This journal is c the Owner Societies 2012 Phys. Chem. Chem. Phys., 2012, 14, 10613–10621 10613

Cite this: Phys. Chem. Chem. Phys., 2012, 14, 10613–10621

Modeling the influence of adsorbed DNA on the lateral pressure and tilt

transition of a zwitterionic lipid monolayer

Klemen Bohinc,aGerald Brezesinski

band Sylvio May*

c

Received 23rd March 2012, Accepted 30th May 2012

DOI: 10.1039/c2cp40923b

Certain lipid monolayers at the air–water interface undergo a second-order transition from a

tilted to an untilted liquid-crystalline state of their lipid hydrocarbon chains at sufficiently large

lateral pressure. Recent experimental observations demonstrate that in the presence of divalent

cations DNA adsorbs onto a zwitterionic lipid monolayer and decreases the tilt transition

pressure. Lowering of the tilt transition pressure indicates that the DNA condenses the lipid

monolayer laterally. To rationalize this finding we analyze a theoretical model that combines a

phenomenological Landau approach with an extension of the Poisson–Boltzmann model to

zwitterionic lipids. Based on numerical calculations of the mean-field electrostatic free energy of a

zwitterionic lipid monolayer–DNA complex in the presence of divalent cations, we analyze the

thermodynamic equilibrium of DNA adsorption. We find that adsorbed DNA induces a 10%

reduction of the electrostatic contribution to the lateral pressure exerted by the monolayer. This

result implies a small but notable decrease in the tilt transition pressure. Additional mechanisms due

to ion–ion correlations and headgroup reorientations are likely to further enhance this decrease.

1 Introduction

Non-viral gene transfer typically1 employs the complexation

of DNA using a cationic agent. For cationic lipids, the

resulting condensates are referred to as lipoplexes.2 The physical

principle underlying lipoplex formation is evident:3 negatively

charged DNA molecules interact electrostatically with cationic

lipids. The resulting compact structures can be taken up by

living cells where they eventually disassemble, thus allowing the

DNA to enter the nucleus.4 The two major drawbacks of using

lipoplexes are low transfection efficiency and cytotoxicity of the

cationic lipids.5 In contrast to cationic lipids, zwitterionic lipids

are biocompatible and thus non-toxic. Hence, there is a

substantial incentive to replace cationic by zwitterionic lipids.

However, the lack of strong electrostatic attraction renders the

initiation and control of complex formation between DNA

and zwitterionic lipids challenging. A method to initiate

formation of zwitterionic lipid–DNA complexes is to add

divalent cations6–10 such as Ca2+or Mg2+. Although with

low efficiency, these complexes are viable nonviral gene delivery

vectors.11,12 To control and improve their stability, it is

beneficial to better understand divalent cation-mediated binding

of DNA and its influence on the structure and energy of

zwitterionic lipid layers.

Langmuir monolayers at the air–water interface are a

suitable system for studying the binding between DNA and

zwitterionic lipids.13 To motivate the present theoretical study,

we briefly discuss a recent experiment by Gromelski and

Brezesinski14,15 in which the influence of DNA and added

divalent cations (5 mM Ca2+or Mg2+) on the pressure–area

isotherm of a monolayer consisting of the zwitterionic lipid

dimyristoylphosphatidylethanolamine (DMPE) was measured.

To this end, Gromelski and Brezesinski have employed Infrared

Reflection Absorption Spectroscopy (IRRAS), X-ray Reflec-

tivity (XR), Grazing Incidence X-ray Diffraction (GIXD), and

Brewster AngleMicroscopy (BAM). Generally, upon increasing

the lateral pressure a monolayer passes through a number of

phases:16 the gas phase, the liquid-expanded phase, and various

liquid-condensed phases, which exhibit in-plane structures

known from smectic liquid crystals.17 In these crystallo-

graphically ordered phases, the hydrocarbon chains are fully

stretched (i.e., residing in their all-trans conformation) and

tightly packed. Depending on the phase type and the molecular

chemical structure, the cross-sectional chain areas are between

0.185 and 0.205 nm2. The mismatch between the in-plane area

requirement of the hydrophilic headgroups and the hydro-

phobic chains leads to a non-vanishing tilt angle of the chains

with respect to the monolayer normal direction. Further

compression of the film has an influence on the head-

group area by changing the headgroup orientation and/or

hydration. Therefore, the tilt of the hydrocarbon chains can

a Faculty of Health Sciences, University of Ljubljana,SI-1000 Ljubljana, Slovenia. E-mail: klemen.bohinc@fe.uni-lj.si

bMax Planck Institute of Colloids and Surfaces, Science ParkPotsdam-Golm, Am Muehlenberg 1, 14476 Potsdam, Germany.E-mail: gerald.brezesinski@mpikg.mpi.de

cDepartment of Physics, North Dakota State University, Fargo,ND 58108-6050, USA. E-mail: sylvio.may@ndsu.edu

PCCP Dynamic Article Links

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10614 Phys. Chem. Chem. Phys., 2012, 14, 10613–10621 This journal is c the Owner Societies 2012

be diminished by compression. In certain cases, compression

can even lead to the non-tilted state. This transition is signified

in the isotherms by a kink, indicating a second-order transition

in contrast to the first-order transition from the disordered

fluid state to the ordered condensed state, which is characterized

by a plateau region in the isotherms. The transition from the

tilted to the untilted phase and the corresponding tilt angle y as

a function of the applied lateral pressure P are schematically

illustrated in Fig. 1.

