Article pubs.acs.org/Langmuir
Surface-Lattice Model Describes Electrostatic Interactions of Ions and Polycations with Bacterial Lipopolysaccharides: Ion Valence and Polycation’s Excluded Area Norman H. Lam†,‡,§ and Bae-Yeun Ha*,† †
Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada Department of Physics, University of Toronto, Toronto, Ontario M5S 1W4, Canada
‡
S Supporting Information *
ABSTRACT: The bacterial outer membrane (OM) is compositionally distinct and contains polyanionic lipopolysaccharide (LPS) in the outer layer as a main component. It has long been known that the cation-binding ability of LPS is one of the key determinants of OM permeability. Here we present a two-dimensional lattice model of the outer LPS layer, in which the lattice is decorated with bound ions or polycations; while small ions can occupy single binding sites, polycations, modeled as (charged) rods, compete for binding sites through their area exclusion, a consequence of their multisite binding. Our results suggest that in the parameter space of biological relevance, the effect of area exclusion is well-reflected in the competitive binding of Mg2+ and polycations onto LPS; by reducing the apparent binding affinity of polycations, it enhances Mg2+ binding. Despite simplifications, our results are generally consistent with the common view of Mg2+ as OM-stabilizing and polycations as OM-perturbing agents. They will be useful for understanding how cationic antimicrobials can gain entry into the cytoplasmic membrane. We also outline a few strategies for extending our model toward a more realistic modeling of OM permeability.
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INTRODUCTION Unlike eukaryotic cells or Gram-positive bacteria, Gramnegative bacteria (e.g., E. coli) are surrounded by two membranes: outer and inner.1−4 The outer membrane (OM) consists of two compositionally distinct layers, forming a highly asymmetrical bilayer: the outer layer mainly populated with polyanionic lipopolysaccharides (LPSs) and the inner layer containing phospholipids.1−4 Nominally, the OM serves as a protective coat and is resistant to potentially-harmful foreign molecules such as antibiotics and lysozyme.3−5 Intriguingly, the cation-binding ability of LPS has both favorable and adverse effects on the OM, as illustrated in Figure 1A. If the binding of Mg2+ (or Ca2+) is essential for the integrity of the OM, polycations including antimicrobial peptides (AMPs) bring about opposite (OM-perturbing) effects, despite their similar cationicity.3−10 In addition to its impact on OM permeability, electrostatic modification of LPS is relevant in a variety of contexts. For instance, (cationic) antimicrobials can selectively attach to and disrupt bacterial membranes, carrying a large fraction of anionic lipids.9,10 Neutralization of LPS charges is often a common © 2014 American Chemical Society
strategy for Gram-negative bacteria to reduce their susceptibility to antimicrobials.9−11 Similarly, they can weaken the adverse electrostatic LPS repulsions in an Mg2+-limiting environment.12,13 Finally, it was shown that divalent cations diminish water penetration through LPS layers, decrease LPSchain mobility,14,15 or enhance lipid-tail ordering as manifested by an increase in the melting temperature from the gel-like to liquid crystalline phases.2,3 Despite its relevance in various contexts discussed above, a clear picture of the physical effects of ions and polycations on LPS has been elusive. Ion (K+ or Mg2+) distributions around LPS layers have been the focus of a few earlier studies.16,17 How the binding of ions and polycations influences LPS would, however, remain unclear until examined quantitatively. In an intuitive picture, Mg2+ is often viewed as linking adjacent LPS charges (see, for instance, ref 5). For a better understanding of Received: July 29, 2014 Revised: September 29, 2014 Published: October 23, 2014 13631
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Figure 1. Electrostatic modification of (A) LPS and outer membrane permeability and (B−D) our modeling. (A) The upper and lower panels in this figure summarize the competing effects of Mg2+ vs amphiphilic polycations on the outer membrane, which have long been recognized in the literature3−10 (also see Figure 7 in ref 12). (Upper panel) At a high surface coverage of Mg2+, polycation binding is hindered and the outer membrane is resistant to polycations. (Lower panel) At a high concentration in bulk, polycations can displace Mg2+ competitively from the LPS surface, increasing the permeability of the membrane. In contrast, EDTA (polygon in gray) displaces Mg2+ by chelation in bulk, possibly leading to release of LPS from the LPS layer. (B−C) Molecular structure of LPS and our lattice model of the LPS layer. (B) Each hexagon represents a sugar molecule: glucosamine in light blue, Kdo in yellow, etc. The illustration in (B) is adapted with permission from ref 19. Copyrighted by the American Physical Society. (See refs 1−3 for more details.) Our LPS model is based on a simplified picture of LPS Re as shown in (B): using the simplification illustrated on the top right corner, we treat LPS charges as forming discrete lattice sites as shown in (C), where the numbers 1, 2, 3, etc. are to keep track of charges belonging to the same LPS molecule. (D) Ion pairing/binding on the resulting LPS lattice and excluded area of a polycation. Besides the transverse interaction between two opposite charges (e.g., Mg2+ and an LPS charge right below), the lateral interaction with other surrounding (backbone or bound) charges also influences ion binding on the surface. Here, two different lines are used to describe the lattice: solid and dashed. The dashed line or a short stretch of lattice sites is too short to accommodate a polycation in the same orientation. The illustrations in this figure are reproduced with modifications from ref 18 by permission of The Royal Society of Chemistry.
