Experimental Determination of High-Order Bending Elastic Constants

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Experimental Determination of High-Order Bending Elastic Constants of Lipid Bilayers Liliana G. Toscano-Flores, Damián Jacinto-Méndez, and Mauricio D Carbajal-Tinoco J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.6b03983 • Publication Date (Web): 07 Jun 2016 Downloaded from http://pubs.acs.org on June 11, 2016

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The Journal of Physical Chemistry

Experimental Determination of High-Order Bending Elastic Constants of Lipid Bilayers Liliana G. Toscano-Flores, Damián Jacinto-Méndez, and Mauricio D. Carbajal-Tinoco∗ Departamento de F´ısica, Centro de Investigaci´ on y de Estudios Avanzados del IPN, Apartado Postal 14-740, 07000 Cd. de México, Mexico (Dated: June 2, 2016)

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2 I.

ABSTRACT

We present a method to describe the formation of small lipid vesicles in terms of three bending elastic constants that can be experimentally measured. Our method combines a general expression of the elastic free energy of the bilayer and the thermodynamic description of molecular aggregation. The resulting model requires the size distribution of liposomes, which is determined from the X-rays scattered intensity spectra of vesicular dispersions. By using two different preparation methods, we studied a series of vesicular solutions made of distinct lipids and we obtained their corresponding bending elastic constants that are consistent with known bending rigidities.


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3 II.

INTRODUCTION

Lipids are very important molecules for life as they are the main constituents of the cell membranes. A variety of biological processes occur at the membranes or are mediated by them, in coordination with membrane proteins. Among them, we can mention transport of ions and molecules, signalling, immunological reaction and enzymatic activity. Membranes that mimic cells can be used to encapsulate medicaments and deliver them in a controlled way, even in regions of difficult access. Otherwise, a new class of antibiotics is designed to destroy the cell membranes of resistant bacteria. Thus, it is of crucial importance to have a better understanding of the mechanical properties of lipid bilayers that form membranes, in order to predict and eventually control their shape and shape fluctuations. Although, in a first stage, the studied bilayers are only made with phosphatidylcholine lipids and without inserted molecules. During the last decades, several methods have been developed to determine the membrane bending rigidity. There are review papers that describe and analyze the most important of them.1–3 Here, we would like to mention only a few methods according to their classification. In first instance and based on the direct observation of giant vesicles, the bending elastic constant can be determined from the examination of their thermal fluctuations.4 This method is not very demanding since it only requires the observation by video microscopy of the giant vesicles and also an algorithm of image analysis. Another popular approach is the micropipette aspiration technique, which induces a mechanical deformation on the liposome.5 There is a class of methods based on X-rays scattering techniques including the diffuse X-rays scattering on highly oriented lipid bilayer stacks.6 Some other approaches take advantage of the size distribution obtained from the analysis of Transmission Electron Microscopy (TEM) images.7–9 In this paper, we present a method that is, in some sense, a combination of the two last types of studies. Starting from an X-rays scattering spectrum, we determine the size distribution of liposomes, which is analyzed in an analogous, but more general way. In order to carry out such analysis, we propose a model that combines a geometric approach for the bending energy of the membranes together with the thermodynamic description of the lipid aggregation in vesicles of different sizes, as explained in the next sections.



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4 III. A.

THEORETICAL BACKGROUND

Bilayer elastic free energy and self-assembly

Lipid bilayers form fluid membranes of a certain thickness that can be modeled in terms of a Hookean bending energy. For certain length scales, however, the Hookean approach does not provide a sufficiently accurate description of the system and terms of higher order (also called non-Hookean) are utilized to explain some properties including the fact that vesicles cannot be of an arbitrarily small size. In the absence of both a spontaneous curvature and a chiral deformation, Helfrich used quartic invariants under rotation to introduce a fourthorder equation describing the free energy per unit area of a weakly curved lipid bilayer,11 1 ¯ K + κ2 J 4 + κ ¯ 2 K 2 + κ′ (∇J)2 + κ′′ ∇2 J 2 + κ ¯ ′′ ∇2 K 2 + κ ¯4K 4. g = κJ 2 + κ 2

(1)

