INTRODUCTION

TOP ABSTRACT INTRODUCTION EXPERIMENTAL METHODS DEUTERIUM NMR RESULTS THEORY STRUCTURAL RESULTS DISCUSSION REFERENCES

A comprehensive understanding of structure-function relations of biomembranes ultimately relies on knowledge of the properties of lipid bilayers in relation to lipid-protein interactions. The complex structural dynamics of membranes involve a balance of forces characteristic of amphiphilic systems, as manifested by both the protein and lipid structural parameters and their connection to specific biomembrane functions (Brown, 1994; Epand, 1998; Bezrukov et al., 1998; White and Wimley, 1999). One quantity that describes the bilayer microstructure with regard to molecular packing is the average interfacial area per lipid molecule A, which is defined in reference to the average hydrocarbon thickness D_C and the hydrocarbon chain volume V_C. The precise values of these lipid structural parameters are determined by the interactions between the various lipid bilayer constituents, and their measurement provides a practical way to acquire information on the intermolecular interactions. By studying the interaction of lipid molecules, one is better positioned to further explore more complex systems, such as lipid-protein assemblies, in terms of the effect of membrane components on the bilayer thermodynamic parameters, and, in particular, on the area per lipid.

The question is thus how to obtain accurate estimates of structural parameters for the disordered, but biologically relevant fluid state as a mean of relating the bilayer microstructure to the characteristic force balance responsible for lipid self-assembly. In this regard, the most powerful biophysical techniques are small-angle X-ray scattering (Tardieu et al., 1973; Worthington et al., 1973; Lewis and Engelman, 1983; McIntosh and Simon, 1986; Rand and Parsegian, 1989), small-angle neutron scattering (Zaccai et al., 1979; Wiener and White, 1992; Lemmich et al., 1996), Fourier transform infrared spectroscopy (FTIR) (Mendelsohn et al., 1989; Davies et al., 1992; Tuchtenhagen et al., 1994), and deuterium (²H) NMR spectroscopy (Seelig and Seelig, 1974; De Young and Dill, 1988; Ipsen et al., 1990; Thurmond et al., 1991; Koenig et al., 1997). Moreover, computer simulations of lipid bilayers are useful in revealing important details of the structural dynamics in terms of the underlying force fields (Pastor and Feller, 1996). The structural parameters D_C and A can be directly determined from scattering experiments in special cases where relatively high positional ordering of membrane components is present (Nagle and Wiener, 1988; Tristram-Nagle et al., 1993; Sun et al., 1994). More often though, especially in the fluid state at full hydration, the direct measurement of the area per lipid using scattering techniques is a much more difficult task, even for pure lipid bilayers (McIntosh and Simon, 1986; Nagle et al., 1996; Koenig et al., 1997; Petrache et al., 1998a; Tristram-Nagle et al., 1998).

Deuterium NMR spectroscopy has been a powerful tool for investigations of biological membranes. Although it measures orientational ordering as opposed to positional order, it reveals detailed aspects with regard to the lipid chain packing from which structural parameters can be obtained. Since its early use (Seelig and Seelig, 1974; Mely et al., 1975), ²H NMR has provided a wealth of information regarding the organization of membrane components, especially in regard to the lipid polar headgroups (Brown and Seelig, 1977, 1978) and acyl chains (Seelig, 1977; Salmon et al., 1987). The ²H NMR order profile of the acyl chains is very sensitive to physicochemical conditions, such as temperature, concentration, and hydration level, and therefore it has been used to measure the response of the membrane to these various factors (Bloom et al., 1991; Jansson et al., 1992; Morrow et al., 1992; Koenig et al., 1997). Another important use of ²H NMR spectra of lipid acyl chains is to describe the influences of membrane components, such as cholesterol (Oldfield et al., 1978; Trouard et al., 1999) and hydrophobic peptides (De Planque et al., 1998; Koenig et al., 1999), on the lipid bilayer. For a complete understanding of these effects one needs to relate the measured ²H NMR spectra to physical properties of the lipid chains.

