J. Phys. Chem. B 2000, 104, 6059-6064
6059
Line Shape Analysis of NMR Powder Spectra of 2H2O in Lipid Bilayer Systems P.-O. Westlund Department of Chemistry, Biophysical Chemistry, Umea˚ UniVersity, S-901 87 Umea˚ , Sweden ReceiVed: NoVember 15, 1999; In Final Form: February 28, 2000
A NMR line shape/spin relaxation model is developed for 2H2O studies of structure and dynamics of the lipid-water interfaces of phosphatidylcholine bilayers. A line shape function describing the orientational dependence of 2H2O is derived. In addition, also expressions of the observed quadrupole splitting and the spin-lattice relaxation rate are derived within the same dynamic model. The model comprises two chemically interchanging fractions of water namely, “free”, and “bound”. There are four molecular parameters characterizing the “bound” water of the lipid water interface, namely, (1) the fraction water molecules bound dD to lipid molecules, (2) the local water order parameter SPd 0 , (3) the order parameter S0 which describes the averaged “bound” water, and an effective correlation time (4) τc, characterizing water translational diffusion at the interface. This model allows for analyzing quadrupole splittings, spin-lattice relaxation rates, and water powder line shapes. Thus, dynamics as well as structural information about the water molecules residing in the water lipid interface may be extracted. In the reinterpretation of 2H2O powder spectra obtained for lamellar phases of dipalmitoylphosphatidylcholine (DPPC), the results clearly indicate that SdD 0 (LR) > SdD (L ) when comparing the liquid crystalline phase with 10-11 water molecules per lipid molecule and the β′ 0 gel phase with 3.5-4.2 water molecules per lipid molecule. Whereas the order of the perturbed water is Pd similar in both phases, SPd 0 (LR) ≈ S0 (Lβ′). The lateral diffusion is characterized by a correlation time τc > 60 ns but cannot be determined without measuring the spin-lattice relaxation measurements.
1. Introduction In the description of the 2H NMR powder line shape in solid state systems, relaxation effects can usually be ignored.1 However, this is not a good approximation when examining physical properties of 2H2O in liquid crystals and some other systems. For example, in the liquid crystalline LR phase of dimethyldodecylamine oxide (DDAO) and water,2 or in microheterogeneous systems such as polycrystalline tert-butyl alcohol,3 or when this alcohol is absorbed on zeolites,3 the powder spectra clearly display relaxation effects. In these systems the powder spectra display a broadening of only 1-2 kHz with a characteristic dip at the magic angle. The latter is caused by spin relaxation. A typical theoretical water powder spectrum of a lamellar phase under extreme narrowing conditions is shown in Figure 1. Recently, the origin of the dip was analyzed within a simple model. It was shown that the presence of the dip is due to the relative magnitude between the diagonal and the offdiagonal matrix elements of the full Liouville matrix. The Liouville matrix is composed by the static Zeeman and the static quadrupole interaction super matrices, together with the Redfield relaxation supermatrix. The dip disappears under nonextreme narrowing conditions, i.e., slow dynamics, because then the Redfield relaxation matrix becomes approximately diagonal.4 In this paper a line shape/relaxation analysis approach is presented suitable for studies of heavy water NMR powder spectra of bilayer systems. The spectra discussed are obtained from the gel and the liquid crystalline phase of dipalmitoylphosphatidylcholine (DPPC). The relaxation model comprises two sites of water hereafter denoted as “bound” and “free”. The former water molecules refer to the hydration of the molecular interface, whereas the latter are nonperturbed molecules with a zero quadrupole splitting. Within this model, the orientation-
Figure 1. Theoretical powder spectrum of heavy water for a lamellar phase bilayer displaying a dip at the magic angle for the dominant dynamics, τc, being fast as compared to the inverse of the Larmor frequency ω0.
