J. Phys. Chem. B 2009, 113, 12277–12282
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Binding of DNA to Zwitterionic Lipid Layers Mediated by Divalent Cations Demmelash H. Mengistu,† Klemen Bohinc,‡,§ and Sylvio May*,† Department of Physics, North Dakota State UniVersity, Fargo, North Dakota 58108-6050, and Faculty of Health Sciences, Poljanska 26a, UniVersity of Ljubljana, 1000 Ljubljana, SloVenia ReceiVed: May 28, 2009; ReVised Manuscript ReceiVed: July 9, 2009
Divalent cations, i.e., calcium, magnesium, and others, are able to enhance the ability of DNA to interact with membranes that are composed of zwitterionic lipids such as phosphatidylcholine. The resulting condensed complexes offer potential applications as nontoxic gene delivery vehicles. The present study suggests a generic theoretical model to describe the energetics and structural features of a zwitterionic lipid-DNA complex in the presence of divalent cations. Specifically, we consider the adsorption of a single molecule of doublestranded DNA onto a planar zwitterionic lipid layer. Our theoretical model is based on the continuum Poisson-Boltzmann formalisms, which we modified so as to account for the two opposite charges and orientational freedom of the zwitterionic lipid headgroups. We find a substantially more favorable adsorption free energy of the DNA if divalent cations are present. In addition, our model predicts the divalent cations to preferentially interact with the phosphate groups of the zwitterionic lipids, given these lipids are located in close vicinity to the DNA. This is accompanied by a small but notable reorientation of the zwitterionic headgroups toward the DNA. We demonstrate that the binding of DNA onto a zwitterionic lipid layer is not driven by the release of counterions. Instead, the binding leads to a partial redistribution of the divalent cations, from the phosphate groups of the DNA (prior to the binding) to the phosphate groups of the zwitterionic lipids (after the binding). Our results thus suggest a general physical mechanism underlying complex formation between DNA and zwitterionic lipids in terms of mean-field electrostatics, i.e., neither involving correlations nor specific interactions of the divalent cations. Introduction Complexes between lipid bilayers and DNA have attracted considerable interest in the past because of their potential use as nonviral carriers of genetic material into living cells.1 The bilayer commonly contains a fraction of cationic lipids that interact electrostatically with the negatively charged phosphate groups of the DNA. The two major drawbacks of using these so-called polyplexes are the relatively low transfection efficiency and the cytotoxicity of the cationic lipids.2 Naturally occurring uncharged lipids are biodegradable and nontoxic, but they are incapable of condensing DNA into stable aggregates. However, upon the addition of divalent metal cations even uncharged lipid layers acquire the ability to form complexes with DNA. Experimental evidence exists particularly for the calcium-induced formation of complexes between DNA and lipid bilayers consisting of dipalmitoylphosphatidylcholine (DPPC).3-5 Here the bilayer-DNA interaction appears strong enough to form an ordered rectangular columnar phase.6 Yet, also with other lipids, such as dioleoylphosphatidylcholine (DOPC) and dioleoylphosphatidylethanolamine (DOPE), or with other divalent cations, including magnesium (Mg2+), manganese (Mn2+), and others, has complex formation been observed.4,7,8 Complex formation is not restricted to bilayers; also uncharged lipid monolayers on the air-water interface have been observed to interact with DNA in the presence of divalent cations.9-11 We note that depending on the choice of the uncharged lipid and divalent cation, there is considerable diversity in structure, * E-mail:
[email protected]. † North Dakota State University. ‡ University of Ljubljana. § Second address: Faculty of Electrical Engineering, Trzˇasˇka 25, University of Ljubljana, 1000 Ljubljana, Slovenia.
stability, and phase behavior of the complexes.5 For example, replacing DOPC by DOPE can induce the transition from a lamellar to an inverse hexagonal structure.7,8,12 Moreover, lipid-DNA complexes coexist with uncomplexed lipid layers above a certain lipid-to-DNA ratio.13 Finally, the stability of the complexes depends on the chemical type of divalent ion and thus involves, to some extent, specific interactions. Still, the very fact that for a diverse set of systems (bilayers and monolayers) various divalent cations enhance the interaction between uncharged lipids and DNA suggests the existence of a general underlying physical mechanism. Uncharged lipids such as phosphatidylcholine or phosphatidylethanolamine are zwitterionic. That is, they possess a large headgroup dipole which results from the spatial separation of the negatively charged phosphate group from a positively charged moiety (i.e., choline for phosphatidylcholine and ethanolamine for phosphatidylethanolamine). The presence of two individual charges of opposite sign in each lipid headgroup suggests a possible mechanism for the interaction with DNA: binding of the divalent cations to the phosphate groups renders zwitterionic lipids effectively cationic. This hypothesis has been put forward by McManus et al.,3 who have extracted from smallangle X-ray scattering data a stoichiometric binding model. According to that model, calcium binds to DPPC much stronger in the presence of DNA. That is, calcium bridges the phosphate groups between two neighboring lipid molecules, effectively leaving the two lipids each with a single cationic charge. As a result, the involved lipids are able to interact with the DNA through their positively charged moieties. This interaction is further enhanced by a reorientation of the headgroup toward the DNA charges.
