INTRODUCTION

Top Abstract Introduction Discussion Concluding Remarks Appendix References

It is difficult to imagine a biological structure or process in which electrostatics do not play a significant role. This is because of the charge carried by virtually all proteins, polynucleotides (e.g., DNA), and cell membranes. Accordingly, it is not surprising that attempts to understand the interaction of specific proteins with DNA, and with cell membranes, have inspired researchers to focus a great deal of theoretical effort on practical ways of determining the distribution of mobile counterions and their consequent screening effects in aqueous solution (Honig and Nicholls, 1995). Similarly, in recent discussions of liposomal vectors for gene delivery, i.e., targeting of extracellular DNA into cell nuclei, fundamental electrostatic issues arise immediately because of the strong interactions between the DNA and cationic lipids used to complex it (Felgner et al., 1987, 1996; Felgner and Ringold, 1989; Gershon et al., 1993; Gustafsson et al., 1995; Lasic et al., 1997; Zuidam and Barenholz, 1997; Hui et al., 1996; Mok and Cullis, 1997).

A most compelling example is provided by the studies of Rädler et al. (Rädler et al., 1997; Salditt et al., 1997), who report the existence of highly novel DNA-cationic liposome complexes, as determined by high-resolution synchrotron x-ray diffraction and optical microscopy. In particular, the lipoplex is shown to consist of multilayer lamellar stacks of charged bilayer, each consisting of a mixture of charged DOTAP (dioleoyltrimethylammonium-propane) and neutral DOPC (dioleoylphosphatidylcholine) lipid, with the parallel DNA strands intercalated between.

Quite different morphologies are expected to arise for other choices of neutral ("helper") lipid; in the case of DOPE (dioleoylphosphatidylethanolamine), for example, inverted hexagonal ("honeycomb") organization of the lipid, with single strands of DNA in aqueous solution regions, is implicated (Felgner et al., 1987; Tarahovsky et al., 1996). "Spaghetti" structures have also been reported, in which each DNA strand is coated by a cylindrical bilayer of the cationic/neutral lipid mixture (Sternberg et al., 1994; Sternberg, 1996). Both of these honeycomb and spaghetti-like structures have recently been investigated theoretically (May and Ben-Shaul, 1997; Dan, 1998).

In the present paper we treat in detail the electrostatics and self-assembly characteristics of the multibilayer lamellar stacks of intercalated DNA, structures that we shall refer to henceforth as L^c complexes (see Fig. 1). We address them within the general context of the statistical thermodynamics of aqueous solutions of DNA and mixtures of neutral and cationic lipids (see Theory). Mobile counterions are described by the nonlinear Poisson-Boltzmann (PB) equation, which is solved numerically. Although we neglect elastic deformations of the DNA strands and bilayers, we do allow for the possibility of spatial inhomogeneities in the membrane surface charge density, in response to interactions with the anionic DNA. This effect turns out to be significant, and reflects the "extra" degree of freedom associated with cationic lipids in mixed, fluid bilayers. In solving the PB equation, then, we need to treat the (Gauss law) boundary conditions at the membrane surface in a fully self-consistent way, because the charge density there varies along the direction normal to the DNA strands, and does so in a way that depends on the distribution of counterions (electrostatic potential), which in turn depends on the charge at the surface. We do this in the Results section for a wide range of DNA-DNA spacings, overall lipid composition, and added salt concentrations. We then determine the phase evolution of the system by calculating free energies and solving the equations that express equilibrium between the L^c complex and, alternately, excess DNA and excess lipid. In this way we establish how DNA-DNA spacings d vary with the ratio of charged lipid to DNA, for each of several different lipid compositions (ratio of neutral to cationic lipid). In agreement with experiment, we find that for a lipid mixture of given composition, the spacings are constant throughout the low  range, where the complex coexists with excess DNA. In the high  range, where the complex coexists with excess lipid, the spacings are nearly constant as well. Throughout the "single-phase" region, however, where all of the DNA and lipids are accommodated by the complex, the DNA-DNA spacings increase linearly with , as implied by material conservation. This region is found to include the special ("isoelectric") point at which the total charges carried by DNA and lipid are equal. Moreover, at the isoelectric point the free energy of the complex is minimal.

