INTRODUCTION

TOP ABSTRACT INTRODUCTION THEORY RESULTS AND DISCUSSION CONCLUDING REMARKS REFERENCES

Mixing aqueous solutions containing DNA and cationic liposomes results in the spontaneous formation of composite, typically micron size, complexes containing both DNA and lipid molecules (Rädler et al., 1997; Koltover et al., 1998; Lasic et al., 1997; Templeton et al., 1997; Sternberg et al., 1994; Tarahovsky et al., 1996; Hübner et al., 1999; Boukhnikachvili et al., 1997; Pitard et al., 1999). These complexes are of great current interest as gene-delivery vectors, in which context they are sometimes called "lipoplexes" (Felgner et al., 1987; Felgner, 1997; Lasic, 1997; Hope et al., 1998). In general, the liposomes used for complex formation contain at least two kinds of lipid molecules. The key component are the cationic lipids (CL), which serve as the condensing agents of the negatively charged DNA strands. Also important are the neutral helper lipids (HL), which play a crucial role in determining the structure of the composite condensates. They also seem to affect the DNA transfection efficiency, yet their operation mechanism is not entirely clear (Hui et al., 1996; Zuidam et al., 1999).

The preferred equilibrium geometry of a lipid-DNA condensate is dictated by the surface charge density and the elastic properties of its constituent lipid layers. Both of these characteristics depend, in turn, on the nature and composition of the CL/HL mixture. Double-stranded DNA, being a rather rigid molecule (of large persistence length, l_P  500 Å for B-DNA), imposes constraints on the possible lipoplex geometries because it retains its essentially linear structure in all complexes. In contrast, the lipid layers are soft self-assembled membranes that can adapt their structure to optimize the complexation geometry.

Indeed, several different lipoplex morphologies have been observed, corresponding to different lipid mixtures. Some of these structures may correspond to metastable intermediates, e.g., the spaghetti-like aggregates that consist of a (possibly supercoiled) double-stranded DNA wrapped around by the CL/HL bilayer (Sternberg et al., 1994; Sternberg, 1996; May and Ben-Shaul, 1997). Two condensate symmetries have been unambiguously identified as equilibrium ordered phases. These are the lamellar, L^C, and the hexagonal, H_II^C, aggregates, whose structural and thermodynamic characteristics have been quantitatively determined by x-ray diffraction and complementary measurements (Rädler et al., 1997; Salditt et al., 1997; Koltover et al., 1998; see also Lasic et al., 1997; Templeton et al., 1997; Tarahovsky et al., 1996; Hübner et al., 1999; Boukhnikachvili et al., 1997; Pitard et al., 1999).

The L^C (or "sandwich") phase is a smectic-like array of stacked lipid bilayers with DNA monolayers intercalated within the intervening water gaps. The DNA strands within each gallery are parallel to each other, exhibiting a definite repeat distance d. Although d depends on the CL/DNA and CL/HL concentration ratios, the spacing between two apposed lipid monolayers is nearly constant, l  26 Å, corresponding to the diameter of a double stranded B-DNA (2R_D  20 Å) surrounded by a thin hydration layer. The L^C phase is stabilized by the electrostatic attraction between the negatively charged DNA and the cationic lipid bilayer. Without DNA, the lamellar lipid phase (L) is unstable owing to the strong electrostatic repulsion between the charged bilayers.

Similarly, the H_II^C, or "honeycomb" (May and Ben-Shaul, 1997), structure may be regarded as an ordinary inverse-hexagonal (H_II) lipid phase with DNA strands intercalated within its water tubes. Here too, the diameter of the water tubes is just slightly larger than the diameter of the DNA rods. The presence of DNA is crucial for stabilizing the hexagonal structure. Without it, strong electrostatic repulsions will generally drive the lipids to organize in planar bilayers.

