INTRODUCTION

TOP ABSTRACT INTRODUCTION MODEL RESULTS AND DISCUSSION CONCLUSIONS APPENDIX REFERENCES

The development of liposomes as drug carriers was facilitated in the early 1990s, when it was demonstrated that the inclusion of small percentages of lipid-bonded polymers (named polymer-lipids) in the liposome formulation increased its circulation time in vivo, favoring the uptake in the target site versus their elimination by the RES (Blume and Cevc, 1990; Allen et al., 1991). These liposomes are commonly referred to as stealth liposomes, and their resistance to blood stream elimination is due to steric stabilization by the polymer layer attached to the lipid bilayer (Klibanov et al., 1991; Needham et al., 1992; Torchilin et al., 1994a-c; Blume and Cevc, 1993). The most common polymer used for this purpose is poly(ethylene glycol) (PEG), also referred to as poly(ethylene oxide) (PEO). The effect of PEG in the stability of stealth liposomes has been extensively studied during the last decade. The protective effect of PEG layers is believed to prevent protein adsorption on the lipid bilayer (Ceh et al., 1997; Bradley et al., 1998) and, at the same time, to act as a steric barrier for inhibiting liposome fusion (Holland et al., 1996). As a result, successful lipid/PEG-lipid formulations have been developed for drug delivery. The amount of PEG in the formulations must be optimized, because too little PEG exhibits low protective effects in the bilayer, while too much PEG may lead to micellization of the system.

Most of the preparation procedures of PEG-stabilized liposomes reported in the literature involve nonreversible processes, such as dialysis, pH cycling, sonication, and extrusion. In contrast, there are few studies on spontaneous liposomes (Joannic et al., 1997; Szleifer et al., 1998). However, thermodynamically stabilized liposomes are expected to be more sensitive to environmental changes than kinetically trapped liposomes, which is a desirable property to explore for many uses in biological and pharmaceutical applications. Moreover, if the properties and behavior of spontaneous liposomes can be predicted with a model that includes the main features of a lipid/polymer-lipid bilayer, then one could control the state of the aggregate.

To determine the optimal characteristics of a liposome for experimental purposes, a quantitative theoretical approach is needed to provide not only trends, but also comprehensive and practical guidelines. To meet this goal, we further investigate the origins and mechanisms of spontaneous PEG-stabilized liposome formation and the range of PEG-lipid compositions where liposomes form. Specifically, this study is focused on determining the conditions for energetically driven, isolated liposome formation. Energetically driven liposomes refer to those in which the free energy of the isolated aggregate in spherical geometry is lower than that of the planar bilayer. We devote special attention to understanding the molecular structure of the formed aggregates, which is often overlooked. Furthermore, the molecular driving forces for the formation of spontaneous liposomes, their range of stability, and the possible formation of small micellar aggregates is investigated.

The free energy of bilayers is usually described in terms of their elastic constants. Theoretical approaches are complex, because bending properties of these bilayers depend on the details of the molecular structure of the components and, for multicomponent formulations, their mutual interactions. The origin of spontaneous vesiculation was described by Safran et al. (1990) for bilayers of surfactants with identical hydrophobic regions but different polar groups. In this early study, it was shown that coupling between curvature and composition leads to vesicle formation when non-ideal mixing of surfactants occurs. Spontaneous vesicle formation was also predicted by Wang (1992) for a one-component bilayer composed of diblock copolymers. The composition of the diblocks must be sufficiently asymmetric with shorter chains in the core of the bilayer. Formation of polymer-based vesicles has been recently proven experimentally in water-containing solutions (Discher et al., 1999; Luo and Eisenberg, 2001) and in organic solvents (Ding and Liu, 1997). The formation of spontaneous vesicles was also predicted for mixtures of diblock copolymers (Dan and Safran, 1993). According to this study, the lamellar layer can be destabilized by the addition of small quantities of copolymers of different composition, with small fractions of shorter chains than the main component of the bilayer having stronger effects on the spontaneous curvature than a small fraction of longer chains.

