Modeling selectivity of antimicrobial peptides: how it depends on the presence of host cells and cell density

Antimicrobial peptides (AMPs), naturally-occurring peptide antibiotics, are known to attack bacteria selectively over the host cells. The emergence of drug-resistant bacteria has spurred much effort in utilizing optimized (more selective) AMPs as new peptide antibiotics. Cell selectivity of these peptides depends on various factors or parameters such as their binding affinity for cell membranes, peptide trapping in cells, peptide coverages on cell membranes required for membrane rupture, and cell densities. In this work, using a biophysical model of peptide selectivity, we show this dependence quantitatively especially for a mixture of bacteria and host cells. The model suggests a rather nontrivial dependence of the selectivity on the presence of host cells, cell density, and peptide trapping. In a typical biological setting, peptide trapping works in favor of host cells; the selectivity increases with increasing host-cell density but decreases with bacterial cell density. Because of the cell-density dependence of peptide activity, the selectivity can be overestimated by two or three orders of magnitude. The model also clarifies how the cell selectivity of AMPs differs from their membrane selectivity.


Introduction
Antimicrobial peptides (AMPs) are naturally-occurring peptide antibiotics used in the host defense of living organisms (e.g., animals, plants, .). 1,2 They are relatively short, typically consisting of 20-50 amino acids.In the bulk, they oen resemble random coils, but when inserted in membranes, they assume compact, amphiphilic structures (e.g., a helices), as required for their antimicrobial activity (e.g., membrane perturbation).2][3] Cationic AMPs preferentially attach to and rupture microbial membranes over host cell membranes; in the latter case, anionic lipids are segregated to their inner layer (see Fig. 1).Once they gain entry into the cytoplasm, they can target key intra-cellular components (e.g., DNA and proteins), leading to intra-cellular killing of microbes. 1,2[3][4]6 They act via physical mechanisms such as pore formation [1][2][3][4]6 or anionic-lipid clustering 3 in membranes, which bacteria cannot easily avoid. I addition to rupturing bacterial membranes, they act as metabolic inhibitors 1,2 and/or immunomodulators.5 Fig. 1 Origin of peptide selectivity.Cationic antimicrobial peptides interact more strongly with bacterial membranes enriched with anionic lipids.Bound peptides can form pores in the membrane when the bulk concentration is at or above the MIC or MHC. In he figure, the membranes are only schematically illustrated, leaving out such details as membrane proteins and the presence of cholesterol in the host-cell membrane. Te figure is inspired by ref. 1, 2 and 10.
[3][4]6 The resulting selectivity can be quantied by the ratio of two concentrations: the minimum hemolytic concentration (MHC) and the minimum inhibitory concentration (MIC). 10,11At or beyond this concentration, peptides can form pores in their binding membranes, as illustrated in Fig. 1.The larger the ratio MHC/ MIC is for a given peptide, the more selective the peptide is.In a sizeable range of peptide concentration (∼mM) between MIC and MHC, the peptide is active against bacteria while leaving the host cells unharmed.
0][21] Let P/L denote the molar ratio of bound peptides to lipids.At the MIC or MHC, P/L reaches a threshold value, P/L*.The value of P/L* depends on the type of peptide and lipid [19][20][21] and is typically larger for membranes containing lipids with smaller headgroups such as phosphatidylethanolamine (PE) as in bacterial membranes.[16][17][18] The cell-density dependence is oen referred to as an inoculum effect [12][13][14][15] and is known to enhance population survivability. 147][18] For instance, it can be obtained by combining MIC and MHC measured separately from bacteriaonly and host-cell-only solutions, respectively.In this work, the resulting selectivity is referred to as "noncompetitive" selectivity.More realistically, it can be measured from a mixture of both types of cells: "competitive" selectivity.In this case, the presence of host cells raises the MIC and inuences the ratio MHC/MIC. 13,16,22These two approaches generally lead to different levels of selectivity.This implies that the selectivity reects the biological setting of infected sites (e.g., the degree of infection, .).
According to what is discussed above, peptide selectivity not only reects peptide's intrinsic properties such as peptide charge and hydrophobicity, but it also depends on external parameters such as cell density and the presence of host cells.Does this mean that the selectivity should be measured for a wide range of cell density and various combinations of host cell and bacterial cell density?Recent modeling efforts, however, suggest that these two aspects (intrinsic and extrinsic) are well separated. 16,18With an appropriate model, one can gure out the selectivity with varying cell density, once it is known at a low cell-density limit or at conveniently-chosen density.0][21] How does the resulting membrane selectivity differ from cell selectivity measured for cells (bacteria versus host cells)?Peptide trapping is one of the determining factors in the latter [12][13][14]16 but is expected to be insignicant in the former.
Recently, we examined theoretically peptide selectivity and claried the effects of peptide trapping on the selectivity, MHC/ MIC. 16This effort is relevant in the presence of an excess amount of host cells or for a homogeneous solution of either bacteria or host cells.Here, we extend this effort and offer a more complete picture of the activity and selectivity of AMPs, which can be used to interpret selectivity measurements or to assist with our endeavor in nding optimized peptides.3][14][15][16] The results reported in this work, which are relevant for melittin-like peptides, suggest a rather nontrivial dependence of the selectivity on the presence of host cells, peptide trapping, and cell density.Peptide trapping can enhance or reduce the selectivity depending on how cell (host and bacterial) density is chosen.In most cases, it works in favor of the host cells, enhancing the selectivity.The presence of an excess amount of host cells (5 × 10 9 cells per mL) as in whole blood can raise the MIC more than 10-fold, proportionally with the density of bacterial cells.The resulting MIC still falls in a low-mM range as long as the bacterial cell density is somewhat smaller than 5 × 10 7 cells per mL.
Let C B and C H be the density of bacteria and the density of host cells, respectively, and N p the number of peptides trapped per cell.As we raise C B and C H coherently so that C B = C H , the selectivity decreases in both noncompetitive and competitive cases.Similarly, in the presence of an excess amount of host cells, the selectivity decreases with increasing C B in both cases, more so for larger N p .In contrast, when the bacterial cell density is xed at C B = 5 × 10 4 cells per mL or C B = 10 8 cells per mL, the selectivity increases with increasing C H , more rapidly for larger N p .Compared to the competitive one, the noncompetitive selectivity can be overestimated by more than two orders of magnitude, depending on how C B and C H are chosen (see refs.11 and 16 for related discussions).
We also clarify how the cell selectivity of AMPs differs from their membrane selectivity.While the selectivity based on model membranes is typically larger than the corresponding cell selectivity, the (relative) difference between competitive and noncompetitive selectivity is generally larger in the latter.Except for some differences, membrane selectivity and cell selectivity of AMPs are qualitatively similar to each other.If interpreted with care, the former can provide useful information about the latter.
In this work, we will focus our effort on presenting a selectivity model in a pedagogical but yet systematic manner.In our consideration, one of the main differences between model membranes and cells comes from peptide trapping in the latter.Nevertheless, we will use membrane density and cell density interchangeably; also MICs and MHCs refer to peptide concentration beyond which membranes are ruptured, whether they are model membranes or cell membranes.
This paper is organized as follows: in Section 2, we present a simple picture of how the activity and selectivity of AMPs vary with cell density for a noncompetitive and competitive medium.Section 3 introduces a Langmuir model of peptide binding.Section 4 summaries the results for peptide activity and selectivity as a function of cell density; the effect of peptide trapping is highlighted, and membrane selectivity and cell selectivity are compared.All the symbols and acronyms are dened in Table 1.

