1 Introduction
The adhesion of biomimetic membranes and vesicles is governed by various generic and specific interactions [1]. The simplest systems are provided by lipid bilayers which contain only one or a few lipid components and which have a laterally uniform composition. The generic inter- actions between one such membrane and another surface (or between two such membranes) can be of enthalpic or entropic origin. The enthalpic contribution arises from the intermolec- ular forces such as hydration, van der Waals, and elec- trostatic forces. This contribution, called the direct mem- brane interaction, can be measured if the membrane is essentially flat or planar. Experimentally, a flat state can be prepared by immobilizing the membrane on a solid sub- strate. Theoretically, this state corresponds to the limit of a large membrane rigidity. In aqueous solution, bilayer membranes are often quite flexible and then exhibit thermally-excited undulations which act to renormalize their direct interaction. If the direct interaction is purely repulsive, the undulations lead to a free-energy contribution which can be interpreted as an entropic or fluctuation-induced force as proposed in reference [2]. If the direct interaction contains an attrac- tive potential well, the renormalized interaction leads to an unbinding transition as predicted in reference [3]. Biomembranes contain a large number of different lipids and anchored macromolecules. The attractive forces between two membranes are usually mediated by receptors or adhesion molecules which are anchored in the mem- branes [4–6]. These specific interactions govern the highly selective binding of cells which is essential for many bio- logical processes such as immune response or tissue devel- opment [7]. From a theoretical point of view, the adhesive behavior of these rather complex biomembranes can be modelled, to a certain extent, by two-component mem- branes containing "generic" lipids and anchored stickers [8,9]. The interplay of generic and specific interactions has also been investigated experimentally. Adhesion-induced lateral phase separation into domains with small and large membrane separations has been found to occur in sev- eral biomimetic systems. The formation of blisters has been observed in membranes containing cationic lipids in contact with a negatively charged surface [10], and be- tween membranes containing both negatively and posi- tively charged lipids [11]. The coexistence of tight and weak adhesion states has been found for membranes with biotinylated lipids bound to another biotinylated sur- face via streptavidin [12], membranes with homophilic csA-receptors from the slime mold Dictyostelium dis- coideum [13], and membranes containing specific ligands of integrin molecules adsorbed on a substrate [14]. At- tractive membrane-mediated interactions between bound csA-receptors of adhering vesicles have been inferred from membrane tension jumps induced by the micropipet as- piration technique [15]. In addition to the receptors, the membranes studied in references [12–15] also contain re- pulsive lipopolymers to prevent non-specific adhesion. These observations indicate the existence of possible phys- ical mechanisms for the aggregation of receptors in biolog- ical membranes which has been found during cell adhesion and results in the formation of focal contacts [7]. In this article, we present a detailed study of one possi- ble mechanism for adhesion-induced phase separation con- cerning membranes with generic repulsive and specific at- tractive interactions. For these membranes, tracing over the sticker degrees of freedom in the partition function leads to an effective trans-membrane interaction which exhibits a potential barrier. Such trans-membrane inter- actions have been previously studied in references [16, 17]. The phase separation results from the line tension between bound and unbound membrane regions due to this potential barrier. Membrane fluctuations reduce the barrier and, therefore, decrease the tendency for phase separation. The mechanism is thus clearly distinct from entropic, fluctuation-induced mechanisms for adhesion- induced phase separation as discussed in references [9, 18]. Similar mechanisms for phase separation due to an effec- tive potential barrier have also been studied recently in references [9, 19, 20]. The mechanism studied here is rather general as will be demonstrated by considering two different cases. In the first case, we assume that the generic trans-membrane in- teraction is repulsive for short separations and attractive for large separations. The generic trans-interaction is then approximated by a harmonic potential centered at the po- tential minimum at l = l 0 . The specific trans-interaction is expanded up to linear order in l − l 0 , and can be thought to arise from restoring forces of extensible sticker molecules which are irreversibly bound to the membrane and the substrate. In the absence of membrane shape fluctuations, the lateral phase behavior can be determined exactly. Us- ing Monte Carlo simulations, we furthermore show that the membrane fluctuations reduce the potential barrier and the tendency for lateral phase separation. In the second case, we consider a generic trans- membrane repulsion which is modeled as a square-barrier potential. In addition, the trans-interaction of the stickers is modeled as a square-well potential [8,9,18]. The stick- ers are bound for small separations from the substrate, and unbound otherwise. The square-well potential is a simple model for short-ranged lock-and-key interactions of ligands and receptors. After partial summation over the sticker degrees of freedom, we find again an effective potential barrier if the generic repulsion is longer ranged than the specific attraction of the stickers. As in the first case, this barrier leads to lateral phase separation, and is effectively reduced by membrane fluctuations.
