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How Cell Concentrations Are Implicated in Cell Selectivity of Antimicrobial Peptides Azadeh Bagheri,† Sattar Taheri-Araghi,‡ and Bae-Yeun Ha*,† †

Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada Department of Physics, University of CaliforniaSan Diego, La Jolla, California 92093, United States

‡

S Supporting Information *

ABSTRACT: Antimicrobial peptides (AMPs) are known to selectively bind to and kill microbes over host cells. Contrary to a conventional view, there is now evidence that AMP’s cell selectivity varies with cell densities and is not uniquely determined. Using a coarse-grained model, we study how the cell selectivity of membrane-lytic AMPs, defined as the ratio between their minimum hemolytic (MHC) and minimum inhibitory concentrations (MIC), depends on cell densities or on the way it is measured. A general picture emerging from our study is that the selectivity better captures peptide’s intrinsic properties at low cell densities. The selectivity, however, decreases and becomes less intrinsic as the cell density increases, as long as it is chosen to be the same for both types of cells. Importantly, our results show that the selectivity can be excessively overestimated if higher host cell concentrations are used; in contrast, it becomes mistakenly small if measured for a mixture of both types of cells, even with similar choices of cell densities (i.e., higher host cell densities). Our approach can be used as a fitting model for relating the intrinsic selectivity to the apparent (cell-density-dependent) one.

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clarify the parameters that control the selectivity of AMPs.11−13 (See ref 14 for a review.) The notion of peptide selectivity (defined as MHC/MIC) has been widely used in the literature as a standard “ruler” for assessing their potency as therapeutic agents.10,11,14 Despite its significance in a variety of contexts (e.g., the biomedical application of optimized AMPs and the design of experimental protocols), the cell-concentration dependence of AMP activity or selectivity has not yet been criticality or systematically examined. As pointed out in ref 14, some confusion remains to be resolved because the selectivity was often measured with different values of host cell and bacterial densities. Host cell densities used in some hemolytic assays are sometimes 3 orders of magnitude larger than those of bacteria. The selectivity is thus overestimated as correctly referred to as an “experimental illusion”.14 If so, to what extent is it overestimated? In a more general perspective, how strongly does the selectivity depend on cell densities or on the way it is measured? If MHC/MIC is

INTRODUCTION

There has been much appreciation of antimicrobial peptides (AMPs) as novel antibiotics.1−7 They rapidly exert antimicrobial activity against a wide range of microbes via various mechanisms. Many cationic peptides such as magainin 2 and melittin kill microbes by rupturing the microbial membranes.8 What makes AMPs effective as therapeutic agents is their cell selectivity, which enables them to preferentially kill microbes over the host cells.1−5 Cationic AMPs, especially membranetargeting ones, achieve their selectivity through their strong electrostatic attraction with the bacterial membrane, containing a large fraction of anionic lipids.1−5 They rupture membranes in an “all-or-none” concentration-dependent manner.6,7,9 As a result, there exists a concentration window at which they are active against bacterial membranes only. AMP’s cell selectivity is often measured by the so-called therapeutic index defined as the ratio between their minimum hemolytic (MHC) and minimum inhibitory concentrations (MIC): MHC/MIC.10,11 The higher the therapeutic index (MHC/MIC), the more effective the AMP would be as an antibiotic. Accordingly, there have been many attempts to © 2015 American Chemical Society

Received: April 27, 2015 Revised: June 16, 2015 Published: July 2, 2015 8052

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Figure 1. Bacterial membrane perturbation by antimicrobial peptides prior to membrane rupture and our coarse-grained model. (a) Peptides are typically unstructured in the bulk but become compact on the membrane surface, as assumed in our disk model. Because of the peptide’s amphiphilic structure, they can be hydrophobically incorporated into the membrane in a parallel orientation as shown in the figure, while some others remain close to the membrane; peptide binding is initially limited to the outer layer but will be symmetrized. Also shown is the hexagonal arrangement of bound peptides on the surface (top view). (b) If peptides in binding mode S are electrically bound to the surface, then peptides in binding mode I are inserted hydrophobically into the interface between lipid headgroups and tails in a parallel orientation. Here, peptides are evenly distributed between the inner and outer monolayers. To capture lipid demixing, we approximate the spatial lipid distribution by a stepwise function by dividing each WSC (Wigner−Seitz cell) into two zones: zone 1 (including a single peptide at its center) and zone 2. (c) Top view of (b). For clarity, the peptide and counterions are not shown. In (c), two WSCs are included.

Figure 2. Noncompetitive (i) versus competitive (ii): in the latter case, bacterial and host cell membranes in a mixture compete for peptide binding. In (a), the cell density is the same for both types of cells (i.e., CH = CB). In (b), CH ≫ CB. In (b), how peptide selectivity is measured (noncompetitively (i) or competitively (ii)) has a profound impact on the selectivity. It can be excessively overestimated in (i) or underestimated in (ii). See Figure 4(c)−(d) for details.

lethanolamine), bacterial cell walls, and the trapping of peptides in the outer bacterial membrane.1,2,5,14 Thus, our work is more relevant to pure-lipid (model) membranes or lipid bilayers. Here we focus our consideration on membrane disrupters1−4 or “pore formers”.2 As shown in Figure 1, AMPs can reside in close proximity to the membrane surface or can be hydrophobically incorporated into the interface between headgroups and lipid tails in a parallel orientation (with respect to the interface); at a higher population on the surface, they will eventually orient themselves perpendicularly to form pores or disrupt the membrane.6,7,10,17−19 The membrane-perturbing activity of AMPs (prior to pore formation or membrane rupture) is often quantified by a single parameter P/L, defined as the molar ratio of peptides in the “membrane-perturbing mode” to lipids (excluding free lipids in solution)6,7,10,17,18 (peptides in this mode are hydrophobically incorporated in a parallel orientation). Beyond a threshold value P/L*, they can create pores on their binding membrane surface6,7,10,17,18 or disrupt the membrane in a “carpet”-like manner.2,10 The corresponding peptide concentration in the

not a unique measure of a peptide’s intrinsic quality, can it be used in a meaningful way? Here, we offer guiding principles that underlie the cellconcentration dependence of AMP’s membrane-perturbing activity and selectivity. A large degree of coarse graining is inevitable because the parameter space for this kind of consideration would otherwise be prohibitively large. For instance, the concentrations of peptides and cells have to be changed by several orders of magnitude. Thus, experimental efforts in this direction would be costly and tedious. It is thus desirable to develop a physical model for the cell selectivity of AMPs and to make experimentally testable predictions. To this end, we use a coarse-grained model that captures essential molecular details such as lipid composition (neutral vs anionic) and the peptide’s amphiphilicity charge as well as the peptide area occupied on the membrane surface and lipid headgroup area. (See refs 15 and 16 for an earlier coarse-grained model.) However, we largely ignore other biological details such as peptide molecular structure (beyond their amphiphilic structure), the varying content of the lipid PE (phosphatidy8053