Cell and membrane selectivity of antimicrobial peptides
In this section, we present a pedagogical approach to peptide activity and selectivity, which shows how peptide selectivity depends on cell density and peptide trapping in cells.We start with a homogeneous system of either bacterial or host cells, referred to as a noncompetitive case, and turn to a mixture of both types of cells, referred to as a competitive case.
Before proceeding further, we introduce several parameters relevant for peptide activity and selectivity.A key "extrinsic" parameter is the number density of peptides, denoted as C p ; so is the density of cells, C cell . 12,14,17,18The surface area of each cell, A cell , matters. 17In terms of the number of membrane-bound peptides, doubling A cell for given C cell is equivalent to doubling C cell for given A cell .[18] Membrane rupture occurs in an all-or-none C cell -dependent manner. 19,20,23Recall that P/L is the molar ratio of membranebound peptides to lipids.At a certain value of C p , i.e., C * p ; P/L reaches a threshold value required for membrane rupture, P/ L*; 10,19-21 C * p is either MIC or MHC.Finally, N p denotes the number of trapped peptides per cell.This needs to be taken with caution.Below C * p ; we assume that N p = 0.In this case, penetration of peptides into a cell is expected to be a rare event, since it involves overcoming a large free energy barrier for crossing an otherwise intact cell membrane.At C * p ; half of the cell membranes are ruptured.Thus, N p can be interpreted as the number of peptides trapped in each dead cell.Alternatively, it can be considered as the "average" number of peptides trapped per cell at C * p : N * p : Here, we employ this denition of N p , which is half of the number of trapped peptides in a dead cell.Beyond, N p can be larger than N * p : But we ignore the possible weak dependence of N p on C p (see Section 2.2 for further discussion).As a result, for C p $ C * p ; we use N p and N * p interchangeably, unless otherwise indicated.Finally, the subscript 'B' or 'H' will be used to refer to bacteria and host cells, respectively, as in N pB , N pH , ðP=LÞ * B ; and ðP=LÞ * H (see Table 1).Similarly, C B is the bacterial cell density and A B is the bacterial cell surface area; a B and a H are the lipid headgroup Trapping energy of a peptide in a bacterial (host) cell area of bacterial and host-cell membranes, respectively; the binding energy w B and w H can be interpreted similarly.

Homogeneous case
Fig. 2 illustrates how C * p depends on cell density C cell in a homogeneous or noncompetitive case, consisting of either bacteria or host cells.Here, two concentric circles represent cells (membrane bilayers enclosing cells), whereas small circles stand for peptides; if lled ones are free or trapped, unlled ones are membrane-bound.The fraction of bound peptides is controlled by the balance between entropy and energy 24 (also see ref. 25).At a low peptide concentration, peptides are mostly free, because of a large entropic penalty for binding even in a single-cell limit (Fig. 2(i)).As the peptide concentration C p increases, the balance is swayed toward energy, which favors binding.As a result, the surface coverage of peptides P/L (molar ratio of bound peptides to lipids) also increases.Eventually, C p reaches C * p (either MIC or MHC), at which P/L = P/L*.Even in the single cell limit shown in (i), C * p .0: As the cell density increases, different cells compete for peptides.Even though the binding is driven by energy, this competition is entropic in origin and does not involve cell-cell interactions.This is responsible for the cell-density dependence of C * p : It can be worked out progressively as shown in Fig. 2. Now imagine introducing a second cell in Fig. 2(i), converting the system into the one in Fig. 2(ii).Because of the presence of the rst cell, there will be less peptides for the second one: at C p ¼ C * p ; the number of peptides the rst cell consumed is equal to ½ðP=LÞ * Â A cell =a l þ N p ; where a l is the area of each lipid; recall A cell is the surface area of each cell.The presence of a second cell in (ii) is equivalent to removing [(P/L)* × A cell /a l + N p ] peptides in (i), which is at C * p : In order to remain at P/L*, an extra number of peptides should be supplied.The required number of peptides is equal to [(P/L)* × A cell /a l + N p ].This will raise C * p by [(P/L)* × A cell / a l + N p ]/V, where V is the volume of the system: Here, C * p ð1 cellÞ is either MIC or MHC in the single-cell case.The progression from (i) to (iii) shows how this analysis can be extended to the N cell -cell case: where the second equality holds if N cell [ 1, as is oen the case.This equation can be applied to a homogeneous system of either bacteria or host cells.Eqn (2) becomes Here MIC 0 and MHC 0 are, respectively, the MIC and MHC in the low-cell density limit:  As the peptide concentration C p increases, their surface coverage P/L (molar ratio of peptides to lipids) also increases and eventually reaches a threshold P/L* at C * p : Even in the single-cell limit shown in (i), C * p .0; because of the entropy of peptides, which favors unbinding.Imagine introducing a second cell in (i), converting the system into the one in (ii).The number of peptides the first cell consumed is equal to (P/L* × A cell /a 1 + N p ), where a l is the area of each lipid.In order to remain at P/L*, the same number of peptides should be supplied.This will raise C * p by P/L* × A cell /a l + N p /V, where V is the volume of the system: at P/L = (P/L)*.This is larger for larger N p ; peptide trapping in cells makes C * p increase more rapidly with C cell .The 'y'-axis intercept, either MIC 0 or MHC 0 , is set by the interaction of peptides with membranes among others (see Section 3).0][21] It is larger for PE (phosphatidylethanolamine)containing bacterial membranes, which tend to develop a negative curvature.However, this does not change P/L* by an order of magnitude.For the peptide melittin, for instance, ðP=LÞ * B z 0:02 and ðP=LÞ * H z 0:01: 19-21 Imagine combining MHC and MIC values obtained separately for homogeneous solutions.The ratio MHC/MIC increases with C H : the larger C H is, the larger the selectivity is.As evidenced below, this does not correctly represent the selectivity in a biological-relevant medium (e.g., a mixture of host cells and bacteria) but tends to overestimate it.