In their above-mentioned experimental work,14,15 Gromelski

and Brezesinski have studied the transition from the tilted to the

untilted phase. Specifically, in the absence of both DNA and

divalent cations the second-order phase transition was observed

to occur at a lateral pressure of about 32 mN m�1. Upon

addition of both DNA and divalent cations, the transition from

the tilted to the untilted ordered phase was shifted to a smaller

lateral pressure of about 28 mN m�1. We note that neither

DNA nor divalent cations alone caused a significant shift in

the transition pressure. Moreover, the shift in the transition

pressure was very similar irrespective of whether Ca2+ or

Mg2+was used. These experimental observations indicate,

first, that divalent cations and DNA cooperatively lower the

lateral pressure (at fixed cross-sectional area of the lipids) and

thus tend to cause a condensation of the monolayer and,

second, that this is a non-specific effect.

Generally, the interaction of ions with solute molecules –

including lipid layers – exhibits specificity and depends in a

characteristic manner, known as the Hofmeister series, on the

chemical identity of the ions. For example, Aroti et al.18,19

investigated the influence of different anions on a monolayer

composed of the zwitterionic lipid dipalmitoylphosphatidyl-

choline (DPPC) and found that moderate concentrations of

chaotropic anions, such as I�, do not significantly change the

conformation and packing properties of the hydrocarbon

chains. The lattice parameters remain essentially unaffected,

even at quite high electrolyte concentrations. However, anions

partition into or bind to the disordered liquid-expanded phase,

thus providing a stabilization of that phase, but they do not

penetrate into or bind to the domains of the liquid-condensed

phase. We note that the mechanisms for the stabilization of the

disordered phase may be related to the higher degree of water

ordering in the vicinity of the anions when residing in the bulk

aqueous phase as compared to the monolayer-associated state.

This suggests the stabilization of the disordered phase to be

driven by an increase in entropy.18

From a basic physical viewpoint it is not obvious that

adsorbed DNA is able to condense a zwitterionic lipid mono-

layer. In fact, the adsorbed DNA array in itself is expected to

increase the lateral pressure, unless the presence of the divalent

cations induces attractive interactions between the adsorbed

DNA strands via ion–ion correlations.20 Another potential

mechanism leading to the condensation of the monolayer

would be the formation of complexes between a divalent

cation and the phosphate groups of two neighboring lipids.6

The role of the DNA would then be to facilitate penetration of

divalent cations into the lipid layer. We note that both

mechanisms depend on the presence of ion–ion correlations.

In mean-field theory such correlations are absent so that, at

least on this level, one would expect the condensing effect of

the DNA to be absent. As we will demonstrate in the present

work, even on the mean-field level (i.e., in the absence of

ion–ion correlations) the divalent cation-mediated adsorption

of DNA onto a zwitterionic lipid layer leads to a condensing

effect.

In the present theoretical work we study the influence of

adsorbed DNA on a zwitterionic lipid monolayer at the

air–water interface. Specifically, we model the transition from

the tilted to the untilted ordered phase of the lipid’s hydro-

carbon chains upon increasing the applied lateral pressure.

Our objective is to rationalize the experimental finding that

adsorbed DNA shifts the tilt transition to smaller lateral

pressure. The mechanism we suggest is based purely on

electrostatics and involves a decrease in the electrostatic con-

tribution to the lateral pressure of the monolayer upon DNA

adsorption. That is, divalent cations, which mediate the

adsorption of DNA, enter into the headgroup region of the

monolayer where they interact with the phosphate groups of

the lipids, thus decreasing the electrostatic stress. Our theore-

tical model pursues a minimalistic approach that aims at

retaining only the most relevant structural features and energy

contributions. While such an approach benefits from exposing

the underlying physics, we note that it is often not obvious

what the system’s relevant interactions are. Hence, phenomeno-

logical modeling must be guided by experimental findings or

by results from molecular-level simulation methods such as

all-atom molecular dynamics. Here, we employ a mean-field

description of the system, thereby combining a Landau approach

with a Poisson–Boltzmann model for the electrostatic

interactions. The Poisson–Boltzmann approach is modified

according to a previously introduced model to account for the

presence of zwitterionic lipid headgroups.21 We calculate

the electrostatic free energy of a DNA–monolayer complex

(in the presence of 5 mM divalent cations) and use it to

compute the thermodynamic adsorption equilibrium of the

DNA. Multiple calculations for different lateral areas of the

monolayer yield the electrostatic contribution to the lateral

pressure. We demonstrate that our mean-field model predicts a

10% decrease in the lateral pressure upon adsorption of the

DNA. Hence, even on the mean-field level and without

invoking any further DNA-induced structural changes in the

monolayer, DNA induces a small but notable condensing

effect – and thus a reduction in the tilt transition pressure.

Fig. 1 Schematic illustration of a lipid monolayer residing in the

tilted (a) and untilted (b) ordered phase. The tilt angle y reflects the

in-plane area a per lipid, which is controlled by the applied lateral

pressure P. As illustrated in diagram c, a second-order phase transi-

tion occurs for increasingP from a tilted to an untilted ordered phase.

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2 Electrostatic free energy

We consider a planar monolayer consisting of zwitterionic

lipids at the air–water interface. The average cross-sectional

area a per lipid is controlled by the applied lateral pressure P.

The aqueous solution contains divalent cations of bulk

concentration m0 as well as monovalent anions of bulk

concentration 2m0. All ions are assumed to be point-like.

Adsorption of DNA onto a zwitterionic monolayer is facili-

tated by the presence of divalent cations. Here, we model the

electrostatic contribution to the adsorption of (double

stranded) DNAmolecules onto a zwitterionic lipid monolayer.

