Prediction of the Dependence of the Line Tension on the Composition

Article pubs.acs.org/Langmuir

Prediction of the Dependence of the Line Tension on the Composition of Linactants and the Temperature in Phase Separated Membranes Benoit Palmieri,†,‡ Martin Grant,† and Samuel A. Safran*,‡ †

Department of Physics, McGill University, 3600 rue University, Montréal, Québec Canada H3A 2T8 Department of Materials and interfaces, Weizmann Institute of Science, Rehovot 76100, Israel

‡

S Supporting Information *

ABSTRACT: We calculate the line tension between domains in phase separated, ternary membranes that comprise line active molecules (linactants) that tend to increase the compatibility of the two phase separating species. The predicted line tension, which depends explicitly on the linactant composition and temperature, is shown to decrease significantly as the fraction of linactants in the membrane increases toward a Lifshitz point, above which the membrane phase separates into a modulated phase. We predict regimes of zero line tension at temperatures close to the mixing transition and clarify the two different ways in which the line tension can be reduced: (1) The linactants uniformly distribute in the system and reduce the compositional mismatch between the two bulk domains. (2) The linactants accumulate at the interface with a preferred orientation. Both of these mechanisms have been observed in recent experiments and simulations. The second one is unique to line active molecules, and our work shows that it is increasingly important at large fraction of linactants and is necessary for the emergence of a regime of zero line tension. The methodology is based on the ternary mixture model proposed by Palmieri and Safran [Palmieri, B.; Safran, S. A. Langmuir 2013, 29, 5246], and the line tension is calculated via variationally derived, self-consistent profiles for the local variation of composition and linactant orientation in the interface region. to GUVs, giant plasma membrane vesicles (GPMVs)20,34,35 are self-assembled from cell membrane extracts and, hence, from a composition point of view, are more closely related to real cells. Recent Förster resonance energy transfer (FRET) experiments on GUVs8,22,29,31 and small-angle neutron scattering (SANS) experiments on 60 nm diameter large unilamelar vesicles (LUVs)8,31 revealed the presence of equilibrium nanoscale domain (with length scale comparable to postulated rafts sizes) in model membranes in a specific range of lipid composition and temperature. In parallel with the experimental studies, several theoretical models have been proposed to explain the formation and stability of nanoscale domains in model membranes. These are based, for example, on membrane curvature,5,7,36−38 interleaflet couplings,6,7,39 height mismatch between domains,38,40,41 electrostatic interactions,42,43 or a combination of these effects.11 Our linactant based model extends the earlier work of Brewster et al.4 which focused on membranes comprising saturated lipids (which have no double bonds along their

1. INTRODUCTION The line tension between coexisting domains in monolayers and bilayers plays an important role in many applications1 such as surface patterning processes2 and lung surfactant models.3 It is also thought to be crucial to the formation of nanoscale lipid domains4−11 in mixed bilayers that are analogous to lipid rafts12−15 in real cells. Hence, understanding the physical interactions that control line tension is of primary importance. This work focuses on line active molecules (also called linactants16) which reduce the line tension between domains and therefore promote lateral heterogeneities in membranes. A survey of recent theoretical efforts to characterize the effects of line active molecules in membranes can be found in refs 17 and 18. The basic idea of our work applies, in general, to phase separation in 2D, but the model was first motivated by the lipid raft hypothesis12−15 which claims that some cell functions require the presence of nanoscale domains (of the order of 10− 100 nm19), rich in cholesterol and sphingomyelin, within the membrane bilayer. The hypothesis has further motivated numerous experiments on simpler model membranes. Example of such model systems are giant unilamelar vesicles (GUVs)8,20−32 and suspended lipid monolayers.33 In contrast © 2014 American Chemical Society

Received: June 15, 2014 Revised: August 26, 2014 Published: September 3, 2014 11734

dx.doi.org/10.1021/la502347a | Langmuir 2014, 30, 11734−11745

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linactant composition and temperature. This is due to the fact that our model is based on microscopic chain−chain interactions that are captured by only one phenomenological parameter which is fully determined by the mixing temperature. Second, the explicit dependence of the calculated line tension suggests that a regime of zero line tension arises close to, but below the mixing temperature (i.e. at temperatures that may be relevant to experiments on model membranes) when the fraction of linactants is large enough. More precisely, we predict a second regime of zero line tension in addition to the usual one that occurs when the mixing temperature coincides with the critical temperature. This latter prediction is analogous to our earlier studies of the uniform phase (above the mixing transition) where a large amount of linactants was required to significantly decrease (increase) characteristic fluctuation length9 (time10) scales. Another more technical but novel aspect of our work is the use of approximate and self-consistent profiles for the membrane composition and linactant orientations in the vicinity of the interface. Using a variational principle together with the analytic expression for the interface free energy that results from these profiles allows one (1) to calculate the line tension with a small numerical effort and (2) to obtain an analytical expression for the line tension valid close to the mixing transition. Moreover, the explicit dependence of the line tension predicted with our model allows for a direct comparison with recent experiments by Hassan-Zadeh et al.32 that focus on the specific case of mixtures of saturated and hybrid lipids (the linactant) with an without unsaturated lipids. Our findings support and clarify their experimental/molecular dynamics observations that attributed the observed line tension reduction to a “partitioning and loosening” ability of the hybrids. The analytical expression for the line tension we derive below clearly shows the two mechanisms of action by which the line active molecules reduce the line tension: (1) They “partition” equally in the two bulk phases and reduce the mismatch between their physical properties (“loosening”); and (2) they accumulate at the interface (at a higher local concentration than in the bulk) with a preferred orientation. In our language, line active molecules are those that use mechanism (2) to reduce the line tension. Nonlinactants can also reduce the line tension, but through mechanism (1) only. The paper is organized as follows. Section 2 reviews the phase behavior of our ternary mixture model and sets the stage for the calculation of the line tension. Section 3 contains the main results of the paper. The line tension is calculated by a variational approximation for the composition profile and linactant orientation in the vicinity of the interface. The main results are summarized in Figure 4 which predicts a temperature and linactant composition regime where the line tension is reduced to zero. It also highlights how this reduction is correlated with the local composition and orientation of the linactants at the interface. Section 4 compares our predictions to recent experiments and contains concluding remarks. In particular, it highlights the extra line tension reduction that arises due to the line activity on top of the “partitioning and loosening ability” of the linactant as described above.