Within this approach, c1 and c2 are the principal curvatures of the membrane and thus J = c1 + c2 and K = c1 c2 are the total and Gaussian curvatures, respectively. The moduli κ, κ ¯ , κ2 , κ ¯ 2 , κ′ , κ′′ , κ ¯ ′′ , and κ ¯ 4 depend on the specific features of each lipidic system. Of course, not all the terms of Eq. (1) provide a significant contribution to the free energy, as explained in following lines. In the case of symmetric membranes, the contribution of odd powers of the principal curvatures can be neglected and this is also the case of the elements containing ∇2 , whose respective integrals on a closed surface are zero because of the divergence theorem. A term proportional to KJ 2 is neither included in Eq. (1), because it provides an insignificant contribution to the whole expression.11 It can be mentioned that Helfrich et al. gave an estimate for some of these moduli,11,12 i.e., κ = 2γb2 /3, κ ¯ 2 = −8γb4 /9, κ′ ∼ γb4 /30, and κ ¯ 4 ∼ γb8 , with b = h/2. Here, b and h are the monolayer and the bilayer thicknesses, respectively, and γ is the interfacial tension at the water-lipid interface. Assuming that flat bilayers correspond to the stable state, it can be demonstrated that 2κ + κ ¯ > 0 with κ and κ ¯ being the bending and saddle-splay moduli, respectively. Moreover, in the case of spherical deformations, κ ¯ < 0.13 There is limited experimental information about κ ¯ because it is difficult to obtain a signal if there is no change in the topology or in the boundary of the membrane. By using molecular dynamics simulations, different groups have established the approximate relation κ ¯ ≈ −0.5κ for the spherical geometry and this result can be consulted in Ref. 14 and references therein. If there are small gradients of the total curvature and according to the previous estimates,


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5 the term κ′ (∇J)2 can be neglected on the total account of g. On the other hand, the contribution of the term κ2 J 4 was studied by Harmandaris et al.,15 who performed simulations of cylindrical vesicles.15 In such case, the elements containing K are eliminated since K ≡ 0 for the cylindrical geometry. These authors found no deviations from the Hookean behavior, implying a very small, if any, contribution of the term κ2 J 4 while J 6= 0. Such result is in agreement with the experimental findings on tethers pulled from lipid membranes.16 At this point, we mention that fluid membranes can adopt varied shapes, according to the applied stress profiles and even in the case of leaflets made of two different lipid compositions (as found in biological membranes), there is a spontaneous curvature that the bilayer might have. In the absence of important stresses and provided that certain shapes (like the cylindrical one) are not favorable, it is reasonable to suppose that our liposomes are of spheroidal shape. Moreover, and as a working approximation, we assume that the lipid bilayers form spherical vesicles, i.e., c1 = c2 = 1/R, with R being the radius of the spherical shell and it is defined as R = r − b, where r is the external radius of the vesicle. Thus, from Eq. (1) and considering an area per vesicle of 4πR2 , we obtain an expression for the total free energy per liposome, κ ¯2 κ ¯4 G = 4π 2κ + κ ¯+ 2 + 6 . R R

(2)

In the case of considerably larger vesicles (R ≫ h), the last two terms of Eq. (2) provide a negligible contribution to the free energy and the well-known form g = κJ 2 /2 + κ ¯ K is recovered. We want to emphasize that, besides a formulation for the bilayer bending energy, it is necessary to furnish a thermodynamic description of the assembled molecules that form the vesicular aggregates. In a more detailed level, the number of molecules aggregated in a single vesicle, N , is given by the expression N = 8πR2 /Al , where Al is the mean area per lipid. The aggregation number N and its corresponding radius R are taken as equivalent from now on. Considering that the chemical potential of identical molecules in different aggregates is the same,17,18 XN = N

N M XM M 0 0 , exp (µ − µN ) M kB T M

(3)

with XL being the mole fraction and µ0L is the standard part of the chemical potential per molecule in vesicles of aggregation number L (L = N, M ), kB is Boltzmann’s constant, and T is the absolute temperature. In Eq. (3), the aggregation number M is associated with a



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6 reference state of the characteristic radius R0 , which can be chosen to be an extreme value and thus M = 8πR02 /Al . According to Israelachvili,18 N µ0N = N µ0M +GM N , with GM N being the total free energy between the reference state M and an arbitrary state of aggregation number N . Combining this expression with Eqs. (2), (3), and assuming that κ ¯ = −κ/2, we obtain the following equation, ln

XN N

N − ln M

XM M

3 = −4πβ κ 2

2 2 1 R 1 R2 R +κ ¯4 , −1 +κ ¯2 − − R02 R04 R2 R08 R6 (4)

where both sides of Eq. (4) are identical to N β(µ0M − µ0N ), and β = (kB T )−1 . A similar approach was first used by Denkov et al. to determine a bending elastic constant of purple membrane bacteriorhodopsin vesicles.7 These authors, however, only kept (and also adapted) the Hookean term of Eq. (4), which allowed them to estimate a term proportional to κ. The same kind of approximation was also used by other authors to study their vesicular systems, mainly made of surfactants.8,9 On the other hand, we can note in Eq. (4) that XN ∼ D(N ) and D(N ) ≡ D(r) is the size distribution of a collection of non-interacting liposomes of external radius r and such property can be extracted from experimental data, as explained in the following sections.

B.