A quantitative interpretation of the order parameters in terms of membrane thermodynamic parameters and, in particular, the membrane structure has been particularly challenging. This is because the complexity of the lipid bilayer (i.e., the large number of degrees of freedom) makes statistical mechanical calculations starting from first principles intractable (see Nagle, 1980, for a review). As a result, simplified chain conformation models have been considered, which attempt to capture the most relevant degrees of freedom in terms of the appropriate distributions. One example is the diamond lattice model (Seelig and Seelig, 1974; Salmon et al., 1987), from which one obtains a simple linear relationship between the ²H NMR order parameter S_CD⁽ⁱ⁾ of the chain segment (i), and the corresponding projection D_i along the membrane normal (Schindler and Seelig, 1975; Salmon et al., 1987; Thurmond et al., 1991; Nagle, 1993; Douliez et al., 1995), namely,

(1)

Here, D_i is the distance between carbons i + 1 and i  1, projected on the bilayer normal, and D_M = 2.54 Å is the maximum possible projection. The brackets in Eq. 1 represent an ensemble average over chain conformations. By summing over consecutive chain segments (i), one can calculate the full chain projection, or partial distances between different carbon atoms along the chain. Knowing the chain length, the area per lipid can then be calculated by using information on the chain volume (Thurmond et al., 1991; Nagle, 1993; Brown, 1996). Convenient as it is, Eq. 1 is nonetheless an unexpected result, because the order parameter is a quadratic function of the segment tilt (the second-order Legendre polynomial). Clearly, Eq. 1 is a linear approximation of the real underlying relationship between D_i and S_CD⁽ⁱ⁾. This linear form has proved to be quite effective in the past, giving results in fairly good agreement with scattering experiments that measure structural parameters more directly. But with newer and more precise measurements, discrepancies between the analysis of ²H NMR and X-ray data regarding the area per lipid have become increasingly noticeable (Koenig et al., 1997; Petrache et al., 1999). Consequently, Eq. 1 and other similar linear relationships are open to reevaluation.

An alternative to the diamond lattice model was recently proposed by Petrache et al. (1999). Rather than considering a discrete set of segmental conformations, corresponding to the most populated internal dihedral angles, the new model is based on a continuum distribution of segmental orientations. This continuum distribution, here denoted by f(D), is generated by the combined effect of internal degrees of freedom (dihedral angles) and chain motion as a whole (chain tilt). Based on this model, a new analytical relation was derived (Petrache et al., 1999), namely

(2)

where the brackets indicate the ensemble average generated by the orientational distribution function f(D). This expression has been shown to work better than the diamond lattice result for the calculation of chain segment projections (Petrache et al., 1999; Smondyrev and Berkowitz, 1999). However, when further applied to estimate the area per lipid using ²H NMR data, it gave discrepancies with accepted X-ray results (Petrache et al., 1999).

The calculation of the area per lipid A from ²H NMR data is unfortunately not straightforward, starting with the definition of the area itself. Although various moments of the distribution f(D), in particular the average projection D, can be defined and calculated in a straightforward manner, the same is not true for the average area A. One possible approach to connecting the geometrical properties along the bilayer normal D and perpendicular to the bilayer normal A is to define an instantaneous area A  4V_CH2/D, where V_CH2 denotes the volume of the chain segment. Then, as pointed out by Jansson et al. (1992), the calculation of the average area A amounts to calculation of 1/D. The evaluation of 1/D requires knowledge of the functional form of the distribution function f(D), or alternatively, knowledge of the moments of f(D), as discussed by Jansson et al. (1992) and Petrache et al. (1999). Here we propose a methodology for area calculation that takes advantage of the continuum description of segmental conformations. It is important to note that continuum models have been extensively used in the analysis of NMR relaxation data (Brown, 1982; Halle, 1991; Trouard et al., 1994; Brown and Chan, 1995; Althoff et al., 1996; Nevzorov et al., 1998), whereas the analysis of order parameters has tended to involve discrete descriptions. It is therefore desirable to combine the relaxation and order parameter analyses into a more consistent picture.

In what follows, we present the results of an extensive series of experimental ²H NMR measurements of disaturated phospholipids in the fluid lamellar (L) state. Motivated by this ²H NMR data set, and by recent X-ray scattering measurements (Nagle et al., 1996; Koenig et al., 1997; Petrache et al., 1998a), we then reformulate the continuum theory in terms of an effective orientational potential (mean-torque potential) (Brown, 1996). The latter is related to the distribution function for the segment orientations, and provides insight into the intermolecular interactions that govern the microstructure of phospholipid/water dispersions. Finally, the experimental data are interpreted in terms of the theory to yield a new conceptual framework for the analysis of bilayer structural properties.

