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Kinetics of the Micelle-to-Vesicle Transition: Aqueous Lecithin-Bile Salt Mixtures

The University of Edinburgh, School of Physics, King's Buildings, Edinburgh EH9 3JZ, United Kingdom

Correspondence: Address reprint requests to Stefan U. Egelhaaf, Tel.: +44-131-6505291; Fax: +44-131-6505902; E-mail: s.u.egelhaaf{at}ed.ac.uk.

	ABSTRACT

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS EXPERIMENTAL RESULTS MODEL CONCLUSION AND OUTLOOK APPENDIX A APPENDIX B ACKNOWLEDGEMENTS REFERENCES

 
Important routes to lipid vesicles (liposomes) are detergent removal techniques, such as dialysis or dilution. Although they are widely applied, there has been only limited understanding about the structural evolution during the formation of vesicles and the parameters that determine their properties. We use time-resolved static and dynamic light scattering to study vesicle formation in aqueous lecithin-bile salt mixtures. The kinetic rates and vesicle sizes are found to strongly depend on total amphiphile concentration and, even more pronounced, on ionic strength. The observed trends contradict equilibrium calculations, but are in agreement with a kinetic model that we present. This model identifies the key kinetic steps during vesicle formation: rapid formation of disklike intermediate micelles, growth of these metastable micelles, and their closure to form vesicles once line tension dominates bending energy. A comparison of the rates of growth and closure provides a kinetic criterion for the critical size at which disks close and thus for the vesicle size. The model suggests that liposomes are nonequilibrium, kinetically trapped structures of very long lifetime. Their properties are hence controlled by kinetics rather than thermodynamics.

	INTRODUCTION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS EXPERIMENTAL RESULTS MODEL CONCLUSION AND OUTLOOK APPENDIX A APPENDIX B ACKNOWLEDGEMENTS REFERENCES

 
Vesicles, in particular lipid vesicles (liposomes), have an impact on a variety of areas, which range from fundamental science to biotechnology. Vesicles serve as models for cell membranes and allow the study of the basic mechanisms of membrane function, such as fusion (Lichtenberg, 1995; Lasic and Barenholz, 1996). Furthermore, liposomes of controllable size are used as biocompatible and protective structures to encapsulate labile molecules, such as proteins, nucleic acids or drugs, for pharmaceutical, cosmetic or chemical applications; they are also vital to the study of membrane proteins, including determining their structure via two-dimensional crystallization (Lasic, 1993; Lasch, 1995; Lasic, 1997; Rosoff, 1996; Ollivon et al., 2000).

Different methods are used to prepare lipid vesicles using detergent removal techniques: dilution, dialysis, gel exclusion chromatography, adsorption onto polymeric materials, temperature changes, or biochemical reactions (Ollivon et al., 2000). All these methods rely on the very high solubility of detergents compared to lipids. A reduction of the monomeric detergent concentration, for example by dialysis, removes detergent from the aggregates that are present initially, typically spherical or elongated micelles. This change in composition may then induce vesicle formation. Although techniques based on detergent removal are widely used, only limited information is available on the mechanism by which mixed micelles transform into vesicles. A better knowledge of the nonequilibrium behavior could help to optimize the detergents, conditions, and procedures used for vesicle formation.

More generally, little is known about the nonequilibrium behavior of surfactant aggregates, whereas their equilibrium properties are well studied (Evans and Wennerström, 1994). From a physicochemical point of view, particularly interesting are transformations between different monolayer and bilayer topologies (Lipowsky, 1991; Hyde et al., 1997), with the micelle-to-vesicle transition being a classic example. The properties of vesicles are extensively studied theoretically and experimentally in a number of different systems (Schurtenberger et al., 1985; Kaler et al., 1989; Hjelm et al., 1990; Long et al., 1994; Schönfelder and Hoffmann, 1994; Lin et al., 1994; Egelhaaf and Schurtenberger, 1994; Pedersen et al., 1995; Oberdisse et al., 1996; Danino et al., 1997; Cantu et al., 1997; Safran et al., 1990, 1991; Andelman et al., 1994; Fattal et al., 1995). Nevertheless, there has been only limited understanding of their formation and of the sequence of any intermediate structures (Almog et al., 1986, 1990; Edwards and Almgren, 1990, 1991; Walter et al., 1991; Edwards et al., 1993; Silvander et al., 1996; O'Connor et al., 1997; Campbell et al., 1998; Brinkmann et al., 1998; Egelhaaf and Schurtenberger, 1999; Chen et al., 1999; Xia et al., 2002; Schmölzer et al., 2002). It is still not conclusively decided what determines the final ("end-state") properties of vesicles formed by detergent removal; equilibrium calculations (Safran et al., 1990, 1991) give the wrong trends for the dependence of liposome size on bilayer composition (Schurtenberger et al., 1985; Hjelm et al., 1990; Long et al., 1994; Egelhaaf and Schurtenberger, 1994; Kozlov and Andelman, 1996), although they are in agreement with experiments on catanionic surfactant vesicles (for example Kaler et al. (1989)). This suggests that the end-state liposomes are metastable structures that cannot achieve thermal equilibrium on observable timescales. There is, however, no clear consensus yet on whether liposomes represent a true equilibrium state or a metastable state of very long lifetime.

