formula (9), rotational correlation time of a macromolecule tumbling in solution. It's important to remember, however, that derived expressions are valid only when motions differ significantly on time scale. This may not be the case at very high viscosities where shear forces may interfere with the fast motion, which is believed to be affected solely by temperature changes. At these conditions motion of nitroxide cannot be regarded as 'fast' anymore.

#### **2.2. The uniform method for resolving an EPR spectroscopy reverse problem**

Combining the EPR spectra simulation using TwIM model for motion of spin label, and experimental approach of TVD, the following scheme can be used for uniform spectra interpretation. It allows unequivocally treat the line shape modulated by different dynamic effects. It calls for stochastic Liouville equation approach for calculation of slow motional EPR spectra. The flowchart is shown on Fig. 1, and individual steps are described in the following sub-sections. It contains two feed-back optimization cycles: one on the right relies on TVD data (experimental), but it corresponds to axially symmetrical case. The one on the left is more general, and deals with rhombic components of magnetic tensors (more than one order parameter is possible to describe fast motion). Right cycle **1** is responsible for locking the BOP separation in simulated spectra to experimental values, and for distinguishing between fast and slow motional contribution to BOP shift (ca. expression (1)). The generalized cycle **2** allows obtaining finer details of the motion (additional order parameters) from fitting the XY manifold (central region of EPR spectrum). The procedure is considered complete when both cycles are consistent.

**Figure 1.** The scheme of uniform EPR spectra interpretation based on TwIM model and TVD experiment. SLE states for stochastic Liouville equation-based procedure for calculation of slow motional spectra.

### *2.2.1. Determination of magnetic tensors components*

290 Nitroxides – Theory, Experiment and Applications

of *a* is close to 1.1 ns, and parameter b ≈ 1.3.

In this case expression (9) takes a form:

rotational motion of a sphere in a viscous media):

Hence, the dependence between the BOP shift, correlation time *τ* of a macromolecule tumbling, and order parameter *S* of fast spin label motion, should be found in the following form:

> 2 0

*Δ S* 

Empirical parameters *a* and *b* can be found from a set of simulated EPR spectra at various values of *S* and *τ* . The formula (9) is deduced in the dimensionless form, therefore parameter *a* has the dimension of nanoseconds. Parameters *a* and *b* are dependent on changes of initial magnetic tensors and individual line width. But their mutual dependence (they always derived simultaneously, and are not independent) makes splitting (1) weakly affected by these changes. For the most of experiments with spin-labeled samples the value

To establish a link with experimentally measurable values – temperature *Т* and viscosity of a solution *η* , the straightforward way is to call for Stokes-Einstein relation (describing

1 1

2A 2 *'*

*akT b b akT <sup>Δ</sup> <sup>=</sup> <sup>Δ</sup> <sup>S</sup> = A <sup>Δ</sup> <sup>S</sup> SVη SVη* 

The overall result here is that the separation between BOPs is a linear function of

Experimental measurement of this dependence by varying the temperature and the viscosity, it is possible to find 2*A* , and corresponding order parameter *S*, and, according to the formula (9), rotational correlation time of a macromolecule tumbling in solution. It's important to remember, however, that derived expressions are valid only when motions differ significantly on time scale. This may not be the case at very high viscosities where shear forces may interfere with the fast motion, which is believed to be affected solely by temperature

2 0

changes. At these conditions motion of nitroxide cannot be regarded as 'fast' anymore.

**2.2. The uniform method for resolving an EPR spectroscopy reverse problem** 

Combining the EPR spectra simulation using TwIM model for motion of spin label, and experimental approach of TVD, the following scheme can be used for uniform spectra interpretation. It allows unequivocally treat the line shape modulated by different dynamic effects. It calls for stochastic Liouville equation approach for calculation of slow motional EPR spectra. The flowchart is shown on Fig. 1, and individual steps are described in the following sub-sections. It contains two feed-back optimization cycles: one on the right relies on TVD data (experimental), but it corresponds to axially symmetrical case. The one on the

*0*

*<sup>V</sup><sup>η</sup> <sup>τ</sup> <sup>=</sup> kT* (10)

*Δ*

*τS=a*

*b*

(9)

(11)

1 *T / η b* .