Vesicles size distribution

The scattering intensity of an ensemble of non-interacting spherical particles of different sizes can be determined by,19,20 Ic (q) = C

Z

rmax

D(r)V ′2 Fv (qr)dr,

(5)

rmin

where the ideal limiting values are rmin → 0 and rmax → ∞. Such scattering intensity is a function of the wavevector magnitude q = 4πns sin(θ/2)/λ0 , with θ being the observation angle, λ0 is the incident photon wavelength, and ns is the refraction index of the solvent at the given wavelength (i.e., ns = 1.333 for visible light and ns = 1 for X-rays). The constant C depends on both the concentration of liposomes as well as the scattering length density difference between particle and solvent, but here it is used as an adjusting parameter. Fv (qr) is the form factor of a spherical vesicle of external radius r, and V ′ is a term proportional to the volume, i.e., V ′ = r3 − (r − h)3 . In the approximation of a homogeneous scattering


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7 length density of the bilayer, Fv (qr) can be modeled as a spherical shell21 and hence, 3 2 1 3 r Ks (qr) − (r − h) Ks (q(r − h)) . Fv (qr) = 3 r − (r − h)3

(6)

It can be noticed that the previous expression is valid only in the case of an identical solvent inside as well as outside of the spherical membrane, as in the present experiments. Here, Ks (qρ) =

3 [sin(qρ) − qρ cos(qρ)] , (qρ)3

(7)

and the form factor of a sphere of radius ρ is thus Fs (qρ) = [Ks (qρ)]2 . In a regular scattering experiment, the range of radii r that can be sampled in Eq. (5) is actually bounded by the limits rmin and rmax , which are fixed by the conditions rmin ≥ π/qmax and rmax ≤ π/qmin with qmax and qmin being the limiting q-values of the X-rays scattering spectrum under analysis. The size distribution function D(r) can be determined by divers procedures that can be divided into analytical and numerical methods.20 Among them, we selected a Monte Carlo (MC) algorithm that makes no assumption on the analytical form of D(r). Such method has the following features.22 The function D(r) is treated as a histogram of bin size ∆r = (rmax − rmin )/Np with Np being the number of mesh points. Starting with a flat histogram of null values, the size distribution function is fulfilled in a progressive way, by following a sequence of MC steps. In each basic step, we use the discrete version of Eq. (5) to compute a scattering intensity Ic (q) that contains the contribution of a test vesicle of fixed thickness h and a certain radius r, which is randomly chosen within the interval rmin ≤ r ≤ rmax . The variable C is adjusted to match the area under the curve of Ic (q) to the corresponding integral of the experimental spectrum Iexp (q) that is taken as a reference. The MC step is accepted only if the computed scattering intensity gets closer to Iexp (q), according to the relative error measure, Z qmax Iexp (q) − Ic (q) dq, ǫ= I (q) exp qmin

(8)

where the parameter ǫ provides a convergence criterion. If ǫ decreases (increases) then the contribution of the test particle is retained (rejected) and a new test vesicle is probed. The algorithm is stopped when ǫ reaches a minimum value. As a result, we obtain both Ic (q) and D(r). The intensity Ic (q) is useful to verify the fidelity of the reconstruction scheme and D(r) is the most important result of this MC method. The calculation of D(r) requires the



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8 examination of experimental data of the systems of interest. In order to test our method, we first determined the size distribution function of a reference system (a dispersion of previously analyzed nanoparticules) and then we characterized a series of lipid solutions.

IV.

EXPERIMENTAL RESULTS

A.

Vesicles formation

We prepared vesicular suspensions of various nonionic lipids by two complemental methods, namely, extrusion and sonication. Our list of lipids include 1,2-dilinoleoyl-snglycero-3-phosphocholine (DLPC), 1,2-dimyristoyl-sn-glycero-3-phosphocholine (DMPC), 1,2-dioleoyl-sn-glycero-3-phosphocholine (DOPC), Egg L-α-phosphatidylcholine (PC), and 1-palmitoyl-2-oleoyl-sn-glycero-3-phosphocholine (POPC) all of them from Avanti Polar Lipids, Inc. (Alabaster, AL, USA). In general, the formation of vesicles is done under carefully controlled conditions to avoid any possible contamination of dust and dirt. Such conditions comprise the use of meticulously cleaned vials, glassware, and sample holders. The main steps of the two methods of preparation are described in the following lines. The original lipids are dispersed in chloroform and a particular lipid is left under a gentle flow of Ar for about 15 min, in order to evaporate as much chloroform as possible. After that period, it is placed in a vacuum chamber for a minimum of 2 hr at a pressure of ∼1 Pa to eliminate any trace of chloroform that could alter our systems. Keeping the temperature above the transition temperature Tc , the dry lipid is dissolved in distilled deionized water of 18.2 MΩ·cm (Barnstead) with no added sugar or electrolyte. The two methods differ beyond this point. In the first method, the solution is extruded (using an Avanti Miniextruder) for a total of 20 full cycles through a polycarbonate filter of pore size 0.2 µm to ensure the formation of unilamellar vesicles at least around such size. As a result, we obtain a suspension of reference concentration ce = 11.3 g/l, which takes into account the lipid trapped in the filter. Here, the subindex e is related to the protocol of extrusion. On the other hand, the preparation of sonicated liposomes requires the addition of an extra volume of water for a final concentration cs = 3 g/l. In this case, the sample is first mixed in a vortex and then it is sonicated in an ultrasonic bath (Branson 2510) for two periods of 10 min each, while keeping the bath cooled to avoid overheat. The method of sonication reduces the size


