Dr Khalid elhasnaoui

SOP TRANSACTIONS ON THEORETICAL PHYSICS
ISSN(Print): 2372-2487 ISSN(Online): 2372-2495
DOI: 10.15764/TPHY.2014.04002
Volume 1, Number 4, December 2014
Lateral Phase Separation between
Phospholipids and Adhesion
Macromolecules on Two Adhering
Membranes
T. El hafi1*, K. ELhasnaoui1, A. Maarouf1*, N. Hadrioui1, M. Ouarch1,3, M.
Benhamou1,2, H Ridouane1
1 LPPPC, Sciences Faculty Ben M’sik, P.O.Box 7955, Casablanca
2 ENSAM, Moulay Ismail University P.O.Box 25290, Al Mansour, Meknes
3 CRMEF, 298 Avenue Al Alaouiyine, P.O Box 24000, EL Jadida, (Morocco)
*Corresponding author: elhasnaouikhalid@gmail.com; abdelwahad.maarouf@gmail.com
Abstract:
We study analytically the lateral phase separation produced between phospholipids and adhesion
macromolecules on two adhering membranes from a static point of view. In this letter, the
adhesion macromolecules are assumed as long-flexible polymer chains anchored by two
extremities on the inner monolayer of the two adjacent plasma membranes by anchors-segments,
which are big amphiphilic molecules. The aim is to quantify how these polymer chains can
be driven from a dispersed phase to a dense one, under the variation of a suitable parameter
such as temperature and membrane environment.... To investigate this demixtion transition,
we elaborate a new field theory that allows us to derive the expression for the mixing free
energy. From this, we extract the complete shape of the associated phase diagram in the
composition-temperature plane. The essential conclusion is that the anchored polymer chains
experience the significant attractive forces directly result from the shape deformations of two
parallel membranes a mean-distance apart. Also, the solvent quality and the structure (length)
of adhesion macromolecules have a strong influence on the compatibility domain of the mixture.
Keywords:
Membrane Adhesion; Adhesion Macromolecules; Polymers; Phase Separation; Phase Diagrams
1. INTRODUCTION
Adhesion of membranes and vesicles has attracted considerable experimental and theoretical interest
because of its prime importance to many bio-cellular processes [1, 2]. Theoretical treatments of mem-
branes composed of single component lipid bilayers have revealed that generic interactions such as van
der Waals, electrostatic or hydration interactions govern the adhesive properties of interacting membranes.
It is also worthwhile to mention that related phenomena are found in unbinding transition of nearly flat
membranes [3] or adhesion of vesicles to surfaces [4].
13