concentrations. This implies that the lipid headgroups are swollen by the electrostatic repulsion between LPS charges. In other words, the new equilibrium size of lipid headgroups is larger. The binding of Mg2+ onto an LPS charge, however, inverts the sign of the charge (see Figure 1 on the right). This makes the LPS surface charge heterogeneous with positive and negative charges arranged approximately alternatively. The correlation between opposite charges tightens the LPS layer (ΔΠ < 0). Our finding of enhanced lateral packing of LPS in the presence of Mg2+ offers a physical basis of what might have remained as a postulate (i.e., Mg2+-tightening of LPS).3−10 It is also well-aligned with the observations that the presence of divalent cations diminishes water penetration through LPS layers, decreases LPS chain mobility,14,15 or enhances lipid-tail ordering.2,3 Our results also suggest that polycations can displace previously-bound Mg2+ ions competitively from the LPS layer. As they continue to do so, ΔΠ becomes positive eventually. Despite their similar cationic nature, the electrostatic effects of Mg2+ and polycations are opposite. This is consistent with the general picture of polycationic molecules as LPS-perturbing agents.3−10 But our picture of polycations is not so conclusive, since their hydrophobic interaction with LPS is not included. Instead, it implies that the initial perturbation of the LPS layer can be mediated by electrostatic effects. This electrostatic perturbation can be considered as a prerequisite for LPS or OM permeabilization it follows. While it is not entirely clear whether this will ease the hydrophobic insertion of polycationic amphiphiles into the LPS layer, this picture is well-aligned with the observation that Mg2+ displacement (competitively or by chelation) from the LPS layer leads to increased OM permeability.3−10 Additionally, our results imply
how LPS can be modified electrostatically, a systematic approach will be needed (see ref 18 for some primitive effort). Here, we present a coarse-grained approach to electrostatic modification of the LPS layer, as described in Figure 1 (panels B and C). To this end, we extend our earlier two-dimensional lattice model of the LPS layer18 in order to take into account the effect of area exclusion for polycations (see ref 20 for this effect in a nonionic system); for simplicity, here we focus on LPS Re, the simplest form of LPS required for bacterial growth,1 as illustrated in Figure 1B. In our approach, the LPS layer is viewed as a two-dimensional electrostatic lattice formed by LPS charges, which can be “decorated” with opposite charges; small cations can occupy single binding sites but polycations, modeled as (charged) rods or rectangles, can occupy several sites. In this discrete binding-site model, area exclusion arises from the multisite binding property of polycations, since such a molecule is excluded from a short stretch of binding sites21 (see Figure 1D). Using this, we derive the free energy of the decorated LPS layer. This enables us to examine the competitive binding of Mg2+ and polycations onto the LPS surface and its impact on the lateral packing of LPS molecules. To this end, we calculate their surface coverage as well as the electrostatic contribution to the lateral pressure ΔΠ. Our results show that the effect of area exclusion is wellreflected in ion binding (thus in ΔΠ as well) in the parameter range of biological relevance. This effect hinders the binding of polycations onto LPS, especially around their “working” (biologically-relevant) concentrations and thus decreases their apparent binding affinity for LPS. This points to the significance of this effect in modeling outer membrane permeability, which is left out in our earlier work.18 In the presence of monovalent ions only in bulk, we find that ΔΠ > 0 over a wide or practically-entire range of ion 13632
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Figure 2. Our scheme for calculating the lateral electrostatic free energy of the LPS system Flat in the presence of Mg2+ and polycations. We first rearrange charges on the lattice into an energy-minimizing distribution in which the charges alternate in sign except on neutral ones (Na+-paired or polycation-paired sites), as shown on the left top corner. We then use as a reference a “perfect” lattice shown in (i), where equal numbers of positive and negative charges are alternatively arranged. Since this does not correctly represent the original one to the left of (i), we remove some of the charges until the perfect lattice becomes the original alternating distribution and calculate the resulting free-energy cost. (This illustration is reproduced with modifications from ref 18 by permission of The Royal Society of Chemistry.)
