Bending Moduli of Charged Membranes Immersed in Polyelectrolyte Solutions

0.16 MB·

by  Adi Shafir

Bending Moduli of Charged Membranes Immersed in
Polyelectrolyte Solutions
7 Adi Shafir∗and David Andelman†
0
0 School of Physics and Astronomy
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Raymond and Beverly Sackler Faculty of Exact Sciences
n
a Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel
J

8
Received 1st September 2006, corrections 28th December

]
DOI: 10.1039/
t
f
o
s
.
t
a
We study the contribution of polyelectrolytes of such interactions in biology. Understand-
m
in solution to the bending moduli of charged ing the interaction between charged macro-
-
d membranes. Using the Helfrich free energy, molecules such as DNA, RNA and various pro-
n
and within the mean-field theory, we calcu- teins and biological cell membranes (modeled
o
c late the dependence of the bending moduli as charged and flexible interfaces) sheds light
[
on the electrostatics and short-range interac- on many important cellular processes. Besides

2
tions between the membrane and the polyelec- its biological significance, the adsorption of PEs
v
6 trolyte chains. The most significant effect is ontochargedmembranesraisesinteresting ques-
6
seen for strong short-range interactions and low tions about the interplay between short-range
1
9 amounts of added salt where a substantial in- and electrostatic (long range) interactions in
0 crease in the bending moduli of order 1k T these multi-component charged systems. Re-
B
6
is obtained. From short-range repulsive mem- cent works include numerical andanalytical cal-
0
/ branes, the polyelectrolyte contribution to the culations [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11] and
t
a
bending moduli is small, of order 0.1k T up to scaling arguments [1, 4, 5, 12, 13].
m B
atmost 1k T. Forweakshort-rangeattraction,
- B
d the increase in membrane rigidity is smaller and In our previous work [14], three regimes
n have been found for polyelectrolyte adsorption.
of less significance. It may even become neg-
o
(i) When the short-range interaction between
c ative for large enough amounts of added salt.
: the membrane and the PE chains is repulsive,
v Our numerical results are obtained by solving
i the surface charge is low and the ionic strength
X the adsorption problem in spherical and cylin-
of the solution is high, the polymers deplete
r drical geometries. In some cases the bending
a fromthechargedmembrane. (ii)Forhighersur-
moduli are shown to follow simple scaling laws.
face charges, or lower ionic strength, the poly-
electrolytes adsorb on the charged membrane
1 Introduction and screen the surface charges. (iii) When the
short-range interactions between the membrane
and the PE chains are attractive, the PE chains
The study of interactions between charged and
adsorb on the membrane, and the adsorbed
flexible membranes and polyelectrolytes (PEs)
layer carries a higher charge than that of the
in solution has generated a lot of interest in re-
bare membrane. In this situation the polyelec-
cent years, partly motivated by the importance
trolytes over compensate the bare surface —
∗E-mail:shafi[email protected] a phenomenon of practical importance in the
†E-mail:[email protected] build-up ofmultilayers ofalternating cationand
1
2 Mean Field Equations
anion polyelectrolytes [15].
The flexibility of fluid surfaces and membranes
2.1 Free-Energy Formulation
hasbeenstudiedinvariouscases,anditdepends
on the lipid composition, tail length and molec- Consider a bulk aqueous solution containing
ular tilt. In the case of charged and flexible polyelectrolyte (PE) chains, along with their
membranes immersed in a pure ionic solution counter ions and added salt. A curved and
(no macromolecules) [16, 17, 18, 19, 20, 21, 22, charged membrane is placed at the origin |r| =
23, 24, 25, 26, 27], the electrostatic contribu- 0. In order to extract the membrane elastic
tion was found to increase the membrane rigid- moduli, we take the membrane shape to be ei-
ity, and the addition of salt to decrease it. In a ther spherical orcylindrical with radius R . The
different set ofstudies, the adsorptionofneutral free energy for the combined system has been
polymers on membranes has been investigated formulated before [1], and can be written as a
theoretically [28, 29, 30, 31, 32, 33, 34] and in sum of four contributions F = F + F +
tot pol ions
experiments [35, 36, 37, 38]. It was found that F +F .