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9 of liposomes with time,23 and it typically produces small, unilamellar vesicles (SUVs) with diameters between 150 and 500 ˚ A. The same procedures are repeated for all the lipids under analysis. The vesicular solutions are stored at 4◦ C and they are discarded after four days of preparation. During such period of time, the resulting dispersions are assumed to be in thermodynamic equilibrium and they are mainly studied by means of X-rays scattering and one additional technique.

B.

Experimental characterization

We performed small-angle X-rays scattering (SAXS) experiments to characterize vesicular and colloidal suspensions on a MicroMax 002+ apparatus (Rigaku) supplied with a copper filament that delivers a beam of wavelength λ0 = 1.54 ˚ A. The beam is collimated with an array of two pinholes and a third pinhole eliminates any remaining diffraction pattern. The resulting spot is small and homogeneous. We used a multi-wire chamber detector (Gabriel type of 800×800 cells with a pixel size of 152 µm). In the sample holder, a colloidal suspension is encapsulated between two kapton windows separated by 0.75 mm and the temperature is controlled during the full experiment at 25◦ C. We mention that the scattering intensity of our vesicles Iexp (q) is obtained by using the expression,24 Iexp (q) =

Itot (q) − Idc (q) Iw (q) − Idc (q) − , Ttot Tw

(9)

where Itot (q) is the total intensity of the system, Idc (q) is the dark-count intensity of the detector, Iw (q) is the intensity of purified water, Ttot and Tw are the transmittances of the whole sample (vesicles or nanoparticles immersed in water) and the solvent alone, respectively. A regular experiment requires about 8 hr and we only averaged consistent results of the same extrusion as well as results of different preparations. In order to verify our MC algorithm, we also studied a colloidal dispersion of gold nanospheres with a core radius of 75 ± 10 ˚ A, which are sterically stabilized with PEG 5000 (Sigma-Aldrich). The nanoparticles are immersed in an electrolyte of 10 mM of NaCl to avoid any possible electrostatic effect and the concentration of suspended beads is cn = 0.02 g/l. In Figure 1, we present the SAXS scattering intensity of the colloidal suspension together with the scattering intensities of liposomes dispersions made of the lipids POPC and DOPC that we prepared with the protocols of extrusion (E) and sonication (S).



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10 Otherwise, we opted to use experimental information from the literature to obtain two relevant parameters that are required in our analysis, namely, the bilayer thickness h and the mean area per lipid Al .25,26 Such parameters are reported in Table 1 and they provide valuable information for the analysis of our experimental data. The vesicular formation by lipid extrusion was tested with static light scattering (SLS) experiments on a goniometer-based 3D DLS Spectrometer (LS-Instruments), using a HeNe laser operating at a wavelength of 632.8 nm. For vesicular solutions of DMPC and concentration ce /1000, we fitted the spectrum part corresponding to very small q-values ˚−1 ) with the form factor of a spherical shell of thickness h = 37 (5.7 × 10−4 . q . 0.0012 A ˚ A, as defined in Eq. (6). As a result, we obtained an extrusion radius a = 976 ˚ A. It can be noticed that the corresponding diameter coincides with the pore size of the filter membrane. Likewise, the results for the remaining lipids are similar to these ones. We also mention that this method excludes the presence of liposomes larger than the pore size. However, smaller vesicles can appear and it is difficult to detect them with SLS, since their scattered intensity is proportional to r4 . The SUVs size distribution is obtained through the analysis of SAXS experiments.

C.

Results and discussion

Besides an initial test on a suspension of colloidal nanoparticles, we examined SAXS spectra of vesicular solutions of our lipids of interest, which were prepared by following the methods described in previous sections. The typical concentration of extruded liposomes was ce , but we obtained identical results with the dilutions 0.75ce and 0.5ce . The examination of the averaged spectra was achieved according to the basic algorithm of analysis. For each system, we studied a series of at least 100 intensity curves Ic (q) and its corresponding size distribution functions D(r) that were obtained from the reconstruction algorithm described in Sec. II B. Firstly we should mention that Ku˘cerka et al. employed various models to describe the structure of lipid bilayers.27 Among them, they found that the model of homogeneous lipid density, such as in Eq. (6), gives good results in a comA−1 , which is the q-range of our interest. The parison with experimental data for q . 0.1 ˚ reconstruction algorithm makes use of the averaged spectrum and the thickness h. In first stage, we interpolated a total of Np = 240 points to the experimental spectrum and each


