Lipid Segregation on Cylindrically and Spherically Curved Membranes

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1 Lipid Segregation on Cylindrically and Spherically Curved Membrane where n and u are unit vector repreenting, repectively, lipid orientation and membrane normal, repreent the two-dimenional derivative on a curved urface[9, K i the play coefficient, and C i a coefficient meaurarxiv: v [cond-mat.oft 4 Mar 03 Fangfu Ye,, Robin L. B. Selinger, and Jonathan V. Selinger School of Phyic, Georgia Intitute of Technology, Atlanta, GA 3033, USA Liquid Crytal Intitute, Kent State Univerity, Kent, OH 444, USA (Dated: March 3, 03) We invetigate how an externally impoed curvature influence lipid egregation on two-phaecoexitent membrane. We how that the bending-modulu contrat of the two phae and the curvature act together to yield a reduced effective line tenion. On largely curved membrane, a tate of multiple domain (or raft) form due to a mechanim analogou to that cauing magneticvortex formation in type-ii uperconductor. We determine the criterion for uch multi-domain tate to occur; we then calculate repectively the ize of the domain formed on cylindrically and pherically curved membrane. PACS number: 87.6.dt, St, 6.30.Dk Introduction Lipid membrane play important role in maintaining cell integrity and intracellular trafficking [. At high temperature, the three main component of the membrane (aturated lipid, unaturated phopholipid and choleterol) form a homogeneou mixture; below a critical demixing temperature, the three component egregate into two coexitent fluid phae, a aturated-lipid-enriched liquid-ordered (L o ) phae and unaturated-phopholipid-enriched liquid-diordered (L d ) phae. Small L o domain formed on cell membrane are alo referred to a raft [. The L o phae (or raft) ha a larger bending modulu than the L d phae [3 5. There have been a large body of tudie [6 6 invetigating how membrane curvature influence the egregation of membrane lipid. All thee tudie are baed on a phyical mechanim that the L o phae becaue of it larger reitance to bending prefer flatter region and drive the L d phae to more curved region, i.e., thee tudie have implicitly aumed what matter i the variation of curvature rather than the curvature itelf. In thi letter, we invetigate how a uniform externally impoed curvature may influence lipid egregation on an L o -L d coexitent membrane. We how that, with the preence of a bending-modulu contrat, the curvature of the membrane induce the lipid to tilt away from the membrane normal and yield a reduced effective line tenion between the lipid domain; when the curvature i larger than a certain critical value, lipid egregation lead to the formation of multiple L o /L d domain (or raft) of microcopic lengthcale rather than a complete eparation of the two phae. We determine the criterion for the multi-domain pattern to occur; we then calculate repectively the ize of the domain formed on cylindrically curved and pherically curved lipid membrane. We point out that although inpired by lipid ytem the reult obtained in thi letter apply to any membrane that contain two ditinct mectic phae. General Picture Nonzero curvature of lipid membrane can be either induced by their intrinic ponta- a b FIG. : Two type of lipid configuration of curved membrane. The dimer repreent lipid, either lipid in bilayermembrane or lipid in unilamellar ytem. In (a), the lipid are perpendicular to the membrane urface; in (b), the lipid are parallel to each other. neou curvature or impoed externally. We conider the latter cae[7. Example of lipid membrane with an externally impoed curvature include in vivo membrane attached to BAR domain of protein [8 and in vitro membrane attached to rigid ubtrate. Fig. how two extreme type of lipid configuration reulting from an externally impoed curvature. In Fig. (a), the lipid are perpendicular to the membrane urface and the energy cot originate from the play energy cot of lipid orientation; in Fig. (b), the lipid are parallel to each other, and the energy penalty i induced by the deviation of lipid orientation from the membrane normal, or more microcopically, by the relative liding of the lipid with repect to one another. In general, the lipid configuration of a curved membrane hould be a uperpoition of the aforementioned two type. The correponding elatic energy denity thu include two part, the play energy and the tilt energy penalty, f = K( n) C(n u), ()

