INTRODUCTION

TOP ABSTRACT INTRODUCTION THE THEORETICAL MODEL RESULTS DISCUSSION REFERENCES

The major structural component of biomembranes is the lipid bilayer, which not only serves as a permeability barrier, but modulates the activities of embedded proteins and anchored glycoproteins through its collective physical properties (Singer and Nicolson, 1972; Biltonen, 1990; Mouritsen and Biltonen, 1993). One potentially important property of the lipid bilayer is the tendency of chemically or structurally similar lipids to cluster (or phase separate) within the plane of the bilayer (Huang et al., 1993c; Jørgensen and Mouritsen, 1995; Jerala et al., 1996). This clustering can, in turn, promote localization or colocalization of reactive components altering the potential for protein-protein interaction, rates of reaction, and diffusion rates (Thompson et al., 1995; Dibble et al., 1996; Hinderliter et al., 1997; Gil et al., 1998; Sabra and Mouritsen, 1998). The physical basis of this type of functional modulation can be traced, in part, to differences in the energetics of interactions between chemically or structurally distinct lipids (Hønger et al., 1996; Hinderliter et al., 1998; Sugár et al., 1999). Thus the development of an understanding of the relationship between lipid structure and membrane function requires knowledge of the magnitude of these interactions and their structural consequences. The purpose of this study is to establish a macroscopic description of the gel-fluid phase transition in thermodynamic and structural terms of two general classes of lipids as a step in that direction.

Model systems consisting of a few natural or synthetic phospholipids are well defined and can be conveniently studied by physical techniques and theories to gain information on bilayer properties (Melchior and Stein, 1976; Mabrey and Sturtevant, 1978; Seelig and Seelig, 1980; Jørgensen and Mouritsen, 1995; McMullen et al., 1999; Korlach et al., 1999). The gel to fluid or main phase transition has been a major focus of lipid research for decades (Cevc and Marsh, 1987). Since phosphatidylcholines (PCs) are the most abundant lipids in biomembranes and widely used in model membrane studies, there is a large body of information available (Koynova and Caffrey, 1998). In the past two decades, considerable experimental work has been performed on mixed-chain PCs with different acyl chains at the sn-1 and sn-2 positions (Huang and Li, 1999).

It was suggested (Mason et al., 1981; Huang et al., 1983) that when the chain mismatch of the two chains exceeds ~3 methylene units, the bilayer forms a partially interdigitated gel phase (shown schematically in Fig. 1 A). In such a phase the mismatched region of one leaflet is matched with that of the other leaflet, so that the bilayer thickness is approximately the sum of the lengths of the two acyl chains and the area/lipid is about twice the area/chain. As the chain asymmetry increases to the point that the length of the longer chain is about twice that of the shorter one, the bilayer forms a mixed interdigitated gel phase (shown schematically in Fig. 1 B). In this phase the mismatched region of one leaflet is matched with the shorter chain of the other leaflet so that the bilayer thickness is approximately the length of the longer chain, and the area/lipid is about three times the area/chain (McIntosh et al., 1984; Hui et al., 1984). Other studies (Shah et al., 1990; Halladay et al., 1990; Lewis et al., 1994b; Zhu and Caffrey, 1994) further suggested that the mixed interdigitated gel phase undergoes a transformation to the partially interdigitated fluid phase at the phase transition temperature. Huang and co-workers have developed empirical equations to correlate the transition temperatures to chain properties for lipid bilayers existing in non, partially, and mixed interdigitated gel states (Huang, 1991; Huang et al., 1993a, b; Li et al., 1994). Marsh (1992, 1999) attempted to provide some thermodynamic understanding of the work of Huang and co-workers by analyzing the transition temperature data assuming the transition enthalpy and entropy changes to be linear functions of chain-length and chain-asymmetry.

View larger version (13K): [in this window] [in a new window]   FIGURE 1   Schematic diagram to show two lipid bilayer packing structures and their associated structural quantities. (A) Partial interdigitation; (B) mixed interdigitation.

A number of theoretical models have been proposed to gain insight into the mechanism of lipid bilayer gel-fluid phase transitions and to interpret experiments (Nagle, 1980; Caillé et al., 1980; Pink, 1982; Cevc and Marsh, 1987). Models that are microscopic in nature include lattice models (Nagle, 1973; Doniach, 1978; Pink et al., 1980), molecular mean-field models (Marelja, 1974; Meraldi and Schlitter, 1981a, b), Monte Carlo simulations (Mouritsen et al., 1983, 1992; Sugár et al., 1999), and molecular dynamics simulations (van der Ploeg and Berendsen, 1982; Tu et al., 1995). Models that are macroscopic in nature include variations of Landau theory (Owicki et al., 1978; Priest, 1980; Jähnig, 1981), assuming the Gibbs free energy to be a polynomial of some order parameter. These theoretical models only provide descriptions of lipids with identical acyl chains.

