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Molecular order and T1-relaxation, cross-relaxation in nitroxide spin labels
Accepted Manuscript Molecular Order and T 1-relaxation, Cross-relaxation in Nitroxide Spin Labels Derek Marsh PII: DOI: Reference:
S1090-7807(18)30074-0 https://doi.org/10.1016/j.jmr.2018.02.020 YJMRE 6265
To appear in:
Journal of Magnetic Resonance
Received Date: Revised Date: Accepted Date:
10 January 2018 26 February 2018 27 February 2018
Please cite this article as: D. Marsh, Molecular Order and T 1-relaxation, Cross-relaxation in Nitroxide Spin Labels, Journal of Magnetic Resonance (2018), doi: https://doi.org/10.1016/j.jmr.2018.02.020
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Molecular Order and T1-relaxation, Cross-relaxation in Nitroxide Spin Labels
Derek Marsh Max-Planck-Institut für biophysikalische Chemie, 37070 Göttingen, Germany
Tel. +49-551-2011285; e-mail: [email protected]
1
Abstract Interpretation of saturation-recovery EPR experiments on nitroxide spin labels whose angular rotation is restricted by the orienting potential of the environment (e.g., membranes) currently concentrates on the influence of rotational rates and not of molecular order. Here, I consider the dependence on molecular ordering of contributions to the rates of electron spinlattice relaxation and cross relaxation from modulation of N-hyperfine and Zeeman anisotropies. These are determined by the averages cos 2 θ
and cos 4 θ , where θ is the
angle between the nitroxide z-axis and the static magnetic field, which in turn depends on the angles that these two directions make with the director of uniaxial ordering. For saturationrecovery EPR at 9 GHz, the recovery rate constant is predicted to decrease with increasing order for the magnetic field oriented parallel to the director, and to increase slightly for the perpendicular field orientation. The latter situation corresponds to the usual experimental protocol and is consistent with the dependence on chain-labelling position in lipid bilayer membranes. An altered dependence on order parameter is predicted for saturation-recovery EPR at high field (94 GHz) that is not entirely consistent with observation. Comparisons with experiment are complicated by contributions from slow-motional components, and an unexplained background recovery rate that most probably is independent of order parameter. In general, this analysis supports the interpretation that recovery rates are determined principally by rotational diffusion rates, but experiments at other spectral positions/field orientations could increase the sensitivity to order parameter.
Key words: EPR; saturation recovery; order parameter; orientation dependence; spin-lattice relaxation; rotational diffusion; correlation time
2
1. Introduction
Saturation-recovery (SR) EPR is finding increasing application for studying rotational diffusion of spin-labelled molecules, including in partially ordered environments such as membranes (see, e.g., 1; 2; 3; 4; 5; 6). Contributions to the recovery rate constant 1 TSR ,e from modulation of the N-hyperfine and g-value (i.e., Zeeman) anisotropies of the nitroxide spin Hamiltonian by rotational diffusion are characterised by Lorentzian spectral densities. For rotational correlation times τ c 〉〉 1 ω e ( 1.8 × 10 −11 s at 9 GHz), the rate of saturation recovery 1 TSR ,e then becomes proportional to 1 τ c . In this range, contributions from spinrotation interaction are relatively unimportant. However, experiments at longer correlation times in homogeneous solvents reveal an extra contribution to recovery, of uncertain origin, that depends only weakly on correlation time (7; 8; 9; 10). Saturation-recovery rate constants for spin labels undergoing molecular ordering − either lipid chains in membranes or site-directed covalent attachments to proteins − are interpreted mostly in terms of rotational mobility, i.e., correlation times, with less emphasis on order parameters. For site-directed labelling both of T4 lysozyme and lipid chains in membranes, SR-rates increase with increasing rate of rotational diffusion, i.e., with 1 τ c (4; 5; 11). In contrast, order-parameter profiles for a range of membranes are inverted relative to the profiles for saturation-recovery rates (12; 5; 13; 14). This might be anticipated, because decreasing amplitude would decrease the ability of angular fluctuations to induce relaxation. Nevertheless, the influence of molecular ordering on the N-hyperfine (END) and Zeeman mechanisms for saturation recovery does not appear to have been investigated. Here, I treat the effect of ordering on spin-lattice relaxation and cross relaxation by using perturbation theory, i.e., in terms of transition probabilities. An alternative and
3
equivalent approach is to use Abragam’s spin-operator commutator method (15), which I shall refer to briefly. The perturbation approach is possible because rotational diffusion, with correlation times in the region of nanoseconds or less, lies in the fast regime compared with spin-lattice relaxation times that are close to the microsecond range. A particular feature of the SR-experiment is that the rate constant from electron relaxation is determined by the average of the spin-lattice and cross relaxation rates taken over all N-hyperfine lines (16, and see also ref. 10). It does not depend on which nuclear manifold is excited and observed. On the other hand, transverse relaxation ( T2 ) is known to depend on orientation of the magnetic field, relative to the director of ordering (17). This is predicted also to be the case here for T1 relaxation and cross relaxation. The usual protocol in SR-experiments is to excite and observe at the maximum of the absorption spectrum. Thus we can expect mixed degrees of orientational selection from axial powder patterns, which change depending on the frequency band chosen for SR-EPR. We need to take this into account in any comparison with experiment. We find that contributions to the SR-rate constant from END and Zeeman mechanisms can increase or decrease with increasing order parameter, and the dependence on order parameter can be small or appreciable, depending on the magnetic-field orientation and on the EPR frequency. This complexity arises because relaxation rates depend on two angular averages, cos 2 θ and cos 4 θ , where θ is the angle between the nitroxide z-axis and the static magnetic field. Additionally, only the Zeeman mechanism contributes a cos 4 θ -term to the observed SR-rate. Mostly, predictions here support the prevalent view that recovery rates are determined primarily by rotational diffusion rates. But uncertainties remain in the interpretation of high-field SR-EPR. Also, SR-experiments at other spectral positions/field orientations could increase the sensitivity to order parameter.
4
2. Transition Probabilities. Spin-lattice relaxation and cross relaxation are related directly to the probabilities W m′m that transitions are induced between electron and/or nuclear states by time-dependent
fluctuations in local fields. The rate of transition from initial state m to final state m′ in response to a time-dependent random perturbation, H1( q ) F ( q ) (t ) , is:
Wm(′qm) =
2 m′ 2
(q) 1
m
2
j ( q ) (ω m′m )
where an ensemble average over m′
(1)
(q) 1
m
2
is implied. The spectral density, j ( q ) (ωm′m ) ,
is the Fourier transform of the time correlation function of the random function F ( q ) (t ) : +∞
j ( q ) (ω m′m ) =
∫
F ( q )* (t ) F ( q ) (t − τ ) exp( −iω m′mτ ).dτ
(2)
0
The factor of two appears in Eq. 1 because we start the integration at zero, instead of at −∞. For stationary random functions with (reflected) exponential correlation functions, which includes Brownian rotational diffusion, the spectral densities are Lorentzian:
j ( q ) (ω ) = F ( q ) (0)
2
τc 1 + ω 2τ c2
(3)
where τ c is the correlation time, and angular brackets indicate an ensemble (or time) average.
3. Nitroxide spin Hamiltonian and spectral densities
With uniaxial molecular ordering, the time-independent Hamiltonian for a nitroxide with unpaired electron spin S and nitrogen nuclear spin I is:
o
= g o β e Bo S z + ao I ⋅ S +
1
(θ (t ),φ (t ) )
(4)
5
where g o = 13 ( g xx + g yy + g zz ) and a o = 13 ( Axx + Ayy + Azz ) are the isotropic g-value and nitrogen hyperfine coupling constant, respectively, and Bo is the static magnetic field oriented along the laboratory z-axis. The time-dependent part of the Hamiltonian, whose average appears in Eq. 4, contains the following terms
(q) 1
F ( q ) (t ) (18):
( 0) 1
F ( 0) (t ) = 13 (∆gβ e Bo + ∆AI z )(3 cos 2 θ (t ) − 1)S z
(5)
(1) 1
F (1) (t ) = 12 ∆A sin θ (t ) cosθ (t )( I + e − iφ (t ) + I − e iφ (t ) ) S z
(6)
( 2) 1
F ( 2) (t ) =
(3 ) 1
F ( 3) (t ) = 14 ∆A sin 2 θ (t )(I + S + e −i 2φ (t ) + I − S − e i 2φ ( t ) )
(8)
( 4) 1
F ( 4) (t ) = − 121 ∆A(3 cos 2 θ (t ) − 1)(I + S − + I − S + )
(9)
1 2
(∆gβ e Bo + ∆AI z )sinθ (t ) cosθ (t )(S + e −iφ (t ) + S − e iφ (t ) )
(7)
where θ(t) and φ(t) are the instantaneous polar angles that the nitroxide z-axis makes with the laboratory
z-direction.
For
simplicity,
we
assume
axial
symmetry
with
∆g = g zz − 12 ( g xx + g yy ) and ∆A = Azz − 12 ( Axx + Ayy ) .
We get complete azimuthal averaging from the axially symmetric motion, but the nitroxide z-axis is partially ordered about the director axis N, with which it makes angle θ z (see Fig. 1). Using polar coordinates in the director axis-system, θ(t) is related to θ z (t ) by: cos θ (t ) = cos θ z (t ) cos γ + sin θ z (t ) sin γ cos φ z (t )
(10)
where the origin for the azimuth of the nitroxide z-axis is that of the static magnetic field B o which is inclined at fixed angle γ to the director N. Averaging over angles θ z (t ) and φ z (t ) in Eqs. 5−8, we then get:
1
(θ (t ), φ (t ) )
(
)
= 13 3 cos 2 γ − 1 P2 (cosθ z ) (∆gβ e Bo + ∆AI z )S z
6
(11)
where P2 ( x) =
1 2
(3x
2
)
− 1 is a second-order Legendre polynomial, and P2 (cosθ z ) is the
order parameter of the nitroxide z-axis. We ignore the I ± S ∓ non-secular terms from Eq. 9, because they contribute only in second order. Eq. 11 gives the angular dependence of the nitroxide hyperfine splittings and gvalues, in the motional-narrowing regime. For the magnetic field parallel and perpendicular to the director axis, respectively, we get:
A// = ao + 23 ∆A P2 cos(θ z )
(12)
A⊥ = ao − 13 ∆A P2 cos(θ z )
(13)
and similarly for the g-values if we assume axially symmetric ordering, i.e., cos 2 θ x = cos 2 θ y =
1 2
(1 −
)
cos 2 θ z .