10.1021/jp904986j CCC: $40.75 2009 American Chemical Society Published on Web 08/17/2009
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Figure 1. Schematic illustration of a single (double-stranded) DNA molecule, modeled as a long cylinder of radius R and uniform surface charge density σ that is bound to a zwitterionic lipid layer. The DNA contacts the headgroup region, 0 e z e l, of the lipid layer. The positive charges of the zwitterionic headgroups are allowed to move within the headgroup region, whereas the negative charges remain anchored to the polar-apolar interface, z ) 0. Monovalent salt ions (positive and negative) and divalent cations are present in the aqueous phase, including the headgroup region. The average cross-sectional area per lipid is denoted by a, and l is the thickness of the headgroup region.
The objective of the present work is to investigate the underlying physical mechanism that leads to the enhancement of interaction between zwitterionic lipids and negatively charged macro-ions (i.e., DNA) by divalent cations. Our hypothesis is that the role of the divalent cations can be understood on the basis of mean-field electrostatics alone. Hence, the model we choseinthepresentworksamodifiednonlinearPoisson-Boltzmann modelsdoes neither include electrostatic correlations (which are often essential to understand divalent ions14 but become negligible in the small concentration limit15,16) nor any specific interactions involving the divalent cations. Instead, we incorporate into the Poisson-Boltzmann formalism a model for zwitterionic lipids that was developed17 and analyzed18-20 in recent work. This lipid model accounts, in a simplified manner, for the two charges and orientational freedom of typical zwitterionic headgroups such as phosphatidylcholine or phosphatidylethanolamine. We use it here to study a zwitterionic lipid-DNA complex in the presence of divalent cations. Our results demonstrate that the Poisson-Boltzmann model is consistent with the interpretation of McManus et al.:3 divalent cations render zwitterionic lipids effectively cationic but only in the presence of DNA (or similar negatively charged macroions). We also show that, in contrast to cationic lipid-DNA complex formation, the release of counterions is not the mechanism that drives the association of zwitterionic lipid-DNA complexes. Instead, the presence of the DNA enables some divalent cations to associate with the phosphate groups of the zwitterionic lipids; this is energetically more efficient than the interaction of these divalent cations with the phosphate groups of the DNA prior to complex formation. Model We consider a single (double-stranded) DNA molecule adsorbed on a planar lipid bilayer consisting of only zwitterionic lipids such as phosphatidylcholine. The DNA molecule is modeled as a rigid rod of radius R ) 1 nm and charge density σ ) -e/(1.07 nm2), corresponding to an average charge-charge separation of 0.17 nm for B-DNA (e denotes the elementary charge). As indicated in Figure 1, we place the z-axis of a Cartesian coordinate system along the membrane’s normal direction and the x-axis within the polar-apolar interface normal to the DNA. On the mean-field level all system properties are then invariant along the DNA (the y-axis). Our model of the
Mengistu et al. membrane’s headgroup region is based on simple structural assumptions for the zwitterionic lipids as discussed in previous work.17 Briefly, a zwitterionic headgroup is described by two opposite elementary charges at fixed distance l ) 0.5 nm away from each other. The negative charge, representing the phosphate group of the headgroup, is anchored at the polar-apolar interface (i.e., at position z ) 0 in Figure 1). The corresponding positive charge, modeling the choline moiety for phosphatidylcholine or the ethanolamine moiety for phosphatidylethanolamine, possesses orientational freedom: it is allowed to move on the surface of a half-sphere of radius l about the negative charge. This simple “rigid-rod” model for the headgroup leads to a constant density of states along the z-axis within 0 e z e l, where l ) 0.5 nm specifies the thickness of the headgroup region. We denote by P(z|x) the (yet undetermined) conditional probability density to find the positive charge of the headgroup within a small region between heights z and z + dz if the corresponding lipid is located at lateral position x. The conditional probability density fulfills the normalization condition
