SPECTROCHIMICA ACTA

PART A

Spectrochimica Acta Part A 51 (1995) Ll -L6

Paramagnetic relaxation enhancement in continuous wave saturation EPR experiments with nuclear relaxation

Derek Marsh

Abteilung Spektroskopie, Max-Planck-Institut fiir biophysikalische Chemie, am Fapberg, D-37077 Giittingen, Germany

Received 2 December 1994; accepted 19 January 1995

Abstract

Complications that may arise from nuclear relaxation when determining enhancements in spin-lattice rates of “N-nitroxyl spin labels from continuous wave EPR saturation experiments are analysed. General expressions are presented for the effective spin-lattice relaxation times for all hyperfine manifolds, and the errors in the relaxation enhancements derived that are consequent upon neglecting nuclear relaxation are estimated. Additionally, the concentration dependence of paramagnetic relaxation enhancement and the saturation of the integrated spectral intensities are considered.

1. Introduction

The enhancement of the spin-lattice relaxation rate of nitroxyl spin labels by fast relaxing paramagnetic species has found wide ranging application in biological systems [ 11. Such methodology includes measurement of local oxygen concentrations by spin label oximetry [2], determination of spin label location by using a combination of hydrophobic and polar relaxation agents [3,4], determination of secondary structure and orientation for transmembrane protein domains by using hydrophobic relaxation agents [5], and the estimation of interatomic separations by using paramagnetic ions [6,7]. New impetus to such studies has come recently from the introduction of site-directed spin-labelling [8], and from the use of saturation transfer EPR techniques for estimation of T, relaxation times [9].

In principle, paramagnetic relaxation enhancements are amenable to direct determina- tion by pulsed EPR techniques [lo], provided that the time-resolution and signal-to-noise ratio are sufficient to distinguish competing pseudo-secular processes and cross relax- ation, such as nuclear relaxation and Heisenberg exchange, respectively. Methods by which this may be done in time-resolved experiments are reviewed in Refs. [l l] and [12]. However, many studies on paramagnetic relaxation enhancement are preformed instead by continuous wave (CW) saturation [l] for which the effective spin-lattice relaxation time, T$‘(M,), that is measured contains contributions from processes other than electron spin-lattice relaxation, most notably nuclear relaxation. Only in one such CW saturation study of paramagnetic relaxation enhancement has nuclear relaxation been considered explicitly [ 131.

The purpose of the present paper is to consider in detail the possible effects of nuclear relaxation on the effective paramagnetic relaxation enhancements measured in CW EPR

SSDI 0584-8539(95)01413-6

L2 D. Mursh I Speciroclknicu ACIU Part A 51 (1995) LI-L6

saturation experiments. In many cases, nuclear relaxation may be faster than the intrinsic electron relaxation, see Refs. [l l] and [14]. General results are presented for all nuclear manifolds and for the saturation behaviour of the total integrated spectral intensity. The latter is of especial interest because it can be used to correct for inhomogeneous broadening in progressive saturation EPR experiments [15], and is the method for estimating spin-lattice relaxation times from saturation transfer (ST) EPR experiments

[9,16]. A particular aim of this work is to examine the conditions under which neglect of nuclear relaxation, as is often done in extracting paramagnetic relaxation enhancements from CW saturation experiments, is a reasonable approximation for this frequently used

class of experiments.