	EXPERIMENTAL METHODS

TOP ABSTRACT INTRODUCTION EXPERIMENTAL METHODS DEUTERIUM NMR RESULTS THEORY STRUCTURAL RESULTS DISCUSSION REFERENCES

Phospholipid synthesis and sample preparation

High purity (99%) fatty acids were obtained from Sigma (St. Louis, MO) and were perdeuterated by catalytic exchange of ²H for ¹H over a 10% Pd-charcoal catalyst (Aldrich, Milwaukee, WI) at 200°C. The perdeuterated fatty acids were more than 99% pure by gas-liquid chromatography on 10% SP-2330 (Supelco, Bellafonte, PA), and were typically 92-96% labeled with ²H according to mass spectrometry. Symmetric (like chain) phospholipids were synthesized from the cadmium chloride adduct of sn-glycero-3-phosphocholine and the corresponding perdeuterated fatty acid anhydrides, as described by Salmon et al. (1987). Representative yields based on sn-glycero-3-phosphocholine were 75-90%. The asymmetric phosphatidylcholine, 1-palmitoyl-2-perdeuteriopalmitoyl-sn-glycero-3-phosphocholine (DPPC-d₃₁), in which the sn-2 chain was perdeuterated, was synthesized as follows. First, DPPC was reacted with snake venom phospholipase A₂ (Sigma); the resulting 1-palmitoyl-sn-glycero-3-phosphocholine was purified by column chromatography, and finally the lyso phosphatidylcholine was reacylated at the sn-2 position using the anhydride of perdeuteriopalmitic acid. After purification by column chromatography on silicic acid, all phospholipids gave single spots upon thin-layer chromatography in CHCl₃:MeOH:H₂O (65/35/5) followed by charring with 40% H₂SO₄ in EtOH.

Multilamellar samples were prepared by vortexing ~100-150 mg phospholipid together with an equal weight of ²H-depleted ¹H₂O (Aldrich) above the bilayer phase transition temperature. The 50 wt % multilamellar dispersions were then centrifuged and sealed in cutoff ~6 or 10-mm-diameter Kimax- or Pyrex-type glass culture tubes with Teflon plugs. The samples were stored at 85°C when not in use.

Deuterium NMR methods

Deuterium NMR spectroscopy of the multilamellar lipid dispersions involved generation of the solid (quadrupolar) echo followed by Fourier transformation. Measurements were performed at 55.43 MHz with a phase-cycled, quadrupolar echo sequence, (/2)_x  2  (/2)_y  2  acquire. A homebuilt ²H NMR probe was used having a horizontal solenoidal radiofrequency coil design together with high-voltage capacitors (Polyflon, Norwalk, CT). A kilowatt radiofrequency boost amplifier (Model Tempo 2006, Henry Radio, Los Angeles, CA) was used in series with the spectrometer output to enable 90° pulse durations <3-4 µs for the 10-mm radiofrequency coil. The transient NMR signals were shunted to the preamplifier (Miteq, Hauppauge, NY) using a Lowe-Tarr series crossed diode circuit, and were acquired with a fast digitizer. Typical spectral acquisition parameters involved a pulse spacing (₂) of 40 µs, a dwell time of 2 µs (spectral width of ±250 kHz), and collection of 2048 data points. Recycle times were generally 1 s, and typically 600-2400 transients were collected, apodized by exponential multiplication (100-Hz line-broadening), and Fourier transformed beginning at the maximum of the solid echo. Both quadrature channels were used and the spectra were not symmetrized. The sample temperature was monitored before and after each measurement with a thermistor inserted directly above the radiofrequency coil, and was usually found to vary by less than 0.5°C during a given run. The temperatures are estimated accurate to within ±1°C of the reported values. All samples were subsequently checked by thin-layer chromatography and revealed no contamination or degradation.

Data analysis and reduction: Spectral assignments

The ²H NMR spectral powder patterns were numerically deconvoluted (de-Paked) to yield the  = 0° oriented spectra as described (Bloom et al., 1981). The most useful characteristic of the de-Pakeing algorithm is that it leads to increased spectral resolution. In addition, it reveals the "Me-2-3-2" spectral pattern (see below) and the large "plateau" peak, which are characteristic of perdeuterated disaturated phosphatidylcholines (see Fig. 1). Spectral assignments for multilamellar dispersions of 1,2-diperdeuteriomyristoyl-sn-glycero-3-phosphocholine (DMPC-d₅₄) and 1,2-diperdeuteriopalmitoyl-sn-glycero-3-phosphocholine (DPPC-d₆₂) were made by comparison to ²H NMR results for the corresponding specifically deuterated phospholipids (Seelig and Seelig, 1974; Oldfield et al., 1978). For DPPC-d₆₂, assignments of the splittings due to the sn-1 and sn-2 chains entailed comparison of the de-Paked ²H NMR spectra of DPPC-d₆₂ (both chains deuterated) to DPPC-d₃₁ (sn-2 chain deuterated). It was assumed that similar splittings occur for the other phosphatidylcholines, viz. 1,2-diperdeuteriolauroyl-sn-glycero-3-phosphocholine (DLPC-d₄₆), DMPC-d₅₄, and 1,2-diperdeuteriostearoyl-sn-glycero-3-phosphocholine (DSPC-d₇₀). Further peak assignments were made assuming the order parameters decrease in magnitude from C₂ ( carbon) to the methyl end, and that the area under each peak is proportional to the number of deuterons generating the peak (Williams et al., 1985). The only exception was the C₂ segment of the sn-2 chain, for which the splittings were assigned in the case of DPPC-d₆₂ based on literature data for specifically deuterated lipids (Seelig and Seelig, 1974); it was assumed that the other lipids yielded similar C₂ splittings. Additionally, the assignments were made from the methyl terminus toward the plateau region. The terminal methyl groups possess the smallest quadrupolar splittings, and hence are the simplest to assign. The "2-3-2" pattern is then readily assigned, followed by the remainder of the fatty acyl chains.

