Fourth-Order Curvature Energy Model for the Stability of

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Fourth-Order Curvature Energy Model for the Stability of Bicontinuous Inverted Cubic Phases in Amphiphile-Water Systems David P. Siegel* Givaudan Inc., 1199 Edison Drive, Cincinnati, Ohio 45216 Received December 22, 2009. Revised Manuscript Received March 6, 2010 The bicontinuous inverted cubic (QII) phases of amphiphiles in water have many practical applications. It is necessary to understand the stability of these phases as a function of composition and ambient conditions in order to make the best use of them. Moreover, many biomembrane lipids and some biomembrane lipid extracts form QII phases. The stability of QII phases in a given lipid composition is closely related to the susceptibility of that composition to membrane fusion: changes in composition that stabilize QII phases usually increase the rate of membrane fusion. However, the factors determining QII phase stability are not fully understood. Previously, an expression was derived for the curvature free energy of QII phases with respect to that of the lamellar (LR) phase using a model for the curvature energy with terms up to fourth order in curvature as formulated by Mitov. Here this model is extended to account for the effects of water content on QII phase stability. It is shown that the observed LR/QII phase-transition temperature, transition enthalpy, and transition kinetics are all sensitive to water content. The same observables also become sensitive to small noncurvature energy contributions to the total free-energy difference between the QII and LR phases, especially the unbinding energy in the LR phase. These predictions rationalize earlier observations of QII phase formation in N-monomethylated dioleoylphosphatidylethanolamine that otherwise appear to be inconsistent. The model also provides a fundamental explanation of the hysteresis typically observed in transitions between the LR and QII phases. It is an accurate model of QII phase stability when the ratio of the volume fraction of the lipid in the QII phase unit cell is e0.5.

Introduction Lipids in bicontinuous inverted cubic (QII) phases have several biotechnological applications. They are used as media for the crystallization of membrane proteins.1-3Bulk cubic phase or dispersions of particles of the modified cubic phase are used for the controlled release of pharmaceuticals or food bioactives, as described in recent reviews.4-9 QII phases have been used as media for chemical reactions that benefit from a large specific interfacial area or a large amount of interfacial curvature.10 QII phases are also an important part of the phase repertoire of biomembrane lipids. Some biomembrane polar lipid extracts form QII phases under circumstances that are not far from physiological conditions.11 It is important to study the stability of QII phases versus that of lamellar liquid crystalline (LR) phases E-mail: [email protected]. (1) Johansson, L. C.; W€ohri, A. B.; Katona, G.; Engstr€om, S.; Neutze, R. Curr. Opin. Struct. Biol. 2009, 19, 372–378. (2) Caffrey, M; Cherezov, V. Nat. Protoc. 2009, 4, 706–731. (3) Kors, C. A.; Wallace, E.; Davies, D. R.; Li, L.; Laible, P. D.; Nollert, P. Acta Crystallogr., Sect. D: Biol. Crystallogr. 2009, 65, 1062–1073. (4) Larsson, K. Curr. Opin. Colloid Interface Sci. 2009, 14, 16–20. (5) Amar-Yuli, I.; Libster, D.; Aserin, A.; Garti, N. Curr. Opin. Colloid Interface Sci. 2009, 14, 21–32. (6) Sagalowicz, L.; Mezzenga, M.; Leser, M. E. Curr. Opin. Colloid Interface Sci. 2006, 11, 224–229. (7) Sagalowicz, L.; Leser, M. E.; Watzker, H. J.; Michel, M. Trends Food Sci. Technol. 2006, 17, 204–214. (8) Yaghmur, A.; Glatter, O. Adv. Colloid Interface Sci. 2009, 147-148, 333– 342. (9) Efrat, R.; Kesselman, E.; Aserin, A.; Garti, N.; Danino, D. Langmuir 2009, 25, 1316–1326. (10) Vauthey, S.; Milo, C.; Frossard, P.; Garti, N.; Leser, M. E.; Watzke, H. J. J. Agric. Food Chem. 2000, 48, 4808–4816. (11) Koynova, R.; MacDonald, R. C. Biochim. Biophys. Acta 2007, 1768, 2373– 2382. (12) Chernomordik, L. V.; Kozlov, M. M. Nat. Struct. Mol. Biol. 2008, 15, 675– 683.

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in order to understand how biomembranes fuse and undergo fission, as in the processes of exocytosis and endocytosis.12 The mechanism of membrane fusion is closely related to the mechanism of the LR-to-QII phase transition.13,14 Moreover, the freeenergy difference between lipid in the QII phases and in the LR phase and the energy of intermediates in the process of lipid membrane fusion are both controlled by the value of the Gaussian curvature elastic modulus of the membrane.13,15,16 The value of the Gaussian curvature elastic modulus can be most accurately determined by experiments on QII phases.15,17 Finally, one way that fusion-mediating proteins may lower the energy of membrane fusion intermediates is by altering the curvature moduli of the lipid monolayers at the fusion site.18,19 The effects of isolated fusogenic peptides on these moduli can be tested by appropriate experiments on the QII phase and HII phase stability in lipid-peptide mixtures.17 An interpretation of the data from such experiments requires a thorough understanding of the factors determining QII phase stability in lipid systems. Our understanding of QII phase behavior and phase-transition kinetics is incomplete. The transitions between the LR and QII phases are hysteretic, and experimental values of the transition temperatures, transition kinetics, and transition enthalpies differ in significant details even for a well-studied pure lipid that forms (13) Siegel, D. P.; Kozlov., M. M. Biophys. J. 2004, 87, 366–374. (14) Siegel, D. P. The Relationship Between Bicontinuous Inverted Cubic Phases and Membrane Fusion. In Bicontinuous Liquid Crystals; Lynch, M. L., Spicer, P. T., Eds.; Surfactant Science Series; Taylor and Francis: Boca Raton, FL, 2005; Vol. 127, pp 59-98. (15) Siegel, D. P. Biophys. J. 2008, 95, 5200–5215. (16) Kolzovsky, Y.; A. Efrat, A.; Siegel, D. P.; Kozlov, M. M. Biophys. J. 2004, 87, 2508–2521. (17) Siegel, D. P. Biophys. J. 2006, 91, 608–618. (18) Zimmerberg, J.; Kozlov, M. M. Nat. Rev. Mol. Cell Biol. 2006, 7, 9–19. (19) Campelo, F.; McMahon, H. T.; Kozlov, M. M. Biophys. J. 2008, 95, 2325– 2339.