A (or endotoxin) but otherwise vary in size and structure.1−3 LPS Re is the simplest form of LPS required for bacterial growth.1 Thus, we mainly focus on LPS Re, each carrying four ionizable groups (see Figure 1B): two phosphate groups (in light blue) and two carboxyl groups (in yellow). Here we introduce a simplified model of our LPS system shown in Figure 1A (upper panel) that captures a few important details such as charge discreteness, ion paring, and area exclusion. To this end, we generalize our earlier model18 by incorporating the effect of area exclusion between bound polycations. As a result, the LPS layer is viewed as discrete electrostatic binding sites on a square lattice for opposite charges, as illustrated in Figure 1 (panels B and C). Small ions can occupy single binding sites (forming ion pairs with LPS charges), but polycations on the LPS surface, modeled as charged rods, compete for their binding sites not only through Coulomb repulsions but also through their area exclusion, a consequence of their multisite binding. Indeed, a number of theoretical studies highlight the significance of charge discreteness, especially in the presence of multivalent ions.22−24 Furthermore, the effect of area exclusion has been shown to reduce the apparent binding affinity of nonionic objects for a “sticky” surface.20,21 Below, we capture the essence of our theoretical scheme for calculating the free energy of our LPS system. For the intermediate steps leading to the entropic free energy with areaexclusion effects incorporated, refer to the Supporting Information. Also the detailed derivation of the electrostatic free energy can be found elsewhere.18 (For a complete discussion, we outline the derivation.) First, a meanfield picture, which amounts to smearing out surface charges, washes out important details: charge discreteness, ion pairing, and nonuniform charge surface distributions, especially in the presence of Mg2+. Ion-pairing (or transverse-interaction) energy can be readily obtained. Let δ be the distance between two charges, this energy can be expressed as S BΔ/δ (in units of the thermal energy kBT with kB the Boltzmann constant).25 The Bjerrum length S B is given by S B = e2/4πε0εrkBT (see for instance ref 21), where e is the electronic charge, ε0 is the permittivity of free space, and εr is the dielectric constant of a solvent (εr = εw for water). The symbol Δ accounts for the effect of a dielectric discontinuity between water and LPS. For practical purposes, it can be approximated as Δ ≈ 2.18
that the required concentration of polycations to reverse the Mg2+ tightening effect is typically much higher (thus more acceptable) if their area exclusion is taken into account; see below for relevant discussions. This highlights the significance of this effect in modeling OM permeability. A related electrostatic phenomenon is LPS release by EDTA treatment,7 as schematically shown in Figure 1. EDTA is polyanionic (typically carrying four negative charges) and can displace Mg2+ from the LPS layer by chelation. In contrast to polycations, EDTA does not directly perturb LPS. But it can be combined with Mg2+ into an Mg2+−EDTA complex in bulk. As a result, the chemical potential of Mg2+ in bulk will be lowered and some of the previously-bound Mg2+ will be released from the LPS layer. If combined with our results for ΔΠ, this may explain in physical terms why EDTA is effective in releasing LPS molecules from the LPS layer7 (also see refs 3−5). EDTAinduced Mg2+ displacement will create an optimal-area mismatch between the outer and inner layers, possibly leading to LPS release7 (see the bottom panel in Figure 1, where EDTA is described by a polygon in gray). (Polycations may induce similar effects but are expected to be less effective, owing to their favorable interaction with LPS molecules.) Because of the complexity of LPS and amphiphilic polycations (e.g., cationic AMPs), our consideration here is limited to electrostatic effects on LPS that ions and these molecules bring about. The resulting approach may offer a guiding principle for developing a more realistic model of LPS modifications. In the presence of polycationic amphiphiles, these effects can be considered as a precursor to the next step: hydrophobic association with and resulting perturbation of the LPS layer. This is a plausible picture, considering the possible kinetic barrier to their hydrophobic insertion into the LPS layer. It will thus be useful for clarifying to what extent the LPS layer or the OM can be modified electrostatically. In Modeling the LPS Layer, we present a coarse-grained model of the LPS layer. Our results for the competitive binding and its effects on LPS are presented in Results. In Conclusions and Discussions, we briefly discuss how our approach can be improved by including other details (e.g., hydrophobicity of polycations).