hydrocarbon chains) and unsaturated lipids (which have double bonds along both hydrocarbon chains). The idea proposed by Brewster et al. is that a class of naturally occurring lipids act as linactants and thus promote lateral heterogeneities when added to the binary mixture membrane they considered. Previous linactant based theories (i.e., such as in ref 4) only treated the limit in which the fraction of these molecules is small. In this paper, we show that the reduction of the free energy of the interface (that separates bulk domains with different compositions) at temperatures close to the mixing temperature is much more effective for relatively large linactant concentrations. We thus have generalized the theory to treat arbitrary concentration of linactants and to include linactant− linactant interactions that depend on their orientation9 (in analogy to the interactions used in the lattice models proposed in refs 44−46). In general, the model describes the phase behavior of membranes comprising linactants for which the following conditions are satisfied: (1) The molecule that reduces the line tension behaves as a 2D amphiphile and has a preferred orientation at the domain boundaries. (2) The line active molecule has two regions which, respectively, interact favorably with one of the other two classes of molecules that tend to phase separate. In the context of model lipid membranes comprising saturated and unsaturated lipids, the line active molecule is a hybrid lipid which has one saturated and one unsaturated hydrocarbon chain. Hence, it can reduce the interfacial free energy between domains rich in saturated lipids and domains rich in unsaturated lipids when it resides at the interface with the right orientation (i.e., when its saturated [unsaturated] tail points toward the saturated [unsaturated] rich domain). In refs 9 and 10, the model was used to predict the effects of hybrid lipids on fluctuation domains length scales and lifetimes in mixed membranes. One of the goals was to connect with the experimental work of Veatch et al.34 who reported that composition fluctuations in GPMVs can be described by the scaling laws of the 2D Ising model and, hence, that real cell membrane composition is close to a critical point at room temperature (see refs 17 and 47 for a review of the phase behavior in membranes). Hence, in refs 9 and 10, we used our ternary membrane model with linactants to predict the sizes and lifetimes of fluctuation domains, which were shown to, respectively, decrease (down to the nanoscale)9 and increase (up to the milliseconds range)10 with increasing hybrid fractions. The effect of the line active molecules was significant at large fraction of linactants; this occurs when that fraction tends toward a Lifshitz point above which modulated composition fluctuations are predicted by the model. In this paper, we consider the phase-separated regime and predict the effects of the linactants on the free energy of the interface that separates equilibrium domains. We will show that the lowering of the line tension to very small values at temperatures near the mixing temperature is not consistent with the dilute limit used by Brewster et al.4 Indeed, ref 4 predicted that zero line tension between domains could only be achieved at low temperatures. Our more general model shows that, by increasing the amount of linactants, the regimes of zero line tension can be found at increasingly higher temperatures. Zero or very small line tension between phase separated domains in 2D and in the presence of linactants have been predicted in the past (see refs 4 and 48), but the new predictions of our model are the following. First, the line tension predicted with our model is an explicit function of

2. MODEL The model introduced in ref 9 is now used to study the phase separated (coexistence) regime and in particular, the interfacial properties, of a ternary mixture membrane model. The theory applies to systems comprising two species that phase separate 11735


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where ϕ describes local deviations (from the membrane average) of the saturated-unsaturated composition difference and ψ describes local deviations (from the membrane average) of the hybrid lipid composition. Note that eq 1 guarantees that ϕs + ϕu + ϕh = ϕ̅ s + ϕ̅ u + ϕ̅ h = 1 (the lipid composition is conserved). We will see later that this representation is convenient since ϕ is related to the order parameter which characterizes the membrane bulk phases. Positive (negative) values of ϕ characterize local regions rich in saturated (unsaturated) lipids and depleted in unsaturated (saturated) lipids. The hybrid lipids carry an extra orientation vector degree of freedom, σ⃗ = x̂σx + ŷσy (see Figure 1A). As shown in ref 9, in the mixed regime, the membrane has a uniform composition (0) and no preferred orientation of the hybrids (i.e., σ(0) x = σy = 0). In terms of these new variables, the membrane free energy can be written as

below a certain temperature plus an additional line active species. The model free energy is general but it was originally derived for the specific case of membranes comprising saturated, unsaturated and line active hybrid lipids. We thus continue the discussion using the language appropriate to this specific system. The membrane components are illustrated in Figure 1A. We recall that the model free energy was derived by

- = , − T:

(2)

where T is the temperature and , and : are the energetic and entropic contributions, respectively. The expression for the interaction energy is given by eq 15 in ref 9 and we reproduce it here, 2J dx dΔ⃗g (x , Δ⃗){4ϕ(x)ϕ(x + Δ⃗) a2 − 2Δ̂·σ (⃗ x)[ϕ(x + Δ⃗) − ϕ(x − Δ⃗)] − [Δ̂·σ (⃗ x)][Δ̂·σ (⃗ x + Δ⃗)] + [Δ̂ × σ (x)] ·[Δ̂ × σ (⃗ x + Δ⃗)]} ⃗

,=−

Figure 1. (A) Building blocks of the membrane (viewed from the top) composed of saturated lipids (ϕs, red circle), unsaturated lipids (ϕu, blue circle), and line active hybrid lipids (ϕh, red-blue semicircle). The hybrid lipid orientation is denoted by the vector σ⃗ that points in the direction of the unsaturated tail. (B) Examples of configurations that minimize each of the interaction energy terms in eq 3. Configurations I, II, III, and IV minimize the first, second, third, and fourth term on the right-hand-side of eq 3, respectively.

(3)

where 1 δ(Δ − a) g (x , Δ⃗) = 2πa

(4)

is a kernel that describes nearest-neighbors interactions. J in eq 3 is an interaction parameter and a in eqs 3 and 4 is a molecular size. Note that eq 3 is the most general form of a two body interaction (up to second order in the terms of composition and/or orientation deviations from their average values). On the other hand, the coefficients in front of each term are in general all different. As shown in ref 9, the advantage of considering the specific case of linactants (e.g., hybrids) that have one molecular group that is identical to one of the species (e.g., saturated chain) and another that is identical to the other species (e.g., unsaturated chain) is that these coefficients are expressed in terms of a single parameter (J). The physical meaning of all terms in eq 3 can be understood with four types of primitive interactions illustrated in Figure 1B. They are applicable to the general case of linactants, but we discuss them for the specific case of saturated/unsaturated/ hybrid mixtures. Briefly, the first term on the right-hand side of eq 3 favors large domains of saturated and unsaturated lipids. Example of local configurations that minimize that term are shown in Figure 1 B. I. The next term reduces the free energy of the interface when the hybrid orientation is aligned with the saturated/unsaturated composition gradient (Figure 1B. II). This term is largely responsible for the line activity of the hybrids. The last two terms favor hybrid orientations that change sign on the molecular scale, a, in the direction parallel to their orientation vector (see. Figure 1B. III) and that are uniform in the perpendicular direction (see Figure 1B, IV). These last two terms arise from chain−chain interactions between hybrid neighbors (these were neglected in ref 4) and

considering three types of nearest-neighbor chain−chain interactions: saturated−saturated, unsaturated−unsaturated, and saturated−unsaturated. In the general case, this is equivalent to the assumption that the line active molecule has one group (e.g., one chain) that is identical with one of the phase separating species and another group that is identical with the other phase separating species. In ref 9, we derived the free energy of the membrane in the single, mixed phase from a lattice model and showed how the resulting expression can be recast in a way that does not depend on the details of the underlying lattice. A summary of that procedure, applied to the phase-separated regime, is given in part A of the Appendix. The phase behavior of the membrane is strongly influenced by its composition, which is determined by the average membrane fraction of saturated (ϕ̅ s), unsaturated (ϕ̅ u), and hybrid lipids (ϕ̅ h). Throughout the paper, a bar on top of any symbol refers to quantities that are averaged over the entire membrane. We now focus on local deviations from these membrane averages and write them in terms of two new variables, ϕs − ϕs ̅ = ϕ − ψ ϕu − ϕu̅ = −ϕ − ψ ϕh − ϕh̅ = 2ψ

∫ ∫

(1) 11736


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Figure 2. (A) The dependence of the critical temperature (black curve) on the fraction of line active hybrids. The dotted part of the green line corresponds to the Lifshitz line (see eq 7 in the text). The vertical green line indicates a first order transition from a uniform phase (for small hybrid fraction) to a modulated (stripe) phase (for large hybrid fraction). The lines separate four distinct regions labeled by i, ii, iii, and iv that are shown in the pictures in the box to the right. (B) Same as panel (A), but the phase diagram is reported in terms of the reduced temperature, τ(ϕ̅ h), defined in eq 6. The two short-dashed horizontal lines indicate the value of ϕ̅ h and τ at which the wavenumbers of the most probable fluctuation mode (in the mixed regime) and the lowest free energy mode (in the phase separated regime) are, respectively, reported in panels (C) and (D) (refer to the text and the Supporting Information for more details). In the rectangular box: Snapshots of composition fluctuations in the mixed phase are shown at composition below (i) and above (ii) the Lifshitz line following a procedure explained in ref 9, and equilibrium configurations in the phase separated regime are shown at composition below (iii) and above (iv) the line of first order transitions (fluctuations about these equilibrium configuration was neglected, for simplicity). The four panels correspond to the four regions labeled i, ii, iii, and iv in (A) . Regions in red (blue) correspond to local domains enriched (depleted) in saturated lipids.