View larger version (74K): [in this window] [in a new window]   FIGURE 1   Schematic illustration of the lamellar (Lc) lipid-DNA complex.

All of the above results can be qualitatively accounted for by a simple model described in the Discussion, in which the electrostatic effects enter only via the "excess charge" that measures the extent of deviation from the isoelectric point. In this way one can understand the constancy of DNA-DNA spacings at low and high , i.e., at large deviations from the isoelectric point, directly in terms of the mutual repulsions between like-charged DNA strands or lipid bilayer surfaces, respectively. We include in the final section a brief account of the theory of the L^c complex presented independently by Bruinsma (1998), who interprets the observed structural evolution (d versus ) via approximate analytical solution of the nonlinear PB theory. His analysis of the free energy (which is restricted to low cationic lipid contents) is based on a physical picture that is quite similar to ours; his conclusions regarding the phase evolution of the system are somewhat different. We also discuss there the quite different approach suggested by Dan (1996, 1997), who, in contrast, ascribes the preferred d spacing at low  to a competition between short-range electrostatic repulsions and longer-ranged DNA-DNA attractions mediated by the elastic deformation of the bilayer membranes.

	THEORY

In this section we outline our model for calculating the free energy of the L^c complex, and derive the thermodynamic relationships dictating the complex structure and phase behavior in lipid-DNA solutions, as a function of the overall lipid-to-DNA ratio and the cationic/neutral lipid composition.

Model

Ignoring edge effects, we shall treat the complex as an infinite periodic lamellar array consisting of alternating lipid bilayers and DNA monolayers, as schematically illustrated in Fig. 1. The DNA strands are assumed to be infinite, parallel, and equidistant rigid rods, thus forming a one-dimensional (1D) lattice. As noted in the previous section, the existence of a well-defined interaxis distance d (which depends on lipid composition and lipid-to-DNA ratio) has been unequivocally confirmed by x-ray diffraction studies (Rädler et al., 1997). Theoretical support for this finding will be given in the following sections. The naked DNA strands in solution will be treated as infinite cylindrical rods, and the liposomal membranes as perfectly planar infinite bilayers.

In modeling the DNA strands as infinite rods, we ignore the effects associated with their flexibility, in particular curvature fluctuations and undulation forces (Podgornik et al., 1989, 1994; Strey et al., 1997). This approximation is justified in view of the fact that the DNA persistence length (  500 Å) is significantly larger than all of the other relevant length scales in the L^c complex, namely, the DNA radius R  10 Å, the interaxial distance d  20-70 Å, the thickness of the interbilayer water gap h  25 Å, the bilayer thickness w  30 Å, and the average linear dimension of a lipid headgroup a^1/2, where a  70 Å² is the average cross-sectional area per lipid molecule in the membrane. It should be noted that any curvature fluctuation of an individual DNA strand within the monolayer implies a change in d extending over a distance of order . From the calculations presented in the next section, it will become apparent that such changes involve an electrostatic free energy penalty of several k_BT's, indicating that curvature and interaxis fluctuations in the complex are rather small (k_B is Boltzmann's constant and T is the absolute temperature).

Another assumption that will be made in this work is that the lipid bilayers are perfectly planar, and their thickness, w, is constant and independent of their lipid composition. In general, one cannot exclude the possibility of membrane curvature modulations induced by the DNA lattice (as schematically illustrated in Fig. 1). For lipid bilayers of high bending rigidity (Helfrich, 1973), these modulations are expected to play a minor role in determining the complex stability. On the other hand, when "soft" bilayers are involved in complex formation, these curvature modulations may become increasingly important, possibly leading to structural phase transformations involving, say, the inverted hexagonal/honeycomb states mentioned in the Introduction. The assumption of constant w is justified for bilayers whose cationic and neutral lipid components are of similar chain length. This is the case for the neutral lipids DOPC and DOPE, as well as the cationic lipid DOTAP, mixtures of which are known to form lamellar complexes with DNA (Rädler et al., 1997). The extension of our model to cases where w varies with the lipid composition is, in principle, straightforward.

