6724
J. Phys. Chem. 1991,95,6124-6133
*H NMR Reiaxatlon in Phospholipid Bilayers. Toward a Consistent Molecular
I nterpretation
Bertil HaUe
Physical Chemistry 1. University of Lund, Chemical Center, P.O. Box 124, S-22100 Lund, Sweden
(Received: January 29, 1991)
Recent 2H NMR relaxation data from phospholipid bilayers in vesicle suspensions and aligned lamellar samples are analyzed
in terms of an analytical model, which explicitly treats restricted rotational diffusion of phospholipid molecules and also
accounts for internal motions and for vesicle tumbling and phospholipid lateral diffusion. It is concluded that the vesicle
R, dispersion below 10 MHz reflects vesicle tumbling and lateral diffusion, while, above 10 MHz, restricted tumbling (wobbling)
of individual phospholipid molecules with respect to the bilayer normal is responsible for the frequency dependence. The
RI dispersion in vesicles and the anisotropy of RIZand R ~ in
Q aligned bilayers can be quantitatively accounted for by the
model with closely similar parameter values. A resolution of the longstanding linewidth controversy is also suggested: lateral
diffusion over the inner half of the vesicle bilayer produces a narrow Lorentzian, which dominates the apparent line width
and, hence, obviates the need to invoke a drastic curvature-induced reduction of orientational order.
1. Introduction
NMR techniques have been widely used to investigate orientational order and dynamics in phospholipid bilayers. 2H NMR
of specifically deuterated acyl chains, in particular, has proved
to be a powerful experimental
While 2H NMR has played
an important role as a probe of microscopic behavior in lipid
bilayers, a unique and quantitatively consistent picture of phospholipid dynamics has not yet emerged. This is due mainly to
the molecular complexity of the system and the consequent
problem of constructing sufficientlyrealistic, yet tractable, models
to account for the observed 2H spin relaxation behavior.
The various dynamic processes that have been invoked as
sources of spin relaxation of acyl chain deuterons in liquidcrystalline phospholipid bilayers may be broadly classified as (i)
internal motions, (ii) lipid-molecule reorientation, (iii) collective
director fluctuations, and, in the case of small vesicles, (iv) vesicle
tumbling and lipid lateral diffusion. As these classes of motion
span a wide range of time scales, spin relaxation in phospholipid
bilayers is expected to be highly dependent on the resonance
frequency. Here we shall be concerned mainly with 2H relaxation
rates measured in the frequency range 1-100 MHz. In particular,
we a~nsiderthe frequency dependence, in the range 2.5-61.4 MHz,
of the longitudinal relaxation rate Rl from small phospholipid
vesicles, recently reported by Brown et al.? and the orientation
dependence, at 30.7 MHz, of the relaxation rates R l Zand RjQ
from aligned bilayers, reported by Jarrell et ala8 The molecular
interpretation of the observed relaxation behavior was left as an
open question by the authors of these two important studies.
In a series of paper^^*^'^ Brown and co-workers have argued
that 2H (and 13C) relaxation in the megahertz frequency range
from phospholipid bilayers is caused mainly by collective (hydrodynamic) fluctuations in orientational order, a phenomenon
which was first studied in the context of thermotropic nematic
(1) Scelig, J. Q.Reu. Biophys. 1977, 10. 353.
(2) Griffin, R. G.Merh. Enzymol. 1981, 72, 108.
(3) Brown, M. F. J. Chem. Phys. 1982, 77, 1576.
(4) Davis, J. H. Biochim. Biophys. Acta 1983, 737, 117.
( 5 ) Bloom, M.; Smith, 1. C. P. In Progress in Prorein-Lipid Interactions;
Watts, A., De Pont, J. J. H. H. M., Eds.; Elsevier: Amsterdam, 1985; p 61.
(6) Davis, J. H.Chem. Phys. Lipids 1986. 40,223.
(7) Brown, M. F.;Salmon, A.; Henriksson, U.; SWerman, 0. Mol. Phys.
1990.69, 379.
( 8 ) Jarrell, H. C.;Smith, 1. C. P.; Jovall, P. A.; Mantsch, H. H.; Siminovitch. D. J. J . Chcm. Phvs. 1988. 88. 1260.
(9) Brown, M. F. J . Mign. Reson. 19%, 35, 203.
(IO) Brown, M. F.; Ribeiro, A. A,; Williams, G. D. Proc. Narl. Acad. Sei.
U.S.A. 1983. 80.4325.
(1 1) Brown, M. F. J . Chem. Phys. 1984.80, 2808.
(12) Williams, G.D.;Beach, J. M.; Dodd, S.W.; Brown, M. F. J . Am.
Chem. Soc. 1985, 107.6868.
(13) Brown, M. F.; Ellena. J. F.; Trindle, C.; Williams, G. D. J . Chem.
Phys. 1986, 84,465.
liquid ~rystals.'~This interpretation has, however, been challenged
on several grounds. First, the orientation dependence of the *H
relaxation rates from multilamellar dispersion~'~J~
and from
macroscopically aligned bilayer^^.'^*'^ is not consistent with a
collective order fluctuation mechanism. Second, field-cycling 'H
relaxation dispersion studiesIgof multilamellar dispersions reveal
an approximately linear frequency dependence of TI,
indicative
of a smectic-type collective order fluctuation mechanism,20but
only in the frequency range 1-100 kHz. In view of these experimental facts, it seems safe to dismiss collective order fluctuations as a significant source of 2H relaxation in the megahertz
frequency range.
A bewildering variety of dynamic models have been used to
describe the effect on spin relaxation of internal and overall
motions of phospholipids in bilayers. The internal motions, essentially trans-gauche rotational isomerization, have been modeled
as diffusion of kink rotamers along the acyl chains2' or as
large-angle jumps among a small number of discrete bond orientation~.'~.'~*'~,"~~
The reorientation of the entire phospholipid
molecule has also been treated as large-anglej u m p ~ , ' ~but,
J ~ more
commonly, has been described as a continuous, but restricted,
rotational diffusion p r o c e s ~ . ~ ~ - ~ ~
More detailed treatments of rotational isomerization dynamics
in liquid-crystalline phospholipid acyl chains were recently
presented by Pastor et alaBJ"and Ferrarini et ala3, These authors
(14) de Gcnnes, P.G. The Physics of. Liquid
Crystals; Clarendon Press:
.
Oxford, U.K., 1974.
(IS! Siminovitch, D. J.; Ruocco. M. J.; Olcjniczak, E. T.; Das Gupta, S.
K.: Griffin. R. G. Chem. Phvs. Lett. 1985. 119. 251.
(16! Siminovitch, D.J.; R u m , M. J.; Olejniczak, E. T.; Das Gupta, S.
K.; Griffin, R. G. Biophys. J . 1988, 51,373.
(17) Pop, J. M.; Walker, L.; Comell, B. A.; Separovic, F. Mol. Crysr. Li9,
Crysf. 1982, 89, 137.
(18) Maver. C.: Grabner.. G.:. Mliller.. K.:. Weisz.. K.:. Kothe. G. Chcm.
Phys. 'Lerr:1990,'165,
15s.
(19) Rommcl, E.;Noack, F.; Mcier, P.; Kothe, G. J . Phys. Chem. 1988,
92,298 1.
(20) Marqusee, J. A.; Warner, M.; Dill, K. A. J . Chsm. Phys. 1984,81,
6404.
(21) Kimmich, R.; Schnur, G.; Scheuermann, A. Chem. Phys. Upids 1963,
32, 271.
(22) Wittebort, R. J.; Szabo,A. J . Chrm. Phys. 1978. 69,1722.
(23) Huang. T. H.; Skarjune, R. P.; Wittebort. R. J.; Griffin, R. G.;
Oldfield, E. J. Am. Chem. Soc. 1980, 102,7377.
(24) Torchia, D.A.; Szabo,A. J . Magn. Reson. 1982, 49, 107.
(25) Mcier, P.;Ohmes, E.; Kothe, G. J. Chem. Phys. 1986, 85, 3598.
(26) Brainard, J. R.; Szabo,A. Biochemistry 1981, 20,4618.
(27) Fuson, M.M.; Prestegard, J. H. J . Am. Chem. Soc. 1983,/OS,168.
(28) Szabo,A. J . Chem. Phys. 1984,81,150.
(29) Pastor, R.W.; Venable, R. M.; Karplus, M. J . Chcm. Phys. 1988,89,
1112.
(30) Pastor, R. W.; Venable, R. M.; Karplus, M.; Szabo,A. J . Chem.
Phys. 1988, 89, 1128.
