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Copyright © 2007 The Biophysical Society. All rights reserved.
Biophysical Journal, Volume 93, Issue 5, 1591-1607, 1 September 2007

doi:10.1529/biophysj.107.109264

Membranes

Effects of Monovalent Anions of the Hofmeister Series on DPPC Lipid Bilayers Part II: Modeling the Perpendicular and Lateral Equation-of-State

E. Leontidis^*, ,  , A. Aroti^*, L. Belloni^†, M. Dubois^† and T. Zemb^†

^* Department of Chemistry, University of Cyprus, Nicosia, Cyprus
^† CEA/Laboratoire Interdisciplinaire sur I’Organisation Nanomé trique et Supramolé culaire at Département de Recherche sur I’État Condensé, les Atomes, et les Molécules Service de Chimie Moléculaire at Commissariat à I’Energie Atomique, Saclay, F91191 Gif sur Yvette, France

Address reprint requests to E. Leontidis.

In the companion article in this series 1 we described osmotic stress experiments on dipalmitoylphosphatidylcholine (DPPC) bilayers (above the chain melting temperature) in the presence of a range of concentrations of various sodium salts. The objective was to use DPPC bilayers in water (L_α phase in equilibrium with excess water) as a model system for the investigation of the mechanism of action of monovalent anions belonging to the Hofmeister series 2,3,4. Here we undertake a quantitative analysis, by fitting both the “perpendicular” (osmotic pressure versus bilayer distance, logΠ vs. d_w) and the “lateral” (area per DPPC headgroup, a_L, as a function of salt type and concentration) equation-of-state (EOS) data to interaction models that take into account the effects of salts.

Osmotic stress experiments 5,6,7 are currently the principal method to measure the osmotic pressure EOS of phospholipid bilayers 5,6,7,8,9,10,11,12,13,14,15. Much less work has been done on the measurement of the lateral compressibility of bilayers or the lateral EOS 16. Correspondingly, theoretical attention has mostly focused on the perpendicular EOS 17,18,19,20,21,22,23,24 and seldom on the lateral EOS 24,25,26,27. The area per headgroup is however obtained from a combination of complementary measurements, and the lateral EOS provides an additional opportunity (and a challenge) to test models for specific ion effects. A successful model should in fact be capable of reproducing both the perpendicular and lateral EOS of lipid bilayers. The simplest way to simultaneously work on both EOS is to consider a free energy expression for the bilayer system, which contains both in-plane and out-of-plane components, and also, inevitably, cross-terms. Minimization of the free energy with respect to the two principal geometric parameters, d_w and a_L, eventually provides the two EOS. In this approach it is necessary to have available a complete free energy expression for a system of interacting bilayers, which must contain all terms that depend on d_w and a_L.

The interaction between bilayers is supposed to be the sum of three to four “independent” forces, some of them empirical in nature, all of which introduce adjustable parameters. In this work we use the four standard interactions that appear in almost all recent related investigations 8,9,10,11,12,13,14,15,16,17,18,19,23, namely the Van der Waals attractive force, the hydration force, the fluctuation force and the electrostatic force. The effect of electrolytes on the Hamaker constant and bending rigidity of DPPC bilayers is here treated according to literature information, which suggests that the Hamaker constant could be reduced by 50% or more 23,28,29 and the bending rigidity should also decrease 30 in specific ways in the presence of electrolytes. In addition, we make the usual assumption 8,17,19,23,31,32 that the preferential association of one or both ions of the electrolyte with the neutral phospholipid headgroups generates charged layers at the lipid-water interfaces, which create an additional electrostatic repulsion over what is observed in the absence of electrolytes. There are several alternative ways to model this ion-lipid association however, as we recently discussed 33. In this attempt we use both the local binding approach, and also a “penetration” model, which assumes that anions penetrate uniformly the lipid headgroup layer, although alternative models have been proposed 22,24. These two models were found to be nonequivalent in the case of DPPC monolayers at the air-water interface (34; E. Leontidis, L. Belloni, and A. Aroti, unpublished data). The penetration model is not a new idea, since it has been used in the past to model the surface pressure of insoluble monolayers at the air-water interface 35 and also the surfaces of soft colloidal particles like micelles 36. However, it is used here in the spirit of the more general picture of an “active interface”, according to which ions may preferentially partition inside a lipid layer provided that this layer is disordered enough and that the system gains free energy from the liberation of water molecules of the first hydration shells 33,34,37.