In addition to general non-specific interactions mentioned above, it is known from the works of Bell
and coworkers [5, 6] as well as others [7], that highly specific molecular interactions play an essential
role in biological adhesion. This interaction acts between complementary pairs of proteins such as
ligand and receptor, or antibody and antigen. A well-studied example of such coupled systems is the
biotin-avidin complex. The avidin molecule has four biotin binding sites, two on each side, and forms
a five-molecule biotin-avidin-biotin complex. The resulting specific interaction is highly local and
short-ranged. Measurements by surface force apparatus [8] or atomic force microscopy [9, 10] have
shown that the force required to break a biotin-avidin bond is about 170pN. In related experiments
measuring chemical equilibrium constants [11], it was found that the biotin-avidin binding energy is about
30 − 35kBT is larger than thermal fluctuations. Other coupled systems are those of selectins and their
sugar ligands where the bond is much weaker, of the order of 5kBT [12, 13].
More recently several models taking into account thermal fluctuations in membrane adhesion have
been proposed. Zuckerman and Bruinsma [12, 13] used a statistical mechanics model which is mapped
onto a two-dimensional Coulomb plasma with attractive interactions. They predicted an enhancement of
the membrane adhesion due to thermal fluctuations. In another work, Lipowsky considered the adhesion
of lipid membranes which includes anchored stickers, i.e., anchored molecules with adhesive segments
[14, 15]. It was shown that flexible membranes can adhere if the sticker concentration exceeds a certain
threshold.
Currently, the adhesion of two adjacent plasma membranes is provided by bound pairs of such adhesion
macromolecules which form bridges between the membranes. We distinguish three types of adhesion
depending on the structure of bridges : i) Bolaform-sticker adhesion where each bridge molecule consists
of a single sticker having two sticky ends. One sticker end is anchored to one membrane while the other
end is adhering directly to the second membrane. ii) Homophilic-sticker adhesion where the bridges are
formed by two stickers of the same type. Each sticker is anchored on one of the membranes, while their
free ends bind together to form the bridge. iii) Lock-andkey adhesion where the bridges consist of two
different stickers forming a ligand-receptor type bond. This case represents an asymmetric adhesion due
to the lack of symmetry between the ligand and receptor.
A model system for these complex interactions is provided by systems containing lipid bilayers and
polymers. From the physical point of view, polymers can be characterized by several length scales. First
of all, they have a certain length, Nb, where N and b are the number of monomers and the length of these
monomers, respectively. Secondly, linear polymers are characterized by a certain persistence length, ξp :
the polymer is hard and easy to bend on scales which are smaller and larger than ξp respectively.
Many biopolymers seem to have a relatively large persistence length ξp which is comparable or
exceeds its total length Nb; in this case, the polymer behaves as a worm-like chain which exhibits an
average direction. On the other hand, if Nb ξp, polymers crumple or fold up in order to increase their
configurational entropy. This leads to a more compact 3-dimensional structure with a gyration radius,
Rg ∼ Nb. In good solvents, these structures are random coils and Rg ∼ Nε for large N with the Flory
estimate ε 3/5 (in 3-dimensional systems). In bad solvents, the polymers collapse and become densely
packed with Rg ∼ N1/3.
The problem of multi-component membrane adhesion, including lipids and adhesion macromolecules,
is intimately related to that of formation of domains or ‘raft’ (a lateral phase separation). This has
been observed by several experiments. For example, the biotin-avidin interaction occurring during
vesicle-vesicle adhesion was investigated by a micropipette technique [16]. The adhesion between one
avidin-coated vesicle and a second biotinylated vesicle is followed by an accumulation of biotin-avidin
14

Lateral Phase Separation between Phospholipids and Adhesion Macromolecules on Two Adhering Membranes
complex in the contact zone. This accumulation of cross-bridges between the two vesicles is found to be
a diffusion-controlled process.
It is generally believed that multi-component biomembranes in physiological conditions are close to
their critical point, and membrane functions are partially governed through phase separation processes.
Moreover, volume fraction and Flory interaction parameter fluctuations in the vicinity of the critical
point may affect biophysical properties of membranes and can be of importance in regulating membrane
processes.
In this work we provide a general phenomenological approach for the adhesion of multi-component
membranes. Using a mean-field theory, we investigate how the lateral phase separation is drastically
affected by the significant influences of the structure (length) of adhesion macromolecules (cases of
monodisperse and polydisperse systems), the solvent quality (the presence of a good solvent or a theta
solvent) and of the undulations of two coupled membranes separated by a mean-spacing. To model the
adhesion between membranes, we follow a simplified view considering the adhesion macromolecules as
long-flexible polymer chains anchored reversibly into the hydrophobic interior of the two parallel lipids
bilayers by big amphiphilic molecules termed anchors-segments (see Figure 1). In addition, for the sake
of simplicity, we have ignored primitive interactions between two parallel membranes, as van der Waals,
hydration and effective steric interaction. Under certain physical circumstances, these interactions can
be neglected (rigid polymer chains). An important consequence of our model is that the lateral phase
separation is enhanced.
The scheme of this paper is as follows. In a heuristic way in the next section, we present a theoretical
formalism allowing a general expression of the induced interactions between anchors segments. We
compute, in section 3, the expression for the mixing free energy of the phospholipids-anchors segments
mixture. To investigate the phase diagram shape is the aim of section 4. Finally, we draw our conclusions
in the last section.
Figure 1. Sketch showing the adhesion of two adjacent plasma.
2. THEORETICAL FORMULATION
The system of interest is a lamellar phase composed of two roughly parallel fluid membranes (neutrals)
made of identical lipid molecules. The cohesion between these bilayer membranes is ensured by long-
flexible polymer chains. These ones are anchored by two extremities on the inner monolayer of the two
fluctuating fluid membranes via big amphiphilic molecules termed anchors-segments. These latter are
assumed to be of different chemical nature as the host lipid molecules forming the two membranes and
must have a hydrophobic part in order to insert into the bilayer membrane and a hydrophilic head group
15