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11 intensity curve starts with a distinct random seed. Each procedure requires a minimum of 105 MC steps, but we only kept the cases satisfying the following criteria. The parameter ǫ is within the 20% of the lower values of the whole series and the histograms of D(r) have the higher possible values. Such criteria ensure that the average of the selected size distributions converges to a single curve. The SAXS spectra of Figure 1 are shown together with their computed intensity curves Ic (q). As it can be noticed, there is a good agreement between the experimental and the modeled intensities in the plotted range. For the two lipids of such figure, their corresponding spectra are qualitatively similar for preparations made with the same protocol and there are detectable differences between the results of the two distinct methods, especially for the range 0.04 . q . 0.075 ˚ A−1 . In comparison with all of them, the spectrum of the colloidal nanoparticles is more homogeneous and it has a less pronounced upturn in the region of ˚−1 ), provided that larger structures like aggregates are basically small q-values (q . 0.03 A inexistent and thus have a negligible contribution to the scattering intensity of such region. It is important to mention that, according to their size, the extruded dispersions have at least two populations and one of them is related to the pore size. Provided that the sonicated samples do not have a population of large vesicles, they can be used as a reference to estimate the scattering intensity coming from larger vesicles. Therefore, we determined the difference between the areas under the curve (q . 0.04 ˚ A−1 ) of the intensity spectra of both extruded and sonicated samples. For the lipids of Figure 1, the relative differences between the integrated intensities obtained by the two techniques correspond to 19% (POPC) and 11% (DOPC) with respect to the extrusion integrated spectra. In other words, there is relatively minor influence of the large vesicles in the q-range of the intensity profiles under study and this trend is verified in the analysis of the size distributions. In Figure 2, we present the most relevant parts of the size distributions of the systems shown in Figure 1 and obtained by SAXS. For instance, the size histogram of the large vesicles formed by the porous membrane is out of range with respect to the radii under consideration. In Figure 2a, we display a TEM image (from a JEOL JEM-2010 equipment) of the gold nanospheres. An image analysis of a few beads reveals a mean radius of 77 ± 4 ˚ A. The size distribution for the colloidal nanoparticles is plotted in Figure 2b together with a fit function that is proportional to the Schultz distribution function (SD)28 , i.e., ∼ (r − r¯)z exp[−(z + 1)(r − r¯)/¯ r], which properly describes polydisperse spheres systems.



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12 Here, r¯ and z are fitting parameters with r¯ being the mean sphere radius. The root-meansquare deviation from the mean size is given by σr¯ = r¯(z + 1)−1/2 . As a result, r¯ = 74 ± 14 ˚ A, in consistency with the previous measurements. The size distribution histograms of the two lipids of Figure 1 are shown in Figures 2c (POPC) and 2d (DOPC). Specifically, each subfigure contains the histograms D(r) obtained through the reconstruction algorithm for vesicular solutions prepared within the protocols E and S. As it can be observed, extruded liposomes give rise to taller histograms, as a consequence of the condition ce > cs . In comparison with them, sonicated suspensions of vesicles do not yield a population of large vesicles, but produce broader and somewhat noisier histograms for radii larger than the local maximum. For smaller radii, there are no more peaks than those shown in Figures 2c and 2d, which is the well-known result indicating the formation of liposomes beyond a minimum size. At this point, we should mention that Lauf et al. observed spherical lipid vesicles of a few nanometers, which were prepared by extrusion and characterized by freeze-fracture electron microscopy.29,30 After a storage of four weeks at 13◦ C, such small vesicles can form tubular agglomerates.29,30 In Figures 2c and 2d, we also include the corresponding curves that emerge from our model and their construction is explained below. The functions D(r)(= D(N )) are used to determine the left side of Eq. (4). The maximum of the size distribution function is located at the external radius r0 , which can also be expressed in terms of the characteristic radius R0 = r0 − b to facilitate the conversion to the aggregation number M , i.e., D(r0 ) = D(M ). An intermediate step is necessary, provided that the model makes use of mole fractions instead of size distributions. In order to obtain the left side of Eq. (4), we assume that XN = KD(N ) with K being a constant determined P by the normalization condition N XN = 1. In such condition and compared to the main

peak, we can neglect the contribution of the secondary peaks (including the distribution of excluded vesicles, eventually present). The constant K allows us to compute the experimental quantities XN and XM . An example of the symbols corresponding to the left side of Eq.