DNA has a persistence length22 of about x = 50 nm. Hence, it

will exhibit thermally excited bending deformations on length

scales larger than x. The spatial extension of lipid molecules is

much smaller than x. On these length scales, DNA remains a

stiff molecule. It is thus appropriate to represent DNA as an

array of parallel straight rods, each of radius R = 1 nm with

uniform negative surface charge density s = �0.92 e nm�2

where e denotes the elementary charge. Recall that DNA

carries two negative charges per base pair; in B-DNA base

pairs are separated by a distance of 0.34 nm. Hence the

average charge-to-charge separation along the long axis of

the molecule is (0.34/2) nm = 0.17 nm. Indeed, our choices of

R and s ensure the average charge-to-charge separation along

the DNA to be 0.17 nm. The rods are sufficiently long so that

end effects can be neglected. The yet unknown center-to-center

distance d of neighboring DNA strands will be determined

below from the analysis of a thermodynamic adsorption model.

We locate a Cartesian coordinate system with the z-axis normal

to the monolayer (originating at the monolayer’s polar/apolar

interface and pointing to the aqueous phase) and the x-axis

within the polar/apolar interfacial plane normal to the DNA’s

long axis. On the mean-field level all system properties are

invariant along the y-axis; i.e., parallel to the DNA’s long axis.

A cross-section of the DNA–monolayer complex (at fixed y),

including a unit cell (the light-shaded region), is shown in Fig. 2.

We model the zwitterionic lipid headgroup as suggested and

analyzed21 previously as two opposite elementary charges,

both residing at a given lipid’s lateral position x. The negative

headgroup charge is firmly attached to the polar–apolar inter-

face z = 0, and the positive charge resides at variable distance

z within the headgroup region 0 r z r l where l = 0.5 nm

denotes the thickness of the headgroup region. This model

is motivated by the generic structure of zwitterionic phospho-

lipids: the negatively charged phosphate group is linked (via a

backbone) to the hydrocarbon tails, whereas the positive charge

(representing the ethanolamine moiety for phosphatidylathanol-

amine, the choline group for phosphatidylcholine, etc.) has con-

formational flexibility that allows it to reside at different positions

z within the headgroup region. In the following it is convenient to

introduce the conditional probability density P(z|x) to find the

positive headgroup charge at position z, given the corresponding

lipid (i.e., the negative charge of the headgroup) is located at

position x. This probability density is normalized according toR ‘0 PðzjxÞdz ¼ ‘, for any fixed position x.

The charges of the DNA and the lipid monolayer perturb

the local concentrations of the mobile salt ions (i.e., the

monovalent anions and the divalent cations). We denote the

local concentrations of monovalent anions and divalent

cations by n� = n�(x,z), and m = m(x,z), respectively. Note

that both functions are only defined within the aqueous

solution, z Z 0, excluding the DNA cylinders. The corre-

sponding volume charge density r(x,z) at a given point (x,z)

can be expressed as

re¼ 2m� n� þ 1

a‘PðzjxÞ; 0o z � ‘2m� n�; ‘o zo1:

�ð1Þ

The additional contribution eP(z|x)/(al) within the head-

group region 0 r z r l accounts for the positive head-

group charges of the zwitterionic lipids. Starting point of

our model is the following mean-field electrostatic free

energy per unit cell (expressed in units of the thermal energy

kBT, where kB is Boltzmann’s constant and T the absolute

temperature)

Fel

kBT¼ 1

8p‘B

ZV

dvðrCÞ2

þZV

dv n� lnn�2m0

� �� n� þ 2m0

� �

þZV

dv m lnm

m0

� ��mþm0

� �

þ 1

a

ZA

da1

‘

Z‘

0

dzPðzjxÞ lnPðzjxÞ:

ð2Þ

Each integral in this expression describes one free energy

contribution. The first integral accounts for the energy stored

Fig. 2 Schematic illustration of a zwitterionic lipid monolayer–DNA

complex. The DNA molecules (shaded circles) are modeled as long

cylinders of radius R and uniform surface charge density s. The DNA

cylinders contact the headgroup region, 0 r z r l, of the lipid

monolayer. 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. The

aqueous solution contains monovalent anions and divalent cations.

A unit cell, within which we solve the modified Poisson–Boltzmann

equation, is indicated (light-shaded region). The unit cell contains one

single DNA cylinder and extends from x=�d/2 to x= d/2 and z=0

to z-N. The lateral size of the unit cell, d, equals the DNA-to-DNA

distance. Also indicated are the width of the headgroup region l, the

average cross-sectional area per lipid a, and a non-vanishing tilt angle

of the lipid tails.

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in the electric field. Here rC is the gradient of the dimension-

less electrostatic potential C = C(x,z), which is related to the

electrostatic potential F through C = eF/kBT. Furthermore,

lB = 0.7 nm denotes the Bjerrum length in water. This length

is defined such that at a separation lB two elementary charges

experience an interaction energy equal to the thermal energy

kBT. The volume integrationRVdv runs over the aqueous

region (thus excluding the DNA) of the unit cell. The second

and third integrals in eqn (2) are ideal mixing free energies of

the monovalent anions and the divalent cations, respec-

tively. Finally, the fourth integral in eqn (2) accounts for

the orientational entropy of the zwitterionic headgroups.

Here, the integrationRAda runs over the polar–apolar inter-

face (i.e., the surface z � 0) of the unit cell.

Thermal equilibrium of the system is determined by the

minimum of the electrostatic free energy Fel = Fel(n�, m, P).

Minimization results in the Boltzmann distributions m =

m0 exp(�2C), n� = 2m0 exp(C) and P = exp(�C)/q, with

the x-dependent partition sum q ¼ ð1=‘ÞR ‘0 expð �CÞdz.