el int
the addition of polymers reduces the membrane The polymer free energy, F , is:
pol
rigidity. In all those cases, the contribution of
thesolutiontothebending rigiditywasfoundto
a2
ibseonoftoortdheer 0in.1tr−in1sikcBmTo,nwohlaicyherisbleonwdiinngcroimgipdaitry- Fpol = ZV dr" 6 |∇φ|2 +Σ(φ)−Σ(φb)+
of approximately 10kBT. We note that in spe- µ φ2 −φ2 , (1)
b
cial cases, by adding a co-surfactant (alcohol), #
(cid:16) (cid:17)
the membrane bending rigidity can be brought
where a is the monomer size, φ2 is the monomer
down to values of roughly 1k T [39]. b
B bulk concentration, φ2 is the local monomer
concentration. The free energy is given in terms
In the present work, we study the combined
system of charged chains interacting with op- of kBT, and the integration is carried out over
the entire volume outside the sphere/cylinder,
positely charged and flexible membranes. Our
|r| > R. The first term in Eq. (1) accounts for
study is similar in spirit to that done for DNA-
the chain elasticity. The second and third terms
lipid systems [40, 41], where the DNA was mod-
account for the excluded volume interactions in
eled as rigid and charged rod. The main dif-
the solution and in the bulk, respectively:
ference is that our charged macromolecules are
flexible. The above mentioned three regimes
dictateadifferentcontributiontothemembrane
Σ(φ)a3 = 1−a3φ2 log 1−a3φ2 −
rigidity and stability. For short-range repulsive
(cid:16)1 (cid:17) (cid:16) (cid:17)
membranes, the membrane rigidifies due to its v −a3 1−a3φ2 φ2 (2)
2
charges while its rigidity decreases with the in-
(cid:16) (cid:17)(cid:16) (cid:17)
crease of the polyelectrolyte charge. For weak µ = log 1−a3φ2 +
b
short-rangeinteractions, thecontribution to the 1 (cid:16) (cid:17)
va−3 +1 − v −a3 φ2 (3)
membrane rigidity decreases and may become 2 b
slightly negative, while for strong short-range (cid:16) (cid:17) (cid:16) (cid:17)
where v is the excluded volume coefficient. For
attraction the membrane becomes rigid again.
φ2 ≪ a−3 theabove expression canbeexpanded
In most cases, the magnitude of the contribu- to the well known 1v(φ2 −φ2)2, which is com-
tion to the bending rigidity is of the same or- 2 b
monly used forthe excluded volume interaction.
der of magnitude as that of neutral polymers or
The small ion contribution to the entropy is:
puresaltsolutions, namely0.1k T upto1k T.
B B
However, we show in this paper that for strong
enoughshort-rangeattractionamoresignificant F = dr c log ci(x) −c (x)+ci
contribution, of order kBT or higher can be ob- ions i=±ZV " i cib i b#
X
tained. (4)
2
where c and c± are the local and bulk concen- and k−1 = (4πl φ2f)−1/2 is the corresponding
± b m B b
trations of the ± small ions, respectively. The length for the potential decay due to counteri-
third contribution to the free energy, the elec- ons. Note that the actual decay of the electro-
trostatic free-energy, is: static potential is determined by a combination
of salt, counterions, and polymer screening ef-
fects. The excluded volume interaction is taken
F = dr c −c +fφ2 ζ −
el + − into account using the function:
ZV (cid:16) (cid:17)
1
dr|∇ζ|2 + dAσζ (5)
8πlB ZV Z|r|=R
Λ(η) = log(1−φ2a3)−log(1−φ2a3η2)+
where ζ ≡ eψ/k T is the renormalized elec- b b
B
v −a3 φ2 η2 −1 (9)
trostatic potential, f is the fraction of charged b
monomers and l ≡ e2/εk T is the Bjerrum (cid:16) (cid:17) (cid:16) (cid:17)
B B
which represents the full excluded volume inter-
length. The first integral accounts for the in-
action.
