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 ( 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 order parameter SMF = S D F / S D M , the square of which scales all 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.