The negative charges on the DNA surface are densely spaced; the average spacing between these charges along the axis of B-DNA is l = 1.7 Å. We shall assume that these charges form a continuous and uniform charge distribution over the DNA surface, which will be regarded as a perfect cylindrical envelope. This approximation is supported by numerical studies revealing that the electrostatic potential around the DNA surface is not different from that produced by a continuous charge distribution, except for a narrow region in its immediate vicinity (Wagner et al., 1997). In all of our calculations, we shall use R = 10 Å for the radius of this cylinder, implying a uniform charge density  = e/2Rl  0.15 Cm², corresponding, approximately, to one elementary charge, e, per 110 Å².

We shall also assume that the cationic and neutral lipids constituting the membrane are ideally mixed. In the free bilayer this implies, on average, a uniform and continuous charge distribution. The charge density is ⁺ = e/a, where  is the mole fraction of the cationic lipids and a is the average area per lipid headgroup. On the other hand, in the bilayers of the complex we shall allow for spatial modulations of the cationic charges, while assuming that ideal mixing applies locally. In all calculations we shall use a = 70 Å² (implying  = ⁺ when   0.65) for both lipid components, in both the free and the complexed bilayer.

Finally, the naked DNA, the free lipid bilayer, and the lipid-DNA complex will be treated as macroscopic phases, i.e., we ignore the free energy contributions associated with their overall translational and rotational degrees of freedom. These free energies are on the order of 1k_BT per particle, much less than their "internal" (electrostatic and mixing) free energies.

Free energies

We define a unit cell of a complex as a rectangular box of dimensions d × b × s, where d is the distance, along the x axis, between two neighboring DNA strands; b = h + w is the distance between two bilayer midplanes along the y axis; and s is the "depth" of the unit cell along the z (the DNA axis) direction. Because the complex is translationally invariant along the z axis, the calculation of the complex free energy is a 2D problem, and the choice of s is arbitrary (Fig. 2). Our numerical results will be reported for s = 1 Å. For the numerical evaluation of the complex free energy, it is convenient to consider only one-quarter of a unit cell, as shown in Fig. 2.

View larger version (49K): [in this window] [in a new window]   FIGURE 2   Schematic representation of one-quarter of the complex's unit cell. The Poisson-Boltzmann equation is solved in the aqueous interior subject to boundary conditions appropriate for surfaces I-V (see text).

Formation free energy

(1)

Complex

(2)

(3)

(4)

1

(5)

Bare bilayer, naked DNA

(6)

(7)

1

View larger version (16K): [in this window] [in a new window]   FIGURE 3   The free energy per molecule in the bare lipid bilayer (of area a = 70 Å2 per molecule) as a function of the cationic lipid mole fraction. Also shown is the charging energy D = afD/2Rs of a naked DNA of surface area a. The solid and dashed curves correspond to lD = 10 Å and 50 Å, respectively.

Phase behavior

Consider an aqueous solution containing DNA strands of total length sD, N⁺ cationic lipids, and N⁰ neutral (helper) lipids; N⁺ + N⁰ = N. The total length of DNA associated in complexes will be denoted as sD_C, (D_C  D). Note that D_C is also the number of unit cells in the complex. The length distribution of the DNA strands is irrelevant, as both the naked DNA and the complex are treated as (immobile) macroscopic phases.

As the concentration of monomeric lipids in solution is generally negligible, we can safely assume that all lipids are organized in bilayers which, in both the free and complexed states, are assumed to be planar. We find it convenient to express the total bilayer area, A = Na, in the form A = sL, so that L is the total "length" of the bilayer, if regarded as a strip of width s. We shall use L_C = L and L_B = (1  )L to denote the total length of the complexed and free bilayer, respectively. Note that L_C = dD_C, where d is the distance between DNA strands in the complex. Also, using N_C⁺ to denote the number of cationic lipids in the complex, we define L_C⁺ = (a/s)N_C⁺. Similarly, we define L_B⁺ = (a/s)N_B⁺, L_C⁰ = (a/s)N_C⁰, L_B⁰ = (a/s)N_B⁰ and L_C⁺ + L_B⁺ = L⁺, L_C⁰ + L_B⁰ = L⁰. The mole fractions of cationic lipid in the complexed and free bilayer are given by _C = N_C⁺/N_C = L_C⁺/L_C and _B = N_B⁺/N_B, respectively. These two lipid compositions are generally different, but related to each other by the conservation condition ("lever rule")

(8)

where m  N⁺/N = L⁺/L is the overall mole fraction of cationic lipid in solution.