teractions between the positive ions, negative
The solution of Eqs. (7) and (8) requires four
small ions and the monomer charges with the
boundary conditions. Two of the boundary
electrostatic potential. The second integral ac-
conditions are taken far from the membrane,
counts for the self-energy of the electrostatic
where the monomer concentration and elec-
field and the third to the interaction between
trostatic potential retrieve their bulk values
the surface charge and the electrostatic poten-
η(|r| → ∞) → 1 and ζ(|r| → ∞) → 0. The
tial. Note that the third integral is taken only
other two boundary conditions account for the
over the charged body surface, |r| = R.
interaction with the charged membrane, and
The last part of the free energy is the short-
can be obtained by integrating Eqs (7) and (8)
range interaction of the PE chains with the
from |r| = R to a small distance from the mem-
membrane:
brane, yielding:
a2
F = − dAφ2 (6)
int 6d Z|r|=R s nˆ · ∇ζ| = −4πl σ (10)
where φ ≡φ(|r| = R) is the monomer concen- |r|=R B
s
tration on the surface. This term stands for a nˆ · ∇η||r|=R = −d−1ηs (11)
general short-range interaction, where the in-
Equation (10) is the usual electrostatic bound-
teraction strength is determined by the phe-
nomenological constant d−1, which has dimen- ary condition for a given surface charge density,
whileEq.(11)istheCahn-deGennesboundary
sions of inverse length.
condition [42], which is often used for calculat-
Minimization of the total free energy F =
tot
ing polymer profiles [30, 31, 29].
F +F +F +F yields the following mean-
pol ions el int
For large R, the total free-energy can be ex-
field equations, expressed in terms of a dimen-
panded around its flat surface value in the fol-
sionless variable η ≡ φ/φ :
b
lowing way [44]:
∇2ζ = λ−2sinhζ +k2 eζ −η2 −
D m
1 2 2
4πl σδ(cid:16)(|r|−R(cid:17)) (7) F = dA f + δκ c +c − +
B tot 0 1 2
a2 Z|r|=R " 2 (cid:18) R0(cid:19)
∇2η = Λ(η)η−fζη
6 δκ c c (12)
G 1 2
a2 #
− δ(|r|−R)η (8)
6d
where F is the total free energy (in units
tot
where λ = (8πl c )−1/2 is the Debye-Hu¨ckel of k T), f is the free energy per unit area
D B salt B 0
length scale for the screening of the electro- of a solution in contact with a planar surface
static potential in the presence of added salt (R → ∞) that hasthe same system parameters,
3
andc , c aretheradiiofcurvatureforthemem- The contributions of the solution to the curva-
1 2
brane. For spherical surfaces the radii of cur- ture moduli are given by:
vature are c =c =1/R, while for a cylindrical
1 2
surfacec =1/R, c =0. Theparametersδκand
1 2
δκ are the contributions of the PE (and salt) δκ = 2B (15)
G C
solution to the mean and Gaussian curvature δκ = B −4B . (16)
G S C
moduli of the membrane, respectively, in units
ofk T. Namely, κ = κ0+δκandκ = κ0 +δκ A surface is stable under long wave bending
B G G G
include the intrinsic values as well as the contri- fluctuations only when κ>0, and against spon-
butions coming from the solution. Throughout taneous vesiculation (topological change) when
this paper we will consider only the changes in 2κ+κ > 0 [24]. In the following we show that
G
the elastic moduli κ,κ with respect to their thecontributionofthePEsolutiontothestabil-
G
bare values. An increase in the mean curva- ity of a charged surface depends on the amount
ture modulus δκ increases the membrane rigid- of added salt, and has a non-monotonic depen-
ity, while an increase in the Gaussian modulus dence on the short-range interactions between
δκ makes saddle pointsonthemembrane more the membrane and the polyelectrolyte.
G
favorable.