Published on Web 03/29/2010

DOI: 10.1021/la904838z

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QII phases: N-monomethylated DOPE.20-24 Successful models of QII phase stability must rationalize the observed range of stability of QII phases versus that of the LR phase and the observed changes in the QII phase cell constant in response to changes in lipid composition, water content, and temperature. There is extensive literature on modeling the relative stability of inverted phases in amphiphile systems. Recently, self-consistent field theory (SCFT) has been used to rationalize the phase behavior of amphiphiles by Schick and colleagues25,26 and Fredrickson and colleagues.27,28 These are ambitious efforts to understand large portions of the lipid-water phase diagram in terms of molecular properties and the details of the intermolecular interactions, and these workers have rationalized many features of inverted phase behavior in some lipid-water systems. The present work has the more modest goal of explaining the behavior of bicontinuous inverted cubic phases in the comparatively waterrich portions of the phase diagram for phospholipids (g50% by volume of water). A continuum model for the curvature energy of lipid monolayers is used. This approach has been taken previously by many authors in modeling the stability of bicontinuous QII phases.29-40 The curvature energy model does not explain the phase diagram in terms of the details of lipid molecular structure. However, it can rationalize and accurately predict the range of QII phase stability with respect to the LR and HII phases and the dependence of the QII phase lattice constants on temperature and water content in the water-rich region of the phase diagram (>50% by volume). All of the required elastic constants are rather precisely fixed by experiments on the LR, HII, and QII phases as a function of temperature and water content. Previous workers41 used a curvature energy model to reproduce a phospholipid-water phase diagram accurately across a substantial range of water content. Another virtue of the curvature energy model is that it can easily describe the behavior of lipids with complex molecular structure (e.g., phospholipids) without the extensive calculations required for an SCFT description.25-28 Recently, a fourth-order curvature energy model formulated by Mitov42 was applied to the problem of QII phase stability, and a simple expression was derived for the curvature free energy in terms of the curvature elastic parameters of the lipid composition.17 The (20) Gruner, S. M.; Tate, M. W.; Kirk, G. L.; So, P. T. C.; Turner, D. C.; Keane, D. T.; Tilcock, C. P. S.; Cullis, P. R. Biochemistry 1988, 27, 2853–2866. (21) Siegel,D. P.; Banschbach, J. L. Biochemistry 1990, 29, 5975-5981. (22) van Gorkum, L. C. M.; Nie, S.-Q.; Epand, R. M. Biochemistry 1992, 31, 671–677. (23) Cherezov, V.; Siegel, D. P.; Shaw, W.; Burgess, S. W.; Caffrey, M. C. J. Membr. Biol. 2003, 195, 165–182. (24) Siegel, D. P.; Tenchov, B. Biophys. J. 2008, 94, 3987–3995. (25) M€uller, M; Schick, M. Phys. Rev E 1998, 57, 6973–6978. (26) Li, X.; Schick, M. Biophys. J. 2000, 78, 34–46. (27) Lee, W. B.; Mezzenga, R.; Frederickson, G. H. Phys. Rev. Lett. 2007, 99, 187801–1-4. (28) Lee, W. B.; Mezzenga, R.; Frederickson, G. H. J. Chem. Phys. 2008, 128, 074504–1-10. (29) Helfrich, W.; Rennschuh, H. J. Phys., Colloq. 1990, 51, 189-195. (30) Ljunggren, S.; Eriksson, J. C. Langmuir 1992, 8, 1300–1306. (31) Templer, R. H.; Seddon, J. M.; Warrender, N. A. Biophys. Chem. 1994, 49, 1–12. (32) Templer, R. H.; Turner, D. C.; Harper, P.; Seddon, J. M. J. Phys. II 1995, 5, 1053–1065. (33) Engblom, J.; Hyde, S. T. J. Phys. II 1995, 5, 171–190. (34) Templer, R. H. Langmuir 1995, 11, 334–340. (35) Templer, R. H.; Seddon, J. M.; Duesing, P. M.; Winter, R.; Erbes, J. J. Phys. Chem. 1998, 102, 7262–7271. (36) Schwarz, U. S.; Gompper, G. J. Chem. Phys. 2000, 112, 3792–3801. (37) Schwarz, U. S.; Gompper, G. Langmuir 2001, 17, 2084-2096. (38) Schwarz, U. S.; Gompper, G. Phys. Rev. Lett. 2000, 85, 1472-1475. (39) Shearman, G. C.; Khoo, B. J.; Motherwell, M. L.; Brakke, K. A.; Ces, O.; Com, C. E.; Seddon, J. M.; Templer, R. H. Langmuir 2007, 23, 7276–7285. (40) Shearman, G. C.; Ces, O.; Templer, R. H. Soft Matter 2010, 6, 256–262. (41) Kozlov, M. M.; Leikin, S.; Rand, R. P. Biophys. J. 1994, 67, 1603–1611. (42) Mitov, M. D. C. R. Acad. Bulg. Sci. 1978, 31, 513-515.