Inserting the Boltzmann distributions into eqn (1) and using

Poisson’s equationr2C=�4plBr/e gives rise to the modified

Poisson–Boltzmann equation

r2C8p‘B

¼ m0ðeC � e�2CÞ �e�C2‘aq; 0o z � ‘0; ‘o zo1;

�ð3Þ

where r2 denotes the Laplacian. The term �e�C/(2laq) in

eqn (3) is present only within the headgroup region, where it

accounts for the positive headgroup charges. Eqn (3) is a

partial, non-linear, and non-local differential equation for the

potential C = C(x,z) that must be solved within the unit cell.

Symmetry implies the boundary conditions (qC/qx)x=�d/2 =(qC/qx)x=d/2 = 0. Moreover, in the bulk the electric field

vanishes, implying (qC/qz)z-N = 0. Finally, the negative

charges of the phosphate groups fixed at the DNA and

the lipid monolayer entail the two remaining boundary

conditions

@C@n

� �DNA

¼ �4p‘Bse;

@C@z

� �z¼0¼ 4p‘B

a; ð4Þ

where (q/qn)DNA denotes the derivative in the normal direction

of the DNA cylinder (pointing to the aqueous region). We

assume that the hydrophobic interiors of both the DNA

cylinder and the lipid monolayer’s hydrocarbon regions

have a sufficiently low dielectric constant so that the electric

field in these regions is negligibly small. It should also be

mentioned that electroneutrality of the system is ensured, even

without explicitely adding counterions of the DNA to our

system. This is due to the presence of the aqueous bulk

reservoir with concentrations m0 and 2m0 of divalent cations

and monovalent anions, respectively. Mobile ions will be

recruited from the bulk so as to neutralize the charge of

the DNA.

We have numerically solved eqn (3) by transforming the

non-linear differential equation into a sequence of linearized

differential equations that were solved using a Newton–

Raphson iteration scheme.23 Once the potential C(x,z) is

known, the electrostatic free energy of the unit cell Fel according

to eqn (2) can be calculated. To this end, it is convenient to

insert the equilibrium distributions for n�, m, and P back into

eqn (2). This leads to the expression

Fel

kBT¼ s

2e

ZDNA

daC� 1

2a

ZA

daC

þm0

ZV

dvfCðeC � e�2CÞ � ð2eC þ e�2CÞ þ 3g

� 1

2al

ZA

da

Z‘

0

dzCe�C

q� 1

a

ZA

da ln q; ð5Þ

where the surface integrationRDNAda runs over the surface of

the DNA cylinder. Using eqn (5) is most convenient in a

numerical computation because only the potential C(x,z) but

not any (numerically less accurate) derivatives of the potential

need to be known.

The tilt transition of a zwitterionic lipid monolayer depends

on the cross-sectional area a per lipid (which is controlled by

the lateral pressure). Adsorbed DNA influences the tilt transi-

tion and thus makes it dependent on the DNA-to-DNA

distance d (which below we determine from a chemical

adsorption equilibrium). Hence, we need to compute the

electrostatic free energy Fel = Fel(d,a) per unit cell as a

function of both d and a. Numerical results are shown in

Fig. 3 for a unit cell of unit width L = 1 nm along the y-axis.

(Throughout this work we shall use the symbol L for a unit

length of 1 nm. Recall that the y-direction points along the

principal axis of the DNA. Hence, the total electrostatic free

energy for an extended monolayer is equal to Fel times the

total length of the adsorbed DNA, measured in nm.) All

results in Fig. 3 are computed for a divalent ion bulk concen-

tration of m0 = 0.003 nm�3 (5 mM). Each curve in Fig. 3

Fig. 3 The electrostatic free energy per unit cell (of unit width L = 1

nm along the y-axis) Fel = Fel(d,a) as a function of the DNA-to-DNA

distance d for a=0.35 nm2, a=0.40 nm2, a=0.45 nm2, a=0.50 nm2,

a = 0.55 nm2, a = 0.60 nm2, a= 0.65 nm2, a= 0.70 nm2 (curves from

top to bottom). The remaining parameters are m0 = 0.003 nm�3

(5 mM), lB = 0.7 nm, l= 0.5 nm, R= 1 nm, s= �0.92 e nm�2. Theinset re-displays data of the main figure together with the optimal

DNA-to-DNA distance d(a), indicated by the symbol 1 for each cross-

sectional area a. The relation d(a) follows from the analysis of the

DNA adsorption equilibrium.

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refers to Fel as a function of d, different curves correspond to

different a. Let us briefly analyze our findings: first, larger

cross-sectional area a at constant DNA-to-DNA distance d

(i.e., constant size of the unit cell) leads to smaller electrostatic

free energy because fewer headgroup charges – both positive

and negative – remain in the unit cell. In addition, the area

density of zwitterionic headgroups in the unit cell is reduced,

implying weaker electrostatic repulsion between the head-

group dipoles. Second, in the large d regime the electrostatic

free energy increases linearly for increasing d at constant a.

Since for large d the DNA cylinders do no longer repel each

other electrostatically, increasing d merely adds more unper-

turbed zwitterionic lipids (all with the same cross-sectional

area a) to the monolayer. Hence, the constant slopes of the

curves in Fig. 3 at large d yield the corresponding area density

of the electrostatic free energy (in kBT nm�2) for an unper-

turbed zwitterionic monolayer, and the intercept at d = 0 is

the excess energy due to the presence of the adsorbed DNA

(per DNA unit length L = 1 nm). Third, for small d the

electrostatic free energy passes through a minimum as a

function of d for fixed a. The increase in Fel at decreasing d

is a consequence of the electrostatic repulsion between neigh-

boring DNA cylinders for small d.

