Driving Force for Spontaneous Perforation of Bilayers Formed by Ionic

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Driving force for spontaneous perforation of bilayers formed by ionic amphiphiles in aqueous salt Ksenia Alexandrovna Emelyanova, and Alexey Victorov Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.7b02885 • Publication Date (Web): 24 Oct 2017 Downloaded from http://pubs.acs.org on October 25, 2017

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Driving force for spontaneous perforation of bilayers formed by ionic amphiphiles in aqueous salt Ksenia A. Emelyanova, Alexey I. Victorov* St. Petersburg State University, 7/9 Universitetskaya nab., 199034 St. Petersburg, Russia

ABSTRACT. Spontaneous perforation of amphiphilic membranes is important in both living matter and technology because of an impact on functions of biological membranes and shape transitions of self-assembling structures. Nevertheless, no definite molecular mechanism has been established so far even for simple ionic surfactant systems. We show that spontaneous perforation of a bilayer formed by an ionic amphiphile is driven by electrostatics. Creation of large pores with a concave-convex geometry of the rim is promoted by lower electrostatic free energy than that for a flat non-perforated bilayer. The opposite effect comes from the elasticity of the hydrocarbon tails of the amphiphile that prefer flat geometry of a non-perforated bilayer. The balance between electrostatics and tail deformation controls the appearance of pores; this balance is modulated by added salt that screens the electrostatic interactions. We illustrate the proposed mechanism with the aid of classical aggregation model that has been extended by including an analytical description of the electrostatic contribution for the toroidal rim of a pore. Numerical solution of the linearized Poisson-Boltzmann equation confirms the role of electrostatic forces in

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formation of pores. For the ionic surfactants of CnTAB family, we predict shape transitions including bilayer perforations and formation of branched micellar networks induced by changing salinity or temperature and demonstrate the effect of surfactant’s molecular parameters on these transitions.

INTRODUCTION Formation of pores in biological membranes has crucial effect on functions of living cells including ionic transport, transmembrane transport of drugs, neural signaling and viral activity. 1 -4

Molecular mechanism of bilayer perforation is one of the challenging issues in studies of

phospholipid membranes. 5 - 7 Molecular simulations show an important role of electrostatics 3 - 6 in this process. However it is difficult to specify a general mechanism because there are many important factors 3,6,7 for biological membranes of complex chemical structure. Even for much simpler amphiphilic membranes formed by classical ionic surfactants, there is yet no clear physical picture of pore formation. Perforated bilayers are found in a variety of systems containing classical surfactants and block copolymers. 8 Numerous cryo-TEM images of such systems show the sequence of aggregate’s shape transitions where perforated structures appear between the non-perforated bilayers and the spatial networks of branched wormlike micelles. 9, 10 Description of these structural transformations is very important per se and in view of many engineering applications. 11, 12 The classical aggregation models 13 - 16 are widely used to describe aggregates of regular shapes, e. g., spheres, cylinders or lamellae. More difficult is to model aggregates of complex geometry,


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for example, a perforated bilayer or a spatial network of branched micelles. This problem is particularly hard for ionic aggregates, owing to the lack of reliable analytical description of the electrostatics in a complex 3D-geometry. Previous models of perforated bilayers 17,18 employed the electrostatic potential obtained from the linearized Poisson-Boltzmann equation (LPB) for the two-dimensional model of a charged plane with a circular hole. In this work we consider a 3D-model of pore in a bilayer formed by classical ionic surfactant in an aqueous solution of salt. We apply approximate analytical description of the LPBelectrostatics for the toroidal rim of the pore. We show the crucial role of the electrostatics and propose the mechanism of spontaneous perforation of the bilayer. The mechanism suggested by our analytical model is supported by numerical results obtained from the LPB-equation for the geometry of the pore. Transitions from non-perforated bilayer to spatial network of wormlike micelles induced by adding salt have been described for ionic surfactants of CnTAB family. Perforated bilayers appear as intermediate structures in these transitions. THEORETICAL BASIS Our goal is to extend the classical aggregation model 13 - 16 and apply it to a perforation in a bilayer and to a spatial network of wormlike micelles. The perforation is a pore with toroidal rim, 19

Figure 1A. When three such pores come close, they form an element of a spatial network (Y-

shaped junction, Figure 1B) that connects three wormlike micelles. For brevity, the toroidal rim of a perforation is called "torus" and the non-perforated bilayer is called "lamella" throughout this paper.