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most important temperature-dependent curvature parameter is the spontaneous curvature, which is known across a range of temperature for DOPE-Me.13 The model is consistent with the observed temperature dependence of the QII phase cell constant for DOPE-Me.17 Here the same model is extended to predict the free energy, enthalpy, and unit cell constant of QII phases as a function of sample water content and provide a fundamental explanation for the hysteresis in phase transitions between LR and QII phases. Expressions are also derived for the influence of the noncurvature energy contributions on the LR/QII phasetransition temperature and transition enthalpy. It is shown that the model resolves most of the apparent contradictions in the observations concerning DOPE-Me. This confirms that the model is an accurate description of the stability of phospholipid QII phases across a range of water content where the assumptions of the model are valid.

Theoretical Results We consider only systems in which the lipid molecules are insoluble in water and have no net electrostatic charge. Let μQ be the total free energy difference between lipid in the QII and LR phases. First, we investigate the case where μQ is determined solely by the curvature free energy difference between the QII and LR phases, μQc. In the literature describing the curvature energy differences between lipid phases and fusion intermediates, the curvature energy of a lipid assembly of the lipid monolayer, μc, has almost always been calculated with the Helfrich equation:43 " μc ¼ Am

km k m Js 2 ðJm - Js Þ2 þ KKm 2 2

# ð1Þ

Am is the monolayer area. km and κ are the bending (splay) modulus and the Gaussian (saddle splay) modulus of the lipid monolayers, respectively. Jm, Js, and Km are the monolayer mean curvature, spontaneous curvature, and Gaussian curvature as defined at the neutral plane, respectively. The third term in Js2 sets the zero of curvature energy as the flat bilayers of the LR phase (Jm = 0). The terms in eq 1 are second order in curvature: in the absence of lipid molecule tilt, as assumed in this work, Jm = (1/R1 þ 1/R2) and Km = 1/R1R2, where R1 and R2 are the two principal radii of curvature of the surface. The most important temperature-dependent curvature parameter in eq 1 is Js. Equation 1 may not be an accurate description of μc under two circumstances. First, this equation was derived to describe the energy for small deformations of the monolayer, when the radii of curvature of the lipid monolayers are large compared to the length of a lipid molecule. As the radii of curvature decrease to molecular dimensions, the equation may become inaccurate. In this case, terms that are third and fourth order in curvature may be necessary to calculate μc. These terms can be thought of as correction terms that make insignificant contributions when the curvature is small enough for eq 1 to be valid. Second, for IPMS-based lipid phases, κ is 30 nm are stable in excess water.23,24,45 Obviously some other factor minimizes μQ at these large values of the cell constant. Recently it was shown17 that a fourth-order curvature energy expression formulated by Mitov42 accurately predicts the observed temperature dependence of a QII phase cell constant. A fourth-order curvature energy model has also been used by others.29-31,37 However, in contrast with earlier work with fourth-order models, the Mitov model was applied and a method of experimentally measuring the contributions of the higher order terms was demonstrated.17 Because the higher-order curvature model is extended to explain other phenomena in the present work, it is summarized below. In a bicontinuous QII lipid phase, the bilayer midplane lies on a surface corresponding to one of three mathematically well-characterized infinite periodic minimal surfaces (IPMS). The three bicontinuous QII phases that have so far been observed in lipid systems37 belong to the symmetry groups Pn3m, Im3m, and Ia3d (with the corresponding IPMS designations D, P, and G according to the nomenclature of Schoen,46 respectively). The bilayer midplane lies on a minimal surface, so the mean curvature of the surface, J, is zero. The bilayer thickness is assumed to be constant in both the LR and QII phases, and the neutral planes of the monolayers are displaced normally by a constant distance δ from the bilayer midplane. It is also assumed that there is no energy of interaction between the two monolayers across the bilayer midplane. Let the mean curvature of the monolayer neutral planes in the QII phase be Ji, where the two monolayers are labeled with the 6 J and hence Ji 6¼ 0. Likewise, the Gaussian index i. In general, Ji ¼ curvature of the neutral planes, Ki, is different than the value at the bilayer midplane, K. Finally, if one defines an area A on the bilayer midplane, then the area of the segment of the neutral surface of a monolayer that is normally displaced from it will have a different area, Ai. Ji, Ki, and Ai of the monolayers can be expressed in terms of the bilayer midplane values J, K, and A by using a general relationship between parallel surfaces37,47,48 Ji ¼

Ki ¼

( J þ 2δK ð1 ( δJ þ δ2 KÞ K ð1 ( δJ þ δ2 KÞ

¼

¼

2δK ð1 þ δ2 KÞ K ð1 þ δ2 KÞ

Ai ¼ Að1 ( δJ þ δ2 KÞ ¼ Að1 þ δ2 KÞ

ð2Þ

ð3Þ

ð4Þ

The first equality in eqs 2-4 is general for any midplane surface. The positive signs in the J and δJ terms are valid for the outer monolayer, and the negative signs in the J and δJ terms are valid for the inner monolayer. The second equality in each relationship is valid for J = 0, which is the case for bicontinuous QII phases. The values of Ai, Ji, and Ki for the two monolayers in the QII phase are equivalent. The expression for the monolayer curvature energy is the second-order expression of Helfrich43 with additional third- and (45) Tenchov, B.; Koynova, R.; Rapp, G. Biophys. J. 1998, 75, 853–866. (46) Schoen, A. H. Infinite Periodic Minimal Surfaces without Self-Intersections; NASA Technical Note D-5541; Natl. Tech. Inf. Serv.: Springfield, VA, 1970; Document N70-29782. (47) Shemesh, T.; Luin, A.; Malhotra, V.; Boerger, K. N. J.; Kozlov, M. M. Biophys. J. 2003, 85, 3813–3827. (48) do Carmo, M. P. Differential Geometry of Curves and Surfaces; PrenticeHall: Englewood Cliffs, NJ, 1976: p 212.