Positive and negative values for τ(ϕ̅ h), respectively, correspond to the single phase and to the phase separated regime. In the mixed regime (T > Tc(ϕ̅ h) or τ(ϕ̅ h) > 0), the minimum free energy state is uniform and hence ϕ = ψ = σx = σy = 0; there is no deviation in the lipid composition or in the hybrid orientation relative to their average values. Figure 2A and B (see the dashed green curve) and the Supporting Information further show the locus of a Lifshitz line (it extends the Lifshitz point at τ = 0 into the single phase) which is given by

stabilize the hybrids at the interface. The entropic contribution to the free energy is given in eq 22 in part A of the Appendix. The most important point is that local deviations of any degree of freedom from its membrane average costs entropy. Energetically, the saturated/unsaturated lipids and the hybrid lipids are only coupled through the hybrid orientation (i.e. their interactions depend on the hybrid composition, ψ, implicitly through the entropic terms). For more details on the derivation of eq 2 and on the physical meaning of all terms, please refer to ref 9. Throughout our analysis, we focus on the special case where ϕ̅ s = ϕ̅ u = (1 − ϕ̅ h)/2; the membrane has equal amounts of saturated and unsaturated lipids. This simplifies the analysis because, for ϕ̅ s = ϕ̅ u, the mean field solution of our model predicts that the mixing transition is second order,9 and hence, the mixing temperature is also a critical temperature that we denote by Tc(ϕ̅ h). For ϕ̅ h < 2/3, the model predicts a linear relationship between the critical temperature and the average hybrid lipid fraction, Tc(ϕh̅ ) = 4J(1 − ϕh̅ )

τL(ϕh̅ ) =

(5)

T − Tc(ϕh̅ ) Tc(ϕh̅ )

2ϕh̅ − 1

(7)

for 1/2 < ϕ̅ h ≤ 2/3. In the one phase regime (where τ > 0), this Lifshitz line separates regions of the phase diagram in the single phase where the most dominant fluctuation mode is uniform (for τ < τL and τ > 0) from regions where it is “stripelike” (for τ > τL and τ > 0). In the latter case, the modulation wavenumber increases with the hybrid fraction as shown in Figure 2C, i and ii. The phase separated regime, defined by T < Tc(ϕ̅ h) (τ(ϕ̅ h) < 0), can also be separated into two regions. In the first one, for small hybrid fractions, the phase separated domains are macroscopic and uniform (see Figure 2, iii). At large hybrid fraction, the membrane equilibrium structure is characterized by stripes with different compositions (see Figure 2, iv). The hybrid lipid fraction determines the modulation wavenumber (i.e., the stripe wavelength) as shown in Figure 2D. The uniform and modulated stripe phases are separated by a first order transition at ϕ̅ h ≈ 2/3. The fact that the transition is first order can be seen in Figure 2D which shows that the modulation wavenumber changes discontinuously from zero to a finite value near ϕ̅ h = 2/3. Also note that there is no longrange translational order in all phases shown in Figure 2 due to the Mermin−Wagner theorem.49,50 While the saturated-rich

For ϕ̅ h > 2/3, the critical temperature depends nonlinearly on the hybrid lipid fraction, as shown in Figure 2A and in Ref.9 When T > Tc(ϕ̅ h), the membrane is in the single phase, mixed regime, while for T < Tc(ϕ̅ h), the membrane is in the phaseseparated, coexistence, regime. The special composition ϕ̅ h = 2/3 corresponds to a Lifshitz point9,49 (the physical meaning of the Lifshitz point is explained in the next paragraph). Our results are conveniently expressed in terms of a reduced temperature,

τ(ϕh̅ ) =

1 − 3ϕh̅ /2

(6) 11737


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phases are denoted as “liquid ordered”,51,52 the correlations of the translational order in these phases are relatively shortranged. Parenthetically, we would also like to point out that similar phase diagrams are obtained in a variety of systems where interactions that favor long and short wavelength modulations are in competition. Examples are block copolymer systems53 and membranes where macroscopic phase separation competes with curvature54 and/or electrostatic interactions.55 The Lifshitz point at τ = 0 and ϕ̅ h = 2/3 occurs, as usual,49 at the intersection of a line of second order transitions and a line of first order transitions. Note that the experiment of Konyakhina et al.27,29 reported a transition from a uniform phase to a stripe phase in GUVs comprising hybrid lipids that follows the trend illustrated in Figure 2. However, in those experiments, the stripes had a macroscopic modulation wavelength (in contrast to ours which are predicted to occur at the nanoscale). Details on how the phase diagram in Figure 2 was calculated are given the Supporting Information. One important result obtained in refs 9 and 10 (and here shown in the Supporting Information) is that only local deviations in the difference of the saturated and unsaturated concentration, ϕ about ϕ = 0, are unstable at T = Tc(ϕ̅ h) for ϕ̅ h < 2/3 (whereas deviations in ψ and σ⃗ from their average values remain bounded). This instability in the saturated−unsaturated composition implies the coexistence of a saturated-rich phase and an unsaturated-rich phase that are separated by an interface. Hence, for T smaller but close enough to Tc(ϕ̅ h), the mean-field composition can be obtained by minimizing eq 2 as a function of ϕ (which we assume to be uniform at this point) and setting the other fields to zero. The result is ϕbulk = ±m0 /2

exponent predicted by the 2D Ising model (that includes fluctuations) and not with the mean-field exponent. Also, early 2D lattice models for microemulsions in the presence of a linactant have studied the roles of fluctuations.45,46,48 These studies, together with a recent one that focused on membrane phase behavior and established a parallel with lipid rafts,54 show that on top of the different scaling laws, fluctuations in 2D can significantly change the position of the transitions/Lifshitz lines in the phase diagram. However, our aim here is to report the explicit dependence of the line tension on the linactant composition when the mixing transition is approached with a particular focus on the regime where the line tension tends to zero. Hence, for simplicity, we neglect the effects of fluctuations and proceed with mean-field theory. Also note that one of the main assumptions of our model is that cholesterol (which is an important component of real cell membranes and a common constituent of model membranes) is only included implicitly. In fact, the phase transition of binary mixtures of saturated and unsaturated lipids (without cholesterol) has been studied experimentally,56 and coexistence between a liquid-ordered and gel phase was observed at low enough temperature. Such a transition is analogous to a solid− liquid transition and, hence, does not display a critical point. In that sense, our model applies to membranes that contain cholesterol and that display liquid−liquid coexistence. This point is discussed in more details in ref 17. We now move to the main focus of the paper, which are the properties (composition, hybrid orientation, and free energy) of the interface separating the two coexisting bulk phases: one saturated-rich and the other unsaturated-rich. The presence of the interface allows local variations of the hybrid composition and orientation (defined by ψ and σ⃗, which are both uniform in the bulk). In fact, the hybrid lipids can accumulate at the interface and their orientation can be correlated with the saturated/unsaturated composition gradient in the interface region. In the next section, we show that, at large hybrid fraction, the combination of these effects, together with the fact that the difference of the order parameter in the two bulk phases is reduced, significantly decreases the free energy associated with the interface. For the remainder of the paper, we will focus on the case ϕ̅ h ≤ 2/3. For larger hybrid fraction and τ(ϕ̅ h) < 0, the membrane is in the modulated stripe phase where the presence of the interface is not necessary to ensure that the membrane average of saturated/unsaturated lipids vanishes.