Here we study the nonequilibrium behavior of aqueous lecithin-bile salt mixtures, which are prime examples of mixed amphiphile solutions that exhibit a spontaneous micelle-to-vesicle transition (Schurtenberger et al., 1985; Hjelm et al., 1990; Long et al., 1994; Egelhaaf and Schurtenberger, 1994; Pedersen et al., 1995). They are of direct importance in biochemistry, physiology, and pharmacy, with the micelle-to-vesicle transition exploited in studies as mentioned above, and implicated in gallstone formation and digestion (Lichtenberg, 1995). In addition, they are well-controlled model systems, and we use them as such in this study of liposome reconstitution.

In aqueous lecithin-bile salt mixtures, different structures are observed with decreasing total concentration: spherical micelles—elongated, polymerlike micelles—vesicles. This sequence can be rationalized based on the concept of spontaneous curvature. The average spontaneous curvature of a monolayer comprising lecithin and bile salt depends on its composition: lecithin alone forms aggregates of low spontaneous curvature whereas bile salt alone forms highly curved (spherical) micelles. At high bile salt content, therefore, spherical or elongated mixed micelles form. Because bile salt is far more soluble than lecithin, a subsequent dilution causes the composition of the aggregates to change, so that the bile salt content is reduced and the spontaneous monolayer curvature decreased. With increasing dilution factor progressively longer, flexible cylindrical micelles are observed, until at higher dilution factors the end state comprises near-monodisperse, unilamellar vesicles whose size decreases with dilution factor (Schurtenberger et al., 1985; Hjelm et al., 1990; Long et al., 1994; Egelhaaf and Schurtenberger, 1994; Pedersen et al., 1995; Arleth et al., 2003).

Time-resolved light and neutron scattering experiments suggest that on a sudden dilution spherical or elongated micelles very quickly change into disklike micelles (within 1 s), which then transform into vesicles in a much slower process, typically 1 h (Egelhaaf and Schurtenberger, 1999). In this system, vesicle formation seems thus to occur along the following path: spherical or elongated micelles—disklike micelles—vesicles. Disklike intermediates have also been suggested under different conditions and for different systems (Walter et al., 1991; Edwards and Almgren, 1991; Luk et al., 1997; O'Connor et al., 1997; Schmölzer et al., 2002).

Note that when a series of samples of varying composition is prepared by dilution, different end-state structures can arise. Each end state might either represent a true equilibrium state, whose structure forms reversibly and does not depend on the preparation method, or a nonequilibrium metastable state, whose structure is generally path dependent and might have a very long life time. (Note that full equilibration is not guaranteed merely by the fact that a structure forms spontaneously.) As mentioned above, the end states observed in the present system range from spherical micelles via progressively longer cylindrical micelles to vesicles. The cylindrical micelles therefore represent compositional intermediates between spherical micelles and vesicles; such intermediates have been extensively studied. This paper mainly addresses kinetic intermediates that form dynamically during the process between an initial state of micelles and an end state comprising vesicles. For the system described here, disklike micelles have been found to arise as kinetic intermediates in this process (Egelhaaf and Schurtenberger, 1999). However, because they remain reactive (with a lifetime of hours), they are not seen as compositional intermediates in the sequence of end states created by varying composition.

We have performed new time-resolved static and dynamic light scattering experiments to elucidate the pathway of vesicle formation and the role of kinetics in determining the end-state properties of the liposomes. Crucially, we not only varied the final total amphiphile concentration c, but also investigated the dependence on salt (NaCl) concentration c_s. (Note that all "global" parameters are collected in Table 1.) We expect c_s to control the electrostatic interactions between negatively charged bile salt molecules and to influence bile salt solubility (Small, 1973), while having only marginal effects on the properties of the neutral lecithin within the range of c_s studied here (Meyuhas et al., 1997). We use a bile salt (taurochenodeoxycholate) with a very low solubility, which shifts the vesicular region to very low lipid concentrations. Under these conditions, interactions between aggregates have a negligible effect on the light scattering results.

	MATERIALS AND METHODS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS EXPERIMENTAL RESULTS MODEL CONCLUSION AND OUTLOOK APPENDIX A APPENDIX B ACKNOWLEDGEMENTS REFERENCES

 
Sample preparation
Lecithin (egg yolk lecithin (grade 1), Lipid Products, South Nutfield, Surrey, UK) and bile salt (taurochenodeoxycholic acid sodium salt, Fluka, Gillingham, Dorset, UK) were dissolved in ethanol in a lecithin-to-bile salt molar ratio of 0.9 and dried under low pressure (Small et al., 1969; Egelhaaf and Schurtenberger, 1994). Then buffer (50 mM Tris, pH 8.0) was added to obtain a stock solution with a total lipid concentration of 50 mg/ml, which corresponds to lecithin and detergent (bile salt) volume fractions and , respectively. The stock solution was flushed with nitrogen and equilibrated for a few days at a temperature T = 23°C.