(4) is shown in Figure 4. The model given by the right side of Eq. (4) requires information about the characteristic radius R0 and it is slightly larger in the case of sonicated liposomes in comparison with extruded ones (see Figures 2c and 2d). The interesting result is that the three fitting parameters are very similar for each lipid, independently of the preparation method (with


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13 differences no larger than 10%), and the same conclusion applies for our other lipids. As a consequence, we combined the experimental information of liposomes dispersions obtained by the two preparation methods. In each case, we performed a weighted average of the extruded and sonicated intensity spectra and the weight is a term proportional to the corresponding concentrations ce and cs . In Figure 3, we present the averaged intensity spectra for the five lipids of interest together with their reconstructed intensities Ic (r), which properly follow the experimental data. We can mention that such level of agreement cannot be obtained with arbitrary size distribution functions. For instance, in Figure 3 we included the predicted intensity that was obtained by assuming an infinitely narrow distribution of vesicles sizes and also using Eqs. (5)-(9). In Figure 4, we plot the symbols corresponding to the left side of Eq. (4) for the lipids under study, while the continuous lines are fits to the function given by right side of the same equation, which is a term proportional to the chemical potential difference ∆µ = µ0N − µ0M . As a result, we have an estimate of the bending elastic constants κ, κ ¯ 2 , and κ ¯ 4 , whose numerical values are presented in Table 1, together with the theoretical predictions for them, as well as an average of experimental measurements performed with other techniques for the constant κ. On the one hand, such averages were done for systems with experimental conditions close to ours and the data were obtained from two main compilation papers, namely, Ref. 3 (for the lipids DLPC, DMPC, and DOPC) and Ref. 2 (for POPC). There are important variations between the results obtained within different techniques and experimental conditions, which are reflected in considerable standard deviations. On the other, the theoretical predictions emerge from the expressions presented in Sec. II A. Within experimental errors, we can mention that our results for the bending elastic moduli κ are consistent with both experimental measurements of other techniques and most of the theoretical predictions, as it can be observed in Table 1. On the other side, there is scarce information in the literature about the elastic constants κ ¯ 2 and κ ¯ 4 . From our experimental data, we notice, as in the case of the bending modulus κ, a clear reliance on the lipid thickness for the two remaining moduli, as established in the theoretical model of Helfrich. However, there are systematic differences between the experimental (x) and the theoretical (t) moduli. According to our measurements, κ ¯ 2x has the right negative sign predicted by the model, but |¯ κ2x | > |¯ κ2t |. In a similar way, κ ¯ 4x >



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14 κ ¯ 4t . Considering that κ ¯ 4 is of higher order than κ ¯ 2 , the mentioned differences are more accentuated with the increasing order of the term, which is a direct consequence of the higher power-law exponents of the thickness dependence. Although the moduli κ ¯ 2 and κ ¯4 have small values, they define, together with the modulus κ, the free energy that shapes the distribution of radii in the range of 60 to 100 ˚ A, for the lipids under study. The known free energy is of course recovered for larger radii.

V.

CONCLUSIONS

In conclusion, we have presented a method to determine bending elastic constants of lipid bilayers that is a generalization of the approach first introduced by Denkov et al.7 Our method is based on a theoretical model combining a more general expression for the free energy of a lipid vesicle11,12 together with a thermodynamic equation describing the chemical potential of molecules in liposomes of different aggregation numbers.17,18 We mention that our approach does not require any further approximation (as in the case of Denkov et al.) and we can obtain information about two additional constants. Our model makes use of two characteristic parameters (h and Al ) extracted from experimental data from the literature , as well as the experimental size distribution function, which is inferred from regular SAXS experiments done on vesicular solutions that were prepared by two different and complemental methods. Such size distribution function is determined from the scattered intensity spectrum through a MC algorithm. As a result, we obtain three bending elastic constants that fully describe the formation of lipid vesicles of external radius found between about 60 and 100 ˚ A. The numerical values of such constants are in good agreement with theoretical estimates and also with experimental results of other techniques (for the modulus κ). We want to stress that, to the best of our knowledge, no previous experimental measurements have been reported before for the higher-order terms (¯ κ2 and κ ¯4) and they are quite relevant to define the free energy and thus the distribution of liposomes in the mentioned regime. In general, more experiments are necessary to investigate other effects such as a variation of the temperature, the presence of sugar and/or electrolyte or mixtures of lipids.


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15 VI.

AUTHOR INFORMATION

A. ∗

Corresponding Author

E-mail: [email protected]. Telephone: +52 55 57 47 38 00, Ext. 6173.

VII.

ACKNOWLEDGMENTS

We want to thank useful discussions with Carmen Granados and Markus Drechsler. The SLS and TEM experiments were done with the assistance of Gualberto Ojeda and Jaime Santoyo, respectively. This work was supported by CONACyT under Grant No. 152532 and a fellowship given to L.G.T.-F. The calculations were performed at the cluster Xiuhcoatl of Cinvestav.

VIII.

1

REFERENCES

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2

Dimova, R. Recent Developments in the Field of Bending Rigidity Measurements on Membranes Adv. Colloid Interf. Sci. 2014, 208, 225-234.

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Marsh, D. Elastic Curvature Constants of Lipid Monolayers and Bilayers Chem. Phys. Lipids 2006, 144, 146-159.