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Table 1. Values of χ and SN for Different QII Phasesa Pn3m (D)

Im3m (P)

Ia3d (G)

χ S0 S1 S2

-2 -4 -8 1.91889 2.34510 3.09144 -4π -8π -16π 100.294 328.270 996.071 a Each column heading is the space group of the bicontinuous QII phase followed by the letter designation (in parentheses) of the corresponding infinite periodic minimal surface upon which the structure is based, according to the nomenclature of Schoen.46

fourth-order curvature terms as formulated by Mitov42 and applied in ref 47. The curvature free energy for each lipid monolayer, fi, is fi ¼ Ai

 km ðJi - Js Þ2 þ KKi þ η1 Ji 3 þ η2 Ji Ki þ η3 Ji 4 2 i þ η4 Ji 2 Ki þ K Ki 2

ð5Þ

ηj and κC are the elastic moduli of the respective third- and fourthorder curvature terms. These moduli have values that depend on the lipid composition, as do km, Js, and κ. It will be shown below that only two of these higher-order moduli affect the value of μQc in this model and that their contribution is determined by a single experimentally determined constant. As shown previously,17,24 the curvature energy per unit area of the bilayer at the midplane can then be written by substituting eqs 2-4 into eq 5, retaining terms up to fourth order in curvature. This yields fB ¼ ð1 þ δ2 KÞð2fi Þ ¼ K1 K þ K2 K 2

ð6Þ

K1 ¼ 2K - 4δkm Js þ δ2 km Js

ð7Þ

K2 ¼ 4δ2 km þ 4δη2 þ 2K

ð8Þ

where

It is convenient to make the substitutions M ¼

K δ2 J s 2 , K1 ¼ 2km ðM - xÞ , x ¼ 2δJs km 2

ð9Þ

The area average of eq 6 over the area of the unit cell will be the curvature energy per unit area of the bilayer in the QII phase. The integrals of KN over the areas of the unit cells of the different QII phases (including N = 0) are Z K N dA ¼ unit cell

SN ð2N c - 2Þ

ð10Þ

where c is the cell constant of the QII phase and the SN coefficients are geometric constants that have been calculated numerically for each of the relevant infinite periodic minimal surfaces.37,49 Table 1 lists the values of SN for the three bicontinuous QII phases. Via eqs 6 and 10, the area-averaged curvature energy per unit area at the bilayer midplane, fB, is given by  fB ¼ K1

   S1 S2 þ K 2 S0 c 2 S0 c4

ð11Þ

(49) Anderson, D. M.; Gruner, S. M.; Leibler, S. Proc. Natl. Acad. Sci. U.S.A. 1988, 85, 5364–5368.

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In the temperature range where QII phases are thermodynamically stable, the two terms in brackets in eq 11 are of opposite sign. In excess water, the QII phase cell will swell to the value (ceq) that minimizes eq 11. Minimization of fB with respect c at a constant value of Js yields the following expression for ceq:17

ceq

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K2 S2 ¼ km S1 ðM - xÞ

ð12Þ

Let the area per lipid molecule at the neutral plane of each monolayer be a. We make the approximation that the area/ molecule is a at the bilayer midplane. The curvature free energy per lipid molecule at the bilayer midplane, μQc, is μQc

"    # a S1 S2 þ K2 ¼ K1 2 S0 c2 S0 c4

ð13Þ

At temperatures where the QII phase is thermodynamically stable, let the value of μQc when there is enough water for the QII phase to swell to c = ceq be denoted μQc0. An expression for μQc0 is derived by inserting eqs 7, 8, and 12 into eq 13 μ0Qc ¼ -

a Ωkm 2 ðM - xÞ2 2 K2

ð14Þ

where Ω = S12/S2S0. Ω is a constant with the same value (= 0.820545) for all three of the QII phase geometries discussed here. All of the curvature elastic constants in eq 14 can be determined experimentally. The value of km can be measured by appropriate experiments on HII phases.50 The values of M and -(κ2/km)(S2/S1) can then be determined by fitting eq 12 to a plot of the values of ceq at different temperatures versus x or T. x is calculated from experimentally determined values of Js and δ via eq 9. For DOPE-Me, we use the experimentally determined values of Js as a function of temperature and δ (1.3 nm).13 The value of Js decreases with increasing temperature in a linear fashion in the range where the spontaneous formation of the QII phase is observed for DOPE-Me. Js is obtained using the temperaturedependent change in the dimensions of a fully swollen HII phase prepared in the presence of excess long-chain alkane.50 A temperature dependence of Js similar to that of DOPE-Me is found in the X-ray diffraction data for several species of phosphatidylethanolamines41,51-53 and for a mixture of phosphatidylethanolamine and phosphatidylcholine.52 It is likely to hold for other biomembrane lipids as well. Figure 1 is a plot of the experimentally determined values of ceq versus T for the Pn3mI phase of DOPEMe23 comparing the data with the fitted curve for QII-Pn3m. With the assumption that M is constant over the temperature range, the fitted values are M = -0.901 ( 0.004 and κ2/km = 2.44 ( 0.29 nm2. The good fit of ceq to the form predicted in eq 12 and the fact that the same value of M is obtained if the points above 65 C are omitted13 support the assumption that M is constant between 50 and 90 C.17 Theoretical plots for the Im3m and Ia3d phases in excess water are also shown; they were (50) Rand, R. P.; Fuller, N. L.; Gruner, S. M.; Parsegian, V. A. Biochemistry 1990, 29, 76–87. (51) Epand, R. M.; Fuller, N.; Rand, R. P. Biophys. J. 1996, 71, 1806–1810. (52) Tate, M. W.; Gruner, S. M. Biochemistry. 1989, 28, 4245–4523. (53) Harper, P. E.; Mannock, D. A.; Lewis, R. N. A. H.; McElhaney, R. N.; Gruner, S. M. Biophys. J. 2001, 81, 2693–2706.