∫0l P(z|x) dz ) 1
1 l
(1)
at any lateral position x. We also introduce local concentrations of positively and negatively charged monovalent salt ions, n+ and n-, and their corresponding bulk concentration n0. The local concentration of divalent cations is m, and the corresponding bulkconcentrationm0.WiththatwecanwritethePoisson-Boltzmann free energy (expressed in units of the thermal energy kBT, where kB is the Boltzmann constant and T the absolute temperature) of the zwitterionic membrane-DNA complex as
F 1 ) kBT 8πlB
]
n0 +
[ ()
∫ dV(∇Ψ)2 + ∫ dV n+ ln
[ (
n
∫ dV n- ln n0 +-2m0
[ ( )
∫ dV m ln mm0 1 a
∫
)
n+ - n+ + n0
]
- n- + n0 + 2m0 +
]
- m + m0 +
1 da l
∫0 dz P(z|x) ln[P(z|x)] l
(2)
The first term in eq 2 is the total electrostatic energy of the system, expressed in terms of the dimensionless potential Ψ (related to the electrostatic potential Φ through Ψ ) eΦ/kBT) and the Bjerrum length lB ) 0.7 nm in water. The second, third, and fourth terms in eq 2 are ideal mixing contributions of positive monovalent salt ions (second term), negative monovalent salt ions (third term), and divalent cations (fourth term). We note that with each divalent cation two monovalent salt anions are added to the bulk solution to maintain overall electroneutrality. The integrations ∫dV run over the entire aqueous space, excluding the hydrophobic interiors of the DNA and membrane. The last term in eq 2 accounts for the orientational entropy of the zwitterionic headgroups; the integration ∫da runs over the lateral area of the lipid layer, and a ) 0.65 nm2 denotes the (fixed) cross-sectional area per lipid. The local ion concentrations, n+, n-, and m, determine the local volume charge density F ) e(n+ + 2m - n-) in the aqueous space outside the headgroup region. Inside the headgroup region, 0 e z e l, we add the additional term eP(z|x)/(la) to F, which accounts for the positively charged moieties of the zwitterionic lipid headgroups. In thermal equilibrium, F ) F(m,n+,n-,P)
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adopts its minimum for m ) m0 exp(-2Ψ), n+ ) n0 exp(-Ψ), n- ) (n0 + 2m0) exp(Ψ), and P ) exp(-Ψ)/q, where q ) (1/ l)∫0l exp(-Ψ) dz is a partition sum ensuring the normalization condition in eq 1. Inserting these distributions into the Poisson equation ∇2Ψ ) -4πlBF/e yields the following modified Poisson-Boltzmann equation
∇2Ψ e-Ψ ) n0 sinh Ψ + m0(eΨ - e-2Ψ) 8πlB 2laq
(3)
where ∇2 denotes the Laplacian and where the last term appears only within the headgroup region, 0 e z e l, but is absent outside (z > l). Solutions of eq 3 are subject to the boundary conditions
( ∂Ψ ∂n )
DNA
σ ) -4πlB , e
( ∂Ψ ∂z )
) z)0
4πlB a
(4)
In the first boundary condition (∂Ψ/∂n)DNA denotes the derivative of the dimensionless electrostatic potential in normal direction to the DNA, measured at the DNA surface. This boundary condition accounts for the negative charges on the DNA. The second boundary condition describes the negative charges of the lipid’s phosphate groups. Note that the corresponding positively charged ends of the zwitterionic headgroups are not included in the boundary condition; they enter directly into the modified Poisson-Boltzmann equation, the last term in eq 3. Also, the interiors of the two macro-ions (DNA and membrane) do not contribute to the boundary conditions; this accounts for the large ratio in dielectric constants between the aqueous phase and the hydrophobic regions of the two macro-ions. We finally note that the potential Ψ vanishes far away from any of the two macro-ions. For our discussion below we also point out that the lateral density of the phosphate groups on the surface of the DNA is smaller, almost by a factor of 2, than that of the lipid layer; i.e., |aσ/e| ≈ 0.6. Results and Discussion We have solved the modified Poisson-Boltzmann equation, eq 3, numerically for the dimensionless electrostatic potential Ψ. As specified above, in our numerical calculations, we use the cross-sectional area per lipid a ) 0.65 nm2, Bjerrum length lB ) 0.7 nm, thickness of the headgroup region l ) 0.5 nm, radius of the DNA R ) 1 nm, and surface charge density of the DNA σ ) -e/(1.07 nm2). Our results for Ψ (and thus for n+, n-, m, and P) are inserted back into eq 2, leading to a numerical expression of the free energy