2 FIG. : Illutration of external-curvature-induced lipidtilting in the phae-boundary region. The olid and hollow dimer repreent, repectively, L o and L d lipid. The dahed line repreent the membrane normal at the phae boundary. ing how trongly n i locked to u. The combination of K and C give a characteritic length ξ = K/C, which i the penetration length of lipid tilting. For a inglephae membrane, given that ξ i uually much maller than the membrane ize, mot of the lipid are perpendicular to the membrane urface [ee Fig. (a), and the bending modulu of the membrane i motly determined by the play coefficient K. A bending-modulu contrat in literature thu correpond to a play-coefficient contrat. Note that we have ignored in Eq. () a divergence term becaue including it would not change our reult qualitatively. For an L o -L d -coexitent membrane, the total energy include the contribution from both phae. We will ue ubcript o and d to ditinguih the quantitie of thee two phae. For convenience in the expreion, we will alo ue the ubcript i, with i = repreenting the phae with maller volume fraction and i = repreenting the one with larger volume fraction; in a cae of no ambiguity, we will ue a ymbol without ubcript to repreent the correponding quantitie of both phae. The total energy of a two-phae-coexitent membrane hould alo include a phae-boundary energy cot, which equal the product of the line tenion t and the phae-boundary length. To minimize the boundary energy, the two phae on a flat membrane completely eparate from each other, with each phae forming a large ingle domain. However, for a curved membrane, uch a ingle domain may become untable and plit into multiple domain. We now preent the reaon for the plitting of large ingle domain and determine the correponding intability criterion. A illutrated in Fig., when ubjected to an externally impoed curvature, the L o lipid tilt away from the membrane normal and become more parallel to each other o a to reduce the play energy cot, given that K o i larger than K d. The anchoring energy, i.e., the econd term in Eq. (), contraint the tilting to occur within a narrow region of ize ξ around the phae boundary. The energy change induced by uch tilting can be computed. To implify calculation, we aume that the ize of the large ingle domain reulting from a complete phae eparation i much larger than ξ and that the impoed curvature i much maller than /ξ. Thu, in the aforementioned phae-boundary region, the play n i can be approximated by H +( ) i γ i / i, where H i the mean curvature, i i the ditance between the phae boundary and the lipid location in phae i (i.e., the arc length of the geodeic perpendicular to the phae-boundary line), and γ i i the angle between the membrane normal and the orientation of lipid of phae i. Note that the lipid now tilt in the plane defined by the membrane normal and the tangential vector of thee geodeic. The factor ( ) i i added into the expreion of n i becaue of the oppoite directionality of and. Subtituting the approximate expreion of n i into Eq. () and taking the functional derivative yield γ i = γ b exp( i /ξ i ), where γ b repreent the deviation angle at the phae boundary. The energy change reulting from the lipid tilting can then be obtained by minimizing over γ b the um of the energie of L o and L d. The combination of thi energy change and the line tenion t produce a reduced effective line tenion: t e = t H (K o K d ) Co K o + C d K d. () Eq. () clearly how t e can be lowered either by increaing H or the contrat between K o and K d or by decreaing C o and C d. The intability criterion of the ingle-domain tate can then accordingly be obtained by imply etting t e = 0. We can further implify thi criterion to an approximate form: t/h Kξ. Given that K i of order 0 9 J for mot lipid ytem [9 and ξ can be aumed to be comparable to the membrane thickne and i thu of order nm [0, the criterion i then t/h 0 8 J m. Depending on the compoition and temperature, the line tenion t on a lipid membrane uually varie between pn and 0.0pN (or even maller) [4, 5,,. For t pn, the critical curvature H c i thu of order 0 8 m ; for t 0.0pN (for example, in ytem with preence of hybrid lipid [3), H c i of order 0 7 m. A t e become negative, the large ingle domain reulting from a complete phae eparation become untable and plit into multiple mall domain. In the following ection, we calculate the ize of the lipid domain formed on cylindrically curved and pherically curved membrane, repectively. Lipid Segregation on a Cylindrically Curved Membrane For a lipid membrane ubjected to a uniform cylindrical curvature, we aume there i tranlational ymmetry along e z, the direction of the long axi of the cylinder, and that the lipid align perpendicular to e z. In thi cae, a domain i a trip along e z, and the threedimenional problem become a two-dimenional problem. The play of the lipid orientation can then be expreed a n i = co γ i [ + γ i / θ i /R c, where R c i the radiu of the cylinder and the θ are the azimuthal angle repreenting the lipid location in a domain (ee Fig. 3). In the center of a domain, i.e., at θ i = 0, the lipid align along the membrane normal with γ i = 0.