The lateral EOS is obtained from the minimization of the free energy with respect to the headgroup area. There is no established way to partition the intralayer free energy into “independent” components. The in-plane terms are assumed here to be: a), the lipid-water “contact” free energy, modeled as an interfacial tension term; b), the contribution of the conformational entropy of the lipid tails; and c), the nonelectrostatic repulsive energy between headgroups 24,25,26,27,33,38,39,40,41. Cross-terms, describing interactions between two bilayer sheets, also contribute to the lateral EOS. These are the electrostatic energy and the Van der Waals interaction, since both depend on the headgroup area of the lipids.

We believe that the use of two alternative electrostatic models for the ion-lipid association, the examination of the effect of both anion and sodium binding on the lipid headgroups, and—most importantly—the combined modeling of perpendicular and lateral EOS information presented in this work provide a comprehensive modeling platform to improve our understanding of specific salt effects on lipid systems. To our knowledge, an investigation of such a wide range does not exist in the relevant literature to date.

In the companion article 1 we have described the experimental system of bilayer stacks swollen by water and immersed in a solution of polymeric chains that cannot penetrate the lipid phase. In the osmotic stress experiment it is generally assumed that the polymer solution acts as an infinite reservoir of water and ions for the bilayer phase, its osmotic pressure remaining largely unaffected as salt is added 1,5,34. With this basic assumption we focus only on the free energy of the lipid bilayer phase. To properly formulate EOS for this phase one can start either from a partition function of the bilayer system 38,42,43 or from an empirical free energy expression 21,22,24,25,26,27. Although it is a very difficult problem to dissect the bilayer free energy into physically distinct components, an empirical break-up of the free energy is adopted by most workers in the area, and will be attempted in this work as well. We will write the free energy as a function of two principal geometric parameters, which we choose to be the interbilayer water thickness, d_w, and the area per lipid headgroup, a_L. Experimentally, d_w is obtained from the measured period of the lamellar phase, D, and the lipid volume fraction in that phase, φ_L1,5,6,7,8,9,10,11,12,13.

(1)

The area per headgroup is also indirectly obtained from the same data through the bilayer thickness, b_L, as explained in the previous article 1:

(2)

(3)

(4)

(5)

(6)

The intrabilayer free energy contains contributions from the lipid-water interfacial energy, the headgroup nonelectrostatic repulsion, and the conformational entropy of the tails. We initially assume that these depend only on a_L, which is a good approximation in the absence of salts. However, in the presence of salts these terms may depend on d_w, through the ionic adsorption taking place, as will be discussed below. The hydration and undulation forces are considered as pure interbilayer interaction terms depending only on d_w. This is a usual assumption in the literature, but it must be pointed out that the nature of the hydration force and its effect on the undulation force are still poorly understood 5,6,7,10,11,44,45,46,47,48,49,50,51,52,53,54,55. Electrostatic and dispersion interactions depend on both d_w and a_L and are viewed as cross-terms. The osmotic pressure is obtained from the free energy difference via:

(7)

If the in-plane free energy terms of Eq. (4) do not depend on d_w, they do not contribute to the osmotic pressure, and Eq. (7) leads to the usual expression for the osmotic pressure, with the four terms (hydration, undulation, dispersion, electrostatics) currently used by most investigators, as mentioned before. Also starting from ΔF_tot and minimizing with respect to a_L at fixed d_w, T, n_L, we obtain the lateral EOS of the bilayer system, assuming that the bilayers are laterally free to adopt the optimal area per molecule:

(8)

The ΔF_inter terms of Eq. (5), which depend only on d_w, will not contribute to the lateral EOS.

