Biomedical EPR PartB Methodology Instrumentation and Dynamics  Sandra R. Eaton
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DEREK MARSH ET AL. 
relaxation enhancements and slow exchange processes. Both of these aspects have assumed considerable importance in stuctural and dynamic studies of biological membranes. The former is an essential part of sitedirected spinlabelling methodology.
Approximate solutions of the Bloch equations that explicitly include the Zeeman modulation field (see section 4 and Páli et al., 1996) yield the following approximate expressions for the different outofphase dispersion and absorption displays, where n is the harmonic with respect to the Zeeman modulation frequency and the prime indicates outofphase detection. To illustrate simply the dependence on and times, the effects of molecular motion are neglected.
The amplitude of the firstharmonic dispersion measured at the centre of the resonance is (Marsh et al., 1997):
where is the amplitude of the modulation field, that of the microwave field and is the electron gyromagnetic ratio. The sensitivity of the amplitude to therefore does not extend beyond that of the conventional saturation factor The amplitude of the firstharmonic absorption measured at a distance from the resonance position that is equal to the zeroth harmonic linewidth is (Marsh et al., 1997):
where the amplitude at the centre of the line is zero. This displays a considerably greater sensitivity to than does or the conventional inphase adsorption spectrum, The amplitude of the second harmonic outofphase absorption spectrum is given by (Marsh et al., 1997):
This spectral display corresponds to the standard secondharmonic saturation transfer spectroscopy. Again, as for the firstharmonic, the secondharmonic outofphase absorption signal possesses an additional sensitivity to
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beyond that expressed simply by the saturation factor. The first and secondharmonic nonlinear absorption ESR spectra therefore both possess a sensitivity to processes superior to that of conventional progressive saturation experiments, which are performed on the inphase As seen from Fig. 8, the firstharmonic outofphase has advantages over the secondharmonic outofphase for determining enhancements. The depends far less on the time than does the As will be seen in Section 7.2, it is also insensitive to molecular motion. This makes it an almost pure and, as seen below, this is the nonlinear display of choice for studies of spinlattice relaxation enhancement.
Figure 8. Dependence of the amplitude of: A. the firstharmonic (Eq. 35) and B. the secondharmonic (Eq. 36) outofphase EPR absorption signals on time, for the values of indicated. Amplitudes are normalised to the inphase absorption signal. (and for ) (see Marsh et al., 1997).
Below follows a description of studies of the of the different nonlinear spinlabel EPR displays. Particular attention is paid to the use of integrated intensities in the analysis, and also to the effects of molecular motion in the progressive saturation and outofphase first harmonic absorption experiments. For the latter two methods, detailed spin
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lattice relaxation time calibrations of the spectral intensities are presented in the Appendix to this chapter.
7.1Progressive Saturation Experiments
The saturation behaviour of the integrated intensity of the zeroth harmonic absorption, spectrum is independent of the degree of inhomogeneous broadening, because integration over the entire absorption lineshape eliminates the effect of saturation broadening (Páli et al., 1993). Spectral simulation using the methods described in Section 4 confirms that this is also the case in the presence of molecular motion and at 100 kHz Zeeman modulation frequencies (Livshits et al., 1998b). Practically, and in the simulations, the integrated intensity is obtained by double integration of the conventional firstharmonic spectrum. The of the integrated absorption intensity, S, is given by:
where 
is a scaling factor and P is the saturation factor that depends upon 

With slow passage conditions and in the absence of molecular motion 


The 
of this factor in the presence of molecular 
motion and Zeeman modulation can thus be expressed to a first
approximation in terms of an effective 
time, 
More precise 
calibrations are given in the Appendix. 


