Curvature-driven effective attraction in multicomponent membranes
Matthew F. Demers,1 Rastko Sknepnek,2,* and Monica Olvera de la Cruz2,†1Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, Illinois 60208, USA
2Department of Materials Science and Engineering, Northwestern University, Evanston, Illiinois 60208, USA(Received 23 March 2012; revised manuscript received 19 July 2012; published 15 August 2012)
We study closed liquid membranes that segregate into three phases due to differences in the chemical andphysical properties of its components. The shape and in-plane membrane arrangement of the phases are coupledthrough phase-specific bending energies and line tensions. We use simulated annealing Monte Carlo simulationsto find low-energy structures, allowing both phase arrangement and membrane shape to relax. The three-phase system is the simplest one in which there are multiple interface pairs, allowing us to analyze interfacialpreferences and pairwise distinct line tensions. We observe the system’s preference for interface pairs thatmaximize differences in spontaneous curvature. From a pattern selection perspective, this acts as an effectiveattraction between phases of most disparate spontaneous curvature. We show that this effective attraction is robustenough to persist even when the interface between these phases is the most penalized by line tension. This effectis driven by geometry and not by any explicit component-component interaction.
Multicomponent liquid membranes are pervasive in natureand their phase behavior is important in many biologicalprocesses. Such membranes are of interest not only asconstituents of complex biological systems but also as richpattern-forming systems in themselves. They provide a fertileground for exploring the role of geometry in surface patternformation at the nanoscale, which is an important aspect offunctional materials. They have been studied as basic modelsfor biological systems [1,2] and as promising candidatesin the rational design of biocompatible materials [3]. Theirpromise is due in large part to their versatile phase behavior.In a liquid membrane, the membrane’s constituents are freeto rearrange. Although mixing is entropically favorable, thisfreedom has energetic consequences when there are multiplekinds of membrane components since different componentsmay have different chemical and physical properties [4]. Whenthe system adapts its arrangement to its conditions, varyingshapes and component patterns can result.
Patterns of segregated domains have been directly observedin experimentally created multicomponent giant unilamellarvesicles [5–10]. These experiments agree with theoreticalresults in which patterns arise in response to differing bendingproperties of the membrane’s segregated phases [11–16].These bending properties can include bending rigidities,saddle-splay moduli, and spontaneous (preferred) curvatures.When these properties are phase specific, the phase arrange-ment is coupled to the membrane shape.
A key aspect of surface pattern selection is the system’sresponse to geometric constraint. Constraints include fixedsurface area, since the energy required to compress a liquidmembrane is orders of magnitude larger than the energy tobend it, and closure, since tears or holes in the membrane wouldlead to energetically unfavorable exposure of hydrophobic tails
to surrounding water. For a closed membrane that does notexchange material with its environment, there may also bea constraint on enclosed volume. This does not apply in allcases; for example, lipid bilayers have been observed to adjusttheir volume by forming transient pores [17]. We use a constantvolume assumption here for the sake of simplicity and note thatrelaxing this constraint does not qualitatively affect reportedresults. Under these constraints, a multicomponent membranemay not be able to fully indulge the preferences of all of itsconstituents simultaneously.
Thus a strongly segregated membrane may face thisfrustration: On the one hand, immiscibility (realized as aline tension between differing component domains) favorsmacroscopic segregation and interfaces with minimal length[18]; on the other hand, when subject to geometric constraints,component bending preferences may be more favorablyaccommodated by arrangements with extended interfaces,including arrangements with multiple domains. In this senseit is the matter of the length and location of interfaces thatbecomes the battlefront between these competing preferences.
Most studies so far have focused on two-phase systems.In two-phase systems there is only one phase pair, henceonly one type of interface. Introducing a third phase is anontrivial extension. The presence of a third phase providesthree phase pairs and thus introduces the crucial feature ofinterfacial preferences: The three-phase system has freedomnot only over the length and location of interfaces, but alsoover which phase pairs are brought into contact. This turnsout to be an important avenue through which geometry caninfluence the surface pattern. Furthermore, since the linetension between differing phase pairs need not be the same,a three-phase system allows for an additional type of controlparameter, one that can produce qualitatively different patternbehavior.