3 DNA adsorption equilibrium

We use the results for Fel(d,a) in Fig. 3 to analyze the

thermodynamic equilibrium of DNA adsorption onto the

zwitterionic monolayer. To this end, we consider a lipid

monolayer (at the air–water interface) that occupies a fixed

total lateral area A = aN where a is the cross-sectional area

per lipid and N the total (and fixed) number of lipids; see

the schematic illustration in Fig. 4. We also consider the

presence of DNA, either adsorbed onto the monolayer or free

in solution. We express the total amount of DNA as M

cylindrical segments, each of unit length L = 1 nm. If Mm

of the cylindrical DNA segments are adsorbed onto the

zwitterionic lipid monolayer (mediated by the presence of

divalent cations) and Mf = M � Mm cylindrical segments

remain free in solution, then we can express the total electro-

static free energy of the system as

F totel = (M � Mm)m + MmFel(d,a), (6)

where m is the chemical potential of the DNA (per unit length L)

and Fel(d,a) is the electrostatic free energy per unit cell as specified

in eqn (5). Because area conservation implies Na = MmdL, the

DNA-to-DNA distance d = Na/(MmL) depends on Mm.

In thermal equilibrium the amount Mm of adsorbed DNA

follows from the condition in dF totel /dMm = 0. This gives rise

to the equilibrium condition

Felðd; aÞ � d@Felðd; aÞ

@d

� �a

¼ m: ð7Þ

Solving eqn (7) yields d for given m and Fel(d,a) where a is fixed.

However, to numerically compute d we first need to determine

the chemical potential m of the DNA. When a single DNA

cylinder is immersed in an aqueous solution (with bulk con-

centrations m0 of divalent cations and 2m0 of monovalent

anions), the electrostatic free energy can be computed within

the standard Poisson–Boltzmann formalism. This, in fact, is

analogous to the formalism leading to eqn (5), yet in the

absence of the zwitterionic monolayer and in the limit d-N.

The Poisson–Boltzmann equation r2C = 8plBm0(eC � e�2C)

can most conveniently be solved in cylindrical coordinates,

and the chemical potential is given by

mkBT

¼ s2e

ZDNA

daC

þm0

ZV

dvfCðeC � e�2CÞ � ð2eC þ e�2CÞ þ 3g; ð8Þ

to be calculated for a cylindrical segment of unit lengthL=1 nm.

For m0 = 0.003 nm�3 (corresponding to 5 mM) we obtain

m = 13.8 kBT per unit length L. Using this value allows us to

numerically solve eqn (7). The resulting equilibrium distance d

is marked by the open circles in the inset of Fig. 3. Our model

generally predicts a DNA-to-DNA distance in the range of

4 to 6 nm for 0.35 r a/nm2 r 0.7. The smaller values of d for

smaller cross-sectional area a per lipid result from the stronger

adsorption free energy. Distances of order d E 4 nm are

typically observed in experiment.15

We remark that our thermodynamic analysis in eqn (6)

neglects all contributions arising from the entropy of the DNA

on the monolayer or in solution. This is justified because

double stranded DNA has a large persistence length of x E50 nm. Adsorption of DNA segments of length x (or longer)

involves a correspondingly large adsorption free energy com-

pared to which entropic contributions become small. More

specifically, if we account for the translational entropy based

on a simple lattice model, we would find the additional

contribution (L/x) ln[vb (d � 2R)/(2R)] on the right-hand

side of eqn (7). Here vb is the volume fraction of the DNA

Fig. 4 Schematic illustration of DNA adsorption onto a zwitterionic

monolayer. The total amount of DNA the system contains is expressed

as M cylindrical segments, each of unit length L = 1 nm. Of these

M cylindrical segments, Mm are adsorbed onto the monolayer, and

M �Mm remain free in the bulk of the aqueous phase. The center-to-

center distance of the adsorbed DNA is d. The monolayer consists of

N lipids, each one with a cross-sectional area a. The total lateral area

occupied by the lipids is A = Na. Note that divalent cations and

monovalent anions are present in the aqueous solution, but they are

not shown. Also, the illustration indicates a non-vanishing tilt of the

lipids but, depending on a, the tilt may also vanish.

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in the bulk solution. For example d = 4 nm and vb = 10�10

yield (L/x) ln[vb (d � 2R)/(2R)] E �0.4 which is a small

correction to the chemical potential m and leaves the results

reported below virtually unaffected.

Using the result of the equilibrium DNA-to-DNA distance

d = d(a) as a function of the cross-sectional area per lipid a in

the inset of Fig. 3, we can plot the electrostatic free energy

per lipid

felðd; aÞ ¼ ½Felðd; aÞ � m�Mm

N¼ a

L

@Fel

@d

� �a

; ð9Þ

where the last equality follows from eqn (7). Recall that

N/Mm = Ld/a is the number of lipids per unit cell. The result

fel = fel[d(a),a] is shown in Fig. 5 (lower curve).

It is useful to compare fel = fel[d(a),a] in the presence of

DNA with the corresponding result in the absence of DNA.

We denote the latter electrostatic free energy per lipid as

f (0)el = f (0)el (a). Upon removal of the DNA array (while

maintaining the bulk concentration of divalent cations m0 at

5 mM), the Poisson–Boltzmann equation (see eqn (3)) is not

affected (but, due to the translational symmetry along the

x-axis, turns into a one-dimensional equation for the potential

C=C(z). Numerical solutions subject toC0(z= 0) = 4plB/aand C(z - N) = 0 can be used to compute the electrostatic

free energy per lipid f (0)el (a) = aFel/(Ld) where Fel corresponds

to eqn (5) with the first term being absent. The result for

f (0)el = f (0)el (a) is displayed in Fig. 5 (upper solid line). Clearly,

the electrostatic free energy per lipid is significantly lower in

the presence of DNA. In fact, the difference between f (0)el and

fel yields the average gain in electrostatic free energy per lipid

upon adsorption of the DNA onto the zwitterionic monolayer.