Competitive case
The homogeneous-case analysis in Fig. 2 can be extended to a mixture of bacterial and host cells, referred to as a competitive case, as shown in Fig. 3.If the concentric circles in blue represent bacterial cells, the pink ones stand for the host cells.Fig. 3(i) shows a single bacterial cell at the MIC.The introduction of a host cell in Fig. 3(ii) will reduce the amount of peptides for the bacterial cell.The extra number of peptides to maintain C p at the MIC is equal to [(P/L) H × A H /a H + N pH ]; similarly, in Fig. 3(iii), the number of peptides that should be added is ½ðP=LÞ The progression from (i) to (iii) suggests that If N p is set to zero as for model membranes, the second equation in eqn ( 4) can be obtained from the rst one by  swapping the role of bacteria with that of host cells.Here (P/L) H in eqn (4a) is the surface coverage of peptides on the host cells evaluated at C p = MIC, whereas (P/L) B in eqn (4b) is the surface coverage of peptides on bacteria evaluated at C p = MHC.Note that these two lines of equations in eqn ( 4) are not fully symmetric with respect to the exchange in role between host cells and bacteria for the obvious reason: as C p increases, the MIC will be reached rst.This explains why the last term in eqn (4a) does not contain N pH .In other words, ðP=LÞ H \ðP=LÞ * H in eqn (4a).In contrast, ðP=LÞ B .ðP=LÞ * B in eqn (4b).As a result, over a sizeable C p range, the peptide under consideration is active against bacteria only and is thus selective.
Also, eqn (4b) needs to be understood with caution.Beyond ðP=LÞ * B ; some of bound peptides start to rupture the membranes by forming pores, for instance.The last term in these equations may be interpreted as the total amount of bound peptides whether on the membrane surface or in pores.As a result, the binding energy w B needs to be interpreted accordingly.As it turns out, the term inside [.] in eqn ( 4) is dominated by N p (see below).Furthermore, w H , which governs peptide binding and inuences (P/L) H , is not constant but can vary with (P/L) H .The main source of this dependence is the electrostatic interaction between bound peptides.But this dependence is generally weak, since the distance between bound peptides for (P/L) # (P/ L)* z 0.01 is typically larger than the Debye screening length, r D , beyond which the electrostatic interaction is exponentially screened. 25At (P/L)* = 0.01, the typical distance between the adjacent peptides is z ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 100 Â 70 p Å z 80 Å: This is appreciably larger than the screening length under physiological conditions (e.g., in the presence of 100 mM of monovalent salts): r D z 10 Å.Finally, in eqn (4b), N pB is the number of trapped peptides in each cell above the MIC.The value of this parameter will eventually be determined by chemical equilibrium between trapped peptides and those on the membrane or in the bulk.The energetics of this is unknown and can be inuenced by a number of factors such as peptide's interaction with cellular components and crowding in the cell.As mentioned in Section 2, in our consideration, we ignore this complexity and approximate N pB in eqn (4b) by N * pB ; i.e., N pB at MIC.It is worth noting that eqn (3) and ( 4) are a special case of the following relations: Here, v p is the volume occupied by each peptide in the bulk and A p is the peptide area on the membrane surface.The rst term in each line is inspired by eqn (11); recall w B and w H are, respectively, the binding energy of a given peptide on bacterial and host-cell membranes (for details, see the ESIof ref. 17 or Section 3).If evaluated at ðP=LÞ B ¼ ðP=LÞ * B ; the rst term on the right hand side of eqn (5a) is MIC 0 ; the rst term in eqn (5b) can be interpreted similarly.Strictly speaking, both w B and w H have a weak dependence on P/L.At the relevant range of P/L around P/L*, however, this dependence can be neglected as discussed above.
The meaning of N p in eqn ( 5) is somewhat different from that in eqn (4).As noted above, eqn ( 5) is more general in the sense that C p on the le hand side does not have to be equal to C * p ; which is either MIC or MHC.As a result, N p in eqn (5) varies with P/L and is generally different from N * p ; N p ¼ N * p for (P/L) = (P/L)* and N p = 0 for (P/L) < (P/L)* Accordingly, N pH = 0 in eqn (5a), when C p = MIC (<MHC).Eqn (5a) then reduces to the MIC expression in eqn (4a).Similarly, eqn (5b) becomes the MHC expression in eqn (4b) in an appropriate limit.
For given values of C p and cell density (C B and C H ), the two equations in eqn ( 5) can be solved simultaneously for P/L: (P/L) B and (P/L) H . Initially, we set N p = 0 and increase C p gradually from zero.At some value of C p , (P/L) B reaches ðP=LÞ * B : The resulting value of C p with N pB set to N * pB is the MIC.We then increase C p further until ðP=LÞ H ¼ ðP=LÞ * H : The resulting C p with N pH ¼ N * pH is the MHC.In this step, (P/L) B in eqn ( 5) is larger than ðP=LÞ * B : In reality, pore formation in bacterial membranes can complicate the energetics of peptide binding to the membrane.But this complication will not change the MHC in any signicant way, since (P/L) B (A B /a B ) ( N pB at or above the MHC, as discussed in ref. 16 (also see below); the main source of inoculum effects is the trapping of peptides in cells rather than peptide adsorption to membranes.For model membranes, however, this reasoning is not applicable.In our coarse-grained model, all the details governing peptide binding are subsumed into the parameter w (w B and w H ). As noted above, w has a weak dependence on P/L and can also be inuenced by pore formation.In a Langmuir-type model such as the one employed here, w is oen approximated by its representative value.With a similar spirit, we will use a standard value of N p , as discussed in Section 3.
Let's analyze the relative signicance of peptide trapping in determining the cell-density dependence of MIC or MHC.For this, we essentially repeat the earlier analysis in ref. 16.Compare the two terms with each other inside [.] in eqn (4): the number of membrane-bound peptides and the number of absorbed peptides per cell.For the representative bacterium E. coli, A B z 12 mm 2 , which is twice the area of the inner or outer layer of the cytoplasmic membrane. 16,17Since a B z a H z 70 Å 2 , A B /a B z 1.7 × 10 7 .For the peptide melittin, ðP=LÞ * B z 0:02 and ðP=LÞ * H z 0:01: [19][20][21] We thus nd ðP=LÞ * B ðA B =a B Þ z 3:4 Â 10 5 : This number is much smaller than N pB z 10 7 to 10 8 . 14For the outer E. coli membrane, ðP=LÞ * B is several fold larger, 10,26 but this does not change the picture.For human red blood cells as representative host cells, A H z 17A B and A H /a H z 2.9 × 10 8 .As a result, we obtain ðP=LÞ * H ðA H =a H Þ z 2:9 Â 10 6 : This is smaller than N pH z 10 7 . 12,13The main source of inoculum effects is the trapping of peptides inside dead cells at or above (P/L)*.The analysis above implies that only the last term in eqn (4a) has a noticeable, explicit dependence on the binding energy w H for given MHC 0 .As a result, the MIC in eqn (4a) can be sensitive to w H , whereas the MHC in eqn (4b) is not.For similar reasons, both the MIC and the MHC in eqn ( 4) and ( 3) are not sensitive to w B for a xed value of MIC 0 .For the homogeneous case in eqn (3), none of the MIC and the MHC is "explicitly" sensitive to w B or w H .
Similarly to what was observed in the homogenous case in Section 2.1, peptide trapping in cells (the main inoculum effect) makes C * p increase more rapidly with C cell .It makes steeper the slope of a C * p curve versus C cell .

Limiting cases
It proves instructive to take some mathematical limits and simplify eqn (4).First, consider the case C B = C H .In the low celldensity limit, i.e., C B = C H / 0, the MIC and MHC in eqn (4) reduce to MIC 0 and MHC 0 , respectively, as there is no competition between different cells (or membranes) to bind peptides.As a result, the distinction between the competitive and noncompetitive cases disappears in this limit.
In the high-cell-density case, for simplicity, let's assume that A B = A H and N p = 0, as is oen the case for lipid bilayers, and a B = a H z 70 Å 2 , which is a good approximation (if A H s A B , this analysis is applicable to the case: To understand the origin of the inequality, note that (P/L) B in the numerator is larger than ðP=LÞ * B in the denominator, whereas (P/ L) H in the denominator is smaller than ðP=LÞ * H in the numerator.Thus MHC/MIC in this limit will get saturated at some constant larger than 1.
In the noncompetitive case with C B = C H , however, the ratio MHC/MIC approaches the following constant: ðP=LÞ * H =ðP=LÞ * B : The threshold P/L is better known for lipid bilayers mimicking cell membranes than for cell membranes.As noted in Section 2.1, because of the presence of PE (phosphatidylethanolamine) in bacterial cell-membrane mimics, ðP=LÞ * B is somewhat larger than ðP=LÞ * H : In the large cell-density limit in the noncompetitive case, we thus have MHC/MIC ( 1.There is a noticeable difference between the competitive and noncompetitive cases in the large cell-density limit; the selectivity is higher in the former case. If C H [ C B , eqn (4) can be simplied as MIC z (MIC) 0 + A H / a H × (P/L) H C H and MHC z ðMHCÞ 0 þ A H =a H Â ðP=LÞ * H C H : Note that the MIC in this case is much larger than the MIC for the corresponding bacteria-only case and the MHC here is approximately equal to the MHC for the corresponding host-cell-only case, as illustrated in Fig. 4. Accordingly, the ratio MHC/MIC is roughly independent of C B and approaches a constant of order 1, as C H / N (while C B is held xed).
Imagine combining two sets of data: one set for bacteria only and one set for host cells only, i.e., two homogeneous cases in eqn (3).If C H [ C B , MHC/MIC / N as C H / N.This limiting behavior in the homogeneous case is opposite to the one obtained for the corresponding competitive case (see Fig. 4).It explains how the selectivity can be excessively overestimated.
When N p s 0 and A B s A H , our analysis should reect these inequalities.But the difference caused by them is oen quantitative rather than qualitative, as evidenced in Section 4.
A full analysis of eqn ( 4) is involved.As discussed earlier, 16 in some relevant limits, we can simplify eqn (4) (see ref. 16).This is particular the case for C H [ C B as in whole blood.In this case, eqn (4) can be approximated as Here, (P/L) H is to be evaluated at C p = MIC.Notice the obvious difference the competitive MIC in eqn (6a) and the noncompetitive one in eqn (3a).As discussed earlier in Section 2.3 and in Fig. 4, for C H [ C B , the competitive MIC is much larger than the noncompetitive one.In contrast, the MHC is approximately the same for both cases.This results in much larger selectivity in the noncompetitive case compared to the corresponding competitive case.This nding is consistent with the analysis above with N p set to zero.
For the case C H [ C B , the ratio MHC/MIC becomes Eqn (7) suggests that peptide trapping in the host cells enhances peptide selectivity; it works in favor of the host cells.This is a natural consequence of the MHC that increases with N * pH : Since the second term inside [.] in the numerator of eqn ( 7) is larger than the rst term roughly by an order of magnitude (see Section 2.2), the effect of peptide trapping on the selectivity is up to about 10-fold.
So far, we have used a simple biophysical picture, based on Fig. 2-4 to explore how peptide selectivity depends on cell density (C B or C H ) and on the way it is measured (i.e., competitive versus noncompetitive).The y-intercepts, MIC 0 and MHC 0 , may be considered as tting parameters.They can also be related to more microscopic parameters.In the next section, we recapture the main results in this section; we then relate MIC 0 and MHC 0 to the biophysical parameters of peptides and membranes.