The structural differences between the L^C and H_II^C phases imply significant differences between the electrostatic (charging) energies and the lipid elastic energies of these two geometries. In the H_II^C phase, each DNA molecule is surrounded by a highly (negatively) curved lipid monolayer, of radius R  13 Å (Koltover et al., 1998). This cylindrically concentric geometry provides efficient neutralization of the DNA charges by the cationic surface charges, especially at the isoelectric point, where the total cationic charge exactly balances the total DNA charge (May and Ben-Shaul, 1997). In contrast, the strongly bent lipid monolayer may inflict a significant curvature deformation energy penalty. The lower the bending rigidity of the monolayer, k, the smaller the deformation free energy price (Helfrich, 1973). More favorable is the case where c₀, the spontaneous curvature of the monolayer, conforms to the curvature of the DNA rod, namely, c₀  1/R_D, (the minus sign signifying that the monolayer curvature is opposite that of the DNA). Under these circumstances, the hexagonal complexes are expected to be more stable than the lamellar ones. It must be noted, however, that charged lipids generally prefer the planar bilayer geometry (c₀  0), whereas the inverse-hexagonal geometry is preferred by (some) neutral lipids. Thus, the stability of H_II^C complexes is expected to depend sensitively on lipid composition. Similar qualitative considerations imply that lipid mixtures characterized by a high bending stiffness (k k_BT where k_B is Boltzmann's constant and T the temperature) and/or small spontaneous curvature (|c₀| 1/R_D) will favor the formation of the L^C phase (May and Ben-Shaul, 1997; Harries et al., 1998; Koltover et al., 1998). In this geometry, charge matching is somewhat less efficient than in the hexagonal packing, yet the lower curvature energy overrides this difference.

These qualitative notions were elegantly corroborated by recent experiments in which the elastic properties of the lipid monolayers were controlled by changing the nature of the lipid mixture (Koltover et al., 1998). The cationic lipid in these experiments, dioleoyl trimethylamonium propane (DOTAP), is characterized by a very small spontaneous curvature. Using mixed-lipid vesicles composed of DOTAP as the cationic lipid and dioleoyl phosphatidylethanolamine (DOPE) as the helper lipid, it was found that the preferred aggregation geometry is the H_II^C phase. In contrast, using dioleoylphosphatidylcholine (DOPC) as the helper lipid promotes the formation of L^C complexes. These findings are consistent with the fact that pure DOPE self-organizes into an H_II phase, i.e., the spontaneous curvature of this lipid is negative, whereas DOPC molecules prefer the formation of planar bilayers. In these experiments, one tunes the spontaneous curvature of the lipid layer by controlling the composition of the lipid mixture. Based on many experiments in microemulsion systems, it is known that one can also control the bending rigidity of amphiphilic films. For example, by adding short chain alcohols to the mixture, it is possible to reduce the bending rigidity by about one order of magnitude (Safinya et al., 1989; Szleifer et al., 1988). Indeed, the addition of hexanol to the DOTAP/DOPC-DNA system results in a clear, first-order, L^C  H_II^C phase transition (Koltover et al., 1998).

The qualitative considerations outlined above regarding the relative stabilities of different CL-DNA aggregates apply to one, given, CL/HL composition. Furthermore, they are only valid if all lipids and DNA molecules participate in complex formation. Different considerations apply when the mixture is nonstoichiometric. Taking into account that aqueous solutions containing DNA and two kinds of lipids are multicomponent systems, they are expected to exhibit rich and complex phase behaviors.

For a given salt concentration (chemical potential) the aqueous solution can be treated as a large reservoir embedding the condensed phases (i.e., complexes, bare bilayers, and naked DNA), allowing one to count out the water and salt. This leaves us with three relevant chemical species (CL, HL, and DNA) which, by Gibbs' phase rule, corresponds to (a maximum of) five thermodynamic degrees of freedom. Fixing the temperature and assuming that the lipid layers are incompressible (in all four lipid-containing phases), we eliminate two more degrees of freedom. Still, the phase rule implies that (up to) three condensed phases can coexist in solution, e.g., two kinds of complexes and uncomplexed DNA. The experimental observation of a first-order L^C  H_II^C transition (Koltover et al., 1998), i.e., two coexisting phases, is in line with this conclusion. As we shall see, these systems are also expected to exhibit three-phase equilibria.

Our goal in this paper is to analyze theoretically the major determinants of the phase behavior of lipid-DNA solutions. To this end, we have studied in detail several representative systems, corresponding to lipid mixtures of different elastic characteristics. As we shall see, the phase behavior is quite simple for lipid layers which, in the absence of DNA, show strong propensity to form planar bilayers. Much richer and more complex phase diagrams, involving a multitude of transitions and coexistence regimes, are predicted for flexible and/or curvature loving lipid layers.