To obtain samples with the desired dilution factor d (defined as the concentration of the stock solution divided by the sample concentration) and salt concentration c_s, the stock solution was diluted with buffer that also contained sodium chloride (NaCl). The samples were flushed with nitrogen, sealed, and kept at 23°C for at least two weeks. In the following we include the ionic strength originating from the buffer, 28.2 mM "effective" salt concentration, in the total salt concentration c_s. We neglect, however, the contribution from counterions of bile salt (Na⁺), because their concentration, less than 1 mM, is much smaller than the concentration of added salt (c_s 50 mM).

Before the light scattering measurements, 1 ml of sample is transferred into cylindrical scattering cells (10 mm inner diameter) and centrifuged at 5000 rpm and 23°C for 1 h to remove dust particles from the scattering volume. Samples for time-resolved experiments were prepared as follows: a small amount of the initial solution (d = 2, prepared as described above) is transferred into a scattering cell and centrifuged at 5000 rpm and 23°C for 30 min. It is then rapidly diluted with buffer of a given ionic strength, which has been repeatedly filtered using a Millipore filter (pore size 0.1 µm) to remove dust particles. Subsequently the sample is gently shaken and put into the light scattering instrument. The time from mixing until the first measurement is accurately determined and is typically 30 s.

Light scattering
Static (SLS) and dynamic (DLS) light scattering experiments were performed with an ALV goniometer modified to use fiber-optical detection (Gisler et al., 1995) and equipped with an ALV-5000 correlator and an argon ion laser (Coherent, Innova 90, = 514.8 nm). Measurements were made at 23°C and five different scattering angles (30°, 50°, 70°, 90°, and 110°). For the DLS measurements, several individual autocorrelation functions were determined at each angle. They were individually analyzed using a second-order cumulant analysis (Koppel, 1972), which yields the average decay rate (q) and a polydispersity index where q = (4n_ref/) sin (/2) is the scattering vector and n_ref the refractive index of water. The average decay rate is then converted to the collective diffusion coefficient D = (q)/q² and hydrodynamic radius R_h = kT/6D where k is Boltzmann's constant and = 10^-3 Pa s the solvent viscosity. The results were subsequently averaged for each angle. SLS was used to determine the average scattered intensity as a function of scattering vector I(q). The extrapolation to zero scattering vector, I(q 0), was based on the form factor for a suspension of polydisperse shells with average radius R = R_h, thickness 2 = 50 Å (Small, 1967; Pedersen et al., 1995) and a Gaussian size distribution; the radius and its polydispersity were deduced from DLS. In the kinetic measurements, the time dependences of the average scattering intensity I(t) and intensity autocorrelation function, from which D(t) and R_h(t) are obtained, were monitored at one scattering angle ( = 90°) with an individual measurement time of 5 s.

	EXPERIMENTAL RESULTS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS EXPERIMENTAL RESULTS MODEL CONCLUSION AND OUTLOOK APPENDIX A APPENDIX B ACKNOWLEDGEMENTS REFERENCES

 
General phase behavior
First we examine a range of salt concentrations c_s and dilutions d to determine the conditions that lead to vesicles only. Samples were kept at T = 23°C for at least two weeks before they were visually inspected and investigated by static and dynamic light scattering. These measurements yield the hydrodynamic radius R_h, polydispersity index , and the scattered intensity extrapolated to zero scattering vector I(q 0), which is proportional to the average molar mass of the aggregates and the concentration (or 1/d).

Fig. 1 shows the c_s-dependences for samples with d = 50, which is also typical for other dilutions. At low c_s, R_h, and I(q 0) show a pronounced increase whereas is approximately constant with 0.1. The data is consistent with the formation of near-monodisperse vesicles, which has, under similar conditions, already been reported (Schurtenberger et al., 1985; Egelhaaf and Schurtenberger, 1994; Cohen et al., 1998; Degovics et al., 2000). In the range c_s 200–700 mM a modest, approximately linear increase in the detected average hydrodynamic radius R_h is observed (see also open symbols in Fig. 6) with a concomitant, significant rise in polydispersity. At the same time a dramatic drop in the scattered intensity is detected. This indicates the existence of another type of aggregate of lower scattering power, probably micelles that might coexist with vesicles (Long et al., 1994; Egelhaaf and Schurtenberger, 1994; Pedersen et al., 1995). This is expected for vesicular bile salt-to-lecithin ratios exceeding the maximal amount of bile salt that can be accommodated by vesicles (the "saturation concentration") (Lichtenberg, 1995; Roth et al., 2000). At even higher c_s 700 mM, both, R_h and I(q 0) suddenly drop, indicating that only a small fraction of small aggregates are present in the scattering volume. Consistent with these results, the onset of bulk phase separation can be detected. By polarized light microscopy, this can be identified as lamellar phase coexisting with excess buffer. Here we are interested in the range of salt concentrations c_s and dilutions d where only vesicles are present (Fig. 2, hatched area), which is determined from the dramatic change in I(q 0) and the change in slope of R_h (solid line, Fig. 1).