dependent line shape function is derived in a closed analytical form. Our aim is to demonstrate that a combined spin-lattice relaxation and line shape analysis of 2H2O is a valuable approach which improves the molecular picture of lipid hydration. For instance, the results of a gradual hydration study enables determination of the number of water molecules per lipid.5 We may discriminate the hydration of the liquid crystalline phase and the gel phase in terms of an order parameter, SdD 0 and an effective correlation time, τc. Thus, our relaxation model provides a more detailed molecular picture of lipid hydration than other models based on solely static quantities. The present work is also motivated by the necessity of sufficiently detailed molecular models of lipid hydration, in order to make it possible to analyze the NMR relaxation data and quadrupole splittings
10.1021/jp994037l CCC: $19.00 © 2000 American Chemical Society Published on Web 06/06/2000
6060 J. Phys. Chem. B, Vol. 104, No. 25, 2000
Westlund
of the intermediate rippled phases of DPPC or dimyristoylphosphatidylcholine (DMPC). The two-site and the dynamic models presented here follow closely the ideas of Halle and Wennerstro¨m,8 although a slightly different spectral density is used. The model comprises a bound anisotropic site (i.e., water residing close to the interface) involved in chemical exchange with the “free” bulk water. The parameter space of the model, described in the following sections, is rather extensive, but can be effectively reduced to Pd four parameters (PB, SdD 0 , S0 , τc). The fraction (PB) of “bound” water molecules may be determined from monitoring the quadrupole splitting as a function of hydration as was recently demonstrated by Dufourc et al. for DMPC.5 In addition, we may characterize this fraction of “bound” water molecules in Pd terms of two order parameters SdD 0 and S0 and two correlation Pd times τb and τc. The order parameter S0 refers to the averaged due to fast local anisotropic water reorientation at the interface. This motion is characterized by a correlation time τb ranging between 5 and 50 ps and considered to be independent of the lipid bilayer phase. The second order parameter, SdD 0 , on the other hand characterizes an averaged “binding” site of the lipidwater interface and depends on the molecular details of the lipid bilayer phase. The water dynamics at the interface is expected to be complex, involving lateral diffusion along the surface and chemical exchange or what we could describe as an “in” and “out” motion along the normal direction of the surface. Hence the bound water molecules are diffusing among different binding sites thereby averaging over different site order parameters into one effective order parameter, SdD 0 . This motion is characterized by the correlation time τc. The dynamics of free water is described by a correlation time τf ≈ 1-5 ps which can be determined from a separate experiment. It should be noticed that neither τb nor τf is crucial for the line shape analysis. The chemical exchange rate (λ) is kept constant in the fast regime on the NMR time scale. Notice that what we refer to as chemical exchange is included in the translational diffusion processes which also cause spin relaxation. These complex phenomena are treated within the simple two-site chemical exchange model and a correlation time τc. The spin-lattice and the quadrupole splitting together with the line shape function imply three independent mathematical Pd relations involving the unknown parameters PB, SdD 0 , S0 , and τc. The measured spin-lattice relaxation time contains contributions from bound as well as free water according to
1 1 1 ) PB + Pf T1 T1B + τexch T1f
linearly with PB, we reach the fully hydrated lipid state, or the swelling regime. As the fraction of free water molecules is being built up and involved in fast chemical exchange with bound water, the observed quadrupole splitting decreases. Thus, the observed quadrupole splitting decreases linearly with 1/(n/ND).
