Dr Khalid elhasnaoui

SOP TRANSACTIONS ON THEORETICAL PHYSICSISSN(Print): 2372-2487 ISSN(Online): 2372-2495

DOI: 10.15764/TPHY.2014.04002Volume 1, Number 4, December 2014

SOP TRANSACTIONS ON THEORETICAL PHYSICS

Lateral Phase Separation betweenPhospholipids and AdhesionMacromolecules on Two AdheringMembranesT. 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, Casablanca2 ENSAM, Moulay Ismail University P.O.Box 25290, Al Mansour, Meknes3 CRMEF, 298 Avenue Al Alaouiyine, P.O Box 24000, EL Jadida, (Morocco)

*Corresponding author: [email protected]; [email protected]

Abstract:We study analytically the lateral phase separation produced between phospholipids and adhesionmacromolecules on two adhering membranes from a static point of view. In this letter, theadhesion macromolecules are assumed as long-flexible polymer chains anchored by twoextremities 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 canbe driven from a dispersed phase to a dense one, under the variation of a suitable parametersuch 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 freeenergy. From this, we extract the complete shape of the associated phase diagram in thecomposition-temperature plane. The essential conclusion is that the anchored polymer chainsexperience the significant attractive forces directly result from the shape deformations of twoparallel 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 interestbecause 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 vander 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 flatmembranes [3] or adhesion of vesicles to surfaces [4].

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In addition to general non-specific interactions mentioned above, it is known from the works of Belland coworkers [5, 6] as well as others [7], that highly specific molecular interactions play an essentialrole in biological adhesion. This interaction acts between complementary pairs of proteins such asligand and receptor, or antibody and antigen. A well-studied example of such coupled systems is thebiotin-avidin complex. The avidin molecule has four biotin binding sites, two on each side, and formsa five-molecule biotin-avidin-biotin complex. The resulting specific interaction is highly local andshort-ranged. Measurements by surface force apparatus [8] or atomic force microscopy [9, 10] haveshown that the force required to break a biotin-avidin bond is about 170pN. In related experimentsmeasuring chemical equilibrium constants [11], it was found that the biotin-avidin binding energy is about30− 35kBT is larger than thermal fluctuations. Other coupled systems are those of selectins and theirsugar 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 havebeen proposed. Zuckerman and Bruinsma [12, 13] used a statistical mechanics model which is mappedonto a two-dimensional Coulomb plasma with attractive interactions. They predicted an enhancement ofthe membrane adhesion due to thermal fluctuations. In another work, Lipowsky considered the adhesionof 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 certainthreshold.

Currently, the adhesion of two adjacent plasma membranes is provided by bound pairs of such adhesionmacromolecules which form bridges between the membranes. We distinguish three types of adhesiondepending on the structure of bridges : i) Bolaform-sticker adhesion where each bridge molecule consistsof a single sticker having two sticky ends. One sticker end is anchored to one membrane while the otherend is adhering directly to the second membrane. ii) Homophilic-sticker adhesion where the bridges areformed by two stickers of the same type. Each sticker is anchored on one of the membranes, while theirfree ends bind together to form the bridge. iii) Lock-andkey adhesion where the bridges consist of twodifferent stickers forming a ligand-receptor type bond. This case represents an asymmetric adhesion dueto the lack of symmetry between the ligand and receptor.

A model system for these complex interactions is provided by systems containing lipid bilayers andpolymers. From the physical point of view, polymers can be characterized by several length scales. Firstof all, they have a certain length, Nb, where N and b are the number of monomers and the length of thesemonomers, 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 orexceeds its total length Nb; in this case, the polymer behaves as a worm-like chain which exhibits anaverage direction. On the other hand, if Nb� ξp, polymers crumple or fold up in order to increase theirconfigurational 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 Floryestimate ε ' 3/5 (in 3-dimensional systems). In bad solvents, the polymers collapse and become denselypacked 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 hasbeen observed by several experiments. For example, the biotin-avidin interaction occurring duringvesicle-vesicle adhesion was investigated by a micropipette technique [16]. The adhesion between oneavidin-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 bea diffusion-controlled process.

It is generally believed that multi-component biomembranes in physiological conditions are close totheir 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 criticalpoint may affect biophysical properties of membranes and can be of importance in regulating membraneprocesses.