which provides the attachment site for the flexible polymer.
For simplicity, we suppose that the two extremities of a given chain remain along the same axis that is
perpendicular to the two membrane surfaces. We denote by −→rα the position-vector of anchoring extremities
of chain α on the two fluid membranes. We suppose that the grafted chains are in the presence of good or
theta solvents. It is then natural to regard the anchors segments as soft inclusions permanently moving
on the two membrane surfaces. In particular, these experience an interaction force mediated by the
undulations of the two adhering membranes, so-called induced force [17, 18].
In this section, we present a theoretical formalism allowing a general expression of all induced
interactions. As we will see below, such an expression can be derived taking advantage of some method
based on field theory techniques [19, 20]. We shall use the notation −→r = (x, y) ∈ R2 to mean the position
of the representative point on the two surfaces. The position of the two (almost flat) membrane surfaces
labeled by i = 1, 2 are specified through the displacement field, h1 (x, y) and h2 (x, y) of the upper and
lower membrane, respectively. The two interacting membranes fluctuate around the reference plane z = 0,
so that the height-function hi (x, y) may take either positive or negative values.
The separation (or relative displacement field), l = h2 −h1 > 0 of these two membranes is governed by
the configurational energy (or effective Hamiltonian) [3, 21, 22]
H0 [l]
kBT
=
ˆκ
2
d2
r(∆l)2
, (1)
With kB is the Boltzmann constant, T the absolute temperature and the rescaling parameter ˆκ = κ/kBT,
where κ is the common bending rigidity constant of the two membranes. But in the case of two bilayers
of different bending rigidity constants κ1 and κ2, we have
κ = κ1κ2/(κ1 +κ2) . (2)
The limiting case in which the second membrane represents a rigid surface or wall with κ2 = ∞ [23, 24],
is not included here since (2) reduces to κ = κ1 in this limit. Withim the rigidity-dominated regime, we
have ignored the effective surface tension.
To take into acconnt the presence of adhesion macromolecules between membrane pair, one must
introduce an extra Hamiltonian. In this investigation, however, the adhesion macromolecules are long-
flexible polymer chains attached by their two extremities to anchors segments. If we believe that each
polymer chains acts as a local perturbation of the geometrical properties of the two membranes, then, the
adopted form for the additional Hamiltonian is
Hint [l]
kBT
= −
M
∑
α=1
W (rα ) . (3)
Here, M denotes the total number of the anchored polymer chains and W (rα ) stands for a composite
field operator. The latter is a polynomial constructed with the derivatives of field l with respect to the
position vector. Of course, the form of this polynomial depends on the model containing connections via
long-flexible polymer chains.
With these considerations, the total Hamiltonian describing physics of two membranes linked together
reads
16

H [l] = H0 [l]+Hint [l] , (4)
where H0 [l] and Hint [l] are those contibutions given by relations (1) and (3), respectively. The various
contributions to the interaction potential will be defined below.
Now, to determine the free energy of the system, F, where the attached polymers positions, (r1,...,rM ),
are fixed in space, we shall need the expression of the physical quantities, in particular, the partition
function, Z . The latter is defined by the following functional integral
Z (r1,...,rM ) = Dle−H [l]/kBT
. (5)
The latter is performed over all possible cofigurations of l-field.
Let us introduce the following bulk expectation mean-value of an arbitrary functional X [l], calculated
with the bare Hamiltonian H0 [l],
X 0 =
1
Z0
DlX [l]e−H 0[l]/kBT
, (6)
with the partition function of the two membranes in the absence of adhesion macromolecules
Z0 = Dle−H 0[l]/kBT
. (6a)
In terms of this mean value, the averaged partition function may write
Z (r1,...,rM ) = Z0 e−H int [l]/kBT
0
= Z0 exp
M
∑
α=1
W (rα )
0
. (7)
We note that the bulk mean-value in the above expression can be computed using the standard cumulant
method usually encountered in Statistical Field Theory [19, 20], which is based on the approximative
formula
eX
0
= e X 0+(1/2!)[ X2
0
− X 2
0]+...
, (8)
for any functional X [l] of the l-field. Applying this formula to the quantity ∑M
α=1 W (rα ) yields
Z (r1,...,rM ) = Z0
exp
M
∑
α=1
W (rα )+
1
2! ∑
α,β
W (rα )W rβ 0
+
1
3! ∑
α,β,γ
W (rα )W rβ W rγ 0
+... .
(9)
It will be convenient to rewrite the above partition on following form
17