3 3 dD Pd nD* Pd ∆νQ ) χPB|SdD 0 S0 | ) χ|S0 S0 | 4 4 n/Nd
According to Dufourc et al.,5 the decrease of the quadrupole splitting in the swelling regime is different for the liquid crystalline phase compared to the gel phase. Why is it so? Obviously the quadrupole splitting of bound water molecules to DMPC molecules ∆νd is different in the gel phase as compared to the liquid crystalline phase. In their more simple model, no possible explanation can be found because the model lacks the concepts which characterize the bound water. One needs a more elaborate model. Let us consider the derivity of eq 2 with respect to 1/(n/Nd). In our model, this slope is Pd determined by the order parameter product -|SdD 0 S0 |nD* dD where the most interesting order parameter is S0 which also depends on the lipid phase. In addition, one should also mention that the quadrupole splitting increases with increasing temperature.15 Consequently, a more detailed model is needed to explain all observed aspects of hydration of lipids. The relaxation/line shape approach presented here is a first attempt to quantify the difference between model parameters describing the hydration of lipids in the liquid crystalline phase and the gel phase. In order to illustrate the approach, the 2H2O powder spectra of DPPC are analyzed in the gel, and the liquid crystalline phases. The hydration is assumed to be about 1011 water molecules per lipid molecule in the LR phase6,7 and 3.5-4.2 for the gel phase.5 The results of the line shape analysis Pd provide the order parameters SdD 0 and S0 . The spin-lattice relaxation times are also calculated for different correlation times τc. However, one needs (which we do not have in this study) experimental T1 in order to determine τc. Consequently, this approach provides new molecular insight about the bound water molecules of LR and Lβ′ phases. In brief, the line shape analysis is performed as follows: First, Pd the product PBSdD 0 S0 is determined from the measured quadrupole splitting obtained from the powder spectra according to eq 11. Pd SdD 0 S0 )
(1)
where T1 is described in terms of the model parameters given by eq 28. In principle a fourth mathematical relation can be established from double quanta relaxation experiments, or from measurements at other Larmor frequencies. The number of bound water molecules per lipid nD* and consequently PB can be determined by monitoring the quadrupole splittings at different levels of hydration.5 In brief, the fraction of bound water per lipid is obtained by measuring the quadrupole splitting as a function of the mole fraction of water to lipid molecule. Using the same notation as in ref 5 the fraction “bound” water is given by PB ) nD*/(n/ND), where ND denotes the number of lipids, n is the number of water molecules, and nD* is the number of water molecules per lipid of a fully hydrated lipid. In the hydration regime, PB ) 1 or (n/ND < nD*). As the observed quadrupole splitting starts to decrease
(2)
4∆νQ 3χPB
(3)
The fraction PB is obtained from independent measurements. Thus, the two order parameters become dependent. The shortest possible τc can be estimated from the disappearance of the line shape dip. The most important spectral density is determined by the slowest dynamics and given by τc and the amplitude 2 2 factor [(SPd 0 ) - (4∆νQ/3χPB) ] (see eq 17) which depends on , which is left to determine from the one order parameter, SPd 0 line shape analysis. For a number of correlation times τc > 60 ns the corresponding spin-lattice relaxation rates are also calculated. By combining the line shape analysis with T1 measurements, the value of τc can in principle be determined. 2. Theory of Nuclear Spin Relaxation The interaction with the molecular environment of a system of independent nuclear spins is described by Zeeman interaction
NMR Spectra of 2H2O in Lipid Bilayer Systems
J. Phys. Chem. B, Vol. 104, No. 25, 2000 6061
and quadrupolar interaction HSL(t) (rad s-l) according to
Htot ) -γTB0ˆIz + HSL(t)
(4)
The quadrupolar interaction is given by the scalar product 2 between spin operators, A-n , of rank 2 and components of the time-dependent field gradient tensor Vn(t). For the nuclear spin quantum number I ) 1
∑
πeQ HSL(t) ) h
2 (-1)nA-n
VLn (t)
(5) used.8
Here second rank irreducible spin tensor operators are To identify different dynamic processes that contribute on different time scales to the time dependence of the field gradient tensor, four coordinate systems are introduced. These coordinate systems are denoted with the superscripts L, D, d, and P. Let “L” denote the laboratory fixed frame with its zL axis defined by the static magnetic field Bz. The “D” frame refers to the director of the multilamellar phase. The third frame “d” refers to the local director of a bound water molecule site of which the zd axis is in some orientation cos(θ(t)) ) b zDb zd relative to the director D. Finally, the “P” frame is the molecule fixed principal frame of the field gradient tensor. For 2H2O molecules zP is approximately along the O-2H bond. The lab-frame component VLn (t) of the quadrupolar interaction is here expressed in terms of Wigner rotation matrix elements13 and the principal frame components VPi .8,9 The Euler angles, ΩdD(t) and ΩPd(t) are random variables specifying the relative orientation of the coordinate systems D, d, and P. These angles link the spin relaxation theory to the geometric properties of interest in this work, according to