In this work we analyze energy-minimizing structures ofclosed three-phase membranes. We observe that the systemfavors interface pairs that maximize differences in spontaneouscurvature. A pronounced effect of this preference is an effective
DEMERS, SKNEPNEK, AND OLVERA DE LA CRUZ PHYSICAL REVIEW E 86, 021504 (2012)
2
2
2
22
2
FIG. 1. (Color online) Cartoon representation of curvature-driveninterfacial preference. Blocks of type I, II, and III [red (medium),green (lightest), and blue (darkest)] represent constituents with differ-ent preferred curvatures CI, CII, and CIII. In this example |CI − CIII| >
|CI − CII| > |CII − CIII|. Possible four-interface arrangements ofthese constituents are shown. The arrangement that maximizes thecurvature difference at the boundaries (bottom middle) is able to mostclosely accommodate its constituents’ preferences when closed.
attraction between phases of most disparate spontaneouscurvature, an indirect interaction driven by a mutual responseto geometric constraints.
Figure 1 illustrates the basic mechanism. Blocks of typeI, II, and III [red (medium), green (light), and blue (dark)]represent constituents with different preferred curvatures CI,CII, and CIII. Possible four-interface arrangements are shown.The arrangement that maximizes the curvature differenceat the boundaries is able to most closely accommodate itsconstituents’ preferences when closed.
We present cases of equal line tension and cases when linetension is pairwise preferential, showing that the geometry-induced attraction is robust enough to persist even in caseswhen it is most penalized by line tension. This study is focusedon characterizing the effective attraction. A full description ofthe wide pallet of observed patterns is beyond the scope of thepresent paper [19].
II. MODEL
A. Continuum model
We model the membrane as a two-dimensional surface.This approximation is appropriate for structures whose lateraldimensions are much larger than the membrane thickness. Thesystem is considered in the strong-segregation regime, witheach surface element identified with one of three phases. Eachphase has a specific, predetermined spontaneous curvatureand interfaces between different phases are penalized witha line tension. The membrane is assumed to retain a closedspherical topology. We assume that the volume enclosed bythe membrane is conserved, as is the area occupied by eachphase.
Our simulation searches for minimum-energy configura-tions, that is, shape and phase arrangements that satisfy thearea and volume constraints and minimize the total energy.This energy is written as the sum of a bending term Fb and aline tension term F�.
The bending energy takes the form of a Helfrich functional[20], written as
Fb =∑
j
∫2κ(H − Cj )2dSj , (1)
where j = 1,2,3 counts phases, H is the mean curvature,Cj is the spontaneous curvature of phase j , and κ isthe bending rigidity. This term penalizes surface shapeswhose local curvature deviates from its preferred value Cj .Since Cj depends on the local composition j , the bendingenergy couples shape and composition. Note that we do notgive an index to bending rigidity κ since we are assuming thatthe phases are equally bendable.
Note also that the Helfrich functional typically includesa Gaussian curvature term weighted by the saddle-splaymodulus κ . However, as with κ , we are assuming that all κ’s areequal. Then, since the topology of the surface remains fixed,by the Gauss-Bonnet theorem [21] the Gaussian curvatureterm integrates to a constant independent of the particularshape and therefore does not affect the energy. If the κ’s werenot equal, the term would need to be retained. Phase-specificdifferences in bending rigidity κ or saddle-splay modulus κ
provide additional mechanisms for pattern formation [12,22]and domain budding [23].
In monolayers, the spontaneous curvature has an intuitivephysical origin related to the properties of the head andtail groups [4]. Roughly speaking, molecules with largereffective head sizes and smaller effective tail sizes tend topack into surfaces of negative mean curvature, much as coneswould pack. While the actual curvature properties depend onmore than steric considerations, modeling lipid types withspontaneous curvatures has been shown to agree well withexperiments [24]. In lipid bilayers, spontaneous curvaturecan arise from the effects of asymmetry between the innerand outer layers [25–27]. Our model is therefore directlyapplicable to monolayers, such as emulsions in which oildroplets are surrounded by multiple surfactant types, and tobilayers whose phases with distinct spontaneous curvaturesare conserved, such as vesicles whose inner layer is uniformbut whose outer layer is multiphase [28]. To extend the modelto general multicomponent bilayers would require treatmentof separate bilayer properties.
It is worth contrasting our deformation energy with thatof solidlike membranes. There energy penalties associatedwith stretching and shear deformations can lead to buckling[29,30] and nonlinear conformation fluctuations [31] in thehomogeneous case, and in the two-component case elongatedinterconnected domains can arise [22,32,33]. In lipid vesicleswith one fluid and one solidlike phase, complexes of stripes andpolygonal domains have been observed experimentally [34].These are outside the scope of our current study since noneof our three phases include resistance to shear; it would beinteresting to explore interfacial preferences in these systems.
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Line tension energy is given by Ref. [35]
F� =∑i �=j
∫∂ij
λij d�, (2)
where i,j = 1,2,3; λij is the line tension between phases i
and j ; d� is a line element along i-j interfaces; and the lineintegral is calculated along the boundary ∂ij between phases i
and j . This term favors segregation since it penalizes interfacesbetween domains.