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Using a coarse-grained fitting model like the one proposed here, the apparent selectivity can be calculated once the intrinsic one is known. Conversely, the intrinsic selectivity can be extracted from an apparent one. Finally, it is important to note that the difference between noncompetitive and competitive cases can be significant, especially when the cell concentrations are different for bacteria and host cells. The relevance of this in a biological context remains to be explored. This article is organized as follows. We present our theoretical model of a peptide−membrane system and derive a free energy of the model in the section Theoretical Approach. Our results for peptide selectivity are presented and discussed in detail in the next section, Results and Discussion.

bulk is known as an MIC (for bacterial cells) or MHC (for host cells).6,7,10,17 Indeed, P/L* is influenced by various factors: lipid composition (e.g., charged vs neutral), lipid headgroup area, peptide charge and size, and other known and unknown biological details.6,7,17,18,20,21 Here, we do not attempt to calculate P/L* but use a commonly accepted value. (The value of P/L* is specific to both peptides and lipids, and has not been well understood theoretically.) In particular, we note that this quantity is well characterized for the peptide melittin (e.g., refs 6, 7, and 17). We first derive a general free-energy approach to a peptide−membrane system, which holds possibly for a large class of pore-forming cationic AMPs, and apply it to melittin as a representative peptide. Using our approach, we first calculate P/L as a function of the total concentration of peptides Cp. With a standard (experimental) value of P/L* for melittin,6,7,17 we then extract MIC and MHC as well as MHC/MIC. This analysis offers a physical picture of how peptide selectivity depends on cell (lipid) concentrations. Here, we use cell selectivity and lipid selectivity interchangeably; even though the latter describes our model system better, the former is more suggestive and better related to the relevant literature. Also we consider the two following distinct cases: (i) noncompetitive (homogeneous) and (ii) competitive (heterogeneous)the selectivity is obtained separately for our model bacterial and host cell membranes in (i), but it is obtained for a mixture of both types in (ii), as illustrated in Figure 2. In both cases, we find that both MIC and MHC increase as the cell density increases, assuming that in (ii) the cell density is chosen to be the same for the bacteria and host cells (Figure 2(a)). This is a natural consequence of a stiffer competition between cells in recruiting peptides at a higher cell density. Peptide selectivity (measured by MHC/MIC), however, decreases as the cell density increases, as long as the cell density is chosen to be the same for both cells; in this case, the difference between cases (i) and (ii) is insignificant. If the host cell concentration is chosen to be larger, however, the difference between the noncompetitive and competitive cases can be highly nontrivial (main point illustrated in Figure 2(b)): the selectivity is excessively overestimated in the noncompetitive case, as correctly referred to as an “experimental illusion”;14 in sharp contrast, it is deceptively small in the corresponding competitive case. Thus, the selectivity depends on how it is measured. Furthermore, using a much simplified Langmuir-type model for peptide binding, we recapture approximately the cellconcentration dependence described above for both the noncompetitive and competitive cases (details in the Supporting Information). A general physical picture emerging from our analysis is that peptide selectivity defined as MHC/MIC does not necessarily measure the “intrinsic” property of AMPs. As a result, it is not a unique representation of their intrinsic quality. If measured at different cell concentrations, the selectivity would be different. One possible way to make relevant the concept of peptide selectivity is to define it in a single-cell limit. In practice, this can be realized at a sufficiently low cell or lipid density. Our results suggest that for melittin this limit is reached for the concentration range of 103 cells/mL or less; the selectivity becomes roughly insensitive to cell concentrations. However, the apparent (cell-concentration-dependent) cell selectivity of AMPs at a larger cell concentration will be different (smaller).

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THEORETICAL APPROACH Coarse Graining. In this section, we present our coarsegrained model, in which important details are captured. The key determinants of peptide activity and selectivity include the cationic charge Q, size, and amphiphilicity of AMPs, the average fraction of anionic lipids α̅ , and the lipid packing properties (e.g., small or large headgroups).2,6,7,10,21−23 Accordingly, we explicitly specify Q and the hydrophobic energy of the peptide’s parallel insertion (denoted as εI) as well as the area each peptide occupies on a membrane surface Ap and α̅ . On the other hand, the lipid headgroup area is influenced by the degree of hydration and will be implicitly taken into account through Ap, as detailed below. Because of the polymeric nature of AMPs, some complications arise: they are often unstructured in the bulk but are folded into a more compact structure (e.g., an alpha helix) on a membrane surface,22−24 as illustrated in Figure 1(a). The formation of an alpha helix or other compact structures is easier in a hydrophobic environment than in water because water tends to weaken the hydrogen bonding within each peptide that would otherwise stabilize its secondary structure. We thus model peptides as random coils in the bulk and as compact “disks” on the membrane. (See refs 15 and 16 for a similar model.) A “cylinder” might be a more appropriate model for the peptide. But a circular geometry, as assumed in our disk model, fits into our approach better, as illustrated in Figure 1(c). Because of the nonspecific nature of Coulomb interactions, this simplification will not appreciably influence the electrostatics of peptide binding. The hydrogen-bonding energy of compact (e.g., helical) structures will be, however, absorbed into εI because both of them reflect single-peptide properties. On the other hand, the lipid membrane is considered to be a two-dimensional binary fluid mixture of anionic and neutral lipids. The average fraction of anionic lipids α̅ determines the average surface charge density. The membrane is immersed in an ionic solution containing monovalent ions (e.g., Na+ and Cl−). The difference between different lipids (e.g., PE (phosphatidylethanolamine) and PC (phosphatidylcholine)) can be mimicked in part by adjusting their headgroup areas. In our approach, bound peptides can be in any of the two binding modes: electric adsorption and hydrophobic incorporation, both in a parallel orientation (Figure 1), indicated by modes S and I, respectively. (The latter mode is not to be confused with the I state often used for peptides that form pores.6,7,17) Even though peptides in mode I are the main players, in our approach, we treat both binding modes on an equal footing. This is to ensure that bound peptides in any binding mode can induce lipid demixing. (See for instance refs 8054