The coupling between curvature and composition was further studied by Porte and Ligure (1995). Generalizing the ideas of Safran et al. (1990), they predicted a softening of the mean curvature modulus due to internal degrees of freedom when calculated at fixed chemical potential, which can lead to vesicle formation. Porte and Ligure (1995) extended their model to lipid bilayers having adsorbed polymer brushes, which they described in terms of a mean-field theory. Allowing the polymer to relax on both sides of the bilayer, they predicted that vesicle formation may occur at sufficiently high adsorption densities. However, the surface coverage they treated was much higher than the one commonly found in PEGylated liposomes for drug delivery. The effect of grafted polymer on the elastic constants of lipid bilayers has also been studied by Hristova and Needham (1994) and Marsh (2001), although none of them took into account the lipid/polymer-lipid relaxation. Those studies used scaling and mean-field theories to describe the polymer layer. These approaches are not expected to give quantitative predictions for low and moderate polymer coverage (Szleifer, 1996). Thus, they are not always applicable in the relevant experimental range of surface densities used in PEGylated liposomes.

Curvature-composition coupling has been shown to play a major role in the stability of bilayers and the resulting tendency to form spontaneous vesicles (Safran et al., 1990). However, there are no systematic studies that provide a deep understanding of the size and structure of thermodynamically stabilized aggregates formed as a result of that coupling. For this purpose, the application of a quantitative molecular theory to polymer-grafted liposomes is of great interest. A reliable theory would significantly reduce the experimental effort required to develop stable formulations with favorable biological and pharmacological properties. In addition, it would improve our understanding about how the molecular structure of the polymer and lipid layers determines the behavior of the layers.

In the present study, molecular mean-field (MMF) theory (Ben-Shaul et al., 1985; Szleifer and Carignano, 1996) is used to describe both lipid and PEG layers. MMF is applicable at experimentally relevant regimes of surface densities. Furthermore, the theory has been successfully used to provide quantitative predictions for several systems involving hydrocarbon tail packing in bilayer environments (Ben-Shaul et al., 1985) and PEG-grafted layers, such as adsorption isotherms of protein on PEG-grafted surfaces (McPherson et al., 1998; Satulovsky et al., 2000). MMF theory has also been applied to study the stability of PEGylated liposomes (Szleifer et al., 1998). In that study, the minimal loading of polymer necessary to destabilize planar bilayers was predicted as a function of polymer molecular weight. The predictions were successfully compared with experimental data. The limitation of that work, however, was that it only predicted the lack of stability of the planar film; the equilibrium size and structure of the spontaneous forming aggregates were not addressed. In the study presented here we extend that work. Our purpose is to predict not only the composition range where spontaneous liposomes form, but also the spontaneous curvature and the optimal structure of the aggregates.

In the next section we introduce our theoretical approach and a short description of the molecular theory used in this study. The following section introduces and discusses the results obtained. The last section presents the concluding remarks.

	MODEL

TOP ABSTRACT INTRODUCTION MODEL RESULTS AND DISCUSSION CONCLUSIONS APPENDIX REFERENCES

Free energy

The focus of this study is lipid/PEG-lipid bilayers. The lipid molecules are insoluble and have a double hydrocarbon chain tail that prefers the core of the bilayer to avoid contact with the surrounding water solution. The lipid tails are attached to a hydrophilic headgroup that lies on the water-lipid interface. A certain percentage of those headgroups are bonded to PEG chains that extend away from the interface toward the bulk solution. We assume that the lipid in the polymer-lipid molecule is the same as in the pure lipid. Fig. 1 shows a qualitative representation of the system.

View larger version (22K): [in this window] [in a new window]   FIGURE 1   Schematic representation of a lipid/PEG-lipid bilayer. The qualitative shape of the stresses acting on the film is also represented. PEG, lipid tails, and lipid headgroups have a repulsive interaction (i.e., (z) > 0), while the surface tension is the only attractive contribution ((z) < 0).

In this study, we only consider the case of spherical liposomes of radius R. The aggregate is assumed to have a fixed number of lipids and polymer chains. We are interested in equilibrium aggregates. As a consequence, the only relevant case is that of free exchange of molecules across the bilayer, i.e., we assume that for all aggregates the chemical potential of the lipid molecules and the chemical potential of the PEG-lipid molecules are the same at both sides of the bilayer. The partition of the components between the two monolayers is expressed as x_I = N/(N + N), where I represents lipid or PEG molecules and N represents the number of I molecules in the outer monolayer of the bilayer. Thus, the relevant variables of the system are the partition of lipids and polymer-grafted chains between both sides of the membrane (x_lipid and x_PEG, respectively), together with the curvature (c = 1/R).