The parameter R is the radius of spontaneous
0
3 Results
curvature for the membrane, which is depen-
dent on the exact chemical composition of the
We find large contributions, of order 1k T,
membrane. For a positive R , the membrane B
0
to the surface bending moduli for the case
bends towards the external solution, while for
of strong short-range attractive surfaces. For
negative R it bends away from the solution.
0
weaker short-range interactions and for repul-
In the case of a bilayer membrane, where the
sive surfaces, the contribution is smaller, of or-
same solution is in contact with both leaflets of
der 0.1 − 1k T. We discuss first the strong
the membrane, the spontaneous curvature van- B
repulsive and strong attractive surface limits,
ishes due to symmetry. In this paper, we do not
and then turn to the intermediate case, where
calculate the spontaneous curvature but rather
the contribution to the bending rigidity is less
focus on the changes to the bending moduli δκ
significant. We present analogies and scaling
and δκ .
G
calculations to explain the different regimes.
2.2 Numerical Procedure
3.1 Strong Short-Range Attrac-
Equations (7)-(11) are solved numerically for
tive Membranes
the cases of a charged sphere and a charged
cylinder. The numerical procedure follows the In previous publications [5, 6], the adsorbed
relaxation scheme [43], as was described in pre- amount of PEs as well as the layer width were
vious publications [4, 5]. For each solution, we studied in detail for short-range attractive sur-
calculate the total free energy per unit area faces. The adsorbed PEs charge was shown
for the cases of a spherical membrane f and to exceed the bare surface charge significantly
S
a cylindrical membrane f . The contributions for strong short-range interactions, and to ex-
C
of the polyelectrolyte solution to the mean and ceed it mildly for weak short-range interactions.
Gaussian curvatures are then calculated by ex- The width of the adsorbed layer, on the other
panding f , f to second order in 1/R: hand, depends on the shorter of the two ad-
C S
sorption length scales: (i) d, the length scale for
short-range attraction from Eq. (11), and (ii)
A B
S S
fS = f0 + R + R2 (13) ξ ≡ a/ vφ2b, the Edwards correlation length for
A B neutralqpolymer adsorption. In our model, we
C C
fC = f0 + R + R2 (14) use an almost theta solvent 0 < vφ2b ≪ 1, so we
4
assume that the shorter length scale is always by analogy to ionic solutions. There are two
d ≪ ξ. The increase in short-range attractive regimesfor2δκ+δκ inpureionicsolutions[24],
G
interactions, inthiscase,decreasestheadsorbed depending on the surface charge. For weakly
2
layer width [4, 5]. The combined charge of the charged surfaces c ≫ l |σ| (or λ ≪
salt B D
PE-membrane complex is, therefore, opposite λ where λ ≡ (2πl |σ|)−1 is the Gouy-
GC GC B
to the initial surface charge, and its magnitude Chapman length), δκ as well as 2δκ + δκ
G
may be much higher than the initial membrane are positive [18, 20], while for highly charged
2
charge. This charge is distributed within a layer surfaces c ≪ l |σ| we get δκ > 0 and
salt B
of width d close to the surface. 2δκ + δκ < 0 [21, 22]. In our case of poly-
G
Our focus in this section is on the low salt case. electrolyte solutions, for low enough attractive
In Fig. 1 we show the dependence of κ and κ short-range interactions the renormalized sur-
G
on the short-range interaction parameter d−1 face charge is low, and 2δκ + δκ > 0. For
G
for low salt conditions. For strong short-range stronger short-range interactions, the renormal-
attraction, namely d < 2˚A, the calculated δκ ized surface charge increases and 2δκ+δκ < 0.