The goal of the current work is to present a macroscopic model to describe the chain-length and chain-asymmetry dependence of lipid bilayer gel-fluid phase transitions. The model parameters are obtained by fitting the transition temperature and enthalpy data of selected PCs to the model so that the model can accurately represent the data. The model is tested by the comparison of its predictions with the transition temperature and enthalpy data of other PCs not used in the fitting procedure. The model also provides macroscopic estimates of other thermodynamic and structural quantities such as interchain van der Waals energy, number of gauche conformers, chain orientational order parameter, and bond rotational and excluded volume entropies. The agreement between model calculations and existing data obtained with various techniques provides a further test of the accuracy of the model and its parameters. Where experimental information is not yet available, the model provides estimates of unknown quantities. The model is derived from and applied to saturated lipids only, but can be easily modified to account for biological lipids that are unsaturated. This work is inspired by the classic van der Waals theory on liquid-gas phase transitions.

	THE THEORETICAL MODEL

TOP ABSTRACT INTRODUCTION THE THEORETICAL MODEL RESULTS DISCUSSION REFERENCES

A spontaneously formed lipid bilayer in excess water is a complicated structure, even for a single component system. A large number of intra and intermolecular interactions of lipid-lipid and lipid-water need to be considered. These include headgroup electrostatic interactions, hydration, and steric interactions, van der Waals attraction between chains, hard-core repulsions between chains and between different segments of chains, and trans-gauche isomerization of chain segments. In order to provide a description of lipid bilayers, it is necessary to approximate the important interactions and ignore the minor ones, as has been done in a large number of theoretical models on lipid bilayers. The most important interactions are the hard-core repulsion or excluded volume interaction, van der Waals attraction, trans-gauche isomerization, and headgroup steric interactions. The thermodynamic quantities describing these interactions in lipid bilayers are assumed to be related to a single-order parameter.

The order parameter

Since the lipid bilayer has a preferred axis, the bilayer normal, the motion of lipid chains, is anisotropic. Let n denote the unit vector along the bilayer normal. Following Seelig and coworkers (Seelig and Niederberger, 1974; Seelig and Seelig, 1974), a vector, v, is assigned to each chain segment, whose direction is given by the normal to the plane spanned by the two C-H bonds so that when the chain is in an untilted all-trans crystal state, v is coincident with n and its length is a (= 1.27 Å), the projection of one C-C bond on the bilayer normal. When the chain is in motion, v makes an instantaneous angle  with n. The segmental order parameter, Q, is defined as the time average of the second Legendre polynomial, Q = (3cos²  1)/2 and can be calculated from deuterium NMR measurements as Q = 2S_CD, where S_CD is the order parameter of a C-D bond associated with the segment (Seelig and Niederberger, 1974). Q has been shown to vary along the lipid chain (Seelig and Seelig, 1974).

The above description is microscopic in nature. Having a preferred axis, the lipid bilayer may be treated as a uniaxial system from a thermodynamic point of view. Such a uniaxial system can be characterized by a single-order parameter (de Gennes, 1971). To macroscopically describe the lipid bilayer phase transition, we define a chain orientational order parameter, q, as the arithmetic average of the segmental order parameter Q_i along the chain. This is equivalent to assigning a vector to the chain as having an angle  with n, where cos² is calculated as the average of segmental cos²_i, even for a flexible chain, so that we can write

(1a)

(1b)

where n_s is the number of segments in the chain (brackets denote the time average in this work). q = 0 represents the chain in a completely random state and q = 1 in an untilted perfect crystal state, so that q retains the meaning of a normalized order parameter.

The thermodynamic potential

Starting with the molecular mean-field theory of Meraldi and Schlitter (1981a, b) and the statistical mechanical calculation of Priest (1980), the Gibbs potential of lipid bilayers is:

(2a)

(2b)

(2c)

(2d)

(2e)