Recently a novel class of molecules called linactants hasbeen synthesized, which behave as two-dimensional analogs ofsurfactants: They migrate laterally in a fluid layer to boundariesbetween immiscible phases, mediating between the immiscibleconstituents. Small amounts of linactant added to the fluid layerhave been found to reduce the line tension between immisciblephases [36]. This suggests that the line tension between phasescan be viewed as a tunable design parameter.
The surface area and volume constraints can be writtenas
∫dSj = Aj (j = 1,2,3) and
∫dV = V0, where Aj is the
prescribed surface area of phase j and V0 is the prescribedsystem volume. Note that there is an area constraint for eachphase; the system’s total surface area and its compositionfractions are both conserved.
It is convenient to work with dimensionless parameters.Therefore, we measure energy in units of bending rigidity κ ,lengths in units of R0, the radius of a reference sphere whosevolume is V0, curvatures in units of 1/R0, and line tension inunits of κ/R0. In these units, there remain nine independentinput parameters: three spontaneous curvatures, three areas,and three line tensions. While this may seem like a dauntingparameter space, we note that there is a great deal of symmetryand physical analogy and we believe our choice of parameterpairs for the simulation is representative and can provide usefulinsight into the phenomena at hand.
B. Discrete model
Our numerical method is based on a triangulation of thesurface. Each vertex is identified with a phase type and surfacearea elements are represented by polygonal patches centeredat vertices. The discrete analog of the bending energy [Eq. (1)]is computed as a sum over vertices, with the discrete meancurvature near each vertex v computed as Ref. [37]
Hdiscrete(v) =14
∑i |ei |ψi∑j
13Aj
, (3)
where i indexes edges for which v is an end point, |ei | is thelength of the ith edge, ψi is the dihedral angle between twotriangles sharing the ith edge, j indexes triangles containingv, and Aj is the area of the j th triangle.
To compute discrete line tension [Eq. (2)] we sum overall adjacent vertex pairs of different phase types [35]:∑
〈i,j〉 λij (|c(1)ij − mij | + |c(2)
ij − mij |), where 〈i,j 〉 denotespairs of adjacent vertices i and j that have different phasetypes, c(1)
ij and c(2)ij are the centers of the two triangles that have
i and j as vertices, and mij is the midpoint of their sharededge. Note that this differs from the approach used by Huet al. [12], as we take into account local triangle distortionswhen computing interface length.
The surface area constraint is enforced by penalizingdifferences between A(i), the surface area associated witha vertex i, and the prescribed per-vertex value A0(i),Fa = ∑
iσ2 [A(i) − A0(i)]2. The volume constraint is also
enforced by a harmonic potential penalty Fv = ν2 (V − V0)2,
where V is the system’s volume and V0 is the targetedvolume.
Throughout all simulations, we set σ = 2 × 105 and ν =104. These values were tuned to ensure that, by the end of a run,the total volume and surface areas differed from their targetvalues by less than 1%. Using φi to denote the compositionfraction of phase i, the prescribed surface area of phase i is1.05 × 4πφi . This is 5% larger than the area of a sphere ofidentical volume, excess being necessary to avoid the trivialcase of an undeformable sphere.
To find low-energy configurations, simulated annealingMonte Carlo simulations were performed following a linearcooling protocol. Three types of Monte Carlo moves wereused: (i) a surface move, which attempts to perturb theposition of a vertex by |��r| = 2.5 × 10−3, (ii) a phase-swapmove, which attempts to exchange the phase type for tworandomly selected vertices, and (iii) an edge flip, which cutsan edge shared by two triangles and reattaches it so that itspans the opposite previously unattached vertices [38,39]. Allmoves were accepted according to the Metropolis rules. Foreach parameter set, 4 × 106 sweeps, with a sweep definedas an attempted move of each vertex, were performed.Phase-swap moves were performed every five sweeps andedge-flip moves every ten sweeps. Initial configurations wererandom triangulations of a sphere, constructed from regular,Caspar-Klug triangulations [40] such that vertex moves wereconstrained to a sphere and edge flips were performed at avery high temperature to ensure high acceptance rates. Anymoves or edge flips that would result in unphysical edgecrossings were rejected. The resulting configuration had alarge number of vertices with coordination different from 6.Finally, vertex types were randomly permuted. Each vertexwas assumed to have a hard core of diameter lmin = 0.093and each edge was endowed with a tethering potential withmaximum length lmax = 0.157 such that lmax/lmin = 1.688.These values were chosen to be tight enough to preventmembrane self-intersection but slack enough to allow edgeflips [35,41].