Based on Eqs. (3) we find the following expression for the applied osmotic pressure to the lipid bilayers, Π_TOT, as a function of the water bilayer separation, d_w:

(9)

At large distances between the bilayers, the equilibrium spacing (maximum swelling) is determined by the balance between the attractive Van der Waals forces and either electrostatic or specific hydration repulsive forces. Because DPPC is a zwitterionic lipid, electrostatic interactions exist only when the lipid layers are charged by ion adsorption. At separations shorter than ∼20Å the logΠ−d_w curve provides valuable information about the hydration interaction, which overwhelms electrostatics in this distance range 5,6,7,53,54,55. In Eq. (9), is the so-called “hydration” repulsive force. This is usually empirically modeled 5,6,7,10,11,12,13,14,15,16,17,18,19,56,57,58,59,60,61 as an exponentially decaying function of the form:

(10)

(11)

(12)

Π_vdw is the Van der Waals attractive force, modeled here with an equation routinely used by all investigators today 8,9,14,15,23,59,60, although it is strictly valid for two interacting bilayers, whereas more complicated equations exist for bilayer stacks 29,68,69:

(13)

Here H is the Hamaker constant, and b_L is the bilayer thickness, which is inversely proportional to the lipid headgroup area, according to Eq. (2). Π_ele is the electrostatic contribution to the osmotic pressure. This is computed using the osmotic pressure at the midplane between two bilayers facing each other 54,55:

(14)

Here, C_i,∞ is the concentration of ionic species i in the reference solution and C_i,med and φ_med are the ionic concentration and the electrostatic potential at the midplane between the bilayers. Equation (14) implies that both the solution between the bilayers and the reference solution behave ideally and follow Van t’ Hoff's law. Although this is a standard assumption in the literature 8,14,15,54,55, it is a point that must be remembered when discussing the results. In addition, the last equality in Eq. (14) assumes a Poisson-Boltzmann-type description for the diffuse double layer forming between two bilayers. The computation of Π_ele can be carried out numerically and requires the solution of the electrostatic problem between the bilayers and the calculation of the midplane potential, φ_med. Alternative models for the double layer will provide different values for φ_med, hence different Π_ele contributions.

The part of the free energy that depends on a_L is, according to Eqs. (4) and (6):

(15)

The free energy contribution (in J (mol lipid)⁻¹) arising from Van der Waals forces is simply the integral of Eq. (13) over d_w multiplied by the area per mol of lipid:

(16)

The electrostatic free energy expression depends on the model assumed for the ionic adsorption at the lipid-water interface and will be discussed below. For the first three contributions we use models that were effective in the prediction of the area per surfactant molecule in lipid aggregates such as monolayers, bilayers, and micelles in the absence of electrolytes 24,25,26,27,33,38,39,40,41. We thus set:

(17)

Equation (17) describes the penalty for creating a lipid-water interface per mol of lipid; γ is the interfacial tension of this interface and a₀ is the “incompressible” headgroup area of a lipid molecule calculated from the cross section of a molecular model for double-chain lipids. For DPPC we will set a₀ equal to 42Ų and γ equal to 50mNm⁻¹, values generally acceptable for lipid-water interfaces 53,54,70,71,72. The nonelectrostatic (e.g., steric) repulsions between headgroups have been modeled with various empirical equations in the past 25,26,33,38,39,40,41,54,73, but there is no rigorous theory for this term. Stigter and Dill 74,75,76 provided a complex formalism for the evaluation of the headgroup repulsion term, but it is not clear how to generalize this treatment in the presence of salts. Mbamala et al. 27 have recently presented an interesting model for the lipid-water interface for a mixture of zwitterionic and cationic lipids, which assumes that salts change the average tilt of the lipids in a quasicontinuous way. The model assumed in their work is difficult to generalize in the case of charging by adsorption treated here. In this investigation we adopted a hard-disk-like formalism used by several authors in the past 26,39,40,41, according to which:

(18)

A more elaborate equation based on the integration of the surface pressure equation for hard disks was presented by Yuet et al. 25. It has been tried in test calculations without providing improved results. An additional possible equation for the headgroup repulsion can be found in the literature 33 and has also been examined in this work:

(19)

B(μ_w) is a parameter, which depends on the chemical potential of water, and can be considered roughly constant at a fixed salt concentration. The configurational free energy of the lipid chains in bilayers has been calculated using statistical mechanical theories by a number of investigators 26,38,77,78,79. The work by Fattal et al. 26 explicitly provided calculations for DPPC-like lipids with two C₁₆ tails, albeit at 300K. It is expected that this free energy term may not change too much with temperature, apart from the linear “kT” scaling (A. Ben-Shaul, personal communication, 2007). Because we could not find any results for 50°C in the literature, we decided to use the results of Fattal et al. 26. We have, therefore, fitted the DPPC bilayer curves provided by these authors as a function of area per molecule with the following expression, valid for 50°C:

(20)

In this work we have used quite high electrolyte concentrations. If the bilayer lipids were charged, the electrostatic interactions would have been screened at high salt, and the precise treatment of electrostatics would not matter much. However, the bilayers acquire charge through ionic adsorption, so it is not clear at which salt concentration screening starts to dominate the enhanced charging by adsorption. This being the case, the proper treatment of electrostatics is crucial in our systems. We will assume that the mean-field Poisson-Boltzmann equation suffices to describe the diffuse part of the double layer created between two lipid bilayers. We will ignore the coupling between water polarization and electrostatics 32,53,83,84,85, local dielectric saturation effects 86,87,88,89, and ion-ion interactions 90,91 and will concentrate only on the mechanism of ion adsorption at the lipid-water interface. Two alternative models are used for this adsorption process.

Ion binding to the headgroups of the lipid molecules DPPC (Figure 1a) has often been modeled as a chemical reaction at the surface between an ion (here a monovalent anion) A⁻ and a neutral lipid L⁰ (DPPC) to form a charged lipid complex, LA⁻8,17,18,36,42,43,92,93,94,95,96,97,98: L⁰+A⁻ LA⁻. The binding constant K_A of the above reaction is defined as:

(21)

(22)

(23)

Display large version of this figure

Figure 1

(a) Model of local binding. Binding of the anions on the headgroups of the DPPC molecules at a bilayer. (●) Anions; (○) cations. (b) Model of partitioning into a diffuse lipid layer. Partitioning of the anions between the diffuse lipid layer and the bulk water at a bilayer. (●) Anions; (○) cations.

The binding model can be generalized to treat the case of independent sodium ion adsorption on the lipid headgroups, although this can be expected to be weak, since the experimental result of Aroti et al. 1 was that the osmotic pressure curves in the presence of NaCl and NaBr are almost identical to those obtained in the absence of electrolytes. Previous attempts to fit phospholipid bilayer osmotic pressure have consistently avoided treating sodium adsorption 23,92,93,94,95,96,97,98. Recent computer simulation studies of DPPC bilayers in the presence of NaCl have revealed that sodium interacts with the lipids more strongly than chloride 102,103,104,105,106,107,108,109,110. In fact, the simulations suggest that sodium interacts with the carbonyl groups of the lipids with a complexation mechanism, each sodium ion coordinated by one to four different lipid molecules 103,106,108. Given this evidence it appears reasonable to examine the case of separate sodium binding. We will however assume that sodium binds to the lipids less strongly than the large chaotropic anions ( I⁻, SCN⁻), a notion supported by the experimental finding that the effect of NaI and NaSCN on the headgroup area and the interbilayer distance is very pronounced, whereas that of NaCl or NaBr is almost undetectable. In this case two binding constants are needed:

(24)

(25)

(26)

(27)

The possibility of the simultaneous binding of an anion and a sodium ion on the same lipid molecule was not examined in this work.

This model is based on the concept of an active diffuse interface 33,34, as shown in Figure 1b for two approaching bilayers. In this model, the interface (water-lipid) is divided in two regions. For x≥δ the classical nonlinear Poisson-Boltzmann equation (PBE) applies, the analytical solution of which for 1:1 electrolytes is as follows 53,54,55:

(28)

(29)

By setting U₊→∞ for 0≤x<δ we exclude the cations from the lipid layer. The resulting equation has then an analytical solution, which is as follows for monovalent anions (34,111; E. Leontidis, L. Belloni, and A. Aroti, unpublished data):

(30)

(31)

C₀ is the anion concentration at x =0, given by:

(32)

(33)

At x =δ the electrostatic potentials and electric fields () calculated in the two regions must match. The following two conditions can then be derived:

(34)

(35)

For a given U₋ using a combination of Eqs. (31), (34), and (35) one can compute φ_δ, C₀, and Y, and then the complete electrostatic potential profile in the region between two bilayers. The electrostatic contribution to the total osmotic pressure exerted between the lipid bilayers is calculated using Eq. (14) as before. The electrostatic potential ϕ_med at the midplane can be calculated from Eq. (28), once ϕ_δ is known. Finally, the electrostatic contribution to the free energy per mol lipid in the context of this model is as follows (34; E. Leontidis, L. Belloni, and A. Aroti, unpublished data):