VLn (t) )
D2nm(ΩDL)D2mk(ΩdD(t))VP0 (D2k0(ΩPd(t)) + ∑ k,m 2 (ΩPd(t)) + D2k2(ΩPd(t)))) (6) ηP(Dk-2
If VLn (t) is averaged over all molecular motions, it is important to distinguish the fast local anisotropic motions of water present (ΩPd(t)) and the slower translational diffusion modulation condensed in the Euler angles (ΩdD(t)). Taken together, these two dynamic processes yield an averaged quadrupolar interaction and an observable quadrupole splitting. Here a time scale separation and two kinds of order parameters are assumed. In terms of these order parameters the static or residual field gradient anisotropy8-10 reads
) VL,static n
P Pd Pd Pd D2mn(ΩDL)SdD ∑ m V0 (S0 + ηPd(S2 + S-2)) ) m
2 dD 2 2 dD VP0 SPd 0 [D0n(ΩDL)S0 + D2n(ΩDL) + D-2n(ΩDL)S2 ] (7)
In eq 7 it is assumed that the asymmetry parameter ηP ≈ 0. It 2 should be noted that the order parameter SdD m ) 〈D0m(ΩdD(t))〉 is an averaged order parameter of a “bound” site. Because translational diffusion among different water sites occurs on a fast time scale compared with the difference in splitting, this order parameter is given by the weighted average.8,9
effect because bound water molecules with different average 2 (ΩjdD)〉. orientations give rise to different signs of 〈D0m 11 In the secular approximation, the quadrupole splitting ∆ωQ is proportional to the static field gradient given by relation ∆ωQ ) (x6πeQ/h)VL,static . Thus the quadrupole splitting depends on 0 the angle (βLD) between the static magnetic field and the director D.
3 ∆νQ ) χ|SPd d2 (β )SdD| 2 0 00 LD 0
(9)
For a sample where all orientations are equally distributed (i.e., powder sample) eq 9 can be written as
3 ∆νQ(powder) ) χ|SPd SdD| 4 0 0
(10)
In eqs 9 and 10 the quadrupolar coupling constant of water is χ ) eQVzz/h ≈ 220 kHz. As already mentioned, the secondorder parameter SPd 0 is related to the fast local anisotropic motion of water in the binding head group sites. For fast chemical exchange the fraction of bound water molecules PB influences the quadrupole splitting as follows:
3 SdD|P ∆νQ(powder) ) χ|SPd 4 0 0 B
(11)
2.1. Model of Time-Correlation Functions. The stochastic time dependent part of the quadrupolar interaction causes solely relaxation. In order to treat the relaxation, field gradient components with zero mean are defined:
∆VLn (t) ) VLn (t) - VL,static n
(12)
The spin relaxation caused by translational diffusion depends on the curvature of the lipid bilayer, the kind of binding sites, and the translation diffusion coefficient D. These properties are taken into account by the time correlation function in terms of the integral correlation time τc, and by the mean-square value 〈∆VLn (0)2〉. The time correlation functions (and the corresponding spectral densities) relevant for describing water spin relaxation at the lipid-water interface are derived below. The fluctuating field gradient is modulated by translational diffusion and local reorientational motions. To treat these a time scale separation of them is assumed and the correlation functions are modeled as single exponential functions. This model follows the assumption of a time scale separation in the modulation of the field gradient of quadrupole ions and water molecules in microheterogeneous systems.9,10 The correlation functions of eq 18 are obtained as follows: Starting from eq 12 the field gradient correlation functions read
〉2 (13) 〈∆VLn (t)* ∆VLn (0)〉 ) 〈(VLn (t)* VLn (0)〉 - 〈VL,static n By inserting eqs 6 and 7 in eq 13, one obtains
〈∆VLn (t)* ∆VLn (0)〉 )
|d2nm(ΩDL)|2{〈D2/ ∑ mk(ΩdD(t)) k,m
〈VP0 〉2
(8)
2 Dd Pd 2 D2mk(ΩdD(0))〉〈D2/ k0 (ΩPd(t))Dk0(ΩPd(0))〉 - δk0(Sm S0 ) } (14)
Here fj denotes the fraction of jth kind of “bound” water molecules. This order parameter may thus contain a cancellation
Here, the time correlation function is decomposed into a product of two time correlation functions: one is modulated by local restricted reorientational motions, whereas the second is modu-
SdD m )|
2 (ΩjdD)〉| ∑j fj 〈D0m
6062 J. Phys. Chem. B, Vol. 104, No. 25, 2000
Westlund
lated by translational diffusion among different sites at the lipid-water interface. In the single exponential approximation the local restricted reorientational diffusion correlation function is now given by 2 2 2 〈D2/ k0 (ΩPd(t)) Dk0(ΩPd(0))〉 ) [〈|Dk0(ΩPd)| 〉 2 -t/τb 2 + δ0k(SPd δ0k(SPd 0 ) ]e 0 ) (15)
whereas the translational diffusion modulated time correlation is
〈D2/ mk(ΩdD(t))
D2mk(ΩDd(0))〉
)
[〈|D2mk(ΩdD)|2〉 2 -t/τc + δ0m(SdD 0 ) ]e
(
Rf + λPBE -R2 ) -λP BE
Now the field gradient time correlation function of eq 14 can be rewritten by using eqs 15 and 16 and the assumption that τc >> τb.