0022-365419112095-6124$02.50/0 0 1991 American Chemical Society
2H Relaxation in Lipid Bilayers
introduced a hybrid approach where the internal dynamics (in
a fixed phospholipid molecule) were obtained numerically from
Brownian dynamics (BD) simulationsB* or by solving the master
equation (ME) for transitions among conformational states?' while
the effect of overall lipid reorientation was described in terms of
a restricted rotational diffusion model in a single-exponential
a p p r o x i m a t i ~ n ~ involving
~ * ~ z ~ ~three adjustable parameters. The
elaborate modeling of internal dynamics in the BD and ME
approaches is particularly well suited for rationalizing the observed
dependence of Rl on acyl chain p ~ s i t i o n . ~ * ' ~ ' ~
Until recently, the experimental material on 2H relaxation in
phospholipid bilayers was rather limited and certainly not sufficient
to provide discriminating tests among alternative motional models
or even to uniquely determine all the parameters in reasonably
realistic models. With the recent reports of frequency-dependent
R , from unilamellar vesicles7and orientation-dependentRIZand
RIQfrom macroscopically aligned bilayers,8J8 the situation has
improved. In the following we address the question whether the
observed frequency and orientation dependence in the megahertz
range can both be accounted for by restricted rotational diffusion
of phospholipid molecules, or whether additional dynamic processes
have to be invoked.
In section I1 we derive the spectral density functions for a model
which describes phospholipid reorientation as rotational diffusion
of a symmetric top subject to a mean-field torque exerted by the
anisotropic bilayer environment. In the case of vesicles, we include
in our model also the effect of vesicle tumbling and lipid lateral
diffusion, processes which dominate the transverse relaxation rate
R2and also may contribute to R1 at the lower investigated 2H
frequencies. To encourage a wider application of the model, we
present in explicit analytical form all the relations needed to
calculate the 2Hrelaxation rates measured in isotropic (vesicle
suspensions) and macroscopically aligned bilayer systems. The
model is confronted with the experimental data7v8in section 111,
where we demonstrate that it does indeed account simultaneously
for the orientation dependence of R I Zand RIQfrom aligned
bilayers and the frequency dependence of R1 from vesicles.
Previously, the analysis of spin relaxation data from phospholipid
bilayers either has been limited to semiempirical curve fitting7*10*1a13
without explicit connection to a molecular description
or has invoked specific models to rationalize a more limited set
of experimental data. The ability of the present model to quantitatively account for a wider range of 2H NMR data suggests
that it correctly identifies and adequately describes those structural
and dynamic features of phospholipid bilayers that determine the
NMR observables.
11. Theory
A. Quadrupolar Hamiltonian and Motional Averaging. We
consider a system of spin -1 nuclei whose quadrupole moments
eQ are coupled to the molecular environment via the time-dependent electric field gradient (efg) components Vk(t). The
quadrupole coupling is described by the H a m i l t ~ n i a n(in
~ ~frequency units)
where AkLand VkLare spherical components, in the lab-fixed
frame, of the second-rank irreducible spin tensor operator and
efg tensor, respectively, defined as in ref 35. Here and in the
following, all efg components are reduced by VoF= (6'j2/2) V z / ,
where Vz/ is the major principal efg component (along the C d H
bond) in the F frame (vide infra). In (2.1) x is the (static)
quadrupole coupling constant, defined as x = eQVzzF/h.
(31) Ferrarini, A,; Nordio, P. L.; Moro, G. J.; Crepeau, R. H.; Freed, J.
H.J. Chem. Phys. 1989,91,5707.
(32) Moro, G.; Nordio, P. L. Chem. Phys. Lett. 1983, 96, 192.
(33) Ferrarini, A.; Moro, G.; Nordio, P. L. In The Molecular Dynumics
of Liquid Crysrals; Luckhurst, G . R., Ed.; D. Reidel: Dordrecht, in press.
(34) Abragam, A. The Principles of Nuclear Mugnetism; Clarendon Press:
Oxford, U.K.,1961.
(35) Halle, B.; Wennerstr6m, H. J. Chem. Phys. 1981, 75, 1928.
The Journal of Physical Chemistry, Vol. 95, No. 17, 1991 6725
TABLE I: Definitions of Reference Frames
frame
symbol
definition
laboratory
L
zL axis parallel to external magnetic field
director
D
zD axis parallel to (local) bilayer normal
molecule
M
principal frame for rotational diffusion tensor
internal
I
principal frame for internal order tensor
field gradient
F
principal frame for static efg tensor
Phospholipid
,
zn projection
---
Figure 1. The reference frames L, D, M, and I, and the relevant Euler
angles associated with the successive transformations L D M
I. (The principal frame F for the static efg tensor is not shown.) The
angles dMIand dl are fixed in the model, while the angles t$D, dDM, and
4Mflucutate in time as a result of restricted rotational diffusion of the
phospholipid molecule. In the case of vesicle suspensions, also dm is time
dependent.
For the motional model considered here, it is convenient to
transform the lab-frame efg components VkLto the F frame fixed
in the C-2H bond via the intermediate frames D and M defined
in Table I and illustrated in Figure 1. (An additional frame, I,
will be introduced later.) In terms of the second-rank Wigner
rotation matrix,36we thus have
VkL(t)= ( - l ) k C C C o z - k m [ Q L D ( t ) l o z , n [ n D M ( t ) l ~ [ Q M F ( t ) l
m n Q
(2.2)
The zero index on the last Wigner function is a consequence of
the essentially 3-fold symmetry of the efg on the deuteron in a
C2H bond.
The time dependence in (2.2) is due to (i) internal motions
(trans-gauche isomerization), modulating the Euler angles Q M F
between the C d H bond and the molecular frame M, (ii) restricted
rotational diffusion of the (conformationally averaged) phospholipid molecule, modulating the Euler angles QDM between the
bilayer normal (director) and the molecular frame M (in which
the rotational diffusion tensor is diagonal), and (iii) lateral diffusion of the phospholipid molecule over a curved bilayer and
reorientation of the entire bilayer, modulating the Euler angles
QLD between the external magnetic field and the director (cf.
Figure 1 and Table I). We assume that these three kinds of motion
are statistically independent and time-scale separatedM$' Further,
we consider only Larmor frequencies wo such that the time scale
(36) Brink, D. M.;Satchler, G. R. Angular Momentum, 2nd ed.; Clarendon Press: Oxford, U.K.,1968.
6726 The Journal of Physical Chemistry, Vol. 95, No. 17, 1991
of internal motions is short compared to l/oo. The internal
motions will then give an additive frequency-independent contribution to the relaxation rates (vide infra), while the lipid motion
will modulate the residual efg components
VkL(r) 5 ( VkL(r)) M F
(2.3)
= (-l)kCCo?-km[QLD(t)loZmn[QDM(r)l
(@O(QMF)) (2.4)
m n
In a macroscopically aligned bilayer system QLD is time-independent and one observes a quadrupolar line splitting, AUQ which,
in the high-field limit, is given by the secular (k = 0) part of the
quadrupolar Hamiltonian (2. l).34 In frequency units,
AVQ= 722%
( VOL) DM
(2.5)
For planar and spherical bilayers there is cylindrical symmetry
around the director on the time scale defined by the splitting. For
simplicity, we assume that also the conformationally averaged
phospholipid molecule is cylindrically symmetric (or that the zM
axis is, at least, a 3-fold axis). This is admittedly an approximation,
but without it the analysis of the relaxation behavior would become
unduly complicated. On account of these symmetries, only one
(m = n = 0) of the 25 terms in (2.4) survives the averaging in
(2.5) and we obtain
A ~ Q 72P2(cos ~LD)XSDMSMF
(2.6)
where S D M is the molecular order parameter with respect to the
bilayer normal (the zD axis)
SDM (P2(co~~ D M ) )
(2.7)
and S M F is the internal order parameter with respect to the lipid
long axis (the zMaxis)
SMF
= (P2(cos ~ M F ) )
(2.8)
It should be noted that what is commonly referred to simply as
the order parameter corresponds, within the approximations made
here, to the product of the molecular and internal order parameters, i.e.
= (p2(cos e D F ) )
(2.9)
B. Spectral Density Functions and Relaxation Rates. In the
motional narrowing regime, where the spin system evolves according to Redfield’s equation of motion,34the response of the
spin system to external magnetic perturbations (as in a spin relaxation experiment) is governed by the three labframe spectral
density functionsjkL(u),k = 0, 1, 2, which are cosine transforms
of the corresponding time autocorrelation functions, gkL(
t ) , of the
fluctuating part of the lab-frame efg components,
SDF
jkL(w)
SDMSMF
= Jmdf cos (ut) g k L ( t )
(2.10)
= ([VkL(0) - (VkL)l*[VkL(t) - (VkL)I) (2.11)
Note that the efg Components VkLin (2.1 1) have already been
averaged over internal motions.