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corresponding to S, as schematically shown in Figure 1(b)−(c). Lipid demixing within each WCS is captured in an approximate way. As illustrated in Figure 1(b), each WS cell is divided into two zones: (i) zone 1, consisting of a bound peptide and the surrounding lipids with a fraction of anionic lipids α1; (ii) zone 2, consisting of lipids with a smaller fraction of charged lipids α2 ( α̅ > α2. Nevertheless, these two quantities are not totally independent of each other and can be found by minimizing the free energy subject to the constraint that the total number of charged lipids in each WSC is conserved

25 and 26 for the significance of lipid demixing.) The main effect of peptides in mode S is to reduce the availability of anionic lipids for those in mode I. The relative populations of peptides in the two modes will be determined by their electrostatic energy scales and εI. For a larger negative εI, as is often the case, mode I will be favored (see below for the value of this parameter for melittin). The initial peptide distribution will be asymmetrical with the outer layer mainly populated, but it will be eventually symmetrized between the two layers.27 In this work, as illustrated in Figure 1(a), we consider only the symmetric distribution. Note that this will not alter the qualitative picture of peptide activity/selectivity. In the absence of bound peptides, neutral and charged lipids tend to be homogeneously mixed in the membrane. The membrane can be well characterized by a uniform charge density of eσ0 = −eα̅ /al , where e is the elementary charge and al is the area occupied by each lipid. However, peptide binding can induce lipid demixing. In the next sections, we present our approximate scheme for taking this into account. Free Energy. Lipid demixing generates a major barrier to the theoretical analysis of peptide binding, which involves longrange Coulomb interactions.15,16,26 An adequate treatment of lipid demixing would nominally require solving the nonlinear Poisson−Boltzmann equation subject to hard boundary conditions.15,16,26 Here, we propose a new scheme for calculating the electrostatic free energy of our peptide−lipid system, which incorporates lipid demixing and peptide discreteness; peptide charges will not be smeared out in the plane of the membrane surface as in a mean-field-type approach. To keep peptide discreteness and allow lipid demixing, we modify the original approach employed recently28 on the basis of the two-dimensional Wigner−Seitz cell (WSC) model of molecular binding.15,16,26 At a high surface coverage, bound peptides, especially with a large valence Q, tend to be organized into an energy-minimizing arrangement, as assumed in the WSC model in which the peptides form a hexagonal lattice. (See Figure 1(a) and a related point below.) In our approach, regardless of their binding mode, bound peptides define WSCs, as illustrated in Figure 1(b)−(c); in (c), two WSCs are shown. On average, each peptide experiences a radially symmetric distribution of other peptides on the membrane. We thus use a circular-WSC model for peptide binding:26 a peptide modeled as a uniformly charged circular disk is placed at the center of a circle of radius RWS (radius of a WSC). In a numerically oriented analysis, the interaction between bound peptides is taken into account through the boundary condition on the WSC boundary.15,26 In contrast, we introduce an analytical model for calculating the free energy of each WSC. Let Ap be the area increase per inserted peptide and AWS be the area of each WSC. For symmetric binding, AWS can be expressed as AWS =

∫A

(2)

where (x, y) is the position vector in the plane of the membrane surface, assumed to lie in the x−y plane: α(x, y) = α1 (α2) in zone 1 (zone 2). This integral is carried over the surface of the WSC. However, the lower limit depends on the binding mode of the peptide in the center of the cell, which is 0 or Ap for S and I, respectively. One can establish α2I = α2S =

α̅(AWS − A p) − α1I(A1 − A p) AWS − A1 α̅AWS − α1SA1 AWS − A1

(3)

Here and later in the article, combinations of two subscripts are used to refer to different zones and binding modes. For instance, 2I (2S) refers to zone 2 and binding model I (S); recall that Ap is the area occupied by a bound peptide. We thus consider α1S and α1I to be independent parameters but α2S and α2I to be derivable quantities. By free-energy minimization, we determine the independent parameters. The area of zone 1 (A1) is essentially determined by the peptide area and the area occupied by the surrounding lipids that effectively interact with the peptide. It proves useful to introduce the Bjerrum length, denoted as S B,29,30 SB =

e2 4π ϵ0ϵrkBT

(4)

where ϵ0 is the permittivity of free space, ϵr is the relative permittivity of the solvent, kB is the Boltzmann constant, and T is the temperature. (Recall that e is the elementary charge.) On this length scale, the Coulomb energy of two elementary charges is equal to the thermal energy kBT. On the membrane surface, the peptide charge is neutralized by the surrounding anionic lipids and salt ions. For sufficiently large α̅ , as is often the case for the bacterial membrane, the neutralization or screening is dominated by lipid charges. What becomes relevant is then the two-dimensional Debye screening length,31 given by

1 + σIA p σI + σS

[α(x , y) − α̅ ] da = 0 WS

(1)

where σI (σS) is the planar density of the peptides in binding mode I (S). Hydrophobically bound peptides stretch the membrane, as indicated by eq 1 and as illustrated in Figure 1(a). In our approach, the WSC area is the same for the two binding modes (S and I), but there are more lipids in a WSC 8055