Langmuir binding model
In this section, using a Langmuir-type model for molecular binding, 25 we derive the main results presented in Section 2 and relate MIC 0 and MHC 0 to the biophysical parameters of peptides and membranes.Note that such a model was already considered recently. 17,18Here, we recapture the essence of this consideration and generalize it to include peptide trapping in a cell.It suffices to focus on the homogeneous case, since the dependence of peptide activity on cell density in the competitive case is already obvious from eqn (4).
In this model, peptides are either "free" (in the bulk) or "bound"; bound peptides are further classied as adsorbed to the cell surface or trapped inside a cell (see Fig. 2); trapped ones can bind to intracellular components.Initially, peptide binding occurs on the outer membrane layer or the outmost one in the case of Gram-negative bacteria.Adsorbed peptides will be eventually symmetrically distributed between the two layers aer or even prior to membrane rupture 27,28 (also see ref. 17).For simplicity, we ignore peptide trapping below C * p within typical experimental time scales.Indeed, it was shown that a large amount of trapped peptides were observed in dead bacterial cells, but not in dividing cells. 14At and beyond C * p ; the amount of bound peptides is determined by chemical equilibrium between free and bound states.
Let w and u be the adsorption and trapping energy, respectively.The value of w is typically more negative for bacterial membranes containing a large fraction of anionic lipids.It is worth noting that w is an effective parameter in which the effect of lipid demixing and peptide-peptide interactions on the membrane surface are subsumed (see ref. 18 for details).Similarly, u takes into account the interactions of trapped peptides with intracellular components as well as their mutual interactions inside the cell; it is also inuenced by molecular crowding in the cell. 29,30et C p be the total concentration of peptides whether free or bound, s p [=(P/L)/a l ] the planar density of adsorbed peptides and A p the area occupied by a bound peptide; n p the number density of trapped peptides, and v p the volume of each peptide; n p = 0 when P/L < P/L* and n p = N p /V cell when P/L = P/L*, where V cell is the volume of each cell.In our Langmuir model, the chemical potential of bound peptides m bound at and above P/L* can readily be obtained as Here and below, k B is the Boltzmann constant and T the temperature.The logarithmic term is related to the number of ways in which bound peptides are distributed on the membrane surface or inside the cell.The second equality holds in chemical equilibrium between adsorbed and trapped peptides.
The chemical potential of free peptides is Note that the expression inside [.] is the concentration of free peptides and the term inside (.) is the inoculum size.By equating the two chemical potentials in eqn ( 8) and ( 9), we obtain In this nal expression, we eliminated s p in favor of P/L via the relation s p a l = P/L (with a l as the lipid head-group area).In the absence of peptide trapping in cells (or below C * p ), eqn (10a) with n p = 0 describes chemical equilibrium between free and adsorbed peptides; eqn (10b) becomes irrelevant.At C * p ; i.e., either MIC or MHC, Here n * p V cell ¼ N * p is the (average) number of peptides trapped in each cell at C p ¼ C * p : Comparison between eqn (11) and (3) leads to the following relation Here the subscript 'B' and 'H' refer to bacteria and host cells, respectively.Both MIC 0 and MHC 0 are exponentially sensitive to w or u but they are not as sensitive to other quantities.This energy scale is the main origin of peptide selectivity.The results in eqn ( 12) can be used in eqn (4) (competitive) or in eqn (3) (noncompetitive).
It is worth mentioning that we will not attempt to solve eqn (10) for n p , partly because the energetics involved in peptide trapping (i.e., u) is not well known.3][14] With this simplication, eqn (10) can readily be extended to the competitive case shown in Fig. 3.The cell-density dependence of C * p is already obvious in light of the discussion in Section 2.2; one can readily write down eqn (5).

Results
In this section, we present the results for peptide activity and selectivity obtained for model membranes (Section 4.1) and cells (Section 4.2).Recall that one of the main differences between the two comes from peptide trapping in the latter case.As detailed below, MIC 0 and MHC 0 are chosen differently for the two cases.If calculated values of these quantities are used for model membranes, they are chosen appropriately for cells.