Inserting VkL from (2.4) into (2.11) and making use of a
powerful symmetry theorem for time correlation functions of
irreducible tensor component^?^ we obtain for the case where QLD
is time-independent
gkL(r) =
[d2-km(eLD)12gmD(t)
(2.12)
gkL(t)
m
gmD(t)
=
(%(QMF)
)12gmnDM(t)
(2.13)
(2.14)
= (o2,’,[QDM(o)lofn[QDM(f)] ) - &$‘,&,DM2
It can readily be shown that g-mD(t)
= gmD(r)= [gmD(t)]*
and,
hence, that there are three distinct real-valued director-frame
correlation functions. Consequently, (2.10) and (2.1 2) yield for
the orientation-dependent lab-frame spectral density functions
cm2
1
= -(m2
14’12
cm4= -(35m4
1
- 2)
(2.17a)
- 155m2 + 72)
(2.17b)
12(7O1I2)
In a macroscopically aligned sample of planar bilayers, the
orientation-dependent relaxation rates RIZand Rlq, associated
with the dipolar (Zeeman) and quadrupolar magnetic polarizations
of a system of spin -1 nuclei, are given by4
RIZ(WO;eLD)
= Rint + 3/2r2X2blL(@O;eLD)
+ 4j2L(2WO;eLD)l
(2.18)
RIQ(WO;~LD)
= 78int
+ 9/2r2x2jlL(oo;e~~) (2.19)
The relaxation rate Rht, due to internal motions, will be regarded
as a model parameter. As indicated, it is assumed to be independent of bilayer orientation” and Larmor frequency (in the
megahertz range considered here).2’31
In curved bilayers, the orientation 0, of the local bilayer normal
fluctuates in time as a result of lipid lateral diffusion and bilayer
reorientation. Depending on the time scale of these fluctuations,
(i) the orientationdependentspectral densities may be isotropically
averaged (for motions on a time scale < 1 t 2 s), (ii) the residual
quadrupole coupling may be further averaged and may vanish for
an isotropic director distribution (<loa s), and (iii),8 modulation
may give direct contributions to the transverse and (low-frequency)
longitudinal relaxation rates (<lod s). In multilamellar dispersions, the average radius of curvature of the liposomes is
sufficiently large that only (i) is relevant.38 In suspensions of
small vesicles, however, vesicle tumbling and lipid lateral diffusion
take place on a time scale of microseconds3+” so that all three
effects occur. In this case, the longitudinal and transverse relaxation rates are given by
+
+
Rl(wo)= Rint 3/4r2x2[2jL(oo) 8jL(2w0)] (2.20a)
Rz(w0)
= Rint + 74/4*2X2[3jk(o)+ 5 k ~ ( W o ) 2jh(2Wo)l
(2.20b)
with the isotropic spectral density function
The second term in (2.21) is obtained by isotropically averaging
(2.19, making use of (2.16) and the orthogonality of the Legendre
polynomials. The first term in (2.21) is relevant only for small
vesicles and is readily shown to be of the form (assuming spherical
vesicle shape)
(2.22)
where S D F is the overall order parameter in (2.9) and 7”- is the
joint correlation time for vesicle tumbling and lipid lateral diffusion
(cf. section IVB). If the vesicle shape is nonspherical, the effects
of these two motions are no longer additive and cannot be described
by a single correlation time.37
C. Dinctor-Frme spectrpl lkmity Functioas. Returning now
to (2.1 3) and using a symmetry property of the Wigner functions,%
gmnDM(r)
2
jkL(WieLD)
=
m=O
(2 - 6mO)Fkm(eLD)jmD(u)
(2.15)
(37) Halle, B. J. Chem. Phys. 1991, 91, 3150.
(38) Brown, M. F.; Daivs, J. H. Chem. Phys. Lcrr. 1981, 79, 431.
(39) Stockton,G. W.; Polnaszek, C. F.; Tullcch, A. P.; Hasan, F.; Smith,
I. C . P. Biochemistry 1916, 5, 954.
(40)Bloom, M.; Bumell, E. E.; MacKay, A. L.; Nichol, C. P.; Valic, M.
I.; Weeks, G. Biochemistry 1978. 17, 5150.
(41) Parmar, Y. I.; Waasall, S. R.; Cushley, R. J. J . Am. Chem. Soc. 1984,
106, 2434.
2H Relaxation in Lipid Bilayers
The Journal of Physical Chemistry, Vol. 95, No. 17, 1991 6727
we obtain for the three director-frame spectral density functions
It is seen that the rotational spectral density functionsjmnDM(u)
are weighted by factors that are related to the internal order tensor.
Denoting the components of the Cartesian internal order tensor
by Sa,,we have42
I(%(QMF))I
I(%(QMF))I
= S,,
E
SMF
2
6i/2[S"'2+ S,,211"
(2.24a)
(2.24b)
jmnDM(u)= gmnDM(o).7mnDM(w)
(2.29)
The mean-square fluctuations gmDM(0)
are fully determined by
the equilibrium orientational distributionj(8DM),while the reduced
spectral density functions jmDM(u)
depend on the details of the
reorientation process. Using (2.14) and a contraction formula
for products of Wigner functions,36and expressing the resulting
Clebsch-Gordan coefficients in algebraic form, we obtain
4
gmnDM(o)=
YS + CC~,C$%A- ~ o ~ ~ o ( S D(2.30)
M)~
k=l
where the coefficients are given by (2.17) and
(2.31a)
If the internal order tensor is diagonal in the molecular frame M
(in which the rotational diffusion tensor is diagonal), then clearly
l(#o(QMF))l
= 0. If, furthermore, the zM axis is a 3-fold (or
higher) axis of symmetry for the internal order tensor, then also
l(&(QMF))l
= 0. If this is the case, only tumbling of the lipid
long axis with respect to the bilayer normal will contribute to
jmD(w), since motion around the molecular axis only affects rotational modes with n # 0 (vide infra).
For the purpose of interpreting the quantities l(&,(QyF))l in
terms of molecular structure, it is helpful to diagonalize the internal
order tensor (and, simultaneously, the residual efg tensor) and
to express the I(l&(QMF))l in terms of the time-independent Euler
angles QMI for the transformation from the M frame to the
principal frame I of the internal order tensor (cf. Figure 1 and
Table I) and the two quantities SIF
and vi that characterize this
(traceless) tensor in its principal frame. Letting a = sin2OMI and
fi = cos (2&), we obtain
&F2[(1 - Xa)'
I(&(QMF))12
+ ~ ( -172a)fi~i+ 1/4a2B2ti21
(2.25a)
I(D~O(QMF))I~ =
SIF2[3/2a(
1
- a) - a( 1 - CY)&I + h(a - a2B2)v~2]
(2.25b)
(2.3 1 b)
The kth-rank molecular order parameters Sgh in (2.30) are defined as
= (pk(cos ODM))
(2.32)
When no ambiguity can arise, we omit the rank superscript on
the second-rank order parameter, i.e., S D M =
Next we consider the reduced spectral density functions
7mDM(u)
for the model of rotational diffusion of a symmetric top
in an odd uniaxial potential of mean torque w(ODM), which tends
to align the phospholipid molecule along the bilayer normal with
the polar head group at the aqueous interface. The rotational
diffusion tensor is diagonal in the M frame (fixed in the conformationally averaged molecule) with distinct components D,
and D,, referring respectively to the spinning motion of the
phospholipid around its long axis and to the restricted tumbling
(wobbling) motion of this axis relative to the bilayer normal.
While this rotational diffusion problem can be solved numerically to desired a c c u r a ~ y ,we
~ ~shall
+ ~ ~adopt a Lorentzian approximation
a2h.
(2.33)
I(%(QMF))12
S I F ~ [-t%
!ha(]
~ ~- Xa)fivi + 1/6(1 - a + f/4a2b2)v?1
(2.25~)
where the effective correlation times,
T,
are given by
28*32.33
Here we have defined the principal internal order parameter
SIF = (p2(cOs @IF))
(2.26)
and the asymmetry parameter for the diagonal internal order
tensor (and for the residual efg tensor)
v1
= %(sin2 OIF cos (2$iF))/SIF
(2.27)
-
As defined, is restricted to the range [0, I]. Further, it is seen
that the relations (2.25) are invariant with respect to the transformations OMI
180 - OM1 and
180 - $,. Hence, these
angles cannot be uniquely determined. Since the I-frame description replaces the three parameters l(d,(QMF)by
)Ithe four
parameters SI^, vI, OMI, and $I, it is redundant and, hence, not
suitable for fitting purposes. For a spherical top (Dll = D J the
spectral density functions jmnDM(u)
are independent of the molecular-frame projection index n, whence (2.23) and (2.25) reduce
to the expected result
-
jmD(w)
= sIF2(
+ )/3oI2)jmODM(w)
(2.28)
It now remains to specify the spectral density functions
(2.14), which describe
the restricted rotational diffusion of the phospholipid molecule
with respect to the bilayer normal. It is convenient to first make
the decomposition
jmDM(o),defined as the cosine transform of
(42) Zannoni, C. In The Molecular Dynamics ofLiquid Crystals; Luckhurst. G. R., Ed.; D. Reidel: Dordrecht, in press.