Finally, the spectral densities are obtained from a FourierLaplace transform of the field gradient time correlation functions:
Jn(nω0) )
Jfn(nω0) ∞
∫
+
Jsn(nω0)
〈∆VLn (t)* 0
{
)
(21)
( ) (
F1,f 1 (ω) 0.0 ) F1,B 1 (ω) 0.0 -λPf
i∆ -λPB 0
0 -λPf
R2f +iδωf+λPB
0
0 -λPB
R1B+iδωB+λPf 0 0
( )
R2B+iδωB+λPf F1,f1(0) 0.0 F1,B1(0) 0.0
2 -t/τb + 〈∆VLn (t)* ∆VLn (0)〉 ) 〈VP0 〉2([1 - (SPd 0 ) ]e dD 2 -t/τc 2 ) (17) (SPd 0 ) [1 - (S0 ) ]e
)
where E is the unit matrix. The corresponding matrix equation reads
R1f +iδωf+λPB i∆
2 (SdD 0 ) (16)
-λPfE Rb + λPfE
)
×
(22)
In eq 22 a shorthand notation is used: δωf ) (ωf - ω) and δωB ) (ωB - ω). The relaxation rate of the spin vector component is given by
R1B )
3 (χπ)2(3JB0 (0) + 5JB1 (ωB) + 2JB2 (2ωB)) 20
(23)
which is the same as the spin-spin relaxation rate 1/T2. The relaxation rate of the rank 2 spin component is given by
∆VLn (0)〉e-inω0t dt )
τb 2 + 〈VP0 〉2 ([1 - (SPd 0 ) ]) 1 + (τbnω0)2 τc 2 dD 2 (SPd 0 ) [1 - (S0 ) ] 1 + (τcnω0)2
}
R2B )
3 (χπ)2(3JB0 (0) + JB1 (ωB) + 2JB2 (2ωB)) 20
(24)
(18)
Furthermore, the units of the spectral density of eq 18 is here reduced to seconds, reading:
with one contribution from the correlation time τb which refers to the slightly hindered water motion outside the lipid interface. The second correlation time, τc, refers to translational diffusion of water molecules in the vicinity of the lipid-water interface. 2.2. A Two-Site Model. In this section the BlochWangsness-Redfield relaxation theory11 is applied to derive the orientation-dependent two-site line shape function.12 Chemical exchange is assumed between the “bound” (B) anisotropic site of water and a free (f) isotropic site of water. The fractions are denoted by PB and Pf, respectively. Here the equation of motion is given in a statistical tensor representation12,13 and the orientation-dependent line shape function obtained is given by eq 29. Starting with the equation of motion for the reduced spin density operator
τc 2 Pd 2 dD 2 JBn (nω) ) (1 - (SPd 0 ) )τb + (S0 ) (1 - (S0 ) ) (1 + (nωτc)2) (25)
dF(t) i ) - L0F(t) - R2F(t) dt p
)
Taken together, there is one contribution caused by a single exponential decay, (τb (≈50 - 100 ps) as compared to that of free water (τf ) 3-5 ps).14 The fast local averaging reduces 2 the time correlation function to a value, (SPD 0 ) . A slower motion, modulated by the translational diffusion, described by τc, further reduces the time correlation function to the value of 2 dD 2 (SPd 0 ) (S0 ) which determines the quadrupole splitting. For the free site where extreme narrowing conditions prevail, the relaxation rates are given by
3 R1f ) (χπ)2τf 2
(26)
9 (χπ)2τf 10
(27)
(19) R2f )
Equation 19 is transformed to the frequency domain by the Fourier-Laplace transform of the density operator according to F(ω) ) ∫∞0 F(t)eiωt dt. Thereby the equation of motion reads
The observed spin-lattice relaxation rate of eq l is given in terms of the spectral densities by
(20)