Z (r1,...,rM ) = Z0
exp d2
rρ (r) W (r) 0 +
1
2
d2
r d2
r ρ (r)ρ r W (r)W r 0
+... ,
(10)
with the local anchored polymer chains density
ρ (r) =
M
∑
α=1
δ2 (r −rα ) . (10a)
Here, δ2 (−→r ) denotes the two dimentional Dirac-function. Then, the term W (r)W (r ) 0 accounts for
the connected two-point correlation function constructed with the field operator W (r) ,
W (r)W r 0
= W (r)W r 0
− W (r) 0 W r 0
. (10b)
In relation (10), only one and two-body interactions are taken into account. High order terms describing
the three-body interactions and more are then ignored.
To compute the desired free energy, we start from the standard formula FM = −kBT lnZ and find
FM (r1,...,rM ) = F0 +U (r1,...,rM )−TS , (11)
where F0 = −kBT lnZ0 represents the free effective energy of the two membranes free from anchors
segments, S the entropy we will specify below, and U is the contribution of the effective interactions
that can be written as
U (r1,...,rM )
kBT
= − d2
rρ (r) W (r) 0 −
1
2
d2
r d2
r ρ (r)ρ r W (r)W r 0
+... . (12)
Then, this formula is a combination of a one-body potential, W (r) 0, and a two-body one,
W (r)W (r ) 0. On the other hand, comparing the above formula with the general expression of the
p-body interaction potential,
U [ρ] = d2
rρ (r)U1 (r)−
1
2
d2
r d2
r ρ (r)ρ r U2 r,r +... , (13)
we get
Up r(0),...,r(p−1)
kBT
= − W r(0)
,...,W r(p−1)
0
. (14)
The latter indicates that the nature of the p-body interaction potential is determined according to the sign
of the expectation mean value of a product of p composition field operators W ,s.
In the present theoretical model, the two almost parallel membranes connected by polymeric linkers
can be viewed as a two dimensional mixture of phospholipids and anchors segments. The lipid molecules
(phospholipids) are free to move more rapidly in the plane of the membrane than the anchors segments,
18

which are big amphiphilic molecules. Therefore, the corresponding diffusion coefficients are not the
same, because the head groups of the anchors segments have a size larger than the head groups of the
phospholipids, and in addition, the chemical structure of anchors segments may not allow any rotation
at all, contrary to the phospholipids, which can exhibit free rotations around their principal axis. As a
consequence, we state that, even in the homogeneous state, there exists some regions within the two
membrane surfaces that are rich in anchored polymer chains. Under a sudden change of a suitable
parameter, such as temperature, pressure or membrane environment, one assists to the appearance of rafts
[25, 26]. The fact that the phospholipids and the anchors segments have not the same diffusion coefficients
means that the system is out equilibrium. Physically speaking, the mixture is not governed by an annealed
disorder but by a quenched one (the positions in space of the anchors segments are distributed at random).
The natural random variable to consider is the density fluctuation δρ (r) = ρ (r)−ρ0, where ρ (r) is
the local density, and ρ0 = M /∑ the mean adhesion macromolecules density where ∑ is the same lateral
area of the two planar membranes. Then, we shall need to precise how this random variable is distributed.
According to the known central limit theorem, these random density fluctuations around the mean-value
ρ0 can be assumed to be governed by a Gaussian distribution, that is
[δρ (r)] = P0 exp −
1
2ρ2
0
[δρ (r)]2
, (15)
with P0 the normalization constant. Then, the first and second moments of this distribution are as follows
δρ (r) = 0 , (16)
δρ (r)δρ (r ) = ρ0δ r −r . (17)
Therefore, we are concerned with a non-correlated disorder. This means that a change of the density
fluctuation at some point r does not affect its value at another point r of the medium.
As we shall see, if we restrict ourselves to the second-order virial expansion, with respect to the mean
density ρ0, the results are identical to those obtained with a uniform distribution (if the contributions
linear in ρ0 are ignored). Of course, beyond this second order, physics is sensitive to the particular choice
of the random distribution.
Since the disorder is quenched, the physical quantity formulated in (13) is rather the average interactions
energy. Then we have
U [ρ] d2rρ (r)U1 (r)+ 1
2 d2r d2r ρ (r)ρ (r )U2 (|r −r |)+...
= ρ0 d2rU1 (r)+ ρ0 ∑
2 U2 (0)+ 1
2 ρ2
0 ∑ d2rU2 (r)+... .
(18)
This implies that the average of the logarithm of the partition function, lnZ = −FM /kBT, writes as
FM [ρ] = F0 +U [ρ]−TS . (19)
In the following section, we shall be concerned by the computation of the mixing free energy of the
considered mixture.
19