G
is positive and increases strongly, reaching val- The results in Fig. 1 were presented in the low
ues of several k T. The δκ is negative, and added salt case, d ≪ λ . When more salt is
B G D
shows a stronger increase with d−1 than the cor- added into thesolutiontheelectrostatic interac-
responding δκ. The contribution to the vesicu- tions between the polymers become weaker. In
lationstability2δκ+δκ inthiscasecanbeseen caseofahighamountofaddedsalt,d ≫ λ ,the
G D
to be negative for high d−1, leading to destabi- charges of the adsorbed polymers are screened
lization of the surface. For lower (but positive) over smaller length scales than the adsorbed
d−1, the contribution to 2δκ + δκ is positive, layer width, and the analogy to a renormalized
G
and thus enhances the membrane stability. charged surface breaks down. We further dis-
These results can be explained by the following cuss this case in Sec. 3.3.
argument. When the adsorbed layer width is
smaller than the electrostatic screening length, 3.2 Short-Range Repulsive
the membrane-PE complex can be viewed as a
Membranes
renormalized charged surface, containing both
the bare surface charges and the adsorbed PE Without the PE adsorption, the charges on the
charges, in a weak ionic solution. The renor- membrane increase the membrane rigidity, κ,
malized surface interacts with the ionic so- due to the long-range repulsion between them.
lution in the same manner as a bare mem- For strongly repulsive surfaces, the adsorption
brane [16, 17, 18, 19, 20, 21, 22, 23, 24]. The ofPEchainstothemembranescreensitssurface
strong increase in the surface charge and the charges and causes κ to decrease. Note that the
lack of small ion screening (low salt) cause the PE charges do not fully compensate the mem-
surface fluctuations to be strongly unfavorable, brane bare charges like they do in the strongly
making δκ positive. This allows substantial in- attractive regime, leading to the difference in
crease in the magnitude of δκ, δκ , amount- the corresponding system behaviors.
G
ing to several k T, which is a significant con- Therearetworegimesforthecaseofshort-range
B
tribution to the membrane curvature moduli, repulsive membranes: (i) the low-salt regime
as can be seen in Fig. 1. We note that as where the surface charges are mainly balanced
the renormalized surface charge increases, the by the adsorbed monomer charges, and (ii) the
value of δκ should approach the low salt limit high-salt regime where the salt ions screen the
for charged surfaces in pure ionic solutions (no surface charges and the monomers deplete. The
PE) δκ → λ /(2πl ) ≃ 7k T [24] (See Fig. 1). bending moduli in the latter regime are simi-
D B B
However, this limit is still higher than our nu- lar to those of a strong ionic solution with no
merical results for the PE-membrane complex. added PE, as were discussed in detail in other
The crossover between positive and negative papers [16, 17, 18, 20, 24].
values of 2δκ + δκ can also be explained In Ref. [4] we addressed the PE adsorption
G
5
close to a flat charged surface and showed that by the fact that the polyelectrolytes replace the
the screening length of the electrostatic poten- salt ions in screening the surface charges, thus
tial depends strongly on the amount of added allowing greater membrane flexibility.
salt. Forlowsalt conditions, thesurface charges In the high salt regime, the membrane interacts
are screened mainly by the adsorbed monomer onlywiththesmallions,andthepolymerchains
charges. In this case the free-energy per are depleted. The free energy in this case is
unit area scales like f ∼ |σ|5/3f−1/3a2/3l2/3 similar to that of a weakly charged surface in
ads B
and the screening length scales like D ∼ an ionic solution [24]. Namely, the screening
a2/3/(fl |σ|)1/3 [4]. For high amounts of added length is λ and the free energy per unit area
B D
salt, the screening is mainly done by small ions for the case of a flat surface is f ∼ |σ|2l λ .
dep B D
and the PE chains deplete. The transition from The curvature moduli are the same as in a pure
depletiontoadsorptionregimesoccurswhenthe ionic solution [16, 17, 18, 20, 24], where:
screening lengthduetomonomeradsorptionbe-
comes similar to the one for small ion adsorp- δκ, δκ ∼ f λ2 ∼ |σ|2l λ3 . (18)
G dep D B D
tion, i.e., when λ ∼ D.