View larger version (15K): [in this window] [in a new window]   FIGURE 1  (a) Average hydrodynamic radius Rh, (b) polydispersity index , and (c) average scattered intensity I(q 0) as a function of salt concentration cs for samples diluted to d = 50 and left at 23°C for at least two weeks. The different regimes are separated by vertical lines.

View larger version (12K): [in this window] [in a new window]   FIGURE 6  Hydrodynamic radius Rh measured in the end state as a function of salt concentration cs for different final dilutions d (•: 40, : 80, : 120). Solid symbols correspond to vesicular samples and open symbols to samples beyond the vesicular region.

View larger version (16K): [in this window] [in a new window]   FIGURE 2  Diagram indicating the range of salt concentrations cs and dilutions d for which only vesicles are observed (hatched area). The data points refer to changes in the light scattering behavior (Fig. 1, solid lines), while the line is a guide to the eye.

View larger version (11K): [in this window] [in a new window]   FIGURE 3  Time evolution of (a) relative scattered intensity I(t)/I(0) and (b) diffusion coefficient D(t) after a rapid dilution step from an initial dilution d = 2 to a final dilution d = 60 for two salt concentrations cs (solid line: 180 mM, dashed line: 230 mM). The individual measurement time was 5 s.

Initial rate
The initial slopes of the time dependences of the scattered intensity I(t) and diffusion coefficient D(t) were determined as a function of salt concentration c_s and final dilution d. The slopes were obtained by a second order polynomial fit and are converted into rates according to:

(1)

Theoretically, the rate can be related to the rate at which initial, intermediate aggregates coalesce to form aggregates of twice the mass (see "Growth"). Based on intermediate disklike aggregates (Egelhaaf and Schurtenberger, 1999) with a hydrodynamic radius of R_h 60 Å (see "Size of the intermediate aggregates"), we obtain the theoretical values ß_I = 1 and ß_D = 0.38 (Appendix A), which are used to convert the experimentally determined slopes to experimental rates (Eq. 1). Consistent values for the initial rate (Figs. 4 and 5) are obtained. On increasing c_s, shows a steep increase spanning about two decades, which is more pronounced for lower final dilutions d (Fig. 4). This concurs with the increased screening of electrostatic interactions at higher c_s, and also with the presence of higher charge on the disk, i.e., higher bile salt content, at lower d. In contrast, a much weaker dependence of on d is observed, whose absolute level, however, heavily depends on c_s (Fig. 5), consistent with the strong dependence of on c_s.

View larger version (13K): [in this window] [in a new window]   FIGURE 4  Rate obtained from the normalized initial slopes of the scattered intensity (•) and diffusion coefficient () as a function of salt concentration cs for different final dilutions d (a: 40, b: 60, c: 100). Model predictions are shown as lines with electrostatic interactions based on constant potential (solid line) and constant charge (dashed line), respectively. Parameters for calculations: aD = 200 Å2, m = 10 kT and .

View larger version (13K): [in this window] [in a new window]   FIGURE 5  Rate obtained from the normalized initial slopes of the scattered intensity (•) and diffusion coefficient () as a function of final dilution d for different salt concentrations cs (a: 130 mM, b: 230 mM). Model predictions are shown as lines with electrostatic interactions based on constant potential (solid line) and constant charge (dashed line), respectively. Parameters for calculations: aD = 200 Å2, m = 10 kT, and

	MODEL

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS EXPERIMENTAL RESULTS MODEL CONCLUSION AND OUTLOOK APPENDIX A APPENDIX B ACKNOWLEDGEMENTS REFERENCES

 
We now examine theoretically the transition from micelles to vesicles and develop a simple model that describes the kinetic pathway, including the properties of the end-state vesicles. Our kinetic model (Fig. 7) assumes that the key kinetic steps during vesicle formation are those leading from the rapidly formed, intermediate disklike micelles to vesicles. First the disklike micelles grow by coalescence and then in a second stage the enlarged disklike micelles close to form vesicles. We also consider stacking of large disklike micelles, which competes with closure and could lead to a lamellar phase. For each step the theoretical predictions will be compared to our experimental data.

View larger version (21K): [in this window] [in a new window]   FIGURE 7  Schematic representation of our kinetic model of the micelle-to-vesicle transition. The fundamental steps and their typical timescales are shown: rapid formation of disklike intermediate micelles, successive growth of these micelles up to the critical radius r* followed by their closure to form vesicles. Ripening of these vesicles to their equilibrium size was not observed, but might occur on a very long timescale. Under certain conditions growth and closure might become slower than stacking, which could lead to the formation of lamellar phase as the end state.