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Figure 1. Three pores with toroidal rims in a bilayer (A) and a junction of three wormlike micelles (B). b , c are the minor and the major radii of a torus, respectively; ϕ , θ are the angles. Within the classical molecular thermodynamic models, the standard free energy of aggregation,

g , is a sum of several contributions : 13 - 16 g = g tr + g def + g int + g rep + g ion

(1)

describing the transfer of the hydrophobic surfactant’s tail from the aqueous solution to the aggregate’s core ( g tr ) , the elastic deformation of surfactant tails in the core ( g def ) , the steric repulsion of surfactant heads ( g rep ) , the core/corona interfacial term ( g int ) , and the ionic term

( gion ) that includes the electrostatic energy and the entropy of the mobile charges around the aggregate. The first four terms of eq 1 are calculated using standard formulae of the classical model, see ref. 20 for details. The hydrophobic term in Eq 1 does not take into account the effects of the tail’s partial hydration 21 and is independent of the aggregate’s shape. Because g tr does not depend on the aggregate's shape, it is discarded in our study of shape stability. To describe electrostatics for perforations, we use here an approximate analytical solution of the LPB-equation for charged torus with non-overlapping electrical double layers 20


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b K p ( xκ D ) u ( x) = t ( ) p x K p+1 (bκ D )

(2)

r where u(x) is the dimensionless electric potential, x is the radial distance along b (Figure 1B), t = 4πl Bσ / eκ D is the dimensionless surface charge density, l B is the Bjerrum length,

σ is the

surface charge density, e is the elementary charge, κ D is the inverse Debye length, K p is the modified Bessel function of the 1-st kind of order p ; p = (m −1) / 2 , where m = (c + 2b cosθ ) /(c + b cosθ ) is a geometrical parameter. Eq 2 reduces to previously known exact

solutions of the LPB-equation for plane ( m = 0 ), cylinder ( m = 1) and sphere ( m = 2 ), respectively. For the electrostatic free energy of the torus, we have 20 g ion aκ D K p (bκ D ) =t2 kT 8πl B K p+1 (bκ D )

(3)

where a is the surface area per surfactant molecule. The free energy of a micellar junction ( g jun ) is estimated as a sum of its toroidal and planar bilayer parts, Figure 1B 20 g jun = g torη tor + g plη pl

(4)

Here gtor and g pl are the aggregation free energies per molecule in the torus and lamella, respectively; ηtor , η pl are the volume fractions of toroidal and planar parts of the junction. The surface areas of these toroidal and planar parts are πb(πc − 2b) and 2c 2 ( 3 − π 2), respectively, and the volumes of these parts are πb 2 (πc 2 − 2b 3) and 2rplc2 ( 3 − π 2), where

2 r pl

is the bilayer

thickness.


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From eqs 1 - 4 we calculate the free energy for the aggregates of different shapes and minimize it with respect to the dimensions of an aggregate. The aggregate of the stable shape has the lowest free energy. For a perforation in a charged bilayer, we compare our electrostatic description with the previously known analytical results. 17,18 The electrostatic problem has been solved for a circular hole in an infinitely thin membrane charged on both sides and immersed in a salt solution. The LPB-equation has an exact analytical solution: the electrostatic potential at any point is equal to the difference between the electrostatic potential of a charged non-perforated plane and that of a uniformly charged disk. The in-plane distribution of the electrostatic potential is given by 17,18 ∞

u ( x ) = t − tκ D r ∫ 0

J 0 (kx) J 1 (kr )

κ D2 + k 2

(5)

dk

where x is the radial distance from the center of the hole of radius r , J 0 is the Bessel function of the 1-st kind of zero order. The results from eqs 2 and 5 are compared with numerical calculations in the next section. RESULTS AND DISCUSSION We apply our model to study shape transitions of aggregates induced by adding salt to aqueous solution of C16TAB. Figure 2 shows the free energies for the aggregates having optimal dimensions and different shapes at progressively increasing salinity. In the left graph cylindrical aggregates have the lowest free energy. These cylindrical aggregates are branched because the free energy of the spherical endcaps is higher than that of a junction. The right graph shows stabilization of lamellae (i.e., the non-perforated bilayers in our terminology) at high salinity.