D
Note that the depletion condition λ < D im-
The flat surface results can now be easily ex- D
plies that the bending moduli for the high salt
tended to curved surfaces. In the low-salt
(depletion) case are always lower than for low
regime, monomers are adsorbed to the mem-
salt (adsorption) case, despite the depletion of
brane, and the length scale for the adsorp-
the polymers. Both δκ and δκ in this case are
tion is D. We expect the free energy of the G
of order 0.01 − 0.1k T for physiological range
curved membrane (per unit area) to scale like B
2
of system parameters, and scale like |σ| .
f = f h(D/R). Expanding f to second
tot ads tot
order in R−1 and comparing to Eq. (12) shows For both low and high salt regimes, the contri-
bution to the Gaussian bending modulus δκ is
that both curvature moduli scale like: G
negative, resulting from the electrostatic repul-
|σ|a2 sion between the membrane constituents [19].
δκ, δκ ∼ f D2 ∼ , (17)
G ads f ThecontributionofthePEsolutiontothevesic-
ulation stability 2δκ + δκ is always positive,
which is of order 0.1 − 1k T for physiological G
B
evenforverylowsaltconcentrations, incontrast
range of system parameters.
to the pure ionic solution results [21, 22, 23].
In Fig. 2 we present the values of δκ and δκ
G
The low-salt regime of the ionic solutions, in
as a function of the surface charge |σ|, in the
which the contribution to 2δκ + δκ is nega-
case of strongly repulsive membranes d−1 = G
tive, is replaced here by the adsorption regime,
−20˚A−1 and monomer size of a = 10˚A. For
in which this contribution is still positive.
low amounts of added salt we find a scaling re-
lation δκ, δκ ∼ |σ|β with β ≃ 1.2. Note that
G
the numerically calculated exponent β ≃ 1.2 is 3.3 Weak Short-Range Interact-
slightly larger than β = 1 derived in Eq. (17).
ing Membranes
This discrepancy seems to occurs because the
high monomer size used here makes the full ex- In Fig.3 we present an enlargement of Fig. 1 for
cludedvolumeinteractionsubstantial. Forcom- surfaces having only a weak short-range inter-
parison, we plot the corresponding δκ, δκ for action with the PE in solution, d−1 ∼ 0. As can
G
the case of a pure ionic solution, namely when be seen, the magnitudes of both δκ and δκ
G
no polyelectrolytes are added to the solution. decrease substantially for low |d−1|, and may
In this case the magnitudes of both δκ, δκ are become negative for high values of added salt.
G
very high. This shows that the addition of poly- The decrease in the magnitude of the bending
electrolytes into low salt solutions can cause a moduli can be attributed to the strong screen-
strong reduction of the bending moduli, in the ing of the surface charges by the adsorbed PEs,
order of several k T as compared with the pure which makes δκ smaller. We note that the con-
B
salt solution. This reduction can be explained tribution to δκ and δκ is negligible in compar-
G
6
4 Conclusions
ison to the membrane intrinsic bending moduli,
and probably cannot be observed experimen-
tally. This regime is presented only in order
to complete the d−1 dependence picture.
In this paper we show that the interaction be-
tween charged and flexible polymers (PE) and
For low and positive d−1 and high amounts of
oppositely charged and flexible membrane de-
added salt, we find a negative contribution to
pends both on their electrostatic and short-
the mean curvature modulus due to the poly-
range interactions. A non-monotonic depen-
mer adsorption δκ < 0, as well as in 2δκ+δκ .
G dence of the curvature moduli is obtained as
This surprising result can be explained by anal-
function of the short-range interaction between
ogy to the neutral polymer case. In past pub-
the membrane and the PE chains. We find a
lications [29, 30, 31] it was shown that neu-
significant contribution to the bending moduli,
tral polymer solutions have negative δκ, posi-
of order of several k T, in the case of strong
tive δκ and negative 2δκ+δκ . We find here B
G G short-range attraction between the PE chains
similar results.
and the surface. For weak attractive interac-
tions, the contribution of the PE solution to the
The analogy to neutral polymer adsorption is
membrane curvature moduliis small(in unitsof
important and can be understood in the follow-
k T), and for repulsive interactions it increases
B
ing way. The high amount of added salt screens
back, and may reach values of 0.1−1k T.