Based on Helfrich's seminal work (Helfrich, 1973), the classical description of the bending free energy for a symmetric bilayer expanded around the planar film (c = 0) contains terms on curvature up to quadratic order

(1)

where F(c, x_lipid, x_PEG) is the free energy of the aggregate at curvature c and lipid and PEG partition equal to x_lipid and x_PEG, respectively. A(0) is the area at the surface of inextension, which corresponds to the area of the planar film. There are no linear terms in curvature because the expansion is made around the symmetric planar film. K corresponds to the second derivative of the free energy with respect to c, evaluated at c = 0. In terms of Helfrich's definition of the elastic constants we have K = k_b + /2, where k_b is the bending constant and is the saddle-splay constant. In the cases of interest here, there is no need to define two independent elastic constants because we only treat spherical geometries. However, we mention their relations for completeness.

The free energy in Eq. 1 is expanded only as a function of c. The dependence of F(c, x_lipid, x_PEG)/A(0) with x_lipid and x_PEG, therefore, is included in K. Expanding the composition variables around the symmetric planar bilayer (i.e., x = x = 1/2) we have

(2)

where (dⁱx_lipid(PEG)/dcⁱ)_c=0 is the ith derivative of x_lipid(PEG) with respect to c, evaluated at the planar geometry. For the case in which the lipid and polymer compositions can be varied independently, which are the only relevant cases as long as the percentage of PEGylated lipid is sufficiently low, the bending constant K is exactly given by (Ben-Shaul, 1995; Szleifer et al., 1998)

(3)

where f_I² = d²f/dI² and f_I,J = d²f/dIdJ, while _lipid = dx_lipid/dc and _PEG = dx_PEG/dc. Thus, the elastic constant depends only on the first-order derivative of the composition variables with curvature. For the mixture in equilibrium, _lipid and _PEG are the ones that minimize K and hence the free energy, i.e., _lipid = (f_c,xlipid/f_{x) and PEG} = (f_c,xPEG/f_x).

When K > 0 the free energy is minimum for planar symmetric bilayers. Thus, the lamellar phase is the energetically preferred structure. In those cases, giant liposomes may form due to edge effects or favorable entropic contributions, but only for 0  K  1. However, the formation of other lipid aggregates such as liposomes or micelles is favored at K < 0. In those cases, the planar bilayer correspond to a maximum in the free energy. The reduction of the free energy upon formation of a spherical aggregate arises from the coupling between the composition and the curvature, namely, the ability of the molecules to change the ratio between the two monolayers is responsible for the change in sign of K. The problem, however, is that no optimal size and/or structural information of the aggregates can be obtained from the second-order expansion.

Our previous calculations (Szleifer et al., 1998) concentrated on looking at the conditions upon which the elastic constant changes sign. Here we are also interested in predicting what is the optimal size and structure of the spherical aggregates that form when K < 0. To this end, we could follow two different routes. One is to calculate the free energy of the aggregates at all curvatures and compositions for a fixed number of lipids and PEGylated lipids. This is an almost impossible task because it requires the variation of three variables simultaneously. The second way is to continue the free energy expansion along the lines used for the quadratic expression. An expansion to fourth order should provide the optimal aggregate properties for the cases that K < 0. Thus, we could write

(4)

where there are no third-order terms because we are expanding around the planar symmetric film. As for the second-order expansion, Eq. 4 is expressed only in terms of curvature. Therefore, composition-curvature coupling is implicitly included in K and . The coupling between composition and curvature in  goes beyond the linear term coefficients that are needed for K. Actually, one needs terms up to third order in the expansions presented in Eq. 2. The minimization of the free energy will not enable the determination of all the necessary coefficients and, therefore, it is more convenient to directly expand the free energy up to fourth order in curvature, c, and lipid and polymer composition, x_lipid(PEG), the three variables of the system.

The free energy of the lipid/polymer-lipid mixture up to fourth order is given by

(5)

where f_Iⁱ = ⁱf/Iⁱ and f_Iⁱ,J^j = ^i+jf/IⁱJ^j, with all the derivatives evaluated at the expansion point, i.e., c = 0; x_lipid = x_PEG = 1/2. Thus, the determination of the expansion coefficients will allow for finding the optimal size and molecular partition of the aggregates.

The strategy to find the optimal liposomes is the following. First, we use the expansion only up to second order, as given in Eq. 1, to determine the minimum total loading of polymer necessary to have K < 0 for each polymer molecular weight. Second, for loadings where spontaneous liposome formation is predicted, we use Eq. 5 to find the optimal aggregate size and structure. Namely, we find the curvature and composition asymmetry of lipids and polymer that minimize the free energy.