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Figure 1. Plot of the equilibrium QII phase cell constant, ceq, vs T (eq 12). The symbols are the experimentally determined ceq for the QII-Pn3m phase in DOPE-Me.23 The long-dashed line is the fit17 of eq 12 to the data for DOPE-Me, and the dashed vertical line indicates the value of TK obtained from the fit (49.6 C; M = -0.901 ( 0.004; κ2/km = 2.44 ( 0.29 nm2). The other lines are the values of ceq for the two other QII phases plotted versus T as calculated with eq 12 using the same values of M and κ2.

Figure 2. Plot of M - x (eq 9) using values of δ and Js as a function of T from ref 13. The line is a linear regression fit as described in the text.

calculated using the values of M and κ2 obtained from a fit of the Pn3m data and the appropriate values of S1 and S2 from Table 1. The temperature dependence of μQc, μQc0, and ceq can be described more conveniently using the temperature dependence of M - x. Using the (constant) value of M determined from Figure 1 and the experimentally derived values of x as a function of T, the resulting value of M - x for DOPE-Me is plotted as a function of T between 50 and 75 C in Figure 2. It is found that Langmuir 2010, 26(11), 8673–8683

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Figure 3. (A) μQc for the QII-Im3m phase as a function of the cell constant at several temperatures in the vicinity of TK for a lipid with the curvature elastic parameters and lipid monolayer dimensions of DOPE-Me. (B) μQc for each of the three QII phases of DOPE-Me as a function of the lipid volume fraction, φl (eq 17), at several temperatures.

M - x is almost exactly linear in T. A linear regression fit (line in Figure 2) yields (R2 = 0.9997) ðM - xÞ ¼ BðT - TK Þ

ð15Þ -1

where temperatures are expressed in C, B = 0.0042887 C , and TK = 49.81 C. TK is the temperature for which M - x = 0. The significance of TK is clear from an inspection of eq 14: TK is the temperature at which the curvature free energy of the QII phase equals that of the LR phase in excess water. μQc0 decreases with further increases in T because Js decreases. By inserting eq 15 into eq 14, we can write a simple expression for μQc0 in terms of T: μ0Qc ¼ -

aΩk2m B2 ðT - TK Þ2 2K2

ð16Þ

μQc is calculated using a = 0.65 nm2, km = 9kBT,15 and κ = -8.1kBT, as determined from the fitted value of M via eq 9. km was measured at 32 C, so the free energies are reported in units of kBT at 32 C. It is found that μQc is small over a broad temperature range around TK (e.g., at TK þ 10, μQc = -0.00189kBT ≈ -1 cal/mol). Figure 3A is a plot of μQc as a function of c at different temperatures for one QII phase: QII-Im3m. The results for only one phase are plotted to simplify the Figure; qualitatively, the plots for the other two phases are the same. At T < TK, the QII-Im3m phase is unstable at all values of c. At T = TK, μQc decreases asymptotically to zero in the limit of infinite c. At temperatures just above TK, a minimum develops in the plots of μQc at the value c = ceq (eq 12). The minimum in the plots of μQc versus c deepens as T increases. Here we will calculate only μQc for systems with T g TK. At T < TK, a supercooled, metastable QII Langmuir 2010, 26(11), 8673–8683

phase would be expected to swell indefinitely (c f ¥). This is because M - x and κ1 are TK and the completion of the LR/QII phase transition is postponed to a temperature where c has a value such that all of the lipid in the sample can form a stable QII phase. Thus, it is obvious that the observed phase behavior can be sensitive to the water content of a given sample. In excess water at T g TK, the three QII phases are predicted to have the same curvature free energy. However, as described in the Discussion section, this apparent degeneracy is probably lifted by additional, smaller contributions to the free energy, which are not considered here. The three QII phases have different curvature energies at a given temperature in samples with lower water contents. It will be shown that the equilibrium water contents of the three phases are different under these circumstances. To describe the stability of QII phases versus the LR phase and the relative stability of the different QII phases at finite water content, we need an expression for μQc in terms of the water content of the different QII phases. DOI: 10.1021/la904838z

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Dependence of μQc on the Water Content. Let φl and φw be the lipid and water volume fractions in a QII phase, respectively, with φl = 1 - φw. Let the lipid monolayer thickness be l. φl is given by54     l 4πχ l 3 φl ¼ 2S0 þ c 3 c

ð17Þ

where χ is the Euler characteristic of the QII phase. Values of χ for the different phases are given in Table 1. Equation 17 can be inverted to give an expression for 1/c in terms of φl by using the approximation l/c = (φl/2S0) - (4πχ/6S0)(φl/2S0)3 in the cubic term: 2 3  3 !3 1 φ 4πχ φ 4πχ φ l l 5 ¼ l4 l c 2S0 6S0 2S0 6S0 2S0

ð18Þ

With l = 1.9 nm, eq 18 is accurate to within better than 0.1% for φl e 0.5. To calculate μQc as a function of φl, one calculates the value of c corresponding to a given φl with eq 18 and uses that value of c in eq 13. μQc is plotted against φl in Figure 3B for the three QII phases using the same curvature elastic constants as before. At each value of T, with increasing φl the minima of the curves always occur in the order QII-Im3m, QII-Pn3m, and QII-Ia3d. The minima are at values of φl corresponding to c = ceq for each phase. At a given temperature, the φl values of the minima for the three phases are the same to within several percent, although the values of ceq are in the ratio 1:1.279:1.576 for the Pn3m, Im3m, and Ia3d phases, respectively. This ratio is determined by the values of (S2/S1)1/2 for the respective phases (eq 12, Table 1). The equilibrium value of φl is denoted as φeq and is obtained from eqs 12, 15, and 17:

Figure 4. Plot of the values of the equilibrium lipid volume fraction of each of the fully swollen QII phases, φeq, as a function of temperature.