Geometry
The disklike micelles are composed of lecithin and bile salt. These two amphiphiles have very different properties. Lecithin tends to form aggregates of low spontaneous curvature, typically flat bilayers. In contrast, bile salt has a positive spontaneous curvature and self-assembles into highly curved (spherical) micelles when alone in solution. This suggests that the central part of the disk, which is similar to a flat bilayer, is mainly composed of lipids, whereas the bile salt is sequestered at the rim where the curvature is high (Fig. 8).

View larger version (12K): [in this window] [in a new window]   FIGURE 8  Schematic cross section of a disklike micelle. The central bilayer part (radius r, thickness 2) is formed by lipid (L), while the rim also contains bile salt (B). Micellar bile salt is in equilibrium with monomeric bile salt in solution. In contrast, the monomer solubility of lipid is low enough to be neglected.

(2)

and its outer length L is

(3)

As the central part resembles a bilayer fragment, we take for its thickness a typical bilayer thickness, 2 = 50 Å (Small, 1967; Pedersen et al., 1995).

We will argue below (see "Reaction-limited growth of disklike micelles") that the disks grow by coalescence and thus assume that disk radii only exist in discrete steps: r_i = i^1/2r₁ with the radius of the initial disklike micelles r₁ 80 Å as determined based on a hydrodynamic radius R_h(t 0) 60 Å (see "Size of the intermediate aggregates"). The subscript i refers to a disk formed from i initial disklike micelles with radius r₁; we distinguish parameters referring to disks of different radii by subscripts, but suppress them for brevity if only one size of disk or a disk in general is considered.

Composition
The samples are prepared by diluting a stock solution with lecithin and bile salt volume fractions and respectively. For a sample with dilution factor d, this implies a lecithin volume fraction and bile salt volume fraction

As mentioned above, we assume that the central, flat part is formed by lipids with bile salt sequestered at the highly curved rim (Fig. 8). The solubility of lecithin is very small (cmc_L 10^-10 M (Tanford, 1980)) and monomeric lipid in solution is thus neglected. The total area of the central parts of disks and of the vesicles is related to the total amount of lipid _L and assumed constant throughout the kinetic pathway (for these purposes we neglect lipid in the rim):

(4)

where v_L = 1266 ų and a_L = 72 Ų are the volume and headgroup area of a lecithin molecule, respectively (Small, 1967; Huang and Mason, 1978; Cornell et al., 1980), and n_i and n_v are the number densities of disks of radii r_i and vesicles of radius R, respectively. (Because experiments suggest that the vesicles are near monodisperse, we only consider one size of vesicles R.)

Bile salt is much more soluble and we must take bile salt in bulk solution into account (volume fraction _b). Its solubility depends on the ionic strength and lipid concentration, but is for the bile salt we used (taurochenodeoxycholic acid) typically cmc_D 1 mM (Small, 1973; Duane, 1977). In principle, bile salt also enters into the central part of the disk (volume fraction _c). This is driven by the entropy of mixing, but opposed by the curvature elasticity of the mixed bilayer (Kozlov et al., 1997). Based on an estimated equilibrium constant K = _c/(_bL) = 330 for the partitioning of bile salt between bulk and a bilayer (vesicles) (Schurtenberger et al., 1985; Schubert, 1992; Lasch, 1995; Heerklotz and Seelig, 2000), we estimate that for all conditions investigated only a small fraction (0.06) of micellar bile salt is located in the central part. We thus neglect bile salt in the flat, central part of the disks (_c 0). Bile salt is therefore assumed to be partitioned between bulk solution and rims of disks (area fraction _r). Conservation of the total amount of bile salt _D, partitioned between bulk and rims, thus reads:

(5)

where A_t is the total rim area density, v_D the volume of a bile salt molecule with v_D = 660 ų (Matsuoka et al., 1987), and a_D the rim area covered by one bile salt molecule at complete coverage of the rim. Depending on the conditions, a range of values 150 Ų a_D 250 Ų can be found in the literature (Small, 1973; Schurtenberger et al., 1983; Janich et al., 1998); we will use it as an adjustable parameter. Equation 5 considers the most general case of a distribution of disks of different radii r_i. For the initial system, where only disks with radius r₁ are present, it reduces to _D = _b + (v_D/a_D)_rA₁n₁.

The exchange of bile salt between bulk and rim occurs on a timescale of the order of 1 µs (Diamant and Andelman, 1996; Telgmann and Kaatze, 1997). It is thus very fast compared to the processes we aim to describe, which have characteristic times of at least a few seconds (Figs. 3, 4, and 5). We therefore assume local equilibrium. The area fraction _r of bile salt on the rim can then be related to the bulk volume fraction _b through Davies' isotherm, which describes the adsorption of ionic surfactant (Davies, 1958a, 1958b; Diamant and Andelman, 1996):

(6)

where e is the electronic charge, ₀ the electrostatic potential at the interface and the micellization energy _m accounts for the energy gain when one bile salt molecule is added to a disklike micelle. To our knowledge, there is no value for _m available and we thus use it as an adjustable parameter. In using this isotherm, we neglect lateral interactions between bile salt molecules within a monolayer and between different disks, and assume that it remains valid for curved monolayers. We furthermore assume that bile salt in bulk only exists in monomeric form.