Since the system’s energy landscape is complicated, thecomputed configurations are not guaranteed to represent globalminima. However, because independent runs starting fromdifferent random initial configurations reproduce the samequalitative features, we can regard the results as typical.
Patterns are classified using graphs: Each colored domainis identified with a colored node and two nodes are linked ifthey share an interface. Then two configurations are consideredto represent the same pattern if their graphs are isomorphic,respecting color.
III. RESULTS AND DISCUSSION
A sample of the varied patterns is provided in Fig. 2,where we present a phase diagram for a 1
10 : 110 : 8
10 mixture ofcomponents I:II:III (red:green:blue). In this diagram, we varyCIII [spontaneous curvature of the blue (darkest) phase] and
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0.0-0.4-0.8-1.2-1.6-2.0-2.4-2.8Spontaneous curvature of blue (III) phase
Red
-blu
e (I
-III
) li
ne te
nsio
n
0.5
1.0
1.5
2.0
2.5
n = 2
n
2233>3>3>3>2233>3>3>
2233>3>23>2
2
FIG. 2. (Color online) Phase diagram for the composition ratioI:II:III = 1
10 : 110 : 8
10 , varying spontaneous curvature CIII and linetension λI,III. All other parameters are fixed, with CI = 0.0, CII =−1.0, and λI,II = λII,III = 1.0. Phases I, II, and III are shown in red(medium), green (lightest), and blue (darkest), respectively. Eachexample vesicle is shown twice: On the top right, all phases arevisible; on the bottom left, one phase has been removed so that thepattern can be more clearly seen. Bright gray squares in the leftmostcolumn denote assorted structures with multiple green (lightest) andred (medium) domains. All snapshots were generated with the VisualMolecular Dynamics (VMD) package [42] and rendered with theTachyon ray tracer [43].
λI,III [line tension between red (medium) and blue (darkest)phases]. We fix spontaneous curvatures CI = 0 and CII =−1.0 and line tensions λI,II = λII,III = 1.0. Results for eachparameter set are classified according to pattern. We definethe pattern in terms of the number and arrangement of coloreddomains, irrespective of shape.
For the present purposes, we note two features. The firstis that a preferential line tension opens the possibility ofqualitatively different pattern behavior and thus can be animportant control parameter in the system’s pattern selection.Furthermore, we observe a transition near CIII = −1.0, whichis the dividing line where the spontaneous curvature of phaseIII switches from middle to largest in magnitude.
In Fig. 3 we analyze in detail the λI,III = 1.75 row fromthe diagram in Fig. 2. Note that I-III interfaces predominate inthe region where CIII < CII, even though such interfaces arenearly twice as costly as I-II and II-III interfaces.
Figure 4 examines a 13 : 1
3 : 13 composition ratio where all line
tensions are set to 1.0, with CI = −1.6 and CII = −1.0. In thisrow we observe that the spontaneous curvature of phase III
0.0-0.4-0.8-1.2-1.6-2.0-2.4-2.8
Spontaneous curvature of blue (III) phase0-0.4-0.8-1.2-1.6-2.0-2.4-2.8
C
CC
I
IIIII
-3-2-10
0.5
1
I-III
II-IIII-II
Ci
η
FIG. 3. (Color online) Analysis of a row of the diagram shown inFig. 2 for λ = 1.75. The composition fraction is set to 1
10 : 110 : 8
10 andCIII is varied; CI = 0.0, CII = −1.0, and λI,II = λII,III = 1.0 are keptfixed. Top: Configurations with colored squares indicating patterngrouping. Phases I, II, and III are shown in red (medium), green(lightest), and blue (darkest), respectively. Middle: Length of eachinterface type to the total interface length ratio η as a function of CIII.The interface fraction is represented by the height between curves.Bottom: Spontaneous curvatures, with red (medium), green (lightest),and blue (darkest) lines corresponding to the spontaneous curvaturesof phases of types I, II, and III, respectively.
[blue (darkest)] transitions from smallest in magnitude to mid-dle to largest. Green-blue (lightest-darkest, II-III) interfacesare most favored on the left, where blue (darkest) is largest inmagnitude and green (lightest) is smallest. In the middle, green(lightest, II) is smallest in magnitude and red (medium, I) andblue (darkest, III) are close; here red-green (medium-lightest,I-II) and blue-green (darkest-lightest, II-III) interfaces areabout equally favorable and red-blue (medium-darkest, I-III)interfaces are disfavored. On the right, red (medium) is largestin magnitude and blue (darkest) is smallest and the red-blue(medium-darkest) interface is most favored. Red (medium)and green (lightest) are closest in magnitude and red-green(medium-lightest) interfaces are disfavored. The interfacesbetween phases of most disparate spontaneous curvature aremost favored.