(2f)

where G, H, S, T, and R are the Gibbs potential, enthalpy, entropy, absolute temperature, and gas constant; W_s accounts for steric and hydrophobic interactions imposed on the headgroup packing (originally introduced by Marelja, 1974):  is the lateral surface pressure and A the area/lipid; U_g and S_g are the energy and entropy contributions (Priest, 1980), respectively, due to gauche bond formation; E_g is the energy of a single gauche bond relative to that of a trans bond, n_g the number of gauche bonds per molecule, f_g the fraction of gauche bonds, N_c the total number of carbons of the molecule, and s_g the entropy contribution per bond; U_vw and S_pk, respectively, are the energy and entropy contributions of van der Waals attractions and hard-core repulsions (excluded volume or packing) based on the work of Meraldi and Schlitter (1981a, b). [In the work of Meraldi and Schlitter (1981a,b), each segment was treated individually so that U_vw and S_pk were summed over all segments. Meraldi and Schlitter adapted the work of Cotter (1977) and Gelbart and Baron (1977) on nematic liquid crystals to flexible lipid chains. It should also be mentioned that in the calculations to be described, all energy terms are in the units per mole of lipid. The numerical values of some quantities are given in other units (rather than in molar units) only for convenience of communication.] The chain orientational order parameter defined above is indicated by q; B₀, B₂, C₀, and C₂ are chain-length-dependent positive coefficients and are treated in detail in the section on chain-length and chain-asymmetry dependence;  is a factor, originally introduced by Marelja (1974) in his molecular mean-field theory as the ratio of the number of trans bonds to the total number of bonds in a chain. Based on Salem's calculation (1962) of interchain van der Waals interactions, Pink and co-workers (Pink et al., 1980; Pink, 1982) pointed out that ² should be inversely proportional to the 5th power of interchain distance. The interchain distance is equal to the chain diameter and is proportional to the square root of the area/lipid, A. When the untilted all-trans chain in the crystal state is taken as the reference with area/lipid, A₀, one obtains ² = (A₀/A)^5/2. It should be emphasized that all quantities in Eq. 2 are macroscopic averages (e.g., q is averaged over all chains), so that all molecules are treated equally.

To make Eq. 2 useful for the studies of lipid bilayer main phase transitions, we postulate that lipid bilayer gel-fluid phase transitions can be characterized by the single-order parameter, q, as previously described. If the volume is assumed to remain constant, A · l = A₀ · l₀, where l is the (time-averaged) chain-length projection on the bilayer normal and l₀ that of the all-trans extended chain in the crystal state. l = v_i · n = n_sa(1/n_s)cos_i = l₀(1/n_s)cos_i, so that

(3)

which is the arithmetic average of cos_i. However, Eq. 1b is the average of cos²_i. Generally, these two averaging processes will give different results. However, if all cos_i  0, one expects the square root of Eq. 1b to be a good approximation to Eq. 3, so that

(4)

The assumption that cos_i  0 means that in the time-averaged sense, _i  90°. In other words, we ignore the possibility of loop conformations in acyl chains, which is probably reasonable except for chain termini. If all cos_i are the same and equal to cos, Eq. 4 becomes cos = . Thus, the angular fluctuations, (cos²  cos²) > 0, are neglected, so that Eq. 4 is the upper limit of estimates of l/l₀ (see Discussion).

To relate f_g to q, we consider the two extreme conditions: at q = 0, S_g is maximal and at q = 1, S_g = 0. However, when f_g = 1/2, S_g is maximal (Priest, 1980) and at f_g = 0, S_g = 0. The simplest relationship satisfying these two extreme conditions (see Discussion) is

(5)

With Eq. 5, the entropy contribution per bond, s_g, can be expressed as

(6)

which was obtained by Priest (1980, where a factor of 1/2 was missing) using a transfer matrix method following Flory (1969). It should be noted that Eq. 6 is strictly valid only in the limit of infinitely long chains.

Chain-length and chain-asymmetry dependence

First, we would like Eq. 2 to describe a class of saturated lipids that form partially or noninterdigitated (PI) gel phases. This is inspired by Huang's empirical formulation that correlates the transition temperatures of this class of PCs to chain properties (Huang, 1991; Huang et al., 1993a; 1994). Two structural quantities for C(X)C(Y)PC, where X and Y are the number of carbons in sn-1 and sn-2 chains, respectively, are defined as:

(7a)

(7b)

where N is the total number of C-C bonds participating in interchain interactions,  is the number of mismatched bonds between the two chains, and d₁₂ the inherent shortening of the sn-2 chain. Being approximately parallel to the bilayer surface, the first bond of the sn-2 chain is assumed to make no contribution to interchain interactions and is responsible for the shortening of the sn-2 chain.

B₀ and B₂ are related to the van der Waals interaction strength and assumed to be proportional to N  N_0  1f(). N₀ is introduced to account for the minimal chain-length requirement to have the lipid in a bilayer form; ₁f() is introduced to account for the reduced interaction in the mismatched region  (0 < ₁ < 1) with

(8)

Here _th represents the threshold value of : when || < _th, the lipid is assumed to be in a noninterdigitated state and when ||  _th, in a partially interdigitated one. In a noninterdigitated state, the mismatched tail of one lipid molecule may randomly collide with another via thermal motions, so that the interaction strength should be proportional to the probability of a two-body collision ( ²). In a partially interdigitated state, the interaction strength is assumed to be proportional to ||. Imposing continuous and smooth conditions on f gives Eq. 8. Similarly, C₀ and C₂ are related to excluded-volume interactions and assumed to be proportional to N  ₂f() (0 < ₂ < 1). We finally obtain

(9a)

(9b)

(9c)

(9d)

(9e)

(9f)

where b is the sublimation energy of a CH₂ monomer; N₀, c, ₁, ₂, ₂, ₂ are fitting parameters.