This free energy gain per lipid is approximatively equal to

0.1kBT, being somewhat larger for smaller a.

To get yet another perspective we consider again the

electrostatic free energy per lipid in the absence of DNA,

and this time also in the absence of divalent cations. We shall

refer to that electrostatic free energy per lipid by f (1)el = f (1)el (a).

Because the zwitterionic monolayer does not possess any

excess charge, there are no mobile ions (neither anions nor

cations) present in the system. Interestingly, the system is

still equivalent to the standard Poisson–Boltzmann model

of a charged surface in the presence of only counterions:

the charged surface corresponds to the phosphate groups of

the zwitterionic lipids (which are located at fixed position

z = 0), and the role of the counterions is played by the

positive headgroup charges (which are mobile within the

headgroup region). Indeed, for m0 = 0 the Poisson–Boltzmann

equation adopts the familiar formC00(z) = �ke�C(z) where the

constant k merely specifies the reference state of C. That

equation must be solved within the region 0 r z r l (whereas

for z > l the potential vanishes identically), subject to

the boundary conditions C0(l) = 0 and C0(0) = 4plB/a.The solution

CðzÞ ¼ ln cos2B

‘ðz� ‘Þ

� �� �ð10Þ

fulfillsC0(l) = 0 and specifies the reference stateC(l) = 0. To

ensure that the second boundary condition is fulfilled, the

integration constant B must solve the transcendental equation

B tanB = 2plBl/a. Once C(z) is known, we can determine the

electrostatic free energy per lipid according to

fð1Þel

kBT¼ B cotB� 1� ln

sinB cosB

B

� �: ð11Þ

The result f (1)el = f (1)el (a) according to eqn (11), derived for l=

0.5 nm, is included in Fig. 5; see the broken line (right above

the upper solid line). Comparing the results of a bare mono-

layer (i.e., in the absence of DNA) with (m0 = 0.003 nm�3)

and without (m0 = 0) divalent cations we find virtually no

difference. That is, adding mobile salt ions reduces the electro-

static free energy of a zwitterionic monolayer insignificantly.

Indeed, as we shall discuss below, our present model predicts

little penetration of the divalent cations into the lipid mono-

layer if DNA is absent. This is also in agreement with

experimental findings.6,15

The results for fel(a) and f (0)el (a) can be used to calculate

the electrostatic contribution to the lateral pressure Pel =

�dfel/da and P(0)el = �df (0)el /da in the presence and absence

of adsorbed DNA, respectively. The lateral pressure contri-

butions Pel and P(0)el are displayed in the inset of Fig. 5.

Irrespective of a, we find an approximatively 10%

decrease in the lateral pressure (i.e., its electrostatic contri-

bution) upon adding both DNA and divalent cations.

The prediction of this reduction is a major result of the

present study. It implies immediately that upon DNA adsorp-

tion the monolayer condenses laterally so that a lower pressure

for the tilt transition is required. To see this explicitely,

we formulate a simple Landau-type description of the tilt

transition.

Fig. 5 The electrostatic free energy per lipid fel = fel[d(a),a] of a

zwitterionic monolayer–DNA complex (in the presence of 5 mM

divalent cations) as a function of the cross-sectional area a per lipid.

The two upper curves refer to a bare monolayer where DNA is absent:

for f (0)el (upper solid line) divalent cations of concentration 5 mM

(i.e., m0 = 0.003 nm�3) are still present, whereas for f (1)el (broken line),

the aqueous phase does not contain salt ions (i.e., m0 = 0). The inset

shows how the presence of DNA affects the electrostatic contribution

to the lateral pressure. The two curves refer to Pel = �dfel/da and

P(0)el = �df (0)el /da.

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4 Tilt transition

We first model the tilt transition for a zwitterionic monolayer

in the absence of DNA. This will serve us as the reference for

investigating the influence of DNA adsorption. Because we

aim to describe the vicinity of the tilt transition, it is con-

venient to employ a standard Landau model with corre-

sponding free energy fL per lipid. We note that various other

phenomenological models are available;24,25 these aim to

understand how the mismatch between preferred headgroup

area and in-plane chain area leads to the emergence of a chain

tilt. Relating the molecular structure of the lipids to the tilt is

not the subject of the present work so that the simple Landau

model is most appropriate. We formulate the Landau model in

terms of two order parameters, the cross-sectional area per

lipid a and the tilt angle y. The Landau-type mean-field free

energy per lipid fL = fL(a,y) can then be written as

fL ¼B

2

a

a0� 1

� �2

þDPa� ay2a

a0� 1

� �þ c

2y4: ð12Þ

All constants in this model (B, a, a0 and c) must be positive.

The first term in eqn (12) describes the area elasticity of the

monolayer at a certain reference area a0 per lipid; B is the

corresponding compression modulus. This term results from

the interplay of repulsive lipid–lipid interactions and a certain

lateral reference pressure P0. Lateral stability of the mono-

layer requires B > 0. The second term represents the pressure

contribution beyond the reference pressure P0, where DP =

P � P0 denotes the pressure difference between the actual

applied lateral pressure and the reference pressure. The

remaining two terms represent a fourth-order expansion in

terms of the tilt angle y. Specifically, the third term accounts

for the area-dependent tilt elasticity26,27 where a > 0 is a

coupling constant between cross-sectional area and tilt angle.