F. Calculations have been carried out not only for the bound state of zwitterionic lipids and DNA but also for the isolated macro-ions, where they are separated by a large distance. We denote the free energy corresponding to this system of noninteracting DNA and lipid layer by F0. The difference ∆F ) F F0 is the adsorption free energy per unit length, 1 nm, of DNA. Figure 2 shows the adsorption free energy, ∆F, as a function of the bulk concentration m0 of divalent cations. Different curves correspond to different bulk concentrations n0 of monovalent salt. The behavior of ∆F for m0 ) 0 as a function of n0 is plotted in the inset of Figure 2. Even here, in the absence of divalent ions, the adsorption free energy does generally not vanish, as has been discussed in previous work.18 Briefly, the positive sign of ∆F at low salt content indicates unfavorable interaction of the DNA with the zwitterionic lipid layer, which results from
Figure 2. Adsorption free energy, ∆F, of DNA (measured in units of kBT per unit length 1 nm of DNA) onto a planar zwitterionic lipid layer as a function of divalent cation bulk concentration m0. Different curves correspond to different bulk concentration n0 of monovalent salt as indicated. Note the logarithmic scale of the abscissa. Note also that 1 M ≡ 0.6022 nm-3. The inset shows ∆F as a function of n0 in the absence of divalent cations (m0 ) 0). Numerically calculated data points (indicated by the symbol O) are connected by straight lines.
the perturbation of the DNA’s counterion cloud by the lipid layer. However, above a sufficiently large salt content (roughly n0 ) 0.03 nm-3 in the inset of Figure 2) we find ∆F < 0, indicating favorable adsorption due to the attractive interaction between the positively charged moiety of the zwitterionic headgroups and the negative charges on the DNA. Figure 2 demonstrates that adding divalent cations generally renders the adsorption of DNA onto a zwitterionic lipid layer more favorable. The present model predicts a negative sign for ∆F upon adding a sufficient amount of divalent cations, even if initially ∆F > 0 for m0 ) 0. We proceed with several remarks: First, the decrease in ∆F can reach (and even exceed) a magnitude of 1kBT per 1 nm of DNA length. Because doublestranded DNA has a persistence length of about 50 nm, this can amount to a sufficiently large driving force to induce complex formation between zwitterionic membranes and DNA. However, we also note the result of a previous study18 about the absorption of DNA on a charged (i.e., cationic) lipid layer, where a Poisson-Boltzmann approach similar to the present one resulted in the much more favorable adsorption free energy, ∆F ≈ -7kBT per 1 nm of DNA length. Second, effective lowering of ∆F can be achieved already by small concentrations m0 of divalent cations as compared to the initial concentration n0 of monovalent salt. For example, with n0 ) 1 mM adding m0 ) 0.1 mM decreases ∆F by almost 1 kBT. Finally, for very large concentrations of divalent cations, ∆F must ultimately pass through a minimum. The existence of a minimum is a consequence of the limiting behavior ∆F(m0f∞) ) 0 (corresponding to complete screening of all electrostatic interactions). To sum up, our Poisson-Boltzmann model predicts a generic ability of divalent cations to enhance the interaction of DNA with a zwitterionic lipid membrane. In the following, we present and discuss two important structural properties pertaining to the zwitterionic lipid-DNA complex. First, divalent cations preferentially adsorb to the lipid layer in immediate vicinity to the DNA. This is illustrated in Figure 3, which shows the local concentration of the divalent cations m ) m0 exp(-2Ψ) as function of x for z ) 0. Note that in our
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Figure 3. Local concentration of divalent cations, m(x,z)0), for different corresponding bulk concentration m0 ) 100 (a), 10 (b), and 1 mM (c). In all calculations n0 ) 100 mM. Inset: Local concentration of divalent cations, m(x)0,z) (solid line) and m(x)6nm,z) (dashed line), both calculated for n0 ) m0 ) 100 mM. The dotted line, at l ) 0.5 nm, indicates the interface with the DNA. The tick on the right axis of the inset marks the local concentration of divalent cations at the surface of isolated (i.e., uncomplexed) DNA. Note the logarithmic scale on the vertical axis of the inset.