2. Effective relaxation times and nuclear relaxation

The situation considered is depicted by the transition scheme for “N-nitroxyl spin label that is shown in Fig. 1. The population differences in the various spin states are given by nM, = N &, - N;,, where N&, are the absolute spin populations of the M,Y = +i

states for a 14N nuclear spin orientation given by the magnetic quantum number, M,. Electron spin transitions are induced by the microwave magnetic field at a rate W, and the steady state rate equations for the population differences in the various hyperfine states are given by (see Refs. [ 121 and [ 171)

dn,, -= -2(W,+w,)(n”-ng)+2W,,(n+, +n_,)-2Wn,6,,,,=0 dt

dn+, L= -2(W,+to,)(n*, -n~,)+2W,n,,-2Wn,,d‘.,,,,,=0 dt

where n’h, is the value of n,,,,, at Boltzmann equilibrium, and saturation of the transition .

with M, = n7 (I.e. O,,,,,,,, = 1 for M, = 177, and otherwise 0) is assumed. In these equations We and W,, are the intrinsic electron and nuclear spin-lattice relaxation rates, respec- tively, and (0, is the rate of the additional (paramagnetic) relaxation process. Both We and W, are taken to be independent of the nuclear hyperfine manifold [14]. This is consistent with experimental observation for the former [ 11,181, and with theoretical predictions for an END mechanism for the latter [14,17]. The value of o, is linearly proportional to the concentration of the paramagnetic relaxation agent. It is determined by the collisional rate constant and mutual accessibility of spin label and relaxant for a Heisenberg exchange interaction, or by the separation of spin label and relaxant for a dipole-dipole interaction [1,2,8,9]. (It should be noted that corresponding rate equations for cross relaxation resulting from exchange processes, or equivalently rotational diffu- sion, in the presence of nuclear relaxation have been treated previously in Ref. [17].

M,=++ W, W”

M,= -+ W” W”

M,=+l M,=O M,=-I

Fig. I. Spin states and transitions for an S = f, I= 1, “‘N-nitroxyl spin label coupled to a paramagnetic relaxation sink. The intrinsic electron spin-lattice relaxation rate is 2 W, = T,’ and

the “‘N nuclear spin-lattice relaxation rate is 2W, = T,‘. Relaxation to the fast-relaxing sink takes place at a rate 2~0, (dashed lines).

D. Marsh / Spectrochimica Acta Part A 51 (1995) LI-L6 L3

Solution of Eqs. (1) and (2) leads to an expression for the steady-state population difference of the transition irradiated that is of the following form:

n 44,

M’ = 1 + 2 WT;Lf(M,) (3)

This is exactly the expression for the dependence on the microwave power of the saturation of a simple, two-level spin system (see Ref. [19]). For the more complicated multi-level system given in Fig. 1, the effective electron spin-lattice relaxation times given by solution of Eqs. (1) and (2) for the M, = 0 and &- 1 manifolds are, however, respectively

T;;(O, b”‘) = 1 + b + b”’ 1

1 + 3b + b”’ 2 W,(l + b”‘)

and

T’S( f 1 b,,,) = (1 + b”‘)’ + 3b(l + b”‘) + b* le 3

1 (1 + b + b”‘)(l + 3b+ b”‘) 2W,(l + b”‘)

(4)

(5)

where the normalized nuclear relaxation rate is b = W,/ We, and the normalized rate of paramagnetic relaxation enhancement is b”’ = co, / We. A form equivalent to Eq. (4) has been given previously for the MI = 0 manifold by Subczynski and Hyde [13].

3. Extracting relaxation enhancements

The effective electron spin-lattice relaxation times measured by CW saturation EPR experiments therefore depend not only on the rate of paramagnetic relaxation enhance- ment b”‘, but also on the nuclear relaxation rate b. For paramagnetic relaxation that is fast compared with nuclear relaxation (b”’ >> b), and/or nuclear relaxation that is slow compared with electron relaxation (b << l), a simple enhancement is obtained in the effective electron spin-lattice relaxation rate that is given by T;F(O, b”‘)-’ = T$ k 1, b”‘-’ = 2 W,(l + b”‘). Correspondingly, at the other extreme, if nuclear relax- ation is very fast (b D b”‘, 1) then T;T(O, b”‘)-’ = T$ + 1, b”‘)-’ = 6 W,(l + b”‘). In both cases, the enhancement in relaxation rate may then be obtained in a straightforward manner from the effective electron spin-lattice relaxation times measured in CW saturation EPR experiments.