Second, similar to the work of Huang and co-workers (Huang et al., 1993b; Li et al., 1994), another structural quantity is defined for the mixed interdigitated (MI) gel phase bilayers. This structural quantity, , is the number of unmatched bonds between the length of the longer chain plus its terminal van der Waals radius and the sum of the two shorter chains plus the van der Waals distance between the two chain termini.  can be expressed as

(10)

where d_vw is the van der Waals distance between the two opposing chain termini;  and d_vw, together with , are shown schematically in Fig. 1. Consequently, the Gibbs potential of MI bilayers is similar to that of PI bilayers (Eq. 9), except that Eqs. 9e and 9f are replaced by

(11e)

(11f)

where ₃ is a parameter similar to ₁ for PI model, describing the reduced van der Waals interactions in the region ||; ₄ is an MI model-specific parameter for reduced interchain interactions in the unmatched region , and f is given by Eq. 8 with  being replaced by .

Model parameters are obtained by a nonlinear least-square fitting procedure of transition temperature and enthalpy data to Eq. 9 for the PI model or Eq. 11 for the MI model, as described in the Results section. First, however, we will describe the equation of state and Gibbs potential curves for two lipid systems representative of the two kinds of transition processes considered.

Equation of state

Thermodynamic equilibrium requires that at constant temperature the partial derivative of the Gibbs free energy with respect to the order parameter equals zero, i.e.,

(12)

where G(q, T) is given by Eq. 9 or Eq. 11 and Eq. 12 is an equation of state, where q = q(T) at constant . In Fig. 2 the equation of state curve and the Gibbs potential at the transition temperature of C(16)C(16)PC are plotted in panels A and B, respectively. At low (high) temperature, Eq. 12 has only one solution of a large (small) order parameter, implying that the system is in a gel (fluid) phase. At a temperature near the phase transition temperature, T_m, Eq. 12 has two stable solutions separated by an unstable one, indicating that the system can exist in either phase. At T_m, the two stable solutions have the same Gibbs potential (panel B) and the phase transition occurs when the lipid bilayer changes from one state to the other. The dashed line denotes metastable states: between a and b superheated gel states exist, whereas between c and d super-cooled fluid states exist. The region between b and c (dotted line) represents unstable states. When the system is at any point, o, between a and d, the system is inhomogeneous, the fraction of lipids in fluid phase is determined by the lever rule: section ao divided by section ad. The corresponding Gibbs potential that satisfies the global stability condition is the dotted line in panel B. This is the pure first-order phase transition picture for PI lipid bilayers.

View larger version (26K): [in this window] [in a new window]   FIGURE 2   Theoretical curves to illustrate the PI gel to fluid phase transition of C(16)C(16)PC (A and B) and MI gel to PI fluid transition of C(10)C(22)PC (C-F). (A) The equation of state curve of C(16)C(16)PC. The solid line represents the true equilibrium states. The dashed lines represent meta-stable states, whereas the dotted line represents unstable states. (B) The Gibbs potential curve at T = Tm, where the two minima have the same values. (C) The equation of state curve of C(10)C(22)PC (solid line), which consists of MI gel and PI fluid phase curves. (D-F) The Gibbs potential curves of possible PI and MI packing structures at Tm, Tm  10°C and Tm + 10°C, respectively.

For lipids with the length of the longer chain about twice that of the shorter one, the bilayer undergoes a MI gel to PI fluid phase transition (McIntosh et al., 1984; Shah et al., 1990; Zhu and Caffrey, 1994), since the bilayer thickness increases upon the gel to fluid transition. To illustrate this kind of phase transition, the equations of state and the Gibbs potentials of the PI and MI models at three temperatures of C(10)C(22)PC are plotted in Fig. 2, C-F. In panel C, G₁₀(q, T) = 0 is shown for possible PI and MI packing arrangements. At temperatures just below T_m (panel E), the bilayer is in the MI gel phase since its Gibbs potential is lower than that of the PI fluid phase. At T_m, the MI gel phase and the PI fluid phase have the same Gibbs potential (panel D) and the phase transition occurs when the lipid bilayer changes from one phase to the other. At temperatures above T_m (panel F), the bilayer exists in the PI fluid phase because its Gibbs potential is lower than that of the MI fluid phase.