The entire procedure was designed to eliminate as many degrees of freedom as possible while retaining physical sense. The lineshape changes induced by molecular motions are controlled via *τ* and *S*. To set up spectra simulation, magnetic parameters have to be determined in some way. Blind fitting of them would scramble all the averaging effects, and therefore is not suitable here. The most correct way is using diluted crystal, but is production in many cases is not feasible. The compromise is to use values of 0 2a from free label solution EPR spectrum at room temperature and 2A*ZZ* from spectrum of frozen labeled sample at 77K. The rhombic component still remains unknown, and has to be fitted. For hyperfine A tensor, it is small, and the values of *AX* and *AY* are almost equal. This mostly eliminates the A-tensor problem, which is mostly significant as it determines the distance between BOPs used to quantify molecular motion from in TVD experiment. The components of *g* tensor are still subject to fitting, at least its rhombic component which is not negligible as it is in case of hyperfine tensor.

### *2.2.2. Empirical parameters in temperature and viscosity dependences*

Once the magnetic tensor components were determined, it is possible to calculate EPR spectra. The spectra simulation procedure based on tensor averaging for accounting of fast

motion, and SLE for slow rotational diffusion is used for calculation of empirical parameters **a** and **b** in the equation (9). For this purpose, a set of EPR spectra is calculated axially symmetrical case (that is, no rhombic component in hyperfine tensor). The shift of separation between BOPs in theoretical EPR spectra measured with different values of correlation time *τ* and order parameter *S*, if plotted in logarithmic scale versus *τS* according to the formula (9), allows to calculate the parameter *b* from the slope of linear approximation, and parameter *a* from its intersection with ordinate axis (Tkachev, 2010).

**Figure 2.** Theoretical dependence of BOP shift vs. global tumbling correlation time *τ* and fast label motion order parameter *S*. It is used to find *a* and *b* parameters in eqation (9). This particular dependence was calculated for *gxx* = 2.0088, *gyy* = 2.0058, *gzz* = 2.0023, *Axx* = *Ayy* = 6,0 G *Azz* = 35.3 G, and individual line width of 1.1 G. Every black point corresponds to simulated spectrum, and the gray line is least-square approximation.

Values of empirical TVD parameters determined from simulation of set of spectra according to the method described above are dependent on magnetic tensors and line widths used for calculation. High performance in spectra simulation which may be routinely achieved on modern workstations and personal computers makes possible to obtain these parameters interactively, upon changing line width or magnetic tensor components. This is important part of the EPR spectra interpretation method described here, as it may require multiple processing of experimental TVD data with different *a* and *b* values to achieve good coincidence of simulated spectra with experimental ones.

#### *2.2.3. Experimental determination of τ and S values*

When empirical parameters are known, it is possible to use equation (11) for experimental data processing. According to equation (11), the dependence of 2A*'* on 1 *T / η b* should be linear. Therefore, an intersection of linear approximation of experimental results will give the value of 2*A* which encodes the order parameter according to expression (2). Then, from equation (9) it is possible to find the correlation time of slow global tumbling of the macromolecule.

292 Nitroxides – Theory, Experiment and Applications

is least-square approximation.

coincidence of simulated spectra with experimental ones.

*2.2.3. Experimental determination of τ and S values* 

motion, and SLE for slow rotational diffusion is used for calculation of empirical parameters **a** and **b** in the equation (9). For this purpose, a set of EPR spectra is calculated axially symmetrical case (that is, no rhombic component in hyperfine tensor). The shift of separation between BOPs in theoretical EPR spectra measured with different values of correlation time *τ* and order parameter *S*, if plotted in logarithmic scale versus *τS* according to the formula (9), allows to calculate the parameter *b* from the slope of linear approximation, and parameter *a* from its intersection with ordinate axis (Tkachev, 2010).