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The first two terms on the right-hand side describe the electrostatic free energy; the electrostatic (charging) free energy of each zone is given by eq 6. For a WSC containing a peptide in binding mode I, the free-energy gain for hydrophobic insertion is described by εI in the third term. The delta function in the third term is to ensure that this term vanishes for binding mode S. The last two terms account for the entropic penalty for lipid rearrangements induced by peptide binding, with L1 (L2) being the total number of lipids in zone 1 (zone 2). In our consideration, we measure FWS with respect to free peptides in the bulk. The free energy of these peptides can be calculated by considering each to be an unstructured polyelectrolyte (PE) carrying N monomeric units (amino acids) of length b each and total charge Q. The free energy of such a chain is known:36 let R be the end-to-end distance of the chain; the free energy is then given by

λ = al /2π SBα̅ .32 Lipids within this screening length interact with the bound peptide effectively. Let Rp be the radius of each disk on the membrane surface. We choose A1 = π (R p + λ)2 , independently of the binding mode. For α̅ = 0.3, λ ≈ 6 Å. Practically, this means that the peptide charge tends to be neutralized within the first “layer” of anionic lipids around it. Also note that this is somewhat smaller than a typical threedimensional screening length of about 10 Å. As a result, the peptide charge is screened more effectively by the surrounding lipids. This may justify our choice of A1. By our convention, Q is the cationic charge of each peptide in units of e. The charge density of zone 1 (in units of e), denoted as σ1S or σ1I, depends on the peptide-binding mode as follows σ1S =

α Q − 1S A1 al

σ1I =

α1I(A1 − A p) Q − A1 alA1

Fcoil Qκ SB Q 2 SB ⎛ 3 R2 6R ⎞ ⎟ ln⎜1 + = − + 2 ⎝ kBT 2 Nb 2 R Nb2κ ⎠

(5)

Recall that al is the lipid headgroup area. The planar charge density (in units of e) of zone 2 reads σ2I = −α2I/al or σ2S = −α2S/al . Given the charge density of each zone (to be determined later), one can compute the electrostatic free energy per unit area of each zone. For this, we rely on the Poisson−Boltzmann free energy of a uniformly charged surface with a planar charge density of eσ,33 given by ⎤ ⎛ Ψ0 ⎞ Fe(σ ) κ ⎡ = σ Ψ0 − ⎢ cosh⎜ ⎟ − 1⎥ ⎝2⎠ ⎦ π SB ⎣ kBT

The first term on the right-hand side in eq 9 describes the conformational entropy of the PE. The second term arises from the attractive interactions between a PE charge on the chain and the surrounding counterions, and the last term corresponds to the electrostatic repulsion between PE charges.37 The equilibrium size of the PE (denoted as Req) can be obtained by minimizing the free energy in eq 9 with respect to R. The equilibrium free energy Fcoil(Req) should be used as a reference free energy. Equivalently, as a WS free energy, we can use

(6)

Here Ψ0 is the reduced electrostatic potential on the surface (i.e., the surface potential in units of kBT), given by −1

Ψ0 = 2 sinh (2πσlB/κ )

- WS = FWS − Fcoil(R eq)

(7)

(10)

Note that this is the same for both noncompetitive and competitive cases. Finally, hydrophobically incorporated peptides will perturb the surrounding lipid molecules because their shape does not satisfy the lipid’s packing preference. As a result, the lipid tails bend toward the hydrophobic side of the peptide so as to fill any gap, leading to local membrane thinning and bending. If these local deformations propagate beyond the peptide spacing (RWS), as is often the case,38,39 then they overlap on the membrane. In this case, the membrane deformation can be considered to be approximately uniform. The deformation energy per unit area is then given by (1/2)KA(σIA p)2 , where KA is the area stretch modulus.6,38,39 A related point is that these peptides at the tail−headgroup interface tend to be dispersed38,39 (prior to pore formation). While it is not entirely clear to what extent the Coulomb repulsion between the peptides is implicated, this is well aligned with the WS arrangement of bound peptides, assumed in our approach. The electrostatic decoupling between the two layers and the assumption of uniform and symmetric membrane deformation allow us to consider each layer separately. Below, we consider both the noncompetitive and competitive cases and present the corresponding free energy of our peptide−lipid system. (i) Noncompetitive Case. Including all of the relevant terms together, we obtain the total free energy per area:

1/2

and κ = (4π SBn0) , where n0 is the total number density of monovalent salt ions (e.g., Na+ and Cl−).29,30 Beyond the Debye screening length κ−1, the electrostatic interactions are exponentially screened. Note that Fe(σ) in eq 6 is a standard Debye charging free energy and includes the entropy of salt ions around the charged membrane.33 The free energy expression in eq 6 deserves some discussion. The effect of a dielectric discontinuity at the lipid−water interface is approximately taken into account. This can readily b e s e e n i n t h e l i m i t σ SB/κ → 0. I n t h i s l i m i t , Fe/kBT ≈ 4π SBσ 2/κ . This is the electrostatic free energy of a dielectric plate of thickness d and dielectric constant ϵr in the limit ϵw κd/ϵr ≫ 1, as is the case for our membrane system, or equivalently ϵr ≈ 0 or d → ∞. This is twice that of the corresponding plate in the opposite limit ϵwκd/ϵr ≪ 1 or when the dielectric discontinuity is suppressed. Finally, for ϵw κd/ϵr ≫ 1, the coupling between the two (outer and inner) layers of a bilayer membrane can be ignored. (For more details, see ref 34 and references therein.) The free energy of each WSC, denoted as FWS, can be written as35 FWS = A1Fe(σ1) + (AWS − A1)Fe(σ2) + εIδ Ii + kBTL1[α1 ln α1 + (1 − α1)ln(1 − α1)] + kBTL 2[α2 ln α2 + (1 − α2)ln(1 − α2)]

(9)

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In eq 13, σIB (σSB) and σIH (σSH) are the planar densities of the peptides in binding mode I (S) on the bacterial and host cell membranes, respectively. In the third term in eq 13, Nfree is the number of free peptides given by