(36)

Equations (23) and (36) differ only in the second term of the right-hand side, since the first term is the contribution of the diffuse double layer. Sodium partitioning in the diffuse lipid layer can be treated quite easily in this model, but the solution of Eq. (29) must then be found numerically. The osmotic pressure is obtained again from Eq. (14), with properly computed values of the midplane potential, ϕ_med. The electrostatic free energy per mol of lipid is obtained numerically from the integral:

(37)

The detailed fitting analysis required to reduce the parameters of the perpendicular EOS to an absolute minimum starts from the fit of the logΠ−d_w in the absence of electrolytes, hence in the absence of the electrostatic repulsion. This allows the comparison with older literature results 6,51,59,112. The computed fitting parameters are then adjusted in the case of salts, the electrostatic force added, and the binding or partitioning parameters generated. To fit the experimental logΠ−d_w curves one must take into consideration the four parameters P₀, λ, κ_c, and H in Eqs. (10), (11) or (12), and (13). In principle one should use the full model equations containing all the adjustable parameters and carry out a nonlinear regression procedure to simultaneously fit all parameters to the data. However, this would only be feasible if a large number of experimental points were available. We have therefore chosen a different fitting process, which is described in more detail in Appendix I . Several alternative sets of parameters can be used to fit the logΠ−d_w experimental results for DPPC bilayers in pure water equally well. Assuming the power-law expression (Eq. (11)) for the undulation force, and for values of the Hamaker constant, H, in the range of 0.8–1.2 k_BT, one can find reasonable (according to the literature) values for the bending rigidity, which provide excellent fits to the data. In general, it was found that excellent fits were obtained for H=(1.0±0.2) k_BT, λ=(2.55±0.05) Å, P₀=(8.67±0.06)×10⁸ Pa and a wide range of κ_c values (from 9 to 30 k_BT). Similarly, if one uses the exponential expression of Eq. (12) for the undulation force, one can find reasonable values for the bending rigidity and get excellent fits to the data. In Supplementary Fig. S1, a and b, in Supplementary Material we present the best fitting curves for DPPC in water for the two different fluctuation force expressions.

Comparing the results found here and those reported in Table 1 by other researchers we notice that the present parameters do not agree closely with those found by other research groups in the past 6,51,59,112. This may be due to the fact that Lis et al. 112 and Rand et al. 6 calculated the hydration coefficient, P₀, and hydration length, λ, without taking into account the maximum swelling point. McIntosh et al. 51 used an additional exponential steric repulsive force to fit the force curve of DPPC in pure water. McIntosh et al. 51 and Petrache et al. 59 used a different way to define the water bilayer separation (using the electron density profile of the bilayers), and in addition they used the exponential form of the fluctuation force. Thus significant differences can be expected between the parameters reported and those from this work, although the results of Petrache et al. 59 are in fair agreement with the results obtained by us with the exponential fluctuation force. In conclusion, it can be said that the values found for the different parameters over the years are strongly model-dependent, and as a result no perfect agreement can be found.

The experimental logΠ−d_w curves for DPPC in the presence of salt solutions of different concentrations are fitted according to the following procedure: All the parameters that have been obtained (P₀, λ, and κ_c) using the conditional fitting for DPPC in water (as explained before) are kept unchanged. P₀ and λ should not change since the experimental data at all salt concentrations and for all salt types converge at high Π, in agreement with previous observations in the literature 8; κ_c should decrease slightly in the presence of salt solutions 30 but the decrease is expected to be small (∼k_BT) compared to the values actually used in this work, and thus is not taken into account. The Hamaker constant should decrease by ∼50% according to existing theory 28,29, and in agreement with the recent work by Petrache et al. 23, who found a gradual decrease of the Hamaker constant for neutral PC multilamellar vesicles with KBr or KCl concentration to values <50% of that in the absence of salt. A binding constant K_A (M⁻¹) for the anions is introduced that determines the electrostatic repulsive force generated between the lipid bilayers due to anion adsorption. Different values for the binding constant are used until the best fit to the experimental results is found. The results reported below have all been obtained with the power-law form of the fluctuation force. The behavior observed with the exponential fluctuation force was similar, with some differences in the numerical values of the various parameters, and will not be discussed further.