**Figure 2.** Theoretical dependence of BOP shift vs. global tumbling correlation time *τ* and fast label motion order parameter *S*. It is used to find *a* and *b* parameters in eqation (9). This particular dependence was calculated for *gxx* = 2.0088, *gyy* = 2.0058, *gzz* = 2.0023, *Axx* = *Ayy* = 6,0 G *Azz* = 35.3 G, and individual line width of 1.1 G. Every black point corresponds to simulated spectrum, and the gray line

Values of empirical TVD parameters determined from simulation of set of spectra according to the method described above are dependent on magnetic tensors and line widths used for calculation. High performance in spectra simulation which may be routinely achieved on modern workstations and personal computers makes possible to obtain these parameters interactively, upon changing line width or magnetic tensor components. This is important part of the EPR spectra interpretation method described here, as it may require multiple processing of experimental TVD data with different *a* and *b* values to achieve good

When empirical parameters are known, it is possible to use equation (11) for experimental

1 *T / η b* should be

data processing. According to equation (11), the dependence of 2A*'* on

**Figure 3.** Typical plot of temperature and viscosity dependences of BOP separation. This data corresponds to bovine serum albumin (BSA) labeled with 4-(2-chloromercurophenyl)-2,2,5,5 tetramethyl-3-imidazoline- <sup>3</sup> *Δ* -1-oxyl (on inset). It was used as model system for procedure test purposes. For every temperature the linear data approximation crosses ordinate axis on 2*A* , and order parameter value may be derived from it.

#### *2.2.4. Simulation of EPR spectra of spin-labeled macromolecules*

The simulation of EPR spectra of spin-labeled macromolecules used here is also based on the two independent motions (TwIM) model (Dudich et al., 1977; Timofeev 1986; Timofeev & Samaryanov 1993, 1995; Tkachev 2009, 2010). Therefore, fast motion of the spin label relative to the macromolecule is taken into account by effective spin Hamiltonian partial averaging, and the slow motion of the macromolecule is determined by the rotational correlation time of isotropic diffusion.

By means of the generalized model of fast nitroxide oscillations the partially averaged magnetic tensor components can be calculated (Tkachev, 2009, 2010) in such a way that order parameter obtained from TVD experiment can be employed. This is possible because axially symmetrical model presumed in TVD is a special case of the generalized model mentioned. Substitution of averaged tensors into the SLE-based procedure for slow motional nitroxide spectra simulations (Schneider & Freed, 1989) gives the final EPR lineshape with contributions of both fast and slow motions.

On the above basis the simulation program (S\_imult6) was developed. The model of fast limited oscillations (Timofeev and Samaryanov, 1995) gives formulas in analytical form by which components of partially averaged magnetic tensors can be calculated. According to model nitroxide fluctuates around some fixed axis *n* defined in polar coordinates by *<sup>θ</sup>* , angles. Fluctuations are limited by an angular amplitude of oscillation *α* , and all orientations are equally probable in range from – *α* to *α* . However, it was shown (Tkachev 2009, 2010), that partially averaged magnetic tensors aren't diagonal in general case. This implies that tensors have to be diagonalized after averaging, resulting in tilt between principal axes of both partially averaged magnetic tensors to appear. Two parameters defining an axis of oscillation *n* , <sup>2</sup> *S =* 3cos 1 2 *<sup>θ</sup> /* , <sup>2</sup> *<sup>κ</sup> <sup>=</sup>* 1 2cos can be shown to play role of order parameters. The meaning of *S* defined this way is exactly the same as in axially-symmetrical case which arises if one let <sup>0</sup> *α =* 180 . In this case it is equal to experimental McConnell's order parameter. This allows using axially symmetrical case with experimentally determined order parameter as the starting point in scheme shown on Fig. 1, when calculating theoretical EPR spectra.

The general scheme for the resolution of the EPR spectroscopy inverse problem for spinlabeled macromolecules presented as follows. Experimentally determined magnetic tensor components, the correlation time *τ* together with the order parameter *S*, found from temperature-viscosity dependences, are fed into the simulation procedure based on SLE (cycle 1 on Fig. 1). In axially symmetrical case one obtains a theoretical spectrum similar to experimental one only on its wings (position of BOPs). In order to fit calculated spectra to experiment in the central region of magnetic fields, it is necessary to deviate from initial *θ* , and *α* values, breaking axial symmetry (switch to cycle 2 on Fig.1). Additionally, the linewidth and initial tensor rhombic parts may be adjusted as well, to achieve the best fit of a theoretical spectrum to the experimental one. If these are modified, it may be necessary to recalculate empirical TVD parameters (*a* and *b*), and repeat the procedure from cycle 1. Once it is possible to obtain consistent results from both cycles, process is complete. It utilizes the maximum of information from experimental TVD dependence, and yields the set of simulated spectra with the only difference in order parameters and macromolecular tumbling correlation time. The starting conditions are provided by axially symmetrical data obtained directly from experiment. This eliminates most of non-uniqueness which the usual multidimensional fitting procedures is prone to.