The time-dependent perturbation that we need for the relaxation calculation is given by: 1′ (t ) = ∑
(q) 1
F ( q ) (t ) =
1
(θ (t ), φ (t ) ) −
1
(θ (t ),φ (t ) )
(14)
q
We get this from Eqs. 5−9 and 11, if we assume axial symmetry for the A- and g-tensors. The angular averages governing the spectral densities, as defined in Eq. 3, depend directly on the extent of orientational ordering: F ( 2) (0)
F ( 3 ) (0 )
F ( 4 ) (0 )
2
2
2
≡ sin 2 θ cos 2 θ = cos 2 θ − cos 4 θ
(15)
≡ sin 4 θ = 1 − 2 cos 2 θ + cos 4 θ
(16)
(
)
≡ 3 cos 2 θ − 1
2
= 9 cos 4 θ − 6 cos 2 θ + 1
(17)
Thus relaxation in the fast motional regime depends not only on the cos 2 θ average but also on cos 4 θ . Relaxation measurements therefore may provide extra information on molecular
7
ordering, beyond the order parameter that comes from the motionally averaged line positions or splittings.
4. Spin-lattice relaxation
Electron spin transitions are induced by the non-secular terms involving S ± I z and
S ± Bo that correspond to q = 2 in Eq. 7. The electron spin-lattice relaxation rate is given by: 1 T1e,mI = 2We,mI ≡ 2W (∓ 12 , mI → ± 12 , mI ) , where mI is the magnetic quantum number of the nitrogen nuclear spin (see Fig. 2). From Eqs. 1, 3, 7 and 15, we then get: 2 ∆gβ e τc ∆gβ e 2We,mI = Bo + 2 ∆ABo m I + (∆A) 2 m I2 sin 2 θ cos 2 θ 1 + ω e2τ c2
(18)
where τ c is the rotational correlation time for restricted rotation of the nitroxide z-axis, the electron Larmor frequency is ω e ≅ g o β e Bo , and ∆A is in angular frequency units. This relaxation rate has the same dependence on hyperfine manifold mI as do the line widths, but the coefficients are quite different (19). In Eq. 18, θ is the angle that the nitroxide z-axis makes with the static magnetic field. Therefore, the spin-lattice relaxation rate depends on the field direction. However, for the special case when the magnetic field is parallel to the director, we have simply θ = θ z , where
θ z is the angle between the nitroxide z-axis and the director (see Fig. 1). In addition to samples that are specifically aligned normal to the field, this orientation also corresponds to the outer wings of anisotropic powder patterns at 9 GHz. We now use the Maier-Saupe orientation potential, to obtain angular averages such as those appearing in Eqs. 15−17. This potential is given by (20): U (θ z ) = − k B Tλ 2 P2 (cosθ z )
(19)
8
where the strength of the potential λ2 is expressed in units of k BT . The probability of a specific orientation θ z is given by the Boltzmann factor: p (θ z ) = A exp(− U (θ z ) k BT ) , where A normalizes the distribution, such that:
π
∫
0
p(θ z ) sin θ z .dθ z = 1 . We use this to express the
cos 4 θ z average in terms of cos 2 θ z : 1 cos 4 θ z = 1 − λ2
1 cos 2 θ z + 3λ2
(20)
This immediately gives us the average needed for Eq. 18, in terms of the order parameter P2 (cosθ z ) for the nitroxide z-axis:
sin 2 θ z cos 2 θ z =
3 2λ 2
P2 (cos θ z )
(21)
Now we can relate spin-lattice relaxation to the degree of molecular order specified by
P2 (cosθ z ) . The solid line in the upper panel of Fig. 3 shows the dependence of the spin-lattice relaxation rate on Maier-Saupe order parameter, when the static field is parallel to the director. We normalize all values to those for isotropic rotational diffusion We (iso) , given by sin 2 θ cos 2 θ =
2 15
. These normalized values are independent of spin-Hamiltonian
parameters, hyperfine manifold, and rotational correlation time. The ratio is less than unity for moderately high order ( P2 (cos θ z ) > 0.6) , because restricted rotation is less effective at inducing relaxation than is unrestricted rotation. The ratios exceed one (i.e., faster relaxation) at lower order, because the cos 2 θ z and cos 4 θ z averages depend differently on order. Alternatively, for the simple case where the orientation θ z to the director remains fixed, the dependence of spin-lattice relaxation rate on order parameter S zz =
9
1 2
(3 cos
2
θ z − 1)
is given by the dashed lines in the upper panel of Fig. 3. The normalized spin-lattice relaxation rates, for static magnetic field parallel to the director, are then:
We ( S zz ) We (0) = 1 + S zz − 2S zz2
(22)
where we now normalize to the rate, We (0) , for S zz = 0 . A slightly more complicated situation is random wobbling of the nitroxide z-axis within a cone of fixed semi-angle θ C . Angular averages that we need for this model are: n
cos n θ z = ∑ cos k θ C (n + 1) . The normalized spin-lattice relaxation rate then becomes: k =0
We ( S zz ) We (0) = 1 + cos θ C + cos 2 θ C − 32 (1 + cos θ C ) cos 2 θ C
(23)
The dotted line in the upper panel of Fig. 3 shows the dependence on order parameter, when the magnetic field lies along the director, for this case of restricted random walk. Here, S zz = 0 corresponds to isotropic rotation, as for the Maier-Saupe potential.
5. Cross relaxation.
The transition probabilities for cross relaxation, which are given by the dashed lines in Fig. 2, depend on different angular averages from those for spin-lattice relaxation. These combined electron and nuclear spin transitions are induced by the non-secular terms involving S ± I ± with q = 3 and S ± I ∓ with q = 4 in Eqs. 8 and 9, respectively. The rates of the two distinct simultaneous electron and nuclear transitions are given by transition probabilities: Wx1 = W (∓ 12 , mI → ± 12 , mI ± 1) and Wx2 = W (∓ 12 , mI → ± 12 , mI ∓ 1) , as we see from Fig. 2. From Eqs. 1, 3, 8 and 16, we get:
τc 1 W x1 = (∆A) 2 (I ( I + 1) − m I (m I ± 1)) sin 4 θ 8 1 + ω e2τ c2 and from Eqs. 1, 3, 9 and 17:
10
(24)
W x2 =
τc 1 (∆A) 2 (I ( I + 1) − m I (m I ∓ 1) ) (3 cos 2 θ − 1) 2 72 1 + ω e2τ c2
(25)
where we can express the angular averages in terms of cos 2 θ z and cos 4 θ z as in Eqs. 16, 17. For a
15
( I = 12 ) , the numerical values in Eqs. 24, 25 are:
N-nitroxide
I ( I + 1) − m I (mI ± 1) = 1 , in all cases; and correspondingly for a
14
N-nitroxide ( I = 1) :
I ( I + 1) − m I ( m I ± 1) = 2 .
We determine cross relaxation rates with the same three models as used for spinlattice relaxation in Section 4. The lower panel in Fig. 3 shows the dependence of the mean cross-relaxation rate, Wx = 12 (Wx1 + Wx2 ) , on spin-label order parameter, for magnetic field parallel to the director. The combination of angular averages appropriate to the mean is sin 4 θ +
1 9
(3 cos 2 θ − 1) 2
(see Eqs. 24, 25). It is invariably the mean cross-relaxation rate
that enters into the rate which we get in saturation recovery experiments (16). Here it is normalized to Wx (0) , the mean rate for S zz = 0 , which for the Maier-Saupe potential and for wobbling within a cone comes from isotropic averages sin 4 θ =
8 15
and (3 cos 2 θ − 1) 2 =
4 5
.
6. Saturation-recovery EPR.
To first order, the rate constant determined in long-pulse saturation recovery experiments is (16): 1 TSR,e = 2We + 2W x
(26)
where We is the mean spin-lattice relaxation rate, and Wx is the mean Wx -rate for intermanifold cross-relaxations, averaged over all 2 I + 1 hyperfine manifolds, in both cases. This result is independent of which hyperfine manifold it is that is pumped and observed. Eq. 26
11
also applies to short-pulse saturation recovery, when nuclear relaxation is so fast as not to be resolved. We get the combined contributions of direct spin-lattice relaxation and cross relaxation 2We + 2Wx from Eqs. 18, 24, 25. Substituting from Eqs. 15−17 for the angular averages, the saturation recovery rate for a 15N-nitroxide ( I = 12 ) then becomes:
(
)
(
)
1 TSR,e = We, − 1 + We, + 1 + Wx1 + Wx2 = 2
2
2 2 5 (∆A) 2 + ∆gβ e B − 1 (∆A) 2 cos 2 θ − ∆gβ e B cos 4 θ o o 36 12
τ (27) c 2 2 1 + ωe τ c
and for a 14N-nitroxide ( I = 1) :
1 TSR,e = 23 (We, −1 + We, 0 + We, +1 ) + 43 (Wx1 + Wx2 ) = 2 2 10 (∆A) 2 + ∆gβ e B − 2 (∆A) 2 cos 2 θ − ∆gβ e B cos 4 θ o o 27 9
τ (28) c 1 + ωe2τ c2
Using Abragam’s spin-operator method, as in ref. 15, the general result for arbitrary nuclear spin I is:
1 TSR,e = 2We + 2Wx = 2 2 5 I ( I + 1)(∆A) 2 + ∆gβ e B − 1 I ( I + 1)(∆A) 2 cos 2 θ − ∆gβ e B cos 4 θ τ c o o 27 1 + ωe2τ c2 9 (29)
For isotropic rotation, where cos 2 θ z =
1 3
and cos 4 θ z = 15 , Eq. 29 corresponds with the
result obtained in ref. 15 (see also ref. 10). Note that the quantities calculated as W x1 and W x2 in ref. 15 are the sums of the cross-relaxation rates for a given value of mI , and not the single-transition probabilities as defined here by Eqs. 24 and 25. An interesting feature of Eqs. 27−29 is that the cos 4 θ term is contributed only by the field-dependent Zeeman anisotropy. Thus, at low operating frequencies (below 9 GHz), we expect the long-pulse recovery rate to depend almost linearly on order parameter. The lower part of Fig. 4 shows the dependence of saturation-recovery rate on order parameter for 12
the three motional models used already in Sections 4 and 5. In this case the normalised rates depend on both spin-Hamiltonian parameters and spectrometer frequency, because they do not depend solely on a single fixed anisotropy, as for spin-lattice relaxation or cross relaxation alone (cf. Eqs. 18, 24, 25). To allow for non-axiality of the tensors, we replace (∆g ) 2 and (∆A) 2 by (∆g ) 2 + 3(δg ) 2 = 3.379 × 10 −5 and (∆A) 2 + 3(δA) 2 = 7.454 mT2 (14.668
mT2 for 15N), respectively, where δg = 12 ( g xx − g yy ) and δA = 12 ( Axx − A yy ) (see refs. 21, 22). Here we use 14N-DOXYL tensors from refs. 23, 24, and correct for the difference in nuclear gyromagnetic ratio for
15
N-DOXYL. From Fig. 4, we see that uniaxial molecular ordering
can make up to a factor of two difference in saturation recovery rates at a microwave frequency of 9.4 GHz, for a fixed correlation time. The rate of saturation recovery decreases with increasing order parameter, for the magnetic field parallel to the director. This is because restricting the amplitude of angular fluctuation reduces its effectiveness as a relaxation process. Differences in normalised recovery rates between
14
N- and
15
N-nitroxides are
relatively small at an EPR frequency of 9.4 GHz. Also, the various motional models do not change the dependence of the rates on order parameter greatly.