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The graph in the middle shows that a toroidal rim of an optimal size becomes the lowest free energy shape promoting proliferation of perforations in the bilayer and leading to the appearance of a bicontinuous structure. We note here that in a real bicontinuous structure all three "arms" of the junction must be bent out of plane in contrast to the geometrical model depicted in Figure 1B. Therefore our simplified model may not delineate the boundary between the bicontinuous structure and a densely perforated flat bilayer. Nevertheless, the model does provide the boundary between the spatial network of wormlike micelles (where cylinder is the most stable shape) and a bicontinuous structure (where Y-junction is the most stable shape).

Figure 2. Calculated free energy per surfactant molecule for C16TAB aggregates of different shapes in aqueous solution of 1:1 salt at 298.15K. The curves for torus show the free energy vs

l the major radius c of the torus (in units of a fully stretched tail of a surfactant molecule s ) at the optimal value of the minor radius. Horizontal lines show the free energies for the aggregates of other shape and optimal size. The head cross-sectional area a p = 0.54 nm2 and ls = 0.22 nm. Formation of a pore in a bilayer is the result of fine balance between different terms in the free energy. Minor increase of salinity leads to disappearance of perforations in the bilayer (the


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transition from torus to lamella in Figure 2). Below we discuss the perforation mechanism suggested by our model. Figure 3 shows combinations of different contributions to the aggregation free energy calculated for lamella and for torus (of an optimal minor radius) as function of an area per surfactant molecule. The lower group of curves shows the combination of interfacial and ionic terms. The dependence of the interfacial term on the surface area is identical for all geometries. Therefore, the difference between lamella and torus comes solely from different dependence of their electrostatic terms on the surface area.

Figure 3. Combinations of interfacial, ionic and repulsion terms of the free energy vs the area per molecule for C16TAB aggregates of two shapes: torus and lamella. T=298.15 K, 324.5 mM. For small surface areas, the combination of the interfacial and the electrostatic terms favors the non-perforated (lamella) over the perforated (torus) bilayers, whereas at larger areas the electrostatic term g ion is lower for torus than for lamella. This is because in contrast to flat lamellae the torus has two curvatures: a negative curvature in the direction of the major radius c that brings the charged surfactant’s heads closer together, and a positive curvature in the


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direction of the minor radius b that takes the charged heads farther apart, Figure 4. The negative curvature is disfavored by electrostatics. For torus, small surface areas imply large negative curvature (perforation of a small diameter); zero-curvature lamellar aggregates are preferred. Large surface areas imply large diameter of perforations; the negative curvature is small. The favorable positive curvature of the torus leads to its stabilization at large surface areas, Figure 3.

Figure 4. The unfavorable negative and the favorable positive curvature for the charged head groups of an ionic surfactant in the torus.

For surfactants with relatively large heads, their steric repulsion produces large penalty (

g rep

) for

densely packed surfaces thus making small surface areas strongly unfavorable for both lamellae and tori. The result is stabilization of torus: see two upper curves in Figure 3. This means that the bilayer would always suffer perforations if it were not for the deformation of tails. Within the classical model the tail deformation free energy ( g def ) is always lower for lamella than for torus as we show below. For the aggregate of any shape, we have 13

g def = С

π2

b ( )2 80 N B a K

(6)

where N B is the number of segments in surfactant’s tail, b is the tail extension, a K is the Kuhn segment length,

C

is a geometry-dependent constant: C = 10 for lamella, C = (15πc − 12b) /(3πc − 4b)


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for torus. Taking equal surface areas per chain in a lamella and in a torus, we express btor in terms of b pl

btor = b pl

πс / b − 2

(7)

πc /( 2b) − 2 / 3

This equation shows that surfactant tails are more extended in a torus (by factor of 1.26 to 2.0) than in lamellae. From eqs 6 and 7 the difference between the deformation free energies of lamella and torus is proportional to 10 − 15πc / b − 12 ( πc / b − 2 3πc / b − 4

πc /( 2b ) − 2 / 3

) 2 . This expression is negative

for all c ≥ b hence lamella has lower g def than torus of the same area per chain.