Finally, we introduce the (dimensionless) quantity

(9)

expressing the ratio between the total number of surface positive (lipid) charges and negative (DNA) charges in the system. Of particular interest is the "isoelectric point,"  = 1. Experiment shows (at least for m  0.5) that at this point all of the lipids and DNA in solution are involved in complex formation (Rädler et al., 1997). In the next section we shall show that this result holds for a wide range of lipid compositions m and, furthermore, that the isoelectric point corresponds to the minimum of the complex free energy f_C.

Experiment also shows that upon increasing the overall lipid-to-DNA ratio (L/D), at constant lipid composition (m), the system evolves through three distinct regions:

1.	When L/D (equivalently,   L/D) is small, the system is biphasic; the solution contains lipid-DNA complexes that coexist with excess, naked DNA. Thus, in this region, D > DC, whereas LC = L (no free bilayer). The DNA-DNA distance in the complex is constant, d = d1(m), independent of  as long as   1(m), which marks the onset of the next region. Once  = 1, all of the DNA is complexed, so that DC = D = L/d1, and hence, from Eq. 9, 1 = md1(l/a). In general, 1 < 1.
2.	Between 1 and a certain 2 = 2(m) > 1, the system is one-phasic: all of the DNA and lipid is involved in complex formation. Thus, LC = L, DC = D, and hence d = L/D = (a/l)(/m) increases linearly with the lipid/DNA ratio, from d1 at 1, through dI = dI(m) = (a/l)/m at the isoelectric point ( = 1), to d2 = d2(m) at 2(m) = md2(l/a), which marks the onset of the third region. In general, 2 > 1.
3.	For large L/D ( > 2) the system is again biphasic, containing complexes that coexist with an excess bilayer phase, DC = D, LC < L. In this region the system possesses an extra thermodynamic degree of freedom, namely, the lipid composition of the complex, C (or, equivalently, B, which is related to C by Eq. 8). Thus, unlike in region 1, C (hence B) need not be equal to m. In other words, for any m and L/D, the system will adjust both d and C so as to minimize its total free energy. Indeed, we shall see that in the excess bilayer region, both C (hence B) and d vary with . It should be noted, however, that experimentally, d  d2(m) appears to be independent of  in region 3. This result will be discussed in more detail in the next section.

In principle, the system may also exhibit three-phase (complex/bilayer/DNA) coexistence as well as bilayer/DNA coexistence. However, these conditions correspond to very narrow regions of the phase diagram (low m values), where the complexes are either unstable or only marginally stable. We shall thus focus on the three-stage scenario outlined above.

Our analysis involves three possible phases: free DNA, free bilayer, and complex. The first two may be regarded as incompressible condensed phases. On the other hand, the complex is "compressible" because both the DNA-DNA spacing, d, and the interbilayer spacing, h, may vary with m and L/D. However, both experiment and our calculations (next section) show that in general, only d varies significantly with m and L/D, whereas h is essentially constant, h  h*. In other words, for most _C and d, the complex free energy f_C(_C, d, h) has a narrow and deep minimum at h*. Thus we can safely treat f_C = f_C(_C, d) = f_C(_C, d, h = h*) as a function of only two variables.

For given m and L/D (and given l_D), the number and nature of the phases in solution are determined by the minimum of the total free energy,

(10)

with respect to D_C, d, and _C (_B depends on these three variables through Eq. 8).