Figure 9 gives the progressive saturation curves for the integrated absorption intensity of first harmonic spectra as a function of the molecular
rotation frequency 
Spectra were simulated as described in Section 

4 with highfrequency Zeeman modulation, 
All saturation 

curves have the dependence on microwave field intensity, 
that is given 
by Eq. 37, irrespective of rotational frequency, or the presence of inhomogeneous broadening. The saturation behaviour depends strongly on molecular motion in the extreme motional broadening region, for rotational correlation times The effective values of that are deduced from the saturation curves are given by the inset in Fig. 9. A
minimum value of 
is achieved for 
that corresponds to the 

frequency equivalent 
of the 
hyperfine 
anisotropy. In this extreme 

motional broadening regime the values of 
are independent of, and much 

shorter than, the intrinsic 
and depend only relatively weakly on 
the 

rotational correlation time, 
This regime 
is readily identified from the 

conventional 
lineshapes from which approximate values of 
are 
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335 
deduced. Spinlattice relaxation times can therefore be determined from progressive saturation experiments in this motional regime without a detailed knowledge of (or ).
Figure 9. Simulated saturation curves for the double integrated intensity ofthe signal at a Zeeman modulation frequency of for different values of the inverse frequency ofmolecular rotation (or rotational correlation time), The intrinsic spinlattice relaxation time is fixed at The inset gives the dependence on rotational correlation time of the effective that appears in the saturation factor: theintrinsic is 29 ns (see Livshits et al., 1998b).
In both the motional narrowing regime and the very slow motion regime the saturation behaviour depends directly on the intrinsic as expected conventionally for progressive saturation experiments. In the extreme motional narrowing regime and quasirigid limit regime the values of are no longer dependent on rotational correlation time and are equal to the intrinsic values, More precise calibrations, which are essential in the intermediate regime, are given in the Appendix. The method of extracting the spinlattice relaxation times from the experimental saturation factors is outlined there.
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7.2First harmonic, outofphase absorption
As already mentioned, the firstharmonic, outofphase absorption spectrum has been identified as a nonlinear display that is sensitive to but relatively insensitive to both molecular motion (Hyde and Thomas,
1973) and 
(Livshits et al., 1998a). Fig. 10 gives the dependence 

of the outofphase to inphase integrated intensity ratio, 
on spinlattice 
relaxation time that is obtained from spectral simulations for different
rotational correlation times 
Clearly the dependence on molecular 

motion is relatively small, throughout the entire correlation time range 

especially for a Zeeman modulation frequency of 25 kHz. The 

dependence on intrinsic 
is also similarly slight (Livshits et al., 1998a; 

Livshits and Marsh, 2000). 