3. MIXING FREE ENERGY
As different components (phospholipids and anchors segments) of the mixture are free to move in the
plane of the two membranes, lateral phase separation phenomenon is constantly observed in two separate
lipids membranes. This is generally accompanied under certain conditions by the appearance of rafts
[25, 26], which are small domains rich in anchored biopolymer chains.
The purpose is precisely to study the significant effects of the structure (length) of anchored long-
flexible polymers (cases of monodisperse and polydisperse systems), the solvent quality (the presence of
a good solvent or a theta solvent) and the undulations of two coupled membranes on the critical phase
behavior of such a phase transition.
To this end, we need the expression of the mixing free energy (per site), F [ϕ], that can be obtained from
σFM [ρ]/∑ defined in Eq. (19), by ignoring the constant F0 and terms proportional to density ρ0. Here,
σ = πd2/4 denotes the common anchors segments (discs) area. In order to precise the form of F [ϕ],
we first introduce the dimensionless anchors segments density (volume fraction of anchors segments),
ϕ = ρ0 × σ, which is assumed to be the same for the two adhering membranes. If we admit that the
interface presents as two-dimensional (2D) Flory-Huggins lattice [27, 28], where each site is occupied by
an anchors segments or it is empty, then, in the canonical ensemble, the fraction ϕ can be regarded as
the probability that a given site is occupied by an anchor segment. Of course, the probability to have an
empty site is 1−ϕ.
In term of these considerations, the desired mixing free energy (per site), can be written as a sum of
several terms detailed below.
The first contribution is simply the mixing entropy (per site) describing all possible rearrangements of
the attached chains in the polymer layer [29], that is
Smix = S (ϕ)−ϕS (1)−(1−ϕ)S (0) = −kB (ϕ lnϕ +(1−ϕ)ln(1−ϕ)) . (20)
We note that, for polymer chains attached by their two extremities to big amphiphilic molecules (anchors
segments), the first term of the right-hand side, ϕ lnϕ, should be divided by some factor q = A/a, where
a and A account for areas lipid molecules and anchors segments, respectively. In the present case, we
have q = 1. Also the above expression is independent from any structural details of the polymer chains
but it depends on the relative molecular weight. In addition, only the number of a sites occupied on lattice
is important and not to know if the polymer is flexible or if the monomers have a particular geometry.
Next, we consider the interaction energy (per site) coming from the undulations of two interacting
membranes. Really, the induced attractive forces due to the undulations of the membrane pair that are
responsible for the condensation of anchors segments. These forces balance the repulsive ones between
monomers along the connected polymer chains. Then, the energy of mixing, which is a function of the
full internal energy U (ϕ) = U [ρ] (with ρ = ϕσ−1), writes as
Fmix [ϕ] =
σ
∑
[U (ϕ)−ϕU (1)−(1−ϕ)U (0)] . (21)
Here, ∑/σ is the total number of sites. In the above formula, U (1) is the internal energy of the lattice
where all sites are occupied by inclusions. There, the substraction is interesting, because it eliminates
trivial contributions, such as ϕ-independents terms (constants), or linear in ϕ. With these considerations,
the above energy reads
Fmix
kBT
= X ϕ (1−ϕ) , (22)
20