Setting D_C = L/d, _C = m in Eq. 10, and minimizing F with respect to d, we find the equilibrium condition for region 1, 

(11)

This equation determines the equilibrium interaxis distance in the complex, d = d₁(m), in the presence of excess free DNA. Based on this equation, we anticipate that d₁ will be smaller than the "optimal" value, d* = d*(m), corresponding to the minimum of f_C(m, d). This follows from the fact that the free energy of a DNA strand in a stable complex must be lower than in solution, and hence f_D  f_C(m, d) > 0, which means (f_C/d)_d=d1 < 0. Physically, the "overcrowding" (d₁ < d*) of DNA strands in the complex results from the partial release of mobile counterions into solution upon bringing more DNA charges into contact with the cationic lipid charges. When d = d₁, this "overcharging" of the complex by DNA is balanced by DNA-DNA repulsion within the complex (the latter of which increases as d decreases).

In region 2, where all of the DNA and lipids are associated in complexes, F = Df_C(m, d), and d = L/D increases linearly with the lipid-to-DNA ratio. (The linear increase reflects our assumption that the bilayer is planar and laterally incompressible.) At some point within this region, generally very close to  = 1, the complex free energy is minimal (i.e., d_I(m)  d*(m)). The uptake of bilayer into the complex continues beyond this point, as long as the added lipids enjoy lower free energy in the complex as compared to that in the free bilayer. Eventually, at some d = d₂(m) > d* (and = ₂(m) > 1), interbilayer repulsion becomes sufficiently large to forbid further accommodation of bilayer in the complex, marking the onset of region 3. To support this qualitative description, let us first consider the hypothetical case of "blocked lipid exchange," where _B = _C = m. (This limit could perhaps be realized experimentally, as a transient state, if the rate of lipid exchange is small compared to that of complex formation.) Setting D_C = D, _C = _B = m in Eq. 10, and minimizing F with respect to d, we find

(12)

which determines d_{2  2}(m) for the case of blocked exchange. For this special case, let ₂(m) denote the value of  at the boundary between regions 2 and 3, corresponding to  = dD/L = 1 in Eq. 8. From Eq. 12 it follows that d = ₂(m) is constant throughout region 3 ( > ₂(m), or 1 >  > 0). Because ₂(m) is also the maximum d in region 2, it follows that ₂(m) = m₂(l/a). Finally, because the bilayer charging energy, _B(m), is positive, it follows from Eq. 12 that ₂ > d*.

In the more general case of free lipid exchange, the values of d, _C, and _B in the bilayer-complex coexistence region are determined by the equilibrium conditions (F/d) = 0 and (F/_C) = 0. Noting that in this region D_C = D and (dD/L)_C + (1  dD/L)_B = m (see Eq. 8), we obtain

(13)

(14)

We could rewrite the last two equations in a slightly different form in terms of _C  f_C/d, the free energy per unit length of the complex, instead of f_C, the free energy per unit cell. Then, if d were constant ("incompressible complex"), Eqs. 13 and 14 would reduce to the familiar "common tangent construction" for _C and _B, representing the coexistence conditions of two incompressible binary mixtures. If this were the case, we would also find that _C and _B are independent of . However, because the complexes are not incompressible, both d and _C (and hence also _B) may vary with , as will be shown in the next section.

	RESULTS AND ANALYSIS

Following the discussion in the previous section, we shall first present and analyze the numerical results for the free energy and structure of an isolated DNA-lipid complex and then discuss the phase behavior of the solution. Comparison with detailed published data for L^c complexes is possible for only one kind of system: a solution containing an equimolar (m = 0.5) mixture of cationic (DOTAP) and nonionic helper (DOPC) lipids and linear (either -phage or plasmid) DNA, without added salt (Rädler et al., 1997). The bulk concentration of mobile ions in this system is low, but the exact concentration is unknown (as it is volume dependent). Thus, in most calculations, we have used n₀  4 × 10³ M, corresponding to a Debye length l_D = 50 Å. Very similar properties and phase behavior of the complex were found for larger values of l_D. Partial results will also be presented for l_D = 10 Å, corresponding to physiological salt concentrations (n₀  0.1 M). In all of the calculations reported below, we have used R = 10 Å for the DNA radius and a = 70 Ų for the average area per (both cationic and neutral) lipid headgroup.