2 The model
A systematic description of a biomimetic membrane with stickers in contact with a substrate or a second membrane has to include a field l(x,y) for the local separation of the membrane(s) and a concentration field n(x,y) of the stickers above a position ( x, y ) on a reference plane. In the following, we apply a theoretical framework which has been introduced in reference [8] and extended in refer- ences [9,18]. Within this framework, the membrane is di- vided into small patches with a linear size a which corre- sponds to the smallest possible wavelength for bending de- formations. According to computer simulations for molec- ular membranes, this size is about 6 nm for lipid bilay- ers with a thickness of about 4 nm [21]. For a membrane which is on average parallel to a planar substrate, this leads to an effective discretization of the reference plane into a two-dimensional square lattice with lattice param- eter a . The sticker positions are described by occupation numbers n i = 0 or 1 where n i = 1 indicates the presence of a sticker at lattice site i, and the local separation is given by l i ≥ 0 [8], see Figure 1. An alternative continu- ous Ginzburg-Landau theory for the sticker concentration field was used in reference [19]. In terms of these variables, the grand canonical Hamil- tonian has the general form
where the elastic term
represents the discretized bending energy of the membrane with bending rigidity κ , and the discretized Laplacian △ d is given by
The term ( ∆ d l i )² corresponds to the leading-order expres- sion for the mean curvature squared of a membrane with vanishing spontaneous curvature [22,23]. The second term of the Hamiltonian (1) contains i) the generic interac- tion potential Vg(l) between the membrane and substrate and ii) the specific adhesion potential V s ( l ) of the stick- ers which only contributes at lattice sites with n i = 1, i.e. at lattice sites where stickers are present. The rela- tive chemical potential of the stickers is denoted by μ . The same description holds for a multicomponent mem- brane in contact with a second, homogeneous membrane. In the latter case, the effective bending rigidity κ is given by κ 1 κ 2 /( κ 1 + κ 2 ), where κ 1 and κ 2 denote the bending rigidities of the two membranes [24]. Note that Vg(l) and V s (l) are the interaction energies per membrane patch where each patch has area a 2 . Thus, the interaction energies per unit area are given by Vg(l)/a2 and V s ( l )/a 2 , respectively. This differs from the conven- tion in reference [8], where the interaction potentials were defined as energies per unit area. In the following, we neglect direct interactions between pairs of stickers which can be described by quadratic terms in the concentration field n [9,18]. The Hamiltonian (1) is then linear in n , and the sticker degrees of freedom in the partition function
can be partially summed or traced over exactly, leading to
with the effective potential
where T denotes the temperature in energy units ( i.e. , the Boltzmann constant kB is absorbed into the symbol T ). The partial summation over the sticker degrees of freedom {n} thus leads to an equivalent problem of a laterally ho- mogeneous membrane with the effective potential (6).
3 Linear sticker potential
In this section, we consider a generic potential Vg between the membrane and the substrate which has a relatively deep minimum at a certain separation lo from the sub- strate. Such a potential arises, e.g. , for electrically neutral surfaces interacting via strong van der Waals forces. Using a Taylor expansion around the minimum, we approximate this generic potential by the harmonic potential
where v2 = a2(d2Vg/dl2)|r0. In addition to this generic potential, we consider ex- tensible sticker molecules which are irreversibly bound to both the substrate and the membrane, and which have an unstretched extension small compared to l 0 . We will fur- ther assume that the corresponding sticker potential Vs(l) has an essentially constant gradient for those values of l for which we can use the harmonic approximation (7) for the generic potential. In such a situation, we may truncate the expansion of the sticker potential in powers of l — lo and use
with α ≡ α∂V ε (l) /∂l | ι 。 > 0. In order to simplify the notation and to reduce the number of parameters, it is convenient to introduce the rescaled variables
The Hamiltonian (1) with the generic potential (7) and the specific potential (8) can then be written as
in terms of the discrete lattice variables h i , n i , and the dimensionless parameters (10)
and the effective potential (6) has the form
Direct inspection of these equations shows that the system considered here depends on three dimensionless parame- ters: i) the reduced coupling constant ã of the specific potential, ii) the reduced (and shifted) chemical potential jj, and iii) the ratio κ/ν 2 of the bending rigidity κ and the strength v2 of the generic harmonic potential (7).