B
both the surface and the PE charges, so their
effective interaction becomes short-ranged lead-
Our work deals only with uniformly charged
ingthewaytoanalmostneutralpolymerbehav-
membranes. In biological membranes, however,
ior. The increase in δκ derives from the effec-
G the membrane is composed of a mixture of neu-
tive attraction between membrane constituents,
tral and charged lipid. The lipid molecules
which results from their short-range attraction
canrearrangeandcancluster aroundoppositely
to the PE chains. The analogy to neutral poly-
charged PE [47]. This in turn can have a strong
mers requires the screening length for the elec-
effect on the overall membrane rigidity. Future
trostatic interactions to be much smaller than
works may offerextensions ofthepresent oneby
the layer width λ < d. This is satisfied in
D calculatingthecontributionstothespontaneous
high salt and low short-range attraction con-
radius of curvature R , especially in the case
0
ditions. For stronger short-range interactions,
of asymmetrical solutions, or when the mem-
the layer width decreases, and the electrostatic
branes are composed of asymmetric inner and
interactions between monomers becomes impor-
outer leaflets. Other potential directions may
tant. In this case, the analogy to neutral poly-
include changes in the effective lipid headgroup
mers breaks down, and δκ, 2δκ + δκ start to
G size and water activity due to the presence of
increase back. For higher d−1, both δκ and
polyelectrolytes.
2δκ+ δκ become positive again, marking the
G
crossover to the strong attraction regime de-
scribed in Sec. 3.1 above. This can be seen in
Fig. 3, where for high amounts of added salt,
the increase in δκ with d−1 indeed begins when
λ ≃ d, as expected from the above analysis. It
D
is also important to note, that the magnitude
of both δκ and δκ are very small, of order Acknowledgments: The authors wish to thank
G
0.01k T, and are negligible in comparison to Michael Kozlov for helpful discussions. Support
B
the intrinsic bending moduli of a membrane, of from the Israel Science Foundation (ISF) under
order 10k T. The decrease shown here cannot grant no. 160/05 and the US-Israel Binational
B
accountforadecreaseinmembranerigiditythat Foundation (BSF) under grant no. 287/02 is
was recently found for DNA adsorption [46]. gratefully acknowledged.
7
Adi Shafir,a David Andelmana [16] D. Bensimon, F. David, S. Leibler, A.
a School of Physics and Astronomy Pumir, J. Phys. (France), 1990, 51, 689.
Raymond and Beverly Sackler Faculty of Exact
[17] A. Fodgen, B. W. Ninham, Langmuir,
Sciences
1991, 7, 590.
Tel Aviv University, Ramat Aviv, Tel Aviv
69978, Israel.
[18] M. Winterhalter, W. Helfrich, J. Phys.
Chem., 1988, 92, 6865.
References [19] M. Winterhalter, W. Helfrich, J. Phys.
Chem., 1992, 96, 327.
[1] I. Borukhov, D. Andelman, H. Orland,
[20] M. Kiometzis, H. Kleinert, Phys. Lett. A,
Macromolecules, 1998, 31, 1665; Europhys.
1989, 140, 520.
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9
4 0
3.5 (a) (b)
−1
3
−2
2.5 ]
] T
T −3 B
B 2 k
k [
[ G
−4
κ 1.5 κ
δ δ
1 −5
0.5 −6
0
−7
−1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1
−1 o−1 −1 o−1
d [A ] d [A ]
Figure 1: The d−1 dependence of (a) δκ and (b) δκ is presented for low added salt concentration
G
(c = 0.1mM). Other parameters are |σ| = 0.001˚A−2, a = 5˚A, f = 0.5, v = 50˚A3, φ2 =
salt b
10−8˚A−3, T = 300K and ε = 80. For large d−1, we see a significant increase in the magnitude of
both δκ and δκ , amounting to several k T, which is very significant in comparison to normal
G B
membrane curvature moduli.
10