The quantities required to calculate T,,, from (2.34) are defined
in (2.17) and (2.30)-(2.32). The symmetry of the model implies
thatg3T+,, = T ~so ,there are 13 distinct reorientational modes.
The approximation embodied in (2.33) and (2.34) may be
regarded alternatively, and equivalently, as a short-time approximation, a mean relaxation time approximation, or a firstorder perturbation approximation. In Szabo's approach2* the
approximate single-exponential correlation functions gmDM(t)are
forced to agree with the exact ones at short times (to linear order
in time). It is readily shown that this is equivalent to replacing
the exact weighted sum of exponentials with a single exponential
with a decay rate defined as the corresponding weighted average
of the individual decay rates. In the approach of Moro and
N o r d i ~ , 'the
~ (symmetrized) rotational diffusion operator is
evaluated in a (nonorthogonal) basis, wherein it is diagonal in the
limits of vanishing order (Sf$!,
= 0) and complete order
=
1). In general, the diagonal elements in this representation yield
(inverse) effective correlation times corresponding to a first-order
perturbation treatment. Remarkably, the resulting expressions
(2.34) are identical with those obtained from the short-time expansion.28 Under the conditions of interest here (SDM > 0.6 and
Dll >> Dl),the approximation (2.34) is expected to be highly
accurate for all modes except the [mn)= (1 1) spinning modeSg2J3
Halle
6728 The Journal of Physical Chemtstry, Vol. 95, No. 17, 1991
8o
200
t
c
4
-.
v1
d
100
0
10
1
0
100
Lannor frequency, vo / MHz
Figure 2. Frequency dependence of 2H longitudinal relaxation rate R ,
in vesicles of 1,2-DMPC-3’,3’-d2at 30 O C . The experimental data were
obtained by Brown et al.’ The solid and dashed curves passing through
the data points resulted from fits A and B, respectively, with parameter
values accordingto Table 111. The other three dashed curves refer to the
individual contributionsto RI from the three types of motion in the model
used for fit A: vesicle tumbling and lipid lateral diffusion (ves), phospholipid restricted rotation (rot), and internal motions (int).
For this mode we shall therefore use the second-order approximation’ZJ3
where T ~ , Ois obtained from (2.34), and
10
+ 5S& + 2Sgh)’
+ 7SgL + 5Sfh + 28Sgh + 16S&)
21 (3S&
5(2
+ 3Sgh + S&)(14
(2.35d)
The approximation scheme defined by (2.33)-(2.35) applies
to any uniaxial potential of mean torque (pmt). Further, within
this approximation, the actual shape of the pmt does not appear
explicitly and only affects the interrelations among the four lipid
order parameters SEA. In the high-order regime, these interrelations should not be sensitive to the detailed shape of the pmt.%
We adopt a simple dipolar form, consistent with the head-to-tail
asymmetry in each half of the lipid bilayer (cf. section IVA), Le.,
W(6DM) -A cos 6DM
(2.36)
The order parameters are then obtained as42
1,d t ~
I
Sgkh =
texp(At)
)
(2.37)
J;dS exp(AS)
with C = cos 6DM. Since SDMS& is directly related to the
quadrupolar splitting [cf. (2.6)], we choose this quantity (rather
than the coupling parameter A) as the independent parameter.
The remaining three order parameters are then fully determined
by (2.37) once S D M is specified.
111. Analysis of Experimental h t a
We now use the relaxation model developed in section I1 to
analyze *H relaxation data for 1,2-dimyristoyl-sn-glycer0-3phosphocholine (DMPC),selectively deuterated at acyl chain
position 3’ or 4’, in liquid-crystalline bilayers at 30 OC. The data
60 70
80
90
OLD I &g
Figure 3. Orientation dependence at 30.7 MHz of 2H relaxation rates
R I Zand Rlq in aligned bilayers of 2-DMPC-4’,4’-d2 at 30 O C . The
experimental data were obtained by Jarrell et aL8 The measurement
5 s-I. The curves resulted from fit C, with
uncertainty is roughly k
parameter values according to Table 111. The dashed line shows the
contribution to R I Zfrom internal motions.
comprise the relaxation dispersion Rl(vo)in the Larmor frequency
range 2.5-6.4 MHz from vesicle bilayers,’ shown in Figure 2, and
the relaxation anisotropy RIz(BLD)and Rlq(BLD)from aligned
bilayers,” shown in Figure 3.
A. Fitting Procedure. The calculation scheme used to obtain
the relaxation rates may be decomposed into four steps as follows.
1. With assumed values for the lipid order parameter S D M and
rotational diffusion coefficients Dlland D,, the 13 spectral density
functions jmaDM(w)are evaluated at w = wo, 2w0, using
(2.29)-(2.37).
2. The three variables related to the internal order tensor are
chosen as the order parameter SMF, defined by (2.8), and the
quantities u1and u2, defined as
un =
t =
20 30 40 50
I(D~(QMF))I~/SMF~
(3.1)
The internal order parameter SMF
is obtained from the assumed
and the measured quadrupolar ~plitting:~which yields the
overall order parameter S D F = SDMSMF[cf. (2.6) and (2.9)]. The
parameters u1and u2 are either fitted or assigned values consistent
with theoretical result^.^^^^^ As noted in section IIC, uI is related
to the orientation of the principal axes system for the internal order
tensor, while u2.essentially is a measure of the deviation of this
tensor from C, symmetry with respect to the molecular zM axis.
3. The three director-frame spectral densitiesjmD(w)at w =
wo, 2w0 can now be obtained from (2.23).
4a. In the case of aligned bilayers, the lab-frame spectral
densitiesjkL(kwo;BLD)
are obtained from (2.15)-(2.17). With an
assumed value for the internal-motion relaxation rate Rint,the
anisotropic relaxation rates &(WO$LD)
and RIQ(~O;BLD)
are then
obtained from (2.18) and (2.19).
4b. In the case of vesicles, the longitudinal relaxation rate
Rl(wo)is obtained from (2.20)-(2.22). To calculate the contribution jk(w)from vesicle tumbling and lipid lateral diffusion,
we need also the correlation time T ” ~ . As shown below, this
parameter can be eliminated by invoking the measured line width
and using (2.20b).
By including all the available experimental data in the analysis,
we can thus reduce the number of adjustable model parameters
to six: S D M , Dll, D,, uI, u2,and Rint. Ideally, the values of these
parameters would be determined by simultaneously fitting the
relaxation dispersion (Figure 2) and the relaxation anisotropy
(Figure 3). However, since the two data sets refer to slightly
different systems, we have chosen to perform separate fits. The
theoretical model will then be considered adequate, provided that
the parameter values derived from the two fits do not differ more
SDM
(43) Oldfield. E.;Meadows, M.; Rice,
17, 2721.
D.;Jacobs, R.Biochemistry 1978,
The Journal of Physical Chemistry, Vol. 95, No. 17, 1991 6729
2H Relaxation in Lipid Bilayers
TABLE II: Tbe Intend Order Tensor
method
B P
MEb
all-trans
position
3’
4’
3’
4‘
Shld
-0.28
-0.30
-0.27
-0.29
-0.5
fJl*
g2*
0.016
0.015
0.01
<0.01
0.25
0.29
0.51
0.66
1.5
0
a Brownian
dynamics simulations by Pastor et al.*9930 bMaster
equation calculations by Ferrarini et al.” CQuotedresults are averages
for the two methylene deuterons. Internal order parameter, defined
by (2.8). These results were obtained with mean-field parameters that
if SDM= 0.7. ‘These
reproduce the experimental SDF= SDMSMF
quantities are defined by (3.1).
than can be justified by the minor system differences. One obvious
difference is the higher bilayer curvature in the vesicles, which
may affect all model parameters to some extent (cf. section IVB).
Further, there are differences in water content and in labeling
site. In the relaxation dispersion study’ the DMPC was deuterated
at the 3’-positions in both acyl chains, whereas in the relaxation
anisotropy study* the DMPC was deuterated at the 4’-position
in the sn-2 chain.
In all calculations we use for the static quadrupole coupling
constant the commonly adopted value x = 170 kHz. The overall
order arameter was deduced from quadrupolar splittings meas u r d on multilamellar dispersions (excess water) of DMPC at
30 OC: SDF = 0.22 and 0.23, respectively, for the 3’- and 4’positions in the sn-2 chain. We use the former value for the
vesicles, neglecting any curvature effect (cf. section IVB), and
the latter value for the aligned bilayers.