(8)

where m0 = (1 − ϕh̅ )[−3τ(ϕh̅ )]1/2 + O(τ 3/2)

(9)

Of course, the membrane average of ϕ must vanish since the volume fraction of each type of lipid is conserved. Hence, for T < Tc(ϕ̅ h) and ϕ̅ h < 2/3, the equilibrium state of the membrane consists of two macroscopic domains of equal sizes within which ϕ = ±m0/2. This mean-field result already has a strong implication for the free energy of the interface. The physical difference between the bulk phases, as determined by the order parameter m0, decreases as hybrid lipids are added to the membrane (in the bulk, our mean field solution of the model predicts that the linactive molecules are uniformly distributed with a random orientation). Note that this can occur without getting closer to the critical temperature (i.e., by keeping the reduced temperature, τ(ϕ̅ h), constant in eq 9). In other words, increasing the uniform background concentration of hybrids reduces the difference between the ratio of saturated to unsaturated tails in the two bulk domains. This is in agreement with recent experimental observations reported in ref 32 where the physical properties of the coexisting phases in membranes with more hybrids were shown to be more similar. On the other hand, this effect only depends on the properties of the bulk phases (and not of the interface) and hence is unrelated to line activity. We would like to re-emphasize that the above arguments ignore the role of fluctuations, which can be particularly important in 2D. Recall that Veatch et al.34 measured composition fluctuations in the mixed phase of GPMVs and showed that the correlation length scaled favorably with the

3. RESULTS: APPROXIMATE PROFILES Physical insight into the composition and free energy of the interface can be obtained from a variational calculation of the interface profile4,57 that uses simple functions to account for the spatial variations of the composition and orientation fields (ϕ, ψ, σ). We analyze the model in the region where the interfacial tension is positive, hence favoring two macroscopic domains with one interface. We then determine the conditions of temperature and linactant concentration for which the formation of the interface cost no energy (and, hence, where the formation of nanodomains may be facilitated). In these zero line tension regions, nanodomains are expected to spontaneously form. Since the membrane is rotationally invariant, we can set the interface to be perpendicular to the x-axis (i.e., along y), with no loss of generality. The simplification of our variational procedure is due to the fact that we use profiles for which the integrals appearing in eqs 3 and 22 can be done 11738


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its membrane average, ψ, over the whole membrane must vanish (the hybrids that accumulate at the interface must be taken from the bulk). This is taken into account by subtracting the following overall constant to ψ in eq 11,

analytically. Hence, we start with a simple linear approximation for the variation the saturated-unsaturated composition near the interface, m mx ϕ(x) = Θ( −x − λ) − Θ(x + λ) Θ(λ − x) 2 2λ m − Θ(x − λ) (10) 2 where m and λ are profile parameters that, respectively, describe the saturated−unsaturated composition in the bulk domains and the width of the interface and Θ is the Heaviside step function. This linear approximation works well to predict the composition profile and interfacial tension in the case of a simple interface in a binary mixture (in a binary mixture, the exact profile for the composition is described by a hyperbolic tangent57 to which a linear profile is a good approximation). Note that the form of the interfacial prof iles for the linactant concentration and orientation are not arbitrary. Rather, they are fixed in a self-consistent manner given the profile for ϕ. More precisely, to lowest order in ϕ and for long wavelength modulations, minimizing the free energy in the presence any profile for ϕ gives ψ ∝ −ϕ2 and σx ∝ −dϕ/dx. Given our linear ansatz for ϕ, we postulate the following profiles for ψ and σ:

C = 2ψ0λ /3Lx

where 2Lx is the membrane size along x. We show below that this term cannot be set to zero from the start, even if it is very small for large systems (i.e., when Lx → 0), since it will contribute to the free energy of the interface. The hybrid lipid orientation is not a conserved field, so its membrane average can have nonzero values. In the Supporting Information, we show how to calculate the membrane free energy from eqs 3 and 22 using the profiles given by eqs 10 and 11. We denote the result by -p which can be written as the sum of two terms that have a distinct physical meaning, -p = -p,bulk + -p,I

(13)

where the bulk contribution, -p,bulk (which scales with area of the membrane), is given by ⎡ ⎛ ⎤ ⎞ -p,bulk T m4T ⎥ − 4⎟⎟ + = Lx̃ ⎢m2⎜⎜ ⎢⎣ ⎝ J(1 − ϕh̅ ) JL̃y 6J(1 − ϕh̅ )3 ⎥⎦ ⎠

⎛ x2 ⎞ ψ (x) = ψ0⎜1 − 2 ⎟Θ(x + λ) Θ(λ − x) − C , ⎝ λ ⎠ σx(x) = σ0 Θ(x + λ) Θ(λ − x)

(12)

(14)

where L̃ y is the length of the system along y (i.e. the length of the interface). -p,I , the interfacial part of the free energy (which scales with the interface width), is a lengthy but simple function of temperature, hybrid fraction, and all variational parameters (m, λ̃, σ0, and ψ0) that we report in part B of the Appendix (see eq 23). Note that the tilde above any variable that has dimension of length means that it has been scaled by the molecular size (i.e., L̃ x = Lx/a). Next, we minimize -p with respect to the bulk saturated/ unsaturated lipid composition, m (the bulk order parameter), and the interfacial quantities λ̃, σ0, and ψ0. As expected, the bulk part of the free energy, -p,bulk , depends only on the bulk order parameter (and not on the variational parameters related to the interface). The lateral dimension of the membrane is much larger than the interface region (L̃ x ≫ λ̃). Hence, m can be approximately determined by minimizing -p,bulk alone. Doing this, we recover the result m = m0 that we obtained earlier without considering the interface (see eq 9). Including -p,I in the determination of m introduces small corrections of order 1/ L̃ x; m = m0 + m1/L̃ x + O(1/L̃ x2). It is important to note that these small corrections, when inserted back into eq 14, could in principle lead to terms of order O(1); comparable to those that appear in -p,I (and hence that could play a role in the determination of λ̃, σ0, and ψ0). On the other hand, corrections to this order vanish. This is seen by expanding -p,bulk in terms

(11)

where ψ0 and σ0, respectively, describe the accumulation and the orientation of the hybrid at the interface (recall that ψ0 = σ0 = 0 implies a uniform distribution of unoriented linactants). The profiles are illustrated in Figure 3. Note that σy vanishes by

Figure 3. Approximate profiles used in the variational calculation of the free energy and the line tension from eqs 3 and 22. The black, full line shows ϕ(x)/|ϕbulk| (recall that ϕbulk = ±m/2); the orange, dashed line shows σ(x)/σ0; and the green, dashed-dotted line shows ψ(x)/ψ0. Note that the x-axis, which gives the position in the direction perpendicular to the interface, has been scaled by λ. For T close to, but below Tc, the profiles for σ(x) and ψ(x) are self-consistently determined from the profile for φ(x) (as described in the main text). The profile for the hybrid composition has a small, negative (here exaggerated for clarity), contribution in the bulk that ensures that the membrane average composition is conserved (see eq 12).

