Coupling between line tension and domain contact angle in heterogeneous membranes

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1 Available online at Biochimica et Biohysica Acta 1778 (008) Couling between line tension and domain contact angle in heterogeneous membranes Kevin B. Towles a,b, Nily an a, a eartment of Chemical and Biological Engineering, rexel University, Philadelhia PA 19104, USA b eartment of Physics and Astrohysics, University of Pennsylvania, Philadelhia PA 19104, USA Received 7 October 007; received in revised form 15 ecember 007; acceted 6 ecember 007 Available online 1 January 008 Abstract The comositional differences between domains in hase-searated membranes are associated with differences in bilayer thickness and moduli. The resulting acking deformation at the hase boundary gives rise to a line tension, the one dimensional equivalent of surface tension. In this aer we calculate the line tension between a large membrane domain and a continuous hase as a function of the thickness mismatch and the contact angle between the hases. We find that the acking-induced line tension is sensitive to the contact angle, reaching a minimum at a secific value. The difference in the line tension between a flat domain (that is within the lane of the continuous hase) and a domain at the otimal contact angle may be of order 40%. This could exlain why revious calculations of the thickness mismatch based line tension tend to yield values that are higher than those measured exerimentally. 007 Elsevier B.V. All rights reserved. Keywords: Phase searation; Line tension; Vesicle 1. Introduction Multi-comonent model membranes have been found to hase searate, resulting in coexistence between domains of different comositions [1 4]. Phase coexistence in these two dimensional systems gives rise to a line tension (the one dimensional equivalent of surface tension), which affects the kinetics of domain growth and equilibrium domain size. An interesting observation found in fluorescence studies is that the domains do not necessarily remain within the lane of the vesicle or continuous hase, but form sherical cas [5 8]. Such shae transitions in hasesearated membranes are thought to be driven by the need to reduce the length of the contact line, while keeing the area of each domain fixed, and arrested by the associated bending enalty [9 11]. esite extensive studies of hase coexistence in model membranes [1 4], few studies measured the line tension in these systems [5,7,1,13]. Initial measurements obtained values of order Corresonding author Chestnut Street, Philadelhia PA 19104, USA. Tel.: ; fax: address: dan@coe.drexel.edu (N. an). 1 N[5,7], but more recent work suggests that the line tension may be much higher in some cases. Blanchette et al. [1] find the line tension in hase-searated 1,-dioleoyl-sn-glycero-3-hoshocholine (OPC)/galactosylceramide (GalCer) membranes to be 1.7 ± 0.5 N, but in OPC/1,-distearoyl-sn-glycero-3-hoshocholine (SPC) it is found to be 4.3±0.5 N. They [1] suggest that the difference in the line tension between the two systems may be due to the differences in the hydrohobic mismatch between the domains, which is 1.8 nm in the SPC case but only 0.9 nm for the GalCer case [14]. Tian et al.[13] investigated the effect of comosition on line tension in three comonent mixtures of OPC, cholesterol and egg shingomyeline (ESM), finding that the average line tension increases, with increasing cholesterol content, from 0.5 to 3.3 N. However, it is interesting to note that several individual samles dislayed much higher values of line tension that may reach 5 N [13]. The line tension in hase-searated membranes is comosed of two contributions. The first is a chemical comonent associated with differences in comosition between the hases, and the second is a acking contribution arising from differences in thickness and bilayer moduli. The chemical contribution may be negligible in many systems due to the relatively minor comositional /$ - see front matter 007 Elsevier B.V. All rights reserved. doi: /j.bbamem