7. Orientation dependence
So far, we have dealt with the special case (γ = 0) where the director axis for ordering lies along the static field. In general, however, the angular averages depend on the angle γ that the magnetic field makes with the director. Therefore, we now use Eq. 10 to obtain the averages cos 2 θ and cos 4 θ as a function of γ: cos 2 θ =
1 2
cos 2 θ z (3 cos 2 γ − 1) + 12 (1 − cos 2 γ )
cos 4 θ =
1 8
cos 4 θ z 3 − 30 cos 2 γ + 35 cos 4 γ − 34 cos 2 θ z 1 − 6 cos 2 γ + 5 cos 4 γ + 38 1 − cos 2 γ
(
)
(30)
(
) ( (31)
13
)
2
where cos φ z = 0 , cos 2 φ z = 12 , cos 3 φ z = 0 and cos 4 φ z = 83 , when averaging over 0 to 2π. By using Eq. 15−17, we then eventually get the dependence of the relaxation rate on magnetic field orientation. This is expressed in terms of the angular averages cos 2 θ z and cos 4 θ z that we get from one of the three models used above to obtain the relaxation rate
when the field lies along the director. It is clear from Eqs. 30 and 31 that the angular variation of the relaxation rate depends not only on cos 2 γ , but also on cos 4 γ . Such an angular dependence is found experimentally for the line widths (i.e., T2 -relaxation rates) of spin labels in ordered systems (17), at least for moderate degrees of order. If the magnetic field is oriented perpendicular to the director, Eqs. 30 and 31 reduce to: cos 2 θ =
1 2
(1 −
cos 2 θ z
) and
cos 4 θ =
3 8
( cos
4
)
θ z − 2 cos 2 θ z + 1 . The upper part of
Fig. 4 shows the dependence of saturation-recovery rate on order parameter for this γ = 90 o field orientation. We use the same three motional models as for the γ = 0 orientation in the lower part of the figure. Most notable is that the recovery rate increases with increasing order parameter, when the magnetic field is perpendicular to the director. The changes in rate are not as large, however, as when the magnetic field is parallel to the director. Again, differences in the normalised recovery rates of
14
N- and
15
N-nitroxides are not large at 9.4
GHz, and differences between the three motional models are not much greater than for the parallel field orientation. If saturation recovery measurements are made on the absorption maximum of a 9GHz powder pattern, contributions will come mainly from field orientations perpendicular to the director axis, in the mI = 0 manifold. Positions in the powder pattern are then specified by g ⊥
which comes from the equivalent of Eq. 13. We then expect the dependence of
14
recovery rate constant on order parameter that is contributed by modulation of the spinHamiltonian anisotropies to be approximated by the upper part of Fig. 4.
8. Discussion
An important general conclusion from this work is that electron spin-lattice relaxation and saturation recovery EPR of nitroxide spin labels in molecularly ordering environments depend on the orientation of the static magnetic field to the director. In contrast to the independence of SR-rates from hyperfine manifold (1; 16; 25), we expect a dependence on spectral position in powder patterns from nitroxides that are subject to orientational ordering. A pronounced dependence on field orientation is found for nitroxide line widths of ordered systems; (26; 27; 17). Saturation-recovery measurements, on the other hand, are usually performed on the most intense peak of the nitroxide absorption powder spectrum, and currently there is little information on the position dependence from ordered systems such as membranes. Figure 4, which is for the conventional microwave frequency of 9.4 GHz, predicts that the dependence of the saturation-recovery rate on order parameter is opposite for magnetic field oriented parallel or perpendicular to the director of ordering. For the perpendicular orientation, which best approximates the usual experimental protocol for non-aligned samples at 9 GHz, dependence of the recovery rate on order parameter is relatively modest. This justifies a posteriori the assumption that saturation recovery is dominated by the rotational correlation time, and that the degree of ordering is less important (e.g., 10; 5). For chainlabelled lipids in membranes with and without cholesterol, Mainali et al. (5) have shown that the dependence of the saturation-recovery rate at 9 GHz on nitroxide position n tracks with that of the rotational diffusion rate R⊥ , and is in the opposite direction to that of the order
15
parameter S zz . For a series of other membrane systems at 9 GHz, Mainali et al. (12; 13; 14) also find opposite dependences of 1 TSR,e and S zz on n. These latter findings are consistent with Fig. 4 for the γ = 90o orientation. As illustration, Table 1 lists saturation-recovery rates at 9 GHz of chain-labelled phosphatidylcholines in fluid bilayer membranes of dimyristoyl phosphatidylcholine containing 0 mol% or 50 mol% cholesterol. The spin-label order parameter S zz decreases with increasing position n down the hydrocarbon chain, and increases on adding cholesterol. Recovery rates are normalised to the reduced spectral density j (ω e ) = τ c (1 + ω e2τ c2 ), where
τ c is deduced from the rotational diffusion coefficient R⊥ that is obtained by simulating the spectral line shapes (5). The normalised rate should reflect directly the dependence on order parameter given in the upper part of Fig. 4 (cf. Eqs. 28−31). We see that
(1 T ) SR ,e
j (ω e )
remains approximately constant, or increases slightly with increasing order parameter (i.e., with decreasing n), as predicted in Fig. 4. However, it should be cautioned that spectra of chain-labelled lipids contain motional components in both fast and slow regimes of 9-GHz nitroxide EPR (28; 29). In particular, the single correlation times from line-shape simulations used in Table 1 are considerably longer than those used to interpret multi-frequency saturation recovery experiments, in the absence of line-shape analysis (10). At high field, corresponding to 94-GHz EPR, the slow-motion components approach the rigid limit, leaving only the fast-motion components that arise from chain rotational isomerism (30; 31; see also 32). For this case, the strongest absorption peak in a 94-GHz powder pattern occurs for the magnetic field directed perpendicular to the director axis, i.e., again the γ = 90 o orientation. Figure 5 shows the dependence of SR-rate on order parameter predicted for the high field of 94-GHz EPR. As expected, this differs considerably from the dependence at 9.4 GHz given in Fig. 4, because the cos 4 θ angular average makes a major 16
contribution at high values of Bo (see Eq. 29). From the lower panel of Fig. 5, we see that the recovery rate for the γ = 90o orientation now barely increases, or slowly decreases, with increasing order parameter, and then decreases strongly at higher S zz -values. Table 1 shows, in contrast, that the normalised experimental recovery rate
(1 T ) SR , e
j (ω e )
remains
approximately constant, or increases with increasing order parameter at 94 GHz, similar to the situation at 9 GHz. Measurements at 94 GHz are taken from Mainali et al. (11), and the single-component values of both R⊥ and order parameter differ from those at 9 GHz, which corresponds to differences in relative contributions of fast and slow motions at the two frequencies (cf. 30; 33). Complications may arise at high field because the frequency dependence of the recovery rate becomes anomalous. One expects the SR-rate to reach a constant value as the EPR frequency increases and modulation of the Zeeman anisotropy comes to dominate. Instead, the rate at 94 GHz increases from that at 35 GHz, which is not explained by current mechanisms (34; 35). As mentioned in the Introduction, we must also remember that mechanisms other than fluctuations in the electron-nuclear dipolar (END) and anisotropic Zeeman (i.e., g-value) interactions make a major contribution to recovery rates at long correlation times (10; 9). These contributions have a weaker dependence on correlation time than do Lorentzian spectral densities. Even so, deuteration of solvent and of the nitroxide shows that the extra contributions correspond neither to a suggested mechanism of solvent-proton spin diffusion, nor to that of methyl group rotation (9; 36; 37; 38; 1). Also, extrapolation from the temperature dependence of recovery rates in glassy solvents shows that the extra contributions cannot be accounted for entirely by solid-state mechanisms, although this would modify the dependence on correlation time of the remaining contributions (39; 38). Whatever the mechanism, it seems unlikely that these additional contributions will depend on the spin-label order parameter. 17
Finally, I consider briefly the situation for powder samples at other EPR frequencies. Below 9 GHz, e.g., at S-band and L-band, Zeeman anisotropy becomes negligible and the absorption maximum is in the centre of the spectrum, at the position specified by g o . This corresponds to the m I = 0 manifold from an unbiassed superposition of director-axes oriented at all angles to the magnetic field direction. The situation then approximates that of an isotropic distribution, and we expect little dependence of SR-rate on order parameter. For an EPR frequency of 34 GHz, intensity from the m I = +1 manifold piles up in the powder pattern at low field, where resonance positions are determined by g ⊥ and A⊥ (see Eq. 13). The absorption maximum corresponds here to the field oriented along all directions perpendicular to the director, with a close shoulder from the field parallel to the director that merges with the two other orientations as the order parameter decreases. Thus we anticipate a
γ = 90 o orientation at very high order that rapidly converts to an approximately isotropic situation with decreasing order parameter.