of 1/L̃ x : -p,bulk = (-p,bulk)m = m0 Lx̃ + (d-p,bulk /dm)m = m0 m1. + O(1/Lx̃ ) Because m0 minimizes -p,bulk , (d-p,bulk /dm)m = m0 = 0 and the O(1) terms vanishes exactly. The higher order corrections to the bulk free energy, of the order 1/L̃ x, are negligible for the line tension calculation in the limit of large system size (L̃ x → ∞). Figure 4 shows the results of a numerical solution of the minimization of the interfacial part of the free energy with

symmetry and, for the same reason, the profiles depend only on x. The free energy will then be minimized with respect to the four variational profile parameters, m, λ, ψ0, and σ0, to determine their equilibrium values. The hybrid lipid composition is conserved. Hence, the integral of the deviation in the local hybrid composition from 11739


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Figure 4. Top panel: the interfacial free energy, γ (see eq 15), that results from a minimization of eq 23 with respect to the variational profiles parameters ψ0, σ0, and λ̃ (see eqs 10 and 11). The units are arbitrary. Regions of positive line tensions are in red. The region in blue is bounded by a contour where γ = 0 (Note that this contour should end at the Lifshitz point, ϕ̅ h = 2/3 and τ = 0. The fact that it appears to terminate at the point ϕ̅ h = 2/3 and τ ≈ −0.004 is due to mesh size and other numerical errors.). Inside the blue region, the assumptions we used to calculate the line tension break down (see text). The region in green indicates a regime of small and flat (small in magnitude and slowly varying with τ and ϕ̅ h) positive line tension. Bottom panels, from left to right: the values of the variational parameters λ̃, ψ0, and σ0 that minimize γ at τ(ϕ̅ h) = −0.005 (blue curve) and −0.2 (dashed black curve) as a function of the membrane composition. Note that, in all three panels, one of the two curves has been scaled up by a factor of 10.

respect to the profile parameters, λ̃, σ0, and ψ0. It predicts the dimensionless line tension γ = min(-p,I/JL̃y )

hence, the formation of multiple interfaces is favored. There, one of the main assumptions of our calculation, that one flat interface separates two bulk phases, breaks down (a modification of our calculation that includes the effect of having multiple interfaces separated by a finite distance and/or that incorporates other mechanisms, such as the one based on the bending energy of hybrids proposed in ref 58, may extend the range of validity of the theory to region where γ < 0 in Figure 4). Hence, for the remainder of the paper, we will focus on the regime γ ≥ 0 where our theory is valid. Although our model must be modified in order to predict the structure of the membrane in regimes where γ < 0, we can predict, as a function of linactant composition and temperature, where γ tends to zero and hence where the single interface in a phase separated membrane becomes unstable. For membranes with a hybrid lipid fraction that is close to the Lifshitz point, Figure 4 shows that zero line tension can even be achieved close to the critical temperature for phase separation, Tc(ϕ̅ h). This is only possible in our model that accounted for linactant−linactant interactions (and not in previous models appropriate to the limit of dilute linactants), since the zero line tension contour occurs at relatively large linactant concentrations. For many membranes comprising saturated/unsaturated and hybrid lipids, the mixing temper-

(15)

which is the free energy of the interface (divided by the interaction energy parameter and by the dimensionless interface length), as a function of the hybrid lipid fraction, ϕ̅ h, and the reduced temperature, τ(ϕ̅ h). Recall that the critical temperature lies on the line τ(ϕ̅ h) = 0 (where the ordinate is equal to zero). Note that decreasing τ(ϕ̅ h) means that the system is at lower temperatures, deeper in the phase-separated regime. For small hybrid fractions, the presence of the interface always results in a free energy cost (characterized by a positive value for the line tension, γ). This prediction agrees with the one made by Brewster et al.4 In contrast, the new feature of the predicted line tension can be seen in the region on the righthand side of the contour plot in Figure 4: there is a temperature−linactant composition contour line where γ = 0 and, hence, where the presence of the interface has no associated free energy cost. Note that the γ = 0 contour encloses a region where the calculated line tension is negative. In that region, the presence of the interface decreases the free energy of the system and 11740


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ature is close to room temperature.8,26,27,32,56,59 Hence, the fact that the predicted line tension vanishes close to Tc(ϕ̅ h) means that hybrid lipids may favor the formation of nanodomains in a regime which is relevant to experiments on model membranes. The bottom panels of Figure 4 show the value of the variational parameters obtained from the minimization of the interfacial free energy. As usual, the width of the interface diverges when T → Tc(ϕ̅ h) (in Figure 4, λ̃ is much larger for τ = −0.005 compared to τ = −0.2). In contrast, the last two bottom panels in Figure 4 show that the amplitude of the local orientation, σ0, and accumulation, ψ0, of the hybrids at the interface increase when the membrane is cooled deeper into the phase separated regime. In particular, the local orientation of the hybrids at the interface increases significantly as the hybrid fraction approaches the line of first order transitions (the black dashed line in the bottom right panel of Figure 4 increases abruptly when ϕ̅ h → 2/3). Recall that the line of first order transitions, which is shown as the green vertical line in Figure 2, corresponds to the right-most boundary at ϕ̅ h = 2/3 in the top part of Figure 4 and separates the lamellar phase from the coexisting liquid phases. The latter two effects are responsible for the reduction of the line tension to zero (note both ψ0 and σ0 have large values where γ = 0). In the vicinity if the Lifshitz point, ϕ̅ h = 2/3 and τ = 0, the line tension is extremely small and flat (as indicated by a small and positive region labeled in green in Figure 4). In that region, inferring the contour of zero line tension from a finite number of mesh points is more sensitive to numerical errors. We now show that the zero contour line shown in Figure 4 in the region τ(ϕ̅ h) < 0 and ϕ̅ h < 2/3 exactly terminates at the Lifshitz point. Close to but below Tc(ϕ̅ h), the minimization of eq 23 can be done analytically. In this regime, it is simple to show that all variational parameters can be expanded in powers of the small parameter that measures the deviation of the temperature from its critical value, τ(ϕ̅ h)1/2. The leading term in the expansion of λ̃ scales like τ(ϕ̅ h)−1/2, as one would expect from mean-field theory.49 Similarly and as shown in eqs 8 and 9, the order parameter scales like τ(ϕ̅ h)1/2 to lowest order. However, the leading term in the expansions of σ0 and ψ0 scales like τ(ϕ̅ h) (this can be seen recalling that ψ ∝ −ϕ2, σx ∝ −dϕ/dx and that the interface width diverges like τ(ϕ̅ h)−1/2 near the transition). Minimizing the interfacial free energy order by order in a series in the small quantity τ(ϕ̅ h)1/2 results in the following expression (which can be obtained after some standard algebraic manipulations):

This last equation together with the numerical evaluation of the interfacial free energy shown in Figure 4 are the main results of the paper. Equation 17 shows that the line tension exactly vanishes as the hybrid lipid fraction tends toward the Lifshitz point (ϕ̅ h → 2/3). Hence, this shows that the contour of zero line tension in Figure 4 indeed tends to τ(ϕ̅ h) = 0 as ϕ̅ h → 2/3. Equation 17 highlights the main difference between our work and the predictions of Brewster et al.4 where the line tension could achieve zero values only deep into the phase separated regime (where τ(ϕ̅ h) → −1). We conclude this section by pointing out that the expressions for the interface widths derived here for small values of the reduced temperature (near the mixing transition) as well as the amplitude of the local accumulation of hybrids at the interface, λ̃ and ψ0 in eq 16, are both accurate throughout the entire temperature range shown in Figure 4 (from τ(ϕ̅ h) = 0 to −0.5). On the other hand, the expression for σ0 is only valid close to τ(ϕ̅ h) = 0. Also note that we verified that the predicted line tension obtained with our postulated approximate profiles (which simplify the analysis and provide more insight by allowing analytical predictions) is very accurate compared with fully numerical calculations which minimize the membrane free energy (eq 2) without invoking the profiles (eqs 10 and 11). Please refer to the Supporting Information for details on this numerical calculation.