Membrane selectivity
Following Section 2, we rst present our results for peptide activity and selectivity without taking into account peptide trapping using peptide parameters relevant for a melittin-like peptide: 17,18 ðP=LÞ * B ¼ 1=48ðP=LÞ H ¼ 1=99; 19-21 v p = 33 3 Å 3 , and A p = 400 Å 2 . 17,18For this peptide, w was mapped out for model membranes, mimicking bacterial and host-cell membranes: w B = −16.6kB T and w H = −6.72kB T. 18 Also, a B = 71 Å 2 (a l for bacterial membranes), a H = 74 Å 2 (a l for host-cell membranes), [19][20][21] A B = 1.2 ×10 9 Å 2 = 12 mm 2 (suitable for E. coli), and A H = A B or A H = 17A B (as for human red blood cells). 17ote here that this value of A B is two times the surface area of E. coli (z6 mm 2 ). 31 This is to reect the symmetrical binding of peptides on the inner and outer layers of the cytoplasmic membrane, as discussed in Section 3. Finally, we set N p = 0 as expected for model membranes.In reality, N p s 0 at or beyond (P/L)*.But practically, it can be set to zero, since the majority of peptides would remain 'free'; trapped peptides in model membranes are outnumbered by those in the bulk.
We have solved eqn (3) for the noncompetitive case and eqn (4) for the competitive case (both together with eqn ( 12)).This is equivalent to solving eqn (5) for P/L at and found C p at which P/L is equal to P/L*, as discussed below eqn (5).In Fig. 5 The results in Fig. 5(A) and (B) suggest that both MICs and MHCs increase with increasing cell density (C H or C B ), as expected from eqn (3) and (4).For A H = 17A B , the presence of a large amount of host-cell membranes (C H = 5 × 10 9 cells per mL) raises the MIC by an order of magnitude as long as C B ( 5 × 10 5 cells per mL, compared to the case C H = 0; for A H = A B , however, its impact on the MIC appears to be minor.There is no essential difference between the three cases in (A): (i) (labelled as (i), (ii), (iii), respectively, in the legend); in these cases, the MIC is insensitive to the presence of an equal amount of host-cell membranes or the value of A H .As C B increases, the MIC curves eventually collapse onto each other.In this case, it is dominated by the C B -dependent term in eqn (4a).This applies to all the curves shown except the one in tangerine for which C B is held xed.
As shown Fig. 5(B), in the presence of a large amount of hostcell membranes (C H = 5 × 10 9 cells per mL), the MHC obtained with A H = 17A B is about ten times larger than MHC 0 z 3 mM (i.e., the y-intercept of the curve labelled as (ii) or (ii')), as long as C B ( 10 8 cells per mL.Similarly, in the other cases shown, the MHC is larger for larger A H = 17A B than for A H = A B , as long as C H T 10 8 cells per mL.For this, compare a curve obtained with A H = A B with the corresponding one obtained with A H = 17A B (e.g., the curves labelled as (i) and (i ′ ) or those labelled as (ii) and (ii ′ )).The difference between (i) and (i ′ ) seems somewhat minor, but the difference between (ii) and (ii ′ ) (in the absence of bacterial membranes) is pronounced.Also, the MHC curve labelled as (i) lies somewhat above the one labelled as (ii), both obtained with A H = 17A B .In this case, the presence of an equal amount of bacterial membranes (C H = C B ) increases slightly the MHC.When A H = A B (see the curves labelled as (i ′ ) and (ii ′ )), however, the presence of an equal amount of bacterial membranes has a more appreciable impact on the MHC.In other words, the presence of an equal amount of bacterial cells increases the MHC more effectively when A H = A B .This is consistent with eqn (3) or 4, which suggests that the MHC is more sensitive to C B if A H is smaller.The presence of 5 × 10 4 cells per mL of bacterial membranes (A H = 17A B ) does not have any noticeable impact on the MHC (the data not shown for simplicity).
The MIC and MHC results in Fig. 5(A) and (B) suggest that the presence of an equal amount of bacterial membranes inuences MHCs more effectively than the presence of an equal amount of host cell membranes inuences MICs.For this, compare the two curves labeled as (i) The two in (A) tend to collapse onto each other, whereas in (B) the curve obtained with C B = 0, A H = A B falls well below the other one.This difference can be attributed to the stronger binding of peptides to bacterial membranes.
In the competitive case with an excessive amount of host cells (C H = 5 × 10 9 cells per mL), however, the MIC in Fig. 5(A) and the MHC in Fig. 5(B) are much larger than in the other cases as long as C B ( C H .This is consistent with what eqn (4) suggests: the presence of a large amount of host cells increases both the MIC and the MHC (see the relevant discussion in Section 2.3).These equation also suggest that the MIC or the MHC is generally larger for A H = 17A B than for A H = A B , unless the A H -independent terms dominate.For this, compare the two curves in blue and cyan in (A) or (B), for instance.As discussed in Section 2, increasing A H is equivalent to increasing C H .This explains the observation of larger MIC and MHC values for larger A H . 16,17 The ratio MHC/MIC measures peptide selectivity.Our results for this ratio are shown in Fig. 5(C) and (D).In (C), C B = C H ; in (D), except for the red dashed line, C H = 5 × 10 9 cells per mL but C B is allowed to vary.In all cases, the selectivity decreases (or remains at), as the cell density increases as discussed in Section 2. In (C), the difference between the competitive and noncompetitive cases for A H = A B becomes obvious when the cell density is T 10 8 cells per mL, in which the selectivity is higher for the former case.This is correlated with the observation that the MHC is higher for the competitive case in this range of cell density as shown in Fig. 5(B).Also, the selectivity is higher for the larger A H case.
The competitive cases in (D), except for the red dashed line, contain a large amount of host cells (C H = 5 × 10 9 cells per mL) in addition to bacterial cells with variable C B .In the noncompetitive measurement, MHCs obtained with the choice C H = 5 × 10 9 cells per mL were combined with MICs.Similarly to what the graphs in (C) suggests, the selectivity in (D) decreases as C B increases.However, the selectivity in the noncompetitive case is overestimated compared to the corresponding competitive case, as long as C B ( 10 7 to 10 8 cells per mL ( C H .For the large A H case, it is overestimated by up to an order of magnitude.When C B is held xed at C B = 5 × 10 4 cells per mL, the (competitive) selectivity remains roughly at in the C H range shown.
This nding is well aligned with the view that the selectivity measured in a noncompetitive manner (with C H [ C B ) can be an experimental illusion. 11This is not the case for the competitive selectivity.Even in the presence of a large amount of host cells, the selectivity measured in a competitive environment is not an experimental artifact.It just reects the celldensity dependence of the selectivity, presented in Section 2.