9. Conclusion
Fast motional theory predicts that the electron spin-lattice relaxation, and crossrelaxation, rates of nitroxide spin labels in ordering environments depend both on the orientation of the magnetic field and on the order parameter of the nitroxide z-axis. For an EPR frequency of 9 GHz, the rate of saturation recovery at the absorption maximum of the EPR powder pattern increases, but maximally by <20%, with increasing order parameter. Qualitatively this is in accord with experiment, and the common assumption that SR-rates are determined predominantly by rotational diffusion rates (i.e., correlation time). A stronger dependence on order parameter is predicted for the magnetic field oriented along the director axis of ordering. A markedly different dependence is predicted for high-field EPR at 94 GHz,
18
although this is not evident experimentally. Comparisons with experiment are complicated by the presence of slow-motional components and a background recovery rate additional to that from modulation of hyperfine and Zeeman anisotropy, and by the anomalous frequency dependence at 94 GHz. Nonetheless, many of the predictions support the view that SR-rates are determined primarily by the rate of rotational diffusion.
Acknowledgement
I thank Christian Griesinger and the Department of NMR-based structural biology for financial assistance.
19
Table 1. Rate constants 1 TSR,e for saturation recovery of n-DOXYL phosphatidylcholine spin labels in bilayer membranes of dimyristoyl phosphatidylcholine with and without 50 mol% cholesterol, at 27 oC.a Data from refs. 5, 11.
(1 T )
n (C-atom)
SR , e
0 mol% cholesterol
j (ω e ) (× 10 −18 s - 2 )
50 mol% cholesterol
9.2 GHz
94 GHz
9.2 GHz
94 GHz
5
2.49
195
3.56
249
7
2.45
174
3.85
196
10
2.50
173
2.27
162
12
2.30
163
2.21
164
14
2.34
160
2.14
159
16
1.80
a
1.62
Recovery rates are normalized to the reduced spectral density: j (ω e ) = τ c (1 + ω e2τ c2 ), where
the correlation time is related to the rotational diffusion R⊥ coefficient by τ c = 1 (6 R ⊥ ) .
20
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[1]
P.W.Percival, J.S. Hyde, Saturation-recovery measurements of spin-lattice relaxationtimes of some nitroxides in solution, J. Magn. Reson. 23 (1976) 249-257.
[2]
J.S.Hyde, J.J. Yin, W.K. Subczynski, T.G. Camenisch, J.J. Ratke, W. Froncisz, Spinlabel EPR T1 values using saturation recovery from 2 to 35 GHz, J. Phys. Chem. B 108 (2004) 9524-9529.
[3]
J.Pyka, J. Ilnicki, C. Altenbach, W.L. Hubbell, W. Froncisz, Accessibility and dynamics of nitroxide side chains in T4 lysozyme measured by saturation recovery EPR, Biophys. J. 89 (2005) 2059-2068.
[4]
M.D.Bridges, K. Hideg, W.L. Hubbell, Resolving conformational and rotameric exchange in spin-labeled proteins using saturation recovery EPR, Appl. Magn. Reson 37 (2010) 363-390.
[5]
L.Mainali, J.B. Feix, J.S. Hyde, W. Subczynski, Membrane fluidity profiles as deduced by saturation recovery EPR measurements of spin-lattice relaxation times of spin labels, J. Magn. Reson. 212 (2011) 418-425.
[6]
W.Subczynski, M. Raguz, J. Widomska, L. Mainali, A. Komovalov, Function of cholesterol and the cholesterol bilayer domain specific to the fiber cell plasma membrane of the eye lens, J. Membrane Biol. 245 (2012) 51-68.
[7]
P.Fajer, D.D. Thomas, J.B. Feix, J.S. Hyde, Measurement of rotational molecular motion by time-resolved saturation transfer electron paramagnetic resonance, Biophys. J. 50 (1986) 1195-1202.
[8]
B.H.Robinson, D.A. Haas, C. Mailer, Molecular dynamics in liquids: Spin-lattice relaxation of nitroxide spin labels, Science 263 (1994) 490-493.
21
[9]
R.Owenius, G.E. Terry, M.J. Williams, S.S. Eaton, G.R. Eaton, Frequency dependence of electron spin relaxation of nitroxyl radicals in fluid solutions, J. Phys. Chem. B 108 (2004) 9475-9481.
[10] C.Mailer, R.D. Nielsen, B.H. Robinson, Explanation of spin-lattice relaxation rates of spin labels obtained with multifrequency saturation recovery EPR, J. Phys. Chem. A 109 (2005) 4049-4061. [11] L.Mainali, J.S. Hyde, W.K. Subczynski, Using spin-label W-band EPR to study membrane fluidity profiles in samples of small volume, J. Magn. Reson. 226 (2013) 3544. [12] L.Mainali, M. Raguz, T.G. Camenisch, J.S. Hyde, W.K. Subczynski, Spin-label saturation recovery EPR at W-band: Applications to eye lens lipid membranes, J. Magn. Reson. 212 (2011) 86-94. [13] L.Mainali, M. Raguz, W. Subczynski, Phases and domains in sphingomyelincholesterol membranes: structure and properties using EPR spin-labeling methods, Eur. Biophys. J. 41 (2012) 147-159. [14] L.Mainali, M. Raguz, W.J. O'Brien, W. Subczynski, Properties of membranes derived from the total lipids extracted from the human lens cortex and nucleus, Biochim. Biophys. Acta 1828 (2013) 1432-1440. [15] D.Marsh, Coherence transfer and electron T1-, T2-relaxation in nitroxide spin labels, J. Magn. Reson. 277 (2017) 86-94. [16] D.Marsh, Cross relaxation in nitroxide spin labels, J. Magn. Reson. 272 (2016) 172180. [17] G.R.Luckhurst, S.W. Smith, F. Sundholm, A spin-labelled liquid crystal. A new spin probe for the study of the orientational order and molecular dynamics of liquid crystals, Acta Chem. Scand. Ser. A 41 (1987) 218-229.
22
[18] H.M.McConnell, Effect of anisotropic hyperfine interactions on paramagnetic relaxation in liquids, J. Chem. Phys. 25 (1956) 709-711. [19] S.H.Glarum, J.H. Marshall, Paramagnetic relaxation in liquid-crystal solvents, J. Chem. Phys. 46 (1967) 55-62. [20] W.Maier, A. Saupe, Eine einfache molekular-statistische Theorie der nematischen kristallinflüssigen Phase. Teil 1, Z. Naturforsch 14a (1959) 882-889. [21] R.Wilson, D. Kivelson, ESR linewidths in solution. I. Experiments on anisotropic and spin-rotational effects, J. Chem. Phys. 44 (1966) 154-168. [22] L.R.Dalton, B.H. Robinson, L.A. Dalton, P. Coffey, Saturation transfer spectroscopy, Adv. Magn. Reson. 8 (1976) 149-259. [23] P.C.Jost, L.J. Libertini, V.C. Hebert, O.H. Griffith, Lipid spin labels in lecithin multilayers. A study of motion along fatty acid chains, J. Mol. Biol. 59 (1971) 77-98. [24] T.I.Smirnova, A.I. Smirnov, R.B. Clarkson, R.L. Belford, W-Band (95 GHz) EPR spectroscopy of nitroxide radicals with complex proton hyperfine structure: Fast Motion, J. Phys. Chem. 99 (1995) 9008-9016. [25] B.H.Robinson, A.W. Reese, E. Gibbons, C. Mailer, A unified description of the spinspin and spin-lattice relaxation rates applied to nitroxide spin labels in viscous liquids, J. Phys. Chem. B 103 (1999) 5881-5894. [26] H.Schindler, J. Seelig, EPR spectra of spin labels in lipid bilayers, J. Chem. Phys. 59 (1973) 1841-1850. [27] S.Schreier-Muccillo, D. Marsh, H. Dugas, H. Schneider, I.C.P. Smith, A spin probe study of the influence of cholesterol on motion and orientation of phospholipids in oriented multibilayers and vesicles, Chem. Phys. Lipids 10 (1973) 11-27.
23
[28] A.Lange, D. Marsh, K.-H. Wassmer, P. Meier, G. Kothe, Electron spin resonance study of phospholipid membranes employing a comprehensive line-shape model, Biochemistry 24 (1985) 4383-4392. [29] M.Moser, D. Marsh, P. Meier, K.-H. Wassmer, G. Kothe, Chain configuration and flexibility gradient in phospholipid membranes. Comparison between spin-label electron spin resonance and deuteron nuclear magnetic resonance, and identification of new conformations, Biophys. J. 55 (1989) 111-123. [30] V.A.Livshits, D. Kurad, D. Marsh, Simulation studies on high-field EPR of lipid spin labels in cholesterol-containing membranes, J. Phys. Chem. B 108 (2004) 9403-9411. [31] V.A.Livshits, D. Marsh, Simulation studies of high-field EPR spectra of spin-labeled lipids in membranes, J. Magn. Reson. 147 (2000) 59-67. [32] Y.Lou, M. Ge, J.H. Freed, A multifrequency ESR study of the complex dynamics of membranes, J. Phys. Chem. B 105 (2001) 11053-11056. [33] D.Marsh, D. Kurad, V.A. Livshits, High-field spin label EPR of lipid membranes, Magn. Reson. Chem. 43 (2005) S20-S25. [34] W.Froncisz, T.G. Camenisch, J.J. Ratke, J.R. Anderson, W.K. Subczynski, R.A. Strangeway, J.W. Sidabras, J.S. Hyde, Saturation recovery EPR and ELDOR at Wband for spin labels, J. Magn. Reson. 193 (2008) 297-304. [35] W.Subczynski, L. Mainali, T.G. Camenisch, W. Froncisz, J.S. Hyde, Spin-label oximetry at Q- and W-band, J. Magn. Reson. 209 (2011) 142-148. [36] S.S. Eaton, G.R. Eaton, Relaxation times of organic radicals and transition metal ions, in: L.J. Berliner, S.S. Eaton, G.R. Eaton (Eds.), Distance Measurements in Biological Systems, Springer, New York, 2000, 29-154.
24
[37] J.R.Biller, V. Meyer, H. Elajaili, G.M. Rosen, J.P.Y. Kao, S.S. Eaton, G.R. Eaton, Relaxation time and line widths of isotopically substituted nitroxides in aqueous solution at X-band, J. Magn. Reson. 212 (2011) 370-377. [38] H.Sato, S.E. Bottle, J.P. Blinco, A.S. Micallef, G.R. Eaton, S.S. Eaton, Electron spinlattice relaxation of nitroxyl radicals in temperature ranges that span glassy solutions to low-viscosity liquids, J. Magn. Reson. 191 (2008) 66-77. [39] H.Sato, V. Kathirvelu, A. Fielding, J.P. Blinco, A.S. Micallef, S.E. Bottle, S.S. Eaton, G.R. Eaton, Impact of molecular size on electron spin relaxation rates of nitroxyl radicals in glassy solvents between 100 and 300 K, Mol. Phys. 105 (2007) 2137-2151.