model the plane z ) 0 corresponds to the position of the phosphate groups of the zwitterionic headgroups; see Figure 1. At x ) 0, where the lipid layer is closest to the DNA, the increase in m is largest. A similar conclusionsenhancement of msfollows from the inset of Figure 3, which displays the local concentration m of the divalent cations along the z-axis for x ) 0 (at the DNA) and x ) 6 nm (far away from the DNA, or, equivalently, for a bare zwitterionic membrane). We note that at x ) 6 nm the function m(z) exhibits a local minimum at a position slightly smaller than z ) l, arising due to the influence of the positively charged headgroup moiety. This qualitative feature, a peak in the concentration of the divalent cations at the level of the phosphate groups and depletion in the outer headgroup region, agrees with observations of calcium binding to zwitterionic membranes in molecular dynamics simulations.21 Second, upon DNA adsorption the lipid headgroups reorient toward the lipid layer’s normal direction. This reorientation establishes closer spatial proximity (and thus a more favorable interaction) of the positive headgroup charge with the DNA. Similar findings have been reported for cationic lipid-DNA complexes, on the basis of molecular dynamics simulations.22 A quantitative measure of the average headgroup tilt angle θ ) arccos(〈z〉/l) is the average position
1 〈z〉 ) l
∫0 dz zP(z|x) ) l
∫0l dz z exp(-Ψ) ∫0l dz exp(-Ψ)
(5)
of the positive headgroup charge along the z-direction. For 〈z〉 ) 0 the headgroup (i.e., the vector connecting the two headgroup charges) is parallel to the lipid layer, whereas for 〈z〉 ) l it points along the normal direction (the z-direction in Figure 1). Figure 4 displays results for 〈z〉 as a function of x from our Poisson-Boltzmann calculations. The corresponding conditional probability density P(z|x) is shown in the inset of Figure 4 for two specific locations, closest to (x ) 0) and far away from (x ) 6 nm) the DNA. If, hypothetically, all electrostatic interactions are absent (Ψ ≡ 0), we find P(z|x) ≡ 1 and thus 〈z〉 ) l/2 ) 0.25 nm. Figure 4 reveals generally 〈z〉 < l/2 which is a consequence of the electrostatic attraction to the negative headgroup charges, located
Figure 4. Average position 〈z〉 of the positive headgroup charge along the x-axis for m0 ) 0 (a), m0 ) 1 mM (b), and m0 ) 100 mM (c). In all cases n0 ) 100 mM. Inset: Conditional probability density P(z|x) at positions x ) 6 nm (a) and x ) 0 (b). Solid and dotted lines correspond to m0 ) 100 mM and m0 ) 0, respectively. In all cases n0 ) 100 mM.
at z ) 0. Our model thus predicts a preferential headgroup orientation more parallel to the membrane, which is a wellestablished experimental finding for phosphatidylcholine and phosphatidylethanolamine.23,24 Addition of divalent cations to a bare (i.e., uncomplexed) zwitterionic membrane increases 〈z〉, although to a small extent only. This increase is also evident from the pair of curves marked a in the inset of Figure 4, which show P(z|x) far away from the DNA both in the absence (dotted line) and presence (solid line) of divalent cations. The presence of divalent cations leads to a small shift of the probability density P(z|x) toward larger z. The corresponding increase in 〈z〉 is a result of the somewhat enhanced concentration of divalent cations at z ) 0, thus screening (and weakening) the attraction between negative and positive headgroup charges. The increase in 〈z〉 for the bare zwitterionic membrane qualitatively agrees with atomistic simulation results.21,25 A quantitative comparison is, of course, beyond the scope of the present model, which neglects additional effects such as water release, adjustment of the cross-sectional area per lipid, and specificity26,27 upon the binding of divalent cations. The presence of the DNA has a significant influence on the headgroup orientation. In the absence of divalent cations (m0 ) 0, dotted curve in Figure 4), headgroups near the DNA tend to be more upright, i.e. pointing more toward the normal direction of the membrane. The additional presence of divalent cations generally increases 〈z〉 even further. Again, this behavior is also evident from the inset of Figure 4, where the pair of curves marked b show P(z|x) at x ) 0 (i.e., close to the DNA). Here, the proximity of the DNA leads to a pronounced increase in P(z|x)0) within the outer headgroup region (0.3 nm j z e 0.5 nm), and this even more so in the presence of divalent cations. In the vicinity of the DNA the zwitterionic headgroups preferentially point either parallel to the lipid layer or in normal direction. Intermediate tilt angles have a reduced probability. Interestingly, the magnitude of the headgroup’s reorientation upon adding divalent cations depends on the distance to the DNA along the x-axis. If m0 is small (m0 ) 1 mM, corresponding to curve b in Figure 4), the increase of 〈z〉 is larger for headgroups close to the DNA than for those far away. For large m0 (m0 ) 100 mM, corresponding to curve c in Figure 4) this trend reverses. Hence, the addition of (a not too large concentration of) divalent cations causes a reorientation of predominantly those lipid headgroups that reside close to the DNA. The large influence of the DNA on the ability of divalent cations to penetrate into the headgroup region is also evident
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Figure 5. Number M(x) of divalent cations found inside the headgroup region at position x ) 0, measured per lipid, for n0 ) 1 mM (a), n0 ) 10 mM (b), and n0 ) 100 mM (c). Curve d corresponds to x ) 6 nm (far away from the DNA) and is virtually independent of n0.