In this work we provide a general phenomenological approach for the adhesion of multi-componentmembranes. Using a mean-field theory, we investigate how the lateral phase separation is drasticallyaffected by the significant influences of the structure (length) of adhesion macromolecules (cases ofmonodisperse and polydisperse systems), the solvent quality (the presence of a good solvent or a thetasolvent) and of the undulations of two coupled membranes separated by a mean-spacing. To model theadhesion between membranes, we follow a simplified view considering the adhesion macromolecules aslong-flexible polymer chains anchored reversibly into the hydrophobic interior of the two parallel lipidsbilayers by big amphiphilic molecules termed anchors-segments (see Figure 1). In addition, for the sakeof 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 canbe neglected (rigid polymer chains). An important consequence of our model is that the lateral phaseseparation is enhanced.

The scheme of this paper is as follows. In a heuristic way in the next section, we present a theoreticalformalism allowing a general expression of the induced interactions between anchors segments. Wecompute, in section 3, the expression for the mixing free energy of the phospholipids-anchors segmentsmixture. To investigate the phase diagram shape is the aim of section 4. Finally, we draw our conclusionsin 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 twofluctuating fluid membranes via big amphiphilic molecules termed anchors-segments. These latter areassumed to be of different chemical nature as the host lipid molecules forming the two membranes andmust 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 isperpendicular to the two membrane surfaces. We denote by−→rα the position-vector of anchoring extremitiesof chain α on the two fluid membranes. We suppose that the grafted chains are in the presence of good ortheta solvents. It is then natural to regard the anchors segments as soft inclusions permanently movingon the two membrane surfaces. In particular, these experience an interaction force mediated by theundulations 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 inducedinteractions. As we will see below, such an expression can be derived taking advantage of some methodbased on field theory techniques [19, 20]. We shall use the notation −→r = (x, y) ∈ R2 to mean the positionof the representative point on the two surfaces. The position of the two (almost flat) membrane surfaceslabeled by i = 1, 2 are specified through the displacement field, h1 (x, y) and h2 (x, y) of the upper andlower 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 bythe configurational energy (or effective Hamiltonian) [3, 21, 22]

H0 [l]kBT

=κ

2

∫d2r (∆l)2 , (1)

With kB is the Boltzmann constant, T the absolute temperature and the rescaling parameter κ = κ/kBT ,where κ is the common bending rigidity constant of the two membranes. But in the case of two bilayersof 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, wehave ignored the effective surface tension.

To take into acconnt the presence of adhesion macromolecules between membrane pair, one mustintroduce 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 eachpolymer chains acts as a local perturbation of the geometrical properties of the two membranes, then, theadopted 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 compositefield operator. The latter is a polynomial constructed with the derivatives of field l with respect to theposition vector. Of course, the form of this polynomial depends on the model containing connections vialong-flexible polymer chains.

With these considerations, the total Hamiltonian describing physics of two membranes linked togetherreads

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 variouscontributions 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 partitionfunction, Z . The latter is defined by the following functional integral

Z (r1, ...,rM ) =∫

D le−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], calculatedwith the bare Hamiltonian H0 [l],

〈X〉0 =1

Z0

∫D lX [l]e−H 0[l]/kBT , (6)

with the partition function of the two membranes in the absence of adhesion macromolecules

Z0 =∫

D le−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 cumulantmethod usually encountered in Statistical Field Theory [19, 20], which is based on the approximativeformula

⟨eX⟩

0 = e〈X〉0+(1/2!)[〈X2〉0−〈X〉20]+... , (8)

for any functional X [l] of the l-field. Applying this formula to the quantity ∑Mα=1 W (rα) yields

Z (r1, ...,rM ) = Z0

exp

{M

∑α=1

W (rα)+12! ∑

α,β

⟨W (rα)W

(rβ

)⟩0 +

13! ∑

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{∫

d2rρ (r)〈W (r)〉0 +12

∫d2r

∫d2r′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 describingthe 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 anchorssegments, S the entropy we will specify below, and U is the contribution of the effective interactionsthat can be written as

U (r1, ...,rM )

kBT=−

∫d2rρ (r)〈W (r)〉0−

12

∫d2r

∫d2r′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 thep-body interaction potential,