In a first round of calculations we have neglected sodium binding on the DPPC headgroups, as has been done in the past by most researchers in the field 23,92,93,94,95,96,97,98, and have based our calculations on Eqs. (21). As a general observation, the logΠ−d_w curves (including the maximum swelling point) of DPPC in the presence of low salt concentrations (0.05 and 0.1M) using the binding model, cannot be fitted at all, unless the Hamaker constant is allowed to increase very substantially. Even thus, for a 0.5-M salt concentration a very large binding constant must be used to fit the experimental logΠ−d_w data. It would appear that an additional repulsive force is required to fit the experimental results for DPPC both in NaSCN and NaI salt solutions of concentration 0.5M (see Supplementary Figs. S2 and S3 in Supplementary Material) . To improve the fit and reduce the binding constant to values comparable to those found for the lower NaSCN and NaI concentrations we must assume a physically unrealistic Stern layer for Na⁺ adsorption (4–8Å) (results not shown). This is not an acceptable solution, since the Pauling radius for Na⁺ is only 1Å 80,81,82. Stern layers are more meaningful at rigid interfaces and not at this fluid interface at 50°C. The fact that the Hamaker constant must increase so much raises serious doubts about the fitting process, because theoretically the low frequency part of the Hamaker constant must decrease in the presence of a salt solution by a factor proportional to exp(−2κd_w), where κ⁻¹ is the Debye length of the solution 23,28,29,54. At high salt concentration, as the water refractive index approaching that of the lipids 34,113, even the high frequency contribution to the Hamaker constant may decrease, as demonstrated by Petrache et al. 23. We thus anticipate that the Hamaker constant will be reduced by at least 50% when salts are present. It is therefore necessary to question the assumption that the osmotic pressure goes to zero at the maximum swelling point. The presence of minute amounts of impurities could increase the osmotic pressure at the maximum swelling point to a value of 100–200Pa, which is extremely hard to detect and measure precisely. There exists also the possibility of equilibrium of the swollen bilayers with a system of vesicles under tension 114,115. During any multilayered vesicle formation process, the outer layers in onion-like structures are under lateral stress. As a result, at “osmotic equilibrium” the compression due to tensile stresses of the outer layers dominate the residual osmotic pressure, which may be of the order of 100–200Pa, as was discussed by Diamant et al. 116. A less likely possibility, given the extremely low solubility of DPPC 117, is the salting-in of lipids by the chaotropic salts, which might increase their solubility in solution. Other possible arguments in favor of a nonzero osmotic pressure at maximum swelling will be discussed later.

As a result, we decided to fit the experimental logΠ−d_w results without taking into account the maximum swelling point. A value half of that found for DPPC in water (H=0.4 k_BT) was assumed for the Hamaker constant. Figure 2ab, show the best fitting curves found for DPPC in the presence of NaSCN of various concentrations using the binding model. Similar fitting curves are obtained for DPPC in the presence of NaI, NaNO₃, and NaBr. Table 2 shows the best binding constants for each salt concentration obtained without the maximum swelling point. The results in Fig. 2 and Table 2 imply that an ever increasing binding constant must be assumed as electrolyte concentration increases to provide good fits to the data. The increase of K_A at 0.5M NaSCN is excessive, corresponding to complete surface saturation with ions. The binding constants reported in Table 2 are clearly not thermodynamic quantities, and only indicate qualitatively that the salts examined here follow the Hofmeister series. An additional attempt to fit the NaSCN data at 0.5M was made by setting K_A equal to 3.5M⁻¹ (the value found at 0.05M) and allowing the Hamaker constant to gradually decrease. The curves in Fig. 3 show that even setting the Hamaker constant equal to zero in the high salt case will not lead to consistency between the low salt and high salt data.

Display large version of this figure

Figure 2

(a) Fitting curves obtained from the binding model for DPPC in the presence of 0.05M (■) and 0.1M (□) NaSCN (excluding the maximum swelling point). Hamaker constant=0.4 k_BT. K_A=3.5M⁻¹ and K_A=4.0M⁻¹ for NaSCN concentration 0.05 and 0.1M, respectively. (b) Fitting curves obtained from the binding model for DPPC in the presence of 0.5M NaSCN (excluding the maximum swelling point). Hamaker constant=0.4 k_BT. K_A=200M⁻¹ (solid line) and K_A=300M⁻¹ (dotted line).