from Figure 5, which shows the number
M(x) ) a
∫0l m(x, z) dz
(6)
of divalent ions, per lipid molecule, located inside the headgroup region, 0 e z e l, close to the DNA (x ) 0, corresponding to curves a-c) and far away from it (x ) 6 nm, curve d). Note that the behavior of M(x)0) saturates with growing m0. The saturation limit M(x)0) ≈ 0.3 is directly related to the condition of charge neutrality. That is, per lipid, 0.3 divalent cations contribute 0.6 positive charges that neutralize 0.6 negative charges from the DNA (as pointed out above, the density of the DNA phosphate groups to that of the lipid phosphate groups is |aσ/e| ) 0.6). The more interesting point than the mere presence of the divalent cations in the headgroup region (at x ) 0) is their spatial distribution along the z-axis (see inset of Figure 3) and the corresponding implications for the mechanism of zwitterionic lipid-DNA complex formation. This is discussed in the following. What mechanism is responsible for the favorable adsorption of DNA onto zwitterionic lipids in the presence of divalent cations? For a similar system, the adsorption of DNA onto cationic membranes, it is the release of counterions that leads to a corresponding gain in entropy.28 Counterion release acts generally as a major driving force for complex formation between oppositely charged macro-ions in aqueous solution.29,30 However, in the present case, for the interaction of DNA with a zwitterionic lipid layer, there is no significant release of counterions. On the contrary, we predict a small number of additional ions to be immobilized upon the adsorption of the DNA onto the lipid layer. We analyze this for the representative example of m0 ) n0 ) 10 mM. In this case, an isolated DNA molecule (i.e., prior to the interaction with the zwitterionic membrane) immobilizes 0.4314 divalent cations, 0.062 monovalent cations, and -0.076 monovalent anions, all measured per phosphate group of the DNA. Hence, neutralization of the negative DNA charges is mostly due to immobilized divalent cations. Note that the negative number of excess monovalent anions indicates a depletion, below the level of uniformly distributed monovalent anions. An isolated zwitterionic membrane (again, prior to the interaction with DNA) immobilizes 0.044 divalent cations, 0.040 monovalent cations, and 0.128 monovalent anions, all measured per lipid. Hence, all ionic species are enriched close to the membrane, although only to a small extent. Upon adsorption of the DNA onto the zwitterionic membrane, the number of additionally immobilized ions, measured per phosphate group of DNA, is 0.009 for the divalent
Figure 6. Schematic illustration of the physical mechanism that underlies the adsorption of DNA onto a zwitterionic lipid layer according to our Poisson-Boltzmann model. Left: Prior to adsorption, the DNA is screened by divalent cations. Right: Upon the adsorption of DNA, some divalent cations redistribute to the phosphate groups of the lipid heads. This enables the headgroups to extend toward the DNA where the positive headgroup charges contribute to the screening of the DNA. This ion exchange mechanism dominates the adsorption energetics.