3 3 L Γ Γ L b Θ Θ ez c FIG. 3: Cartoon of L and L domain on a cylindrical urface. Subtituting the expreion of n i into Eq. () and then differentiating the energy denity with repect to γ i yield γ i = A i inh (θ i R c /ξ i ), where the A are the amplitude and the approximation ξ i R c and γ i have been ued. Approximately peaking, the deviation γ i decay exponentially for domain of ize much larger than ξ i and linearly for thoe of ize maller than ξ i. The amplitude A and A are not independent from each other they are related by the continuity condition γ (θ ) = γ ( θ ), where θ and θ are, repectively, the angular ize of the L and L domain (from the domain center to the domain edge). We then minimize the total energy (per unit length along the long axi of the cylinder) over the A and obtain E = π θ + θ [ t (K K ) /Rc i= Ci K i coth(θi R c/ξ i ), (3) which clearly how a competition between the energy penalty of the line tenion and the energy gain of the director tilting induced by the difference between K and K. The critical line tenion t c, below which the completely eparated tate become untable, i given by the maximum value of the econd term in the parenthei in Eq. (3), which i the ame a the one we can obtain from Eq. () by etting t e = 0 and H = /(R c ). At t < t c, a table multi-domain ground tate i expected. The domain ize can be determined by numerically minimizing Eq. (3) over θ (or θ ) with the contraint θ /θ = φ/( φ) applied, where φ i the volume fraction of phae. The reult are given in Fig. 4, which how that the domain ize decreae with t and quickly reduce to order ξ o. For φ = / [ee Fig. 4(a), becaue of the permutation ymmetry of the two lipid phae, the curve decribing how the domain ize varie with the normalized line tenion t (= t/t c ) for the K with a certain ratio v (i.e., K = vk ) overlap with the curve for the K with a ratio of /v (i.e., K = K /v). For φ < /, thi permutation ymmetry i broken: the L o domain formed on membrane with the L o phae having a volume fraction φ have a larger ize than the L d domain formed on the phae-permutated membrane, i.e., FIG. 4: Relation between the L -domain ize (meaured in unit ξ o, λ = θr c/ξ o) and the normalized line tenion t = t/t c on a cylindrically curved membrane for (a) φ = 0.5, (b) φ = 0.4, (c) φ = 0.3, and (d) φ = 0.0. The (blue) thick olid and dahed line correpond to K = K and K = K /, and the (red) thin olid and dahed line correpond to K =.K and K = K /., repectively. We have et C = C. membrane with the L d phae having a volume fraction φ. Moreover, the ize difference increae a φ decreae [ee Fig. 4(a) (d). The origin of thi ize difference and it variation with φ i given a follow. A can be een from Fig., with the preence of an external curvature, the L o lipid provide tilt-driving force while the L d lipid reit tilt. Membrane with the L d phae having volume fraction φ poe more L o lipid and thu tend to have more phae-boundary region o that tilt can occur. A preference for boundary region mean a preference for maller domain, and therefore lead to the aforementioned domain-ize difference and it variation with φ. In addition to the numerical reult, we alo preent in the following analytical reult for the mall t limit o a to clearly elucidate the dependence of the domain ize (i.e., θ and θ) on the parameter. We conider two cae eparately: (i) the volume fraction of the two phae are of ame order, i.e., φ /; and (ii) one phae ha a much maller volume fraction than the other, i.e., φ /. In the cae of φ /, when t t c, we have θi ξ i/r c for both phae, and the deviation angle γ and γ both decay (approximately) linearly from the phae boundary to the domain center. We thu approximate coth(x) in Eq. (3) a /x + x/3. We then differentiate the energy given in Eq. (3) with repect to θ, with the relation between θ and θ applied, and obtain ( 3t ) /3 [ + rφ/( φ) /3, θ (4) R c C r where C = C + C ( φ)/φ and r = K /K. We mention the following three noteworthy point contained in Eq. (4): (i) a large contrat between the K (i.e., large