For both the PI gel to fluid and the MI gel to PI fluid phase transitions, the transition temperature, T_m, is defined as the temperature at which the gel and fluid phases at equilibrium have the same Gibbs potential, that is, G₁₀(q_f, T_m) = G₁₀(q_g, T_m) = 0 and G(q_f, T_m) = G(q_g, T_m), where q_g and q_f are the order parameter values in the gel and fluid states, respectively. The transition enthalpy or latent heat is calculated as H_m = H(q_f)  H(q_g). When  is placed in front of any quantity F, it is viewed as an operator and defined as F = F(q_f)  F(q_g).

	RESULTS

TOP ABSTRACT INTRODUCTION THE THEORETICAL MODEL RESULTS DISCUSSION REFERENCES

The macroscopic PI model of lipid bilayers (Eq. 9) is used to describe both gel and fluid states and applied to the gel-fluid phase transitions of phosphatidylcholines (PCs) that form PI gel phase bilayers. T_m and H_m values of 39 PCs obtained by differential scanning calorimetry (DSC) experiments are used to fix six adjustable model parameters to achieve a good fit of the model to the data. For PCs forming MI gel phase bilayers, the fluid state is described by the PI model (with all parameters fixed) and the gel state by the MI model (Eqs. 9a-d plus Eqs. 11e, f). T_m and H_m values of 56 PCs are used to fix four MI model parameters to achieve a good fit of the MI model to the data. This provides initial confirmation that the proposed model (PI and MI) can accurately represent existing T_m and H_m data. The predicting ability of the model is then tested by its ability to predict T_m and H_m values of 17 other PCs not used in the original fitting procedure. Finally, the model is used to provide macroscopic estimates of other thermodynamic and structural quantities. These estimates provide a further test of the model by comparison with existing experimental or theoretical values and in other cases provide predictions of unknown quantities.

Experimental observations, data analyses, and theoretical considerations yield the estimates of the constant parameters E_g, b, A₀, , d₁₂, and _th of the PI model (Eq. 9); E_g = 0.5 kcal/mol (Nagle, 1980); b = 1838 cal/mol (Salem, 1962). Since the area/chain in the crystal state is ~19.0 Ų (Pearson and Pascher, 1979), A₀ = 38.0 Å² for PI packing of two chains per headgroup;  = 14.7 dyn/cm (Meraldi and Schlitter, 1981a, b); d₁₂ = 1.3 C-C bonds (Marsh, 1992, 1999). It is expected that in order to form a partially interdigitated phase, || should be at least equal to the van der Waals distance of chain termini on the opposing leaflets, d_vw, so that _th = 2.6 C-C bonds (Li et al., 1993). The variable parameters N₀, ₁, ₂, c, ₂, ₂, are left as empirical constants to be obtained by fitting of the T_m and H_m values of 39 PCs obtained by differential scanning calorimetry (DSC) experiments. These 39 PCs are thought to be in PI gel phase bilayers based on previous work (see references in Table 2). When there are multiple reports for a lipid, an average value is calculated and its associated error estimated. In the case of a single report, that value is used and a maximal error assigned (for T_m, ~1.5°C; for H_m, ~20%). The data with their associated errors are used in a least-square fitting routine (Johnson and Frasier, 1985), where T_m and H_m are calculated numerically. It should be mentioned that in an initial fitting procedure, d₁₂ and _th were allowed to vary and the values obtained (d₁₂ = 1.31 ± 0.10; _th = 2.75 ± 0.63) were very close to values cited in the literature. Therefore, the literature values are assumed as noted above.

For PCs forming the MI gel phase bilayers, the fluid state is described by the PI model with parameters values obtained above and the gel state by the MI model (Eqs. 9 a-d plus Eqs. 11 e, f). For the MI model, the constant parameters, E_g, b, d₁₂, A₀, d_vw, _th, c, ₂, and N₀ are assigned values as follows: E_g, b, and d₁₂ are the same as for the PI model; A₀ = 57.0 Å² for MI packing of three chains per headgroup; d_vw = 2.6 C-C bonds (Li et al., 1993). Reflecting dynamic effects on chain interactions, _th is assumed to be 2.6 C-C bonds, the same value as _th. Since c and ₂ are related to the excluded volume interactions and N₀ reflects the minimal chain-length requirement, these three parameters should be insensitive to the chain packing and their values are assumed to be the same as for the PI model obtained above. The variable parameters, , ₂, ₃, and ₄, are obtained by fitting of T_m and H_m values of 56 PCs that are thought to be in MI gel phase bilayers based on previous work (see references in Table 3). The fitting procedure is similar to that for the PI model, except that q_f is calculated from the PI model while q_g is from the MI model, i.e., G^PI(q_f, T_m) = G^MI(q_g, T_m). Reflecting the headgroup steric interactions,  is allowed to vary for the three chains per headgroup structure of the MI model; ₂ is also allowed to vary, since the different chain packing structure of the MI model should alter the interchain van der Waals interaction strength; ₃ and ₄ are two MI model specific parameters.