U [ρ] =∫

d2rρ (r)U1 (r)−12

∫d2r

∫d2r′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 signof 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 linkerscan 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 thesame, because the head groups of the anchors segments have a size larger than the head groups of thephospholipids, and in addition, the chemical structure of anchors segments may not allow any rotationat all, contrary to the phospholipids, which can exhibit free rotations around their principal axis. As aconsequence, we state that, even in the homogeneous state, there exists some regions within the twomembrane surfaces that are rich in anchored polymer chains. Under a sudden change of a suitableparameter, 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 coefficientsmeans that the system is out equilibrium. Physically speaking, the mixture is not governed by an annealeddisorder 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) isthe local density, and ρ0 = M /∑ the mean adhesion macromolecules density where ∑ is the same lateralarea 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-valueρ0 can be assumed to be governed by a Gaussian distribution, that is

[δρ (r)] = P0 exp{− 1

2ρ20[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 densityfluctuation 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 meandensity ρ0, the results are identical to those obtained with a uniform distribution (if the contributionslinear in ρ0 are ignored). Of course, beyond this second order, physics is sensitive to the particular choiceof the random distribution.

Since the disorder is quenched, the physical quantity formulated in (13) is rather the average interactionsenergy. Then we have

U [ρ]'∫

d2rρ (r)U1 (r)+ 12∫

d2r∫

d2r′ρ (r)ρ (r′)U2 (|r− r′|)+ ...

= ρ0∫

d2rU1 (r)+ρ0 ∑

2 U2 (0)+ 12 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 theconsidered mixture.

19


3. MIXING FREE ENERGY

As different components (phospholipids and anchors segments) of the mixture are free to move in theplane of the two membranes, lateral phase separation phenomenon is constantly observed in two separatelipids 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 ofa good solvent or a theta solvent) and the undulations of two coupled membranes on the critical phasebehavior 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 fromσFM [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 theinterface presents as two-dimensional (2D) Flory-Huggins lattice [27, 28], where each site is occupied byan anchors segments or it is empty, then, in the canonical ensemble, the fraction ϕ can be regarded asthe probability that a given site is occupied by an anchor segment. Of course, the probability to have anempty site is 1−ϕ .

In term of these considerations, the desired mixing free energy (per site), can be written as a sum ofseveral terms detailed below.

The first contribution is simply the mixing entropy (per site) describing all possible rearrangements ofthe 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 (anchorssegments), the first term of the right-hand side, ϕ lnϕ , should be divided by some factor q = A/a, wherea and A account for areas lipid molecules and anchors segments, respectively. In the present case, wehave q = 1. Also the above expression is independent from any structural details of the polymer chainsbut it depends on the relative molecular weight. In addition, only the number of a sites occupied on latticeis 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 interactingmembranes. Really, the induced attractive forces due to the undulations of the membrane pair that areresponsible for the condensation of anchors segments. These forces balance the repulsive ones betweenmonomers along the connected polymer chains. Then, the energy of mixing, which is a function of thefull 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 latticewhere all sites are occupied by inclusions. There, the substraction is interesting, because it eliminatestrivial 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)

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with the standard Flory interaction parameter

X = X0 +X1 +X2 . (23)

The first term, X0, is the interaction parameter describing the chemical segregation between amphiphilemolecules that are phospholipids and anchors segments. The permanent diffusion of the amphiphilemolecules provokes thermal fluctuations of the two bilayer membranes. This means that these latterexperience fluctuations around an equilibrium plane. Therefore, the presence of adhesion macromoleculeshas tendency to increase or to supress the shape-fluctuations amplitude of the two parallel membranes amean-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 standardFlory interaction parameter is caracterized by the second term X1, in the above equality. From generalrelation (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

∫ ′d2r [〈l (0) l (r)〉0]

2 . (25)

Explicity, we have

X1 =M 2K2

σkBT

∫ ′ d2q

(κq4)2 , (26)

where the prime indicates the low-momentum cutoff qmin ∼ 1/ξ‖ (the effects of the high-momentumcutoff qmax will be ignored). By some straightforward algebra, the interaction parameter arising from themean-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 forthe interaction parameter due to the modulations of the two membranes. Its expression may write

X2 =−1

2σkBT

∫d2rU2 (r) . (28)

For isotropic anchors segments, using the general relation (14), we recover the effective pair-potentialexpression [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 isproportional to the square root of the anchors segments area qa. There, the potential amplitude AH > 0plays the role of the Hamaker constant. It was found that the latter decays with the bending rigidityconstant according to [17]