View larger version (28K): [in this window] [in a new window]   FIGURE 1   (a-c) Representative 2H NMR spectra of homologous series of disaturated phosphatidylcholines with perdeuterated acyl chains. Samples were fully hydrated and were in the lamellar fluid phase at 50°C. Powder-type 2H NMR spectra are shown at left, and the corresponding de-Paked 2H NMR spectra at right.

The C-D bond order parameters, S_CD⁽ⁱ⁾, of the de-Paked ²H NMR spectra were calculated from the quadrupolar splittings using the relation (Brown, 1996):

(3)

where _Q  (e²qQ/h) = 167 kHz (Davis, 1983),  represents the angle between the bilayer director axis and the static magnetic field direction, and P₂(cos ) = (3 cos²  1)/2, which, for  = 0°, is equal to unity. Based on geometrical considerations the values of S_CD⁽ⁱ⁾ are assumed to be negative.

	DEUTERIUM NMR RESULTS

TOP ABSTRACT INTRODUCTION EXPERIMENTAL METHODS DEUTERIUM NMR RESULTS THEORY STRUCTURAL RESULTS DISCUSSION REFERENCES

Disaturated phosphatidylcholines in the liquid-crystalline state

Representative ²H NMR spectra at 50°C are shown in Fig. 1 for a homologous series of disaturated phosphatidylcholines having different acyl lengths. Powder-type ²H NMR spectra of the randomly oriented multilamellar dispersions (fully hydrated; 50 wt % H₂O) are shown at left, and the corresponding deconvoluted (de-Paked) ²H NMR spectra due to the  = 0° bilayer orientation are shown at right. Several conclusions can be immediately drawn from the ²H NMR lineshapes in Fig. 1. First, the ²H NMR spectra are indicative of a uniform spherical distribution of the bilayer director axes, and little or no orientation of the bilayers by the relatively high magnetic field strength of 8.48 T is evident. This is supported by the de-Paked ²H NMR spectra at right in Fig. 1, which are of relatively high quality and evince little or no distortion. Second, the ²H NMR spectra are characteristic of axially symmetric motions of the phospholipids about the bilayer normal (director axis), with a distribution of quadrupolar splittings due to the various inequivalent methylene and methyl groups of the perdeuterated acyl chains. Moreover, the sn-1 and sn-2 acyl groups are inequivalent, as revealed by ²H NMR spectra of multilamellar dispersions of DPPC-d₆₂ (both chains perdeuterated) compared to DPPC-d₃₁ (sn-2 chain deuterated) (spectra not shown). This inequivalence is due to variations in the motional behavior of the individual acyl segments. Finally, the magnitudes of the quadrupolar splittings indicate that the phospholipids possess considerable disorder due to trans-gauche rotational isomerizations and other degrees of freedom (see below). The distribution of the quadrupolar splittings is related to the acyl length and cross-sectional area distributions as considered in further detail below.

Of particular interest here is the dependence of the ²H NMR spectra of the homologous phosphatidylcholines on the length of the acyl chains. Both the ²H NMR powder-type spectra (left) of the randomly oriented multilamellar dispersions, and the corresponding de-Paked spectra (right) show that the quadrupolar splittings increase with increasing chain length. In particular, the de-Paked ²H NMR spectra allow for a better comparison between the lipids due to the increased resolution of the various inequivalent quadrupolar splittings. As a rule, disaturated phosphatidylcholines in the fluid state are distinguished by the Me-2-3-2 spectral pattern present at all temperatures above the melting transition, as can be seen clearly in the de-Paked ²H NMR spectra at right in Fig. 1. Specifically, moving away from the central methyl peak, the quadrupolar splittings fall into groups of 2, 3, and 2 peaks, followed by the larger peaks with the greatest quadrupolar splittings. The latter represent the so-called plateau region because it contains the unresolved splittings corresponding to acyl segments close to the glycerol backbone (beginning of the acyl chain), all of which possess a similar degree of disorder. With increasing acyl length at fixed temperature, the plateau peak becomes larger and it shifts toward larger quadrupolar splittings. This behavior indicates that the length of the plateau acyl chain region increases and becomes less disordered as more mass is added in the hydrocarbon region.