In the general case, the enhancement in relaxation rate of the spin label, relative to that in the absence of the paramagnetic sink, is given from Eqs. (4) and (5) by

T;:W,, 0) T$( M,, b “‘)

= F,,,,,(l + b”‘) (6)

where F

0

= 1 + b”‘/(l + 3b)

1 + b”‘i(1 + 6) (7)

and

F = [l + b”‘/U + b)][l + b”‘/(l + 3b)]

” 1 +b”‘(2+3b+b”‘)/(l +3b+b2) (8)

for the M, = 0 and f 1 manifolds, respectively. The value of FM, in Eq. (6) is the factor by which normalized effective relaxation rate in the presence of nuclear relaxation differs from that for a simple relaxation enhancement (i.e. from T:F((M,, b”‘-’ = 1 + b”‘, for b = 0). In principle, this factor may take values between FM, = 1 and F,,,,, = $. However, unless the paramagnetic relaxation enhancement is extremely fast, F,,,,, is restricted to values closer to 1 (see Fig. 2). Thus there exists a range of values of b and b”’ for which neglecting nuclear relaxation may constitute a reasonable approximation, when normal-

L4 D. Marsh / Spectrochimica Acta Part A 51 (1995) LI-L6

I._

0 2 4 6 a 10

b 111

Fig. 2. Dependence of the effective electron spinlattice relaxation rate, T;F(M,, 6”‘)-‘, for a

“‘N-nitroxyl spin label on the normalized paramagnetic relaxation enhancement rate, b”’ = wy/ W, (Eqs. (4) and (5)). The relaxation rate is normalized relative to that (T;r(M,)-‘) in the

absence of enhancement and to the enhancement (I + 6”‘) in the true spin-lattice relaxation rate,

yielding values of FM, as ordinate. Upper curves: F, for the M, = 0 nuclear manifold and the

values of the normalized nuclear relaxation rate b = I+‘,/ IV, are indicated. Lower curves: F, , for _ M, = f I, shifted downwards by 0.5 for clarity.

ization is made with respect to the relaxation rate T$(M,, O))‘, in the absence of paramagnetic enhancement.

In considering whether neglecting nuclear relaxation is acceptable, it is Ab”‘/b”‘, where

Ab”’ is the difference between the true enhancement in relaxation rate and that obtained by this approximation, that is the decisive quantity. This is given from Eq. (6) by the following:

Ab”‘/b”’ = (1 - F,,,,,)( 1 + l/b”‘) (9)

where the values of FM, are given by Eqs. (7) and (8) for the A4, = 0 and + 1 manifolds, respectively. Values of this fractional error function, Ab”‘/b”‘, are given in Fig. 3. Over

g

0.6

$ 0.4

0.2

0.0 1 I I I I I 0 2 4 a a 10

b 111

Fig. 3. Fractional error in the relaxation enhancement rate Ab”‘/b”‘, for a “N-nitroxyl spin label

introduced by neglecting nuclear relaxation. Values are calculated from Eqs. (7)-(9) and given as

a function of the normalized enhancement in relaxation rate 6”’ for the values of the normalized

nuclear relaxation rate b shown. Lower curves: for M, = 0, Upper curves: for M, = k I, shifted

upwards by 0.5 for clarity.

D. Marsh / Spectrochimica Acra Part A 51 (1995) L/-L6 L5

the range of parameters considered, the fractional error for the A4, = f 1 manifold is smaller than that for the M, = 0 manifold, because fewer nuclear transitions are incolved for the former (see Fig. 1). From Fig. 3 it can be seen that the fractional error that is introduced by neglecting nuclear relaxation is in several circumstances acceptably low and seldom exceeds 40%. In many cases, it is much lower than this and comparable to, or smaller than, the likely experimental error.