25
Figure Legends
Fig. 1. Orientation of a spin-labelled moiety in an ordering environment. N is the
director axis for ordering; z is the nitroxide principal axis; B o is the static magnetic field direction.
Fig. 2.
Energy levels and transitions for
15
N- ( S = 12 , I = 12 ; left )
and
14
N-
( S = 12 , I = 1; right ) nitroxides. Spin states are labeled by their electron and nuclear magnetic quantum numbers, M S and mI respectively. Transition probabilities per unit time for leaving a particular state are given by We,mI for spin-lattice relaxation (Eq. 18), and Wx1 ,
Wx2 for cross relaxation (Eqs. 24, 25).
Fig. 3. Top: Spin-lattice relaxation rate We = 1 2T1e from Eq. 18, as a function of order
parameter S zz ≡ P2 (cosθ z ) , where angular averages are determined by the Maier-Saupe orientation potential (solid line; Eqs. 19−21), or correspond to a fixed value of θ z (dashed line; Eq. 22) or restricted random walk within a cone (dotted line; Eq. 23). Bottom: mean cross-relaxation rate Wx = 12 (Wx1 + Wx2 ) for the same three motional models, according to Eqs. 24 and 25. Relaxation rates, We (S zz ) and Wx ( S zz ) , for the static magnetic field oriented parallel to the director are normalized to their values, We (0) and Wx (0) , for S zz = 0 .
Fig. 4. Saturation recovery rate WSR,e = 1 2TSR,e from Eq. 29 for long pulses at 9.4 GHz,
as a function of order parameter S zz ≡ P2 (cosθ z ) , where angular averages are determined by the Maier-Saupe orientation potential (solid line), or correspond to a fixed value of θ z 26
(dashed line) or restricted random walk within a cone (dotted line). Recovery rates WSR ,e ( S zz ) , with the static magnetic field oriented parallel (bottom; γ = 0 ) and perpendicular (top;
γ = 90 o in Eqs. 30, 31) to the director, are given for 14N-nitroxides (black lines, Eq. 28) and 15
N-nitroxides (grey lines, Eq. 27), and are normalized to their values WSR,e (0) for S zz = 0 .
Fig. 5. Saturation recovery rate WSR,e = 1 2TSR,e for long pulses at 94 GHz, as a function
of order parameter S zz ≡ P2 (cosθ z ) . Angular averages are from the Maier-Saupe orientation potential (solid line), or for a fixed value of θ z (dashed line) or restricted random walk within a cone (dotted line). Recovery rates WSR ,e (S zz ) , with magnetic field parallel (top
panel; γ = 0 o ) and perpendicular (bottom panel; γ = 90o ) to the director, for 14N-nitroxides (black lines) and 15N-nitroxides (grey lines), are normalized to WSR,e (0) for S zz = 0 .
27
Fig. 1.
28
Fig. 2.
29
1.4
We (Szz)/W e (0)
1.2 1.0
wobble
0.8
fixed
0.6
Maier-Saupe
0.4 0.2 0.0
W x(Szz )/ Wx (0)
0.9
Maier-Saupe
0.8 0.7 0.6
fixed
0.5 0.4
wobble 0.0
0.2
0.4
0.6
0.8
order parameter, Szz =
Fig. 3.
30
1.0
1.2
14
o
γ = 90 :
N
1.1
W SR,e(Szz )/WSR,e (0)
15
1.0 15
N
0.9 0.8
14
N
Maier-Saupe fixed wobble
N
0.7 0.6 0.5
γ = 0:
0.4 0.0
0.2
0.4
0.6
0.8
order parameter, Szz =
Fig. 4.
31
1.0
1.4 15
N
1.2 1.0 0.8 14
N
WSR,e(Szz)/WSR,e(0)
0.6 0.4 0.2
94 GHz, γ = 0o
0.0 Maier-Saupe fixed wobble
14
N
1.2 1.0 0.8 0.6
15
N
0.4 0.2
94 GHz, γ = 90o
0.0 0.0
0.2
0.4
0.6
0.8
order parameter, Szz =
Fig. 5.
32
1.0
Graphical abstract
33
Highlights to manuscript „Molecular Order and T1-relaxation, Cross-relaxation in Nitroxide Spin Labels” by Derek Marsh
•
Saturation recovery (SR)-rates of spin labels undergoing molecular ordering depend on field orientation
•
At 9 GHz, nitroxide SR-rates increase slowly with increasing order for field perpendicular to the ordering axis
•
At 9 GHz, SR-rates depend predominantly on rotational rate
•
At 94 GHz, SR-rates do not follow dependence on ordering predicted for fast motion
34
S1090-7807(18)30074-0 https://doi.org/10.1016/j.jmr.2018.02.020 YJMRE 6265
To appear in:
Journal of Magnetic Resonance
Received Date: Revised Date: Accepted Date:
10 January 2018 26 February 2018 27 February 2018
Please cite this article as: D. Marsh, Molecular Order and T 1-relaxation, Cross-relaxation in Nitroxide Spin Labels, Journal of Magnetic Resonance (2018), doi: https://doi.org/10.1016/j.jmr.2018.02.020
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Molecular Order and T1-relaxation, Cross-relaxation in Nitroxide Spin Labels
Derek Marsh Max-Planck-Institut für biophysikalische Chemie, 37070 Göttingen, Germany
Tel. +49-551-2011285; e-mail: [email protected]
1
Abstract Interpretation of saturation-recovery EPR experiments on nitroxide spin labels whose angular rotation is restricted by the orienting potential of the environment (e.g., membranes) currently concentrates on the influence of rotational rates and not of molecular order. Here, I consider the dependence on molecular ordering of contributions to the rates of electron spinlattice relaxation and cross relaxation from modulation of N-hyperfine and Zeeman anisotropies. These are determined by the averages cos 2 θ
and cos 4 θ , where θ is the
angle between the nitroxide z-axis and the static magnetic field, which in turn depends on the angles that these two directions make with the director of uniaxial ordering. For saturationrecovery EPR at 9 GHz, the recovery rate constant is predicted to decrease with increasing order for the magnetic field oriented parallel to the director, and to increase slightly for the perpendicular field orientation. The latter situation corresponds to the usual experimental protocol and is consistent with the dependence on chain-labelling position in lipid bilayer membranes. An altered dependence on order parameter is predicted for saturation-recovery EPR at high field (94 GHz) that is not entirely consistent with observation. Comparisons with experiment are complicated by contributions from slow-motional components, and an unexplained background recovery rate that most probably is independent of order parameter. In general, this analysis supports the interpretation that recovery rates are determined principally by rotational diffusion rates, but experiments at other spectral positions/field orientations could increase the sensitivity to order parameter.
Key words: EPR; saturation recovery; order parameter; orientation dependence; spin-lattice relaxation; rotational diffusion; correlation time
2
1. Introduction
Saturation-recovery (SR) EPR is finding increasing application for studying rotational diffusion of spin-labelled molecules, including in partially ordered environments such as membranes (see, e.g., 1; 2; 3; 4; 5; 6). Contributions to the recovery rate constant 1 TSR ,e from modulation of the N-hyperfine and g-value (i.e., Zeeman) anisotropies of the nitroxide spin Hamiltonian by rotational diffusion are characterised by Lorentzian spectral densities. For rotational correlation times τ c 〉〉 1 ω e ( 1.8 × 10 −11 s at 9 GHz), the rate of saturation recovery 1 TSR ,e then becomes proportional to 1 τ c . In this range, contributions from spinrotation interaction are relatively unimportant. However, experiments at longer correlation times in homogeneous solvents reveal an extra contribution to recovery, of uncertain origin, that depends only weakly on correlation time (7; 8; 9; 10). Saturation-recovery rate constants for spin labels undergoing molecular ordering − either lipid chains in membranes or site-directed covalent attachments to proteins − are interpreted mostly in terms of rotational mobility, i.e., correlation times, with less emphasis on order parameters. For site-directed labelling both of T4 lysozyme and lipid chains in membranes, SR-rates increase with increasing rate of rotational diffusion, i.e., with 1 τ c (4; 5; 11). In contrast, order-parameter profiles for a range of membranes are inverted relative to the profiles for saturation-recovery rates (12; 5; 13; 14). This might be anticipated, because decreasing amplitude would decrease the ability of angular fluctuations to induce relaxation. Nevertheless, the influence of molecular ordering on the N-hyperfine (END) and Zeeman mechanisms for saturation recovery does not appear to have been investigated. Here, I treat the effect of ordering on spin-lattice relaxation and cross relaxation by using perturbation theory, i.e., in terms of transition probabilities. An alternative and
3
equivalent approach is to use Abragam’s spin-operator commutator method (15), which I shall refer to briefly. The perturbation approach is possible because rotational diffusion, with correlation times in the region of nanoseconds or less, lies in the fast regime compared with spin-lattice relaxation times that are close to the microsecond range. A particular feature of the SR-experiment is that the rate constant from electron relaxation is determined by the average of the spin-lattice and cross relaxation rates taken over all N-hyperfine lines (16, and see also ref. 10). It does not depend on which nuclear manifold is excited and observed. On the other hand, transverse relaxation ( T2 ) is known to depend on orientation of the magnetic field, relative to the director of ordering (17). This is predicted also to be the case here for T1 relaxation and cross relaxation. The usual protocol in SR-experiments is to excite and observe at the maximum of the absorption spectrum. Thus we can expect mixed degrees of orientational selection from axial powder patterns, which change depending on the frequency band chosen for SR-EPR. We need to take this into account in any comparison with experiment. We find that contributions to the SR-rate constant from END and Zeeman mechanisms can increase or decrease with increasing order parameter, and the dependence on order parameter can be small or appreciable, depending on the magnetic-field orientation and on the EPR frequency. This complexity arises because relaxation rates depend on two angular averages, cos 2 θ and cos 4 θ , where θ is the angle between the nitroxide z-axis and the static magnetic field. Additionally, only the Zeeman mechanism contributes a cos 4 θ -term to the observed SR-rate. Mostly, predictions here support the prevalent view that recovery rates are determined primarily by rotational diffusion rates. But uncertainties remain in the interpretation of high-field SR-EPR. Also, SR-experiments at other spectral positions/field orientations could increase the sensitivity to order parameter.