Order profiles: Effect of acyl length

The above observations are more evident upon reduction of the ²H NMR spectral data in terms of order parameter profiles. The spectral assignments and the corresponding quadrupolar splittings for the  = 0° orientation, (_Q) obtained as described in the Experimental Methods section, are summarized in Tables 1-4. The order parameters S_CD⁽ⁱ⁾, which are model free, are plotted as a function of the acyl segment position in Fig. 2, and provide a quantitative measure of the degree of order along the lipid acyl chains. In each case, the order parameters of the initial segments, close to the glycerol and headgroup region, show a broad plateau followed by a reduction in ordering toward the central region of the bilayer. The characteristic order profile is due to the tethering and alignment of the acyl segments near the aqueous interface, with the increased disorder in the bilayer center due to chain terminations. The order parameters for each lipid decrease with T, reflecting the increased chain disorder resulting from raising the temperature (entropic effect). The largest influence of temperature is on the initial plateau region of the acyl chains, which becomes both more disordered and narrower with increasing T, leading to an overall narrowing of the order profiles.

View this table: [in this window] [in a new window]   TABLE 1   De-Paked 2H NMR spectral assignments and (Q) splittings for DLPC-d46 in the liquid-crystalline (L) phase

View this table: [in this window] [in a new window]   TABLE 2   De-Paked 2H NMR spectral assignments and (Q) splittings for DMPC-d54 in the liquid-crystalline (L) phase

View this table: [in this window] [in a new window]   TABLE 3   De-Paked 2H NMR spectral assignments and (Q) splittings for DPPC-d62 in the liquid-crystalline (L) phase

View this table: [in this window] [in a new window]   TABLE 4   De-Paked 2H NMR spectral assignments and (Q) splittings for DSPC-d70 in the liquid-crystalline (L) phase

View larger version (30K): [in this window] [in a new window]   FIGURE 2   (a-d) Measured C-D bond order parameter profiles as a function of carbon number along the acyl chains for the homologous series of disaturated phosphatidylcholines with acyl lengths from nC = 12 to nC = 18. Data for sn-1 and sn-2 chains are shown separately (but with the same symbols) for each temperature. The double resonances for the C2 carbon of the sn-2 chain are also shown. For each lipid, the order parameters decrease in magnitude with increasing temperature, and the effect is larger for the plateau region.

Further perspective on the order parameter results is gained by comparison of the data for the different lipids at a fixed absolute temperature, as shown in Fig. 3. A first observation is that the plateau regions are larger in magnitude and broader for longer chains. This is basically opposite to the temperature effect described above. However, although the plateau region shows a strong chain-length dependence at any given temperature, the nonplateau segments (chain ends) are practically independent of chain length. This second observation suggests that, at least in a first approximation, the differences in the structure and dynamics of different acyl chain lengths at the same absolute temperature can be obtained just from the analysis of the plateau regions.

View larger version (28K): [in this window] [in a new window]   FIGURE 3   (a-d) Data as in Fig. 2, plotted at the same absolute temperature in each panel. At any given temperature, the plateau order parameters are larger for longer chains, whereas nonplateau order parameters show little variation.

One particular goal of the ²H NMR data analysis is to relate the above order parameter profiles to the average projections of the acyl chain segments onto the bilayer normal, with the purpose of calculating the average lipid chain length and the area per lipid in the fluid state. To achieve this goal, one needs to construct a statistical model for the segmental configurations.

	THEORY

TOP ABSTRACT INTRODUCTION EXPERIMENTAL METHODS DEUTERIUM NMR RESULTS THEORY STRUCTURAL RESULTS DISCUSSION REFERENCES

Segmental motions

The S_CD parameters in ²H NMR spectroscopy comprise experimental observables that are measures of the carbon-deuteron (CD) bond orientations with respect to the static external magnetic field axis. Let _PL be the instantaneous angle between the C-D bond and the direction of the static external magnetic field B₀. In terms of the rotation group parameters, _PL is the polar angle (colatitude) of the Euler angles _PL = (_PL, _PL, _PL) that rotate the quadrupolar coupling tensor from its principal axes system P to the laboratory frame L defined by the magnetic field (see Fig. 4). The quadrupolar splitting _Q measured by ²H NMR spectroscopy detects the ensemble average (Brown, 1996)

(4)