3 1 ) PB (χπ)2(JB1 (ω0) + 4JB2 (2ω0)) + PfR1f (28) T1,exp 10
of which the nondiagonal spin-spin Redfield relaxation matrix is
2.3. The Orientation-Dependent 2H Line Shape L(βLD,rLD,ω) for Two-Site Chemical Exchange. The orienta-
i -F(0) ) - L0 - iω + R2 F(ω) p
(
)
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J. Phys. Chem. B, Vol. 104, No. 25, 2000 6063
tion-dependent complex line shape L (βLD,RLD,ω) is readily obtained from the solution of eq 22 by a matrix inversion:
L(βLD,RLD,ω) ≡
F1,f 1 (ω,βLD)
+
F1,B 1 (ω,
βLD) ) PfR + PB∆X + PBex (29) NAB
The different terms in eq 29 are defined as
PfR ) Pf(R1B + iδωB + λPB) [(R2f + iδωf + λPB)(R2B + iδωB + λPf) - λ2PfPB] (30) PB∆X ) PB(R2B + iδωB + λPf)[2(R2f + iδωf + λPB)(λPf) + (R2f + iδωf + λPB)(R2f + iδωf + λPB) + ∆2] (31) PB∆ex ) -PB(λ2PfPB)[2(λPf) + (R1f + iδωf + λPB)] (32) NAB ) [(R1f + iδωf + λPB)(R2f + iδωf + λPB) + ∆2](R1B + iδωB + λPf)(R2B + iδωB + λPf) - (λ2PfPB)[(R1f + iδωf + λPB)(R1B + iδωB + λPf) + (R2f + iδωf + λPB)(R2B + iδωB + λPf) - (λ2PfPB)] (33) Then the line shape of the powder spectrum is obtained by numerical integration of the real (Re) part of L(βLD,RLD,ω) over all possible orientations:
L(ω) ) Re
∫0πL(βLD,RLD,ω) P(RLD,βLD) sin(βLD) dβLD dRLD
TABLE 1: Model Parameters of the Lr Phase model parameters
data set 2
data set 4
PB SdD 0 SPd 0 τc (ns) τf (ps) τb (ps) η T1 (ms)
(A) nD ) 9.75, PB ) 0.75 0.75 0.75 0.75 0.042 0.048 0.053 0.115 0.10 0.09 60 80 100 5 5 5 50 50 50 0 0 0 117 156 185
0.75 0.064 0.075 150 5 50 0 226
PB SdD 0 SPd 0 τc (ns) τf (ps) τb (ps) η T1 (ms)
(B) nD ) 11, PB ) 0.85 0.85 0.85 0.85 0.040 0.044 0.049 0.105 0.095 0.085 60 80 100 5 5 5 50 50 50 0 0 0 121 155 185
0.85 0.060 0.070 150 5 50 0 226
3. 2H2O NMR Line Shape Analysis of the Lr and Lβ Phases of DPPC The line shape function derived in the previous section (cf. eqs 29-34, combined with spin-lattice relaxation rate (cf. eq 28) and the quadrupole splitting (eq 11), are implemented in a Fortran program. Together they are used to reinterpret the experimental powder spectra of dipalmitoylphosphatidylcholine (DPPC)-2H2O reported by Ulmius et al.15 The lamellar phase LR and gel phase Lβ, are reproduced, and the order parameters Pd SdD 0 and S0 are determined for different τc. The line shape fitting was performed by reproducing the intensity ratio between the maximum intensity of the doublets and the intensity value at the magic angle position of the spectra. Quantitative comparisons of model parameters describing the bound water of these two phases are summarized in Tables 1 and 2. For the best fit values, the spin-lattice relaxation rates are also calculated. The line shape model depends on PB, the fraction of perturbed water molecules; λ, the chemical exchange rate; χ, the quadrupole coupling constant; τb and τc, correlation times of the dD “bound” site; SPd 0 , S0 , order parameters; η, as well as the asymmetry parameter. To ensure fast exchange conditions, we kept λ ) 107. Other parameters kept constant are χ ) 220 kHz,8 τf ) 5 ps, and τb ) 50 ps.14 The line shape fitting is carried out for a couple of correlation times τc > 1/ω0. For fixed τc we explore changes in SdD because of the phase transition 0 LR f Lβ′. The simulated spectrum in Figure 2A agrees very well with the experimental LR spectrum (see Figure la in ref 15). The 2H quadrupole splitting of the LR phase spectrum is about 595 Hz,