Figure 5 shows the total aggregation free energy (in excess of g tr ), eq 1, for torus and for lamella as function of the area per molecule. Perforated bilayer (torus) is formed at a lower salinity where the favorable electrostatic term overwhelms the unfavorable tail-deformation term. An increase of salinity enhances screening and diminishes the role of the electrostatic contribution; the deformation of tails is responsible for stabilization of the non-perforated bilayer (lamella).

Figure 5. The aggregation free energies for torus and lamella vs the area per molecule at 324.55 and 330 mM. T=298.15 K, C16TAB.


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Our findings are based on analytical approximation for the deformation free energy, eq 6. More complex dependence of the deformation free energy on the core radii is found from a numerical self-consistent field approach that takes into account the effect of aggregate shape on the conformational statistics of tails. 22 Nevertheless, for large surface areas dictated by steric repulsion of heads, the conformational free energies are in the same sequence for different shapes, see Figure 3 of ref. 23 Our extended model may be applied to predict the sequence of morphology transitions induced by changing salinity or temperature and to describe the effects of surfactant’s molecular parameters on the aggregate stability. An illustrative example is shown in Figure 6. For stable perforated structures, the salinity intervals broaden (and shift to a higher salinity) with an increase of temperature, an increase in the head’s cross-sectional area, and with a decrease in the length of surfactant’s tail.

Figure 6. Salinity intervals of stable aggregate shapes and shape transitions predicted for classical cationic surfactants. The effects of temperature (C16TAB, A), tail length (CnTAB, B), and cross sectional area

ap

of the surfactant head (C16-surfactant, C).

a p = 0 . 54

nm2 (A, B); T =

298.15 K (B, C)


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The electrostatic contribution that according to our model is responsible for perforation of a bilayer has been derived within two serious approximations: (1) the electrical double layers do not interpenetrate within the pore, eq 3, and (2) the free energy of an aggregate is the sum of contributions from its constituent elements, eq 4. The latter approximation is normally used in molecular thermodynamic theories of micellization. 13 To verify our findings we solve the LPB-equation numerically, using Comsol Multyphysics 4.3b software for charged bodies of two different geometries: flat bilayer with a pore and an isolated toroid immersed in solution of salt. Calculations have been performed for many combinations of parameter values typical of C10-C18 ionic surfactants. The bilayer half thickness b ∈ [0.5 ÷ 2.0] nm, the ratio of the major-to-the minor torus’s radii, c b , is varied from 1.2 to values as large as 20.0, the salinity is from 50 mM to 1 M, the reduced charged density t ∈[2.0 ÷10.0] , and T = 298.15 K. The upper t -boundary corresponds to an area of ca 0.6 to 0.4 nm2 per charge for salinities in the range 200 - 400 mM. Examples below are for the values of parameters that result in perforated bilayers according to the analytical model. Figure 7 shows the surface potentials from the numerical solution of the LPB-equation and from analytical theories, eqs 2, 5. The diameter of the bilayer pore has strong effect on the profile of the surface potential. For the small pore, Figure 7A, the shape of this profile is similar to that for an isolated toroid. For a large pore in the bilayer, the potential profile changes qualitatively, exhibiting a minimum, Figure 7B. In contrast, for an isolated toroid, the shape of the potential’s profile is maintained for different major radii (cf. Figures 7A and 7B). The approximate analytical model of the torus, eq 2, is in excellent agreement with the numerical results for large radii с , Figure 7B, but may only provide an estimate of the average surface potential at small с ,


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Figure 7A. For a large pore in a bilayer, Figure 7B, this model fails to describe the shape of the profile because this model has been derived for the isolated toroid but not for the perforation in bilayer.