Here, _Q = _Q⁺  _Q, where _Q^± = _±  0 with ₀ being the Larmor frequency and _± = ±(E₁  E₀)/h the eigen-frequencies of the single quantum transitions of the I = 1 ²H nucleus. Additionally, _Q  e²qQ/h is the static quadrupolar constant; D₀₀⁽²⁾(_PL) is the Wigner rotation matrix element with angular momentum j = 2 and projections m = m' = 0; and P₂ is the second-order Legendre polynomial. The above result is obtained with the assumption that the coupling tensor is axially symmetric (Brown, 1996). The brackets in Eq. 4 indicate an average over the tensor orientations sampled on the NMR time scale, i.e., over the motions that occur on a time scale comparable to, or less than, the inverse quadrupolar splitting. Because of thermal motion, the segmental tensor orientation with respect to the laboratory frame is time dependent, and this fact is indicated in Eq. 4 above.

View larger version (25K): [in this window] [in a new window]   FIGURE 4   Intermediate frames used to describe motions of a D-C-D group in the lipid acyl chains: principal axis system of a C-D bond (P), internal frame (I), local director frame (N), global (average) director frame (D), and laboratory frame (L). Transformations between frames are carried out by rotations with the Euler angles XY, where X and Y are generic frame indices. The figure serves for illustrative purposes only and therefore is not drawn to scale.

In principle, one expects the presence of contributions from a hierarchy of motions including segmental motions, molecular motions, and collective fluctuations of the bilayer itself. For the treatment of these various motions, it is convenient to expand the Wigner matrix D₀₀⁽²⁾(_PL; t) in a sequence of frame transformations, using the closure property of the rotation group. In a general formulation, one can write (Brown, 1996),

(5)

where all summations run from 2 to 2. The intermediate frames considered here are the internal frame, I, the molecular axis system, M (not shown in Fig. 4), the local director frame, N, and the average director frame, D. The z axis of the internal frame I is taken to be the normal to the D-C-D plane, and therefore _PI = 90°. For each chain segment i, this axis gives the orientation of the virtual bond C_i1-C_i+1, which we designate as the orientation of segment i. The projection of this virtual bond on the bilayer normal is denoted by D_i. Note that these virtual bond vectors differ from the previous definition of Salmon et al. (1987). Clearly, the expansion of D₀₀⁽²⁾(_PL; t) can include any arbitrary number of coordinate transformations, depending on the motional model considered. The expansion in Eq. 5 is a general example that can be reduced by collapsing those transformations not present in the model (see below).

The C-D order parameter for each chain segment i is defined with respect to the bilayer director frame D, as the ensemble average

(6)

which, in terms of Eq. 5, includes the first four transformations on the right-hand side. The last Wigner matrix in Eq. 5 describes the fixed transformation from the average director D, about which there is cylindrical symmetry, to the laboratory frame L, and is accounted for by the de-Pakeing procedure. The above development can be used to relate the spectroscopic observables, viz., the measured C-D bond order parameters, to order parameters that are more convenient to use in describing the structure and dynamics of the acyl chains. How can one compute these ensemble averages? In general, any statistical property () (here  is a generalized Euler angle) can be expressed in terms of a distribution function f() which gives the ensemble average,

(7)

The orientational distribution function f() can be expanded in a series of orthogonal polynomials, e.g., the Legendre polynomials P_j(cos ),

(8)

where the P_j(cos ) comprise the various moments of the distribution, i.e., order parameters. Clearly, knowledge of all the moments is required to specify the distribution function. However in many cases only the lower moments are available. In this formulation, the order parameter S_CD measured by ²H NMR spectroscopy is related to the second moment P₂(cos ) of the orientational distribution function f(). As a rule, f() is a function of both even- and odd-rank order parameters, P_j(cos ), including of particular interest the odd-rank term P₁(cos ), which is related to the acyl chain segmental projection on the bilayer normal. One therefore needs to use a model of segmental conformations to reconstruct P₁(cos ) given P₂(cos ). In other words, one needs to assume a functional form for the orientational distribution function f(). The conventional approach, namely the diamond lattice model, is briefly reviewed in the next section, followed by a detailed description of the continuum model developed in this paper.