The sign of a is positive to ensure that non-vanishing tilt tends

to lower the free energy if a> a0. Note that we also require the

coupling constant a to be sufficiently small so that the condi-

tion a2 o Bc is fulfilled. The last term stabilizes the tilt; the

constant c must be positive to keep the lipid layer stable even

for nonvanishing tilt angle y.To account for thermal equilibrium we minimize the free

energy fL(a,y) with respect to the cross-sectional area per lipid

a and tilt angle y. We need to distinguish two different cases.

First, for DP > 0 we obtain

y ¼ 0; a ¼ a0 �a20BDP; ð13Þ

and the corresponding optimal free energy reads

fL ¼ a0DP 1� a0

2BDP

h i: ð14Þ

This describes the untilted phase, which is adopted for suffi-

ciently large lateral pressure (i.e., forP>P0 in the absence of

DNA). Second, for DP o 0 the minimization yields

y ¼ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�DPa0aBc� a2

r; a ¼ a0 �

a20B� a2=c

DP; ð15Þ

and the corresponding free energy is

fL ¼ a0DP 1� a0

2ðB� a2=cÞDP� �

: ð16Þ

This second case describes the tilted phase, which is adopted

for sufficiently small lateral pressure (i.e., for P o P0 in the

absence of DNA). We use the free energy fL according to

eqn (14) and (16) as our phenomenological model for the tilt

transition of a zwitterionic lipid monolayer in the absence of

DNA. Divalent cations of concentration 5 mMmay be present

or not; the transition pressure remains essentially unaffected;

see Fig. 5.

We write for the total free energy of the zwitterionic

monolayer in the presence of adsorbed DNA

f(a,y) = fL(a,y) + fel[d(a),a] � f (0)el (a). (17)

Here fL is the Landau free energy according to eqn (12), which

describes the total monolayer free energy in the absence of

DNA. The second term, fel[d(a),a], is the electrostatic free

energy of the monolayer in the presence of DNA and divalent

cations (5 mM) according to eqn (9). The final term, f (0)el (a),

corresponds to the electrostatic contribution to the free energy

in the absence of DNA (with 5 mM divalent cations present).

Recall that both fel[d(a),a] and f (0)el (a) are plotted in Fig. 5. The

difference fel[d(a),a] � f (0)el (a) accounts for the influence of

adsorbed DNA on the total free energy of the zwitterionic

monolayer. We can express the differential of this difference as

d[fel[d(a),a] � f (0)el (a)] = DPel(a) da with DPel = Pel � P(0)el

being the change in the electrostatic contribution to the lateral

pressure upon adsorption of the DNA. (Recall Pel = �dfel/daand P(0)

el = �df (0)el /da). Hence, dfL(a,y) remains identical to

df(a,y) if we replace the term DPa in eqn (12) by (DP � DPel)a.

In the absence of DNA the tilt transition occurs at DP = 0.

Analogously, in the presence of DNA the tilt transition occurs

at DP= DPel. From the inset of Fig. 5 we recall that DPel o 0

has a negative sign. We conclude that the DNA lowers the tilt

transition pressure.

To further illustrate the reduction in the tilt transition

pressure upon adsorption of the DNA, we explicitely calculate

the relation y = y(DP) from the minimization of f(a,y)according to eqn (17). In the absence of DNA, where the total

free energy is specified solely by fL, the relation y = y(DP) is

given analytically in eqn (13) and (15). In the presence of DNA

we determine the relation y= y(DP) numerically. To this end,

we chose a0 = 0.4 nm2, B = 10kBT, and c = a = 2kBT. Note

that a0 = 0.4 nm2 is the average cross-sectional area per lipid

where the tilt transition is observed experimentally,14,15 B= Ka0is related to the compression modulus K = 40 kBT nm�2 of a

lipid layer,28 and a has been estimated from previous modeling27

of the tilt modulus and its dependence on the cross-sectional

area per lipid a. The constant c is not known but can be used

to control the magnitude of y in the tilted phase. Fig. 6 shows

the tilt angle y (and the inset the equilibrium value of a) as a

function of the lateral excess pressure DP. In the absence

of DNA (solid line in Fig. 6) the tilt transition is located at

DP = 0. Upon adsorption of DNA onto the monolayer the

tilt angle y decreases and the tilt transition shifts to smaller

lateral pressure. As argued above, the shift is equal to DPel.

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This finding is in qualitative agreement with experimental

observations,15 where the binding of DNA onto a zwitterionic

monolayer composed of DMPE was mediated by either

Ca2+or Mg2+. We also note that the lateral equilibrium area

a (which generally decreases with increasing DP and exhibits a

kink at the position of the second order tilt transition)

becomes slightly smaller when DNA is present. This too is a

manifestation of the DNA-induced monolayer condensation.

5 Discussion and conclusions

The present electrostatic mean-field model predicts a lateral

condensation of the monolayer upon adsorption of DNA.

Divalent cations are needed to render the adsorption process

of the DNA favorable. Indeed, it was demonstrated in pre-

vious work29 that in the absence of the divalent cations our

model predicts no DNA adsorption. How do divalent cations

induce the adsorption process? The mechanism that the pre-

sent electrostatic mean-field model predicts is based on the

migration of divalent cations from the DNA prior to adsorp-

tion to the lipid’s phosphate groups after adsorption. This

releases positive headgroup charges, which move toward the

DNA in order to interact with the phosphate groups of the

bound DNA. The driving force for this migration is the larger

area density of the lipid’s phosphate groups (1.43 nm�2 o1/ao 2.9 nm�2) as compared to the DNA’s phosphate groups

(1.09 nm�2). In essence, it is more favorable for a divalent

cation to neutralize the more densely packed phosphate

groups of the lipids as compared to those of the DNA.