with the standard Flory interaction parameter
X = X0 +X1 +X2 . (23)
The first term, X0, is the interaction parameter describing the chemical segregation between amphiphile
molecules that are phospholipids and anchors segments. The permanent diffusion of the amphiphile
molecules provokes thermal fluctuations of the two bilayer membranes. This means that these latter
experience fluctuations around an equilibrium plane. Therefore, the presence of adhesion macromolecules
has tendency to increase or to supress the shape-fluctuations amplitude of the two parallel membranes a
mean-distance apart. This depends, in particular, on the molecular-weight of connected macromolecules.
Then, the contribution of the mean-separation between the two fluctuating membranes on the standard
Flory interaction parameter is caracterized by the second term X1, in the above equality. From general
relation (14), we obtain the following expression
X1 = 1
2σkBT d2r W (0)W (r) 0
=M 2K2
2σkBT d2r l2 (0)l2 (r) 0
,
(24)
where K denotes the common elastic constant of M vertical polymer chains. Using the Wick theorem
[26, 27] yields
X1 =
M 2K2
σkBT
d2
r[ l (0)l (r) 0]2
. (25)
Explicity, we have
X1 =
M 2K2
σkBT
d2q
( ˆκq4)2
, (26)
where the prime indicates the low-momentum cutoff qmin ∼ 1/ξ (the effects of the high-momentum
cutoff qmax will be ignored). By some straightforward algebra, the interaction parameter arising from the
mean-separation between the two membranes, scales as
X1 ∼ κ−2
. (27)
Come back to Eq. (23) defining the standard Flory interaction parameter, the last term, X2, accounts for
the interaction parameter due to the modulations of the two membranes. Its expression may write
X2 = −
1
2σkBT
d2
rU2 (r) . (28)
For isotropic anchors segments, using the general relation (14), we recover the effective pair-potential
expression [17–28, 30]
U2 (r) =
∞ , r <d
−AH
d
r
4
, r >d
(29)
where r the distance between a pair of anchors segments, d their common hard disc diameter, which is
proportional to the square root of the anchors segments area qa. There, the potential amplitude AH > 0
plays the role of the Hamaker constant. It was found that the latter decays with the bending rigidity
constant according to [17]
AH ∼ κ−2
. (30)
21

For this pair potential, the attraction parameter, X2, can be derived by integrating the equation (28). We
then obtain
X2 =
AH
3kBT
∼ κ−2
. (31)
We note that, it is easy to see that both interaction parameters X1 and X2 defined in Eqs. (27) and (31)
varie as the inverse of the squared common bending rigidity constant of the two undulating membranes.
But these latter are also sensitive to temperature. This behaviour indicates that the effective attraction
phenomenon between species of the same chemical nature is relevant only for those biomembranes of
small bending rigidity constant. Return to the bending rigidity constant κ, and notice that it is of the
order of few kBT. This quantity may be estimated measuring the membrane fluctuations amplitude. On
the other hand, the positivity of X1 and X2 tell us that the undulations of the two coupled membranes
and their mean-separation increase the chemical segregation between unlike species, and then, the phase
separation is accentuated by these effects.
Now, we will include the influence of the solvent quality and the length of polymeric linkers on the phase
separation. Therefore, the volume free energy (per site) owing to the excluded-volume forces between
monomers belonging to the anchored polymer chains [31], formally defined by
Fvol
kBT
∼
N
q
b2
qa
ν−1
ζ (N)ϕν
, (32)
where b represents the monomer size and qa the anchors segments area (a is that of the phospholipids
polar-heads). For monodisperse polymer layers, X2 denotes the common polymerization degree of
anchored polymer chains. But for polydispersed ones, N is rather the polymerization degree of the longest
anchored chains. There, the exponent ν depends on the solvent quality [32]. When the anchoring is
accomplished in a dilute solution with a good solvent or a θ-solvent, its values of are ν = 11/6 or ν = 2,
respectively.
The coefficient ζ (N) can be writes explicitly as
ζ (N) = 1 , (monodisperse systems) , (33)
ζ (N) ∼ Nν/(1−ν)
, (polydisperse systems) . (34)
It is easy to see that ζ (N) is a decreasing function of N in the case of polydisperse systems. This means
that ζ (N) < 1, since, in all cases, ν > 1. Thus, the excludedvolume effect becomes less important for
polydisperse polymer chains, in comparison with the monodisperse ones, where all chains have the same
polymerization degree N.
The total mixing free energy (per site) considered in our model is the sum of (20), (22), and (32).
Explicitly, we have
Ftot
kBT
= ϕ lnϕ +(1−ϕ)ln(1−ϕ)+
N
q
b2
qa
ν−1
ζ (N)ϕν
+X ϕ (1−ϕ) . (35)
In the absence of both additives positives coefficients originating from the undulations and the mean-
spacing of two interacting membranes, the free energy of the grafted layer is naturally expressed as
follows
F0
kBT
= ϕ lnϕ +(1−ϕ)ln(1−ϕ)+
N
q
b2
qa
ν−1
ζ (N)ϕν
+X0ϕ (1−ϕ) . (36)
22