2 K.B. Towles, N. an / Biochimica et Biohysica Acta 1778 (008) differences between the hases [1 4]. Molecular models for liid acking have found a correlation between liid acking and line tension [14]. The couling between a thickness mismatch in hasesearated membranes, which causes a acking erturbation, and the line tension has been reviously calculated [15 17]. As may be exected, the line tension is highly sensitive to the thickness mismatch between the hases [15 17]. Examining the effect of domain size and sacing on the line tension [15 17] showed that the line tension should be maximal, for a single domain, when the domain radius is smaller than order 0 nm. This suggests that small domains are unstable and would either coalesce to larger sizes or dissolve. However, domains larger than aroximately nm are redicted to be meta-stable and may be long-lived due to reulsive interactions between domains at moderate sacing. Substituting tyical values for the moduli of the two membrane hases and the thickness mismatch yields values of 0. to 3.5 kt/nm, or about 1 to 15 N, for the line tension [15 17]. The values calculated by the acking models for the thickness mismatch contribution to the line tension [15 17] are generally higher than those measured exerimentally, desite the fact that they are exected to be a lower limit that may increase when the contribution of the chemical comosition difference is included. Is there another mechanism that can reduce the acking line tension? Models examining the deformation energy imosed by transmembrane roteins (see, for examle, [18 0]) suggest that the enalty due to a thickness mismatch is sensitive to the contact angle between the rotein and the bilayer. We hyothesize here that the formation of a finite contact angle between the domain and the lane of the continuous hase (see Fig. 1) may reduce the value of the line tension due to the thickness mismatch. Such a contact angle would be on lengthscales of order the bilayer thickness, and thus not observable by current domain imaging techniques [5 8]. However, although the micron-scale contact angle observed in some systems [5 8] is likely dominated by the need to reduce the contact line between the domain and the continuous hase, it does not contradict the ossibility of a contact angle on smaller lengthscales that reduces the value of the line tension. Recently, Fournier and Ben Amar [1] examined the effect of a contact angle (or a crease in their terms) between a membrane domain and a continuous hase. They conclude that the stresses at the interface between membrane domains yield a line tension that is contact angle deendent. Fournier and Ben Amar [1] show that the line tension can obtain a shar minimum at a articular value of the contact angle in some cases. The Fournier and Ben Amar [1] analysis clearly suorts our hyothesis that a molecular scale contact angle can reduce the line tension between membrane domains. However, their analysis, which is limited to domains with strong sontaneous curvature such as biological rafts, cannot exlain the low line tension measured in synthetic vesicles [5,7,1,13]. Moreover, their [1] analysis is based on a continuous model where the couling between the thickness and the shae of the membrane was taken to be (somewhat arbitrarily) linear. In this aer we examine the effect of the contact angle between a domain hase and a continuous membrane hase on the acking, or thickness mismatch contribution to the line tension. Our analysis focuses on domains that have zero sontaneous curvature, and uses a meso-scale model which has been found to successfully aly to membrane erturbation by roteins [18 0] to coule between the membrane rofile and deformation enalties. We find that allowing for a contact angle between the domain and the continuous hase may reduce the line tension by u to 40%, even in systems where both hases have zero sontaneous curvature. Thus, it is ossible that the low values of line tension measured exerimentally [5,7,1,13], when comared to the redictions of acking models [15 17] may be due to this additional degree of freedom.. Model Previous studies [15 18] have shown that a thickness mismatch at the boundary of a self-assembled membrane leads to a erturbation in the bilayer rofile and an energetic enalty that can be correlated to the line tension. Taking the unerturbed monolayer thickness to be h, the degree of erturbation may be defined by the normalized thickness of the monolayer at distance Fig. 1. A sketch of our system. The domain hase and the continuous hase differ in their thickness, and ossibly also their bending and area moduli. The energetic enalty due to the formation of a deformed erimeter (for the domain and the continuous hase) is the line tension. The domain may remain in the lane of the continuous hase, or form a contact angle θ with the continuous hase. Note that in our notation, a flat domain is denoted by θ=0.