3.1 Limit of rigid membranes
For large values of the ratio κ/υ 2 , thermally excited shape fluctuations of the membrane can be neglected. The free energy F = −(T/A)ln Z per area A is then given by Vcf/a2. The phase behavior is determined by the mini- mization of the effective potential (12):
First-order phase transitions are found when different minima of V ef coexist. For the effective potential (12), the discussion of the phase behavior is simplified if this potential is expressed in terms of the shifted separation field
For the special line in the ( j , ã ) parameter space as given
As one varies the parameter ã while keeping ji = ji*(ã), the effective potential given by (16) undergoes a continuous bifurcation at the critical value α = α c = 2, see Figure 2. For ã < ã c and ã > ã c , this potential has a single minimum at z = 0 and two degenerate minima at finite values of z , respectively. The critical value ac = 2 of the bifurcation point can be directly inferred from the second derivative of (16) as given by
For z = 0, this expression is equal to 1 − ã²/4 which van- ishes for ã = ã c = 2. In the limit of rigid membranes as considered here, one can ignore the effect of membrane fluctuations and the bifurcation point of the effective po- tential is identical with the critical point of the system which thus lies at ã c = 2 and jjc = − ã c 2 /2 = −2. Thus, for fi = fix(ä) and ä > 2, the effective potential (12) is a symmetric double-well potential with two degen- erate minima. As soon as the chemical potential j  deviates from its coexistence value j = j * , this symmetry is bro- ken and the effective potential exhibits a unique global minimum. Therefore, the system undergoes a discontinu- ous transition as one changes the chemical potential from µ = µ* — є to µ = µ* + є for µ > 2, where є denotes a small chemical potential difference. The positions, say zo, of the extrema of the effective potential are determined by dVcf(z)/dz = 0. Along the coexistence line given by j = ji* = − a 2 /2, this leads to the transcendental equation
This equation has the trivial solution z 0 = 0 for all values of ã which corresponds to a minimum and maximum for ā < ā c = 2 and ā > ā c = 2, respectively. For ā > ā c = 2, equation (18) has two additional solutions corresponding to the two degenerate minima of the effective potential Vef, see Figure 2. For rigid membranes with large κ/ν 2 , the sticker con- centration X ≡ {n,}/a2 is given by
with y 0 ≡ z 0 − α /2 which denotes the position of the min- ima of the effective potential (12). Along the coexistence line with µ = µ ⋆ = − a 2 /2, this expression simplifies and becomes
Inserting the numerically determined solutions of the transcendental equation (18) into (20) leads to the con- centrations of the coexisting phases which determine the phase diagram shown in Figure 3. Inside the shaded two- phase region, a sticker-poor phase characterized by a rela- tively large separation yo of the membrane from the sub- strate coexists with a sticker-rich phase with a relatively small value of yo.
3.2 Flexible membranes
For a fluctuating membrane in a symmetric double-well potential, first-order transitions only exist if the barrier between the two potential wells exceeds a certain critical height [16, 17]. For barrier heights below this critical value, the fluctuating membrane "tunnels" through the barrier and there is no phase transition. Therefore, the critical point of a flexible membrane in the double-well potential (16) will be characterized by reduced coupling constants âc = âc(k/v2) which exceed the bifurcation value âc = 2 as obtained for rigid membranes in the limit of large κ/ν 2 . Using Monte Carlo simulations, the critical point can be determined, for a fixed value of κ/ν 2 , from an evalua- tion of the moments
where
is the spatially averaged order parameter, and 〈・・〉 de- notes averages over all membrane configurations [17,25]. For ã > ã c and correlation lengths ξ which are much smaller than the linear size L of the finite membrane, the moments reach the values C 2 = 1 and C 4 = 1, whereas for 0 < α < α c and ξ ≪ L, we have C 2 = π/2 ≈ 1.57 and C 4 = 3. For L ≪ ξ on the other hand, the moments C 2 and C 4 vary only weakly with the linear size L . The criti- cal value ã c of the reduced coupling constant can then be estimated from the common intersection point of C 2 and C 4 , respectively, as a function of α for several values of L [17, 25], see Figure 4. In Figure 5, we display the obtained values for the critical rescaled coupling constant ã c as a function of the reduced rigidity κ/ν 2 . For large κ/ν 2 , α c approaches the limiting value ã c = 2 which is appropriate for rigid membranes as discussed in the previous section. As one decreases κ/ν 2 , the membrane fluctuations become more pronounced and act to increase the value of ãc. This im- plies that the wells of the effective potential (16) have a finite depth as one reaches the critical point of the system. Lateral phase separation occurs for coupling constants â > âª. In either of the two phases, the entire membrane is located around one of the minima in the effective poten- tial. In the sticker-poor phase, the membrane is found in the minimum with larger separation y 0 from the substrate. This minimum is dominated by the generic membrane po- tential and corresponds to a state of weak adhesion. In the sticker-rich phase, the membrane fluctuates around the minimum with smaller separation y 0 corresponding to a state of tight adhesion. In contrast, there is only a single phase for coupling constants ã < ã c . For example, for 2 < ã < ã c the two minima of the effective poten- tial are both populated by many different segments of the fluctuating membrane which is able to cross the potential barrier between the minima.