The diamond lattice model

The diamond lattice model developed by Schindler and Seelig (1975) has been extensively used to model the measured order parameters. Here we give a short derivation in terms of the orientational distribution function, f(_IM), of the chain segments (virtual bonds) relative to the molecular axis M (Salmon et al., 1987; Jansson et al., 1992; Douliez et al., 1995). In this model, the transformations considered are from the principal axis system P to the internal frame I, then from I to the molecular system M, where M is taken as coincident with the local director frame N (see Eq. 5). The model assumes that the C-D bond orientations fall on a tetrahedral lattice with segment orientations _i = 0°, 60°, 90°, 120°, 180°, where the subscript IM has been suppressed for clarity. For such a model of discrete segmental orientations, the moments of the segmental distribution are given by

(9)

where the probabilities p_i are normalized (_i p_i = 1). Evidently, substitution into Eq. 8 gives the distribution function,

(10)

where  denotes the Dirac delta function. By construction of the model, the distribution function f() is basically a sum of delta functions centered at discrete orientations _i. As noted above, the average value of any angular dependent quantity can then be obtained as

(11)

where

(12)

and

(13)

In this framework, neglecting the contribution from the orientations at 120° and 180° due to backfolding of the chains (Schindler and Seelig, 1975; Salmon et al., 1987), one obtains for the C-D bond,

(14)

where S  p₂(cos ) and

(15)

Given that the probability function is normalized (p₀ + p₆₀ + p₉₀  1), the above are combined to yield Eq. 1 of the Introduction. The acyl chain length in the molecular frame M, relative to the all-trans state, is then calculated by summing over all carbon segments (Salmon et al., 1987), giving

(16)

where the last term of the sum represents the contribution of the terminal methyl. Because the configurational statistics of the segments are only considered with respect to the molecular frame, the above model neglects molecular motions and collective bilayer motions (see below).

The mean-torque model

Using the same framework, one can also consider a continuum model of segmental orientations, which has been shown to be superior to the diamond lattice model result (Petrache et al., 1999; Smondyrev and Berkowitz, 1999). In this model, we retain from Eq. 5 the intermediate frames shown in Fig. 4, namely the internal frame I, the local director frame N, and the average director frame D. Assuming that i) there is cylindrical symmetry about the local director, and ii) the local motions with respect to the local director axis and the director fluctuations are statistically independent, we have that the observed second-rank order parameter is given by

(17)

The first bracket of the right-hand term in the above expression is the molecular order parameter of the ith segment with respect to the local director n(t); as such, it combines internal degrees of freedom (chain isomerization) and molecular motion relative to the local director n(t). The second bracket in Eq. 17 is the order parameter of the local director itself with respect to the global (average) director, n₀, and describes the order director fluctuations (ODF). Similar to Eq. 17, we can write the corresponding relation for the first-rank order parameters,

(18)

For a statistical treatment of the segmental configurations, we assume that the orientational order for each chain segment i, relative to the local director frame N, can be described by a mean-field orientational potential (potential of mean torque), denoted by U(_IN⁽ⁱ⁾). In what follows, we absorb the superscript i and the subscripts IN for clarity, and consider the series expansion of the mean-torque potential U() in terms of Legendre polynomials,

(19)

It should be remarked that the above decomposition includes both even and odd parity terms. In particular, the presence of a nonvanishing odd term is a consequence of tethering of the acyl chains, within a given monolayer, to the aqueous interface (Halle, 1991; Trouard et al., 1992). The energy parameters U₁ and U₂ depend upon various factors such as the chain position i, temperature, pressure, and hydration level. Mathematically, U₁ and U₂ are the moments of the function U() in terms of the Legendre polynomials. Note that U() completely specifies the distribution function f(), and hence all the moments P_j(cos ). The functional form of the distribution function is given by the Boltzmann factor, leading to

(20)

with the partition function Z being

(21)

(Note that the orientational potential U() is defined up to a -independent additive term, and this fact should be taken into account in free energy calculations.) For convenience of notation, we introduce the following dimensionless parameters:

(22)

(23)

(24)

Given the orientational preference of the lipid chains, this particular choice for the sign makes ₁ a positive quantity for the upper monolayer, i.e., the chain segments in the upper monolayer have mostly positive projections on the bilayer normal (see Fig. 4). The opposite is true for the lower monolayer, because the two monolayers are related by inversion (Trouard et al., 1994). Using the above notations, we can rewrite Eq. 21 up to second order as

(25)

It is useful to note the features of the probability distribution generated by U(x) (called the singlet orientational distribution function), namely