φeq

   S1 km BðT - TK Þ 1=2 4πχl 3 S1 km BðT - TK Þ 3=2 ¼ 2S0 l þ S2 3 S2 K2 K2

ð19Þ At a given temperature, the values of φeq for the Pn3m, Im3m, and Ia3d phases are in the ratio 1:1.047:1.070; respectively. The values of φeq for the three QII phases of DOPE-Me are plotted as a function of temperature in Figure 4. The amount of water in a given sample can limit the degree to which a QII phase can swell. Let φs be the lipid volume fraction within a lipid-water sample. We will consider only samples with φs e 0.5, which is well below the LR phase lipid volume fraction of ca. 0.63.20 In a sample where φs > φeq at T > TK, there is not enough water for all of the lipid in the sample to form a QII phase with c = ceq. In Figure 3B, it is clear that μQc becomes larger than μQc0 as φl is increased to above φeq. Therefore, in a sample with φs > φeq, lipid will enter the QII phase until either all of the lipid has formed a QII phase with φl = φs or until φl reaches the value for which μQc = 0 and there is no driving force for further QII phase formation. For sufficiently large φs (corresponding to a small water content), μQc will be >0 for all the three QII phases so that complete transformation of the LR phase into a QII phase is impossible. For example, it is clear from Figure 3B that for the QII-Im3m phase at T = TK þ 10K, because μQc = 0 at φs = ca. 0.45, all of the lipid in samples with φs e 0.45 can enter the QII-Im3m phase but this is not possible at φs = 0.5. Thus, the (54) Turner, D. C.; Wang, Z. G.; Gruner, S. M.; Mannock, D. A.; McElhaney, R. N. J. Phys. II 1992, 2, 2039–2063.

8678 DOI: 10.1021/la904838z

Figure 5. Predicted values of μQc of the three QII phases as a

function of the lipid volume fraction in the phase, φl, at 60 C for DOPE-Me.

value of φs can limit the fraction of the lipid in a sample that forms the QII phase and can determine the value of φl (and c) at the maximum extent of the LR/QII phase transition. The relative stability of the QII-Im3m, QII-Pn3m, and QII-Ia3d phases can change as a function of φs at constant T. Figure 5 is a plot of the predicted value of μQc for the three different QII phases against φl at 60 C (TK þ 10) for DOPE-Me. The three phases have the same curvature free energy for φl < 0.33. However, for larger values of φl the Ia3d phase is the most stable phase. In theory, transitions between the phases can be driven by increasing φs (dehydrating the sample) at constant temperature. However, the differences in μQc among the three phases are very small. At 60 C and φl = 0.45, the difference in μQc between the Im3m and Ia3d phases is about Langmuir 2010, 26(11), 8673–8683

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0.001kBT per lipid molecule (Figure 5), which is 0.6 cal/mol. Hence the thermodynamic driving force for such transitions is small, and kinetic factors may inhibit equilibration. Range of Ol for Which the Model Is Valid. The quantitative predictions in this work rest on the validity of eqs 11 and 14. In the derivation of these equations,17 the value of a at the neutral plane is assumed to be the same as at the bilayer midplane. However, the curvature of the bilayers changes the area per molecule as a function of depth in the monolayers (eq 4). The area/molecule at the bilayer midplane will be different from a by a factor of (1 þ (S1δ2/S0c2))-1, which is >1 because S1 is TK at which φeq = φs. The transition occurs in the temperature interval between TK and T0 , and increasing φs increases the width of this interval. For example, if φs = 0.40 for a DOPE-Me sample, the LR/QIIIa3d phase transition cannot be completed until T0 is ∼63.5 C (Figure 4). The finite lipid content of the samples also makes the phase transition weakly first order, with a finite transition enthalpy. An inspection of eqs 16 and 20 shows that any lipid that transforms from the LR to the QII phase at T > TK experiences a finite change in the curvature free energy and entropy, respectively. Let ΔhQ be the partial molar enthalpy difference between lipids in the QII and LR phases at a given DOI: 10.1021/la904838z

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unbinding energy per lipid molecule, fu > 0. In this case, μQ is given by μQ ¼ μQc 0 þ fu ¼ μQc 0 þ ðfQ - fL Þ

ð24Þ

where fQ and fL are the free energies of interaction of the bilayers in the QII and LR phases, respectively. The value of fL falls in the range of -0.001kBT to -0.01kBT per lipid molecule for lipids such as DOPE-Me and DOPE, respectively.57-60 It is assumed that fu is temperature-invariant near TK and that the principal long-range forces acting between membranes are van der Waals forces. fQ is estimated as follows. The separation between lipid-water interfaces in QII phases, D, can be estimated using an expression for rw, the radius of the water channels in a QII phase:61 0sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 S 0 c - lA D ¼ 2rw  2@ 2πχ

Figure 7. Completion temperature, T0 , (left-hand axis) and total

enthalpy change (right-hand axis) during the LR/QII -Ia3d phase transition as a function of the sample lipid volume fraction, φs.