cations, -0.006 for the monovalent cations, and 0.012 for the monovalent anions. (A negative sign indicates the release of ions.) Hence, a small total number of 0.009 - 0.006 + 0.012 ) 0.015 additional ions, measured per DNA charge, are taken up by the complex upon forming a zwitterionic membrane-DNA complex. Clearly then, counterion release does not act as a driving force. The physical mechanism that leads to the favorable interaction between zwitterionic lipids and DNA is related to a redistribution of the immobilized divalent cations. Prior to the adsorption of the DNA, most immobilized divalent cations interact with the DNA, where they screen the negatively charged phosphate groups. After the adsorption of the DNA, some divalent cations redistribute away from the DNA to the phosphate groups of the zwitterionic lipids, releasing the positive headgroup charges and enabling them to interact with the DNA. This is schematically illustrated in Figure 6. Our present calculations support this mechanism as can be seen in the inset of Figure 3. Here, the local concentrations of divalent cations for the uncomplexed macro-ions, i.e., prior to the adsorption, are larger at the DNA phosphate groups (m ) 3.0/nm3, indicated by the tick on the right axis of the inset in Figure 3) than at the lipid phosphate groups (m ) 1.2/nm3; see the broken line at z ) 0 in the inset in Figure 3). After the adsorption, m is larger at the lipid phosphate groups (m ) 4.4/nm3; see the solid line at z ) 0 in the inset in Figure 3) than at the DNA phosphate groups that are close to the membrane (m ) 1.3/nm3; see the solid line at z ) 0.5 nm in the inset in Figure 3). What drives the re-distribution of the divalent cations to the phosphate groups of the lipids? The reason is, within our Poisson-Boltzmann formalism, the smaller density of phosphate groups on the DNA as compared to the lipid membrane; recall |aσ/e| ) 0.6. Divalent counterions interact less efficiently with surfaces of lower charge density. To qualitatiVely illustrate this, consider two isolated, planar surfaces with surface charge densities σ1 ) σ and σ2 ) 2σ, representing the charged phosphate groups of the DNA and the lipid layer, respectively. The first surface initially interacts with a divalent salt solution, implying a free energy (per unit area) ˆfi1 ) ˆf0/2. Within the linearized Debye-Hu¨ckel approximation the function ˆf0 can be specified; it is ˆf0 ) 2πlBlD(σ/e)2, where
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lD is the Debye screening length. The second (twice as highly charged) surface initially interacts with a monovalent salt solution of the same bulk concentration; the corresponding free energy is ˆf2i ) 4fˆ0. Now, we exchange the salt solutions; i.e., the first surface is in contact with the monovalent salt solution, implying a free energy ˆf1f ) ˆf0, and the second surface interacts with the divalent salt solution; the corresponding free energy is ˆf2f ) 2fˆ0. The change in free energy upon exchanging the salt solutions,
3 ∆fˆ ) ˆf1f + ˆf2f - ˆf1i - ˆf2i ) - ˆf0 2
(7)
is negative. Hence, divalent ions interact more effectively with surfaces of higher charge density. This statement, which is also valid for nonlinear Poisson-Boltzmann theory, is relevant for our present interpretation (as illustrated in Figure 6) if we identify the monovalent salt solution with the positively charged groups of the zwitterionic headgroups. Adsorption of the DNA to the membrane allows the divalent cations and the positive headgroup charges to exchange the surface they neutralize. This exchange constitutes the mechanism by which divalent cations enhance the interaction between DNA and zwitterionic membranes. Conclusions The ability of divalent metal cations to enhance the interaction between DNA and zwitterionic membranes has potential relevance for the design of nonviral, lipid-based gene delivery vectors.31 Also from a physical point of view, the adsorption of DNA onto a zwitterionic lipid layer is interesting because it is not obvious whether purely electrostatic considerations alone suffice to elucidate the role of the divalent cations and whether the release of counterions contributes to the mechanism of complex formation. The present work addresses these questions using a (modified) Poisson-Boltzmann approach that ignores correlations, solvent structure, and all specific interactions and makes simplifying assumptions about the structure of the two macro-ions. Specifically, DNA is modeled as a uniformly charged straight cylinder, and the headgroups of the zwitterionic lipids consist of two opposite point charges that are separated by