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MODELING THE LPS LAYER Figure 1B shows a schematic of amphiphilic LPS molecules, where each hexagon represents a sugar molecule: glucosamine in light blue, Kdo in yellow. etc. All forms of LPS contain lipid 13633
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Ion pairing between Mg2+ and an LPS charge will invert locally the sign of charge on the binding site. Thus, this pair tends to be surrounded by unpaired LPS charges. There is, however, no simple way to systematically include this effect, referred to as lateral correlations, without suppressing other details such as charge discreteness and ion pairing. In the past, a number of approximation schemes for calculating lateral correlations have been developed in the literature. For instance, long-wavelength (Gaussian) charge fluctuations are captured in ref 26. In an alternative approach, bound ions are viewed as forming a strongly-correlated liquid on the uniform background of backbone charges.27 In our approach illustrated in Figure 2 and also detailed in ref 18, we treat both backbone charges and bound ions on an equal footing, and keep track of charge discreteness and ion pairing as necessary. To this end, we first rearrange a given charge distribution into an alternating one, in which charges on the lattice alternate in sign except on neutral sites, paired with Na+ or polycations (see Figure 2). The resulting charge distribution minimizes the electrostatic energy of the LPS surface. We then decompose it into a few simpler ones labeled as (i−iv) in Figure 2 in the following way. The one in (i) shows a “perfect” lattice where equal numbers of positive and negative charges alternate. We then remove some of them or add peptide charges until the perfect lattice becomes the original alternating distribution to the left of (i) and calculate the resulting free-energy cost. Let N0 be the number of lattice sites; N1, N2, and Np the number of Na+ ions, Mg2+ ions, and polycations bound on the lattice, respectively; let N± be the number of positive (green balls) or negative charges (tangerine balls) on the original lattice on the left corner in Figure 2. Obviously, N+ = N2 and N− = N0 − N1 − N2 − QNp. If Σalt is the electrostatic free energy of the perfect, alternating lattice per site in (i), the lateral free energy per site can be written as18 Flat × N0 ≈
N0 Σalt − M Σalt + Frr 2
two discretization schemes converge onto each other in the two limiting cases: a homogeneously-charged lattice and a perfect, alternating lattice. This means that the two lattices are more equivalent to each other than naively indicated above. But the LPS free energy based on the latter one is easier to analyze numerically and will be employed in this work, even though the difference between the two is expected to be minor. Accordingly, Frr can be approximated as Frr ≈
⎞⎛ N ⎞ 2 ⎛ N0 ⎜ − N+⎟⎜ 0 − N −⎟Σalt ⎝ ⎠ ⎝ ⎠ N0 2 2 +
2 π SB (N+ − N −) Δ κ a2 N0
(2)
This can be used in eq 1. As for the electrostatic binding mode of polycations, we view each polycation as a series connection of a few monovalent cations as shown in Figure 1D, since its individual charges are monovalent. Note that many cationic antimicrobial peptides such as magainin 2 assume compact structures (e.g., α-helices) on the membrane surface (see, for instance, ref 10), as assumed in our model of polycations. Finally, a short stretch of free sites ( ρ* = 4.48, they tend to be entropically ordered into a nematic phase (the value of ρ* varies a bit from reference to reference37). Our results in Figure 3B on the right (solid lines), however, imply that QNp/ N0 < 0.3 or ρ < 1.2 < ρ* for Q = 4. Thus, our rod system is well below the required condition for such a transition to occur. Our analysis above is based on our view that the polycations assume compact structures (e.g., α helices) on the LPS surface. If they were stretched linearly, they would be longer. (Unless they are compacted, they will be too flexible to be modeled as rods.) However, the length of pore-forming peptides such as
The surface coverage of AMPs shown in Figure 3 can be considered as the lower bound for the amount of bound AMPs (or P/L) in the sense that their hydrophobicity enhances binding at the expense of a membrane-elastic energy (see Conclusions and Discussions for relevant discussions).34 (For Q = 4 and LPS Re, P/L is identical to the fractional charge occupancy, QNp/N0, except that hydrophobic interactions are not reflected in the latter.) In light of the estimated P/L* ∼ 1/ 10 (or more generally between 1/10 and 1/5) for a magainin− LPS system,33 our results obtained with area exclusion are more acceptable compared to those obtained without taking into account this effect. For this, see the graph on the right in Figure 3B, where the dashed gray line describes P/L* = 1/10. For [Mg2+] = 1 mM, the AMP concentration required for the occupancy to reach this threshold is around [AMP] = 5 μM. This value appears to fall in the range of minimal inhibitory concentrations (MICs) (i.e., the AMP concentration in bulk corresponding to P/L*, which is typically larger than a few μM).36 But this value is not to be taken too literally, since it depends on [Mg2+], peptide−lipid parameters, and cell concentrations used;36 in our approach, we consider a single 13637