3 119 K.B. Towles, N. an / Biochimica et Biohysica Acta 1778 (008) z from the erturbation boundary, such that Δ(z)=h(z)/h 1. The free energy enalty associated with the erturbation may be written, for a hase with zero sontaneous curvature, as [15 17]: dg m ¼ B þ Kh d dr ð1þ γ denotes the free energy enalty er unit length of the erturbation interface, namely, the line tension of one monolayer. B is the area stretch modulus, and K the bending modulus of the membrane. The bending term in Eq. (1) requires a brief discussion: Monolayers and bilayers dislay a bending enalty which scales as 1/C, where C is the curvature. The curvature is comosed of two contributions: One is the global membrane curvature (e.g. the radius of curvature in a vesicle, or membrane fluctuations). The second is due to a local curvature on the molecular scale due to the insertion of a erturbation locus, e.g. by a membrane rotein. Thus, a flat bilayer may dislay local bending due to the insertion an interface (see, for examle, [16,18]). The two curvatures are decouled and additive, for examle when considering the overall curvature in a vesicle with embedded roteins. In Eq. (1) we consider only the contribution of the local bending, neglecting that of the overall vesicle radius and the macroscoic bending of the domain. Therefore, interface curvature is defined here by d h/dz d Δ/dz namely, the second derivative of the thickness as a function of distance from the erturbation center [18 0]. Thus, in our model the curvature term becomes zero when there is no bilayer erturbation. In a hase-searated membrane, the free energy enalty associated with the interface is comosed of two contributions: one for the domain hase () and one for the continuous hase (C). Since the moduli (K, B) ofthetwohases may differ significantly, the line tension is comosed of two contributions Z 0 g ¼ g þ g C ¼ þ K h d! dz dz Z C þ C þ K C h d! C 0 dz dz where z=0 denotes the interface between the hases, is the dimension (along the z axis) of a hase, and the subscrits, C refer to the domain and continuous hase, resectively. Note that here the line tension is comosed of the contributions from both monolayers. At the interface, the thickness of the two hases must match (see Fig. 1), and the sloe of the domain hase is set by the contact angle (see Fig. 1). The remaining boundary conditions are determined by the functional minimization as obtained by calculus of variations (see [17 0]). ðþ 3. Results and discussion Minimization of the line tension with resect to the otimal membrane deformation rofile in both the continuous and the domain hase yields the value of γ as a function of the hase moduli and thickness. In the limit where the domains are large (when comared to the bilayer thickness) and widely saced, the line tension is found to vary with the contact angle as n B 1=4 K1=4 49 ffiffiffiffiffiffiffiffiffiffiffi h B gðhþ ¼ B C K1=4 C ðh C h Þ þ4 ffiffiffi C B1=4 h K 1=4 14 ffiffiffiffiffiffiffiffi h which reduces, when there is no thickness mismatch (h C =h )to B 1=4 K gðhþ ¼ 14 ffiffiffiffiffi h 35B C K1=4 C þ 17B K1=4 K1=4 C ðh C h Þsinh þ ffiffiffiffiffiffi K B C h3= K1=4 C þ B h3= C K1=4 35B C h3= K1=4 C þ 17B h3= C K1=4 o sin h sin h B ð3:bþ C K1=4 C þ B K1=4 If θ=0 so that the domain is flat, Eq. (3.a). reduces to gð0þ ¼ 7B C B K1=4 C K1=4 ðh C h Þ ffiffi ð4þ C h3= K1=4 C þ B h3= C K1=4 as reviously calculated [16]. Thus, the line tension increases with the degree of thickness mismatch h C h. In Fig. we lot the line tension (Eq. (3.a)) as a function of the contact angle. We see that when there is no thickness mismatch between the hases (h =h C ) the line tension is zero when the contact angle is 0, as exected: There is no enalty for interface formation if the hases have identical thicknesses if the domain remains within the lane of the continuous hase. However, forcing a domain without a thickness mismatch to rotrude from the continuous hase (θ 0) induces a acking tilt enalty which is ð3:aþ