AH ∼ κ−2 . (30)

21


For this pair potential, the attraction parameter, X2, can be derived by integrating the equation (28). Wethen 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 attractionphenomenon between species of the same chemical nature is relevant only for those biomembranes ofsmall bending rigidity constant. Return to the bending rigidity constant κ , and notice that it is of theorder of few kBT . This quantity may be estimated measuring the membrane fluctuations amplitude. Onthe other hand, the positivity of X1 and X2 tell us that the undulations of the two coupled membranesand their mean-separation increase the chemical segregation between unlike species, and then, the phaseseparation is accentuated by these effects.

Now, we will include the influence of the solvent quality and the length of polymeric linkers on the phaseseparation. Therefore, the volume free energy (per site) owing to the excluded-volume forces betweenmonomers 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 phospholipidspolar-heads). For monodisperse polymer layers, X2 denotes the common polymerization degree ofanchored polymer chains. But for polydispersed ones, N is rather the polymerization degree of the longestanchored chains. There, the exponent ν depends on the solvent quality [32]. When the anchoring isaccomplished 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 meansthat ζ (N)< 1, since, in all cases, ν > 1. Thus, the excludedvolume effect becomes less important forpolydisperse polymer chains, in comparison with the monodisperse ones, where all chains have the samepolymerization 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)+

Nq

(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 asfollows

F0

kBT= ϕ lnϕ +(1−ϕ) ln(1−ϕ)+

Nq

(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 phasediagram in the plane of variables (ϕ,X ). Indeed, all of the temperature dependence is contained in theFlory interaction parameter X .

4. PHASE DIAGRM

With the help of the above mixing free energy, we can determine the shape of the phase diagramin the (ϕ ,χ)-plane. We restrict ourselves to the spinodal curve, only. Along this curve, the thermalcompressibility become infinite. The spinodal curve equation can be obtained by equating to zero thesecond 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)+12

[ν (ν−1)

Nq

(b2

qa

)ν−1

ζ (N)ϕν−2 +

1ϕ+

11−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 parameterexpression.

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 transitionappears 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 adecreasing function of N. Thus, the polydispersity of loops has a tendency to reduce the compatibilitydomain in comparison with the monodisperse case.

The critical volume fraction, ϕc, can be obtained by minimizing the critical parameter Xc (ϕ) withrespect to the ϕ-variable. We then obtain

ν (ν−1)(ν−2)Nq

(b2

qa

)ν−1

ζ (N)ϕν−3c − 1

ϕ2c+

1

(1−ϕc)2 = 0 . (38)

For good solvents (ν = 11/6), we have

55216

Nq

(b2

qa

)5/6

ζ (N)ϕ−7/6c − 1

ϕ2c+

1

(1−ϕc)2 = 0 . (39)

Therefore, the critical volume fraction is the abscissa of the the intersection point of the curve ofthe 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, itmust be greater than the value 1/2 (for mathematical compatibility). The coordinates of the critical pointfrom which the system splits up into two phase (a dispersed phase and a dense one) are (ϕc,Xc), whereϕc 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, andthe coordinates of the critical point are as follows

ϕc =12, Xc =−2κ

−2 +2+ z(q) , (40)

withz(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 whenthe 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 lowervalues 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 thevolume 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 fixedparameter 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 parameterN. As expected the critical parameter is shifted to higher values when we augmented the typicalpolymerization degree.

Finally, we compare, in Figure 5, the spinodal curves for a polydisperse system for the case of thetasolvents and those for the case of good solvents, at fixed parameters N. All the curves in the figure reflectour discussions made above.

Figure 2. Schematic phase diagrams showing the variations of the critical parameter Xc versus the volume fractionof 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.