4 4 r ) give a mall domain; (ii) although the angular ize θ decreae a R c increae, the linear domain ize (R c θ) increae with R c ; (iii) at φ = /, the permutation ymmetry i conerved, viz, the value of θ doe not change under the replacement r /r. We now turn to the cae of φ /. In thi cae, we have θ ξ /R c but θ ξ /R c ; γ decay linearly and γ decay exponentially from the phae boundary to the domain center. We thu replace coth(θr c /ξ ) and coth(θr c /ξ ) in Eq. (3) by ξ /(θr c ) and coth( ), repectively. Minimizing Eq. (3) then yield ( θ t ) / C K r, (5) which i inverely proportional to r and independent of the radiu R c (the linear domain ize i thu linearly proportional to R c ). In addition, the angular ize i now more enitive to the variation of t than the angular ize given in Eq. (4) i. Lipid Segregation on a Spherically Curved Membrane We proceed to addre lipid egregation on a pherically curved membrane. A complete phae eparation now lead to the formation of a ingle (curved) dik of L lipid embedded in an L -lipid ea. The intability criterion of the ingle-dik pattern an be obtained from Eq. () by etting t e = 0 and H = /R. For a negative t e, a multi-domain pattern form. In the cae of φ /, the hape of the formed multiple domain i complicated, and thi complication make an analytic calculation of the ytem energy inacceible. We will tudy the φ / cae in a numerical approach in a future publication and conider here only the cae of φ /. In thi cae, we have a pattern of multiple L dik embedded in an L ea. We firt give the explicit form of the energy a a function of the deviation angle γ and the equation determining how γ varie from the center of a domain to the domain edge. We adopt a pherical-coordinate decription for the poition of the lipid, with the point of zero polar angle (i.e., θ = 0) correponding to the center of an L domain, and we aume γ only depend on the polar angle θ but not the azimuth angle. Thu, for the total energy, we have E = πn d {R t in θ θ +R [ dθ in θ K ( n ) + C in γ +R 0 θ u θ n i = { [ co γ i (θ) R [ dθ in θ K ( n ) + C in γ }, (6) + γ i(θ) θ } + cot θ in γ i (θ), where R i the radiu of the phere, θ and N d = φ/( co θ ) are repectively the angular ize and the 6 4 FIG. 5: Relation between the L -domain ize (meaured in unit ξ o, λ = θr /ξ o) and the normalized line tenion t = t/t c on a pherically curved membrane for K = K [the (blue) thick olid line, K = K / [(blue) thick dahed, K =.K [(red) thin olid and K = K /. [(red) thin dahed with the aumption φ and C = C. total number of the L domain. The upper limit θ u of the econd integration equal to π in the ingle-domain cae; and it value in the multi-domain cae will be addreed later. Differentiating E with repect to γ, under the aumption that γ i mall, yield γ i θ + γ ( i R ) θ cot θ γ i ξi 4 + cc θ = 0. (7) We now proceed to determine the domain ize, i.e., the ize of the L dik. Given the contraint φ /, the ditance between the dik become much larger the dik ize; and, in the L ea, only a narrow annulu around the dik contribute to the ytem energy. Furthermore, becaue the domain ize i mall, the deviation angle i nonzero only at mall θ. We thu replace cot θ and cc θ in Eq. (7) by /θ, and obtain γ BI (b θ) and γ BK (b θ), where b i = (R /ξ i ) 4 R /ξ i, and BI n (x) and BK n (x) are, repectively, the modified Beel function of the firt and econd kind. To calculate the energy, we then follow a procedure imilar to that for the cylindrical cae and obtain [ R t E = 8πφ θ (K K ) W(θ ), (8a) W(θ) := K b θ BI 0 (b θ) BI (b θ ) + K b θ BK 0 (b θ) BK (b θ ). (8b) Minimizing thi energy yield the domain ize. Fig. 5 how the numerical reult, which are qualitatively imilar to the reult of cylindrically curved membrane with mall φ [ee Fig. 4(d). Concluding Remark We have hown that a curved two-phae-coexitent lipid membrane, compared with a flat one, ha a maller effective line tenion due to lipid tilting. Moreover, a large enough curvature lead to a negative effective line tenion, and thu induce a tate with multiple domain (or raft) of microcopic length cale determined by the play and anchoring coefficient.