Cell selectivity: inoculum effects
We have solved eqn (4) with realistic choices of N p and mapped out various scenarios for peptide activity and selectivity.One of the challenges in this effort is that the parameters in these equations are not well known for real cells.In particular, w B for Gram-negative bacteria is also inuenced by the peptide interaction with their outer membrane (OM); recall that this is an effective parameter, in which microscopic details (e.g., peptide charge, peptide interaction with the OM) are subsumed (see Section 3).This quantity has only recently been mapped out theoretically for the interaction of melittin-like peptides with model membranes. 18For the reasons explained in Section 2, however, the dependence of peptide activity on w B is reected mainly through MIC 0 .Furthermore, the MIC and the MHC in the homogeneous case in eqn (3) do not depend sensitively on w B or w H for given MIC 0 and MHC 0 .
Here we do not attempt to calculate the effective binding energy w (either w B or w H ) for real cells and use it in the computation of MIC 0 and MHC 0 .Instead, we start with conveniently-chosen but biophysically-relevant values of MIC 0 and MHC 0 : MIC 0 = 1 mm and MHC 0 = 5 mm (see ref. 14 and 15, for instance, for MIC 0 ).For simplicity, the number of trapped peptides N p is chosen to be the same for bacteria and host cells: N p = 0, 10 7 , 5 × 10 7 .Otherwise, we choose the same parameters used in Fig. 5: the bacterial cell surface area A B = 12 mm 2 (suitable for E. coli); the host cell surface area A H = 200 mm 2 z 17 × A B (as for human red blood cells); a H = 71 Å 2 and a H = 74 Å 2 ; w B = −16.6 k B T and w H = −6.72 k B T; 18 v p = 33 3 Å 3 and A p = 400 Å 2 . 17,18ig. 6 displays the results for the MIC (A) and the MHC (B) for the noncompetitive and competitive cases, represented by solid lines with unlled symbols or unlled symbols and dashed lines with lled symbols, respectively.As in Fig. 5, when C H (C B ) is held xed, the 'x' axis represents C B (C H ); for the case C H = C B , it stands for both C H and C B .
In all cases shown in Fig. 6(A) and (B), both MICs and MHCs increase linearly with increasing cell density (C H or C B ), similarly to what is shown for model membranes in Fig. 5.This is a natural consequence of the cell-density dependence shown in eqn (4).
As indicated in the graph on the le in Fig. 6(A), the presence of an excess amount of host cells raises the MIC, more so for larger N p as long as C B ( 5 × 10 8 cells per mL; for this, compare the curve obtained with C H = 5 × 10 9 cells per mL with the one obtained with C H = 0. Nevertheless, the MIC remains somewhat smaller than 10 mm if C B ( 5 × 10 7 cells per mL.When C B T 5 × 10 8 cells per mL, the presence of host cells does not have a signicant impact on the MIC; in this case, peptide trapping in bacterial cells is a determining factor.For the same value of N p , different curves representing different values of C H collapse onto each other for sufficiently large C B : C B T 5 × 10 8 cells per mL.Also, the MIC obtained with N p = 5 × 10 7 increases more rapidly with cell density than the corresponding one obtained with N p = 10 7 does, as suggested by eqn (4).Finally, for given N p , there is no noticeable difference between the two cases: C B = C H (competitive) and C H = 0 (bacterial-cell only).The presence of an equal amount of host cells has an insignicant impact on the MIC.At the MIC, the host cells are above the MHC (no trapping in the cells) and their effect on the MIC is expected to be minor (see Section 2.2.for the relative signicance of membrane association of peptides versus peptide trapping in cells).
As shown in the graph on the right in Fig. 6, when C B is held xed at C B = 5 × 10 4 cells per mL, the MIC is insensitive to the value of N p used, as if bacterial cells are in the low-cell density limit (i.e., their presence creates a minimal inoculum effect).At the MIC, the host cells, which are present together with bacterial cells, are below the MHC.As a result, the binding of peptides to the host-cell membrane is responsible for the slow increase of the MIC with C H .The presence of a large amount of bacterial cells (C B = 10 8 cells per mL) increases the MIC about ten-fold as long as C H ( 10 8 cells per mL (the two homogenous MIC curves from the graph in the le are also included for comparison purposes).As shown in the graph on the right in Fig. 6, the presence of a small concentration of bacteria (i.e., C B = 5 × 10 4 cells per mL) does not alter the MHC in any signicant way.For this, compare Fig. 7 shows the results for MHC/MIC.The dashed lines with lled symbols represent competitive selectivity, whereas the solid lines with unlled symbols describe noncompetitive selectivity; in the latter case, MHCs and MICs, obtained for hostcell only and bacteria-only solutions, respectively, are combined into MHC/MIC.
In the graph on the le in Fig. 7, we have chosen C H = C B .In all cases shown in the graph, the selectivity, MHC/MIC, decreases from the initial value MHC 0 /MIC 0 as the cell density increases.It is larger for the competitive case (lled symbols) than for the corresponding noncompetitive case so that the difference between the two cases is more pronounced for larger C H = C B .For C H = C B ( 10 9 cells per mL, the selectivity is somewhat larger when N p is smaller; in this case, peptide trapping works in bacteria's favor by increasing the MIC. As shown in the graph on the right in Fig. 7, the selectivity obtained with C H = 5 × 10 9 cells per mL decreases with increasing C B , more rapidly when N p is larger.In this case, peptide trapping enhances the selectivity for C B ( 5 × 10 9 cells per mL (competitive) or C B ( 5 × 10 8 cells per mL (noncompetitive) but does not seem to have a noticeable impact outside this range, as it approaches MHC 0 /MIC 0 .In contrast, it increases with C H , more so for larger N p , when C B is held xed at C B = 5 × 10 4 cells per mL or C B = 5 × 10 8 cells per mL.The selectivity is smaller for the latter choice of C B .The presence of host cells in the competitive case works in favor of the host cells by enhancing the selectivity, more effectively for larger N p .
The results in Fig. 7 show how the selectivity can be overestimated.With the parameter choices used, the noncompetitive selectivity can be an order of magnitude larger than the corresponding competitive one.Furthermore, depending on how the selectivity is measured, it can be two or three order of magnitude different; for this, compare the solid line with unlled squares in blue (noncompetitive) and the dashed line with lled squares in purple (competitive).
The picture offered by the graph on the right in Fig. 7 is not only consistent with the earlier observation that the selectivity can be excessively overestimated 11 (see ref. 16 for a theoretical basis) but also claries further how peptide selectivity is inuenced by various factors or even the way it is measured: competitive, noncompetitive, the presence of host cells, peptide trapping in dead cells.

Membrane versus cell selectivity
There are both similarities and differences between membrane selectivity (Fig. 5) and cell selectivity (Fig. 7) of antimicrobial peptides.In both cases, the membrane-density or cell-density dependence of the selectivity is well manifested.If we set C H = C B , both membrane and cell selectivity decrease with C H = C B .
In the presence of 5 × 10 9 cells per mL of host cells or neutral membranes (mimicking host cell membranes) as in whole blood, the selectivity decreases as C B increases.In both cases, the selectivity tends to be overestimated in a noncompetitive environment with reference to the corresponding competitive case; when A H = A B , however, the difference between competitive and noncompetitive selectivity against model membranes appears to be minor, especially when C B ( 10 9 cells per mL (Fig. 5).When C B is held xed at C B = 5 × 10 4 cells per mL, the membrane selectivity remains nearly at as a function of C H , whereas the cell selectivity increases up to about 10 folds for N p = 5 × 10 7 ; if N p = 0, the selectivity would remain nearly at (the data not shown).
It is worth noting that the MICs for bacterial membranes in Fig. 5 are much smaller than those for bacterial cells in Fig. 6.In contrast, the MHCs in the two gures are comparable.In the case of E. coli, the outer membrane enclosing the cell tends to raise MIC 0 .In addition, peptide trapping in dead cells is also responsible for the differences between membranes and cells.Nevertheless, the qualitative picture offered from membranes (Fig. 5) is generally consistent with the one obtained for cells.