4 Square-well sticker potential
Let us now turn to stickers with a specific adhesion po- tential
where θ(x) is the Heaviside function θ(x) = 0 for x < 0 and θ(x) = 1 for x ≥ 0. The parameter U has the same dimension as the membrane potential Vs and represents the interaction energy per sticker. Stickers which interact via the square-well poten- tial (23) can attain two different states: A bound state with binding energy U < 0 if the local separation l be- tween the membrane and the substrate is smaller than the potential range l v , and an unbound state otherwise. Be- cause the fluctuating membrane cannot penetrate the sub- strate surface, the membrane separations l are restricted to non-negative values. In contrast to the linear sticker potential of the previous section with the specific poten- tial (8), the stickers characterized by the interaction po- tential (23) have a fixed length and cannot be stretched. Thus, the square-well potential as given by (23) provides a simple model for short-range interactions arising, e.g., from specific ligand/receptor lock-and-key interactions or from screened electrostatic forces for charged stickers in contact with an oppositely charged substrate. The phase behavior of multicomponent membranes containing stickers which interact via the square-well po- tential as given in (23) has been studied previously for the case in which one can ignore generic interactions with the substrate. The membrane was found to undergo lateral phase separation even for purely repulsive cis-interactions between the stickers if these stickers have an increased lateral size [18] or a larger bending rigidity than the lipid matrix [9]. The phase separation is then driven by the shape fluctuations of the membrane. Here, we consider the interplay of the specific sticker potential (23) with a generic repulsive trans-interaction between the membrane and the substrate. If the range of these generic interactions is smaller than the potential range l v of the stickers, the bound state of the stickers is more restricted, but the general entropic phase behavior described above will not be affected. However, for repul- sive generic interactions with a potential range which ex- ceeds l v , a different scenario is possible. For simplicity, we consider here a generic potential of the form
with a barrier height U ba > 0 and range l ba > l v . The effective potential (6) obtained after the summation of the sticker degrees of freedom is shown in Figure 6 and can be written as
and the constant term V 0 = − T ln (1 + e π / T ) which de- pends only on the reduced chemical potential μ/T . Since the sticker binding energy U is negative, we have U we < . Uba In the context of interacting membranes, an effec- tive potential of the form (25) was first studied in ref- erence [16]. More recently, such an interaction potential has also been derived for membranes which contain both stickers and repellers, i.e. non-adhesive molecules which protrude from the membrane surface [9]. The repellers have been modeled by a local square-barrier potential, the stickers by the local square-well potential (23). The generic square-barrier potential (24) thus affects the phase behav- ior in a similar way as repellers with a local square-barrier potential. As previously discussed, the membrane unbinds at a certain critical depth U we of the attractive potential well which is given by [16]
where c is a dimensionless coefficient, because the excess free energy for a membrane confined to a potential well with depth U we is V fi (l v ) ～ T 2 / Kl 2 , and the free-energy difference between the bound and the unbound state of the membrane reads △F = − | Uwe | /α 2 + cT 2 /κ | 2 . The charac- ter of the unbinding transition depends on the height of the potential barrier which induces a line tension between bound and unbound membrane segments. On small length scales, this line tension can be estimated as Ucff L with ef- effective width L ～ ( lba − lv ) √ κ / T of the membrane strip crossing the barrier [16]. Taking also into account the line entropy on larger length scales, the unbinding transition is found to be discontinuous for relatively strong barriers with (28) and continuous for weak barriers with U ba ( l ba − l x ) 2 ≪ L |U* |l 2 . A discontinuous transition implies the coexistence of a bound phase with a high concentration of stickers and an unbound phase with a low sticker concentration. Therefore, sufficiently strong barriers also lead to lateral phase separation and sticker aggregation. In contrast to the entropic mechanisms mentioned above, the phase separation is caused by the line tension between bound and unbound membrane patches. This line tension is induced by the potential barrier. In order to un- derstand the influence of membrane fluctuations, we have to take into account that the transition value | U * e | of the contact energy increases with the temperature T and de- creases with the bending rigidity κ , see (27). As implied by (28), membranes with more pronounced shape fluctu- ations require larger potential barriers for a discontinuous unbinding transition and lateral phase separation. As in the first case studied in the previous section, membrane fluctuations decrease the tendency of the membrane to phase-separate.