4. Concentration dependence

The question of whether the nuclear relaxation rates are such that they do not introduce an appreciable error in deriving the relaxation enhancement rates may also be addressed experimentally by considering the dependence on the concentration of the paramagnetic relaxation agent. For both Heisenberg exchange and magnetic dipole- dipole interactions, the enhancement in relaxation rate b”’ should be linearly dependent on the concentration of the relaxation agent. Therefore, Fig. 2 gives the predicted concentration dependence of the relaxation rate. Only when F,,,,, is constant will the observed relaxation rate be linearly dependent on concentration of the fast relaxing species, as would be expected when nuclear relaxation may be neglected (see Eq. (6)). A linear concentration dependence is therefore a suitable test for validity of the approxima- tion, if data are included at low concentrations (see Fig. 2). If deviations from linearity are observed at low concentration, then the gradient of the concentration dependence at higher concentrations will be reduced by the (approximately constant) factor FM,. In general, the gradient of the concentration dependence is given by (see Eq. (6)).

dR,x, dF,i+ -=F,,,,,+(l +b”‘)= db “’

where Rexp = T$?(M,, O)/T;F(M,, b”‘) is the normalized relaxation rate that is observed. In principle, the true relaxation enhancement (and nuclear relaxation rate) may be obtained by fitting the entire concentration dependence to Eqs. (6) and (7) or (8).

5. Integrated intensities

Often the saturation of the entire integrated spectral intensity, rather than of a specific hyperfine manifold, is analysed. This is done either to allow for inhomogeneous broaden- ing in progressive saturation experiments [15], or to extract effective spin-lattice relax- ation times from saturation transfer EPR spectra [16].

For progressive saturation studies, the power dependence of the integrated intensity is given by (see Ref. [I 51)

Z,,,(P) = 32 1 2

3 [l +c,PT;~(O)]~~~+[~ +ck,PT;;(+l)]‘!* (11)

where P is the microwave power, c,,,,, are constants, and I,, is the value of Z,,, if saturation were absent. If the power dependence of the saturation of the integrated intensity can be fitted well with a single component model, i.e. the saturation of the individual hyperfine components cannot be differentiated, then the difference between T;:(O) and T;F( + 1) must be considerably less than a factor of 5- 10 [15]. This puts some limits on the effects of nuclear relaxation (see Eqs. (4) and (5)). In general, the saturation behaviour of the integrated intensity will be dominated by that of the A4, = + 1 components because these have the greater statistical weight. As noted above, these components are less susceptible to the effects of nuclear relaxation than is the M, = 0 component.

The integrated intensity of the saturation transfer EPR spectrum is given approxi- mately by (see Ref. [9])

L6 D. Marsh 1 Spectrochimica Acta Part A 51 (1995) LI-L6

I = Io T;!30, b”‘) T;;( & 1, b “I) ST

O mo, 0) +(I:, +I%,) pT(fl,O) (12)

where IoM, is the intensity of the M, manifold in the absence of paramagnetic relaxation enhancement. Slow rotational diffusion of the spin label is neglected in this treatment. It is assumed that integration is made over a static distribution of spin label orientations. In the cases where rotational diffusion contributes to the spin label relaxation it is assumed that this may be subsumed into an average effective T,-relaxation time for each hyperfine manifold. Within this approximation, the analysis then follows straightfor- wardly, in principle, from Eqs. (6)-(8), and Fig. 2. Depending on the rotational correlation time, the M, = 0 manifold may take a relatively greater contribution to the total STEPR intensity than it does to the conventional EPR spectrum (see Ref. [20]).

6. Conclusions

Effective spin-lattice relaxation times that include nuclear relaxation and can be used for analysing paramagnetic relaxation enhancements and their concentration dependence have been derived for CW saturation EPR experiments on spin labels. The fractional errors in the relaxation enhancement rates derived by neglecting nuclear relaxation have been estimated and found not always to be negligible. The saturation behaviour on integrating the experimental spectra over all hyperfine manifolds has also been considered for both conventional and saturation transfer EPR.

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