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magainin 2 should match membrane thickness ≈4 nm,10 which does not deviate much from Qa = 4a0 = 3.2 nm for Q = 4. A related point is that before this inequality (ρ > ρ*) is met for a polycation−LPS system, P/L* will be reached, assuming that the surface coverage in Figure 3 serves as the lower bound for P/L. This may justify the neglect of entropic ordering of polycations in our derivation of the entropic free energy presented in the Supporting Information; this subtlety does not appear to be relevant for compact (pore-forming) peptides. How the LPS layer is electrostatically modified especially by Mg2+ and polycations can be quantified in terms of the excess lateral pressure (or tension) ΔΠ arising from electrostatic interactions (both lateral and transverse) on the LPS layer. Along this line, it is worth noting that the optimal headgroup area, a20 in our notation (not to be confused with the optimal headgroup area often denoted as a031), is one of the key parameters that determine the “packing” properties of amphiphilic molecules.31 The electrostatic interactions between charges on the LPS surface can change a0 from its unperturbed value (chosen to be a0 = 8 Å). The resulting a0 will be larger (smaller) than this value if ΔΠ > 0 (ΔΠ < 0). Let F̃LPS (a) be the total LPS free energy (per binding site), minimized with respect to Ni and Np for a given a, then ΔΠ = −∂FLPS/∂a. If ΔΠ > 0 then the electrostatic effect is unfavorable to the LPS layer, while if ΔΠ < 0, it tightens the LPS layer. For a more complete picture, one has to examine how ΔΠ will modify the packing shape of LPS molecules. Here, it is our view that LPS molecules would have to shrink their headgroups (or ΔΠ < 0) in order for them to form a bilayer or a planar structure. This is consistent with the view that Mg2+ links neighboring LPS charges,5 thus enhancing the integrity of the OM. Figure 4 shows our results for ΔΠ, as a function of (A) [Na+] and (B) [Mg2+] or [AMP]. (A) When [Mg2+] = 0, ΔΠ is positive (outward) in the entire range of [Na+] shown because of the repulsion between LPS charges. It is worth mentioning that this is more entropically driven.22 By expanding the layer from a0, some of bound ions are freed or they are more loosely bound. At higher [Mg2+], ΔΠ is more negative (inward). For [Mg2+] = 1 mM, this effect tightens the LPS layer over a wide [Na+] range. In this case and for [Mg2+] = 100 μM, as [Na+] increases from [Na+] ≈ 0, ΔΠ increases initially, since Mg2+ binding is discouraged and the lateral correlation is screened. For small [Mg2+] (≈ 0.10 μM), however, it varies nonmonotonically and eventually decreases as [Na+] increases. In this case, it is mainly Na+ that binds to the LPS layer. The entropic gain of freed Na+ ions is less pronounced for larger [Na+]. This explains the decrease of ΔΠ with increasing [Na+]. On the other hand, ΔΠ decreases as [Na+] → 0, since N1 is roughly independent of a in this limit and the entropic gain is insignificant.39 This explains the aforementioned nonmonotonicity. The graph on the right in Figure 4A shows the adverse effects of AMPs for [Mg2+] = 0.1 mM. As [AMP] increases, the ΔΠ curves shift upward so that ΔΠ becomes positive for larger [AMP]. This can be attributed to diminished Mg2+ binding at higher [AMP]. As [AMP] increases, AMPs continue to displace bound Mg2+ ions from the LPS layer, and the tightening effect of these ions becomes less appreciable. It is worth recalling that the individual charged residues of the AMP are monovalent (see Figure 1D), each neutralizing one LPS charge, similarly to what Na+ does.
The results in Figure 4B (left) show the Mg2+-tightening effect as a function of [Mg2+] for various choices of [AMP] ([Na+] = 100 mM). For given [AMP], a stronger tightening effect is induced at higher [Mg2+], since the LPS layer becomes more populated with Mg2+ but less with AMPs. A weaker tightening effect is observed for larger [AMP]; for higher [AMP], a larger Mg2+ concentration is required for tightening the LPS layer. The graph on the right in Figure 4B shows how the LPSstabilizing effect of Mg2+ seen in the graph on the left can be reversed by increasing [AMP]. For larger [Mg2+], larger [AMP] is required to counteract the favorable effect of Mg2+. Note that the curves for small [Mg2+] become flat as [AMP] increases. This is correlated with the results in the graph on the right in Figure 3B. At high [AMP], the peptide occupancy tends to a constant, especially for small Mg2+. This explains why ΔΠ becomes [AMP]-independent as observed in the graph on the right in Figure 4B. To highlight the effect of area exclusion on ΔΠ, in the graphs on the right in (A) and in both graphs in (B), we have included the dashed line with open squares obtained without taking into account this effect, which corresponds to the solid curve with filled squares in the same color. When this effect is suppressed, on a molar basis, AMPs bind to and perturb the LPS layer more effectively. In (B), the lateral pressure in this case turns into a positive value much faster. This is correlated with the results shown in Figure 3. Also the nonmonotonic dependence of ΔΠ on [AMP] in this case can be understood in parallel with that of ΔΠ on [Na+] in (A) for small [Mg2+]. This comparison suggests that the effect of area exclusion is well-reflected in ion binding and ΔΠ in parameter ranges of biological relevance: a millimolar range of Mg2+ and a micromolar range of AMPs. For this, compare the ΔΠ curves for [Mg2+] = 1 mM with and without this effect in the graph on the right in Figure 4. The Mg2+-tightening effect demonstrated in Figure 4 is consistent with a series of experimental observations such as diminished water penetration through LPS layers or decreased LPS chain mobility, in the presence of divalent cations.14,15 Importantly, it is well-aligned with the view of divalent cations as OM-stabilizing agents, which increase bacterial resistance against cationic AMPs or other membrane-perturbing agents.3−12