5 Conclusions
In this article, we have studied one possible mechanism for adhesion-induced phase separation of biomimetic mem- branes. The mechanism is applicable to membranes which experience both specific attractive and generic repulsive trans-interactions. The specific interactions are taken to arise from sticker molecules which are embedded in the membrane, while the generic repulsion may originate, e.g. , from electrostatic forces between similarly and ho- mogeneously charged membranes. The effective trans- interaction obtained by an explicit summation of the sticker degrees of freedom in the partition function is shown to exhibit a potential barrier. This barrier induces a line tension between bound and unbound membrane re- gions resulting in a coexistence of a sticker-rich phase, characterized by a small separation between membrane and substrate, and a sticker-poor phase, characterized by a larger membrane-substrate separation. Thermally excited shape fluctuations of the membrane are shown to decrease the tendency for lateral phase separation by reducing the potential barrier height. Two different cases have been considered here. In the first case, the generic trans-interaction between membrane and substrate is assumed to have a minimum at a finite membrane separation and is approximated by a harmonic potential centered at the minimum. The specific trans- interaction of the stickers is taken to depend linearly on membrane separation. This model introduced in refer- ence [19] has been studied using a mean-field approach. The adhesion was shown to shift the critical point of the lateral phase transition. In the present article, the con- centration field n of the stickers is described as a lattice gas variable on a discretized elastic membrane. The phase behavior is determined exactly in the absence of shape fluctuations of the membrane. Fluctuations of the mem- brane are subsequently taken into account by Monte Carlo simulations. In the second case, the specific sticker trans-interaction is modeled by a short-ranged square-well potential, and the generic trans-interaction between membrane and sub- strate is assumed to be a purely repulsive step func- tion. This second case turns out to be closely related to biomimetic membranes which contain stickers and re- pellers, i.e. repulsive molecules which protrude from the membrane surface [9], since the effective potential has the same functional form as in equation (25) and exhibits a potential barrier. As shown in references [16,17], the mem- brane fluctuations can "tunnel" through a relatively small barrier but are trapped by a relatively large one. The mechanism for adhesion-induced phase separation as studied in the present article is distinct from several entropic mechanisms which have been identified in pre- vious works [9,18]. Examples of these other mechanisms include stickers with an increased lateral size [18], stickers with an increased rigidity, and stickers with attractive cis- interactions which are renormalized by the shape fluctua- tions of the membrane [9]. These entropic mechanisms de- pend strongly on the rescaled potential range (l λ /α) λ κ / T of the stickers where l v is the range of the square-well po- tential (23), and a is the size of the membrane patches. Thus, for the entropic mechanisms, the tendency for lat- eral phase separation increases with decreasing potential range and/or increasing temperature. This is in contrast to the mechanism described in the present work which is governed by the potential barrier contained in the effec- tive interaction potential of the membrane. In this case, the shape fluctuations of the membrane reduce the po- tential barrier which implies that the tendency for lateral phase separation decreases with increasing temperature. Experimentally, the temperature-dependence of adhesion-induced phase separation has not yet been studied. The presence of repulsive lipopolymers in the biomimetic systems investigated in references [12–15] points towards a barrier-mechanism for adhesion-induced phase separation as emphasized in this article. Entropic mechanisms, on the other hand, might be relevant in the case of membranes containing oppositely charged lipids [10,11] which induce a tight membrane coupling and small membrane separations below 4 nm [11]. One of us (DA) would like to acknowledge partial support from the U.S.-Israel Binational Foundation (B.S.F.) under grant No. 98-00429 and the Israel Science Foundation, Centers of Excel- lence Program.