Two comments should be made at this point. First, the free energy expansions used up to this point assume that the area per molecule at which the molecules are packed is known and does not change upon bending. We will determine this area by minimizing the free energy of the planar film, from which the expansion is made, with respect to the area per lipid molecule, a(0). Details are given below. Second, we are only describing the optimal size of an isolated aggregate. However, one would expect to have a complete size distribution of aggregates. The full determination of the size distribution can be obtained from the full free energy; however, it is beyond the scope of the work presented here.

The next step is the determination of the phenomenological expansion coefficients from a molecular approach. We apply a molecular theory to describe the lipid tails and PEG chains. This theory has been shown to be successful in describing the properties of lipid tails and PEG chains for conformational and thermodynamic properties. We formulate the free energy of the system based on this molecular theory. We differentiate this microscopic free energy with respect to curvature and composition of lipids and polymers. This provides the coefficients that are needed in the phenomenological expansion, Eq. 5. The main advantage of this method is that we can determine all the coefficients using the molecular theory applied only to the symmetric planar film. This is a very efficient way to perform systematic calculations that could not be done if the explicit free energy as a function of c, x_lipid, and x_PEG, needed to be determined for each combination of variables.

We consider that the hydrophilic heads of the lipids are located at the lipid-solvent interface. It is assumed that an a priori determined percentage of the total number of lipid heads are covalently attached to PEG chains. The hydrophobic lipid tails form a continuous isolated phase, i.e., no penetration of solvent and/or PEG chains into the hydrophobic phase is allowed. Fig. 1 shows a schematic representation of the system.

Both hydrophobic tails and polymer-grafted chains are described using MMF theory. The headgroup interactions will be modeled using the opposing forces approach of Tanford (Israelachvili, 1991). A detailed description of MMF can be found elsewhere for the lipid tails (Szleifer et al., 1990; Ben-Shaul, 1995) and for grafted polymer chains (Szleifer and Carignano, 1996). A basic description of the relevant aspects of the MMF model for our system is provided below. The basic idea of the theory is to look at a central chain with its intramolecular interactions taken into account in an exact way, within the chosen molecular model, while the intermolecular interactions are considered within a mean-field approximation. Due to the inhomogeneous character of the system in the direction perpendicular to the interface, z, the mean-field felt by the molecules is inhomogeneous in that direction. We show below a short derivation for the lipid chains and for the polymers. The difference arises from the fact that the hydrophobic lipid tails are assumed to be in a melt (no solvent) environment, while the polymer chains share the volume with the solvent molecules.

The starting point of the theory is to write the free energy of the system in terms of the probability distribution function (pdf) of chain conformations of the polymer (or lipid tails) and the distribution of solvent (absent for the lipid case). Thus, for the polymer case the free energy can be expressed as

(6)

where z denotes the radial distance from the bilayer midplane and G(z) is a geometric factor that is unity for planar films and is the spherical element of volume for the spherical aggregates. N is the number of PEG chains grafted at the interface i = in (out), Pⁱ() is the probability for the PEG chain to be in conformation , () is the internal energy of the PEG chain in the inner (outer) interface in that conformation, ⁱ is the number of chains per unit area at interface i; is the chain surface density of the planar symmetric bilayer, and N_solvent(z) is the number of solvent molecules at z. The first and fourth terms in Eq. 6 account for the conformational entropy of the PEG-grafted chains, while the second and fifth terms describe the translational entropy of the polymer at each monolayer of the bilayer. The third and sixth terms are the z-dependent translational entropy of the solvent. Note that P(), N, N_solvent(z), and ⁱ are all functions of the curvature.

The free energy of the lipid tails, F_lipid, is given by

(7)

which includes the conformational entropy of the chains and the internal energy of the molecules, with (') being the internal energy of a chain in the inner (outer) monolayer in conformation '. The last two terms correspond to the translational entropy of the molecules. There are only terms for the lipid chains because the hydrophobic core of the bilayer is assumed to be dry, i.e., without any solvent.

The repulsive intermolecular interactions are included in the model as hard-core excluded volume. These interactions are accounted for by packing constraints, which assume that the total volume available at each layer parallel (concentric) to the bilayer midplane (which is the origin of the z axis) must be filled with PEG segments or solvent for the polymer layer or with lipid tail segments for the hydrophobic core of the lipid bilayer. They are expressed as

(8)

where A(z) = A(0)(1 + 2cz + c²z²) is the total area at distance z from the bilayer midplane, v_PEG is the volume of a PEG segments, v_solvent is the volume of the solvent, v_lipid is the volume of a lipid segment, and n(, z) is the number of chain segments in z at conformation . For simplicity, we assume that v_solvent = v_PEG = v.