Although the combined relaxation dispersion and anisotropy
data would seem to provide a fairly stringent test of the model,
it would be asking too much to hope that blind fitting could
uniquely determine the values of all six model parameters.
However, independent sources of information are available and
can be used to narrow down the relevant part of the six-dimensional parameter space. The internal motions, in particular, have
recently been investigated theoretically by using realistic mode l ~ The
. ~results
~ of
~ these
~ studies could in principle be used
to fix four of our six model parameters: Rint,uI, u2, and SDM
(=SDF/SMF, with SD, obtained from the quadrupolar splitting and
SMF from the theoretical studies). However, since both the BDBsM
and the ME” calculations rely on somewhat uncertain empirical
rate parameters to fix the time scale, their a priori results for Rint
are not very precise. The same is true for the internal order
parameter SMF, which depends significantly on the somewhat
uncertain mean-field parameters in the single-chain models.
However, the quantities uI and u2, being defined as ratios of
components of the internal order tensor, should be less sensitive
to the mean-field parameters. The theoretical results for the 3’and 4’-positions in the acyl chain are collected in Table 11. As
a reference, the results for a rigid all-trans chain (for which tIMI
= 90° and qI = 0) are also given. It is seen that both calculations
predict that uI is close to zero; Le., the principal frames of the
rotational diffusion tensor and the internal order tensor are nearly
coincident. (However, this does not necessarily imply that the
principal axes zMand zIcoincide. In fact, they are probably nearly
orthogonal.) In the fits, we therefore freeze this parameter to
ul = 0. Further, it will be seen that acceptable tits are obtained
only if the spinning lipid motion is in the extreme-narrowinglimit.
Consequently, the parameters u2 and Dll appear in the theory
essentially as the combination u2/DII.To estimate DIl, we shall
therefore make use of the theoretical results for u2 given in Table
,!
11.
6. Relaxation Dispersion. It is immediately apparent from
the data in Figure 2 that at least two distinct motional correlation
times are required to explain the RI dispersion in the 1-100-MHz
range. Within the framework of our model, these motions might
be the spinning and tumbling rotational motions of the lipid
molecule. However, the low-frequency part of the dispersion might
instead reflect vesicle tumbling and lipid lateral diffusion. This
issue can be settled by appealing to the transverse relaxation rate
TABLE III: Parameter Vahne Derived from Fits to % R e h x a t h
Data from DMPC Bilayers at 30 O C O
parameter
fit A
fit B
fit c
Rint, s-l
(27.4)
25.8
21.4
SDM
0.7 1
0.90
0.69
D,. 1O*s-I
(7.5)
2.4
(7.5)
D,, 107 s-I
2.0
0.064
6.6
0.35
[OI
VI
[OI
0.27
(0.3)
02
(0.3)
“Fit A: RI dispersion with vesicle tumbling (Figure 2, solid curve).
Fit B: R I dispersion without vesicle tumbling (Figure 2, dashed curve).
Fit C: RIZand R anisotropy (Figure 3). The nonlinear least-squares
fits were perform3 according to the Levenberg-Marquardt method.’5
Numbers within parentheses are not uniquely determined by the fit (cf.
text). Square brackets indicate frozen parameter.
TABLE I V Spectral Densities for Restricted Rotrtiod Diffwiua
with S
, D .r ,and D,- from Fit A
K,D*(vo)/nsa
m
n
gnDM(0) 7,/ns
2.5 MHz
30.7 MHz
0.48
0
0
0.066
2.84
0.92
1.31
0.189
4.77
4.43
1
0
0.18
2
0
0.025
2.56
0.32
1.01
0.89
0.189
1.07
0
1
4.04
3.34
1
1
0.618
1.31
0.7 1
2
1
0.156
1.04
0.8 1
0.15
0.14
0.033
0.91
1
-1
0.003
0.01
0.79
0.01
2
-1
0.04
0
2
0.025
0.30
0.04
0.25
0.156
0.32
1
2
0.25
0.816
1.35
0.33
1.36
2
2
<0.01
1
-2
0.003
0.29
<0.01
<0.001
<0.01
0.28
<0.01
2
-2
+
‘ K , , , , , ~ ~ (=VjmDM(uo)
,,)
4jmDM(2u0).
R2. For the same vesicle sample as was used to record the RI
dispersion in Figure 2, the measured line width at 30 O C yields
R2 = 2435 s-’ at 61.4 M H Z . ~The large magnitude of R2 implies
that, at this frequency, (i) wo rw >> 1, and (ii) the relative contributions to R2 from internal motions and restricted rotational
diffusion are entirely negligible. It then follows from (2.20b) and
(2.22) that
(3.2)
With SDF = 0.22, we thus obtain ryq = 0.39 ps. Using (2.20a)
and (2.22), we can now calculate the contribution to RI from
vesicle tumbling and lipid lateral diffusion. At the lowest experimental frequency, 2.5 MHz, we find RIw = 85 s-l, We thus
conclude that the R1dispersion below ca. 10 MHz is essentially
due to vesicle tumbling and lipid lateral dsffusion. It should be
noted that since this contribution is fully determined by the
measured quadrupolar splitting (which yields SDF) and the
measured line width (which yields SDF2rvr), it can be accounted
for without invoking additional adjustable parameters.
The result of fitting the five model parameters Rint,SDM,D,,
D,, and u2 to the R1dispersion data is shown as the solid curve
in Figure 2. The resulting parameter values are given in Table
111 (fit A). In Figure 2 we have also indicated the individual
contributions to RI from the three types of motion in our model.
Vesicle tumbling and lipid lateral diffusion are seen to be responsible for the dispersion below ca. 10 MHz, while phospholipid
reorientation accounts for the frequency dependence in the range
10-100 MHz.
In order to analyze the phospholipid reorientation in more detail,
we give in Table IV the fluctuation amplitudes gmDM(0),correlation times r,,,,,,and spectral density contributions (at two Larmor
frequencies) for each of the 13 rotational modes (mn). The
large-amplitude modes (1 1) and (22)are seen to be virtually pure
spinning modes, Le., r,,,,= (n2D,)-*.Now, according to (2.23)
(44) SWerman, O., personal communication.
Halle
6730 The Journal of Physical Chemistry, Vol. 95, No. 17, 1991
and (3.1). the relative contributions to R I from the (m f 1) and
{m f 2} modes are obtained by multiplying the corresponding
quantities K , ~ ~ ( V ~from
)
Table IV by ul and u2,respectively. As
uI = 0, the Im f 1) modes, most notably the 111) spinning mode,
do not contribute. Further, among the three tumbling modes (mol,
which only involve D,, the (IO)mode is dominant.
We can thus conclude that the R I dispersion in the 10-100M H z range is due to restricted tumbling of the phospholipid
molecule with respect to the bilayer normal. With a tumbling
diffusion coefficient of D, = 2 X IO7 s-I and a mean torque
corresponding to a lipid order parameter S D M = 0.71, the correlation time for the dominant {IO)mode is ca. 5 ns. The {m f
2) modes, dominated by the pure spinning mode (221, all have
correlation times around (4DlI)-l = 0.3 ns and, hence, are in the
extreme-narrowing limit even at the highest experimental frequency. Consequently, the contributions to R , from axial
phospholipid diffusion and from trans-gauche isomerization cannot
be distinguished by the data. In fact, for the given values of SDM
and D,, virtually identical fits can be obtained for any combination
of values for the remaining three parameters that satisfy the
relation RpI-+ 1.985 X I O i o u z / ~ l=l 35.3 s-l, provided that Dll
> 5 X 10 s I so that the spinning motion is in the extremenarrowing limit.
We note that, if vesicle tumbling and lipid lateral diffusion do
not contribute to R1in the investigated frequency range, then the
present model cannot produce an acceptable fit to the R , dispersion
with uI = 0. However, if ul is allowed to vary, an excellent fit
can be obtained, as shown by the dashed curve in Figure 2. The
resulting parameter values (fit B in Table 111) are distinctly
different from those derived from fit A. The low-frequency part
of the dispersion is now due to very slow phospholipid tumbling
in a strongly orienting pmt (SDM = 0.90),while the high-frequency
dispersion reflects the spinning motion. However, the picture
emerging from this fit is problematic in several respects. First,
the small value D, = 6.4 X IO5 s-I for the tumbling diffusion
coefficient is not easily accommodated within the present model
of noncollective lipid reorientation (cf. section IVC). Further,
the value uI = 0.35, implying that the internal order tensor does
not a t all conform to the molecular symmetry, exceeds the theoretical r e s u l t ~by
~ ,more
~ ~ than an order of magnitude. Finally,
it is not possible to obtain a satisfactory fit to the R I Zand RiQ
anisotropies with parameter values resembling those derived from
fit B. In view of all this, we conclude that the R1 dispersion, by
itself, provides independent support for our previous conclusion,
based on the line width, that vesicle tumbling and lipid lateral
diffusion are responsible for the low-frequency dispersion.