data set 1
TABLE 2: Model Parameters of the Lβ′ Phase model parameters
data set 2
data set 4
PB SdD 0 SPd 0 τc (ns) τf (ps) τb (ps) η T1 (ms)
(A) nD ) 3, 5, PB ) 0.50 0.50 0.50 0.50 0.163 0.187 0.201 0.08 0.070 0.065 60 80 100 5 5 5 50 50 50 0 0 0 195 224 239
0.50 0.251 0.052 150 5 50 0 261
PB SdD 0 SPd 0 τc (ns) τf (ps) τb (ps) η T1 (ms)
nD ) 4, 2, PB ) 0.60 0.60 0.60 0.60 0.145 0.168 0.182 0.075 0.065 0.060 60 80 100 5 5 5 50 50 50 0 0 0 191 222 237
0.60 0.218 0.050 150 5 50 0 258
(34) where P(RLD,βLD) is a orientation distribution density function.
data set 1
data set 1
data set 1
at a water content of 13 2H2O molecules per DPPC molecule and T ) 322.3 K. To illustrate the model, we use the hydration value of 10-11 water molecules per lipid molecule, meaning PB ) 0.75 (Table 1A) and 0.85 (Table 1B). In Tables 1 and 2 the model parameters describing the powder line shapes of LR and Lβ′ are summarized. Pd Note that for a fixed value of PB we have SdD 0 < S0 as displayed in both Tables 1A and 1B. However, the order parameters become more similar as τc increases, that is, as the spin-lattice relaxation time increases. This is a trend also observed for the gel phase parameters as shown in Table 2. In Pd this case the reverse relation, SdD 0 > S0 , is obtained. The order parameter describing the fast local average of the water molecules, SPd 0 , seems to be rather similar for the two phases. We observe that the order parameter describing the bound site is larger in the gel phase than in the liquid crystalline phase. 4. Conclusions In order to analyze the water powder line shape, the combined spin-lattice relaxation and line shape analyses were developed. The spin-lattice relaxation rate and the quadrupole splitting of the liquid crystalline phase and the gel phase of DMPC were investigated. The fraction of bound water molecules at the lipidwater interface is described by two order parameters SPd 0 and
6064 J. Phys. Chem. B, Vol. 104, No. 25, 2000
Figure 2. (A) 2H2O powder spectrum of heavy water in fast chemical exchange between a free site (χ ) 220 kHz, τf ) 5 ps) and the “bound” site of the LR phase of a D2O-dipalmitoylphosphatidylcholine system. The quadrupole splitting is about 590 Hz, PB ) 0.75, 0.85, λ ) 107, χ ) 220 kHz, τc ) 60, 80, 100, 150 ns, SdD 0 , and other parameters as in Table 1. The remaining dip at τc ) 60 ns is shown, but it disappears for τc > 65 ns. (B) 2H2O powder spectrum of the Lβ′ phase of a D2ODPPC system. The quadrupole splitting is about 1060 Hz, λ ) 107, χ ) 220 kHz, τf ) 5.0 ps, PB ) 0.50, 0.60, τb ) 50 ps as in Table 2.