The constant parameters used and the adjustable parameters obtained from the fitting procedures for both PI and MI models are listed in Table 1. The experimental T_m and H_m values of the 39 PCs used in the parameter fitting of the PI model are listed in Table 2, while those of the 56 PCs used in the fitting of MI model are listed in Table 3, along with values obtained from the PI and MI model calculations using Eqs. 9 and 11 with parameter values given in Table 1, respectively. The T_m and H_m data listed in Tables 2 and 3 are plotted as a function of /N in the top and bottom panels of Fig. 3, respectively. The horizontal lines are calculated for the PI and MI models with varying  and fixed N (the value is indicated by the number within the figure). The vertical lines are obtained by varying N with fixed  for the PI model or fixed  for the MI model (the  or  value is shown by the numbers within the figure). The agreement between the calculated values and experimental results is excellent for T_m, but less so for H_m. The reason for this larger discrepancy is likely due to the experimental error of H_m being large (~±25%, Lin et al., 1990), whereas T_m is generally accurate to ±1% on the absolute temperature scale. For the 39 PCs in Table 2, the difference between the experimental values and the PI model calculations of T_m are within 1.5°C (<±1% on the absolute temperature scale). For the 56 PCs in Table 3, the difference between the experimental values and the MI model calculations of T_m exceeds 3.0°C (~±1%) only for C(8)C(19)PC, C(8)C(20)PC, and C(8)C(21)PC. Those H_m data with an associated error bar are plotted and found to agree with the calculated curves within their error bars, except for one very short chain lipid, C(8)C(18)PC (see below).

View larger version (62K): [in this window] [in a new window]   FIGURE 3   Tm (top) and Hm (bottom) variation with chain-length N and chain-asymmetry . The circles are DSC data listed in Tables 2 and 3. The diamonds are data from Table 4 that have not been used in the model parameter-fitting procedures. The open symbols represent PCs with one chain having an odd number of carbons and the other an even number of carbons. The lines are our model calculations: the horizontal lines are calculated from fixed N (value indicated), whereas the vertical lines are from fixed  for the PI model or fixed  for the MI model (value indicated). The dashed lines are for odd/even PC series.

It has been shown above that the calculated values of T_m and H_m agree well with the experimental values. However, since the experimental data were used in the parameter fitting procedures, the comparison was not independent. To test the predicting abilities of the model, we compared the model predictions to experimental observations for lipids whose T_m and H_m values were not used in the parameter fitting procedure. Seventeen lipids have been synthesized and their thermodynamic properties determined in the Huang laboratory. The results are listed in Table 4, along with the model predictions. Some of the data have been published (Huang et al., 1993a, 1994; Li et al., 1994), while the others are unpublished (personal communication). It is evident that the model predictions agree well with experimental observations for all 17 lipids of quite different chain-length and chain-asymmetry variations. These 17 lipids are plotted as diamond symbols in Fig. 3, with 14 belonging to the group of PI gel phase lipids and 2 belonging to the group of MI gel phase lipids. The results for C(14)C(24)PC fall on the boundary distinguishing the PI and MI models. Since the experimental value of H_m of C(14)C(24)PC is not available and both the PI and MI models predict the same T_m, we cannot assign it to either group.

Chain-length dependence

The results of the PI and MI model calculations for some of the lipids shown in Tables 2 and 3 are plotted in Fig. 4 to show the chain-length variations of T_m and H_m. Both PI and MI models predict an almost linear relation for H_m and S_m with N, except at short chain-lengths (Fig. 5), indicating that with each CH₂ unit increase, the increase of H_m is almost a constant. The T_m plateaus with increasing N and its limiting value is estimated to be 147°C (N = 10¹⁰⁰ in a model calculation), consistent with the melting temperature of polyethylene (~138-141°C).

View larger version (18K): [in this window] [in a new window]   FIGURE 4   Tm and Hm variation with chain-length N of PI series (A and B) and MI series (C and D). The symbols are experimental observations from DSC: , the average value in the original data set listed in Tables 2 and 3; , not used in the fitting procedure (in Table 4); , Ichimori et al. (1998); , Lewis et al. (1987). The lines are model calculations. A short notation is used for C(X)C(Y)PC as X/Y, which is also used in Figs. 5-7.