5 5 We expect that the theoretical framework addreed in thi paper, if applied to mall intracellular veicle, ha ome biological ignificance. We mention that the multi-domain-inducing mechanim addreed in thi paper i different from that uing elatic repulion to tabilize multiple domain on a lipid veicle [4, 4, but analogou to the mechanim via which a large magnetic field induce the formation of magnetic vortice in type-ii uperconductor [5. The tilt penetration length ξ in thi work i the counterpart of the magnetic field penetration length in uperconductor. What correpond to the uperconducting coherence length i the correlation length of the denity fluctuation of the variou component of lipid membrane [. We have implicitly aumed thi correlation length i maller than ξ, viz., we have aumed a harp boundary between the two coexiting phae. We will preent elewhere how an externally impoed curvature influence lipid egregation on membrane with thi aumption violated, for example, on membrane with the temperature cloe to a critical demixing point. Acknowledgement We thank A. Traveet, P.M. Goldbart, T.C. Lubenky, and A.C. Shi for helpful dicuion. Thi work wa upported by the National Science Foundation via Grant DMR , DMR-0604 (RLBS, JVS) and DMR-0706 (FY). FY alo acknowledge the upport of the Center for the Phyic of Living Cell of the Univerity of Illinoi at Urbana-Champaign and the upport of the Branche Cot Sharing Fund of the Intitute for Complex Adaptive Matter. [ G. van Meer and H. Sprong, Curr. Opin. Cell Biol. 6, 373 (004); G. van Meer et al., Nat. Rev. Mol. Cell Biol. 9, (008). [ K. Simon and E. Ikonen, Nature 387, 569 (997). [3 S. L. Veatch and S. L. Keller, Biophy. J. 85, 3074 (003). [4 T. Baumgart, S. T. He, and W. W. Webb, Nature 45, 8 (003). [5 T. Baumgart et al., Biophy. J. 89, 067 (005). [6 T.-Y. Yoon et al., Nat. Mater. 5, 8 (006). [7 R. Parthaarathy, C. Yu, and J. T. Grove, Langmuir, 5095 (006); R. Parthaarathy and J. T. Grove, Soft Matter 3, 4 (007). [8 B. Rozycki, T. R. Weikl, and R. Lipowky, Phy. Rev. Lett. 00, (008). [9 H. Jiang and T. R. Power, Phy. Rev. Lett. 0, 0803 (008). [0 Q. Liang and Y.-Q. Ma, J. Phy. Chem. B 3, 8049 (009). [ B. Sorre et al., Proc. Natl. Acad. Sci. U.S.A. 06, 56 (009). [ M. Heinrich et al., Proc. Natl. Acad. Sci. U.S.A. 07, 708 (00). [3 T. Stogbauer, M. Hennig, and J. O. Radler, Biophy. Rev. Lett. 5, 53 (00). [4 M. I. Hoope, R. Faller, and M. L. Longo, Langmuir 7, 783 (0). [5 H. J. Rielada, S. J. Marrink, and M. Muller, Phy. Rev. Lett. 06, 480 (0). [6 H. Jiang, Phy. Rev. Lett. 09, 980 (0). [7 More preciely, we are intereted in membrane whoe curvature are different from their pontateou value, including flat membrane with nonzero pontaneou curvature (SC). Although thi paper focue on membrane having zero SC, the reult achieved here can be eaily generalized (with uitable modification) to thoe with nonzero SC. In particular, when applied to flat membrane with nonzero SC, our reult give the ize of raft. [8 H. T. McMahon and J. L. Gallop, Nature 438, 590 (005). [9 Z.-C. Ou-Yang, J.-X. Liu, and Y.-Z. Xie, Geometric Method in the Elatic Theory of Membrane in Liquid Crytal Phae (World Scientific, Singapore, 999). [0 M. Hamm and M. M. Kozlov, Eur. Phy. J. B 6, 59 (998); Eur. Phy. J. E 3, 33 (000). [ D. J. Benvegnu and H. M. McConnell, J. Phy. Chem. 96, 680 (99). [ A. R. Honerkamp-Smith et al., Biophy. J. 95, 36 (008). [3 R. Brewter, P. A. Pincu, and S. A. Safran, Biophy. J. 97, 087 (009). [4 T. S. Urell, W. S. Klug, and R. Phillip, Proc. Natl. Acad. Sci. U.S.A. 06, 330 (009). [5 M. Tinkham, Introduction to Superconductivity (McGraw-Hill, New York, 996).