Biomedical EPR Part-B Methodology Instrumentation and Dynamics - Sandra R. Eaton

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 site-directed 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 out-of-phase dispersion and absorption displays, where n is the harmonic with respect to the Zeeman modulation frequency and the prime indicates out-of-phase detection. To illustrate simply the dependence on and times, the effects of molecular motion are neglected.

The amplitude of the first-harmonic 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 first-harmonic 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 in-phase adsorption spectrum, The amplitude of the second harmonic out-of-phase absorption spectrum is given by (Marsh et al., 1997):

This spectral display corresponds to the standard second-harmonic saturation transfer spectroscopy. Again, as for the first-harmonic, the second-harmonic out-of-phase absorption signal possesses an additional sensitivity to

beyond that expressed simply by the saturation factor. The first and secondharmonic non-linear absorption ESR spectra therefore both possess a sensitivity to processes superior to that of conventional progressive saturation experiments, which are performed on the in-phase As seen from Fig. 8, the first-harmonic out-of-phase has advantages over the second-harmonic out-of-phase 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 spin-lattice relaxation enhancement.

Figure 8. Dependence of the amplitude of: A. the first-harmonic (Eq. 35) and B. the secondharmonic (Eq. 36) out-of-phase EPR absorption signals on time, for the values of indicated. Amplitudes are normalised to the in-phase absorption signal. (and for ) (see Marsh et al., 1997).

Below follows a description of studies of the of the different nonlinear spin-label 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 out-of-phase first harmonic absorption experiments. For the latter two methods, detailed spin-

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 first-harmonic spectrum. The of the integrated absorption intensity, S, is given by:

motion and Zeeman modulation can thus be expressed to a first

Figure 9 gives the progressive saturation curves for the integrated absorption intensity of first harmonic spectra as a function of the molecular

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

deduced. Spin-lattice 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 spin-lattice 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 quasi-rigid 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 spin-lattice relaxation times from the experimental saturation factors is outlined there.

7.2First harmonic, out-of-phase absorption

As already mentioned, the first-harmonic, out-of-phase absorption spectrum has been identified as a non-linear display that is sensitive to but relatively insensitive to both molecular motion (Hyde and Thomas,

relaxation time that is obtained from spectral simulations for different

(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 out-of-phase/in-phase ratio is most sensitive to short spin-lattice relaxation times, The lower modulation frequency of 25 kHz extends this range of sensitivity to the

range, for which the out-of-phase intensities become comparable to those at 100 kHz modulation frequency. Thus the sensitivity of the firstharmonic out-of-phase display can be tuned to different ranges of spin-lattice relaxation time by changing the modulation frequency. Practical examples are given by Livshits et al. (1998a).

Figure 10. Dependence of the ratio, of out-of-phase to in-phase integrated intensities of the first-harmonic absorption spectra on spin-lattice 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 out-of-phase Absorption

Although the second-harmonic out-of-phase absorption spectrum is less optimal for estimation of times than is the first harmonic equivalent, it is included here because it is the classical ST-EPR display and to date is also that most used for determining spin-lattice relaxation enhancements (see e.g., Marsh et al., 1998). There are obvious advantages to determining both rotational mobility and spin-lattice relaxation enhancements in one and the same experiment.

Figure 11 illustrates the application to relaxation enhancements induced by spin-spin interactions (i.e., cross relaxation) for a spin-labelled lipid in gel-phase bilayer membranes. Theoretical dependences of the second harmonic integrated intensity on spin-lattice relaxation time are given in the inset to this figure. The intensity ratio, is strongly dependent on spin-lattice 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

8.SLOW EXCHANGE AND PARAMAGNETIC ENHANCEMENTS

This section describes the different processes, other than slow rotational diffusion, to which nonlinear spin-label EPR spectroscopy is sensitive. It therefore covers the non-classical aspects of saturation transfer EPR spectroscopy (Marsh et al., 1998). There are three basic areas of application: slow physical or chemical exchange processes; weak spin-spin interactions between spin labels; and weak relaxation enhancements induced by paramagnetic species. In the latter two cases, both Heisenberg spin exchange and magnetic dipole-dipole interactions contribute to the relaxation mechanism.

8.1Relaxation enhancement by exchange processes

Exchange at rates comparable to that of the spin-lattice 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:

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 right-hand 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 two-site 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 concentration-dependent, singlelabelling experiment. Depending on spin-label concentration, the relaxation enhancement can then be considerably more efficient than in the two-site case. This more usual single-labelling case is considered after first treating two-site exchange.

8.1.1Relaxation by slow two-site 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 steady-state condition for the spin population difference at site b, including exchange and spin-lattice relaxation, is:

where is the spin population difference at Boltzmann equilibrium, and is the spin-lattice relaxation time in the absence of exchange. A corresponding steady-state 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 spin-lattice relaxation time in the presence of exchange is given by (Horváth et al., 1993):

from which the off-rate, for exchange may be determined. The corresponding expression for Heisenberg spin-exchange between two distinguishable spin-labelled 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):