Display large version of this figure

Figure 3

Fitting curves obtained from the binding model for DPPC in the presence of 0.5M NaSCN, setting K_A=3.5M⁻¹ and allowing the Hamaker constant to decrease and eventually drop to zero. The values for H are 0.8 k_BT (solid line), 0.4 k_BT (long-dashed line), 0.2 k_BT (short-dashed line), and zero (dotted line).

Finally, we have repeated the calculations assuming that sodium binds to the lipids as well and using Eqs. (24) for the electrostatics. Fig. 4 and Table 3 summarize the findings, which were very similar to those in the absence of sodium binding. It is possible to find several pairs of sodium and anion binding constants that produce good fits to the osmotic pressure curves at low electrolyte concentrations, although even then a slight tendency for larger binding constants is observed at 0.1M. When the concentration increases to 0.5M it is no more possible to fit the data in a reasonable way, excessive values of the anion binding constant being needed. We therefore conclude that sodium binding will not explain the high salt results for NaI or NaSCN.

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Figure 4

Fitting curves obtained from the binding model for DPPC in the presence of 0.1M NaSCN with different binding constants for sodium (excluding the maximum swelling point). Hamaker constant=0.4 k_BT.

The previous calculations might be viewed as proof that the local anion binding model does not work in the DPPC bilayers. We have therefore carried out fitting calculations using the model of ionic partitioning into the lipid layer. A fitting procedure similar to that for the binding model was used (maximum swelling was not taken into account). The thickness of the diffuse lipid layer was set equal to δ=4Å, which is roughly the average headgroup size of a DPPC molecule, excluding the glycerol group 6,10,118, but values as large as 10Å have also been examined. The lipid layer thickness was kept fixed (δ=4Å) for all concentrations of all salts to avoid the introduction of an additional adjustable parameter. Calculations using different values of δ provide comparable fits to the data with different values of U_i (see Eq. (38) below). The fitting curves are very similar to those observed using the binding model, as shown in Fig. 5 for DPPC in the presence of 0.05, 0.1, and 0.5M NaSCN solutions. Similar results were also obtained for NaI. The interaction potentials U₋ of the anions, obtained from these fits are summarized in Table 4. These interaction potentials depend on the anion used, but also on the concentration of the electrolyte solution. The binding constants and interaction potentials increase following the Hofmeister series of anions Br⁻<<I⁻<SCN⁻ for the same salt concentration; for different salt concentrations the interaction potentials increase with concentration. Generally, the binding constants of anions estimated using the binding model demonstrate the same behavior as the interaction potentials, U₋. In fact, the binding constants of anions obtained from the binding model can be roughly transformed to interaction potentials using the following approximate expression. This approximate expression is derived by combining the charge regulation expression, Eq. (21), with the U₋ viewed as a partitioning parameter: [A⁻]_LL/[A⁻]_s =exp(−βU₋), where [A⁻]_LL is the mean concentration of ion A⁻ within the lipid layer ([A⁻]_LL =[LA⁻]/δ to use the terminology of Eq. (21)).

(38)

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Figure 5

Fitting attempts with the binding model for the interbilayer water thickness of DPPC in mixtures of NaCl and NaI salt solutions at concentrations 0.1 and 0.5M. A power-law-fluctuation force and no sodium binding is assumed. The osmotic pressure in all these experiments was set equal to logΠ=4.6. Total [NaCl]+[NaI]=0.1M (●) and [NaCl]+[NaI]=0.5M (○).

The partitioning model has also been extended by assuming that sodium partitions in the lipid layers as well. The results are summarized in Table 5 and a plot of fitting curves is provided in the Supplementary Material (Supplementary Fig. S4) . Once more it can be seen that introducing a stronger sodium-lipid interaction does not allow fitting the osmotic pressure results over the entire salt concentration range. The problem observed at 0.5M is thus a genuine effect, which illustrates that some important aspect is missing from the modeling platform adopted here. The problem is apparently not connected to the way that the electrostatic boundary condition at the lipid surfaces is handled.