View larger version (18K): [in this window] [in a new window]   FIGURE 5   Model calculations of symmetric PC series with varying chain-length N at Tm: A, various contributions to the excess enthalpy; B, contributions to the excess entropy; C, the number of gauche bonds; and D, the order parameter. For comparison, the experimental data of Hm, Uvw, ng, and q (listed in Tables 2 or 5) are included as symbols: , DSC average values; , dilatometry of Nagle and Wilkinson (1978); , Raman spectroscopy of Pink et al. (1980); , deuterium NMR of Seelig and Seelig (1974) and Marsh et al. (1983).

For symmetric chain PCs, especially C(14)C(14)PC and C(16)C(16)PC, more information is available. Estimates of U_vw have been provided by Nagle and Wilkinson (1978) based upon dilatometry measurements; n_g have been estimated directly from Raman spectroscopic studies (Yellin and Levin, 1977; Pink et al., 1980) and indirectly from the U_vw. The calculated values are compared with experimental estimates in Table 5. In Fig. 5, the various calculated contributions to the transition enthalpy and entropy, n_g and q at T_m with varying N are shown, along with the limited data in literature for a comparison. The agreement of the PI model predictions with experimental estimates is very good for U_vw and quite good for n_g and q. The n_g values estimated by Pink et al. (1980) from Raman spectroscopy with their 10-state model agree with ours in the gel state, but are higher than ours in fluid state. However, U_vw of Nagle and Wilkinson (1978) plus U_g computed from n_g of Pink et al. (1980) exceeds H_m of DSC measurements. For C(16)C(16)PC at about T_m + 6°C, our model predicts 8.4 gauche bonds/lipid, which compares favorably to infrared spectroscopic measured values of 7.8 (Mendelsohn et al., 1989) and 8.4 gauche bonds/lipid (Lewis et al., 1994b). The model predicts that U_vw makes the major contribution to the transition enthalpy H_m, while the entropy from intrachain trans-gauche isomerization, S_g, makes the major contribution to the transition entropy, S_m, as expected.

It is observed (Fig. 5 D) from the PI model calculation that, at the phase transition temperature, q increases in the gel state but decreases in the fluid state as N increases. The increased order in the gel phase is apparently due to a stronger van der Waals interaction per bond. To overcome this stronger force requires a higher T_m. The decreased order in the fluid phase is likely the result of the increased thermal motions at elevated T_m, which more than compensate the slightly stronger van der Waals interaction per bond. Therefore, a larger change in the order parameter at its phase transition is observed for a lipid with larger N (see Fig. 5 D). This difference is observed in deuterium NMR experiments with C(14)C(14)PC, C(16)C(16)PC, and C(18)C(18)PC (Morrow et al., 1992). The vertex of the curve in Fig. 5 D is the critical point predicted by the PI model, where the difference in order parameter between gel phase and fluid phase vanishes. Direct experimental confirmation of this critical point is not feasible because it lies between C(11)C(11)PC and C(10)C(10)PC, whose T_m is far below the freezing point of aqueous dispersions of lipids. It is well known that when a system is close to a critical point, fluctuations of the system become large. Therefore, it is expected from the model that lipid systems with shorter chains will have larger fluctuations. This is consistent with results of the Monte Carlo simulations (Ipsen et al., 1990; Hønger et al., 1996). It should be emphasized that we do not expect the model to provide an accurate description of short chain lipids for which the chain-chain interactions described by the model become less dominant (also recall that Eq. 6 is strictly valid only in the long chain limit). In Fig. 4, B and D, we see that the enthalpy results of the model calculations tend to deviate from experimental observations at short chain-lengths.

Chain-asymmetry dependence

In Fig. 6, the T_m of C(15)C(15)PC series (N = 27) and C(16)C(16)PC series (N = 29) and H_m of C(15)C(15)PC series are plotted in the top and bottom panels, respectively. Variation of T_m with chain-asymmetry of these two series and some others have been extensively studied by Huang and co-workers (Lin et al., 1991; Huang et al., 1993a). Recall that  is a measure of mismatched bonds between sn-1 and sn-2 chains and  is a measure of the unmatched bonds in the MI packing structure. For a lipid series with constant N, as || increases from || = 0, both T_m and H_m decrease due to decreased van der Waals interactions in the mismatched region in the gel state. As || increases beyond a certain point, lipids such as C(18)C(12)PC prefer the MI gel phase because of its lower Gibbs potential. The T_m and H_m begin to increase due to favored van der Waals interactions in the MI gel phase until the maximal values are reached at || = 0. It is noted that there is a large discontinuous change of H_m at the boundary distinguishing PI and MI packing (indicated by the dotted lines in the bottom panel of Fig. 6), while T_m is continuous across this boundary.