Figure 7. Profiles of the surface potential inside a pore in a flat bilayer (circles) and inside an isolated toroid (triangles) in 400 mM solution of salt, T=298.15 K. The radii c and b for the pore and for the toroid are identical. Angle θ is shown in Figure 1B. Small pore c b = 1.5 (A) and large pore c b = 6.0 (B); t = 8, b = 0.5κ D . Symbols: Numerical solution of LPB equation. Horizontal lines: eq 5 for the hole of radius r in a charged plane 17,18; r = c (dashed) and

r = c − b (dotted). Solid curves: eq 2 for a charged torus. The previous analytical theory of a perforated bilayer, eq 5, 17, 18 does not provide the potential profiles across the bilayer (horizontal lines in Figures 7A, B) since this theory has been developed for a 2-D model of charged plane with a circular hole. Figure 7 shows that this theory, stronger than eq 2, underestimates the surface electrostatic potential of the bilayer pore. For the perforated bilayer, both theories do not give a quantitative description but reflect correctly a


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decrease in the average surface potential with the size of the pore (cf. Figures 7A and 7B); this is important since the average potential determines the electrostatic free energy. Spatial distribution of the electric potential is shown in Figure 8. As expected, the electrical double layers interfere inside small holes, Figure 8 A, B.

Figure 8. Spatial distribution of the electrostatic potential around pore in a charged bilayer (A, C) and around charged toroid (B, D) from the numerical solution of the LPB equation. Shown are the cross-sections by the plane that contains the axis of rotational symmetry; white areas represent the hydrocarbon interior of the aggregates. Small pore c b = 1.5 (A, B) and large pore c b = 6.0 (C, D); t = 8 ( b ≈ 0.2 nm at 400 mM salt and T=298.15 K).

The electrostatic free energy Gion of the computation cell is obtained from the spatial distribution of the electric potential 20


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Gion 1 = kT 8πl B κ D

∫ [(∇u )

~ V

2

]

~ + u 2 dV

(10)

All distances in the integral are in units of the Debye length, the integration is over the cell volume. The height of the cell has been chosen large enough to ensure zero potential values at the top and the bottom, see Figure 8. The cell width is equal to nc , where large enough n ensures vanishing effect of the bilayer pore at the lateral boundaries of the cell, Figure 8A. The surface areas of the aggregates in the cell are 2( nc ) 2 and 2((nc) 2 − πc 2 + πb(πc − 2b)) for nonperforated and perforated bilayers, respectively (we express these surface areas in κ D−2 -units). ~ The reduced electrostatic free energy of the cell per unit area of the aggregate, Gion , is given in

Figure 9 for perforations of different major radii. The horizontal line shows the reduced electrostatic free energy per unit area for lamella (non-perforated). Figure 9 shows a minimum in the electrostatic free energy for the bilayer with a medium-size pore ( c / b ≈ 4 ): this energy is lower than that of a non-perforated bilayer. Thus, the numerical results confirm the crucial role of electrostatics in the perforation mechanism.


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Figure 9. Reduced electrostatic free energy per unit surface area for the bilayer with a pore (solid) vs the pore’s radius. Dashed: calculated for the non-perforated bilayer using cells of varying width. b = 0.5κ D , t =8, n =5. CONCLUSIONS Molecular thermodynamic model of self-assembly has been extended to perforated bilayers of ionic surfactants by including an analytical description of the electrostatic contribution for a toroidal rim of a pore. The model suggests a simple mechanism of perforation of a bilayer. Creation of large pores with a concave-convex geometry of the rim is promoted by favorable electrostatics but is opposed by the elasticity of the lipid tails that prefer flat geometry of a nonperforated membrane. The balance between these opposite trends controls the appearance of pores; this balance is modulated by added salt that screens the electrostatic interactions. The role of steric repulsion of heads is to prohibit small surface areas where lamellae may win over tori owing to a lesser bare interfacial free energy. It is essential that the pore is a 3-D structure. Previous theories of bilayer perforation 17, 18 revert to some additional factors (e.g., anisotropic inclusions 18) to explain formation of a pore because the electrostatic contribution has been derived from a 2-D model of a perforated plane. For ionic surfactants of CnTAB family, we predict the sequence of shape transitions induced by changing salinity or temperature and demonstrate the effect of surfactant’s molecular parameters. The obtained morphology-stability maps show perforated bilayers and branched micellar networks. The sequence of structures in these maps (derived by comparing the standard free energy of aggregation for different shapes) is in line with the general trend observed in experiment. 8, 10 More detailed description of a bilayer-to-network transition requires