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CONCLUSIONS AND DISCUSSIONS It has long been recognized that despite their similar charge property Mg2+ and polycations have competing effects on outer membrane (OM) permeability.3−10 It has not been, however, clear whether the difference is of electrostatic origin or mainly comes from other details. To focus on electrostatic effects, we have introduced an electrostatic model of LPS, which captures important molecular details such as ionic sizes, surface charge heterogeneity, and area-exclusion effects of polycations. While some of these effects have been discussed separately in the literature,22−24 they are combined systematically in our approach. Decoration of the LPS layer with Mg2+ results in charge inhomogeneity on the LPS surface, which creates a negative lateral pressure, a favorable effect that holds LPS molecules together. In a more intuitive picture often used in the literature, Mg2+ is thought to link neighboring LPS molecules.5 Similar Mg2+-tightening effects are manifested in different contexts: reduced water penetration through and lipid tail ordering in 13638
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LPS layers,14 as well as stabilization of reverse hexagonal phases of phosphatidylserine, which would nominally form lamellar phases,22 all induced by Mg2+. In neutralizing LPS charges, polycations such as cationic AMPs (with Q > 2) are expected to be more effective on a molar basis than Mg2+. However, our results suggest that their electrostatic effect is opposite to what we expect from Mg2+. This clarifies the extent of electrostatic perturbations of LPS. The key to resolving this seeming contradiction is an adequate treatment of the discrete charge distribution on each peptide (see Figure 1D). Our results highlight the significance of polycation’s area exclusion. In parameter ranges of biological relevance, this effect is well-reflected in polycation binding and the lateral packing of LPS molecules, as quantified by ΔΠ (the electrostatic lateral pressure). Primarily, this effect reduces the apparent binding affinity of polycations for LPS, thus enhancing Mg2+ binding. It is worth recalling our earlier discussion on this effect in light of a threshold peptide to lipid ratio often expressed as P/ L*, beyond which bound AMPs form pores in their binding membrane,9,10 including an LPS bilayer.33 The corresponding bulk concentration is referred to as a minimal inhibitory concentration (MIC).10,36 For a magainin−LPS system, P/L* ∼ 1/10 (or between 1/10 and 1/5).33 If the effect of area exclusion is included, this threshold can be reached when [AMP] ≈ 5 μM for [Mg2+] = 1 mM. Without this effect, the required [AMP] < 1 μM. It is tempting to compare these [AMP] values with measured MICs, which are typically larger than a few micromolar.36 Considering that the occupancy in Figure 3 is lower bound, the [AMP] value, corresponding to P/ L*, without area exclusion effects appears to be a bit too small. Considering the cell-concentration dependence of MIC measurements,36 experiments under controlled conditions will be useful. For a larger cell concentration, the MIC will be higher; in our approach, there will be more binding sites to be occupied by AMPs in this case. Our results are also consistent with several features of what has been observed for bacterial LPS molecules.3−10,14 But for a more complete quantitative picture, other details need to be added in our electrostatic model. One obvious possibility is the hydrophobic association of amphiphilic polycations with LPS. For this, one has to set up a free-energy model for describing how they insert into and perturb LPS molecules, similarly to what was done for a symmetrical phospholipid membrane.35 This effort will be useful for understanding how AMPs can cross the outer membrane and reach the inner one, where they create pores. In parallel with this endeavor, experiments with model LPS membranes will be desirable (see ref 33 for some early effort). Also, it will be instructive to measure threshold surface coverages of AMPs required for LPS-membrane rupture with varying Mg2+ concentrations. Such an effort will advance our understanding of the role of LPS changes in determining the susceptibility and resistance of Gram-negative bacteria to cationic antimicrobial agents. Furthermore, for wild-type LPS, long polysaccharide chains (green dashed lines shown for a few LPSs in Figure 1) can be taken into account. In a coarse-grained picture, they can be considered as forming a polymer (or polyelectrolyte) brush or end-grafted polymers.40,41 In that case, our approximation of LPS charges as a planar distribution will turn out to be rather crude. Indeed, the presence of divalent cations can collapse the
polysaccharide chains16 (see ref 41 for the collapse of a polyelectrolyte brush). It will thus be essential to capture simultaneously intermolecular and intramolecular charge correlations between polysaccharide chains. A coarse-grained model like the one proposed here needs to be used with some caution, since there will remain some features that may not be captured easily in such a model. In particular, together with the cationicity of OM-permeabilizing agents, their conformations can be implicated in OM permeabilizing activity.5 Also, the hydrophobic interaction among bound AMPs can lead to their oligomerization on a membrane surface. Intriguingly, this effect can be reflected differently in LPS membranes, compared to those composed of phospholipids.42 It will thus be useful to focus on simpler cases where such complexities would not arise. Considering the biological complexity of LPS (the OM as well) and theoretical challenges in formulating even a coarse-grained model of LPS Re, possibly the simplest form of LPS, it will be desirable to design theoretically testable experiments, which will in turn benefit theoretical modeling.