P(), P(), P_lipid('), and N_solvent(z) are determined by minimizing the free energy (Eq. 6 for grafted PEG and Eq. 7 for lipid tails) subject to the packing constraints (Eqs. 8). The minimization is carried out by introducing a set of Lagrange multipliers, _lipid(z) in the lipid region and _PEG(z) in the polymer-solvent region. The results are

(9)

Q = _' exp(_intra(')   _lipid(z)v_lipidnⁱ(', z)dz) is the partition function (normalization constant) of the lipid in monolayer i. Q =  exp(_intra()   (z)vnⁱ(, z)dz is the partition function (normalization constant) of the PEG molecules attached to interface i. _lipid(z) and _PEG(z) are the lateral pressures acting on the central chain at each z to fulfill the packing constraints. The magnitudes of _lipid(z) and _PEG(z) indicate the level of compression of the chains at each distance from the center of the bilayer. A thorough discussion of the physical origin of the Lagrange multipliers can be found in Ben-Shaul et al. (1985) and Szleifer and Carignano (1996).

The values of the Lagrange multipliers are obtained by replacing the explicit expressions for the pdfs and the solvent distribution, Eqs. 9, into the constraint equations, Eqs. 8. The input necessary to solve the equations is A(z) for all z, the set of single chain configurations for the PEG and the set for the lipid chains, the surface coverage of polymer and the thickness of the bilayer, 2h. Then, the equations are solved by standard numerical methodologies from which the lateral pressures are obtained, and from them any desired average conformational and thermodynamic property of the aggregate. A detailed description can be found in Szleifer and Carignano (1996).

Introducing the explicit forms of the pdfs and solvent distribution, Eqs. 9, into the free energy for the PEG, Eq. 6, we obtain

(10)

and for the lipid, Eq. 7, we get

(11)

Note that the geometry of the aggregate enters to the PEG and lipid chain free energies through the lateral pressures, _lipid(PEG)(z), and the area, A(z).

The last remaining element is the treatment of the lipid headgroups. We use the opposing forces model of Tanford (Israelachvili, 1991), in which there are attractive interactions arising from the hydrocarbon-water surface tension and a repulsive term that accounts, in an approximate way, for the steric interactions between the headgroups. There are two contributions arising from the two interfaces that give

(12)

where  is the hydrocarbon-water surface tension and A_h is a phenomenological parameter that measures the strength of the repulsions between the headgroups.

The total free energy of the system is obtained by adding the three contributions from Eqs. 10-12. It turns out to be more convenient to define the free energy per lipid molecule, given by

(13)

with N_lipid(PEG) = N + N. The total free energy is a function of the area per lipid molecule, a(0) = A(0)/N_lipid, the curvature, c, the ratio of PEGylated lipid molecules, r_PEG = N_PEG/N_lipid, the asymmetry of the lipids, x_lipid = N/N_lipid, and the asymmetry of the polymer chains, x_PEG = N/N_PEG. The phenomenological expansion of the free energy in Eq. 5 does not include the area. The reason is that we are assuming that the area per (lipid) molecule does not change upon deformation. That area is the surface of inextension, which for a symmetric planar bilayer is the midplane. Thus, the area a(0) is kept constant upon the formation of spherical liposomes.

The value of a(0) that we use is the one that minimizes the free energy of the planar bilayer. Namely, for each type of lipid molecule and each loading of polymer the total free energy is minimized with respect to area. The area at the minimum is the a(0) that we use throughout the calculations for that lipid and polymer loading. The area per molecule that minimizes the free energy of the aggregate is the correct one to use because we assume that the lipids are insoluble in the solvent. Therefore, all the lipid molecules are involved in aggregate formation. Note that in the case of soluble lipids (or surfactants), the area per molecule is not determined by minimizing the free energy of the aggregate with respect to area. Rather, the optimal area is the one in which the lipids have the same chemical potential as those found dissolved in the solvent. However, as mentioned above we are only interested here in insoluble lipids.

One of the interesting results in insoluble lipids is that because the proper packing is the one in which the area per molecule is the one that minimizes the free energy, the bilayer is therefore a tensionless aggregate. Namely, the stresses acting on the bilayer must cancel. This implies strong restrictions on the number of PEGylated lipids that can be included in the aggregate, because the only attractive interaction in the bilayer is the fixed water-oil surface tension. This interaction must balance exactly the repulsions arising from the packing of the lipid tails, the lipid headgroups, and the polymers.