At the present pH (pH 8.0), lecithin is zwitterionic and thus overall neutral. Bile salt, however, is fully dissociated (Small, 1973) and carries a negative charge. We assume that it is also fully dissociated in the micelles. This leads to a surface charge density = e_r/a_D and thus creates the electrostatic potential ₀ at the interface. For a 1:1 electrolyte, such as NaCl, of molar concentration c_s the surface charge density is related to the potential ₀ of a single micelle by the Gouy-Chapman theory (Russel et al., 1991):

(7)

where is the Debye length and the dielectric constant of water. (Note that c_s accounts for the added NaCl and the ionic strength originating from the buffer, see "Sample preparation"). We assume that the curvature of the interface does not significantly alter the above equation, because for our samples the (smallest) radius of curvature is larger than the Debye length

Solving Eqs. 5, 6, and 7 simultaneously yields the composition of the disklike micelles. This depends on solution conditions, such as salt concentration c_s and dilution factor d, as well as molecular parameters, namely a_D, v_D, and _m. There is no analytical solution of this set of equations, but the general trends are as follows (Fig. 9): For given d, increasing salt concentration c_s progressively screens the electrostatic interactions between bile salt molecules. This favors adsorption of bile salt molecules onto the rim and thus _r is increased (Fig. 9 a) and _b is decreased (Fig. 9 b). On the other hand, for given c_s, upon dilution bile salt molecules leave the aggregates to maintain the monomer concentration and thus _r decreases (Fig. 9 a, inset). Due to the decrease in total concentration, also _b decreases (Fig. 9 b, inset). An increase in c_s also affects the electrostatic potential ₀, whereas dilution has only a weak effect on ₀ (Fig. 9 c).

View larger version (24K): [in this window] [in a new window]   FIGURE 9  Effect of salt concentration cs and dilution d on the composition and properties of disks. (a) Rim area fraction r covered by bile salt and number of bile salt molecules per initial disk rA1/aD (right axis); (b) bulk volume fraction b of bile salt and molar concentration of monomeric bile salt b/vDNA (right axis); (c) electrostatic energy e0; (d) relative line tension /0 and relative vesiculation index Vf/V0 as a function of salt concentration cs for different dilution factors d (solid line: d = 40, dashed line: d = 80, dotted line: d = 120). The dependences on dilution d are shown in the insets (solid line: cs = 50 mM; dashed line: cs = 150 mM; dotted line: cs = 500 mM). The insets have the same y axes as the main figures. Eqs. 5, 6, and 7 were solved simultaneously for the initial system, i.e., monodisperse disks with i = 1, and Eqs. 9 and 21 were used for panel d. Parameters: r1 = 80 Å, = 25 Å, aD = 200 Å2, m = 10 kT, and 0 = 0.3 kT/Å. These parameters imply V0 = 0.6 (Eq. 21).

View larger version (9K): [in this window] [in a new window]   FIGURE 10  Effect of disk radius r on the properties of disks for a constant sample composition. (a) Relative total rim area density At(r)/At(r1) and (b) vesiculation index Vf as a function of normalized disk radius r/r1 for dilution d = 80 and different salt concentrations cs (solid line: cs = 50 mM, dotted line: cs = 500 mM). Eqs. 5, 6, and 7 were solved simultaneously to obtain the composition, and then Eqs. 2, 4, 9, and 21 were used. Disks are assumed to be monodisperse. Parameters: = 25 Å, aD = 200 Å2, m = 10 kT, and 0 = 0.3 kT/Å.

Fromherz (1983) proposed a thermodynamic analogy between surfactant molecules that decrease surface tension and "edge-actant" molecules that decrease line tension. The decrease of upon adsorption of edge-actant molecules is modeled using Gibbs relation for adsorption of bile salt on the rim (which is assumed to reproduce the correct trend also for curved monolayers):

(8)

where A_r/La_D is the number of adsorbed bile salt molecules per unit length of rim. This relation links the change of to the adsorption isotherm, which is harmonious with our description of bile salt partitioning between bulk and rim (Eq. 6). Using the Gibbs relation (Eq. 8) together with the Davies' isotherm (Eq. 6), we obtain:

(9)

where ₀ is the line tension without bile salt (with experimental values ₀ = 0.2–0.8 kT/Å, (Moroz and Nelson, 1997) and references therein). We will use ₀ as an adjustable parameter. The parameter _b = ₀La_D/A is the size-dependent energy gain upon binding of one bile salt molecule to the rim. It characterizes the ability of an edge actant to lower the line tension by providing a cover with high curvature, but also depends on the nature of the adsorbing surface (the rim). In contrast, the micellization energy _m (see "Composition") is related to the energy gain when one bile salt molecule is added to a disklike micelle, which reflects the hydrophobic nature of the molecule. We will see (see "Dependence on salt concentration and dilution") that the interplay between the (surfactant) hydrophobic effect and the (edge actant) ability to cover a highly curved surface determines the capability of a molecule to stabilize disks. Fig. 9 d illustrates how the line tension is controlled by the salt concentration c_s and dilution factor d: Increasing c_s screens electrostatic interactions and thus favors adsorption of bile salt, i.e., increases _r (Fig. 9 a), which relieves packing stress at the rim and hence results in a decrease of the line tension (Eq. 9). On the other hand, increasing dilution d, reduces the bile salt concentration and hence its adsorption (Fig. 9 a, inset), which leads to an increase in line tension (Fig. 9 d, inset). The line tension also depends on the size r of disks through the total rim area density A_t (Fig. 10).