4. DISCUSSION In this paper, we showed how linactants (analogous to surfactants in 2D) can reduce the line tension between equilibrium domains in mixed membranes close to the mixing temperature. The difference between our model and conventional surfactant models is that below, but close to the mixing temperature, the linactant activity (and the resulting reduction in the interfacial free energy) occurs for relatively large linactant concentrations. Indeed, for conventional surfactants that stabilize microemulsions in water-oil-surfactant mixtures, the mixing temperature cannot be attained experimentally (at atmospheric pressure), but is theoretically much higher than room temperature (for metastable fluid phases). Therefore, even a small amount of surfactant can significantly reduce the line tension (for a recent example, see ref 60) since the energetic reduction of the interfacial energy is large compared to the entropy associated with the accumulation of surfactants at the interface. Brewster et al.4 recovered that result in the context of hybrid lipids in a membrane composed of mostly saturated and unsaturated lipids with only a small amount of hybrids (where the saturated chain-unsaturated chain interactions play the role of hydrophobic-hybrophilic interactions) by expanding the free energy about T = 0. In contrast, our predictions, shown in Figure 4 using the model developed in ref 9, allow for arbitrary fractions of linactants and includes interactions among the linactants. By expanding the free energy about the critical temperature T = Tc(ϕ̅ h) and using a variational approximation, we obtained an analytical expression for the interfacial free energy. The variational approximation profile for the composition of the components that phase separate (ϕ in eq 1) was chosen to vary linearly in the interface region. The spatial variations of the linactant composition (ψ in eq 1) and orientation (σ in eq 1) near the interface were determined self-consistently from the profile for ϕ. The analytical expression for the free energy of the interface, eq 23, was then minimized with respect to the approximate profiles parameters. Our main results are the

⎛ ⎞1/2 15 ⎟⎟ ( −τ(ϕh̅ ))−1/2 + O(τ1/2) λ ̃ = ⎜⎜ 16(1 ) − ϕ ̅ ⎝ h ⎠ ⎛ 1 − ϕ ̅ ⎞1/2 h ⎟ ϕh̅ τ(ϕh̅ ) + O(τ 2) σ0 = −⎜ 5 ⎠ ⎝ ψ0 = −

3ϕh̅ (1 − ϕh̅ ) 4

τ(ϕh̅ ) + O(τ 2)

(16)

and yields the following small τ(ϕ̅ h) expansion for the line tension, ⎛ 3 ⎞1/2 γ = 8⎜ ⎟ (1 − ϕh̅ )3/2 (1 − 3ϕh̅ /2)( −τ(ϕh̅ ))3/2 ⎝5⎠ + O(τ 5/2)

(17) 11741


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subsequent predictions for the line tension and they are summarized in Figure 4 and eq 17. Regimes of zero line tension (as shown in Figure 4) are even predicted to occur close to the transition temperature and large fraction of linactants (in eq 17, the line tension γ vanishes when ϕ̅ h = 2/3). In general, the activity of any linactant is characterized by at least one order parameter that vanishes in the bulk, but that has a nonzero value at the interface (and hence reduces the free energy of the interface). In our model, there are two such parameters: the local excess concentration of linactants relative to their bulk average values, ψ, and their local orientation, σ. We now return to the application of our theory to the case of line active hybrid lipids in saturated−unsaturated mixed lipid bilayers. As pointed out recently by Hassan-Zadeh et al.,32 the line tension is reduced by the addition of hybrid lipids even if they do not behave as linactants. To distinguish the line active character of the hybrids from their “partitioning and loosening” ability (using the language of ref 32) that arises from their presence in each of the bulk phases (that makes the chain composition of these phases more similar), we calculate the interfacial free energy following the procedure described in section 3 by enforcing a uniform composition and random orientation for the hybrids throughout the membrane (σ0 and ψ0 are set to zero in eq 23). In this case, the hybrids decrease the composition mismatch between the two bulk domains (see eq 9) and the predicted line tension is ⎛ 3 ⎞1/2 γ0 = 8⎜ ⎟ (1 − ϕh̅ )2 ( −τ(ϕh̅ ))3/2 ⎝5⎠

γ = γ0

1−

3ϕ̅h 2

1 − ϕh̅

(19)

which demonstrates, along with Figure 5, that the extra amount of line tension reduction due to the line active character of the hybrids gets increasingly large when the hybrid fractions tends toward the Lifshitz point (ϕ(0) h → 2/3). Also note that the regime of zero line tension shown in Figure 4 occurs before the first order transition to the modulated phases (at ϕ̅ h ≈ 2/3, recall Figure 2). In that latter phase, composition conservation does not require the presence of an interface. In order to form nanodomains in model membranes with additional interactions, the line tension does not need to identically vanish. But, for these relatively small interactions to be effective, the bare line tension (calculated in their absence, by our theory) must be very small. In fact, phenomenological models show that interactions that favor small domains (i.e., electrostatics55,61 and/or membrane curvature11) can compete with the line tension (which favors large domains) and result in domains whose length scales are determined by the relative strength of the two competing interactions. Our linactant-based theory may provide a microscopic justification for the small line tension values used with these competing interaction models to predict nanodomains. In particular, our model predicts that the region of zero line tension is preceded by a region where the free energy due to the presence of the interface is extremely flat and small (i.e., the green region in Figure 4). In the context of the phase diagram shown in Figure 2, we now recapitulate where nanodomains would form, according to our model. (1) In the mixed phase below the Lifshitz line (region i in Figure 2A), small (nano) fluctuation domains (with a finite lifetime) can be obtained by tuning the temperature above Tc(ϕ̅ h). In ref 9, we showed that, at large fractions of linactants (close to the Lifshitz line), that temperature could be much closer to Tc(ϕ̅ h) (where large length scales are expected without linactants). (2) In the phase separated regime (region iii in Figure 2A), small (nano) domains (that are thermodynamic stable) could arise close to the contour of zero line tension shown in Figure 4 by extending our model to include competing interactions (i.e., electrostatic and membrane curvature as discussed right above). One should also not rule out the blue region in Figure 2A where our line tension calculation breaks down. Note that regions ii and iv in Figure 2A do not predict conventional (circular, on average) nanodomains in our model but rather stripelike fluctuations (ii) and stripelike equilibrium domains (iv) where the stripe wavelength is on the nanoscale. Finally, we note that the experiments of Hassan-Zadeh et al.32 focused on membranes self-assembled from mixtures containing DSPC (a saturated lipid) (with and without DOPC [an unsaturated lipid with a single double bond on both chains]) in the presence of hybrid lipids. The hybrid lipids considered were POPC (one double bond along the unsaturated chain), PLPC (two double bonds along the unsaturated chain), and PAPC (four double bonds along the unsaturated chain). One of their conclusions is that POPC was the only hybrid that behaved as a linactant (POPC decreased the line tension between domains when its fraction in DSPC/ DOPC mixtures was increased and mixtures comprising POPC/DSPC without DOPC phase separated into nanodomains). The authors then claimed that the other two types of hybrids (when individually mixed with DSPC without DOPC)

(18)

to lowest order in τ(ϕ̅ h)1/2. This means that the line tension can be reduced by molecules that do not possess line active abilities (that do not accumulate at and are not preferentially oriented close to the interface) and that simply dilute the bulk domains, as observed in ref 32. On the other hand, zero line tension cannot arise from such a mechanism alone (for hybrid fractions less than unity, as shown in eq 18). Figure 5 predicts the value of the line tension

Figure 5. Lowest order (in a reduced temperature expansion in powers of τ(ϕ̅ h)1/2) contribution to the line tension is scaled by τ(ϕ̅ h)3/2 and reported as a function of the fraction of line active hybrid lipids. Full line: total value of the line tension, γ (see eq 17). Dotted line: value of the line tension when the line active ability of the hybrid lipid is “turned off”, γ0 (see eq 19).