4
2. Transition Probabilities. Spin-lattice relaxation and cross relaxation are related directly to the probabilities W m′m that transitions are induced between electron and/or nuclear states by time-dependent
fluctuations in local fields. The rate of transition from initial state m to final state m′ in response to a time-dependent random perturbation, H1( q ) F ( q ) (t ) , is:
Wm(′qm) =
2 m′ 2
(q) 1
m
2
j ( q ) (ω m′m )
where an ensemble average over m′
(1)
(q) 1
m
2
is implied. The spectral density, j ( q ) (ωm′m ) ,
is the Fourier transform of the time correlation function of the random function F ( q ) (t ) : +∞
j ( q ) (ω m′m ) =
∫
F ( q )* (t ) F ( q ) (t − τ ) exp( −iω m′mτ ).dτ
(2)
0
The factor of two appears in Eq. 1 because we start the integration at zero, instead of at −∞. For stationary random functions with (reflected) exponential correlation functions, which includes Brownian rotational diffusion, the spectral densities are Lorentzian:
j ( q ) (ω ) = F ( q ) (0)
2
τc 1 + ω 2τ c2
(3)
where τ c is the correlation time, and angular brackets indicate an ensemble (or time) average.
3. Nitroxide spin Hamiltonian and spectral densities
With uniaxial molecular ordering, the time-independent Hamiltonian for a nitroxide with unpaired electron spin S and nitrogen nuclear spin I is:
o
= g o β e Bo S z + ao I ⋅ S +
1
(θ (t ),φ (t ) )
(4)
5
where g o = 13 ( g xx + g yy + g zz ) and a o = 13 ( Axx + Ayy + Azz ) are the isotropic g-value and nitrogen hyperfine coupling constant, respectively, and Bo is the static magnetic field oriented along the laboratory z-axis. The time-dependent part of the Hamiltonian, whose average appears in Eq. 4, contains the following terms
(q) 1
F ( q ) (t ) (18):
( 0) 1
F ( 0) (t ) = 13 (∆gβ e Bo + ∆AI z )(3 cos 2 θ (t ) − 1)S z
(5)
(1) 1
F (1) (t ) = 12 ∆A sin θ (t ) cosθ (t )( I + e − iφ (t ) + I − e iφ (t ) ) S z
(6)
( 2) 1
F ( 2) (t ) =
(3 ) 1
F ( 3) (t ) = 14 ∆A sin 2 θ (t )(I + S + e −i 2φ (t ) + I − S − e i 2φ ( t ) )
(8)
( 4) 1
F ( 4) (t ) = − 121 ∆A(3 cos 2 θ (t ) − 1)(I + S − + I − S + )
(9)
1 2
(∆gβ e Bo + ∆AI z )sinθ (t ) cosθ (t )(S + e −iφ (t ) + S − e iφ (t ) )
(7)
where θ(t) and φ(t) are the instantaneous polar angles that the nitroxide z-axis makes with the laboratory
z-direction.
For
simplicity,
we
assume
axial
symmetry
with
∆g = g zz − 12 ( g xx + g yy ) and ∆A = Azz − 12 ( Axx + Ayy ) .
We get complete azimuthal averaging from the axially symmetric motion, but the nitroxide z-axis is partially ordered about the director axis N, with which it makes angle θ z (see Fig. 1). Using polar coordinates in the director axis-system, θ(t) is related to θ z (t ) by: cos θ (t ) = cos θ z (t ) cos γ + sin θ z (t ) sin γ cos φ z (t )
(10)
where the origin for the azimuth of the nitroxide z-axis is that of the static magnetic field B o which is inclined at fixed angle γ to the director N. Averaging over angles θ z (t ) and φ z (t ) in Eqs. 5−8, we then get:
1
(θ (t ), φ (t ) )
(
)
= 13 3 cos 2 γ − 1 P2 (cosθ z ) (∆gβ e Bo + ∆AI z )S z
6
(11)
where P2 ( x) =
1 2
(3x
2
)
− 1 is a second-order Legendre polynomial, and P2 (cosθ z ) is the
order parameter of the nitroxide z-axis. We ignore the I ± S ∓ non-secular terms from Eq. 9, because they contribute only in second order. Eq. 11 gives the angular dependence of the nitroxide hyperfine splittings and gvalues, in the motional-narrowing regime. For the magnetic field parallel and perpendicular to the director axis, respectively, we get:
A// = ao + 23 ∆A P2 cos(θ z )
(12)
A⊥ = ao − 13 ∆A P2 cos(θ z )
(13)
and similarly for the g-values if we assume axially symmetric ordering, i.e., cos 2 θ x = cos 2 θ y =
1 2
(1 −
)
cos 2 θ z .
The time-dependent perturbation that we need for the relaxation calculation is given by: 1′ (t ) = ∑
(q) 1
F ( q ) (t ) =
1
(θ (t ), φ (t ) ) −
1
(θ (t ),φ (t ) )
(14)
q
We get this from Eqs. 5−9 and 11, if we assume axial symmetry for the A- and g-tensors. The angular averages governing the spectral densities, as defined in Eq. 3, depend directly on the extent of orientational ordering: F ( 2) (0)
F ( 3 ) (0 )
F ( 4 ) (0 )
2
2
2
≡ sin 2 θ cos 2 θ = cos 2 θ − cos 4 θ
(15)
≡ sin 4 θ = 1 − 2 cos 2 θ + cos 4 θ
(16)
(
)
≡ 3 cos 2 θ − 1
2
= 9 cos 4 θ − 6 cos 2 θ + 1
(17)
Thus relaxation in the fast motional regime depends not only on the cos 2 θ average but also on cos 4 θ . Relaxation measurements therefore may provide extra information on molecular
7
ordering, beyond the order parameter that comes from the motionally averaged line positions or splittings.
4. Spin-lattice relaxation
Electron spin transitions are induced by the non-secular terms involving S ± I z and
S ± Bo that correspond to q = 2 in Eq. 7. The electron spin-lattice relaxation rate is given by: 1 T1e,mI = 2We,mI ≡ 2W (∓ 12 , mI → ± 12 , mI ) , where mI is the magnetic quantum number of the nitrogen nuclear spin (see Fig. 2). From Eqs. 1, 3, 7 and 15, we then get: 2 ∆gβ e τc ∆gβ e 2We,mI = Bo + 2 ∆ABo m I + (∆A) 2 m I2 sin 2 θ cos 2 θ 1 + ω e2τ c2
(18)
where τ c is the rotational correlation time for restricted rotation of the nitroxide z-axis, the electron Larmor frequency is ω e ≅ g o β e Bo , and ∆A is in angular frequency units. This relaxation rate has the same dependence on hyperfine manifold mI as do the line widths, but the coefficients are quite different (19). In Eq. 18, θ is the angle that the nitroxide z-axis makes with the static magnetic field. Therefore, the spin-lattice relaxation rate depends on the field direction. However, for the special case when the magnetic field is parallel to the director, we have simply θ = θ z , where
θ z is the angle between the nitroxide z-axis and the director (see Fig. 1). In addition to samples that are specifically aligned normal to the field, this orientation also corresponds to the outer wings of anisotropic powder patterns at 9 GHz. We now use the Maier-Saupe orientation potential, to obtain angular averages such as those appearing in Eqs. 15−17. This potential is given by (20): U (θ z ) = − k B Tλ 2 P2 (cosθ z )
(19)
8
where the strength of the potential λ2 is expressed in units of k BT . The probability of a specific orientation θ z is given by the Boltzmann factor: p (θ z ) = A exp(− U (θ z ) k BT ) , where A normalizes the distribution, such that:
π
∫
0
p(θ z ) sin θ z .dθ z = 1 . We use this to express the
cos 4 θ z average in terms of cos 2 θ z : 1 cos 4 θ z = 1 − λ2
1 cos 2 θ z + 3λ2
(20)
This immediately gives us the average needed for Eq. 18, in terms of the order parameter P2 (cosθ z ) for the nitroxide z-axis:
sin 2 θ z cos 2 θ z =
3 2λ 2
P2 (cos θ z )
(21)
Now we can relate spin-lattice relaxation to the degree of molecular order specified by
P2 (cosθ z ) . The solid line in the upper panel of Fig. 3 shows the dependence of the spin-lattice relaxation rate on Maier-Saupe order parameter, when the static field is parallel to the director. We normalize all values to those for isotropic rotational diffusion We (iso) , given by sin 2 θ cos 2 θ =
2 15
. These normalized values are independent of spin-Hamiltonian
parameters, hyperfine manifold, and rotational correlation time. The ratio is less than unity for moderately high order ( P2 (cos θ z ) > 0.6) , because restricted rotation is less effective at inducing relaxation than is unrestricted rotation. The ratios exceed one (i.e., faster relaxation) at lower order, because the cos 2 θ z and cos 4 θ z averages depend differently on order. Alternatively, for the simple case where the orientation θ z to the director remains fixed, the dependence of spin-lattice relaxation rate on order parameter S zz =
9
1 2
(3 cos
2
θ z − 1)
is given by the dashed lines in the upper panel of Fig. 3. The normalized spin-lattice relaxation rates, for static magnetic field parallel to the director, are then:
We ( S zz ) We (0) = 1 + S zz − 2S zz2
(22)
where we now normalize to the rate, We (0) , for S zz = 0 . A slightly more complicated situation is random wobbling of the nitroxide z-axis within a cone of fixed semi-angle θ C . Angular averages that we need for this model are: n
cos n θ z = ∑ cos k θ C (n + 1) . The normalized spin-lattice relaxation rate then becomes: k =0
We ( S zz ) We (0) = 1 + cos θ C + cos 2 θ C − 32 (1 + cos θ C ) cos 2 θ C
(23)
The dotted line in the upper panel of Fig. 3 shows the dependence on order parameter, when the magnetic field lies along the director, for this case of restricted random walk. Here, S zz = 0 corresponds to isotropic rotation, as for the Maier-Saupe potential.