Complex structure and stability

The electrostatic (charging) free energy per unit cell of the complex, f_C, is shown as a function of d for several values of _C in Fig. 4 (for s = 1 Å, l_D = 50 Å). Similarly, Fig. 5 shows f_C as a function of _C for several values of d.

View larger version (24K): [in this window] [in a new window]   FIGURE 4   The free energy per unit cell of the complex as a function of the DNA-DNA spacing, for several different mole fractions of the cationic lipid: C = 0.23 (), 0.39 (), 0.50 (), 0.62 (), 0.78 (). The inset shows the optimal interbilayer distance, h*, versus the optimal DNA-DNA spacing, d*, for a low lipid composition, C = 0.15 (). For all C larger than ~0.2, h* is constant ().

View larger version (24K): [in this window] [in a new window]   FIGURE 5   The free energy per unit cell of the complex, as a function of the lipid composition, for several values of the DNA-DNA spacing: d = 23 (), 33 (), 43 (), 73 (), 93 () Å. The inset shows how the optimal spacing, d* (---) and isoelectric spacing, dI () vary with the lipid composition in the complex, revealing that d* and dI are essentially identical.

All of the results shown in Figs. 4 and 5 were obtained using h = h* = 26 Å, corresponding to a minimal distance of 3 Å between the DNA and bilayer surfaces. This is the value of h* observed experimentally for the L^c complex by Rädler et al. (1997). It should be noted, however, that h* is larger than the minimal value of the interbilayer spacing, h_min = 2R = 20 Å. In fact, for most values of _C, our calculations show that the electrostatic free energy of the complex decreases monotonically as h decreases, including the region h* > h > h_min. Thus we treat h*  26 Å as the effective range of a "hard-wall" potential, representing the short-range repulsive forces arising from hydration, protrusion, and other excluded volume interactions (Israelachvili, 1992; Israelachvili and Wennerström, 1990). Subject to this condition, we find that for all _C larger than ~0.2, the minimum in f_C(_C, d, h) is always at h = h*, regardless of d. For very low values of _C (less than 0.2), we find, for low d's, that the optimal value of h increases as d decreases, as demonstrated for _C = 0.15 in the inset to Fig. 4. Note, however, that for these low _C's, the minimum of f_C occurs at large d*'s, where again, h = h*. More generally, our conclusions regarding the complex structure and stability or the phase behavior of the system are not sensitive to small variations in h*.

In Fig. 4 we see that the optimal DNA spacing in the complex, d*, is a decreasing function of _C. Similarly, Fig. 5 shows that the optimal complex composition ^*_C is a decreasing function of the DNA-DNA distance.

Qualitatively, these results are easily understood. The minima in the electrostatic free energy are expected to occur when the fixed negative charges on the DNA surface are balanced by the same number of positive charges on the bilayer surface, i.e., at the isoelectric point. At this point, the complex will remain electrically neutral, even if all of the mobile ions in its interior would be released into the bulk solution, thus increasing their translational entropy and consequently lowering the free energy of the system. Of course, some counterions will always remain within the complex water gaps, as dictated by the bulk value of their chemical potentials. However, the concentrations of these mobile ions will be much smaller than in the diffuse layers near the surfaces of the noncomplexed DNA and membrane. Now the total charge on the bilayer surface is proportional to d × , whereas the total charge on the DNA surface is constant. Thus, at the isoelectric point d_I(_C) = (a/l)/_C, explaining the decrease in d_I  d* with _C. The inset to Fig. 5 shows how d_I and d* vary with _C. The two curves are essentially identical, confirming that the complex free energy is, indeed, minimal at the isoelectric point. Thus, hereafter, we set d_I  d*.

Figs. 4 and 5 also reveal that the minimum value of the complex free energy f^*_C  f_C(_C, d*(_C)) varies rather weakly with _C. More generally, we note that as _C (or d) is changed, the complex can change its d (or _C), i.e., "cross" to a neighboring free energy curve, without significantly changing its free energy. This ability of the complex to change its composition (and d) at minimal free energy cost is manifested when complexes coexist with an excess bilayer phase, in which case _C and _B are determined by the minimum of F (rather than f_C), as will be demonstrated in the next section (Phase evolution).