View larger version (26K): [in this window] [in a new window]   FIGURE 6   Chain-asymmetry dependence of Tm (top) of C(15)C(15)PC and C(16)C(16)PC series and Hm (bottom) of C(15)C(15)PC series.  is the DSC average value listed in Tables 2 or 3, mainly from Huang and co-workers (Lin et al., 1991; Bultmann et al., 1991; Huang et al., 1993b). The lines are model calculations from this work.

Based on their extensive synthetic and DSC studies, and molecular mechanics study of the C(14)C(14)PC series, Huang and co-workers suggested that lipids with C/CL < 0.41 prefer the PI gel phase, while those with C/CL > 0.41 prefer the MI gel phase (Lin et al., 1991; Li et al., 1994), where C = ||, and CL is the effective length of the longer chain. The ||/N boundary value between PI and MI packing can be estimated by our model in terms of the two structural quantities,  and N. The results from C(13)C(13)PC series to C(26)C(26)PC series are listed in Table 6, along with the equivalent C/CL values and the discontinuous changes in H_m. Generally speaking, when ||/N is less than the boundary value, the lipid is anticipated to exist in the PI gel phase and when ||/N is greater than the boundary value, in the MI gel phase. But when ||/N value of a lipid is very close to the corresponding boundary value, its gel phase structure cannot be predicted by the values of  and N alone. However, comparison of the experimentally measured H_m with the model values allows such a distinction to be made. For an illustration, see C(11)C(19)PC and C(18)C(12)PC in Fig. 6. Both lipids lie close to the boundary with their T_m values close to each other and model predictions. However, their H_m values differ by ~4 kcal/mol. Based upon the model calculations, one would anticipate that C(11)C(19)PC is in the PI gel phase, while C(18)C(12)PC is in the MI gel phase. It is noted that the boundary value decreases monotonously with increasing N, while the change of H_m initially increases and then decreases slightly, remaining essentially constant.

View this table: [in this window] [in a new window]   TABLE 6   Variation of boundary values of ||/N, C/CL, and the changes in Hm with the chain-length for lipids from C(13)C(13)PC series (N = 23) to C(26)C(26)PC series (N = 49) predicted by the model

For the C(16)C(16)PC series (N = 29), the various contributions to the excess enthalpy and entropy, n_g and q, have been calculated at the transition temperature and are shown in Fig. 7, along with limited data from the literature for a comparison. It is apparent that both the PI and MI model calculations agree well with the limited experimental measurements available. Both the PI and MI models predict that U_vw makes the dominate contribution to H_m, while S_g makes the dominate contribution to S_m. As || increases from zero, q decreases in the gel phase and increases in the fluid phase at the T_m. As || increases further, q increases discontinuously in the gel phase as the lipids shift from PI to MI packing, but decreases continuously in the fluid phase, resulting in a larger change of q. This discontinuous change of q is due to the structural change from PI to MI packing in the gel phase, resulting in a 2.7 kcal/mol discontinuous change in H_m. With further increase of ||/N until || = 0, q decreases rapidly in the fluid phase due to the increase of T_m.

View larger version (19K): [in this window] [in a new window]   FIGURE 7   Model calculation of C(16)C(16)PC series with varying chain-asymmetry  at Tm: A, various contributions to excess enthalpy; B, contributions to excess entropy; C, the number of gauche bonds; and D, the order parameter. For comparison, the experimental data of Hm, Uvw, ng, and q (listed in Tables 2, 3, 5, or 7) are included as symbols: , DSC data; , dilatometry of Nagle and Wilkinson (1978); , Raman spectroscopy of Pink et al. (1980); , deuterium NMR of Seelig and Seelig (1974) and Lewis et al. (1994b). The DSC data with an error bar are averages and those without an error bar are from a single experimental report, and tend to have large errors.

Temperature effect on chain order and gauche conformers

Above T_m, the chain order decreases and gauche conformers increase with increasing temperature due to increased thermal motions. The chain order is usually studied by deuterium NMR (Seelig and Seelig, 1974). Recently, gauche conformers have been studied by FTIR (Mendelsohn et al., 1989; Lewis et al., 1994b) and by NMR with the kink and jog model (Douliez et al., 1995, 1996). Some results are listed in Table 7 and compared to our results. Douliez et al. studied separately the sn-1 and sn-2 chains of C(14)C(14)PC with and without 30% cholesterol at various temperatures. Since the first bond of the sn-2 chain is approximately parallel to the bilayer surface, we feel that the results of the sn-1 chain represent the bilayer interior more accurately and is thus used for comparison in Table 7. The difference between experimental and calculated values are within ±11% for the order parameter and ±4% for the total gauche conformers.

	DISCUSSION

TOP ABSTRACT INTRODUCTION