(scaled by τ(ϕ̅ h)−3/2) to lowest order in reduced temperature with and without the linactant character of the hybrids. Of course, the complete expression for the line tension, γ, can be written in terms of the one obtained without the line active abilities of the hybrids, γ0. The result is 11742


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nearest-neighbor, rotationally invariant one given by eq 4. This procedure is identical to taking an average over all possible orientations of the original lattice. For the entropic part, the expression derived in ref 9 is

did not behave as linactants since the observed domains were macroscopic. We believe this observation does not provide sufficient information to make such a claim. Of course, in mixtures that contain hybrids and saturated lipids (and cholesterol) in the absence of unsaturated lipids, the hybrid lipids do not accumulate at the interface since they are already there. Hence, the only criteria for line activity is the preferred orientation at the interface. If there is only a slight preference for the saturated tail of the hybrids at the interface to point toward the saturated domain, the line tension reduction will be small and large domains will form. On the other hand, the observation of macroscopic domains (and not nanodomains) does not necessarily imply that the orientation of the hybrids at the interface is completely random. In fact, we note that the molecular dynamics simulations of Rosetti and Pastorino62 focused on membranes comprising large amounts of saturated and hybrid lipids (no unsaturated lipids) with the same saturated hydrocarbon chain and demonstrated that the line activity of the hybrid was more pronounced when the number of double bonds along its unsaturated chain was increased. Further, Hassan-Zadeh et al. concluded that POPC in membranes comprising DSPC and DOPC was a “weak” linactant. Our theory agrees with this statement, unless the hybrid (POPC) fraction is large (tending toward the Lifshitz point) where the line active ability of the hybrid is considerably enhanced. Recall that our model contains only one parameter, J in eq 3, that controls the strength of the chain−chain interactions and that can be fully determined from measurements of the critical temperature. Future refinements of the model presented here could also include the effects of local chain order as proposed by Yamamoto co-workers.63,64 This would permit the study of systems that, for example, comprise lipids of various chain lengths. In particular, it may facilitate connections with the recent findings of Heberle et al.31 who showed that mixtures of saturated/unsaturated lipids with POPC (a hybrid lipid) and DLPC (a short chain saturated lipids) show similar micrometer-scale modulated phases as well as similar nanodomain sizes. In other words, they showed experimentally that nanodomains can arise in model membranes with no hybrids, but with an additional, short-chain lipid. While at first glance this could exclude the linactant nature of POPC in stabilizing the domains, it is also possible that DLPC does accumulate slightly at the interface (or it may also induce local chain conformation changes in the neighboring lipids at the interface), in which case it would be line active to some degree.

■

:=−

∫

⎧ ⎪ dx⎨ϕs(x) log ϕs(x) + ϕu(x) log ϕu(x) ⎪ ⎩ ϕ h (x)

− ϕh(x) log ϕh(x) +

2

log[ϕh(x)4

− 2ϕh(x)2 (σx(x)2 + σy(x)2 ) + (σx(x)2 − σy(x)2 )2 ] ⎡ σ (x)2 − σ (x)2 − ϕ (x)2 ⎤ y x h ⎥ − σx(x) coth−1⎢ ⎢⎣ ⎥⎦ 2ϕh(x)σx(x) ⎡ σ (x)2 −1⎢ x

−σy(x) coth

⎢⎣

⎫ − σy(x)2 − ϕh(x)2 ⎤⎪ ⎥⎬ ⎥⎦⎪ 2ϕh(x)σy(x) ⎭

(21)

where ϕs, ϕu, and ϕh can be expressed in terms of ϕ, ψ, and the membrane averages as in eq 1. In ref 9, we expanded : to second order in local fluctuations (in terms of ϕ, ψ, and σ⃗). In the phase separated regime, the matrix of coefficients for the quadratic terms is not positive definite. Hence, we will kept terms up to fourth order. : in this form still depends on the underlying lattice from which it was derived. As before, we average over all possible orientations and set ϕ̅ s = ϕ̅ u to obtain ⎧ 4ψ 3(1 − 2ϕh̅ ) 2ψ 2 − d x⎨ 3((1 − ϕh̅ )ϕh̅ )2 ⎩ (1 − ϕh̅ )ϕh̅ ψ 4(1 + 3(1 − 2ϕh̅ )2 ) 2ϕ2 ⎡ 2ψ ⎢1 + + + 3 1 − ϕh̅ ⎢⎣ 1 − ϕh̅ 3((1 − ϕh̅ )ϕh̅ )

:=−

1 a2

∫

⎪

⎪

+

⎡ ⎤ 4ψ 2 ⎤ 2ψ 4ψ 2 ⎥ |σ |⃗ 2 ⎢ ⎥ 1 + − + 2 ϕh̅ ⎢⎣ ϕh̅ (1 − ϕh̅ )2 ⎥⎦ ϕh̅ ⎥⎦

+

⎫ 4ϕ4 |σ |⃗ 4 ⎪ ⎬ + 3⎪ 3(1 − ϕh̅ )3 4ϕh̅ ⎭

(22)

Note that the fourth order Taylor expansion in terms of ϕ, ψ, σx, and σy was performed before the average over the orientation of the lattice was taken. Higher order terms could have been kept, but these are unimportant if the temperature is not too far from Tc. Note that local deviation of the hybrid composition, ψ, can reduce the entropic cost associated with local deviation of the hybrid orientation, σ⃗, and/or saturated− unsaturated lipid composition, ϕ, as can be seen by examining the terms inside the square brackets in eq 22 which have a minimum at ψ ≠ 0. This is important for phase separated membranes where bulk regions rich in saturated or unsaturated lipids cost less entropy with hybrids.

APPENDIX

A. Membrane Free Energy

In ref 9, we proposed a continuous free energy to describe ternary mixture membranes made of saturated, unsaturated, and line active hybrid lipids. The energetic part of the free energy was derived from a lattice model and resulted in eq 3, but with an interaction kernel that contains the details of the underlying (square) lattice, g (x , Δ⃗) = a 2 ∑ δ(x − xi) δ(y − yj )[δ(Δ⃗ − xâ )

B. Interfacial Part of the Free Energy

ij

+ δ(Δ⃗ − yâ )]

1 a2

Details on the calculation of the total free energy obtained from the approximate profiles are given in the Supporting Information. The resulting expression is the sum of two contributions. The first one is given by eq 14 and scales like the system size and the other is,

(20)

where xi and yi are the continuum coordinates that correspond to the lattice site labeled by ij. In order to remove the details of the lattice, in ref 9, we replaced the last expression for g by the 11743