In Figs. 5 and 6 we show that these interfacial preferenceshold over wide slices of parameter space. Figure 5 shows theI-III interface length as a fraction of the total interface for theconfigurations obtained from the parameter sets used in Fig. 2.In the regime where CIII < CII < CI, I-III interfaces predomi-nate until they become twice as energetically expensive as I-IIor II-III.
Figure 6 plots the I-III interface fraction in the systemwith 1
3 : 13 : 1
3 composition ratio, varying CI and CIII, withλI,II = λI,III = λII,III = 1.0 and CII = −1.0 remaining fixed.As expected, symmetry about the line CI = CIII is evident.We observe the fraction of the I-III interface predominating inregions where CI > CII > CIII or CI < CII < CIII. Conversely,I-III interfaces are strongly disfavored in regions where|CI − CIII| < |CI − CII| and |CI − CIII| < |CII − CIII|.
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FIG. 4. (Color online) Top: Configurations with colored squaresindicating pattern grouping. Phases I, II, and III are shown in red(medium), green (lightest), and blue (darkest), respectively. Middle:Length of each interface type to the total interface length ratio η
as a function of CI. The interface fraction is represented by theheight between curves. Bottom: Spontaneous curvatures, with red(medium), green (lightest), and blue (darkest) lines corresponding tothe spontaneous curvatures of phases I, II, and III, respectively. Thecomposition fraction is set to 1
3 : 13 : 1
3 and CI = −1.6, CII = −1.0, andλI,II = λII,III = λI,III = 1.0 are kept fixed.
We point out that for large values of line tension orspontaneous curvatures, well beyond the range used in thisstudy, one observes interesting budding effects. Budding playsan important role in biological systems [44–46] and has beeninvestigated in a number of studies [15,26,35,47–50]. The
−3−2.6
−2.2−1.8
−1.4−1
−0.6−0.20
0.5
1
1.5
2
2.5
0
0.5
1
CIII
Red−blue (I-III) interface as a percentage of total interface
I,III
0.10.20.30.40.50.60.7
λ
FIG. 5. (Color online) Length of the I-III [red-blue (medium-darkest)] interface as a fraction of the total interface, for compositionratio I : II : III = 1
10 : 110 : 8
10 , varying spontaneous curvature CIII andline tension λI,III. All other parameters are fixed, with CI = 0.0, CII =−1.0, and λI,II = λII,III = 1.0.
−3−2.6
−2.2−1.8
−1.4−1
−0.6−0.20 −1.8
−1.4−1
−0.6−0.2
00
0.5
1
CI
Red−blue (I-III) interface as a fraction of total interface
CIII
0.10.20.30.40.50.60.7
FIG. 6. (Color online) Length of the I-III [red-blue (medium-darkest)] interface as a fraction of the total interface, for compositionratio I : II : III = 1
3 : 13 : 1
3 , varying spontaneous curvatures CI and CIII.All other parameters are fixed, with CII = −1.0, and λI,II = λI,III =λII,III = 1.0.
effects of interfacial preference on budding in a three-phaseliquid membrane will not be addressed herein.
IV. CONCLUSION
In conclusion, we have seen that the mutual response togeometry acts as an effective attraction between phases ofmost disparate spontaneous curvature in a three-phase liquidmembrane. This effect arises indirectly through the couplingof deformation and compositional arrangement rather thanthrough a direct component-component interaction. Nonethe-less, it is robust enough to compete with a countervailingline tension. In some cases it results in predominance ofinterfaces between phases least miscible by pure line tensionconsiderations. Therefore, this system provides an examplewhere geometric constraints, rather than direct interactions,can dominate its conformation. Our findings suggest that anintricate interplay between the geometry and composition canlead to a rich phase behavior of complex fluid membranes. Wehope that our results will stimulate further experimental andtheoretical work on these rich systems.
ACKNOWLEDGMENTS
We would like to thank S. Patala, F. Solis, and C.K.Thomas for useful discussions. Numerical simulations werein part performed using the Northwestern University HighPerformance Computing Cluster Quest. M.F.D. and M.O. aregrateful for the financial support of the Air Force Officeof Scientific Research under Grant No. FA9550-10-1-0167.R.S. and M.O. are grateful for the financial support of theUS Department of Energy Grant No. DEFG02-08ER46539and the Office of the Director of Defense Research andEngineering.
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