If we admit that the volume free energy has a slight dependence on temperature, we will draw the phase
diagram in the plane of variables (ϕ,X ). Indeed, all of the temperature dependence is contained in the
Flory interaction parameter X .
4. PHASE DIAGRM
With the help of the above mixing free energy, we can determine the shape of the phase diagram
in the (ϕ,χ)-plane. We restrict ourselves to the spinodal curve, only. Along this curve, the thermal
compressibility become infinite. The spinodal curve equation can be obtained by equating to zero the
second derivative of the mixing free energy with respect to composition ϕ; that is, ∂2Ftot/∂ϕ2 = 0. Then,
we obtain the following expression for the critical Flory interaction parameter
Xc (ϕ) = −(X1 +X2)+
1
2
ν (ν −1)
N
q
b2
qa
ν−1
ζ (N)ϕν−2
+
1
ϕ
+
1
1−ϕ
. (37)
Above this critical interaction parameter appear two phases: one is homogeneous and the other is separated.
Of course, the linear term in ϕ appearing in equality (33) does not contribute to the critical parameter
expression.
We remark that, in usual solvents, this critical interaction parameter is increased due to the presence of
(two or three-body) repulsive interactions between monomers belonging to the anchored polymer layer.
This means that these interactions widen the compatibility domain, and then, the separation transition
appears at low temperature.
Now, to see the influence of the solvent quality, we rewrite the coefficient ζ (N) as ζ (N) ∼ Nν/(1−ν) < 1,
where the particular value ζ (N) = 1 corresponds to a monodisperse system. This mean that ζ (N) is a
decreasing function of N. Thus, the polydispersity of loops has a tendency to reduce the compatibility
domain in comparison with the monodisperse case.
The critical volume fraction, ϕc, can be obtained by minimizing the critical parameter Xc (ϕ) with
respect to the ϕ-variable. We then obtain
ν (ν −1)(ν −2)
N
q
b2
qa
ν−1
ζ (N)ϕν−3
c −
1
ϕ2
c
+
1
(1−ϕc)2
= 0 . (38)
For good solvents (ν = 11/6), we have
55
216
N
q
b2
qa
5/6
ζ (N)ϕ
−7/6
c −
1
ϕ2
c
+
1
(1−ϕc)2
= 0 . (39)
Therefore, the critical volume fraction is the abscissa of the the intersection point of the curve of
the equation (2x−1)/x5/6 (1−x)2
and the horizontal straight line of the equation y = (55/216)z(q),
withz(q) ∼ q−11/6.
Notice that this critical volume fraction is unique, at fixed value of the areas ratio q, and in addition, it
must be greater than the value 1/2 (for mathematical compatibility). The coordinates of the critical point
from which the system splits up into two phase (a dispersed phase and a dense one) are (ϕc,Xc), where
ϕc solves the above equation and Xc = Xc (ϕc). The latter can be determined of the equations (37). For
23

theta solvents (ν = 2), incidentally, the critical volume fraction is independent on the areas ration q, and
the coordinates of the critical point are as follows
ϕc =
1
2
, Xc = −2κ−2
+2+z(q) , (40)
with
z(q) ∼
1
q2
, (41)
And κ is the common bending rigidity constant of the two membranes.
The above relation clearly shows that the polymer chains condensation rapidly takes place only when
the areas ratio q is high enough. The same tendency is also seen in the case of good solvents.
It is straightforward to show that the critical fraction and the critical parameter are shifted to lower
values in the case of polydisperse systems, whatever be the quality of the surrounding solvent.
In Figure 2, we depict a suprposition of curves representing the critical parameter Xc, versus the
volume fraction of anchors ϕ, with and without thermal fluctuations of the two coupled membranes.
We present, in Figure 3, the spinodal curve for monodisperse and polydisperse systems, with a fixed
parameter N. We have chosen the good solvents situation. For theta solvents, the same tendency is seen.
In Figure 4, we present the spinodal curve for a polydisperse system, at various values of the parameter
N. As expected the critical parameter is shifted to higher values when we augmented the typical
polymerization degree.
Finally, we compare, in Figure 5, the spinodal curves for a polydisperse system for the case of theta
solvents and those for the case of good solvents, at fixed parameters N. All the curves in the figure reflect
our discussions made above.
Figure 2. Schematic phase diagrams showing the variations of the critical parameter Xc versus the volume fraction
of anchors ϕ on two adhering membranes (in a good solvent), with (solid line) and without thermal fluctu-
ations (dotted line). For these curves, we have chosen typical values for the parameters N, q, X1 and X 2.
24