Based on the numerical results for f_C, we can estimate the amplitude of interaxis fluctuations, d = [(d  d*)²]^1/2. Imagine that one DNA strand, say of length   500 Å, is displaced toward one of its neighbors by a distance d, thus creating two unit cells of dimensions d = d* ± d. Allowing the lipid composition in the new unit cells to relax (implying  = (d/d*)), the free energy cost of this fluctuation is f = [f^*_C(d  d) + f^*_C(d + d)  2f^*_C(d)] = (²f^*_C/d²)(d)². We find that f  1 k_BT for d  |d|  1 Å.

When d < d*, there is a net negative surface charge on the complex "walls." To ensure electrical neutrality, positive mobile ions must be brought from the bulk solution into the confines of the complex, thus increasing the free energy of the system. As d decreases, the excess concentration of positive counterions increases, for two reasons: the increase in the excess surface charge and the decrease in the inner complex volume. The concomitant increase in the free energy of the complex, and hence the effective DNA-DNA repulsion, is due to the excess charging energy of the DNA surfaces, and the increased osmotic pressure of the counterions within the complex interior. (A simple electrostatic model accounting for this behavior will be described in the Discussion.)

Similarly, as d increases above d*, negative mobile ions must be brought into the complex to balance the excess positive charge on the (lipid bilayer) surfaces. However, unlike in the d < d* region, where counterion confinement depends strongly on d, in this region counterion confinement is mainly due to the finite bilayer spacing h. Because h is constant, f_C is expected to increase linearly with d (in the large d region), as is indeed observed in Fig. 4. The rate of this increase, i.e., f_C/d, is proportional to the electrostatic free energy per unit area of the bilayer in the complex. This free energy is a sum of the bilayer charging energy, which increases with _C (see below) and the interbilayer repulsion energy. For most values of _C considered here, the complex conditions are those of the "Gouy-Chapman regime" (Andelman, 1995), where the interbilayer interaction energy is independent of the surface charge density. Thus the _C dependence of the asymptotic slope of f_C in Fig. 4 is mostly due to the charging energy of the lipid monolayers.

These notions are confirmed in Fig. 6, which shows the formation free energy of the complex, f_C, as a function of d for several values of _C. Note from Eq. 1 that this quantity, which represents the net stabilization energy of the complex, is obtained from f_C after subtracting the charging energy of the noncomplexed DNA and bilayer. Thus the steep variation of f_C at small values of d is dominated by the strong DNA-DNA repulsion (counterion confinement) in this regime. Similarly, the increase in f_C at high d's (d  d*) is due almost exclusively to interbilayer repulsion. From the discussion above it follows that in this region f_C/d should be nearly independent of _C, as confirmed by Fig. 6.

View larger version (16K): [in this window] [in a new window]   FIGURE 6   The formation free energy of the complex, as a function of d, for several values of the lipid composition, C: 0.3 (- · - · -), 0.5 (---), 0.7 (· · · ·), 0.9 ().

From the results in Fig. 6 we also conclude that stable complexes (f_C < 0) can be formed for a wide range of lipid compositions. The complex stabilization energies are on the order of a few k_BT's per unit cell. For a "mesoscopic" complex, containing DNA strands of total length on the order of, say, 1 µm, this implies a total stabilization energy on the order of 10⁴ k_BT.

In the previous section we emphasized the fact that the lateral distribution of the cationic lipid charges in the complex need not be uniform. Indeed, we find that the actual charge distribution is polarized, reflecting a compromise between the tendency to minimize the electrostatic energy on the one hand, and the unavoidable demixing entropy penalty on the other hand. The extent of spatial charge modulations in the complex is demonstrated in Fig. 7. The figure shows the variation in the local charge density (x) between two neighboring DNA strands, for complexes of three lipid compositions (high, low, and equimolar _C = (x)), all at their isoelectric (i.e., optimal) value of d.