Growth
Reaction-limited growth of disklike micelles
We assume a first stage in which disklike micelles, initially monodisperse, grow. The low solubility of lecithin precludes growth by molecular diffusion or Ostwald ripening. Growth is thus likely to proceed by coalescence (or "aggregation"). When aggregation of two particles occurs immediately at contact, the process is limited by diffusion and in general is fast. Under these conditions, the characteristic aggregation time for a suspension of spheres of number density n₁ is 3/4kTn₁ (Russel et al., 1991). For the densities of our solutions (n₁ = 10²⁰–10²² m^-3) and the viscosity of water = 10^-3 Pa s, we obtain 10^-5–10^-3 s. This is much faster than our experimentally observed timescales, which are tens of seconds (Figs. 4 and 5). This rules out a diffusion-limited mechanism and indicates that growth is slowed down by repulsive interactions between disks, which lead to an activation barrier and prevent immediate coalescence. In the limit of a very large repulsive potential (compared to kT), the aggregation is reaction limited and in general the aggregation probability is so small that particles explore all possible mutual configurations before aggregation proceeds (Ball et al., 1987). In principle, we thus have to take into account all possible relative orientations of disks with edge-to-edge, edge-to-face, and face-to-face representing the main classes. However, the experimentally determined initial growth rate strongly depends on salt concentration c_s (Fig. 4) suggesting that electrostatic interactions play a significant role. Because there is little charged bile salt in the flat, central part of the disk (in our model it is in fact neglected), the only configuration involving significant electrostatic interactions is edge-to-edge (Fig. 11). This indicates that this configuration is the dominant one for coalescence. This is corroborated by estimates of the activation barrier (see "Interactions between disklike micelles") which are lower in the edge-to-edge geometry despite the electrostatic contribution.

View larger version (16K): [in this window] [in a new window]   FIGURE 11  Schematic representation of the coalescence of two disklike micelles in a edge-to-edge configuration. (a) Integration over the coordinates of all configurations at separation h hf gives the reaction volume Sij with Sij the reaction surface and the distance over which coalescence can typically occur. (b) The scaling for the reaction surface, Sij rj(ri + rj), is obtained by considering a disk of radius rj sampling all possible edge-to-edge configurations with a disk of radius ri.

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The first term on the right-hand side describes the creation of a k-aggregate by binary collisions of i- and j-aggregates (i + j = k) whereas the second term represents the disappearance of k-aggregates by binary collisions with any other aggregate. Three-body collisions are thus not taken into account and the equation is hence only valid for dilute systems, a condition well satisfied in our experiments. The productive collisions, i.e., those collisions leading to coalescence, between i- and j-aggregates occur with rate coefficients _ij, the kernels of the Smoluchowski equations. They contain all the information on the reaction. Before we can calculate the kernels _ij and examine the growth of disks, a description of the interactions between disklike micelles is required.

Interactions between disklike micelles
For coalescence to occur, two disks must first approach each other. This is controlled by the interactions between disks, which depend on their distance and relative orientation. Then a topological connection between the disks, a "neck," has to be formed before coalescence can be completed. In the following we try to estimate the topological barrier to coalescence (the "bare fusion barrier"), which is related to formation of a neck and is expected to be relatively high, and the DLVO interactions, which comprise electrostatic and van der Waals interactions. We will call the sum of topological barrier and DLVO potential at the fusion distance, which represents the overall barrier, the "fusion barrier."

Topological barrier
Estimating the topological barrier E_t involved in forming a "neck" between two disklike micelles is very difficult. It is, however, similar to the fusion of bilayers, for which several models exist (Leikin et al., 1987; Israelachvili, 1992; Siegel, 1993; Lentz, 1994; Chernomordik et al., 1995; Lee and Lentz, 1997, 1998; Markin and Albanesi, 2002; Kozlovsky and Kozlov, 2002). They suggest that the topological barrier in the face-to-face orientation is 25 kT. Above we argued that the strong c_s-dependence of indicates that the edge-to-edge orientation dominates. Compared to the face-to-face (and also face-to-edge) orientation, less surface of similar curvature is involved in the edge-to-edge orientation and we thus expect a significantly lower energy barrier, which nevertheless amounts to several kT. (The actual height of the topological barrier E_t is contained in the adjustable parameter see "Initial growth".) In the presence of bile salt, the spontaneous curvature will become positive. Because the topological barrier depends on the membrane curvature, with negatively curved intermediates involved in the transition, the energy cost of the deformation and hence E_t will increase. We neglect this composition dependence and we also neglect possible compositional inhomogeneities, such as bile salt-depleted, "sticky" patches on the rim, which might lower E_t.