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ASSOCIATED CONTENT
S Supporting Information *
Entropic free energy: area exclusion effects. This material is available free of charge via the Internet at http://pubs.acs.org.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Present Address §
Department of Physics, Yale University, New Haven, Connecticut, 06511, USA.
Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work was supported by NSERC (Canada) (N.H.L. and B.Y.H.). We acknowledge useful discussions with Matt Smart.
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REFERENCES
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(31) Israelachvili, J. N. Intermolecular and Surface Forces, 3rd ed.; Academic Press: Waltham, MA, 2011. (32) These numbers are not to be taken too literally. The relationship between d and r is not so obvious, if there is any. Hydration and ion binding influence both: if the former enlarges ion sizes, the latter reduces them. It is reasonable to choose δ < r. Our choices of δ and r are typical ones. (See also ref 31.) (33) Ding, L.; Yang, L.; Weiss, T. M.; Waring, A. J.; Lehrer, R. I.; Huang, H. W. Interaction of Antimicrobial Peptides with Lipopolysaccharides. Biochemistry 2003, 42, 12251−12259. (34) More precisely speaking, for P/L > P/L*, AMPs can be in any of the two binding modes: electrostatic binding on the membrane surface captured in our model and hydrophobic incorporation into the membrane, both in a parallel orientation. AMPs in these two binding modes will be in chemical equilibrium. How they will be partitioned between the two binding modes will depend on Q and other peptide and lipid parameters. See ref 35 for a similar issue for a phospholipid membrane. (35) Taheri-Araghi, S.; Ha, B.-Y. Cationic Antimicrobial Peptides: A Physical Basis for Their Selective Membrane-Disrupting Activity. Soft Matter 2010, 6, 1933−1940. (36) Matsuzaki, K. Control of Cell Selectivity of Antimicrobial Peptides. Biochim. Biophys. Acta 2009, 1788, 1687−1692. (37) Almeida, P. F. F.; Wiegel, F. W. A Simple Theory of Peptide Interactions on a Membrane Surface: Excluded Volume and Entropic Order. J. Theor. Biol. 2006, 238, 269−278. (38) de Gennes, P.-G.; Prost, J. The Physics of Liquid Crystals, 2nd ed.; Oxford University Press: New York, 1993). (39) At a low-salt limit, the effective planar charge density can be well-approximated by the meanfield result: −eκ/πlB → 0 as [Na+] → 0 (see for instance ref 26), leading to N1 ≈ N0, independent of a. (40) de Gennes, P. G. Conformations of Polymers Attached to an Interface. Macromolecules 1980, 13, 1069−1075. (41) Rühe, J.; Ballauff, M.; Biesalski, M.; Dziezok, P.; Gröhn, F.; Johannsmann, D.; Houbenov, N.; Hugenberg, N.; Konradi, R.; Minko, S.; Motornov, M.; Netz, R. R.; Schmidt, M.; Seidel, C.; Stamm, M.; Stephan, T.; Usov, D.; Zhang, H. Polyelectrolyte Brushes. Adv. Polym. Sci. 2004, 165, 79−150. (42) Papo, N.; Shai, Y. A Molecular Mechanism for Lipopolysaccharide Protection of Gram-negative Bacteria from Antimicrobial Peptides. J. Biol. Chem. 2005, 280, 10378−10387.