The balance of forces can be seen in more detail by minimizing the free energy, Eq. 13, with respect to the area. In other words, we need to find the area for which
which gives

(14)

where _PEG(z) = (1  exp(_PEG(z)v  _PEG(z)v)/v is the lateral pressure of the polymer-solvent layer (Szleifer and Carignano, 1996). To understand the balance of forces let us take the case in which a_h = 0. Namely, the repulsive contribution of the lipid headgroups is zero. Actually, this is usually the case because a_h a(0) and, thus, it gives only a minor contribution to the repulsion. In this case, the equilibrium area of the bilayer is determined by the equality

(15)

Both _lipid(z) and _PEG(z) are absolute positive quantities because they represent the pressure arising from the repulsions associated with stretching the lipid and polymer chains. Thus, the optimal area is the one in which the lipid and polymer combined exert a repulsive interaction exactly equal to the (attractive) water-oil surface tension (see Fig. 1). As mentioned above, this limits the number of polymers that can be in the bilayer, as a large surface coverage of polymer results in a large repulsive interaction. Under these conditions, the PEG and lipid tail contributions exceed the surface tension attraction, thus breaking the aggregate. For conditions that we are interested in, it turns out that the most dominant contribution determining the area per molecule arises from the lipid tails. Actually, in most cases, neglecting the polymer contribution to determine the optimal area will give an error in the estimated area per molecule of <2%.

In a recent paper, the effect of lateral expansion induced by polymer brushes on the lipid layer was taken into account to study elastic constants of polymer-grafted lipid membranes (Marsh, 2001). The area changes predicted in the range of polymer concentration of experimental interest are <5%. It should be stressed, however, that for the determination of the elastic properties of the bilayers none of the contributions can be neglected, as it will be shown in the Results section.

The free energy of the aggregate from a molecular theory and the optimal packing area have now been described. The next step is to use that molecular free energy to determine the phenomenological coefficients that are presented in Eq. 5. To this end, we follow the same procedure of Szleifer et al. (1990), in which the explicit expression of the free energy from the molecular theory is differentiated with respect to curvature, lipid, and polymer asymmetry. In our case, however, we need to take derivatives up to fourth-order to obtain all the coefficients of the expansion. The results of the expansion are shown in the Appendix, with all the necessary details of how the calculation is carried out. The main result is that all the coefficients can be obtained from the knowledge of the properties of the equilibrium planar film. Therefore, the calculations, even though not trivial, become feasible for a large variety of relevant experimental accessible variables, without the need to perform calculations for each curvature and composition of lipid and polymer.

To summarize, we use the molecular theory to obtain the expansion coefficients that are necessary to determine the optimal size and structure of the spontaneously forming liposomes. Furthermore, if we now have the coefficients of the expansion we can calculate the free energy under all conditions.

In principle, we can determine the complete size distribution of the aggregates by combining the free energy of a single aggregate, Eq. 5, with a mixing term of the aggregates, in the ideal solution limit. The size distribution is then obtained by the minimization of that free energy subject to the constrain that the sum of lipids and the sum of polymers over all aggregates, weighted by the appropriate number of the aggregates, gives the total number of lipids and polymers present in solution. This will be pursued in future work.

All the calculations presented below were carried out for double tail lipid bilayers of varying length. Each lipid tail was of the form (CH₂)_n CH₃, with 11 < n < 15. Each CH₂ group was modeled as a unit and the chains were generated using the rotational isomeric state model (Flory, 1988) with the appropriate bond length and bond angles. Each bond was allowed to have one trans or two gauche configurations. The energetic price of the gauche configuration was taken as 500 cal/mol. Thus, the internal energy of a configuration is given by the number of gauche bonds multiplied by the gauche energy. The volume of a CH₂ group was taken as 27 Å³, while that of the CH₃ was equal to 54 Å³. For the lipid tails, all the possible configurations of the chains were generated and only those that were self-avoiding were used in the calculations. For details of the chain model of the hydrocarbon tails and how the calculations are carried out, see Szleifer et al. (1986, 1990). The model PEG chains had chain lengths in the range 50  N_EG  150. Each CH₂CH₂O was taken as a unit of volume 61 Å³ and the chains were generated using the rotational isomeric state model with the distance between EG groups taken as 3.2 Å, and the three states of each bond were taken to be isoenergetic, i.e., _intra() = 0 for all . This was the model used in Faure et al., 1998, where quantitative agreement was found for the pressure-area isotherms predicted by the theory and experimental observations of polymers containing PEG chains. We have used a sample of 1 × 10⁶ independent self-avoiding conformations that were generated by simple sampling. For more details on the chain model, calculation details, and predictions of the theory for PEG chains, see Szleifer and Carignano (1996, 2000); Szleifer (1996, 1997a); Faure et al. (1998); and Satulovsky et al. (2000).