5. Cross relaxation.
The transition probabilities for cross relaxation, which are given by the dashed lines in Fig. 2, depend on different angular averages from those for spin-lattice relaxation. These combined electron and nuclear spin transitions are induced by the non-secular terms involving S ± I ± with q = 3 and S ± I ∓ with q = 4 in Eqs. 8 and 9, respectively. The rates of the two distinct simultaneous electron and nuclear transitions are given by transition probabilities: Wx1 = W (∓ 12 , mI → ± 12 , mI ± 1) and Wx2 = W (∓ 12 , mI → ± 12 , mI ∓ 1) , as we see from Fig. 2. From Eqs. 1, 3, 8 and 16, we get:
τc 1 W x1 = (∆A) 2 (I ( I + 1) − m I (m I ± 1)) sin 4 θ 8 1 + ω e2τ c2 and from Eqs. 1, 3, 9 and 17:
10
(24)
W x2 =
τc 1 (∆A) 2 (I ( I + 1) − m I (m I ∓ 1) ) (3 cos 2 θ − 1) 2 72 1 + ω e2τ c2
(25)
where we can express the angular averages in terms of cos 2 θ z and cos 4 θ z as in Eqs. 16, 17. For a
15
( I = 12 ) , the numerical values in Eqs. 24, 25 are:
N-nitroxide
I ( I + 1) − m I (mI ± 1) = 1 , in all cases; and correspondingly for a
14
N-nitroxide ( I = 1) :
I ( I + 1) − m I ( m I ± 1) = 2 .
We determine cross relaxation rates with the same three models as used for spinlattice relaxation in Section 4. The lower panel in Fig. 3 shows the dependence of the mean cross-relaxation rate, Wx = 12 (Wx1 + Wx2 ) , on spin-label order parameter, for magnetic field parallel to the director. The combination of angular averages appropriate to the mean is sin 4 θ +
1 9
(3 cos 2 θ − 1) 2
(see Eqs. 24, 25). It is invariably the mean cross-relaxation rate
that enters into the rate which we get in saturation recovery experiments (16). Here it is normalized to Wx (0) , the mean rate for S zz = 0 , which for the Maier-Saupe potential and for wobbling within a cone comes from isotropic averages sin 4 θ =
8 15
and (3 cos 2 θ − 1) 2 =
4 5
.
6. Saturation-recovery EPR.
To first order, the rate constant determined in long-pulse saturation recovery experiments is (16): 1 TSR,e = 2We + 2W x
(26)
where We is the mean spin-lattice relaxation rate, and Wx is the mean Wx -rate for intermanifold cross-relaxations, averaged over all 2 I + 1 hyperfine manifolds, in both cases. This result is independent of which hyperfine manifold it is that is pumped and observed. Eq. 26
11
also applies to short-pulse saturation recovery, when nuclear relaxation is so fast as not to be resolved. We get the combined contributions of direct spin-lattice relaxation and cross relaxation 2We + 2Wx from Eqs. 18, 24, 25. Substituting from Eqs. 15−17 for the angular averages, the saturation recovery rate for a 15N-nitroxide ( I = 12 ) then becomes:
(
)
(
)
1 TSR,e = We, − 1 + We, + 1 + Wx1 + Wx2 = 2
2
2 2 5 (∆A) 2 + ∆gβ e B − 1 (∆A) 2 cos 2 θ − ∆gβ e B cos 4 θ o o 36 12
τ (27) c 2 2 1 + ωe τ c
and for a 14N-nitroxide ( I = 1) :
1 TSR,e = 23 (We, −1 + We, 0 + We, +1 ) + 43 (Wx1 + Wx2 ) = 2 2 10 (∆A) 2 + ∆gβ e B − 2 (∆A) 2 cos 2 θ − ∆gβ e B cos 4 θ o o 27 9
τ (28) c 1 + ωe2τ c2
Using Abragam’s spin-operator method, as in ref. 15, the general result for arbitrary nuclear spin I is:
1 TSR,e = 2We + 2Wx = 2 2 5 I ( I + 1)(∆A) 2 + ∆gβ e B − 1 I ( I + 1)(∆A) 2 cos 2 θ − ∆gβ e B cos 4 θ τ c o o 27 1 + ωe2τ c2 9 (29)
For isotropic rotation, where cos 2 θ z =
1 3
and cos 4 θ z = 15 , Eq. 29 corresponds with the
result obtained in ref. 15 (see also ref. 10). Note that the quantities calculated as W x1 and W x2 in ref. 15 are the sums of the cross-relaxation rates for a given value of mI , and not the single-transition probabilities as defined here by Eqs. 24 and 25. An interesting feature of Eqs. 27−29 is that the cos 4 θ term is contributed only by the field-dependent Zeeman anisotropy. Thus, at low operating frequencies (below 9 GHz), we expect the long-pulse recovery rate to depend almost linearly on order parameter. The lower part of Fig. 4 shows the dependence of saturation-recovery rate on order parameter for 12
the three motional models used already in Sections 4 and 5. In this case the normalised rates depend on both spin-Hamiltonian parameters and spectrometer frequency, because they do not depend solely on a single fixed anisotropy, as for spin-lattice relaxation or cross relaxation alone (cf. Eqs. 18, 24, 25). To allow for non-axiality of the tensors, we replace (∆g ) 2 and (∆A) 2 by (∆g ) 2 + 3(δg ) 2 = 3.379 × 10 −5 and (∆A) 2 + 3(δA) 2 = 7.454 mT2 (14.668
mT2 for 15N), respectively, where δg = 12 ( g xx − g yy ) and δA = 12 ( Axx − A yy ) (see refs. 21, 22). Here we use 14N-DOXYL tensors from refs. 23, 24, and correct for the difference in nuclear gyromagnetic ratio for
15
N-DOXYL. From Fig. 4, we see that uniaxial molecular ordering
can make up to a factor of two difference in saturation recovery rates at a microwave frequency of 9.4 GHz, for a fixed correlation time. The rate of saturation recovery decreases with increasing order parameter, for the magnetic field parallel to the director. This is because restricting the amplitude of angular fluctuation reduces its effectiveness as a relaxation process. Differences in normalised recovery rates between
14
N- and
15
N-nitroxides are
relatively small at an EPR frequency of 9.4 GHz. Also, the various motional models do not change the dependence of the rates on order parameter greatly.
7. Orientation dependence
So far, we have dealt with the special case (γ = 0) where the director axis for ordering lies along the static field. In general, however, the angular averages depend on the angle γ that the magnetic field makes with the director. Therefore, we now use Eq. 10 to obtain the averages cos 2 θ and cos 4 θ as a function of γ: cos 2 θ =
1 2
cos 2 θ z (3 cos 2 γ − 1) + 12 (1 − cos 2 γ )
cos 4 θ =
1 8
cos 4 θ z 3 − 30 cos 2 γ + 35 cos 4 γ − 34 cos 2 θ z 1 − 6 cos 2 γ + 5 cos 4 γ + 38 1 − cos 2 γ
(
)
(30)
(
) ( (31)
13
)
2
where cos φ z = 0 , cos 2 φ z = 12 , cos 3 φ z = 0 and cos 4 φ z = 83 , when averaging over 0 to 2π. By using Eq. 15−17, we then eventually get the dependence of the relaxation rate on magnetic field orientation. This is expressed in terms of the angular averages cos 2 θ z and cos 4 θ z that we get from one of the three models used above to obtain the relaxation rate
when the field lies along the director. It is clear from Eqs. 30 and 31 that the angular variation of the relaxation rate depends not only on cos 2 γ , but also on cos 4 γ . Such an angular dependence is found experimentally for the line widths (i.e., T2 -relaxation rates) of spin labels in ordered systems (17), at least for moderate degrees of order. If the magnetic field is oriented perpendicular to the director, Eqs. 30 and 31 reduce to: cos 2 θ =
1 2
(1 −
cos 2 θ z
) and
cos 4 θ =
3 8
( cos
4
)
θ z − 2 cos 2 θ z + 1 . The upper part of
Fig. 4 shows the dependence of saturation-recovery rate on order parameter for this γ = 90 o field orientation. We use the same three motional models as for the γ = 0 orientation in the lower part of the figure. Most notable is that the recovery rate increases with increasing order parameter, when the magnetic field is perpendicular to the director. The changes in rate are not as large, however, as when the magnetic field is parallel to the director. Again, differences in the normalised recovery rates of
14
N- and
15
N-nitroxides are not large at 9.4
GHz, and differences between the three motional models are not much greater than for the parallel field orientation. If saturation recovery measurements are made on the absorption maximum of a 9GHz powder pattern, contributions will come mainly from field orientations perpendicular to the director axis, in the mI = 0 manifold. Positions in the powder pattern are then specified by g ⊥
which comes from the equivalent of Eq. 13. We then expect the dependence of
14
recovery rate constant on order parameter that is contributed by modulation of the spinHamiltonian anisotropies to be approximated by the upper part of Fig. 4.