(26)

as a function of the mean-torque parameters ₁ and ₂. For the analysis of ²H NMR data, we are interested in the ensemble averages x and (3x²  1)/2, which are precisely the Legendre coefficients (moments) of the orientational distribution f(x). Figure 5 shows semilogarithmic plots of f(x) for an odd-rank parameter ₁ = 3 (a typical value for the lipids considered) and three different values for the even-rank parameter ₂. A positive ₂ increases the probability at x = ±1 versus ₂ = 0. In particular, the increase around x = 1 corresponds to the presence of chain end "upturns" (Nagle, 1993), as measured experimentally by Nuclear Overhauser Effect NMR spectroscopy (Xu and Cafiso, 1986; Huster et al., 1999) and observed in molecular dynamics simulations (Petrache et al., 1999; Feller et al., 1999). Conversely, a negative ₂ gives rise to a maximum of f(x), as shown in Fig. 5, that shifts from x = 1 toward x = 0 as the absolute magnitude of ₂ increases (not shown). This models the situation when the most probable segment orientation is tilted (i.e., x  1) relative to the local z-axis. We emphasize that one should make a distinction between the most probable tilt and the average tilt x, the latter being generally smaller in magnitude than the former (see Pastor et al., 1988, 1990, on lipid "wobbling" motion). This fact is a direct consequence of the strong asymmetry of f(x) about x, and, as we show later, it plays a major role in the calculation of the area per lipid.

View larger version (16K): [in this window] [in a new window]   FIGURE 5   Semilogarithmic plots of orientational distribution f(x) of the chain segment tilt x cos . Results are shown for an odd-rank parameter of 1 = 3 and even-rank parameter of 2 = 2, 0, and 2. The 2 > 0 case models upturns, whereas 2 < 0 models the case when the point of maximum probability shifts away from x = 1.

For a complete description of the orientational potential U(x), one would like to determine the mean-torque parameters ₁, ₂, ... , from which thermodynamic quantities can be calculated. Equivalently, the orientational distribution function f(x) can be completely reconstituted if one knows all its moments

(27)

In practice, the ²H NMR observables give only the second-rank moment x², via its relationship with the S_CD order parameter,

(28)

which is obtained from Eq. 17 assuming a negligible contribution from order director fluctuations. With only one available constraint, i.e., the value of x², the two parameters, ₁ and ₂, cannot be determined independently; rather, one finds a set of solutions in the (₁, ₂) plane. Such solutions, obtained numerically, using Eqs. 25-28, are presented in Fig. 6 a for S_CD values within the experimental range.

View larger version (39K): [in this window] [in a new window]   FIGURE 6   (a) Curves of constant SCD in the (1, 2) plane, obtained by solving Eq. 27 for k = 1 and k = 2 self-consistently. (b) Dependence of x on 2 for given values of SCD. Because the acyl chains are tethered to the water interface, the 1 term in the orientational distribution function f(x) is dominant, especially for the initial segments (plateau). In consequence, the region 2 > 1 is excluded as explained in the text. Part (b) shows that, for large enough |SCD| values (plateau region), the value of x is practically independent of 2.

For the analysis of the acyl chain average structure, we seek to determine the average segment projection x, i.e., the first moment of f(x), given the second moment x², for all segments along the chain (Jansson et al., 1992). As found in molecular dynamics simulations (Petrache et al., 1999), the distributions f(x) are all reasonably well modeled by simple exponential functions, meaning that the ₁ term is the dominant term. We can therefore consider that, in general, |₂| < ₁, which is especially true for the plateau segments, which are closer to the water interface and therefore feel a stronger restoring potential. The excluded region |₂| > ₁ is indicated in Fig. 6 by the gray area. The solutions in Fig. 6 a correspond to the quantity of interest, namely x, which is shown in Fig. 6 b as a function of ₂ for the given values of S_CD. Again, the figure emphasizes the allowed region |₂| < ₁. We observe that, for large values of S_CD corresponding to the plateau region in the ²H NMR profiles, the first moment x is almost insensitive to ₂, as revealed by the plateau regions of x in the vicinity of ₂ = 0. This is an important aspect, because it justifies the use of an ₂ = 0 model for the treatment of plateau carbons. Therefore, as a first approximation, we can set ₂ = 0 when S_CD is safely large, and, in this way, eliminate one unknown variable. The only unknown variable left is ₁, which can now be uniquely determined for each given S_CD by solving Eqs. 25-28 self-consistently (intersection of the constant S_CD curves in Fig. 6 a with the vertical dashed line at ₂ = 0). We will call this the first-order mean-torque model (MT), because it only involves the first term in the Legendre expansion of the mean-torque potential (Eq. 19). In the framework of this model, from each measured S_CD⁽ⁱ⁾, one finds the corresponding mean-torque parameter ₁⁽ⁱ⁾, assuming that ₂⁽ⁱ⁾ = 0. The average chain projections x⁽ⁱ⁾ are then calculated using the values of ₁⁽ⁱ⁾ obtained in this way. Note that we have reintroduced the chain segment index i to emphasize that the mean-torque parameter ₁⁽ⁱ⁾ depends on chain position.

The first-order mean-torque model

Basically, this is the model proposed by Petrache et al. (1999) for the analysis of simulated order parameter profiles. Given the second moment x²