(11) Peschel, A.; Sahl, H.-G. The Co-Evolution of Host Cationic Antimicrobial Peptides and Microbial Resistance. Nat. Rev. Microbiol. 2006, 4, 529−536. (12) Guo, L.; Lim, K. B.; Poduje, C. M.; Daniel, M.; Gunn, J. S.; Hackett, M.; Miller, S. I.; Lipid, A. Acylation and Bacterial Resistance against Vertebrate Antimicrobial Peptides. Cell 1998, 95, 189−198. (13) Groisman, E. A. The Pleiotropic Two-Component Regulatory System PhoP-PhoQ. J. Bacteriol. 2001, 183, 1835−1842. (14) Kučerka, N.; Papp-Szabo, E.; Nieh, M. P.; Harroun, T. A.; Schooling, S. R.; Pencer, J.; Nicholson, E. A.; Beveridge, T. J.; Katsaras, J. Effect of Cations on the Structure of Bilayers Formed by Lipopolysaccharides Isolated from Pseudomonas aeruginosa PAO1. J. Phys. Chem. B 2008, 112, 8057−8062. (15) Garidel, P.; Rappolt, M.; Schromm, A. B.; Howe, J.; Lohner, K.; Andrä, J.; Koch, M. H.; Brandenburg, K. Divalent Cations Affect Chain Mobility and Aggregate Structure of Lipopolysaccharide from Salmonella Minnesota Reflected in a Decrease of Its Biological Activity. Biochim. Biophys. Acta 2005, 1715, 122−131. (16) Pink, D. A.; Hansen, L. T.; Gill, T. A.; Quinn, B. E.; Jericho, M. H.; Beveridge, T. J. Divalent Calcium Ions Inhibit the Penetration of Protamine through the Polysaccharide Brush of the Outer Membrane of Gram-Negative Bacteria. Langmuir 2003, 19, 8852−8858. (17) Schneck, E.; Schubert, T.; Konovalov, O. V.; Quinn, B. E.; Gutsmann, T.; Brandenburg, K.; Oliveira, R. G.; Pink, D. A.; Tanaka, M. Quantitative Determination of Ion Distributions in Bacterial Lipopolysaccharide Membranes by Grazing-Incidence X-ray Fluorescence. Proc. Natl. Acad. Sci. U.S.A. 2010, 107, 9147−51. (18) Lam, N.; Ma, Z.; Ha, B.-Y. Electrostatic Modification of the Lipopolysaccharide Layer: Competing Effects of Divalent Cations and Polycationic or Polyanionic Molecules. Soft Matter 2014, 10, 7528− 7544. (19) Schneck, E.; Oliveira, R. G.; Rehfeldt, F.; Demé, B.; Brandenburg, K.; Seydel, U.; Tanaka, M. Mechanical Properties of Interacting Lipopolysaccharide Membranes from Bacteria Mutants Studied by Specular and off-Specular Neutron Scattering. Phys. Rev. E 2009, 80, 041929−1-041929-9. (20) Chatelier, R. C.; Minton, A. P. Adsorption of Globular Proteins on Locally Planar Surfaces: Models for the Effect of Excluded Surface Area and Aggregation of Adsorbed Protein on Adsorption Equilibria. Biophys. J. 1996, 71, 2367−2374. (21) See for instance Molecular Driving Forces: Statistical Thermodynamics in Biology, Chemistry, Physics, and Nanoscience, 2nd ed.; K. A. Dill and S. Bromberg, Eds.; Garland Science: New York, 2010. (22) Taheri-Araghi, S.; Ha, B.-Y. Electrostatic Bending of Lipid Membranes: How Are Lipid and Electrostatic Properties Interrelated? Langmuir 2010, 26, 14737−14746. (23) Henle, M. L.; Santangelo, C. D.; Patel, D. M.; Pincus, P. A. Distribution of Counterions near Discretely Charged Planes and Rods. Europhys. Lett. 2004, 66, 284−290. (24) Travesset, A.; Vaknin, D. Bjerrum Pairing Correlations at Charged Interfaces. Europhys. Lett. 2006, 74, 181−187. (25) See ref 24 for a more sophisticated model of ion pairs, in which ion pairing is thermodynamically defined and the long-range nature of Coulomb forces is better reflected. (26) Taheri-Araghi, S.; Ha, B.-Y. Charge Renormalization and Inversion of a Highly Charged Lipid Bilayer: Effects of Dielectric Discontinuities and Charge Correlations. Phys. Rev. E 2005, 72, 021508-1−021508-10. (27) Nguyen, T. T.; Grosberg, A. Y.; Shklovskii, B. I. Macroions in Salty Water with Multivalent Ions: Giant Inversion of Charge. Phys. Rev. Lett. 2000, 85, 1568−1571. (28) Ludtke, S.; He, K.; Huang, H. Membrane Thinning Caused by Magainin 2. Biochemistry 1995, 34, 16764−16769. (29) It is not so obvious how to take into account this difference in our derivation of the second term in eqs 8 and 9. It is related to the degree of water structuring around ions.31 It is, however, conceivable that this will not change this term appreciably. (30) Hill, T. L. An Introduction to Statistical Thermodynamics; Dover Publications: Mineola, NY, 1986. 13640
dx.doi.org/10.1021/la502905m | Langmuir 2014, 30, 13631−13640