Discussions and conclusions
We have presented a biophysical model of peptide activity and selectivity by combining a pedagogical approach with a Langmuir-type model.If the former captures the cell-density dependence of peptide activity and selectivity in an intuitivelyobvious way, the latter relates peptide binding (or trapping) to an effective binding (or trapping) energy.
Using the model, we have claried how the presence of host cells and peptide trapping inuence peptide selectivity and how competitive selectivity differs from noncompetitive selectivity.If the competitive selectivity represents a mixture of bacteria and host cells, the noncompetitive one is obtained by combining MICs and MHCs for bacterium-only and host-cell-only solutions, respectively.0][21] and relevant references therein).
The results based on the model suggest a rather nontrivial dependence of the selectivity on the presence of host cells, cell density, and peptide trapping; these factors or effects can enhance or reduce the selectivity depending on how the density of host cells and that of bacterial cells are chosen.When C B = C H , the selectivity is somewhat smaller for larger N p , unless C B = C H is sufficiently large (le graph in Fig. 7).In more general cases (right graph in Fig. 7), however, peptide trapping tends to enhance the selectivity; also the presence of host cells works in favor of the host cells, but it raises the MIC up to about 10-fold (Fig. 6(A)).
When C B = C H , the selectivity decreases from the initial value MHC 0 /MIC 0 , with increasing C B = C H , more rapidly for the noncompetitive case; the selectivity is higher for the competitive case and is not sensitive to the choice of N p .In the presence of a large amount of host cells (C H = 5 × 10 9 cells per mL), the selectivity decreases with increasing C B in both competitive and noncompetitive cases.The noncompetitive selectivity can be one-order of magnitude larger than the corresponding competitive one.When C B is held xed at C B = 5 × 10 4 cells per mL or at C B = 10 8 cells per mL, the competitive selectivity increases with C H ; the selectivity is smaller for the latter choice of C B .Depending on how cell density is chosen, the selectivity can be overly overestimatedalmost by three orders of magnitude.
Our work also claries how the cell selectivity of AMPs differs from their membrane selectivity.The selectivity based on model membranes is typically larger than the one measured for cells.In both cases (membranes and cells), noncompetitive selectivity is typically larger than the corresponding competitive one, except for the case C B = C H .
The results in this work suggest that the selectivity reects not only peptide-membrane parameters but also cell density, peptide trapping, and even the way the selectivity is measured (competitive vs. noncompetitive).This is a natural consequence of MICs and MHCs that vary with cell density and N p .Mapping out possible scenarios of peptide activity and selectivity thus would involve exploring wide ranges of C B and C H , which are not easily realized in experiments.
If the involved peptide-membrane parameters are characterized, our model described by eqn (4) can be used as a predictive model.It enables one to calculate MICs, MHCs, and MHC/MIC, as a function of cell density: C B or C H , the density of bacterial and host cells, respectively.
Alternatively, eqn (3) can be used as a tting model for analyzing MIC and MHC data obtained in a noncompetitive manner: the 'y'-intercept and the 'slope' can be extracted by tting MIC or MHC data to eqn (3a) or (3b), respectively.This enables one to determine MIC 0 or MHC 0 .Eqn (12) shows how these quantities are related to peptide's binding energy w (w B or w H ) and P=L * ððP=LÞ * B or ðP=LÞ * H Þ: It is worth noting that P/L* has been measured for various model membranes [19][20][21] as well as for cells. 10Once P/L* is known, MIC 0 and MHC 0 can be converted into w B and w H , respectively.Conversely, if w is known, P/L* can be estimated.If all this information is used in the 'slope,' the value of N p can be extracted.
The information from the homogeneous analysis above can be used in eqn (4), which represents a competitive case.Accordingly, one can quantify peptide selectivity for a biologically relevant setting, which reects the degree and location of infection.For instance, C B ranges from 1 colony-forming unit (CFU mL −1 ) (in blood stream, where C H z 5 × 10 9 cells per mL) to 10 9 CFU mL −1 (in so tissue or peritonea) (see a recent review 12 and relevant references therein).
To advance our model and to take fuller advantage of its predictive power, computational and experimental methods can be employed to evaluate further the respective roles of host cells, cell density, and peptide trapping in the selectivity of AMPs (see Fig. 6).Because of their complexity, peptide-cell systems are not so amenable to microscopic computational modeling based on molecular dynamics simulations. 32A concerted effort between theoretical modeling, computational approaches, and experiments would be desired.Along the line of what was done in recent studies, 14 in which a number of key parameters including N p were extracted, parameters for multispecies cultures can be mapped out and used in eqn (4) or its variation.
In this work and in a typical experimental setting, the total number of AMPs is treated as a constant.In reality, however, it is inuenced by the expression of AMPs by the host 14 and peptide degradation by protease. 12,22Furthermore, earlier studies highlight the stochastic nature of eliminating bacteria with AMPs and its impact on the survivability of a population. 14t was shown that below the MIC, two sub-populations emerged: one group that stopped dividing and another group that could grow unharmed and divide.To clarify the roles of these population uctuations, stochastic modeling of population dynamics can be employed. 33,34ntimicrobial peptide cecropin A, Biophys.J., 2008, 94, 1667-1680.24 When the peptide concentration is low, the propensity for them to remain unbound is high because of the substantial entropic penalty for binding.Entropy is a measure of the number of microscopic arrangements subject to a macroscopic constraint (e.g., total energy,.).
In the context of our model in this work, in the low peptide concentration limit, the multitude of possible unbound states for the peptides results in a much larger entropy, contrasted with the smaller number of bound states.The entropy of peptides favors unbound states.This argument is most obvious in light of Eqs. 8 and 9, as is particularly the case for C cell = 0.When C cell = 0, these equations suggest that the entropic free energy cost for peptide binging per peptide, denoted as DF ent , is given by The smaller C p is, the C p ðC cell /0Þ z C * p ð1 cellÞ: Eqn (3) can be viewed as a function of C cell : C B or C H .Both the MIC and the MHC increase linearly with the cell density C B and C H , respectively.The slope of the relation in eqn (3), [(P/L)* A cell /a l + N p ], is the total number of peptides consumed per cell

Fig. 2
Fig.2Cell-density dependence of C * p ; i.e., either MIC or MHC: a homogeneous or noncompetitive case.Cells are represented by two concentric circles and peptides by filled (free or trapped) or unfilled circles (membrane-bound).As the peptide concentration C p increases, their surface coverage P/L (molar ratio of peptides to lipids) also increases and eventually reaches a threshold P/L* at C * p : Even in the single-cell limit shown in (i), C * p .0; because of the entropy of peptides, which favors unbinding.Imagine introducing a second cell in (i), converting the system into the one in (ii).The number of peptides the first cell consumed is equal to (P/L* × A cell /a 1 + N p ), where a l is the area of each lipid.In order to remain at P/L*, the same number of peptides should be supplied.This will raise C * p by P/L* × A cell /a l + N p /V, where V is the volume of the system:C * p ð2 cellsÞ ¼ C * p ð1 cellÞ þ ðP=L* Â A cell =a l þ N p Þ=V: The progression from (i) to (iii) shows that C * p ¼ C * p ð1 cellÞ þ ðN cell À 1Þ Â ðP=L* Â A cell =a l þ N p Þ= V z C * p ð1 cellÞ þ ðP=L* Â A cell =a l þ N p Þ ÂC cell : When applied to bacteria, this equation become MIC(C cell ) = MIC 0 + (P/L* × A cell /a l + N p )C cell , where MIC 0 is MIC in the low-cell density limit: C cell / 0. Figure adapted with permission from ref. 17.Copyright 2015 American Chemical Society; Reproduced with modifications from ref. 18 with permission from the Royal Society of Chemistry.
Fig.2Cell-density dependence of C * p ; i.e., either MIC or MHC: a homogeneous or noncompetitive case.Cells are represented by two concentric circles and peptides by filled (free or trapped) or unfilled circles (membrane-bound).As the peptide concentration C p increases, their surface coverage P/L (molar ratio of peptides to lipids) also increases and eventually reaches a threshold P/L* at C * p : Even in the single-cell limit shown in (i), C * p .0; because of the entropy of peptides, which favors unbinding.Imagine introducing a second cell in (i), converting the system into the one in (ii).The number of peptides the first cell consumed is equal to (P/L* × A cell /a 1 + N p ), where a l is the area of each lipid.In order to remain at P/L*, the same number of peptides should be supplied.This will raise C * p by P/L* × A cell /a l + N p /V, where V is the volume of the system:C * p ð2 cellsÞ ¼ C * p ð1 cellÞ þ ðP=L* Â A cell =a l þ N p Þ=V: The progression from (i) to (iii) shows that C * p ¼ C * p ð1 cellÞ þ ðN cell À 1Þ Â ðP=L* Â A cell =a l þ N p Þ= V z C * p ð1 cellÞ þ ðP=L* Â A cell =a l þ N p Þ ÂC cell : When applied to bacteria, this equation become MIC(C cell ) = MIC 0 + (P/L* × A cell /a l + N p )C cell , where MIC 0 is MIC in the low-cell density limit: C cell / 0. Figure adapted with permission from ref. 17.Copyright 2015 American Chemical Society; Reproduced with modifications from ref. 18 with permission from the Royal Society of Chemistry.