An ensemble may be inhomogeneous in sense of averaging parameters. One of the reasons for this is existence of macromolecule's structure fluctuations, or set of different structural conformations of the labelling site. This leads to concept of sub-ensembles of nitroxides, called clusters (Timofeev & Samarianov, 1995). They are clustered according to averaging order parameters (all kinds of equivalent dynamical behaviors, producing identical averaged tensors), and these may be assumed to be normally distributed. The spectrum depends on 0 *α(α ,σ)* , where 0 *α* is an average, and *σ* is a standard deviation of the Gaussian distribution (Tkachev, 2010, Timofeev & Samarianov, 1995). The resulting spectrum is calculated as Gaussian-weighted sum of spectra of individual clusters. This adds single fitting parameter, but in many real-world cases helps to dramatically improve fits quality.

294 Nitroxides – Theory, Experiment and Applications

defining an axis of oscillation *n*

when calculating theoretical EPR spectra.

multidimensional fitting procedures is prone to.

lineshape with contributions of both fast and slow motions.

model nitroxide fluctuates around some fixed axis *n*

order parameter obtained from TVD experiment can be employed. This is possible because axially symmetrical model presumed in TVD is a special case of the generalized model mentioned. Substitution of averaged tensors into the SLE-based procedure for slow motional nitroxide spectra simulations (Schneider & Freed, 1989) gives the final EPR

On the above basis the simulation program (S\_imult6) was developed. The model of fast limited oscillations (Timofeev and Samaryanov, 1995) gives formulas in analytical form by which components of partially averaged magnetic tensors can be calculated. According to

angles. Fluctuations are limited by an angular amplitude of oscillation *α* , and all orientations are equally probable in range from – *α* to *α* . However, it was shown (Tkachev 2009, 2010), that partially averaged magnetic tensors aren't diagonal in general case. This implies that tensors have to be diagonalized after averaging, resulting in tilt between principal axes of both partially averaged magnetic tensors to appear. Two parameters

role of order parameters. The meaning of *S* defined this way is exactly the same as in axially-symmetrical case which arises if one let <sup>0</sup> *α =* 180 . In this case it is equal to experimental McConnell's order parameter. This allows using axially symmetrical case with experimentally determined order parameter as the starting point in scheme shown on Fig. 1,

The general scheme for the resolution of the EPR spectroscopy inverse problem for spinlabeled macromolecules presented as follows. Experimentally determined magnetic tensor components, the correlation time *τ* together with the order parameter *S*, found from temperature-viscosity dependences, are fed into the simulation procedure based on SLE (cycle 1 on Fig. 1). In axially symmetrical case one obtains a theoretical spectrum similar to experimental one only on its wings (position of BOPs). In order to fit calculated spectra to experiment in the central region of magnetic fields, it is necessary to deviate from initial *θ* , and *α* values, breaking axial symmetry (switch to cycle 2 on Fig.1). Additionally, the linewidth and initial tensor rhombic parts may be adjusted as well, to achieve the best fit of a theoretical spectrum to the experimental one. If these are modified, it may be necessary to recalculate empirical TVD parameters (*a* and *b*), and repeat the procedure from cycle 1. Once it is possible to obtain consistent results from both cycles, process is complete. It utilizes the maximum of information from experimental TVD dependence, and yields the set of simulated spectra with the only difference in order parameters and macromolecular tumbling correlation time. The starting conditions are provided by axially symmetrical data obtained directly from experiment. This eliminates most of non-uniqueness which the usual

An ensemble may be inhomogeneous in sense of averaging parameters. One of the reasons for this is existence of macromolecule's structure fluctuations, or set of different structural conformations of the labelling site. This leads to concept of sub-ensembles of nitroxides,

defined in polar coordinates by *<sup>θ</sup>* ,

, <sup>2</sup> *S =* 3cos 1 2 *<sup>θ</sup> /* , <sup>2</sup> *<sup>κ</sup> <sup>=</sup>* 1 2cos can be shown to play