Figure 3. Spinodal curves (in a good solvent) for monodisperse (dashed line) and polydisperse (solid line) systems,
when N = 100, with the typical values for the parameters q, X1 and X 2.
Figure 4. Spinodal curves for a polydisperse system, when N = 50 (solid line), 100 (dashed line) and 150 (dotted
line), with typical values for the parameters q, X1 and X 2. We assumed that the surrounding liquid is a
good solvent. For theta solvents, the tendency is the same.
5. DISCUSSION AND CONCLUSIONS
In this paper, the interplay between thermal ﬂuctuations and lateral phase separation of multi-component
membranes including lipids and adhesion macromolecules is investigated. We have considered the adhe-
sion macromolecules as long-ﬂexible polymer chains which are anchored reversibly by two extremities
25

Figure 5. Superposition of spinodal curves for a polydisperse system for the case of theta solvents (solid line) on
those for the case of good solvents (dashed line). For these curves, we chose N = 100 and with typical
values for the parameters q, X1 and X 2.
on the inner monolayer of the two fluctuating fluid membranes by big amphiphilic molecules termed
anchors-segments. However, we have viewed this adhesion as a two dimensional mixture of phospholipids
and anchors segments. We have also assumed that these different components of the mixture are free to
move in the plane of the two membranes. This is the so-called lateral diffusion [33]. In fact, the lipid
molecules and anchors segments undergo a phase separation driving the mixture from a homogeneous
phase to two separated ones, under a variation of a suitable parameter, e.g. the absolute temperature and
the membrane environment. This is one of the main consequences of our model.
To achieve the investigation of how this separation transition occurs, we have exactly determined
the phase diagram and critical properties of the anchored polymer chains. To do calculations, we first
computed the expression for the phenomenological mixing free energy by adopting the Flory-Huggins
lattice image usually encountered in polymer physics [27, 28]. Such an expression shows that there is
competition between four contributions: entropy, chemical mismatch between unlike species, interaction
energy between monomers belonging to the anchored layer and the interaction energy induced by the
undulations and the mean-spacing between two adhering membranes. Such a competition governs the
phases succession. From this mixing free energy, we drawn the complete shape of the phase diagram.
Recall that we have described the phase diagram in terms of the volume fraction of inclusions ϕ, using
a lattice image. But, this phase diagram may also be drawn considering the inclusion density ρ0. In
conclusion, we can state that the choice of ϕ or ρ0, as description parameters, has no consequence on the
phase diagram architecture.
As we have seen, the main conclusion is that, the phase behavior is essentially controlled by the
interaction parameter segregations which is increased by the both additives terms X1 and X2, scaling
as κ−2. This means that the phase separation is accentuated due to the presence of thermal fluctuations
and mean-separation between the two coupled membranes. We note that the quality of solvent (selective,
good or theta-solvent) where the two lipids membranes are trapped has some relevance for physics of the
26

phase separation. In addition, the length of adhesion macromolecules also affects this phase transition.
Finally, in this study, the problem was examined from a static (phase diagrams point of view. A natural
question to ask would be the extension of the present work, in a straightforward way, to kinetics (relaxation
in time) of phase behavior related to the lateral phase separation.
ACKNOWLEDGMENTS
We are indebted to Professors T Bickel, J-F Joanny and C Marques for helpful discussions during
the First International Workshop on Soft-Condensed Matter Physics and Biological Systems (14–17
November 2006, Marrakech, Morocco). MB thanks Professor C Misbah for fruitful correspondence
and the Laboratoire de Spectroscopie Physique (Joseph Fourier University of Grenoble) for their kind
hospitality during his visit.
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