8. Discussion
An important general conclusion from this work is that electron spin-lattice relaxation and saturation recovery EPR of nitroxide spin labels in molecularly ordering environments depend on the orientation of the static magnetic field to the director. In contrast to the independence of SR-rates from hyperfine manifold (1; 16; 25), we expect a dependence on spectral position in powder patterns from nitroxides that are subject to orientational ordering. A pronounced dependence on field orientation is found for nitroxide line widths of ordered systems; (26; 27; 17). Saturation-recovery measurements, on the other hand, are usually performed on the most intense peak of the nitroxide absorption powder spectrum, and currently there is little information on the position dependence from ordered systems such as membranes. Figure 4, which is for the conventional microwave frequency of 9.4 GHz, predicts that the dependence of the saturation-recovery rate on order parameter is opposite for magnetic field oriented parallel or perpendicular to the director of ordering. For the perpendicular orientation, which best approximates the usual experimental protocol for non-aligned samples at 9 GHz, dependence of the recovery rate on order parameter is relatively modest. This justifies a posteriori the assumption that saturation recovery is dominated by the rotational correlation time, and that the degree of ordering is less important (e.g., 10; 5). For chainlabelled lipids in membranes with and without cholesterol, Mainali et al. (5) have shown that the dependence of the saturation-recovery rate at 9 GHz on nitroxide position n tracks with that of the rotational diffusion rate R⊥ , and is in the opposite direction to that of the order
15
parameter S zz . For a series of other membrane systems at 9 GHz, Mainali et al. (12; 13; 14) also find opposite dependences of 1 TSR,e and S zz on n. These latter findings are consistent with Fig. 4 for the γ = 90o orientation. As illustration, Table 1 lists saturation-recovery rates at 9 GHz of chain-labelled phosphatidylcholines in fluid bilayer membranes of dimyristoyl phosphatidylcholine containing 0 mol% or 50 mol% cholesterol. The spin-label order parameter S zz decreases with increasing position n down the hydrocarbon chain, and increases on adding cholesterol. Recovery rates are normalised to the reduced spectral density j (ω e ) = τ c (1 + ω e2τ c2 ), where
τ c is deduced from the rotational diffusion coefficient R⊥ that is obtained by simulating the spectral line shapes (5). The normalised rate should reflect directly the dependence on order parameter given in the upper part of Fig. 4 (cf. Eqs. 28−31). We see that
(1 T ) SR ,e
j (ω e )
remains approximately constant, or increases slightly with increasing order parameter (i.e., with decreasing n), as predicted in Fig. 4. However, it should be cautioned that spectra of chain-labelled lipids contain motional components in both fast and slow regimes of 9-GHz nitroxide EPR (28; 29). In particular, the single correlation times from line-shape simulations used in Table 1 are considerably longer than those used to interpret multi-frequency saturation recovery experiments, in the absence of line-shape analysis (10). At high field, corresponding to 94-GHz EPR, the slow-motion components approach the rigid limit, leaving only the fast-motion components that arise from chain rotational isomerism (30; 31; see also 32). For this case, the strongest absorption peak in a 94-GHz powder pattern occurs for the magnetic field directed perpendicular to the director axis, i.e., again the γ = 90 o orientation. Figure 5 shows the dependence of SR-rate on order parameter predicted for the high field of 94-GHz EPR. As expected, this differs considerably from the dependence at 9.4 GHz given in Fig. 4, because the cos 4 θ angular average makes a major 16
contribution at high values of Bo (see Eq. 29). From the lower panel of Fig. 5, we see that the recovery rate for the γ = 90o orientation now barely increases, or slowly decreases, with increasing order parameter, and then decreases strongly at higher S zz -values. Table 1 shows, in contrast, that the normalised experimental recovery rate
(1 T ) SR , e
j (ω e )
remains
approximately constant, or increases with increasing order parameter at 94 GHz, similar to the situation at 9 GHz. Measurements at 94 GHz are taken from Mainali et al. (11), and the single-component values of both R⊥ and order parameter differ from those at 9 GHz, which corresponds to differences in relative contributions of fast and slow motions at the two frequencies (cf. 30; 33). Complications may arise at high field because the frequency dependence of the recovery rate becomes anomalous. One expects the SR-rate to reach a constant value as the EPR frequency increases and modulation of the Zeeman anisotropy comes to dominate. Instead, the rate at 94 GHz increases from that at 35 GHz, which is not explained by current mechanisms (34; 35). As mentioned in the Introduction, we must also remember that mechanisms other than fluctuations in the electron-nuclear dipolar (END) and anisotropic Zeeman (i.e., g-value) interactions make a major contribution to recovery rates at long correlation times (10; 9). These contributions have a weaker dependence on correlation time than do Lorentzian spectral densities. Even so, deuteration of solvent and of the nitroxide shows that the extra contributions correspond neither to a suggested mechanism of solvent-proton spin diffusion, nor to that of methyl group rotation (9; 36; 37; 38; 1). Also, extrapolation from the temperature dependence of recovery rates in glassy solvents shows that the extra contributions cannot be accounted for entirely by solid-state mechanisms, although this would modify the dependence on correlation time of the remaining contributions (39; 38). Whatever the mechanism, it seems unlikely that these additional contributions will depend on the spin-label order parameter. 17
Finally, I consider briefly the situation for powder samples at other EPR frequencies. Below 9 GHz, e.g., at S-band and L-band, Zeeman anisotropy becomes negligible and the absorption maximum is in the centre of the spectrum, at the position specified by g o . This corresponds to the m I = 0 manifold from an unbiassed superposition of director-axes oriented at all angles to the magnetic field direction. The situation then approximates that of an isotropic distribution, and we expect little dependence of SR-rate on order parameter. For an EPR frequency of 34 GHz, intensity from the m I = +1 manifold piles up in the powder pattern at low field, where resonance positions are determined by g ⊥ and A⊥ (see Eq. 13). The absorption maximum corresponds here to the field oriented along all directions perpendicular to the director, with a close shoulder from the field parallel to the director that merges with the two other orientations as the order parameter decreases. Thus we anticipate a
γ = 90 o orientation at very high order that rapidly converts to an approximately isotropic situation with decreasing order parameter.
9. Conclusion
Fast motional theory predicts that the electron spin-lattice relaxation, and crossrelaxation, rates of nitroxide spin labels in ordering environments depend both on the orientation of the magnetic field and on the order parameter of the nitroxide z-axis. For an EPR frequency of 9 GHz, the rate of saturation recovery at the absorption maximum of the EPR powder pattern increases, but maximally by <20%, with increasing order parameter. Qualitatively this is in accord with experiment, and the common assumption that SR-rates are determined predominantly by rotational diffusion rates (i.e., correlation time). A stronger dependence on order parameter is predicted for the magnetic field oriented along the director axis of ordering. A markedly different dependence is predicted for high-field EPR at 94 GHz,
18
although this is not evident experimentally. Comparisons with experiment are complicated by the presence of slow-motional components and a background recovery rate additional to that from modulation of hyperfine and Zeeman anisotropy, and by the anomalous frequency dependence at 94 GHz. Nonetheless, many of the predictions support the view that SR-rates are determined primarily by the rate of rotational diffusion.
Acknowledgement
I thank Christian Griesinger and the Department of NMR-based structural biology for financial assistance.
19
Table 1. Rate constants 1 TSR,e for saturation recovery of n-DOXYL phosphatidylcholine spin labels in bilayer membranes of dimyristoyl phosphatidylcholine with and without 50 mol% cholesterol, at 27 oC.a Data from refs. 5, 11.
(1 T )
n (C-atom)
SR , e
0 mol% cholesterol
j (ω e ) (× 10 −18 s - 2 )
50 mol% cholesterol
9.2 GHz
94 GHz
9.2 GHz
94 GHz
5
2.49
195
3.56
249
7
2.45
174
3.85
196
10
2.50
173
2.27
162
12
2.30
163
2.21
164
14
2.34
160
2.14
159
16
1.80
a
1.62
Recovery rates are normalized to the reduced spectral density: j (ω e ) = τ c (1 + ω e2τ c2 ), where
the correlation time is related to the rotational diffusion R⊥ coefficient by τ c = 1 (6 R ⊥ ) .
20
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25
Figure Legends
Fig. 1. Orientation of a spin-labelled moiety in an ordering environment. N is the
director axis for ordering; z is the nitroxide principal axis; B o is the static magnetic field direction.
Fig. 2.
Energy levels and transitions for
15
N- ( S = 12 , I = 12 ; left )
and
14
N-
( S = 12 , I = 1; right ) nitroxides. Spin states are labeled by their electron and nuclear magnetic quantum numbers, M S and mI respectively. Transition probabilities per unit time for leaving a particular state are given by We,mI for spin-lattice relaxation (Eq. 18), and Wx1 ,
Wx2 for cross relaxation (Eqs. 24, 25).
Fig. 3. Top: Spin-lattice relaxation rate We = 1 2T1e from Eq. 18, as a function of order
parameter S zz ≡ P2 (cosθ z ) , where angular averages are determined by the Maier-Saupe orientation potential (solid line; Eqs. 19−21), or correspond to a fixed value of θ z (dashed line; Eq. 22) or restricted random walk within a cone (dotted line; Eq. 23). Bottom: mean cross-relaxation rate Wx = 12 (Wx1 + Wx2 ) for the same three motional models, according to Eqs. 24 and 25. Relaxation rates, We (S zz ) and Wx ( S zz ) , for the static magnetic field oriented parallel to the director are normalized to their values, We (0) and Wx (0) , for S zz = 0 .
Fig. 4. Saturation recovery rate WSR,e = 1 2TSR,e from Eq. 29 for long pulses at 9.4 GHz,
as a function of order parameter S zz ≡ P2 (cosθ z ) , where angular averages are determined by the Maier-Saupe orientation potential (solid line), or correspond to a fixed value of θ z 26
(dashed line) or restricted random walk within a cone (dotted line). Recovery rates WSR ,e ( S zz ) , with the static magnetic field oriented parallel (bottom; γ = 0 ) and perpendicular (top;
γ = 90 o in Eqs. 30, 31) to the director, are given for 14N-nitroxides (black lines, Eq. 28) and 15
N-nitroxides (grey lines, Eq. 27), and are normalized to their values WSR,e (0) for S zz = 0 .
Fig. 5. Saturation recovery rate WSR,e = 1 2TSR,e for long pulses at 94 GHz, as a function
of order parameter S zz ≡ P2 (cosθ z ) . Angular averages are from the Maier-Saupe orientation potential (solid line), or for a fixed value of θ z (dashed line) or restricted random walk within a cone (dotted line). Recovery rates WSR ,e (S zz ) , with magnetic field parallel (top
panel; γ = 0 o ) and perpendicular (bottom panel; γ = 90o ) to the director, for 14N-nitroxides (black lines) and 15N-nitroxides (grey lines), are normalized to WSR,e (0) for S zz = 0 .
27
Fig. 1.
28
Fig. 2.
29
1.4
We (Szz)/W e (0)
1.2 1.0
wobble
0.8
fixed
0.6
Maier-Saupe
0.4 0.2 0.0
W x(Szz )/ Wx (0)
0.9
Maier-Saupe
0.8 0.7 0.6
fixed
0.5 0.4
wobble 0.0
0.2
0.4
0.6
0.8
order parameter, Szz =
Fig. 3.
30
1.0
1.2
14
o
γ = 90 :
N
1.1
W SR,e(Szz )/WSR,e (0)
15
1.0 15
N
0.9 0.8
14
N
Maier-Saupe fixed wobble
N
0.7 0.6 0.5
γ = 0:
0.4 0.0
0.2
0.4
0.6
0.8
order parameter, Szz =
Fig. 4.
31
1.0
1.4 15
N
1.2 1.0 0.8 14
N
WSR,e(Szz)/WSR,e(0)
0.6 0.4 0.2
94 GHz, γ = 0o
0.0 Maier-Saupe fixed wobble
14
N
1.2 1.0 0.8 0.6
15
N
0.4 0.2
94 GHz, γ = 90o
0.0 0.0
0.2
0.4
0.6
0.8
order parameter, Szz =
Fig. 5.
32
1.0
Graphical abstract
33
Highlights to manuscript „Molecular Order and T1-relaxation, Cross-relaxation in Nitroxide Spin Labels” by Derek Marsh
•
Saturation recovery (SR)-rates of spin labels undergoing molecular ordering depend on field orientation
•
At 9 GHz, nitroxide SR-rates increase slowly with increasing order for field perpendicular to the ordering axis
•
At 9 GHz, SR-rates depend predominantly on rotational rate
•
At 94 GHz, SR-rates do not follow dependence on ordering predicted for fast motion
34