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⎛ 8λ ̃ -p,I 1 4 ⎞ 4 2 = m2⎜ + − − σ0 2⎟ ̃ ̃ JLy λ ⎝ 3 9πλ ̃ ⎠ 3π ⎡ 32ψ0 2 ⎛ 2 ⎞ Tλ ̃ ⎢ ⎟ + − 4mσ0⎜1 − ⎝ J ⎢⎣ 15ϕh̅ (1 − ϕh̅ ) 3πλ ̃ ⎠ −

128(1 − 2ϕh̅ )ψ0 3 105(1 − ϕh̅ )2 ϕh̅

2

+

256(1 + 3(1 − 2ϕh̅ )2 )ψ0 4 945(1 − ϕh̅ )3 ϕh̅

3

+

4ψ0 32ψ0 2 ⎞ 2σ0 2 ⎛⎜ ⎟ − + 1 2⎟ ϕh̅ ⎜⎝ 3ϕh̅ 15ϕh̅ ⎠

−

⎞ 8ψ0 16ψ0 2 2m2 ⎛ ⎜⎜1 + ⎟⎟ − 3(1 − ϕh̅ ) ⎝ 15(1 − ϕh̅ ) 35(1 − ϕh̅ )2 ⎠

+

σ0 4 2ϕh̅

3

−

(4) Brewster, R.; Pincus, P.; Safran, S. Hybrid Lipids as a Biological Surface-Active Component. Biophys. J. 2009, 97, 1087. (5) Schick, M. Membrane Heterogeneity: Manifestation of a Curvature-Induced Microemulsion. Phys. Rev. E 2012, 85, 031902. (6) Hirose, Y.; Komura, S.; Andelman, D. Concentration Fluctuations and Phase Transitions in Coupled Modulated Bilayers. Phys. Rev. E 2012, 86, 021916. (7) Shlomovitz, R.; Schick, M. Model of a Raft in Both Leaves of an Asymmetric Lipid Bilayer. Biophys. J. 2013, 105, 1406. (8) Heberle, F. A.; Petruzielo, R. S.; Pan, J.; Drazba, P.; Kučerka, N.; Standaert, R. F.; Feigenson, G. W.; Katsaras, J. Bilayer Thickness Mismatch Controls Domain Size in Model Membranes. J. Am. Chem. Soc. 2013, 135, 6853. (9) Palmieri, B.; Safran, S. A. Hybrid Lipids Increase the Probability of Fluctuating Nanodomains in Mixed Membranes. Langmuir 2013, 29, 5246. (10) Palmieri, B.; Safran, S. A. Hybrid Lipids Increase Nanoscale Fluctuation Lifetimes in Mixed Membranes. Phys. Rev. E 2013, 88, 032708. (11) Amazon, J. J.; Feigenson, G. W. Lattice Simulations of Phase Morphology on Lipid Bilayers: Renormalization, Membrane Shape, and Electrostatic Dipole Interactions. Phys. Rev. E 2014, 89, 022702. (12) Simons, K.; Toomre, D. Lipid Rafts and Signal Transduction. Nat. Rev. Mol. Cell Biol. 2000, 1, 31. (13) Chiang, S.; Baumann, C.; Kanzaki, M.; Thurmond, D.; Watson, R.; Neudauer, C.; Macara, I.; Pessin, J.; Saltiel, A. Insulin-Stimulated GLUT4 Translocation Requires the CAP-Dependent Activation of TC10. Nature 2001, 410, 944. (14) Palazzo, A.; Eng, C.; Schlaepfer, D.; Marcantonio, E.; Gundersen, G. Localized Stabilization of Microtubules by Integrinand FAK-Facilitated Rho Signaling. Science 2004, 303, 836. (15) Suzuki, N.; et al. A Critical Role for the Innate Immune Signaling Molecule IRAK-4 in T Cell Activation. Science 2006, 311, 1927. (16) Trabelsi, S.; Zhang, S.; Lee, T.; Schwartz, D. Linactants: Surfactant Analogues in Two Dimensions. Phys. Rev. Lett. 2008, 100, 037802. (17) Palmieri, B.; Yamamoto, T.; Brewster, R. C.; Safran, S. A. Line Active Molecules Promote Inhomogeneous Structures in Membranes: Theory, Simulations and Experiments. Adv. Colloid Interface Sci. 2014, 208, 58. (18) Komura, S.; Andelman, D. Physical Aspects of Heterogeneities in Multi-Component Lipid Membranes. Adv. Colloid Interface Sci. 2014, 208, 34. (19) Elson, E.; Fried, E.; Delbow, J.; Genin, G. Phase Separation in Biological Membranes: Integration of Theory and Experiment. Annu. Rev. Biophys. 2010, 39, 207. (20) Dietrich, C.; Bagatolli, L.; Volovyk, Z.; Thompson, N.; Levi, M.; Jacobson, K.; Gratton, E. Lipid Rafts Reconstituted in Model Membranes. Biophys. J. 2001, 80, 1417. (21) Samsonov, A. V.; Mihalyov, I.; Cohen, F. S. Characterization of Cholesterol-Sphingomyelin Domains and Their Dynamics in Bilayer Membranes. Biophys. J. 2001, 81, 1486. (22) Feigenson, G. W.; Buboltz, J. Ternary Phase Diagram of Dipalmitoyl-PC/Dilauroyl-PC/Cholesterol: Nanoscopic Domain Formation Driven by Cholesterol. Biophys. J. 2001, 80, 2775. (23) Veatch, S. L.; Keller, S. L. Organization in Lipid Membranes Containing Cholesterol. Phys. Rev. Lett. 2002, 89, 268101. (24) Silva, L.; de Almeida, R. F.; Fedorov, A.; Matos, A. P.; Prieto, M. Ceramide-Platform Formation and -Induced Biophysical Changes in a Fluid Phospholipid Membrane. Mol. Membr. Biol. 2006, 23, 137. (25) Pinto, S. N.; Silva, L. C.; de Almeida, R. F.; Prieto, M. Membrane Domain Formation, Interdigitation, and Morphological Alterations Induced by the Very Long Chain Asymmetric C24:1 Ceramide. Biophys. J. 2008, 95, 2867. (26) Heberle, F.; Wu, J.; Goh, S.; Petruzielo, R.; Feigenson, G. Comparison of Three Ternary Lipid Bilayer Mixtures: FRET and ESR Reveal Nanodomains. Biophys. J. 2010, 99, 3309.

⎤ 2m 4 ⎥ 15(1 − ϕh̅ )3 ⎥⎦

(23)

which varies with the variational parameters that describe the interface.

■

ASSOCIATED CONTENT

S Supporting Information *

Contains technical details on the calculation performed to obtain the membrane phase behavior shown in Figure 2, on the interfacial free energy calculation using the variational profiles in eqs 10 and 11, and on the test performed to check the accuracy of the results obtained with the approximate profiles. This material is available free of charge via the Internet at http://pubs.acs.org.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS We are grateful for discussions with David Andelman, Shige Komura, Gerald Feigenson, Michael Schick, David Ronis, Tetsuya Yamamoto, Robert Brewster, Philip Pincus, Jonathan Amazon, David Ackerman, Sarah Veatch, Matthew Stone, Sarah Keller, Benjamin Machta, Ishi Talmon, Naama Koifman, Lia Addadi and Uri Raviv. The Israel Science Foundation, the Schmidt Minerva Center, the historic generosity of the Perlman Family Foundation, the Natural Sciences and Engineering Research Council of Canada and the Fonds québécois de la recherche sur la nature et les technologies are gratefully acknowledged for funding this research. Also, Benoit Palmieri is grateful to the Azrieli Foundation for the award of an Azrieli Fellowship.

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REFERENCES

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Prediction of the Dependence of the Line Tension on the Composition

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