1 KA(σIA p)2 2

+ kBT[σI ln(σIA p) + σS ln(σSA p)] ⎛ σ + σS ⎞ + kBT (σM − σI − σS) ln⎜1 − I ⎟ σM ⎠ ⎝ ⎤ ⎛ Nfreevp ⎞ kT⎡ + B ⎢Nfree ln⎜ ⎟ − Nfree ⎥ − Fref NtA t ⎣ ⎝ V ⎠ ⎦

Nfree = Np − NBAB(σIB + σSB) − NHAH(σIH + σSH) (15)

The last term (i.e., Fref) describes the total free energy of the reference state (not to be confused with the reference free energy per area Fref in eq 12):

(11)

where -I (-S) is - WS for binding mode I (S), given in eq 10. The first three terms on the right-hand side were introduced earlier in the subsection Free Energy. The fourth−sixth terms in eq 11 describe the standard entropy of mixing within a twodimensional (surface) lattice-gas model, where σM = 1/Ap is the maximum surface coverage of peptides (e.g., ref 29). The seventh term is the entropic free energy of free peptides. Here, Nt is the total number of target cells (either host cells or bacteria), At is the area of each target cell, V is the total volume of the system, vp is the volume of a peptide in the bulk, and Nfree is the total number of free peptides; Nt = NB (NH) for bacterial (host) cells and At = AB (AH) for the bacterial (host) cell surface area. When the solution is not connected to a peptide reservoir as in our system and in a typical experimental setting, the number of free peptides is not fixed and decreases upon peptide binding. The total number of bound peptides can be expressed as NtA t (σI + σS ), and the number of free peptides is then Nfree = Np − NtA t (σI + σS). Finally, the reference free energy per area Fref describes the corresponding system without bound peptides and is given by kBT [α̅ ln α̅ + (1 − α̅ )ln(1 − α̅ )] al ⎤ ⎛ Np vp ⎞ kT⎡ + B ⎢Np ln⎜ ⎟ − Np⎥ NtA t ⎣ ⎝ V ⎠ ⎦

Fref = NHAHFe(σ0H) + NBAbFe(σ0B) ⎛N A ⎞ + kBT ⎜ B B ⎟[αB̅ ln αB̅ + (1 − αB̅ )ln(1 − αB̅ )] ⎝ al B ⎠ ⎡ ⎤ ⎛ Npvp ⎞ + kBT ⎢Np ln⎜ ⎟ − Np⎥ ⎝ V ⎠ ⎣ ⎦

where σ0H (σ0B) is the charge density of the host cell (bacterial) membrane without bound peptides and F( e σ0B) is the e σ0H) or F( resulting free energy per unit area as defined in eq 6. Finally, as in the noncompetitive case, -I(-S) in eq 14 is the WSC free energy of a peptide on the membrane in binding mode I (S), given in eq 10. The total free energy of the peptide−membrane system in eqs 11 and 13 is a function of a few independent parameters: the planar density of bound peptides (σI and σS) and the fraction of anionic lipids (α1I and α1S). The equilibrium values of these variables can be found by free-energy minimization. For a weakly charged membrane such as a host cell membrane (α̅ ≪ 1), lipid demixing is entropically discouraged because of a high entropic penalty for lipid demixing and can be ignored. We then have α1 = α2 = α̅ and A1 = AWS. The resulting free energy would then be a function of two independent variables σI and σS. The weakly charged membrane can be treated as a special case of eq 11 or eq 13.

Fref = Fe(σ0) +

(12)

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This term is contributed by three distinct effects: the charging free energy of a membrane without bound peptides (recall σ0 = −α̅ /al), the lipid entropy of mixing, and the entropy of peptides (all free). In the absence of peptides, the membrane surface is surrounded by a diffusive layer of counterions only. (ii) Competitive Case. Recall that NB is the number of bacteria with a surface area AB and NH is the number of host cells with a surface area AH. The total free energy of a peptide− membrane system in the presence of both host and bacterial cells is given by Ftotal = NBABE B(σIB , σSB) + NHAH E H(σIH , σSH) ⎡ ⎤ ⎛ Nfreevp ⎞ + kBT ⎢Nfree ln⎜ ⎟ − Nfree ⎥ − Fref ⎝ V ⎠ ⎣ ⎦

RESULTS AND DISCUSSIONS By minimizing the total free energy in eq 11 or eq 13, we can obtain the equilibrium densities of bound peptides on the membrane (i.e., σS and σI). The latter can be converted into the molar ratio of peptides in binding mode I to lipids, denoted as P/L. Here, we limit our analysis to pure-lipid membranes (in the absence of membrane proteins or cholesterol) that mimic cell membranes in a noncompetitive and a competitive environment (i.e., host and bacterial cells separately or altogether, respectively). In this kind of coarse-grained approach, parameter choices will be crucial and should reflect several key experimental data. Host cell and bacterial membranes share a set of common features while showing clear differences. First, we have chosen the generic parameters as follows: KA = 0.58kBT/Å2 (∼240 pN/nm),40 κ = 0.1 Å−1, T = 300 K, and ϵr = ϵw = 80 for water; as noted above, the effect of dielectric discontinuities is already taken into account. We have also used Q = 6 and εI = −14kBT, as for the well-known peptide melittin.41 The actual charge can be less than this value and depends on pH, peptide concentration, and the membrane surface potential41,42 (it was claimed to be 6 in refs 7 and 18 or 5 in ref 42 at neutral pH). For simplicity, we choose Q = 6 instead of determining its value vs pH. The volume of each

(13)

where E is defined as E = σI -I + σS -S +

⎛1⎞ 2 ⎜ ⎟K (σ A ) ⎝2⎠ A I p

+ kBT[σI ln(σIA p) + σS ln(σSA p)] ⎛ σ + σS ⎞ + kBT (σM − σI − σS) ln⎜1 − I ⎟ σM ⎠ ⎝

(16)