Overcoming the bare fusion barrier of several kT is thus a strongly limiting step and leads to coalescence being reaction limited under all conditions investigated, even for high salt concentrations where electrostatic repulsion is negligible. Fusion occurs at a distance h_f, which is of the order of a molecular length, typically two hydration layers and thus h_f 10 Å (Leikin et al., 1987).

DLVO interactions
We will first examine DLVO interactions between flat monolayers and then use the Deryaguin approximation to account for the curvature of the monolayers (Israelachvili, 1992; White, 1983). DLVO interactions comprise van der Waals attraction and electrostatic repulsion. The van der Waals interaction energy per unit area is taken as (Israelachvili, 1992)

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where h is the distance between the monolayers, and H the Hamaker constant of the lipid-water-lipid system (H 5 x 10^-21 J (Israelachvili, 1992)). We neglect retardation effects.

The negatively charged bile salt molecules lead to a surface charge density = e_r/a_D and thus control the electrostatic potential ₀ at the monolayer (Eq. 7; Fig. 9 c). This results in an electrostatic repulsion between two monolayers. However, the calculation of the electrostatic interaction is complicated by the fact that the bile salt molecules (and thus the charges) are mobile. When exposed to an electric field, for instance caused by another monolayer, bile salt molecules could either move to a more distant point on the monolayer or leave the monolayer. Furthermore, the degree of bile salt dissociation could change. An exact determination of the electrostatic interaction thus requires knowledge on how the charge on each of two approaching monolayers is regulated in response to their growing electrostatic interaction. Whatever the charge regulation mechanism (Russel et al., 1991; Yaminsky et al., 1996; Dean and Sentenac, 1997; Tsao and Sheng, 2001), we expect it to be bracketed by the two limiting cases of constant surface charge and constant surface potential. A crossover between the two regimes might occur and can depend on the charge density of the monolayer; for higher charge densities and/or lower salt concentration c_s the constant potential limit should provide a better description, whereas for lower and/or higher c_s the constant charge regime should be more appropriate.

We only investigate conditions implying relatively low surface charge densities and/or high salt concentrations c_s. Under these conditions, the electrostatic interaction energy per unit area based on constant charge, and constant potential, respectively, can be approximated by (Russel et al., 1991):

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The above Eqs. 11 and 12 describe DLVO interactions between flat monolayers at a distance h. We now use the Deryaguin approximation to take the curvature of the monolayers into account. Based on the total interaction energy per area V_d(h) = V_a(h) + V_e(h), which can either be based on constant charge, or constant potential, the interaction energy E_d(h) of approaching monolayers with arbitrary curvature and orientation can be calculated using (White, 1983):

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with

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where c_i, c_j, and are the two principal curvatures of the two monolayers and is the angle between the principal axes of the two monolayers (Fig. 11 a). Note that the angle (Fig. 11 a) is an implicit parameter in the curvatures. The Deryaguin approximation is only valid for radii of curvature large compared to the length scale of the interactions. The smallest radius of curvature = 25 Å thus has to be larger than the largest Debye length which is satisfied for salt concentrations c_s 20 mM and thus for all conditions investigated.

The interaction energy E_d(h) as a function of distance h exhibits a typical behavior with a primary maximum due to the electrostatic repulsion that vanishes at high salt concentration c_s and/or low charge, i.e., small _r. More relevant is the interaction energy at the typical fusion distance E_d(h_f). Fig. 12 shows its dependence on c_s for the electrostatic interaction based on constant charge (a) and constant potential (b) respectively. In the case of constant charge, is in the range 0–10 kT, whereas a constant potential results in lower values for 0–3 kT, for the salt concentrations c_s 50 mM studied here. Above 300 mM, electrostatic interactions are essentially screened and the contribution of DLVO interactions to the growth rate is expected to be negligible. (There is, however, still an effect of the salt concentration on the line tension due to the salt-dependent bile salt partitioning between rim and bulk; see Fig. 9.)

View larger version (16K): [in this window] [in a new window]   FIGURE 12  The DLVO interaction energy Ed(hf) between two initial disks in edge-to-edge configuration (with = 0) and for a typical fusion distance hf = 10 Å as a function of salt concentration cs for three dilution factors d (solid line: d = 40, dotted line: d = 80, dashed line: d = 120). The electrostatic interactions are based on (a) constant charge, and (b) constant potential, respectively.

The rate coefficients, or kernels, _ij of the Smoluchowski rate equations are (Appendix B):

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where is the attempt frequency for coalescence (Eq. 45) with D_i the diffusion coefficient, the distance over which coalescence can typically occur (Figs. 11 a and 17 in Appendix B) and S_ij