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Ku˘cerka, N.; Tristram-Nagle, S.; Nagle, J. F. Structure of Fully Hydrated Fluid Phase Lipid Bilayers with Monounsaturated Chains J. Membrane Biol. 2005, 208, 193-202.

7

Denkov, N. D.; Yoshimura, H.; Kouyama, T.; Walz, J.; Nagayama, K. Electron Cryomicroscopy of Bacteriorhodopsin Vesicles: Mechanism of Vesicle Formation Biophys. J. 1998, 74, 1409-1420.



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16 8

Jung, H. T.; Coldren, B.; Zasadzinski, J. A.; Iampietro, D. J.; Kaler, E. W. The Origins of Stability of Spontaneous Vesicles Proc. Natl. Acad. Sci. USA 2001 98, 1353-1357.

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van Zanten, R.; Zasadzinski, J. A. Using Cryo-Electron Microscopy to Determine Thermodynamic and Elastic Properties of Membranes Curr. Opin. Colloid Interf. Sci. 2005, 10, 261-268.

10

Niggemann, G.; Kummrow, M.; Helfrich, W. The Bending Rigidity of Phosphatidylcholine Bilayers: Dependences on Experimental Methods, Sample Cell Sealing and Temperature J. Phys. II (France) 1995, 5, 413-425.

11

Helfrich, W. Bending Elasticity in Fluid Membranes In Giant Vesicles; Luisi, P. L.; Walde P. Eds.; Wiley: Chichester, 2000, pp 51-70.

12

Goetz, R.; Helfrich, W. The Egg Carton: Theory of a Periodic Superstructure of Some Lipid Membranes J. Phys. II (France) 1996, 6, 215-223.

13

Méléard, P.; Gerbeaud, C.; Pott, T.; Mitov, M. D. Electromechanical Properties of Model Membranes and Giant Vesicle Deformations In Giant Vesicles; Luisi, P. L.; Walde P. Eds.; Wiley: Chichester, 2000, pp 185-205.

14

Mingyang Hu, M.; de Jong, D. H.; Marrink, S. J.; Deserno, M. Gaussian Curvature Elasticity Determined from Global Shape Transformations and Local Stress Distributions: a Comparative Study Using the MARTINI Model Faraday Discuss. 2013, 161, 365-382.

15

Harmandaris, V. A.; Deserno, M. A Novel Method for Measuring the Bending Rigidity of Model Lipid Membranes by Simulating Tethers J. Chem. Phys. 2006, 125, 204905-204910.

16

Cuvelier, D.; Derenyi, I.; Bassereau, P.; Nassoy, P. Coalescence of Membrane Tethers: Experiments, Theory, and Applications Biophys. J. 2005, 88, 2714-2726.

17

Israelachvili, J. N.; Mitchell, D. J.; Ninham, B. W. Theory of Self-Assembly of Hydrocarbon Amphiphiles into Micelles and Bilayers J. Chem. Soc. Faraday Trans. 2 1976, 72, 1526-1568.

18

Israelachvili, J. N. Intermolecular and Surface Forces (2nd. Ed.), Academic Press: London, 1986.

19

Guinier, A. Théorie et Technique de la Radiocristallographie, Dunod: Paris, 1956.

20

Glatter, O. Data Treatment In Small Angle X-ray Scattering, Glatter, O; Kratky O. Eds., Academic Press: London, 1982, pp 119-165.

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Higgins, J. S.; Benoˆıt, H. C. Polymers and Neutron Scattering, Oxford Univ. Press: New York, 1997.


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Martelli, S.; Di Nunzio, P. E. Particle Size Distribution of Nanospheres by Monte Carlo Fitting of Small Angle X-Ray Scattering Curves Part. Part. Syst. Charact. 2002, 19, 247-255.

23

Woodbury, D. J.; Richardson, E. S.; Grigg, A. W.; Welling, R. D.; Knudson, B. H. Reducing Liposome Size with Ultrasound: Bimodal Size Distributions J. Liposome Res. 2006, 16, 57-80.

24

Pauw, B. R. Everything SAXS: Small-Angle Scattering Pattern Collection and Correction J. Phys.: Condens. Mat. 2013, 25, 383201.

25

Pan, J.; Tristram-Nagle, S.; Ku˘cerka, N.; Nagle, J. F. Temperature Dependence of Structure, Bending Rigidity, and Bilayer Interactions of Dioleoylphosphatidylcholine Bilayers Biophys. J. 2008, 94, 117-124.

26

Ku˘cerka, N.; Nieh, M.-P.; Katsaras J. Fluid Phase Lipid Areas and Bilayer Thicknesses of Commonly Used Phosphatidylcholines as a Function of Temperature Biochim. Biophys. Acta 2011, 1808, 27612771.