The best sensitivity to 
is obtained at relatively high microwave 

magnetic field intensities, 
(Livshits et al., 1998a). The data in 

Fig. 10 are calculated for 
The dependence of the outofphase to 

inphase ratio on 
can be 
fitted by an empirical expression of the form 
(Livshits et al., 1998a):
where and m are fitting parameters. Calibration values are given in the Appendix.
The most striking feature of Fig. 10 is the dependence on Zeeman modulation frequency. For the outofphase/inphase ratio is most sensitive to short spinlattice relaxation times, The lower modulation frequency of 25 kHz extends this range of sensitivity to the
range, for which the outofphase intensities become comparable to those at 100 kHz modulation frequency. Thus the sensitivity of the firstharmonic outofphase display can be tuned to different ranges of spinlattice relaxation time by changing the modulation frequency. Practical examples are given by Livshits et al. (1998a).
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Figure 10. Dependence of the ratio, of outofphase to inphase integrated intensities of the firstharmonic absorption spectra on spinlattice relaxation time, Data are obtained from spectral simulations according to Section 4 for various rotational correlation times, and for Zeeman modulation frequencies: and 25 kHz (see Livshits and Marsh, 2000)
7.3Second Harmonic outofphase Absorption
Although the secondharmonic outofphase absorption spectrum is less optimal for estimation of times than is the first harmonic equivalent, it is included here because it is the classical STEPR display and to date is also that most used for determining spinlattice relaxation enhancements (see e.g., Marsh et al., 1998). There are obvious advantages to determining both rotational mobility and spinlattice relaxation enhancements in one and the same experiment.
Figure 11 illustrates the application to relaxation enhancements induced by spinspin interactions (i.e., cross relaxation) for a spinlabelled lipid in gelphase bilayer membranes. Theoretical dependences of the second harmonic integrated intensity on spinlattice relaxation time are given in the inset to this figure. The intensity ratio, is strongly dependent on spinlattice relaxation time in the range but also depends strongly on the intrinsic From the correlation with the effective relaxation time product obtained from progressive saturation experiments (see Fig. 11), a consistent value of at 25°C (8.5 ns at 15°C and 6.5 ns at 35°C) is obtained to describe the experimental correlation between
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8.SLOW EXCHANGE AND PARAMAGNETIC ENHANCEMENTS
This section describes the different processes, other than slow rotational diffusion, to which nonlinear spinlabel EPR spectroscopy is sensitive. It therefore covers the nonclassical aspects of saturation transfer EPR spectroscopy (Marsh et al., 1998). There are three basic areas of application: slow physical or chemical exchange processes; weak spinspin interactions between spin labels; and weak relaxation enhancements induced by paramagnetic species. In the latter two cases, both Heisenberg spin exchange and magnetic dipoledipole interactions contribute to the relaxation mechanism.
8.1Relaxation enhancement by exchange processes
Exchange at rates comparable to that of the spinlattice relaxation alleviates saturation in nitroxide ESR spectra. This results either from transfer of spin polarisation, in the case of physical or chemical exchange, or by mutual antiparallel spin flips (i.e., cross relaxation), in the case of Heisenberg exchange (see Fig. 1). For exchange between two sites, both cases are equivalent as is seen immediately below. The analysis uses rate equations for spin population differences following Yin and Hyde (1987).
Consider two sites with total spin populations and between which physical or chemical exchange takes place:
where is the rate of transfer from site i, and that from site j. These rate constants are related by detailed balance:
The rate equation for the spin population difference, at site i is then:
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where 
are the spin populations of the 
states (with 
at site i, and similarly for
Heisenberg spin exchange between the two sites, on the other hand, can be depicted as:
where is the bimolecular rate constant for spin exchange. The corresponding rate equation for the spin population difference at site i is (Yin and Hyde, 1987):
where the righthand side can be rewritten in terms of population differences (Marsh, 1992a):
The exchange frequency defined in the usual bimolecular formalism is:
Thus physical/chemical exchange and Heisenberg spin exchange are kinetically equivalent, with the following identities obtained from Eqs. 40 to 42: and where the fractional population of state i is and
The twosite spin exchange situation is specifically applicable to double labelling experiments (Snel and Marsh, 1994; Páli et al., 1999). Commonly, however, Heisenberg exchange takes place between more than two distinguishable spin label species in a concentrationdependent, singlelabelling experiment. Depending on spinlabel concentration, the relaxation enhancement can then be considerably more efficient than in the twosite case. This more usual singlelabelling case is considered after first treating twosite exchange.
8.1.1Relaxation by slow twosite exchange
Consider exchange between two sites b and f (see Fig. 1) where spin transitions are induced at rate 2W by at site b. The steadystate condition for the spin population difference at site b, including exchange and spinlattice relaxation, is:
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341 
where is the spin population difference at Boltzmann equilibrium, and is the spinlattice relaxation time in the absence of exchange. A corresponding steadystate condition applies to the spins at site f that are not
irradiated (i.e., with W= 0):
Solution of rate equations 44 and 45, together with Eq. 40 (that holds also for population differences), yields the standard expression for saturation of the spin system at site b (see Slichter, 1978):
where the effective spinlattice relaxation time in the presence of exchange is given by (Horváth et al., 1993):
from which the offrate, for exchange may be determined. The corresponding expression for Heisenberg spinexchange between two distinguishable spinlabelled species is (Snel and Marsh, 1994):
which can be deduced from the identities between the exchange rates in the two cases, viz., (see above).
8.1.2Concentration dependence of Heisenberg exchange
In the general case, Heisenberg exchange takes place between many spin labels of the same type that are distinguished either by their different states, or by their different orientations with respect to the static magnetic field. Equation 43 must be summed over all states j with which the state i is exchanging. The steady state rate equation for the population difference of the ith transition is then (Marsh, 1992a):