View larger version (23K): [in this window] [in a new window]   FIGURE 7   Spatial modulations of the cationic lipid charge within a unit cell of the complex. The local charge density profile, (x) (between two neighboring DNA strands), is shown (solid lines) for complexes of three lipid compositions: C = 0.23, 0.50, 0.78. All complexes are at their isoelectric value of d. The horizontal (dashed) lines correspond to uniform charge densities. The dash-dotted line, corresponding to C = 0.5, shows the charge density profile in a (hypothetical) complex in which charge modulation (lipid demixing) does not involve any entropy penalty. Note that in all but the highest C case, cationic lipid is pushed out from between the DNA positions.

When _C is low, d* is necessarily large. To effectively screen the negative DNA charges, cationic lipids must be displaced over a relatively large distance, resulting in a dramatic charge modulation. On the other hand, when _C is large, d is small, and the charge segregation is rather weak. In fact, in this case some of the charged lipids are shifted from the immediate vicinity of the DNA toward the center of the unit cell, as their optimal local concentration near the DNA strands is lower than _C. (Recall that the charge density on the DNA surface corresponds to one elementary charge per ~110 Ų. The average charge density on the bilayer surface is _C/a, which, for _C = 0.78, corresponds to one elementary charge per ~90 Ų.) Intermediate though substantial charge modulation is found for the equimolar lipid mixture, _C = 0.5. For this system we also show, for comparison, the charge density profile in the hypothetical case in which lipid segregation does not involve a demixing entropy penalty. (Namely, we artificially ignore the lipid mixing entropy contribution to f_C. The PB equation is then solved subject to the condition of constant electrical potential on the bilayer surfaces, as if they were conducting sheets.) As expected, the charge modulation in this system is still more dramatic than in the "real" complex.

Phase evolution

In Fig. 8, A and B, we show how d, the DNA-DNA spacing in the complex, varies with  = m(l/a)L/D, the (scaled) charged-lipid to DNA ratio in solution. The d   plots in Fig. 8 a were calculated for a solution of low salt content, l_D = 50 Å, and several different lipid compositions m. Similar calculations are shown in Fig. 8 b for l_D = 10 Å.

View larger version (18K): [in this window] [in a new window]   FIGURE 8   The DNA-DNA spacing d in the complex, as a function of the charged lipid-to-DNA ratio , for several lipid compositions: m = 0.3, 0.4, 0.5, 0.6, 0.8 (solid lines). For each value of m, the dashed curve describes the variations in d for the case of blocked lipid exchange. (A) lD = 50 Å. (B) lD = 10 Å.

These calculations provide the most critical test of our model, because d is an experimentally measurable quantity. The experimental d   data points of Rädler et al. (1997), which were obtained for an equimolar lipid mixture (m = 0.5) and without added salt, are shown in Fig. 9. Also shown in this figure are the theoretical curves corresponding to l_D = 10 Å and 50 Å, both for m = 0.5. The low-salt (l_D = 50 Å) results show reasonable agreement with the experimental data. The inset to Fig. 9 shows how the (calculated) lipid compositions in the complex and free bilayer (in the "excess bilayer" regime) vary with the charged lipid-to-DNA ratio.

View larger version (21K): [in this window] [in a new window]   FIGURE 9   The DNA-DNA spacing, d, as a function of charged lipid-to-DNA ratio, , for m = 0.5; lD = 50 Å (solid line), 10 Å (dashed line). The dots are the experimental data of Rädler et al. (1997). The inset shows the variation in lipid composition in the complex and free bilayer as a function of the charged lipid-to-DNA ratio, for lD = 50 Å.

The d   "phase diagrams" in Figs. 8 and 9 were calculated using Eq. 11 for region 1 (excess DNA), and Eqs. 13 and 14 together with the lever rule (Eq. 8) for region 3 (excess bilayer). Equation 11 yields d₁ = d₁(m) for the complex-DNA coexistence region 1, 0    ₁(m) = md₁(l/a). In the one phase (complex) region 2, d = L/D = (a/l)(/m) varies linearly with . The slope, d/, in region 2 is inversely proportional to the charged lipid mole fraction, m.

For region 3 the calculation is a little more complicated because of the additional lipid composition degree of freedom. For each value of m, the solution of Eqs. 13, 14, and 8 yield d, _C, _B as a function of