The phase diagrams presented in the following sections involve two levels of calculations. First, for a given type of lipid mixture, we calculate, as a function of the lipid composition (CL ratio) and lipid/DNA ratio, the elastic, mixing, and electrostatic charging free energies of all relevant structures, i.e., the L^C and H_II^C complexes, the bilayer and inverse-hexagonal lipid phases, and the uncomplexed DNA, as illustrated in Fig. 1. (The symbols H for H_II^C etc. are used for notational brevity.)

View larger version (72K): [in this window] [in a new window]   FIGURE 1   Schematic illustration of the five macroscopic phases included in our theoretical scheme. The phases denoted by H and S are the HIIC and LC complex structures, respectively. The symbols I and B mark the HII and L phases, respectively. D represents uncomplexed DNA. The shaded regions correspond to the DNA cross sectional area. The lipid layers are mixed, consisting of cationic and uncharged (helper) lipids.

The electrostatic free energies are calculated based on the nonlinear Poisson-Boltzmann (PB) equation using methods described elsewhere (May and Ben-Shaul, 1997; Harries et al., 1998). The elastic terms are evaluated using familiar expressions for the curvature and stretching deformations and simple models for the elastic constants of mixed lipid monolayers. Then, writing the total free energy of the solution as a weighted sum involving all possible phases, we determine the phase diagram by minimizing this free energy with respect to all relevant thermodynamic variables.

	THEORY

TOP ABSTRACT INTRODUCTION THEORY RESULTS AND DISCUSSION CONCLUDING REMARKS REFERENCES

We consider an aqueous salt solution containing N⁺ (monovalent) cationic lipids, N⁰ helper lipids, and double-stranded DNA of total charge eM, e denoting the elementary charge. The lipid and DNA molecules are distributed among the five possible structures shown in Fig. 1. We assume that all these structures, including the naked DNA, are large enough and can thus be treated as macroscopic phases. The total volume fraction of the condensed phases is assumed to be small, enabling us to treat the embedding solution as an infinite reservoir of (monovalent) salt of concentration n₀⁺ = n₀ = n₀ = constant. Under these assumptions, the total volume of the solution is irrelevant for phase transitions involving the condensed phases.

At a given temperature T and salt concentration n₀, our three-component system (DNA, CL, HL) is specified by two composition variables,

(1)

Here, m is the mole fraction of the cationic lipid in the original lipid mixture, and  is the ratio between the total number of cationic and DNA charges in the system. Equivalently, because all lipids, whether in pure lipid phases or DNA-lipid complexes, are organized in monolayers, we can regard  as the ratio between positive and negative macroion charges in the solution (to distinguish from the mobile counterion charges).

The total free energy of the three-component system F = F(N⁺, N⁰, M; n₀, T) is a sum of terms corresponding to the various phases. Each term involves several thermodynamic and structural degrees of freedom. The phase diagram of the system is determined by minimizing F with respect to these variables subject to material conservation conditions. In the two following subsections, we first define the relevant degrees of freedom corresponding to the various phases, and then describe our model for calculating the free energy components of each phase. We end this section with a brief discussion of the approximations and assumptions used in our theoretical model and their possible influence on our conclusions.

Phases

DNA

Lipid bilayer

Inverse-hexagonal phase

(2)

Lamellar complexes

Hexagonal complexes

Degrees of freedom

The DNA/CL/HL mixture can exhibit a variety of phase equilibria. One way to map the phase diagram of this system is to consider all possible two- and three-phase equilibria, solve the relevant coexistence equations, and identify the phase boundaries by matching the chemical potentials of the pertaining components. We adopt here an alternative, computationally more efficient, route. Namely, we express the total free energy of the three-component mixture, F, as a sum of contributions representing all possible phases and minimize it with respect to all relevant variables. For every given lipid/DNA mixture the minimization yields the number and identity of the coexisting phases, their relative proportions, and their compositional and structural characteristics.

Explicitly, our free energy functional involves eleven concentration variables: four N_i⁺ (i = B, I, S, H), four N_i⁰, and three M_i (i = D, S, H). All quantities appearing in F,

(3)

are functions of these variables, e.g., N_H = N_H⁺ + N_H⁰, _H = N_H⁺/N_H, etc. However, not all variables are independent. Furthermore, according to the phase rule, there can be no more than three coexisting phases, implying that (following the minimization of F for a given mixture) some of the concentrations must vanish.