C. Relaxation Anisotropy. We consider now the orientation
dependence of R i Zand R i p As seen from Figure 3, an excellent
fit is obtained with parameter values (fit C in Table 111) that do
not differ much from those derived from the R 1 dispersion (fit
A). It should be noted that, in using our model to analyze orientation-dependent relaxation rates, we assume that the contribution RhI from internal motions is orientation independent. This
approximation is justified by the results of the recent detailed
theoretical studies of the time correlation functions describing
trans-gauche isomerization dynamic^.^'^^
As in fit A, the dominant rotation modes are the {IO)tumbling
mode and the 122)spinning mode, the former being an order of
magnitude slower than the latter. Using (2.15) and (2.23) together
with the parameter values derived in fit C, we may approximate
the ~ L Ddependence of the relaxation rates (at 30.7 MHz) as
RId6LD) = Rinc
+ 30Fli(@LD) + 100F12(6LD)
+ ~OFZZ(~LD)
(3.3a)
R I Q ( @ L=D YSRint
)
+ 85FIl(@LD)
+ 30Fl2(6LD)
(3.3b)
The contributions from terms involving FIO(~LD)
and FZo(6LD) are
small and have been neglected in (3.3). The numerical factors
multiplying the angular functions in (3.3) depend on the parameters SDM,
DI,,
D,, and 6 2 . Although the spinning motion is in
the extreme-narrowing limit, we cannot, as in the fit to the R ,
dispersion, compensate a change in DIlor uz by adjusting Rint,since
0'5
0.4
0.0
3
'
0
10
20
30
40
50
60
70
80
90
OLD I deg
Figure 4. Angular functions determining the anisotropy of R , , and Rip
in aligned bilayers, calculated according to (2.16) and (2.17).
the spinning motion, but not the internal motions, gives rise to
dependence. However, as in fit A, Dlland u2 cannot be
a
individually determined; we only obtain their ratio and the information that Dll> 5 X IO8 s-].
As shown in Figure 4, the three angular functions in (3.3) have
qualitatively different 6LD dependences. Whereas Fi2and FZ2,
respectively, increase and decrease monotonically, FII exhibits a
minimum a t 6 , =
~ 52.24'. With the aid of (3.3) and Figure 4,
the observed relaxation anisotropy can now be rationalized. RlQ
is dominated by FII,with some admixture of Flz,and should thus
display a minimum at eLD= 50'. For R i Zthe situation is more
complex, with substantial contributions from all three angular
functions. The parameter values determining the numerical
coefficients in (3.3) are such that these three angular functions
are combined in just the right proportions to cancel the ~ L deD
pendence in R I Z .(An even more striking cancellation effect of
this nature was recently observed in a 23Na relaxation study of
a reversed hexagonal lyotropic mesophase.&) The effect of
changing the lipid order parameter S D M is essentially a uniform
scaling of all 6LD-dependent terms in R I zand R I Q Hence, if SDM
is increased, the eLD-independentcontribution from internal
motions becomes relatively more important, but the shape of the
anisotropy is qualitatively the same. If the dynamic parameters
D, and D,1/u2are changed, then both the shape of the anisotropy
and the relative magnitude of R I Zand RlQ are affected.
IV. Discussion
A. The Restricted Rotational Diffusion Model. Restricted
rotational diffusion models have been widely used to describe the
effect of phospholipid motion in bilayers on observables related
to time-dependent interaction tensors as measured, e. in spin
relaxation'8,25-27~30~3i
or fluorescence depolarizati~n~**~
48 experiments. Exact results for the relevant time correlation functions
have been obtained for both
and discontinu0us51*s2
(the so-called cone model) potentials of mean torque (pmt).
For the relatively high degrees of orientational order in lipid
bilayers, sufficient accuracy is obtained with single-exponential
approximations to the correlation functions. Such approximations,
valid for an arbitrary shape of the pmt, have been derived by
several a ~ t h o r s . ~Since
, ~ u ~these approximations can be expressed
in simple analytical form, there is now little incentive for using
less accurate approximations.
fV
(45) Press, W. H.; Flannery, B. P.; Teukolsky, S.A,; Vetterling, W. T .
Numrricul Recipes; Cambridge University Press: Cambridge, U.K.,1986.
(46) Furb, I.; Halle, B.; Quist, P.-0.; Wong, T. C. J . Phys. Chem. 1990,
94., 2600.
(47) Kinosita, K.; Kawato. S.; Ikegami, A. Biophys. J . 1977, 20, 289.
(48) Lipari, G.; Szabo, A. Biophys. J . 1980, 30, 489.
(49) Polnaszek, C. F.; Bruno, G. V.;Freed. J. H.J . Chem. Phys. 1973.58,
3185.
(50) Vold, R. R.;Vold, R. L. J . Chem. Phys. 1988, 88, 1443.
(51) Wang, C. C.; Pecora, R. J . Chem. Phys. 1980, 72, 5333.
(52) Kumar, A . J . Chem. Phys. 1989. PI, 1232.
The Journal of Physical Chemistry, Vol. 95, No. 17, 1991 6731
2H Relaxation in Lipid Bilayers
In a recent contrib~tion:~Brown and SMerman (hereafter
referred to as BS) consider the restricted rotational diffusion model
in the context of the orientational (in)dependence of RIZin aligned
bilayers.8 Since we use the same basic model, it is appropriate
to identify and resolve certain discrepancies between our results
and those of BS. While claiming to investigate a model of
“anisotropic rotational diffusion in an ordering potential”, BS
entirely neglect the effect of the ordering potential on the rotational
dynamics; i.e., they set all SgL = 0 in (2.34). As a result, they
underestimate the spinning-mode correlation times and overestimate the tumbling-mode correlation times. (For S D M = 0.7 and
D, = D,, as used in the fits by BS, a consistent calculation of
the correlation time ratio for the dominant modes, T ~ ~ / yields
T ~ ~
7.8 as compared to 1 according to BS.)
In describing the effect of the ordering potential on the equiBS retain only even-rank order palibrium averages gmnDM(0),
rameters “since the bilayer is symmetric upon reflection through
its midplane”. However, on the NMR time scale, a phospholipid
molecule experiences only one side of the bilayer. Since the
phosphocholine head group is constrained to the aqueous interface,
the pmt must have polar character, as in (2.36), and the odd-rank
order parameters must be retained. This complication appears
to have been ignored in several recent
of spin
relaxation in lipid bilayers. For a typical (second rank) lipid order
parameter S D M = 0.7, the difference between the odd pmt - X
cos ODM and the even pmt - X cos2 6DM is not large as far as the
correlation times T,,,,, are concerned. However, the mean-square
fluctuations gmnDM(0)
for nonzero m and n are roughly twice as
large for the odd as compared to the even pmt. Another inconsistency in the BS treatment is the assumed equalit of the second-rank and fourth-rank order parameters. For SbL
Y = 0.7, the
odd and even pmt’s referred to above yield
= 0.31 and 0.35,
respectively.
A further difference between our derivation and that of BS
concerns the quantities I ( d d ( s 2 M ~ ) ) I , related to the internal order
tensor. For reasons of “simplicity”, BS neglect all terms in (2.25)
linear in ql. However, even the quadratic terms retained by BS
are incorrect. As a consequence of this error, the Euler angle c$~
does not appear in their results.
On the basis of explicit calculations, BS conclude that their
version of the restricted lipid rotation model can account for the
observed8 orientation independence of R I ~ .(The substantial
anisotropy in R ~ was
Q not addressed by BS.) While it is not too
surprising that a seven-parameter model can reproduce data that
could also be represented by a single parameter, it should be noted
that one of the two parameter sets used by BS has OMI = 54.7O
and qr = 0, which, according to (2.25a), corresponds to a zero
internal order parameter S M F and, hence, to a vanishing quadrupolar splitting! Further, with the second parameter set, the
theory used by BS predicts an orientation dependence in R!Q that
is precisely opposite to what is observed.8 These contradictions
illustrate the dangers of analyzing the complex dynamic behavior
of phospholipid bilayers with only a very limited set of experimental data in view.
B. Effects of Bilayer Curvature. The validity of the approach
taken here of simultaneouslyanalyzing 2H NMR data from small
unilamellar vesicles (RI dispersion and line width), from multilamellar dispersions (quadrupolar splittings), and from macroscopically aligned planar bilayers (R1zand R ~ anisotropy)
Q
hinges
on the assumption that the NMR parameters are essentially
invariant with respect to changes in bilayer curvature.