T when φeq g φs. ΔhQ is calculated using the general relationship ΔhQ = μQC0 þ TΔSQ and eqs 16 and 20: ΔhQ ¼

aΩkm 2 B2 ðT 2 - TK 2 Þ 2K2

ð23Þ

At T > TK, ΔhQ is >0, so the transition is endothermic. The total enthalpy change between lipid in the LR phase at T = TK and lipid in the QII phase with φl = φeq at T = T0 is pathindependent. Therefore, the value of ΔhQ at T = T0 is the total molar enthalpy of the transition, ΔHQ, which occurs over the interval TK to T0 . As φs increases, T0 increases and ΔHQ increases. The values of ΔHQ for the Im3m, Pn3m, and Ia3d QII phases at a given T are different because the values of φeq are different at any T > TK (Figure 4.) The value of ΔHQ for each QII phase at a given value of φs is calculated by using eq 18 to obtain the corresponding value of ceq. Then eq 12 is used to calculate the corresponding values of M - x and thus T0 (eq 15). This value is then inserted into eq 23. Figure 7 is a plot of T0 and ΔHQ for DOPE-Me in the QII-Ia3d phase as a function of φs. The units are calories/mol to ease comparison with calorimetric experiments. The predicted enthalpy of transition is modest and occurs over an increasing temperature range as φs increases. In comparing experimental values of ΔHQ to the values predicted here, care must be taken that the temperature scan rate is slow enough for the system to come to equilibrium at T0 .23,24 Obviously, these calculations also assume that no third phase (such as HII) forms in the temperature interval between TK and T0 . Effect of Noncurvature Energy Contributions to μQ. Noncurvature energy contributions to μQ that destabilize the QII phase increase the temperature of the LR/QII phase transition and impose a finite enthalpy of transition, even in very dilute samples. As previously shown,24 an important example of a noncurvature energy contribution is the difference in the free energy of bilayer-bilayer interactions in the QII versus the LR phase. For lipids without a net electrostatic charge, in order to form the QII phase, work must first be done to separate the bilayers in the LR phase. This work is referred to as the 8680 DOI: 10.1021/la904838z

ð25Þ

With eq 25, it is found that the lipid/water interfaces in QII phases at φl e 0.5 are separated by more than 7 nm. It is assumed that the forces per unit area of bilayer in the QII phase are the same as those between planar bilayers of equivalent separation. The free energy of van der Waals attraction between the bilayers in the QII phase is then given by62 " #   a H 1 2 1 fQ  þ 2 12π D2 ðDþ2lÞ2 ðD þ 4lÞ2

ð26Þ

where H is a Hamaker coefficient. Using H = 4  10-14 erg, which is typical of phospholipids, one can show that for a QII-Pn3m phase with φl e 0.5, fQ g -7  10-5kBT per lipid. Therefore, |fQ| , |fL|. To predict the influence of fu on μQ, TQ, and ΔHQ, we have to determine the effect of fu on ceq because this could change the magnitude of the curvature energy term in μQ. It can be shown that     ðceq Þ2 ðceq Þ5 S0 dfu ¼ 1 ceq 2 2aK2 S2 dc

ð27Þ

where ceq* is the equilibrium value of c under the influence of fu and ceq is the value at fu = 0. We obtain the value of dfu/dc by differentiating eq 26: dfu dfQ ¼ dc dc 

H ¼a 6π

"

#0sffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 1 2 1 @ - S0 A ð28Þ þ D3 ðD þ 2lÞ3 ðD þ 4lÞ3 2πχ

For a Pn3m phase with φl = 0.5, a numerical analysis of eq 27 shows that ceq* = ceq to within about 0.1%. The change in ceq* is (57) Rand, R. P.; Parsegian, V. A. Biochim. Biophys. Acta 1989, 988, 351–376. (58) Needham, D.; Zhelev, D. V. Use of Micropipitte Manipulation Techniques to Measure the Properties of Giant Lipid Vesicles. In Giant Vesicles; Luisi, P. L., Walde, P., Eds.; John Wiley & Sons: New York, 2000; pp 103-47. (59) Evans, E.; Needham, D. J. Phys. Chem. 1987, 91, 4219–4228. (60) Evans, E.; Needham, D. Faraday Discuss. Chem. Soc. 1986, 81, 267-280. (61) Kraineva, J.; Narayanan, R. A.; Kondrashkina, E.; Thiyagarajan, P.; Winter, R. Langmuir 2005, 21, 3559–3571. (62) Parsegian, V. A. van der Waals Forces: A Handbook for Biologists, Chemists, Engineers, and Physicists; Cambridge University Press: Cambridge, U.K., 2006; p 215.

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Figure 8. Dependence of the LR/QII phase transition temperature in excess water, TQ, (left-hand axis) and the enthalpy change during the LR/QII phase transition, ΔHQ, (right-had axis) on the magnitude of fu, the unbinding energy contribution to μQ (eqs 29 and 30, respectively). The excess-water condition is defined as φs e φeq for T g TQ.

smaller for smaller values of φl. Thus, van der Waals interactions between the bilayers have a negligible effect on ceq and μQc0. For μQ to be e0, the system has to be heated to some temperature >TK to overcome the effect of fu. Let TQ be the temperature at which μQ = 0. It is assumed that φs is e φeq for all three QII phases at T g TQ so that QII phase formation is not limited by the water content of the sample. TQ will then be a firstorder phase-transition temperature because there will be discontinuities in ΔSQ and ΔHQ (eqs 20 and 23). Setting μQ = 0 in eq 24 and using the expression for μQc0 (eq 14), we have sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2K2 fu - 2K2 fL TQ ¼ TK þ ¼ TK þ aΩB2 km 2 aΩB2 km 2

ð29Þ

where we have used fu = -fL in the second equality. Because we have assumed that fu is independent of T in an interval of tens of degrees around TK, by analogy to eq 23, we can write ΔHQ ¼

aΩkm 2 B2 ðTQ 2 - TK 2 Þ þ fu 2K2

ð30Þ

Figure 8 is a plot of the values of TQ and ΔHQ as a function of fu for a lipid with the curvature elastic parameters of DOPE-Me for all three QII phases in excess water at T g TQ. Both TQ and ΔHQ increase rapidly with increasing fu. For samples with φs g φeq at TQ, TQ will be the onset temperature for QII phase formation but the transition will not be complete until a higher temperature T0 is reached for which φs = φeq, and ΔHQ will be given by eq 23 with T = T0 .