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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 (dottedline), with typical values for the parameters q, X1 and X 2. We assumed that the surrounding liquid is agood 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-componentmembranes 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

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Figure 5. Superposition of spinodal curves for a polydisperse system for the case of theta solvents (solid line) onthose for the case of good solvents (dashed line). For these curves, we chose N = 100 and with typicalvalues for the parameters q, X1 and X 2.

on the inner monolayer of the two fluctuating fluid membranes by big amphiphilic molecules termedanchors-segments. However, we have viewed this adhesion as a two dimensional mixture of phospholipidsand anchors segments. We have also assumed that these different components of the mixture are free tomove in the plane of the two membranes. This is the so-called lateral diffusion [33]. In fact, the lipidmolecules and anchors segments undergo a phase separation driving the mixture from a homogeneousphase to two separated ones, under a variation of a suitable parameter, e.g. the absolute temperature andthe 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 determinedthe phase diagram and critical properties of the anchored polymer chains. To do calculations, we firstcomputed the expression for the phenomenological mixing free energy by adopting the Flory-Hugginslattice image usually encountered in polymer physics [27, 28]. Such an expression shows that there iscompetition between four contributions: entropy, chemical mismatch between unlike species, interactionenergy between monomers belonging to the anchored layer and the interaction energy induced by theundulations and the mean-spacing between two adhering membranes. Such a competition governs thephases 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 , usinga lattice image. But, this phase diagram may also be drawn considering the inclusion density ρ0. Inconclusion, we can state that the choice of ϕ or ρ0, as description parameters, has no consequence on thephase diagram architecture.

As we have seen, the main conclusion is that, the phase behavior is essentially controlled by theinteraction parameter segregations which is increased by the both additives terms X1 and X2, scalingas κ−2. This means that the phase separation is accentuated due to the presence of thermal fluctuationsand 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

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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 naturalquestion to ask would be the extension of the present work, in a straightforward way, to kinetics (relaxationin 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 duringthe First International Workshop on Soft-Condensed Matter Physics and Biological Systems (14–17November 2006, Marrakech, Morocco). MB thanks Professor C Misbah for fruitful correspondenceand the Laboratoire de Spectroscopie Physique (Joseph Fourier University of Grenoble) for their kindhospitality during his visit.

References

[1] B. Alberts, D. Bray, J. Lewis, M. Raff, K. Roberts, and J. D. Watson, “Molecular Biology of theCell,” Garland, New York, pp. 139–194, 1944.

[2] E. Sackmann, “Membrane bending energy concept of vesicle-and cell-shapes and shape-transitions,”FEBS Letters, vol. 346, no. 1, pp. 3–16, 1994.

[3] R. Lipowsky and S. Leibler, “Unbinding transitions of interacting membranes,” Physical ReviewLetters, vol. 56, no. 23, p. 2541, 1986.

[4] U. Seifert and R. Lipowsky, “Adhesion of vesicles,” Physical Review A, vol. 42, no. 8, p. 4768, 1990.[5] G. I. Bell, “Models for the specific adhesion of cells to cells,” Science, vol. 200, no. 4342, pp. 618–

627, 1978.[6] G. I. Bell, M. Dembo, and P. Bongrand, “Cell adhesion. Competition between nonspecific repulsion

and specific bonding,” Biophysical Journal, vol. 45, no. 6, pp. 1051–1064, 1984.[7] E. A. Evans, “Detailed mechanics of membrane-membrane adhesion and separation. I. Continuum

of molecular cross-bridges,” Biophysical Journal, vol. 48, no. 1, pp. 175–183, 1985.[8] D. Leckband, J. Israelachvili, F. Schmitt, and W. Knoll, “Long-range attraction and molecular

rearrangements in receptor-ligand interactions,” Science, vol. 255, no. 5050, pp. 1419–1421, 1992.[9] E.-L. Florin, V. T. Moy, and H. E. Gaub, “Adhesion forces between individual ligand-receptor pairs,”

Science, vol. 264, no. 5157, pp. 415–417, 1994.[10] V. T. Moy, E.-L. Florin, and H. E. Gaub, “Intermolecular forces and energies between ligands and

receptors,” Science, vol. 266, no. 5183, pp. 257–259, 1994.[11] W. Muller, H. Ringsdorf, E. Rump, G. Wildburg, X. Zhang, L. Angermaier, W. Knoll, M. Liley,

and J. Spinke, “Attempts to mimic docking processes of the immune system: recognition-inducedformation of protein multilayers,” Science, vol. 262, no. 5140, pp. 1706–1708, 1993.

[12] D. Zuckerman and R. Bruinsma, “Statistical mechanics of membrane adhesion by reversible molecu-lar bonds,” Physical Review Letters, vol. 74, no. 19, p. 3900, 1995.