(14) 8057

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Figure 3. Molar ratio of peptides (in the membrane-perturbing mode) to lipids, denoted as P/L for Q = 6 as for melittin, as a function of the total peptide concentration Cp, for the bacterial membrane (α̅ = 0.3) (a) and the host cell membrane (α̅ = 0.05) (b); various colors represent different bacterial cell densities CB in (a) or host cell densities CH in (b), including the single-cell case (dashed gray line) for comparison. We have also chosen AB = AH = 12 μm2 without a loss of generality; changing AH to a larger value can be mimicked by increasing CH. The onset of binding occurs at higher Cp for larger CB (a) or CH (b). In any case, as Cp increases, P/L eventually collapses onto the single target case. Also superimposed are curves (dashed lines in green) describing the corresponding competitive case for CB = CH = 6 × 105 cells/mL. In this case, bacterial and host cells compete for peptides. As a result, P/L is smaller compared to the noncompetitive case (solid lines), especially for small Cp. (c) P/L as a function of the bacterial cell density CB for a few choices of Cp. For sufficiently small CB, P/L remains unchanged, especially for larger Cp; the system is effectively in a single-cell limit.

complication, as it is approximately captured within a much simplified, Langmuir-type model presented in the Supporting Information, in which this kind of detail is suppressed.) For the host cell membrane (PC only), however, we choose α̅ = 0.05, al = 74 Å2, Ap = 246 Å2, and P/L* = 1/99 ≈ 0.01.6,7,17 Note that the Ap value is somewhat larger for the host cell membrane. The difference in Ap between host cell and bacterial membranes is related to the so-called dehydration effect.6,7,17 Normally, the polar headgroups of lipids are surrounded by loosely bound water molecules, which will be released by peptide binding. The enhanced dehydration results in the smaller area occupied by the bound peptide. In the bacterial membrane with a smaller lipid-headgroup area, more water molecules are released by peptide binding, which leads to a smaller value for Ap.6,7,17 This is well correlated with a larger P/ L* for PE-containing membranes.6,7,17 On the other hand, the more negative spontaneous curvature of PE compared to that of PC does not seem to influence P/L* appreciably.6 Finally, we choose AB = 12 μm2 as the bacterial membrane area. This is 2 times the surface area of the representative bacterium E. coli. The factor of 2 is to reflect the symmetrical binding of peptides on the inner and outer layers of the bacterial (cytoplasmic) membrane. (Here we do not consider the interaction of AMPs with the outer bacterial membrane, enclosing Gram-negative bacteria. See refs 1, 2, and 5.) On the other hand, for the host membrane area, we use AH = AB as well as AH = 17 × AB; the latter one is suitable for human red blood cells chosen as representative host cells. We have first calculated P/L for various parameter choices described above with the choice AB = AH. This choice will not necessarily limit the scope of our consideration because increasing AH is equivalent to increasing CH. (See Figure 4 for the effect of a larger AH on MHCs and selectivity.) Figure 3(a)−(b) summarizes our results for our model bacterial and host cell membranes, respectively, primarily for the noncompetitive case (solid lines), in which the two types of cells are considered separately. In any case, P/L increases as the peptide concentration in bulk Cp increases, as expected. For a sufficiently large Cp, larger for a larger Ct, the P/L curves

peptide in the bulk is given by its size as a random coil (eq 9): vp = R3 = 333 Å3. (The volume of our system V does not have to be specified explicitly. For the competitive case, for instance, we minimize the free energy per volume. What matters is a ratio such as Ct/V or Np/V.) On the other hand, for a given peptide, al and Ap are lipiddependent and should reflect the content of PE (phosphatidylethanolamine). PE has a smaller headgroup compared to that of other lipids such as PG (phosphatidylglycerol) and PC (phosphatidylcholine). The greater the PE content is, the smaller al and Ap are, and Ap is more sensitive to the PE content than al is.6,7,17 As a result, the presence of PE increases P/L* because the membrane undergoes less deformations for a given P/L. Also note that the PE content varies appreciably from species to species, from ∼0 to ∼80%.5 However, the variation of Ap and al was examined for a few PC−PE mixtures: DOPC− DOPE at 2:1 and 3:1 molar ratios for melittin.6,7,17 Here we use their values for a 2:1 mixture of PC and PE:6,7,17 al = 71, Ap = 162 Å2, and P/L* = 1/48 ≈ 0.02.6,7,17 For a larger PE content, Ap will be smaller but is expected to get saturated as the PE context increases. Furthermore, for our study, it is imperative to include the anionic lipid PG. Except for the charge properties, PG is similar to PC. Thus, P/L* will not effectively reflect the difference between PC and PG, but MICs will be smaller for PG-containing membranes because of their stronger interaction with cationic peptides. Some complications arise when anionic PG is included. Highly cationic peptides such as melittin tend to be surrounded by PG lipids in a PG−PE membrane, inducing lipid demixing, which is entirely lacking in the PC−PE counterpart. As a result, the impact of PE on Ap and P/L* will be less pronounced in this case than in the corresponding PC−PE case. Furthermore, the dependence of P/L* on the PG content is not well known. Here we choose the smallest Ap, corresponding to the largest P/L*, reported recently.6,7,17 Also we use α̅ = 0.3 for the bacterial membrane.41 (See also ref 5 for variations.) Together with our choices of Ap and al, this better represents the hypothetical membrane, mimicking the bacterial-cell membrane: PC−PG−PE at a 1:1:1 molar ratio. (It is worth noting that our general finding will be largely independent of this 8058

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Figure 4. (a)−(d) Cell-concentration dependence of MIC, MHC, and MHC/MIC for the noncompetitive and competitive cases, represented by solid and dashed lines, respectively. As indicated below, the meaning of cell density can be different for different graphs. We have chosen Q = 6 (as for melittin) and AB = 12 μm2 (suitable for E. coli) and have used two values of AH: AH = AB and AH = 200 μm2 ≈ 17 × AB (as for human red blood cells). Here MICs correspond to P /L* = 1/48 ≡ P /L B* and MHCs correspond to P /L* = 1/99 ≡ P /L H*. (a)−(b) Both MICs and MHCs increase as the cell density increases coherently (i.e., CH = CB (see curves with squares)) for both the noncompetitive and competitive cases. In contrast, when the bacterial cell density varies while the host cell density is fixed at C H = 6 × 108 cells/mL , MHC and MIC are constant at low cell densities but increase slightly at high densities (curves with triangles). The MHC is higher for a larger AH because more peptides are needed to cover a larger cell area. (c)−(d) MHC/MIC as a function of cell density. In (c), CH = CB for both competitive and noncompetitive cases. The difference between the two cases is minor, and MHC/MIC decreases as the cell density increases. In (d), CH = 6 × 108 cells/mL while the bacterial cell density varies for both cases (solid and dashed lines). The difference between the two cases can be highly nontrivial. If CH ≫ CB, then the selectivity is overestimated in the noncompetitive case but it is almost completely suppressed in the competitive case; in the competitive case, the bacterial cells are effectively in the large-cell density limit. This is responsible for the underestimate and the insensitivity of MHC/MIC to CB (see also Figure 2).