a fixed distance and possess orientational freedom. Our model predicts an increase in the magnitude of the DNA’s adsorption free energy ∆F onto a zwitterionic membrane due to the presence of divalent cations. For zwitterionic membranes the adsorption free energy, |∆F| j 1kBT per 1 nm of DNA length, is generally about an order of magnitude smaller as compared to membranes that also contain cationic lipids. Importantly, our model predicts ∆F to adopt a negative sign (i.e., favorable adsorption) for sufficiently large concentrations of divalent cations. (For example, in the presence of a 10 mM monovalent salt solution we predict a concentration of divalent cations of roughly 0.1 mM to render ∆F negative.) The negative sign of ∆F is a necessary condition for the experimentally observed ability of zwitterionic membranes to form condensed complexes with DNA. Our model also suggests a physical mechanism for the decrease in ∆F; this mechanism is electrostatic in nature and is based on an exchange mechanism between divalent cations and
the positive charges of the zwitterionic headgroups. That is, upon adsorption of the DNA divalent cations migrate from the phosphate groups of the DNA to the phosphate groups of the zwitterionic lipids where their electrostatic screening is more efficient. Concomitantly, lipid headgroups reorient toward the DNA where their positive moieties contribute to the screening of the DNA charges. We finally note that possible extensions of the present work include accounting, first, for DNA-induced changes of the lateral cross-sectional area a per lipid and, second, for DNA-DNA interactions as they emerge for nonvanishingly small concentrations of DNA on the lipid layer. This would allow one to study the influence of the DNA (or, more generally, of electrostatically adsorbed macro-ions) on a lipid monolayer’s phase state and its lateral pressure.9,11 Acknowledgment. This work was supported by NSF through Grant DMR-0605883. References and Notes (1) Karmali, P. P.; Chaudhuri, A. Med. Res. ReV. 2007, 27, 696–722. (2) Dass, C. R. J. Pharm. Pharmacol. 2002, 54, 593–601. (3) McManus, J. J.; Radler, J. O.; Dawson, K. A. J. Phys. Chem. B 2003, 107, 9869–9875. (4) Pisani, M.; Bruni, P.; Caracciolo, G.; Caminiti, R.; Francescangeli, O. J. Phys. Chem. B 2006, 110, 13203–13211. (5) Uhrikova, D.; Lengyel, A.; Hanulova, M.; Funari, S. S.; Balgavy, P. Eur. Biophys. J. 2007, 36, 363–375. (6) McManus, J. J.; Radler, J. O.; Dawson, K. A. J. Am. Chem. Soc. 2004, 126, 15966–15967. (7) Francescangeli, O.; Stanic, V.; Gobbi, L.; Bruni, P.; Iacussi, M.; Tosi, G.; Bernstorff, S. Phys. ReV. E 2003, 67, 011904. (8) Francescangeli, O.; Pisani, M.; Stanic, V.; Bruni, P.; Weiss, T. M. Europhys. Lett. 2004, 67, 669–675. (9) Gromelski, S.; Brezesinski, G. Phys. Chem. Chem. Phys. 2004, 6, 5551–5556. (10) McLoughlin, D.; Dias, R.; Lindman, B.; Cardenas, M.; Nylander, T.; Dawson, K.; Miguel, M.; Langevin, D. Langmuir 2005, 21, 1900–1907. (11) Gromelski, S.; Brezesinski, G. Langmuir 2006, 22, 6293–6301. (12) Tresset, G.; Cheong, W. C. D.; Lam, Y. M. J. Phys. Chem. B 2007, 111, 14233–14238. (13) McManus, J. J.; Radler, J. O.; Dawson, K. A. Langmuir 2003, 19, 9630–9637. (14) Grosberg, A. Y.; Nguyen, T. T.; Shklovskii, B. I. ReV. Mod. Phys. 2002, 74, 329–345. (15) Guldbrand, L.; Jonsson, B.; Wennerstrom, H.; Linse, P. J. Chem. Phys. 1984, 80, 2221–2228. (16) Kjellander, R. Ber. Bunsen-Ges. 1996, 100, 894–904. (17) Mbamala, E. C.; Fahr, A.; May, S. Langmuir 2006, 22, 5129–5136. (18) Haugen, A.; May, S. J. Chem. Phys. 2007, 127, 215104. (19) Mengistu, D. H.; May, S. Eur. Phys. J. E 2008, 26, 251–260. (20) Mengistu, D. H.; May, S. J. Chem. Phys. 2008, 129, 121105. (21) Vernier, P. T.; Ziegler, M. J.; Dimova, R. Langmuir 2009, 25, 1020– 1027. (22) Bandyopadhyay, S.; Tarek, M.; Klein, M. L. J. Phys. Chem. B 1999, 103, 10075–10080. (23) Seelig, J.; Gally, H. U. Biochemistry 1976, 15, 5199–5204. (24) Seelig, J. Biochim. Biophys. Acta 1977, 467, 109–119. (25) Bockmann, R. A.; Grubmuller, H. Angew. Chem., Int. Ed. 2004, 43, 1021–1024. (26) McLaughlin, A.; Grathwohl, C.; McLaughlin, S. Biochim. Biophys. Acta 1978, 513, 338–357. (27) McLaughlin, S.; Mulrine, N.; Gresalfi, T.; Vaio, G.; McLaughlin, A. J. Gen. Physiol. 1981, 77, 445–473. (28) Wagner, K.; Harries, D.; May, S.; Kahl, V.; Radler, J. O.; BenShaul, A. Langmuir 2000, 16, 303–306. (29) Meier-Koll, A. A.; Fleck, C. C.; von Grunberg, H. H. J. Phys.: Condens. Matter 2004, 16, 6041–6052. (30) Safran, S. A. Europhys. Lett. 2005, 69, 826–831. (31) Mozafari, M. R.; Omri, A. J. Pharm. Sci. 2007, 96, 1955–1966.
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