4 K.B. Towles, N. an / Biochimica et Biohysica Acta 1778 (008) Fig.. The acking line tension, θ (in N) as a function of the contact angle θ. For clarity we lot the angle relative to π/ (when the domain is in the lane of the continuous hase). Solid line: No thickness mismatch between the domain and the continuous hase. ashed line: A 0% thickness mismatch. Membrane arameters used are tyical for such systems [18,19]: K C = , K = J, B C =60 mn/m, B =40 mn/m, and h C =4.4 nm (i.e. K is of order 50 kt, and B of order 50 kt/nm). We see that, as may be exected, when there is no thickness mismatch the acking line tension is zero when there is no contact angle between the hases, and the domain lies within the lane of the continuous hase, since in this case both hases are not deformed at the interface. In this case, forcing the domain to adot another contact angle introduces a deformation, manifested by an increase in the line tension. However, when there is a thickness mismatch, the line tension is minimal at a secific contact angle that is not π/ but is a function of the thickness mismatch (see Eq. (5)); yet, even in this otimal contact angle, there is some deformation of the hases so that the line tension is not zero. manifested in a non-zero value of the line tension. May et al. [] defined a membrane tilt modulus k t as df/a=(1/) k t t, where F is the membrane free energy and t defines the degree of tilt. In our notation, t can be reresented by sinθ, and F/A by γ/h. As a result, the effective tilt modulus for a system without thickness mismatch is, in our model, given by k t ¼ B 1=4 K 35B C K1=4 C þ 17B K1=4 14h 3= ffiffi ð5þ C K1=4 C þ B K1=4 substituting tyical numbers for the moduli [3,4] yields values for k t which are of order 0.3 kt/å, quite similar to the 0. kt/å calculated by May et al. []. In systems with a finite thickness mismatch (h C h ) the minimum in the line tension shifts away from θ=0. Also, unlike the thickness-matched case, even at the otimal contact angle the line tension is finite, due to the erturbation induced by the thickness mismatch. Minimization of the line tension with resect to the contact angle yields sinh 1 ffiffi C ¼ B1=4 h K 1=4 C ðh h C Þ ð6þ 35B C h3= K1=4 C K1=4 þ 17B h3= C K1= and the otimal (minimal) line tension is then g h 119B ¼ C B K1=4 C K1=4 ðh C h Þ ffiffi ð7:aþ 35B C h3= K1=4 C þ 17B h3= C K1=4 The ratio between the otimal line tension at the referred contact angle θ and the flat-lying domain where θ=0 is given by g h 17 B gð0þ ¼ C h3= K1=4 C þ B h3= C K1=4 u 17ð1 þ aþ ð7:bþ 35B C h3= K1=4 C þ 17B h3= C K1= ð35 þ 17aÞ where a ¼ B h3= K1=4 = B C h3= K1=4 C. Note that Eq. (7.b) does not strictly aly to thickness-matched hases where both γ(θ ) and γ(θ) are zero. Alying tyical numbers for the moduli and thickness [3,4] yields that the line tension at the otimal contact angle for a membrane with a 0% thickness mismatch between hases may be lower than the line tension of the flat domain by 40%.

5 1194 K.B. Towles, N. an / Biochimica et Biohysica Acta 1778 (008) Conclusions Thickness mismatch between membrane hases gives rise to a acking-induced line tension, arising from the deformation of the liid organization near the interface. Previous studies [15 17] have calculated this line tension for domains embedded in a continuous hase, assuming that the domain remains within the lane of the vesicle, finding values for the line tension that are between 50% and 100% higher than those measured exerimentally [5,7,1,13]. Several studies [5 8] find that hase-searated vesicles undergo shae transitions resulting in a finite contact angle between domains. In this aer we calculate the effect of a finite contact angle between a domain hase and a continuous hase on the acking line tension. As may be exected (Fig. ), we find that the acking line tension is zero when there is no thickness mismatch and the domain contact angle is zero. Forcing the domain to rotrude and form a non-zero contact angle with the continuous hase leads, in this case, to an energetic enalty that is due to liid tilt. However, unlike revious analysis where tilt was assigned its own modulus [may tilt], our model accounts for the tilt enalty through a combination of the area and bending moduli. In systems where there is a thickness mismatch between the domain and continuous hases, we find that the line tension is quite sensitive to the contact angle (Fig. ), achieving a minimum at a finite value of the contact angle that is set by the hase roerties and the thickness mismatch (Eq. (6)). The reduction in the line tension due to the adotion of the otimal contact angle, when comared to the flat domain case, is significant and may be of order 40 50%. Moreover, such reductions in the line tension may be