	RESULTS AND DISCUSSION

TOP ABSTRACT INTRODUCTION MODEL RESULTS AND DISCUSSION CONCLUSIONS APPENDIX REFERENCES

Fig. 1 presents a schematic representation of a lipid/PEG-lipid film. It also includes a representative curve of the lateral stresses acting on the film. The relative contributions are plotted to scale. Polymer, lipid, and headgroups have repulsive interactions ((z) > 0), while water-oil surface tension is the attractive contribution ((z) < 0) that holds the system together. The larger repulsions arise from the packing of the lipid hydrophobic chains. PEG repulsions are smaller in magnitude and act at larger distances from the bilayer midplane. Both surface tension attraction and head-to-head repulsions from the lipid headgroups are assumed to act exactly at a distance corresponding to half-bilayer thickness. The magnitude of the lateral pressures has to be such that the total repulsion exactly balances the attraction, as expressed in Eq. 14.

In our previous study of spontaneous liposome formation from lipid/PEG-lipid mixtures (Szleifer et al., 1998), we used the free energy expansion up to the second order, as shown in Eq. 1. In terms of Helfrich's description of bilayer elastic constants (Helfrich, 1973), we looked at the conditions under which K = k_b + /2 < 0, where k_b is the bending elastic constant and is the saddle-splay constant. k_b is always positive and its value can be significantly reduced by considering the relaxation of the lipids and polymers between the two monolayers (Szleifer et al., 1990; Ben-Shaul, 1995). must be negative and larger than 2k_b to achieve K < 0.  is given by the second moment of the lateral stresses. Therefore, to maintain a tensionless membrane and obtain K < 0, we need to have relatively small pressures at large distances from the midplane. This was the explanation provided in our earlier work for the ability of polymer layers to induce spontaneous formation of liposomes.

Fig. 2, top shows K as a function of PEG loading. As the polymer loading increases, the value of the elastic constant decreases, eventually becoming negative. The different curves represent different chain lengths of PEG. The longer the polymer, the smaller the amount that it is necessary to have spontaneously forming liposomes. This has been shown to be the result of becoming large and negative as the chain length (and surface loading) of PEG increases, with only a small change in the value of the bending constant k_b (Szleifer et al., 1998). Fig. 2, bottom shows a schematic stability curve represented by the onset of the point in which K changes signs. Above the curve, spontaneous liposome formation is expected; lamellar phases are stable below it.

View larger version (18K): [in this window] [in a new window]   FIGURE 2   (Top) Elastic constant, K, as a function of the lipid-PEG loading for molecular mass of the PEG chains of 2.2 kDa (dashed line), 4.4 kDa (dotted line), 5.5 kDa (solid line), and 6.6 kDa (dotted-dashed line). The horizontal line is a guide to the eye for K = 0. (Bottom) Stability range for lamellar phases and for curved aggregates as a function of the PEG molecular mass. The line represents the percentage of PEG as a function of molecular mass at which K = 0. The lipid molecules have two tails each with (CH2)15 CH3 and the lipid headgroup interactions are modeled with  = 0.12 kT/Å2 and ah = 22.3 Å2.

The results presented in Fig. 2, as well as their explanation, are essentially the same as presented in our previous work. The problem is that, while we can analyze why spontaneous liposome formation happens in terms of the bending and saddle-splay constants, two main questions remain unanswered. First, what is the size and structure of the equilibrium aggregates formed, and second, what is the physical molecular driving force for spontaneous liposome formation? To answer these questions we now turn to the predictions of the theory using the free energy expansion, including fourth-order terms in curvature and composition of lipid and PEG.