SdD and a correlation time τc characterizing the dynamics 0 among different bound sites. The lipid hydration of different phases may thus be described with respect to local ordering of hydrated water. The results suggest that the order parameters describing the bound or hydrated water molecules are phase dD Pd Pd dependent with SdD 0 (LR) < S0 (Lβ′) whereas S0 (LR) ≈ S0 (Lβ′). The order parameters in gel Lβ′ are larger possibly because the temperature is significantly lower as compared to the LR phase. However, it is more likely that the gel phase loses certain types of binding sites with negative order parameters which are available in the liquid crystalline phase. This may explain the larger order and the temperature dependence of the quadrupole splitting, but in this context more studies are needed. Consider, for instance, the following example. Assume that there are only two types of water sites. There is a fraction of water deep in the surface region which is preferentially oriented perpendicular to the director. These molecules have their dipole vectors oriented preferentially along the dipoles of the lipid head groups. The result is a negative order parameter. Then there is a fraction of bound water hydrating the phosphate group with a positive order parameter. Such a simplified picture has been suggested8,15 and is also supported by recent results from a molecular dynamics (MD) simulation of DPPC.16 From the MD simulation the order parameter profile clearly displays a region with a negative order parameter for a few water molecules deep in the surface region near the carbonyl group (water density less than 0.3) and a region with positive order parameters for water further
Westlund out (water density less than 0.7). Consequently, if the fraction of bound water with orientation more likely to be perpendicular to the director decreases as we go from L R to Lβ, it may explain the increase in the order parameter SdD 0 (Lβ′). Still, the waterlipid interaction is similar in the two phases and the higher order in the gel phase is only a consequence of cancellation in eq 8. A second factor which can be important in line shape analysis is inhomogeneous broadening. We think that the presence or absence of the powder dip in water powder line shapes indicates that relaxation effects dominate the line shape. To sum up the result: Combining line shape analysis and quadrupole splitting measurements with spin-lattice relaxation dD measurement, we obtained SdD 0 (LR) > S0 (Lβ′). By measuring spin-lattice relaxation rates, also the dynamics of the bound water my be determined. This model is sufficiently complex for analyzing the lipid hydration of liquid crystalline and gel phases. Acknowledgment. I am grateful to Profs. Go¨ran Lindblom and Ha˚kan Wennerstro¨m for references and many valuable discussions and to Prof. Lennart B.-A° . Johansson for critically reading the manuscript. Financial support was given by the Swedish Natural Science Research Council (NFR) and the Faculty of Science and Technology at Umea˚ University. The Fortran program calculating the powder line shape and spinlattice relaxation rates is available upon request from
[email protected]. References and Notes (1) Schmidt-Rohr, K.; Spiess, H. W. Multidimensional Solid-State NMR and Polymers; Academic Press: New York, 1994; Chapter 2. (2) Lindblom, G. AdV. Lipid Meth. 1996, 3, 137. (3) Stepanov, A. G.; Maryasov, A. G.; Romannikov, V. N.; Zamanev, K. I. NMR Chem. 1994, 32, 16. (4) Westlund, P.-O. J. Magn. Reson. 2000, in press. (5) Faure, C.; Bonakdar, L.; Dufourc, E. J. FEBS Lett. 1997, 405. (6) Arnold, K.; Pratsh, L.; Gawrisch, K. Biochim. Biophys. Acta 1983, 728, 121. (7) Demel, R. A.; Jansen, J. W. C. M.; van Dijck, P. W. M.; van Deenen, L. L. M. Biochim. Biophys. Acta 1976, 465, 1. (8) Halle, B.; Wennersto¨m, H. J. Chem. Phys. 1981, 75, 1928. (9) Wennerstro¨m, H.; Lindblom, G.; Lindman, B. Chem. Scr. 1974, 6, 97. (10) Wennerstro¨m, H.; Lindman, B.; So¨derman, O.; Drakenberg, T.; Rosenholm, J. B. J. Am. Chem. Soc. 1979, 101, 6860. (11) Slichter, C. P. Principle of Magnetic Resonance; Springer-Verlag: New York, 1963; Chapter 7. (12) Westlund, P.-O.; Wennerstro¨m, H. J. Magn. Reson. 1982, 50, 451. (13) Brink, D. M.; Satchler, G. R. Angular Momentum; Clarendon Press: 1993. (14) Volke, F.; Eisenbla¨tter, S.; Galle, J.; Klose, G. Chem. Phys. Lipids 1994, 70, 121. (15) Ulmius, J.; Wennerstro¨m, H.; Lindblom, G.; Arvidsson, G. Biochemistry 1977, 16, 6742. (16) Edholm, O.; Lindahl, E. Personal communication, 1999.