bacterial and host cell membranes, respectively.6,7,17 As P/L exceeds P/L*, the membrane will be disrupted. The corresponding bulk concentration of peptides is known as an MIC (for bacterial membranes) or MHC (for host cell membranes),10 which can be extracted from the results in Figure 3. Beyond P/L*, pore energetics will influence the P/L curve, a complication our analysis leaves out. Here our focus in Figure 3 is on extracting MHC or MIC values. This complication will not limit our analysis of these values. We have obtained MIC, MHC, and MHC/MIC and plotted them in Figure 4 for both the noncompetitive and competitive cases, described by solid lines and dashed lines, respectively (both with symbols). For both cases, we vary or choose the cell concentrations CB and CH in two different ways: (a) CH = CB, as in Figure 2(a), and (b) CB is variable while CH is held fixed at CH = 6 × 108 cells/mL, similar to Figure 2(b). However, note that this is irrelevant for MICs and MHCs for the noncompetitive case because they are measured separately, but it can have a nontrivial impact on their ratio; MHCs and MICs at different cell concentrations can be combined into the ratio. With the choice CH = CB, the difference between the noncompetitive case and the competitive case appears to be

converge to the one describing the single-target case (the dashed line in dark gray at the top); recall that Ct is either CB or CH. This is a natural consequence of a much reduced competition to bind peptides between membranes. This can be seen in a more transparent way in Figure 3(c), which shows P/L as a function of bacterial-cell density CB for a fixed Cp. As the cell density increases, the competition becomes stiffer, resulting in the observed behavior in the figure, more so for a smaller Cp. Also included in Figure 3(a)−(b) are our results for P/L for the competitive case (dashed lines) for a fixed cell concentration (6 × 105 cells/mL for both cells). Compared to the corresponding noncompetitive case, there are twice as many cells because both types of cells are present. This explains why P/L is smaller for the competitive case when Cp is low. However, the difference between the two cases becomes insignificant for large Cp values because the resulting P/L approaches P/L for the singe-cell limit. Peptide selectivity can be estimated as follows. For P/L < P/ L*,6,7,10,17,18 membranes remain intact, even though they suffer from deformations. Recall that we chose P/L* = P/LB* = 1/48 ≈ 0.02 and P/L* = P/LH* = 1/99 ≈ 0.01 for the model 8059

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undetermined. (See the Supporting Information for the limitations of this approach.) Let Cp* be either MIC or MHC. Our Langmuir approach leads to

insignificant for MIC (Figure 4(a)) or moderately significant for MHC (Figure 4(b)). For the noncompetitive case and for the competitive case with CH = CB, at high cell densities, more peptides in the bulk will be required in order for P/L to reach P/L*. Consequently, MIC and MHC are higher at higher cell densities, as shown in Figure 4(a)−(b). When the host cell concentration is fixed at CH = 6 × 108 cells/mL but the bacterial density changes in the competitive case, both MIC and MHC curves are nearly flat at low cell densities and increase slightly at high densities (Figure 4(a)−(b)). In this case, the high concentration of host cells in the solution is a dominant factor. This is responsible for the insensitivity of MICs to the bacterial cell concentration. At high densities, however, the competition between bacterial cells requires a higher concentration of peptides in order for P/L to reach P/L* on the surface, as shown in Figure 4(a)−(b). For a similar reason, MHCs are larger for the larger AH case (AH = 17AB) because more peptides are needed in order for P/ L to reach P/L*. To see this, compare the green (AH = 17 AB) and blue (AH = AB) curves; the difference is roughly an order of magnitude (i.e., nontrivial). Figure 4(c)−(d) displays our results for peptide selectivity measured as the ratio MHC/MIC as a function of cell density: the density of host and bacterial cells chosen to be identical in (c) and the density of bacterial cells (with the host cell density fixed at 6 × 108 cells/mL) in (d). As shown in Figure 4(c), peptide selectivity decreases as the concentrations of both cells increase coherently in both the noncompetitive (solid lines) and competitive (dashed lines) environments. The difference between the two cases appears to be minor, as illustrated in Figure 2(a). Selectivity results obtained with CH ≠ CB are displayed in Figure 4(d). For this, we held CH at 6 × 108 cells/mL while varying CB over a sizable range. In contrast to the case CH = CB, the noncompetitive and competitive cases can be profoundly different: If the selectivity is unreasonably overestimated around 104 for AH = AB in the noncompetitive case, then it is almost completely suppressed in the corresponding competitive case (the disparity between the two cases is illustrated in Figure 2(b)). Note that both the overestimate and underestimate are artifacts in the sense that they are largely unrelated to the intrinsic properties of peptides. As CB → CH, the selectivity decreases and the difference between the two cases becomes insignificant, as expected from our results in Figure 4(c). Finally, the peptide selectivity is higher for a higher host cell area, similar to what was observed for a larger CH. This observation is well aligned with the finding that the MHC is larger for a larger AH, as shown in Figure 4(b). The results in Figure 4(d) clearly suggest that how the selectivity is measured (i.e., noncompetitively or competitively) can have a remarkable impact on it. The difference in selectivity between the two cases can be as large as 4 orders of magnitude (for AB = AH). This is, however, an illusion in the sense that the intrinsic selectivity is too exaggerated or suppressed. To offer a simple physical picture of peptide selectivity, in the Supporting Information we present a Langmuir-type approach to the selectivity. In this approach, peptide binding is characterized by a single parameter: the binding energy w, assumed to be different for the host cell and bacterial cell membranes. Obviously, w should be larger in magnitude for the latter. In this approach, however, the electrostatic interaction between bound peptides is not taken into account. As a result, the value of w for the bacterial membrane remains