Optimal size and structure

Fig. 3 shows the free energy, including the fourth-order terms, as a function of curvature for three different loadings of PEG for chains with N_EG = 100, corresponding to PEG of M_W = 4400. The free energy at each curvature is the minimal free energy with respect to the PEG and lipid distribution (i.e., x_PEG and x_lipid). Namely, the calculated free energies are obtained by fixing the curvature and finding the optimal partition of lipid and polymer between the two monolayers. This is repeated as a function of curvature, and then the onsets of minimized free energies are plotted. The three curves show maximal free energy for the planar bilayer. This is actually already known from Fig. 2, where the second-order term becomes negative at PEG loading of 9.12%. However, Fig. 3 also provides the optimal size of the liposomes. As the loading of PEG increases, the optimal radius of the aggregate decreases. The free energy changes over the interesting range of curvatures (namely, c < 0.01 Å) are rather small. For example, for the intermediate loading shown the optimal aggregate has a free energy gain, as compared to the planar bilayer, of ~10³ kT per lipid molecule. This implies that the optimal aggregate will have an overall gain in free energy of ~20 kT as compared to the planar geometry.

View larger version (12K): [in this window] [in a new window]   FIGURE 3   Minimum free energy as a function of curvature for a lipid/PEG-lipid bilayer containing 9.15% (solid line), 9.5% (dashed line), and 9.7% (dotted line) of PEG-lipid. The lipid tails are modeled as in Fig. 2 and the PEG chains have a molecular mass of 4.4 kDa.

Looking at the highest loading shown in Fig. 3, one can see the presence of a maximum at relatively high curvature c  0.0085 Ź. This implies that for this polymer loading spontaneous liposomes represent a local minimum in the free energy. There is a lower minimum at even smaller radii, implying that micelles are likely the preferred structure. Note that we are using a free energy expression that should not be valid for very small aggregates, therefore we will not analyze the structure of small micelles. Nevertheless, we will discuss below why we believe these micelles will form and under what conditions.

At this point it is interesting to look at the different contributions to the free energy to see what factors induce spontaneous liposome formation. Fig. 4 shows the separate contributions to the free energy together with their sum for the intermediate loading shown in Fig. 3. The free energy contributions are evaluated for each curvature at the optimal composition, as is the case in Fig. 3. Recall that we define loading as the total amount of PEG (or polymer) in the aggregate. Composition and/or partition asymmetry refers to the partition of PEG between the two monolayers. Thus, optimal composition means, for a fixed loading, the distribution of molecules between the two monolayers that minimizes the free energy at each curvature.

View larger version (15K): [in this window] [in a new window]   FIGURE 4   Contributions of lipid tails (flipid, dashed line), lipid headgroups (fheadgroup, dotted-dashed line), and PEG chains (fPEG, dotted line) to the aggregate minimum free energy (faggregate, solid line) as a function of curvature. The inset presents a zoom-in of the faggregate curve. Note the difference in free energy scale. The calculations correspond to the same system as Fig. 3 with PEG loading of 9.5%.

The free energy of polymer and lipid chains decreases as the curvature increases. Therefore, the more curved the aggregate, the better the chains are packed. Note that this is the packing of the chains in an asymmetric bilayer as discussed below. The only positive contribution is that of the headgroups. Actually, it is the surface tension term that opposes the increase in curvature.

The three separate contributions to the free energy are relatively large. However, the sum of them results in a small free energy per molecule with a minimal free energy at finite curvature. The small free energy at the minimum and the large separate contributions to it point to the special care required in accounting for all the relevant contributions. A small variation in one of the contributions is enough to result in a large change in the optimal size or the existence of spontaneous liposomes.

To better understand the origin of each contribution to the free energy, it is best to look at the optimal partition of the polymer and lipids as a function of curvature (Fig. 5). The lipid molecules show small asymmetry as the curvature increases. The increase of the number of molecules in the outer monolayer (and consequent decrease in the inner monolayer) is the result of the larger (smaller) available volume for packing of the chains in the outer (inner) monolayer as the curvature increases. This is optimal for the lipid tails, but it results in a high free energy cost to the surface tension term, which is proportional to the total interface area, which increases with curvature. Thus, a positive contribution to the free energy arises from the headgroup term, as shown in Fig. 4. It is interesting to note that the sum of the lipid chains and headgroup will result in an overall positive free energy, and, therefore, no spontaneous liposome formation without the inclusion of PEG.

View larger version (10K): [in this window] [in a new window]   FIGURE 5   Lipid (dotted-dashed line) and PEG (dashed line) mole fraction at the outer monolayer as a function of the curvature corresponding to the equilibrium aggregate (i.e., minimum in faggregate) presented in Fig. 4. The mole fraction at the outer monolayer for PEG (x) calculated as a linear function of curvature (i.e., using Eq. 2 to first order) is also shown (solid line). x as a linear function of curvature overlaps with the curve presented for the equilibrium aggregate and it is not shown.