Discussion In this work, it is shown that the LR/QII phase transition temperature and transition enthalpy are sensitive to water content and to small noncurvature energy effects, especially the unbinding energy in the LR phase. It is shown below that these observations rationalize many observations of QII phase formation and stability in a well-studied system, DOPE-Me. Langmuir 2010, 26(11), 8673–8683

Lr/QII Phase-Transition Temperature. In the present model, the LR/QII phase transition is a second-order phase transition occurring at TK in infinitely dilute samples of noninteracting bilayers. When there is an LR phase unbinding energy, the temperature for the onset of QII phase formation, TQ, is >TK (eq 29). Either the unbinding energy or the water content of the sample can control the observed onset temperature, depending on the size of fu and φs: the QII phase will begin to form at TQ only if there is sufficient water present for the nascent QII phase to swell to a value of φl for which μQ e 0 (Figure 3B). In samples of finite lipid content, the temperature at which the LR/QII phase transition is complete is T0 > TK, where φs = φeq (Figure 4). The value of T0 is usually unaffected by the presence of an unbinding energy. If fu is very large, then it is possible for TQ to become so high that no QII phase forms before a higher-temperature phase such as HII forms.24 Otherwise, because the value of fu has a negligible effect on the value of ceq (eq 27), T0 is determined solely by the criterion φs = φeq(T). For DOPE-Me, an analysis of the temperature dependence of ceq indicates that TK = 50 C.17 By using the value of fu for a related N-monomethylated phosphatidylethanolamine, TQ was predicted24 to be 56 C in excess water. For l = 1.9 nm and 56 C, excess water for all three QII phases would be a sample with φs j 0.27 (Figure 4). There are conflicting observations of the temperature for QII-phase precursors22,63-66 and QII-phase20-24 formation in DOPE-Me. In the following discussion, we use volume percent and weight percent interchangeably because the density of similar phospholipids such as DOPE 52 and DOPC 67 is very close to that of water. Gruner et al.20 were the first to observe the QII phase in DOPE-Me: they found only slow disordering of the LR phase lattice but no QII phase formation during 20 h of incubation at 55 C for a 30 wt % sample and QII-phase formation over 1.5 years in a room-temperature sample, in which lipid hydrolysis had lowered the l-phase transition temperature. However, Cherezov et al.23 found that the QII phase begins to form after 6 h of incubation at 55 C in samples of approximately 30 wt % lipid. No QII phase was observed during 9 h of incubation at 52.8 C. The waiting time for the onset of QII-phase formation decreased rapidly as the sample incubation temperature increased from 55 to 57 C 23. Finally, Siegel and Tenchov24 reported that complete transformation to the QII phase occurs at 55 C after 45 min in samples with a lipid concentration of 16.7% (w/v). No QII phase formed at 55 C in the work of Gruner et al.,20 and Cherezov et al.23 observed some QII-phase formation after a long delay because the lipid concentration in both cases was ca. 0.3 whereas the value of φeq at this temperature is 0.23 to 0.25. A QII phase was observed after a shorter incubation at 55 C by Siegel and Tenchov24 because their value of φs was much smaller than φeq. These observations are thus all in good agreement with the predicted value24 of TQ = 56 C in excess water. Effects of Sample Water Content on QII-Phase Behavior. There are additional observations in the literature showing that the temperature at which the QII phase first appears depends on the water content of the sample. First, QII phases were not detected in 33 wt % lipid samples at 58-60 C whereas the QII (63) Gagne, J.; Stamatatos, L.; Diacovo, T. S.; Hui, W.; Yeagle, P. L.; Silvius, J. R. Biochemistry 1985, 24, 4400–4408. (64) Ellens, H.; Siegel, D. P.; Alford, D.; Yeagle, P. L.; Boni, L.; Lis, L. J.; Quinn, P. J.; Bentz, J. Biochemistry 1989, 28, 3692-3703. (65) Yeagle, P. L.; Epand, R. M.; Richardson, C. D.; Flanagan, T. D. Biochim. Biophys. Acta 1991, 1065, 49–53. (66) Epand, R. M.; Epand, R. F.; Richardson, C. D.; Yeagle, P. L. Biochim. Biophys. Acta 1993, 1152, 128–134. (67) Wiener, M. C.; White, S. H. Biophys. J. 1992, 61, 428–433.

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Pn3m phase was observed at T g 61.0 C.21 Second, in a later study at a lipid concentration of 16.7%24 only QII-Im3m reflections existed in X-ray diffraction patterns taken at 55 C. These two observations are consistent with the prediction here (Figure 7) that the range of T0 for the three QII phases is 59-60 C for 33 vol % lipid whereas it is only 52 C for 16.7 vol % lipid. Traces of the QII phase were probably not observed at 60 C in the earlier study21 because of the use of less-sensitive experimental methods than in the later study24 (rotating-anode X-ray source and photographic film versus a synchrotron source and electronic array detector, respectively). Third, in another study,23 samples with φs ≈ 0.30 incubated at T between 55 and 57.4 C formed coexisting LR and QII phases but only the QII phase at 58.3 and 64.6 C. These observations are consistent with the fact that φeq is 70 C for φs = 0.5: another less-hydrated phase must coexist with QII for TQ < T