Consider the low-frequency (1-10 MHz) RI contribution from
vesicle tumbling and lipid lateral diffusion (Figure 2). As shown
in section IIB, this contribution is completely determined by the
line width and the order parameter SDpIn fact, since w07, >>
1 in this frequency range, it follows that RIVm SDF2!~v, and
R2 SDF’T,,.
whence RIVw S D F 4 for a given line width. On
account of this strong dependence of R I on the order parameter,
the assumption that S D F has the same value in the vesicles (radius
of order 10 nmKS6)as in the liposome dispersions (average radius
a4L
-
-
-
,
of curvature of order 100 nms7) is obviously critical. Moreover,
this is a controversial assumption. A large number of IH,2H,
and I3C line-width measurements on phospholipid vesicles have
been reported, some39*40*s8
of which have been taken to support
the notion of an essentially curvature invariant S D F , while othe r ~ ~have
~ led
, ~to ~the, conclusion
~ ~
that the high curvature of
vesicle bilayers reduces S D F by a factor of 2 or more relative to
effectively planar bilayers. In the following we present further
arguments in support of the former view.
The first argument is based on the observed’ strong frequency
dependence of R I in the range 1-10 MHz. We concluded in
section IIIB that this dispersion is due to vesicle tumbling and
lipid lateral diffusion and, hence, should not be present in multilamellar powder samples. This prediction is essentially confirmed
by the proton R, dispersion from a multilamellar powder sample
of DMPC, recently measured by Rommel et a1.I8 using fieldcycling techniques. While these data agree qualitatively with the
vesicle data in the range 10-100 MHz, they exhibit a plateau in
the range 1-10 MHz, where the vesicle data show a strong dispersion. Further, since RIVB S D F 4 (for a given line width), it
follows that even a modest reduction of the order parameter would
dramatically affect the fit to the 2H RI dispersion. In section IIIB
we showed that the R1 dispersion can be accounted for by lipid
reorientation as the only motion, but that the very different parameter values resulting from this fit are inconsistent with BDm
and ME3’ calculations of the internal order tensor and with the
R I Zand R ~ anisotropy.8
Q
A similar conclusion is reached if S D F
is significantly reduced below the value deduced from the quadrupolar splitting in multilamellar powder samples.
The principal argument in favor of substantially reduced orientational order in vesicle bilayers derives from line-width calculations using (3.2) and the following well-known expressions
for the correlation time T,=
-
(4.1)
By inserting values for the phospholipid lateral diffusion coefficient
Dlot,the vesicle radius R,,, and the solvent viscosity to,T,, can
thus be calculated. Using (3.2) and the observed line width, one
then obtains the order parameter S D p An important point, not
always appreciated in this connection, is that a bilayer has two
sides. For phospholipids in the inner layer, the radius in (4.3)
should not be the outer vesicle radius R, but rather R = R,
- d, where d is the bilayer thickness. Even for a monodisperse
vesicle suspension, the line shape should therefore be a superposition of two Lorentziansassociated with the inner and outer layers.
The second important point is that even if the inner-layer Lorentzian is only half as wide as the outer-layer Lorentzian, the
observed line shape will not appear appreciably non-lorentzian,
but its effective width will be strongly influenced by the narrow
component. This point is illustrated in Figure 5, showing that
the
line width (775 Hz)in DMPC vesicles can be
reproduced by a superposition of two Lorentziansof relative weight
1:2, corresponding to the experimentally d e t e r m i r ~ e d phos~~.~~
pholipid distribution between the inner and outer layers of small
vesicles, and with line widths of 552 and 1067 Hz, respectively.
These line widths are obtained by taking S D F = 0.22 (as in
multilamellar dispersion^^^) for the outer layer and a slightly
reduced value, S D F = 0.19. for the inner layer. The other pa(54) Huang, C. J . Am. Chcm. Soc. 1973, 95, 234.
(55) Chrzeszczyk,A.; Wishnia, A.; Springer, C. S. Biochim. Biophys. Acro
1977,470, 161.
(56) Huang, C.; Mason, J. T. Proc. Narl. Acad. Sct. U.S.A.1978,75,308.
(57) Papahadjopoulos, D.; Miller, N. Blochim. Biophys. Acro 1967,135,
624.
(53) Brown, M. F.; ScMerman, 0. Chcm. Phys. Leu. 1990, 167, 158.
( 5 8 ) Finer, E. 0.J . Magn. Rcson. 1974, 13, 76.
(59) Bocian, D. F.; Chan, S. 1. Annu. Reo. Phys. Chcm. 1978, 29, 307.
Halle
6732 The Journal of Physical Chemistry, Vol. 95, No. 17, 1991
0.0
-2.5
0
2.5
Fnquency / kHz
Figure 5. Hypothetical line shape for a phospholipid vesicle with parameters chosen to reproduce the line width observed for 1,ZDMPC3',3'-d2 at 30 0C.7*uThe parameter values are R, = 10.0 nm, d = 3.5
nm, D,,, = 1 X IO-" m2 s-l, S D F = 0.22 (outer layer), and SDF= 0.19
(inner layer). The dashed line shape is a Lorentzian with the same
half-width and height as the composite line shape.
rameter values, given in the figure caption, are reasonable (vide
infra) but, of course, not unique.
The lateral diffusion coefficient of DMPC in macroscopically
aligned planar bilayers with 20 wt '3% D 2 0 at 30 "C has been
determinedw to Dlat= (3 i 1) X 10-I2 m2 s-I. It is instructive
to compare this value with the prediction of continuum fluid
mechanics. For a cylinder of height h and radius a embedded
in a planar fluid layer thickness h and viscosity 7 , the lateral
diffusion coefficient may be expressed as6I
between 10 and 11 nm have been reported for DPPC vesicles.*%
The value R,, = 10.0 nm used in Figure 5 is thus not unreasonable. Depending on preparation techniques, vesicle size polydispersity may be a complicating factor. It should be noted,
however, that the effective line width will be influenced more by
the smaller vesicles than by the larger ones.
In conclusion, we believe that the assumption of an essentially
curvature independent order parameter S D F (i) is supported by
the R I dispersion data, and (ii) is not at variance with the linewidth data.
C. Model Parameters. In this subsection we discuss the parameter values derived from our fits (Table 111) and compare them
with the results of previous studies. As explained in section 111,
the parameters uz and D, cannot be individually determined by
the data analyzed here. However, the ratio u2/Dlis reasonably
well determined by the relaxation anisotropy data (fit C).
Adopting a value u2 = 0.3, which is close to the BD simulation
result,30we obtain D, = 7.5 X lo8 s-I. Assuming that u2 and D,
have the same values in vesicles and aligned bilayers, we can get
an estimate for Rht(Table 111). The resulting difference between
the Rinlvalues in fits A and C may be ascribed to differences in
labeling site. 13C data from DPPC vesicles at 50 OC reveal a
frequency-independent R I difference of ca. 0.6 s-l between the
3' and 4' acyl chain positions?*I0Due to differences in spin-lattice
coupling
the corresponding value for 2H should be
larger by a factor of 10. This compares well with our results: Rht
= 27.4 s-I for the 3'-position (fit A) and Rint= 21.4 s-l for the
4'-position (fit C).
The value of Dllmay be compared to the prediction of hydrodynamic theory. For the cylinder model described above, one has6'
-I
D,, =
where y = 0.5772 is Euler's constant and the expansion parameter
e is
e = h9/aq
(4.5)
q being the average viscosity of the two media bounding the lipid
layer. Using the reasonable values30h = 1.7 nm, a = 0.45 nm,
q = 2 cP, and q = 1.4 CP(30 "C), we obtain from (4.4) and (4.5)
Dial = 1.4 X 10-lom2 PI. This is 2 orders of magnitude larger
than the measured value and strongly suggests that lateral diffusion
of phospholipids in bilayers is not governed by the viscosity of the
apolar interior of the bilayer. A similar conclusion was drawn
from a molecular dynamics simulation of a bilayer consisting of
decane molecules." Instead, it seems likely that the rate of lateral
motion is limited by strong interactions between the zwitterionic
phosphocholine head groups. Furthermore, as the experimental
Dlptvalue refers to a system with only 8.5 water molecules per
phospholipid, we expect head-group interactions between phospholipids in adjacent layers to be as important as the interactions
within a layer. This view is in accord with the finding@that DLat
increases significantly with increasing water content. On going
from 20 to 40 wt '3% D 2 0 (8.5 to 22.6 D20/DMPC), it can be
estimatedw that D,,,increases by a factor 1.5. An increase by
an additional factor 2 on going to effectively infinite dilution (as
in vesicle suspensions) does not appear unreasonable, in which
case the value Dhl = 1 X lo-" mz s-I used in Figure 5 would be
justified. Phospholipid lateral diffusion coefficients of this order
of magnitude have also been inferred from 'H and "C relaxation
data from DPPC vesicles.63 The picture is further complicated
by the inferenceSSsthat the head-group area is larger in the outer
layer than in inner layer of the vesicle, in which case also Dlatcan
be expected to be larger in the outer layer.