To illustrate both the migration of divalent cations from the

DNA into the headgroup region and the concomitant release

of positive headgroup charges, we have carried out a repre-

sentative calculation for the specific choices m0 = 0.003 nm�3

(5 mM), a = 0.4 nm2 and d = 4.33 nm (which is where our

model predicts the tilt transition; see Fig. 6). Fig. 7 shows the

local number of divalent cations per lipid, defined through

ZðxÞ ¼ aR ‘0 mðz;xÞdz. Note the hypothetical value Z = 0.5 at

which one divalent cation would neutralize the phosphate

groups of two lipids. Solid and broken lines in Fig. 7 refer

to the presence and absence of DNA, respectively. In the

absence of DNA, we obtain a uniform cation-to-lipid ratio

Z(x) = 0.003, which is small but still five times higher than it

would be for m � m0; i.e., Z ¼ aR ‘0 m0dz ¼ a‘m0 ¼ 0:0006.

Hence, an insignificant amount of divalent cations penetrate

into the headgroup region of a bare monolayer. The notion of

weak penetration of calcium ions into purely zwitterionic lipid

layers is in general agreement with experimental studies6,30–32

and molecular dynamics simulations.33 The presence of acidic

lipids enhances the tendency of cation binding.33,34 In contrast,

when DNA is adsorbed the accumulation is significant, especially

in the vicinity of the adsorption region x=0 (see the solid line in

Fig. 7) where a magnitude of Z(x = 0) = 0.16 is reached.

The inset of Fig. 7 shows the average position of the positive

headgroup charges hzi ¼ ð1=‘ÞR ‘0zPðzjxÞdz. Again, solid and

broken lines refer to, respectively, the presence and absence of

DNA. In the absence of DNA, hzi= 0.133 which results from

the interplay of headgroup entropy and electrostatic attraction

between the two headgroup charges in a bare lipid layer. In the

presence of DNA, hzi increases somewhat, indicating the

tendency of the headgroups to reorient toward the DNA. This

reorientation has also been observed in computer simulations.35

The structural features we observe (see Fig. 7) suggest a

mechanism for the reduction in the lateral pressure P upon

DNA adsorption. The penetration of divalent cations into the

headgroup region neutralizes the lipid’s phosphate groups

more efficiently than the positive headgroup charges can do.

This reduces the electrostatic pressure within the headgroup

region, and this reduction is more significant than the simul-

taneous increase in P due to the DNA strands above the

headgroup region (i.e., in the region lo zo l+2R). We also

point out that the magnitude of the predicted decrease in Pis significantly smaller than that observed in experiment.

Fig. 6 Tilt angle y = y(DP) as a function of the excess pressure DP.

The excess pressure is defined so that for DP = 0 a zwitterionic lipid

monolayer undergoes the tilt transition in the absence of DNA. This is

the case for the solid curve, which corresponds to eqn (13) and (15)

and is derived from fL; i.e., the absence of DNA. The presence of DNA

leads to the broken line, which is derived from the minimization of

f(a,y) in eqn (17). Both curves are computed forB=10 kBT, a0= 0.4 nm 2,

and c = a = 2 kBT. The inset shows the corresponding dependence of

the cross-sectional area a per lipid on DP (solid and broken lines in the

main figure and the inset correspond to each other).

Fig. 7 Local cation-to-lipid ratio ZðxÞ ¼ aR ‘0 mðz;xÞdz (main figure)

and position of positive headgroup charges hzi ¼ ð1=‘ÞR ‘0zPðzjxÞdz

(inset). Solid and broken lines refer to the presence and absence of

adsorbed DNA, respectively. All calculations refer to m0 = 0.003 nm�3

(5 mM), a = 0.4 nm2 and d = 4.33 nm. This point is marked in the

inset of Fig. 3 by the symbol 1; see the second curve from top.

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Upon comparing DP E 0.16 kBT nm�2 E 0.7 mN m�1 in

Fig. 6 to the experimental15 value DP E 4 mN m�1, we see

two major reasons for this discrepancy. First, headgroup

reorientations and water hydration within the headgroup

region affect steric interactions and thus contribute to changes

in the lateral monolayer pressure. Steric interactions can

be accounted for phenomenologically.21 Second, mean-field

electrostatics neglects ion–ion correlations,36,37 which tend to

increase the lateral condensation of the monolayer. More

specifically, individual divalent cations can migrate between

the phosphate groups of neighboring lipids, thus forming an

ion bridge that leads to an additional electrostatic attraction

beyond the mean-field level.6 The same mechanism can also

act between DNA molecules.20 One could include such effects

into the present model by introducing a specific binding of the

divalent ions with the phosphate groups of either DNA or

lipid.38 In the present work we have not included any exten-

sions that would introduce additional unknown parameters

(i.e., to account for steric interactions, specific binding, etc.).

Instead, our extended Poisson–Boltzmann model involves

only a minimal set of known structural quantities (namely a,

lB, m0, l, R, s).In summary, we have studied the influence of adsorbed

DNA on the lateral pressure and tilt transition of a zwitter-

ionic lipid monolayer at the air–water interface. Our main

finding is that even on the electrostatic mean-field level, DNA

induces a lateral condensation of the monolayer. This con-

densation results from the (DNA-induced) penetration of

divalent cations into the headgroup region of the zwitterionic

monolayer. The condensation manifests itself in a downshift of

the lateral pressure at which the transition from a tilted to an

untilted ordered phase of the hydrocarbon chains takes place.

This agrees qualitatively with experimental observations.15

Acknowledgements

The authors acknowledge support from the Slovenian Research

Agency through grant No. BI-US/11-12-046.

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