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consideration of a finite size of the lamellar/vesicular aggregates and modeling of the aggregation equilibria. 13 For solutions of individual single-chain ionic surfactants, bilayers are only observed either as transient structures (induced by blotting in preparation of samples for cryo-TEM experiment 9, 24) or at high surfactant concentration in the liquid crystalline domain. Branched wormlike micelles are quite typical of single-chain ionic surfactants in aqueous salt: experiment and modeling for micellar networks have been widely discussed in several reviews. 11, 25, 26

In this paper we mainly focus on perforations in bilayers. Perforated bilayer structures are

observed in phospholipid + surfactant mixtures, 8 in mixtures of classical single-chain anionic+cationic surfactants, 10, 27 and in surfactant + cosurfactant mixtures. 8 Of particular interest is the data on the size of the bilayer pores obtained from SANS and cryo-TEM experiments for mixed surfactants. 27 Extension of our model to mixed surfactant bilayers is in progress. The suggested new mechanism of pore formation has been confirmed by a numerical solution of the LPB equation for a charged bilayer with a pore in aqueous salt. The electrostatic free energy passes through a minimum for a medium-size pore and is lower than that of a non-perforated bilayer. For large pores, the charged plane has strong impact on the surface potential inside the perforation. Small pores and large pores have qualitatively different profiles of the surface potential, in contrast to isolated toroids showing no qualitative change of potential profiles for varying major radius. Our analytical model does not describe the boundary effects for large pores; analytical description of these effects is an interesting perspective. Another interesting perspective is to apply the non-linearized version of the PB-theory (NLPB) to describe the electrostatics of perforation. This will extend the range of applicability of the model to membranes of a higher charge density. Analytical approximations to the solutions of the NLPB


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in a weakly curved geometry are well-known. 28, 29 The major problem however is to develop analytical solution for a strongly curved small pore with toroidal rim. The balance between electrostatics and tail deformation is crucial for self-assembly in the toroid geometry of a pore. This has been illustrated within a simplistic model of an ionic bilayer. The model nevertheless contains the principal features of many self-assembling ionic amphiphiles (tail elasticity, ionic charge, steric repulsion of heads). The suggested mechanism is quite general and must play an important role in formation of pores in a far more complex systems.

ASSOCIATED CONTENT AUTHOR INFORMATION

Corresponding Author * Alexey I. Victorov. E-mail: [email protected]

Author Contributions The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript.

Funding Sources The Russian Science Foundation (project # 16-13-10042). ACKNOWLEDGMENT We thank the Resource center "Computer center" of St. Petersburg State University for the access to the Comsol Multyphysics software.


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ABBREVIATIONS cryo-TEM, cryo-transmission electron microscopy; C16TAB, cetyltrimethylammonium bromide; CnTAB, alkyltrimethylammonium bromide. REFERENCES (1) Angelescu, D.G.; Linse, P. Viruses as supramolecular self-assemblies: modelling of capsid formation and genome packaging. Soft Matter. 2008, 4, 1981-1990. (2) Andreeva-Kovalevskaya, Z. I.; Solonin, A. S.; Sineva, E. V.; Ternovsky, V. I. Pore-forming proteins and adaptation of living organisms to environmental conditions. Biochemistry-Moscow.