For a given mixture, characterized by the total numbers of molecules, M, N⁺, and N⁰, three of the eleven variables are eliminated by the material conservation conditions,

(4)

Yet another variable can be counted out because of the structural-compositional constraint imposed on the H_II^C phase,

(5)

Subject to these conditions, F is now a function of seven independent variables. The thermodynamic state of a given lipid/DNA mixture (M, N⁺, N⁰) is determined by the global minimum of F in the multidimensional space defined by the seven composition variables. Note that specifying M, N⁺, and N⁰ is equivalent to specifying  and m, (see Eq. 1), and one extensive variable that is irrelevant for determining the phase behavior of the mixture. Thus, the phase diagrams presented in the next section will be described in the , m plane.

As a convenient reference point for calculating F, we choose the state where all lipids reside in a planar bilayer and all DNA is uncomplexed. Relative to this state, the free energy of the system is given by

(6)

When all lipids and DNA are associated in one phase, e.g., the lamellar complex, this free energy change may be regarded as the formation free energy of this phase.

Free energies

In this section, we describe the various contributions to the free energy of the different phases, and their dependence on the relevant chemical compositions. In fact, for all phases except the naked DNA (D), the free energy is of the form

(7)

where the three terms on the right-hand side of this equation represent the electrostatic (charging) free energy, the elastic curvature energy, and the 2D mixing entropy of the lipid layers, respectively. In the following, we briefly discuss each of these contributions and its specific form in a given phase.

Electrostatics

(8)

(9)

L.

(10)

Elastic energy

(11)

(12)

Mixing entropy

(13)

Molecular free energies

(14)

Approximations of the model

	RESULTS AND DISCUSSION

TOP ABSTRACT INTRODUCTION THEORY RESULTS AND DISCUSSION CONCLUDING REMARKS REFERENCES

The most interesting and relevant phases in lipid-DNA mixtures are, of course, the lipoplexes. Our model accounts for the two most important structures, namely, the H_II^C and L^C phases. In both phases, the DNA and lipid layers are tightly associated, yet the complexation geometries are qualitatively different. These differences imply different electrostatic stabilization energies and different dependencies on the elastic properties of the lipid layers and their composition.

The goal of the forthcoming analysis is to provide a theoretical scheme for predicting the conditions favoring one lipoplex phase over the other or, possibly, the coexistence of both structures. The term "conditions" refers here to the elastic properties of the lipid monolayers on the one hand, and the relative amounts of HL, CL, and DNA in solution, i.e.,  and m, on the other.

As we shall see below, the phase diagrams of CL/HL/DNA mixtures may exhibit rather complex behaviors, involving a variety of phase transitions and coexistence regimes. To assist the interpretation of these phase diagrams, we begin the discussion with two preparatory subsections. In the first, we compare the electrostatic free energies of the two complex phases as a function of lipid composition and lipid/DNA ratio. The second subsection is concerned with the effects of electrostatic interactions on the relative stabilities of the pure lipid phases, L and H_II.

All the calculations presented below were carried out for n₀ = 4 mM, (l_D = 50 Å). Similar phase behaviors correspond to lower salt concentrations. Significant differences are expected only at very high salt contents, that is, when the Debye length becomes considerably smaller than the dimension of a typical lipoplex unit cell. In this limit, however, the complexes become unstable.

Electrostatics of the H_II^C and L^C phases

In our phase diagram calculations, the radius of the lipid headgroup surfaces in the H_II^C phase is kept fixed at R_H = R_D +  = 13 Å. It is instructive, however, to examine how the electrostatic free energy of this structure varies with  and _H. In Fig. 2, the electrostatic free energy per hexagonal unit cell, _H^es, is shown as a function of _H for four values of the water gap thickness;  = 0.5, 3.0, 8.5, and 15.0 Å. (The lowest value of  is unrealistic, because we must allow for at least a minimal water layer, which we set equal to  = 3 Å. It is shown only for comparison.) Note that

(15)

where f_H^es is the electrostatic energy per lipid molecule in the H_II^C phase. For R_D +  + h = 19 Å and a = 70 Ų, we have _H^es/f_H^es  1.7 Å¹.