27

Ku˘cerka, N.; Nagle, J. F.; Feller, S. E.; Balgav´ y, P. Models to Analize Small-Angle Neutron Scattering from Unilamellar Lipid Vesicles Phys. Rev. E 2004, 69, 051903-051912.

28

Zimm, B. H. Apparatus and Methods for Measurement and Interpretation of the Angular Variation of Light Scattering; Preliminary Results on Polystyrene Solutions J. Chem. Phys. 1948, 16, 1099.

29

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Lauf, U. PhD Dissertation at the Friedrich-Schiller-Universität Jena, 2003.



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18

FIG. 1. (Color online) Scattering intensity spectra of vesicular suspensions made of two lipids and formed by two methods, namely, sonication (S) in dark gray symbols and extrusion (E) in light gray symbols. The lipids are POPC (triangles) and DOPC (squares). The scattering intensity spectrum of gold nanoparticles (Au) in suspension is included (circles). The intensity spectra of different materials are multiplied by constants to enhance the differences between them. The continuous lines are the corresponding reconstructed intensities obtained by means of Eq. (5) (see text).


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19

FIG. 2. (Color online) Size distributions of different colloidal and vesicular suspensions. (a) TEM image of gold nanoparticles of a nominal core diameter of 150 ˚ A. The black bar indicates a distance of 500 ˚ A. (b) Size distribution of the colloidal nanoparticles extracted by the analysis of its corresponding SAXS spectrum. The continuous line is a fit to a function proportional to the Schultz distribution (SD). (c) and (d) Size distribution histograms of the two lipids, (c) POPC and (d) DOPC, obtained from their SAXS spectra. The lipid solutions were prepared by two methods, namely, extrusion (E) in light gray bars and sonication (S) in dark gray bars. The continuous lines (M) are the distribution functions determined through the model proposed in the text.



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20

FIG. 3. (Color online) Experimental SAXS intensity of five distinct vesicular systems. The intensity spectra were multiplied by constants to enhance the differences between them. The lipids under study are POPC (full triangles), PC (squares), DOPC (circles), DMPC (diamonds), and DLPC (empty triangles). The continuous lines are the corresponding reconstructed intensities, which are based on a polydisperse model (PD) and they are obtained by means of Eq. (5) (see text). In the case of the DLPC spectrum and for comparison purposes, we included the best intensity that can be reconstructed with a vesicular model of a single-size diameter (SS, in dashed line, with an external radius of 93 ˚ A).


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21

FIG. 4. (Color online) Chemical potential difference ∆βµ = β(µ0N − µ0M ), multiplied by the parameter N . Here, β −1 is the thermal energy, M and N are numbers of lipid molecules per vesicle with M being the aggregation number of the size distribution maximum. The symbols are experimental data obtained from the left side of Eq. (4) and they depend on the mole fractions XN and XM , which come from a normalized size distribution function D(r) (see text). For clarity purposes, the curves are separated with additive constants for the following lipids: POPC (full triangles), DMPC (diamonds), PC (squares), DMPC (circles), and DLPC (empty triangles). The continuous lines (Model) are the corresponding fits to the function given by the right side of Eq. (4). The fitting parameters are the elastic constants κ, κ ¯ 2 , and κ ¯ 4 , all of them reported in Table 1.



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22 Table 1. Characteristic parameters of vesicles made of the lipids under study, namely, DLPC, DMPC, DOPC, PC, and POPC. The bilayer thickness h and the mean area per lipid Al were determined from Refs. 25 and 26 . Our experimental elastic constants κ, κ ¯ 2 , and κ ¯ 4 are presented in comparison with their corresponding theoretical estimations (in square brackets) that were computed assuming γ = 50 mN m−1 . We also present average values of the constant κ (in parenthesis), obtained by means of other experimental techniques (see text).

Lipid

h (˚ A) Al (˚ A2 )

κ (×10−20 J)

κ ¯ 2 (×10−37 J·m2 ) κ ¯ 4 (×10−71 J·m4 )

DLPC

33

60

7.1 ± 1.0 [9.0] (6.0 ± 2.9)

−23.2 ± 3.3 [-3.3]

27.0 ± 4.0 [0.27]

DMPC

37

59

7.3 ± 0.9 [11.4] (6.9 ± 2.3) −27.0 ± 3.1 [-4.0]

40.8 ± 4.8 [0.69]

DOPC

37

71

7.4 ± 2.7 [11.4] (5.5 ± 3.9) −28.9 ± 9.5 [-5.2]

47.7 ± 15 [0.69]

PC

∼ 38

∼ 63

POPC

39

63

8.5 ± 3.0 [12.0] (9.8 ± 5.6) −32.4 ± 10.3 [-5.8] 52.1 ± 15.8 [0.85] 8.5 ± 0.4 [12.6] (8.6 ± 5.6)

−34.5 ± 1 [-6.4]

FIG. 5. TOC graphic


63.6 ± 3 [1.0]

Experimental Determination of High-Order Bending Elastic Constants

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