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(9) Danino, D. Cryo-TEM of soft molecular assemblies. Current opinion in colloid & interface science. 2012, 17, 316-329. (10) Kakehashi, R.; Karlsson, G.; Almgren, M. Stomatosomes, blastula vesicles and bilayer disks: morphological richness of structures formed in dilute aqueous mixtures of a cationic and an anionic surfactant. J. of Colloid and Interface Science. 2009, 331, 484-493. (11) Dreiss, C. A. Wormlike micelles: where do we stand? Recent developments, linear rheology and scattering techniques. Soft Matter. 2007, 3, 956-970. (12) Victorov, A. I. Molecular thermodynamics of soft self-assembling structures for engineering applications, J. Chem. Technol. Biotechnol. 2015, 90, 1357–1363. (13) Nagarajan, R.; Ruckenstein, E. Self-assembled systems // In "Equations of state for fluids and fluid mixtures". Eds.: Sengers, J. V.; Kayser, R. F.; Peters, C. J.; White, H. J. -Amsterdam: Elsevier Science, 2000.-P. 589-749. (14) Nagarajan, R.; Ruckenstein, E. Theory of surfactant self-assembly: a predictive molecular thermodynamic approach. Langmuir. 1991, 7, 2934-2969. (15) Puvvada, S.; Blankschtein, D. Molecular-thermodynamic approach to predict micellization, phase behavior and phase separation of micellar solutions. I. Application to nonionic surfactants. J. Chem. Phys. 1990, 92, 3710-3724. (16) Yuet, P. K.; Blankschtein, D. Molecular-thermodynamic modeling of mixed cationic/anionic vesicles. Langmuir. 1996, 12, 3802-3818. (17) Betterton, M.D.; Brenner, M.P. Electrostatic edge instability of lipid membranes. Phys. Rev. Letters. 1999, 82, 1598-1601. (18) Fošnarič, M.; Kralj-Iglič, V.; Bohinc, K.; Iglič, A.; May, S. Stabilization of pores in lipid bilayers by anisotropic inclusions. J. Phys. Chem. B. 2003, 107, 12519-12526.


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(19) Wang, S.; Larson, R. G. Water channel formation and ion transport in linear and branched lipid bilayers. Phys. Chem. Chem. Phys. 2014, 16, 7251-7262. (20) Andreev, V. A.; Victorov, A. I. Molecular thermodynamics for micellar branching in solutions of ionic surfactants. Langmuir. 2006, 22, 8298-8310. (21) Stephenson, B. C.;

Goldsipe, A.; Beers, K. J.; Blankschtein, D. Quantifying the

hydrophobic effect. 1. A computer simulation molecular-thermodynamic model for the micellization of ionic and zwitterionic surfactants in aqueous solution. J. Phys. Chem. B, 2007, 111, 1025−1044. (22) Victorov, A. I.; Plotnikov, N. V.; Hong, P.-D. Molecular thermodynamic modeling of morphology transitions in a solution of a diblock copolymer containing a weak polyelectrolyte chain. The Journal of Physical Chemistry B. 2010, 114, 8846-8860. (23) May, S.; Ben-Shaul, A. In Giant Micelles: Properties and Applications, Zana, R., Kaler, E. W., Eds.; CRC Press: Boca Raton, FL, 2007; Chapter 2. (24) Abezgauz, L.; Kuperkar, K.; Hassan, P.A.; Ramon, O.; Bahadur, P.; Danino, D. Effect of hofmeister anions on micellization and micellar growth of the surfactant cetylpyridinium chloride. J. Colloid Interface Sci. 2010, 342, 83–92. (25) Rogers, S. A.; Calabrese, M. A.; Wagner, N. J. Rheology of branched wormlike micelles. Current Opinion in Colloid & Interface Science. 2014, 19, 530–535. (26) Victorov, A. I.; Voznesenskiy, M. A.; Safonova, E. A. Spatial networks in solutions of worm-like aggregates: universal behavior and molecular portraits. Russ. Chem. Rev. 2015, 84, 693–711. (27) Bergström, L. M.;

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sodium octyl sulfate and hexadecyltrimethylammonium bromide. Langmuir. 2014, 30, 3928−3938. (28) Lekkerkerker, H.N.W. Contribution of the electric double layer to the curvature elasticity of charged amphiphilic monolayers. Physica A. 1989, 159, 319-328 (29) Evans, D. F.; Ninham, B. W. Ion binding and the hydrophobic effect. J. Phys. Chem.

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Driving Force for Spontaneous Perforation of Bilayers Formed by Ionic

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