View larger version (24K): [in this window] [in a new window]   FIGURE 2   The electrostatic free energy, Hes, of the HIIC complex per hexagonal unit cell (of length 1 Å) for (a),  = 0.5 Å; (b),  = 3.0 Å; (c),  = 8.5 Å; and (d),  = 15.0 Å. The dotted lines indicate the compositions, *H, for which the HIIC structure is isoelectrical. The dashed curve corresponds to the free energy, cap(H = *H), according to the capacitor model, as given in Eq. 17.

For all , we find that the free energy _H^es(_H) is minimal at, or very near, the isoelectric point. At this point, _H is given by

(16)

as marked by the vertical dotted lines in Fig. 2. The minima of _H^es are more pronounced and occur closer to the isoelectric point for the smaller values of .

The dashed curve in Fig. 2 denotes the electrostatic energy according to the capacitor model mentioned in the previous section. This is the free energy of a concentric cylindrical capacitor, composed of an inner surface of radius R_D = 10 Å and an outer surface of radius R_H = R_D + , with water as the dielectric medium. The charge densities on these two surfaces are e/2bR_D and e_H/2b(R_D + ), respectively. The charging energy per 1 Å of this capacitor is

(17)

with  = a/2b_H  (R_D + h).

For l_D (recall l_D = 50 Å), the minimum in the PB free energy exactly coincides with the simple capacitor model (curves a and b in Fig. 2), indicating that the surface charges are not screened by counterions. Namely, all the excess (diffuse layer) counterions have been expelled into the bulk solution. The capacitor model becomes less adequate as  approaches l_D. Correspondingly, the minimum of _H^es is shifted from ^*_H to _H < ^*_H, i.e., to a lower charge density of the outer surface, thus reducing the charging energy. The minimum of _H^es increases, reflecting the less efficient charge neutralization associated with the increasing value of .

Unfortunately, the simple capacitor model is valid only at the isoelectric point. For _H  ^*_H, we need the PB equation to calculate the electrostatic energy. When the surfaces are not equally charged, counterions must be present in the aqueous gap to ensure electrical neutrality. The reduced entropy of these counterions results in a repulsive interaction (disjoining pressure) between the apposed surfaces (Parsegian and Gingell, 1972). To a good approximation, this energy is equal to the capacitor energy plus the excess charging energy of the lipid surface (when _H > ^*_H) or the DNA surface (when _H < ^*_H), i.e., the charging energy of the relevant surface by the amount of charge _H  ^*_H.

Hereafter, when referring to the H_II^C phase, we shall consistently use  = R_H  R_D = 3 Å. In addition to being the electrostatically most favorable configuration, this  also corresponds to minimal chain stretching (frustration) energy in the inverse-hexagonal symmetry. The isoelectric point corresponding to  = 3 Å, h = 6 Å and a = 70 Ų occurs at _H = ^*_H = 0.345.

Let us now compare the electrostatic energies per unit cell in the H_II^C and L^C complexes. In analogy to _H^es in Eq. 15, we define _S^es as the electrostatic free energy per 1 Å of the L^C unit cell,

(18)

where 2d/a = N_S/(bM_S) = _S/b_S.

In Fig. 3, we show _S^es as a function of the fraction of charged lipid in the complex, _S, for several values of the DNA-DNA spacing d (Harries et al., 1998). Also shown, (broken curve), is the electrostatic free energy of the H_II^C phase for  = 3 Å, (curve b in Fig. 2). The curves marked a-d in Fig. 3 correspond to lamellar complexes containing exactly the same number of lipids per unit cell as those marked a-d, respectively, in Fig. 2, which describes the hexagonal complexes.

View larger version (25K): [in this window] [in a new window]   FIGURE 3   The electrostatic free energy of the LC complex per unit cell (of length 1 Å), as a function of the CL mole fraction, S. The solid curves correspond to five representative values of the DNA-DNA spacing: (a), d = 51.8 Å; (b), d = 59.7 Å; (c), d = 77.0 Å; (d), d = 97.4 Å; and (e), d = 35.0 Å. (The minimal value of d is d = 2RD +  = 23 Å.) For comparison, we also show (broken curve) the electrostatic energy of the hexagonal complex, Hes(H), for  = 3 Å (curve (b) in Fig. 2). Note that a given d implies a g