( )* w/k T e A P * ( ) a L

Ap P L

A ⎛ P ⎞* 1 al C*p = t ⎜ ⎟ C t + ⎝ ⎠ al L vp 1 −

B

p

l

(17)

Equation 17 suggests that C*p is insensitive to Ct for Ct ≈ 0 but it increases approximately linearly with Ct for sufficiently large Ct. Here we analyze eq 17 only for the following simplified case: P/L*, At, al, and Ap are the same for both cell types. In this case, we find that δ ≡ MHC − MIC becomes independent of Ct: 1 δ= vp

A pal 1−

( PL )*

Ap al

*

() P L

(e wH / kBT − e wB / kBT ) (18)

This allows us to write MHC/MIC as MHC MIC + δ = MIC MIC

(19)

Because MIC increases monotonically with the cell density Ct, eq 19 suggests that MHC/MIC decreases monotonically as the cell density increases. Our Langmuir approach qualitatively captures the cell-concentration dependence of peptide selectivity displayed in Figure 4(c). In our Langmuir approach, this is a result of the balance between the two terms in eq 17. We have also extended this model to the competitive case and presented it in the Supporting Information. In summary, if CH ≫ CB and if CH is sufficiently large, then MHC/MIC ≈ 1: the selectivity is close to 1 and remains roughly constant, largely independently of CB. As a result, the intrinsic selectivity is completely hidden. If CH = CB, however, the ratio MHC/ MIC decreases monotonically and approaches 1, as in the corresponding noncompetitive case (CH = CB). Similar to our full analysis in Figure 4, eq 19 suggests that both C*p and MHC/MIC are more intrinsic to AMPs at low cell concentrations. In our Langmuir model, w is the only parameter that takes into account the difference between host cell and bacterial membranes. (Recall that others are assumed to be the same for both types of cells.) The w dependence of MHC/MIC is stronger at low cell concentrations. According to our results in Figure 4 and eq 19, the peptide selectivity reported in the literature may not necessarily reflect the intrinsic properties of AMPs. If the selectivity measurement were repeated at different cell concentrations, then different results would be obtained. The resulting peptide selectivity is then less meaningful than naively thought. One possible way to make peptide selectivity relevant is to define it in a single-cell limit. In practice, this can be realized at sufficiently low cell densities. The results in Figures 3(c) and 4 suggest that for melittin this limit is reached for the concentration range of 103 cells/mL or less. This intrinsic selectivity becomes insensitive to cell concentrations. More discussions along this line can be found in the next section.

■

CONCLUSIONS We have presented a physical model for the cell (lipid) selectivity of antimicrobial peptides (AMPs), measured in terms of the ratio between their minimum hemolytic and minimum 8060

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■

inhibitory concentrations: MHC/MIC. Our coarse-grained model is generally applicable to peptide−membrane systems as long as their biophysical properties are well characterized. To be specific, we have focused on melittin interacting with model (lipid-only) membranes: PC membranes and PE−PG-containing membranes mimicking host cell and bacterial membranes, respectively. Our model offers a quantitative picture of how the selectivity depends on cell concentration and on the way it is measured. The essence of this is illustrated in Figure 2. In a homogeneous or noncompetitive medium (i), bacterial and host cell membranes are considered separately, as in a typical experimental setting;14 in a competitive medium (ii), they are present simultaneously and compete for peptide binding. If the cell concentration is chosen to be the same for bacteria and host cells (Figure 2(a)), then the difference between (i) and (ii) is insignificant: with increasing cell density (CB = CH), both MIC and MHC increase but MHC/MIC decreases monotonically. On the other hand, if the host cell concentration is much higher, the difference between the noncompetitive and competitive cases can be highly nontrivial. In the competitive case, the large host cell concentration is a dominating factor for peptide binding. As a result, MIC ≈ MHC, and the resulting MIC and MHC curves are nearly flat, if plotted against the bacterial cell concentration. Thus, the intrinsic selectivity is almost completely suppressed in MHC/MIC ≈ 1. In sharp contrast, in the corresponding noncompetitive case, the selectivity can be exaggerated far beyond AMP’s intrinsic properties. This offers a physical basis of the earlier selectivity analysis leading to a similar conclusion (i.e., an overestimate).14 The results summarized in Figure 4 can be used to understand how to separate a peptide’s intrinsic properties from external parameters such as cell concentrations and environments (i.e., noncompetitive vs competitive). The most obvious way is to define selectivity in a single-cell or low-celldensity limit, primarily in a noncompetitive environment. For melittin, this limit is reached for the concentration range of 103 cells/mL or less. This limit will likely be realized differently for different peptides and for different membranes. One can then use our approach as a fitting model for estimating an apparent (cell-concentration-dependent) selectivity at different cell concentrations, once the intrinsic selectivity or the selectivity at a given cell concentration is known. How to choose cell concentrations in the competitive case is less obvious. In a biological context, it needs to reflect the degree of infection. Our model system is characterized by several key parameters such as the fraction of charged lipids and lipid headgroup area as well as the charge, size, and hydrophobicity of peptides. It will be useful to map out these parameter values for other peptides and more realistic model membranes (e.g., cholesterol-containing PC membranes and PE−PG at a varying molar ratio). In our approach, biological details can be absorbed into peptide or membrane parameters. For instance, the presence of cholesterol will increase the area stretch modulus KA.43 Finally, in the case of Gram-negative bacteria, the trapping of AMPs in the outer membrane (OM) will effectively lower the selectivity by increasing the MIC. To include this effect, one has to set up a coarse-grained model of the OM at a level similar to what was done here. (See refs 44 and 45 for recent attempts.)

Article

ASSOCIATED CONTENT

S Supporting Information *

Langmuir binding model. The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.langmuir.5b01533.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■ ■

ACKNOWLEDGMENTS This work was supported by NSERC (Canada). REFERENCES

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