obtained at relatively low values of the contact angle. For examle, a reduction of order 40% in the line tension, when comared to the flat domain, may be obtained for a thickness mismatch of order 0% if the domain angle is of order 0.3 rad, or 14. These values are within the same order of magnitude as those calculated by Fournier and Ben Amar [1], although their analysis alies to small domains with high sontaneous curvature (while our domains are large and have zero sontaneous curvature). It should be noted that in our calculations we did not account for saddle-slay deformation [5 7], so that our calculation strictly alies only to strie like domains. However, in systems with sherical symmetry such as circular domains, the bending and Gaussian moduli can be combined into an effective bending modulus whose magnitude is similar to the bending modulus K [8]. As a result, including the Gaussian modulus would lead to only a minor change in the value of α in Eq. (7.b), but would not affect our qualitative findings regarding the deendence of the line tension of the contact angle. Our results suggest an exlanation for the high values of revious estimates of the acking line tension, calculated for flat domains [15 18], when comared to the exerimentally determined ones [5,7,1,13]. It should be emhasized, though, that the calculation resented here focuses on the line tension only. As reviously shown [9,10], the formation of a contact angle between the domain and the continuous hase is associated with an increase in the overall bending energy of the domain (see Fig. 1). The associated bending enalty is likely to be negligible for large domains where the ensuing radius of curvature is low, but for small domains where the radius of curvature is of order the bilayer thickness this enalty may dominate over the reduction in line tension, thereby enforcing a zero contact angle. Acknowledgments Thanks to Tobias Baumgart and Sylvio May for their helful discussions. KBT was suorted by the US eartment of Education GAANN award # P00A0300. References [1] K. Simons, W.L.C. Vaz, Model systems, liid rafts, and cell membranes, Annu. Rev. Biohys. Biomol. Struct. 33 (004) [] Liowsky, R. imova, omains in membranes and vesicles, J. Phy., Condens. Matter 15 (003) S31 S45. [3] S.L. Veatch, S.L. Keller, Searation of liquid hases in giant vesicles of ternary mixtures of hosholiids and cholesterol, Biohys. J. 85 (003) [4] E. London, How rinciles of domain formation in model membranes may exlain ambiguities concerning liid raft formation in cells, Biochim. Biohys. Acta, Mol. Cell Res (005) [5] T. Baumgart, S.T. Hess, W.W. Webb, Imaging coexisting fluid domains in biomembrane models couling curvature and line tension, Nature 45 (003) [6] K. Bacia, P. Schwille, T. 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6 K.B. Towles, N. an / Biochimica et Biohysica Acta 1778 (008) [3] J.F. Nagle, S. Tristram-Nagle, Structure of liid bilayers, Biochim. Biohys. Acta 1469 (000) [4] W. Rawicz, K.C. Olbrich, T. Mcintosh,. Needham, E. Evans, Effect of chain length and unsaturation on elasticity of liid bilayers, Biohys. J. 79 (000) [5] S.T. Milner, T.A. Witten, Bending moduli of olymeric surfactant interface, J. Phys. Fr 49 (1988) [6].P. Siegel, M.M. Kozlov, The Gaussian curvature elastic modulus of N- monomethylated dioleoylhoshatidylethanolamine: relevance to membrane fusion and liid hase behavior, Biohys. J. 87 (004) [7] R.H. Temler, B.J. Khoo, J.M. Seddon, Gaussian curvature modulus of an amhihilic monolayer, Langmuir 14 (1998) [8] The Helfrich free energy for a monolayer is classically written as [see, for examle, [1 3]]. dg m ¼ df 0 þ 1 Kh ðc 1 þ c c 0 Þ þ K P h c 1 c, where df 0 is the energy of a flat membrane (a function of B, the area modulus), c 0 the sontaneous curvature, c 1 and c are the rincial curvatures, and K is the Gaussian slay modulus. r defines the radial distance. In our notation, if the deformation is sherically symmetrical so that c 1 = c and the sontaneous h icurvature is zero (which alies, as h i a rule to liids). dg m ¼ B þ Kh d P h i dr þ K h d dr ¼ B þ K h d dr,where K =K+ K. Thus, accounting for the Gaussian slay deformation would change the numerical value of the effective bending modulus, but not the deendence of the line tension on the contact angle. Moreover, it has been shown that in many cases [1 3] K ~ (/3)K,sothatK (4/3)K K.