[13] D. M. Zuckerman and R. F. Bruinsma, “Vesicle-vesicle adhesion by mobile lock-and-key molecules:Debye-Huckel theory and Monte Carlo simulation,” Physical Review E, vol. 57, no. 1, p. 964, 1998.

[14] R. Lipowsky, “Adhesion of membranes via anchored stickers,” Physical Review Letters, vol. 77,no. 8, p. 1652, 1996.

27


[15] R. Lipowsky, “Flexible membranes with anchored polymers,” Colloids and Surfaces A: Physico-chemical and Engineering Aspects, vol. 128, no. 1, pp. 255–264, 1997.

[16] D. A. Noppl-Simson and D. Needham, “Avidin-biotin interactions at vesicle surfaces: adsorptionand binding, cross-bridge formation, and lateral interactions,” Biophysical Journal, vol. 70, no. 3,pp. 1391–1401, 1996.

[17] V. Marchenko and C. Misbah, “Elastic interaction of point defects on biological membranes,” TheEuropean Physical Journal E: Soft Matter and Biological Physics, vol. 8, no. 5, pp. 477–484, 2002.

[18] M. Goulian, R. Bruinsma, and P. Pincus, “Long-range forces in heterogeneous fluid membranes,”EPL (Europhysics Letters), vol. 22, no. 2, p. 145, 1993.

[19] C. Itzykson and J.-M. Drouffe, Statistical Field Theory: Volume 2, Strong Coupling, Monte CarloMethods, Conformal Field Theory and Random Systems. Cambridge University Press, 1989.

[20] J. Zinn-Justin, “Quantum Field Theory and Critical Phenomena,” tech. rep., 1989.[21] R. Lipowsky, “Lines of renormalization group fixed points for fluid and crystalline membranes,”

EPL(Europhysics Letters), vol. 7, no. 3, p. 255, 1988.[22] R. Lipowsky, “Europj1ys. lett. 7 (1988) 255; lipowsky, r,” Phys. Rev. Lett, vol. 62, p. 704, 1989.[23] T. El Hafi, M. Benhamou, K. Elhasnaoui, and H. Kaidi, “FLUCTUATION SPECTRA OF SUP-

PORTED MEMBRANES VIA LONG-FLEXIBLE POLYMERS,” International Journal of Aca-demic Research, vol. 5, no. 3, p. 5, 2013.

[24] T. R. Weikl, “Dynamic phase separation of fluid membranes with rigid inclusions,” Physical ReviewE, vol. 66, no. 6, p. 061915, 2002.

[25] K. Simons, E. Ikonen, et al., “Functional rafts in cell membranes,” Nature, vol. 387, no. 6633,pp. 569–572, 1997.

[26] R. G. Anderson and K. Jacobson, “A role for lipid shells in targeting proteins to caveolae, rafts, andother lipid domains,” Science, vol. 296, no. 5574, pp. 1821–1825, 2002.

[27] P. J. Flory, Principles of Polymer Chemistry. Cornell University Press, 1953.[28] P.-G. De Gennes, Scaling Concept in Polymer Physics. Cornell university press, 1979.[29] M. Benhamou, F. Elhajjaji, K. Elhasnaoui, and A. Derouiche, “Colloidal Aggregation in Critical

Crosslinked Polymer Blends,” Chinese Journal of Physics, vol. 51, no. 4, pp. 700–717, 2013.[30] M. Benhamou, I. Joudar, and H. Kaidi, “Phase separation between phospholipids and grafted polymer

chains onto a fluctuating membrane,” The European Physical Journal E: Soft Matter and BiologicalPhysics, vol. 24, no. 4, pp. 343–351, 2007.

[31] M. Benhamou, I. Joudar, H. Kaidi, K. Elhasnaoui, H. Ridouane, and H. Qamar, “An extended studyof the phase separation between phospholipids and grafted polymers on a bilayer biomembrane,”Physica Scripta, vol. 83, no. 6, p. 065801, 2011.

[32] M. Manghi and M. Aubouy, “Validity of the scaling functional approach for polymer interfaces as avariational theory,” Physical Review E, vol. 68, no. 4, p. 041802, 2003.

[33] P. F. Almeida, W. L. Vaz, and T. Thompson, “Lateral diffusion and percolation in two-phase, two-component lipid bilayers. Topology of the solid-phase domains in-plane and across the lipid bilayer,”Biochemistry, vol. 31, no. 31, pp. 7198–7210, 1992.

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