As seen from (3.2) and (4.1)-(4.3), the line width depends
sensitively on the outer vesicle radius R,. While different
techniques give somewhat different results,u hydrodynamic radii
(60)Kuo, A.-L.; Wade, C . G. Biochemistry 1979, 18, 2300.
(61) Hughes, 8. D.;Pailthorpe, B. A.; White, L. R.1.Fluid Meeh. 1981,
I IO. 349.
(62) van der Ploeg, P.;Berendsen, H. J. C. Mol. Phys. 1981, 49, 233.
(63) Bdlet, P.;McConnell, H.M. Prw. Nut/. Acud. Sci. U.S.A. 1975, 72,
1451.
kBT [I + 16 + o ( 4 ]
47rqha2
3*t
(4.6)
with e given by (4.5). Using dimensional parameters as for the
above calculation of Dhl, we find that our value D, = 7.5 X lo8
s-l corresponds to an internal bilayer viscosity of ca. 1 cP. This
is a reasonable value if head-group interactions are not rate limiting, as they seem to be for the lateral diffusion (vide supra).
The values of 2.0 X IO7 s-l and 6.6 X lo7 s-I obtained for the
tumbling diffusion coefficient D, correspond to dynamic anisotropies DI,/D, of 10-40, far exceeding the range 2-3 expectedu
from the geometrical shape of a phospholipid molecule. This
suggests that head-group interactions are important for controlling
the tumbling rate. (The difference in the D , values derived from
vesicles and aligned bilayers may then be ascribed to minor
curvature-induced differences in head-goup area.) The correlation
time i I O
for the dominant restricted tumbling mode was found
from fits A and C to be 2-5 ns (cf. Table IV). It is instructive
to relate this time scale to that for lateral diffusion. With Dhl
= (3-10) x 10-l2 m2 s-I (vide supra), the lateral head group
displacement during a time i l ois found to be 0.2-0.4 nm. Such
values are consistent with the motional model used here, where
the correlation time i I O
refers to orientational fluctuations of the
lipid long axis in the local orienting pmt exerted by the neighboring
phospholipid molecules.
The phospholipid order parameter was found to be S D M = 0.70,
with little variation between vesicles and aligned bilayers. This
parameter is rather accurately determined as it, for a given SDF
(derived from the quadrupolar splitting), determines the internal
= S D F / S D M , the square of which scales all
order parameter SMF
contributions to the relaxation rates from phospholipid reorientation.
We now comment briefly on the relation of our results to those
of some recent analyses of longitudinal spin relaxation data from
phospholipid bilayers. Pastor et
analyzed 13CRl datal0 in
the range 15-126 MHz from DPPC vesicles at 50 OC using a
model for the phospholipid reorientation similar to the present
one, but treating the internal motions with the aid of BD simu(64) Huang, C. Biochemistry 1%9,8, 344.
(65) Sbderman, 0. J. Mugn. Reson. 1986, 68, 296.
(66) Perrin, F. J . Phys. Rudium (Paris) 1934, 5, 497.
2H Relaxation in Lipid Bilayers
The Journal of Physical Chemistry, Vol. 95, No. 17, 1991 6133
lations. In accordance with our results, they concluded that the
spinning motion is in the extreme-narrowing limit, while the
observed frequency dependence can be ascribed to restricted
tumbling (wobbling) of the lipid molecule. Further, our results
for S D M and D, are in the range where acceptable fits to the I3C
dispersion for acyl carbons 3' and 4' could be obtained.
In a recent contribution,Is Mayer et al. reported 'H R,z(@LD)
data at 46.1 MHz from aligned bilayers of 1,2-DMPC-6',6'-d2
at 35 OC. These data, which show the same remarkable
invariance as reported earlier by Jarrell et a1.,8 were interpreted
in terms of internal motions and restricted lipid reorientation. The
model25used for the latter motion is similar to the one used here
(except for the nonpolar pmt). However, rather than employing
approximate analytical solutions, Mayer et a!. compute the spectral
densities numerically by discretizing the angular space and treating
internal motions (three-site jump model) and overall lipid motions
within the same formalism. The main conclusion, that the R,z
anisotropy can be accounted for by restricted lipid reorientation,
is in accordance with our results (which also account for the R ~ Q
anisotropy). The parameters describing this motion were determined18 to SDM
= 0.68, Dll = 3.3 X lo8 s-I, and D, = 3.3 X
lo7 s-I, in good agreement with our results (fit C in Table 111).
[Mayer et al. define reorientational correlation times as 7Ru =
(6D,)-I and i R ,= (6D,)-'.] These authors also investigated the
effect of incorporating 40 mol % cholesterol into the bilayer,
concluding that lipid reorientation is dramatically slowed down
(D,l = 9.3 X IO6 s-I, D, = 1.9 X lo5 s-I). In our opinion, this
is a surprisingly large effect. According to the present model,
the parameter values obtained by Mayer et al. imply that the only
lipid motion that affects RIZis the (22)spinning mode (722 = 27
ns), while the tumbling modes (710 = 270 ns) are too slow to
contribute at 46.1 MHz. It would then be impossible to determine
D,. On the other hand, using the present model, we find that
the R I Zanisotropy can be accurately reproduced with virtually
no change in the reorientational dynamics: D,, = 3.1 X lo8 s-l,
D, = 5.1 X lo7 s-l, SDM
= 0.95, Rint= 9 s- , uI = 0 (as also
assumed by Mayer et al.), and u2 = 0.15. This suggests that the
RIZ(BLD)data alone are insufficient to uniquely determine the
model parameters and that R,q(@LD) measurements may help to
resolve the issue.
Additional information about molecular dynamics in phospholipid bilayers can be obtained from the 'H relaxation of
specifically deuterated cholesterol incorporated into the bilayers.
Bonmatin et al. thus recently reported 'H RIZ(6LD) and Rlq(e,D)
data at 30.7 MHz from aligned DPPC bilayers at 30 OC with 50
mol % cholesterol labeled in different positions.67 The rigidity
of the sterol framework considerably simplifies the analysis. In
the present model, Rint= 0 and the quantities I&(QMF)l
can be
(67) Bonmatin, J.-M.;
Smith, 1. C. P.; Jarrell, H. C.; Siminovitch, D. J.
J . Am. Chcm. Soc. 1990, 112, 1697.
obtained from (2.25) by setting SIF= 1 and V I = 0 (the I frame
is then superfluous). Using the present model, we essentially
confirm the results of Bonmartin et al.: the relaxation is dominated
by the {I 1) and (221spinning modes (DIl= 3.4 X 108 9') with little
contribution from restricted tumbling. However, mainly on the
basis of a temperature dependence, these authors favored6' a (less
intuitively appealing) three-site large-angle jump model with an
order of magnitude slower axial rotation rate.
V. Conclusions
In the foregoing we have used a relatively simple motional model
to analyze recent 'H relaxation data from phospholipid bilayers.
The model includes the three motional degrees of freedom that,
a priori, are most likely to significantly affect the relaxation in
the megahertz frequency range: internal motions, phospholipid
reorientation, and joint vesicle tumbling and lateral diffusion. For
the present analysis, it is not necessary to model the internal
motions explicitly. The emphasis is instead on the phospholipid
reorientation, which is described in a consistent way using a fairly
realistic model.
The main conclusions emerging from our analysis are as follows.
1. The R , dispersion from vesicle suspensions in the range 1-10
MHz is due to vesicle tumbling and lipid lateral diffusion.
2. The R , dispersion in the range 10-100 MHz is due to
restricted tumbling (wobbling) of individual phospholipid molecules.
3. There is no need to invoke collective reorientation mechanisms to account for RI (or R2) in the range 1-100 MHz.
4. Reorientation of individual phospholipid molecules also
accounts for the anisotropy of RIZand R ~ from
Q aligned bilayers.
5. A resolution of the line-width controversy is suggested:
lateral diffusion in the inner phospholipid layer of the vesicle
produces a narrow line which strongly affects the observed line
width. There is no need to invoke a substantial reduction of
orientational order in vesicle bilayers.
We have emphasized the importance of simultaneously addressing a wide range of *H N M R data in order to minimize
interpretational ambiguity. Yet, even in the present analysis, all
model parameters could not be uniquely determined. Only by
supplementing the N M R data with theoretical results related to
the internal order tensor could we estimate the values of the
remaining parameters. Hopefully, further 'H relaxation studies
will improve the situation. More complete information about the
two-dimensional functions RIZ(W();BLD) and RIQ(c&D) in macroscopically aligned lamellar samples would be particularly
helpful.
Acknowledgment. This work was supported by the Swedish
Natural Science Research Council. I am grateful to Olle
Sijderman for communicating the line-width data.
Registry No. DMPC, 18194-24-6.