Fig. 3
Fig. 3 Cell-density dependence of MIC (A) and MHC (B): competitive case.Cells are represented by two concentric circles and peptides by filled (free or trapped) or unfilled circles (membrane-bound); if the blue circles represent bacterial cells, the pink ones stand for host cells.Let A cell = A B or A H be the bacterial or host cell surface area, respectively; a B and a H the lipid headgroup area of the bacterial or host-cell membranes, respectively; N pB and N pH the number of trapped peptides in each bacterial and host cell, respectively; (P/L) B and (P/L) H are the molar ratio of bound peptides to lipids on the bacterial and host-cell membranes, respectively.(A) The progression from(i) to (iii) suggests that MICðC cell Þ ¼ ðMICÞ 0 þ ½A B =a B Â ðP=LÞ * B þ N pB C B þ A H =a H Â ðP=LÞ H C H : (B) Using a similar line of reasoning, we arrive at MHCðC cell Þ ¼ ðMHCÞ 0 þ ½A H =a H Â ðP=LÞ * H þ N pH C H þ ½A B =a B Â ðP=LÞ B þ N pB C B :Figure adapted with permission from ref. 17.Copyright 2015 American Chemical Society; Reproduced with modifications from ref. 18 with permission from the Royal Society of Chemistry.
Fig. 3 Cell-density dependence of MIC (A) and MHC (B): competitive case.Cells are represented by two concentric circles and peptides by filled (free or trapped) or unfilled circles (membrane-bound); if the blue circles represent bacterial cells, the pink ones stand for host cells.Let A cell = A B or A H be the bacterial or host cell surface area, respectively; a B and a H the lipid headgroup area of the bacterial or host-cell membranes, respectively; N pB and N pH the number of trapped peptides in each bacterial and host cell, respectively; (P/L) B and (P/L) H are the molar ratio of bound peptides to lipids on the bacterial and host-cell membranes, respectively.(A) The progression from(i) to (iii) suggests that MICðC cell Þ ¼ ðMICÞ 0 þ ½A B =a B Â ðP=LÞ * B þ N pB C B þ A H =a H Â ðP=LÞ H C H : (B) Using a similar line of reasoning, we arrive at MHCðC cell Þ ¼ ðMHCÞ 0 þ ½A H =a H Â ðP=LÞ * H þ N pH C H þ ½A B =a B Â ðP=LÞ B þ N pB C B :Figure adapted with permission from ref. 17.Copyright 2015 American Chemical Society; Reproduced with modifications from ref. 18 with permission from the Royal Society of Chemistry.

Fig. 4
Fig. 4 Peptide selectivity for a noncompetitive (homogeneous) (i) versus competitive (heterogeneous) case (ii).It is assumed that C H [ C B .In this case, whether the selectivity is measured noncompetitively (i) or competitively (ii) has a profound impact on the selectivity.It can be excessively overestimated in the noncompetitive case (i) with reference to the corresponding competitive case (ii), since the MIC is much larger for the latter case.The opposite is true if C H ( C B . Figure adapted with permission from ref. 17.Copyright 2015 American Chemical Society; Reproduced with modifications from ref. 18 with permission from the Royal Society of Chemistry.

Fig. 5
Fig. 5 Cell (membrane) density dependence of MIC, MHC, and MHC/MIC for the noncompetitive and competitive cases, represented by solid lines with unfilled symbols and dashed lines with filled symbols, respectively.When C H (C B ) is held fixed, the 'x' axis represents C B (C H ); for the case C H = C B , it stands for both C H and C B .We have chosen the parameter as follows: the bacterial cell surface area A B = 12 mm 2 (suitable for E. coli); the host cell surface area A H = A B and A H = 200 mm 2 z 17 × A B (as for human red blood cells); a B = 71 Å 2 and a H = 74 Å 2 ; P=L * B ¼ 1=48 and P=L * H ¼ 1=99; 19-21 v p = 33 3 Å 3 and A p = 400 Å 2 ; 17,18 w B = −16.6 k B T and w H = −6.72 k B T 18 as for the peptide melittin.(A)-(B) In all cases, both MICs and MHCs increase with increasing C H or C B , as expected from eqn (4a).Also, the presence of a large amount of hot-cell membranes (C H = 5 × 10 9 cells per mL) raises both the MIC and the MIC, almost by an order of magnitude for the case A H = 17A B as long as C H [ C B .There is no essential difference between the three cases labelled as (i), (ii), and (iii) in the legend in (A): the presence of an equal amount of host-cell membranes (C H = C B ) or the value of A H does not influence the MIC in any noticeable way.As shown in (B), the MHC is larger for larger A H (i.e., A H = 17A B ).For this, compare a curve obtained with A H = A B with the corresponding one obtained with A H = 17A B .Also, the MHC curve labelled as (i) lies somewhat above the one labelled as (ii), both obtained with A H = 17A B .In this case, the presence of an equal amount of bacterial membranes (C H = C B ) increases slightly the MHC.When A H = A B represented as (i') and (ii'), however, it has a more appreciable impact on the MHC.The competitive MHC in the presence of 5 × 10 4 cells per mL of bacterial membranes with A H = 17A B is almost identical to the corresponding noncompetitive one (i.e., C B = 0) (data not shown for simplicity).The selectivity in (C), as measured by MHC/MIC, decreases as the membrane density increases; in both competitive and noncompetitive cases, we chose C H = C B .The difference between the competitive and noncompetitive cases becomes obvious when the cell density is T 10 8 cells per mL, in which the selectivity is higher for the former case.Also the selectivity is higher for the larger A H case as long as C B = C H T 10 8 cells per mL.In (D), except for the red dashed curve with inverted filled triangles, C H = 5 × 10 9 cells per mL but C B varies.Similarly to what the graph in (C) suggests, the selectivity in (D) decreases as C B decreases.Compared to the competitive case represented by the blue dashed curve with filled squares, the corresponding noncompetitive case overestimates the selectivity by about one order of magnitude at a low C B range of C B ( 10 5 cells per mL.The red dashed line with inverted triangles obtained with C B = 5 × 10 4 cells per mL is nearly flat in the C H range shown.

Fig. 6 (
B) shows how the MHC varies as a function of cell density: C B or C H .In all cases, the MHC increases with increasing cell density.When C H = 5 × 10 9 cells per mL, the MHC is large and remains roughly at as C B increases up to C B Paper RSC Advances = 10 9 cells per mL.This is consistent with eqn (4b), which suggests that the MHC is roughly independent of C B , as long as C H is sufficiently larger than C B .The MHC is obviously larger for the larger N p case (squares or diamonds).Finally, the MHC is somewhat larger in the presence of an equal amount of host cells (C B = C H ) compared to the host-cell only case C B = 0. Peptide trapping in the bacterial cells is responsible for this.

Fig. 6
Fig. 6 Cell-density dependence of MIC (A) and MHC (B) for the noncompetitive and competitive cases, represented by solid lines with unfilled symbols (or unfilled symbols) and dashed lines with filled symbols, respectively.When C H (C B ) is held fixed, the 'x' axis represents C B (C H ); for the case C H = C B , it indicates both C H and C B (as indicated in the legends of (A), (B), and (C)).We have chosen the parameters as follows: MIC 0 = 1 mm and MHC 0 = 5 mm; w B = −16.6 k B T and w H = −6.72 k B T, as for a melittin-like peptide; the bacterial cell surface area A B = 12 mm 2 (suitable for E. coli); the host cell surface area A H = 200 mm 2 z 17 × A B (as for human red blood cells); the lipid headgroup area a B = 71 Å 2 and a H = 74 Å 2 ; v p = 33 3 Å 3 and A p = 400 Å 2 .In all cases shown in (A) and (B), both the MIC and MHC increase with increasing cell density (C H or C B ), as expected from eqn 4(a) and (b).(A) (Left) The presence of a large amount of host cells as in whole blood increases the MIC up to ten-fold, as long as C B ( 5 × 10 8 cells per mL; for this, compare the curve obtained with C H = 5 × 10 9 cells per mL with the corresponding one obtained with C H = 0. Also the MIC increases more rapidly, if N p is larger.The MIC remains ( 10 mm if C B ( 5 × 10 7 cells per mL.If C B T 5 × 10 8 cells per mL, the presence of host cells does not have a significant impact on the MIC; in this case, peptide trapping in bacterial cells is a determining factor.For the same value of N p , different curves representing different values of C H collapse onto each other for sufficiently large C B : C B T 5 × 10 8 cells per mL.There is no noticeable difference between the two cases: C B = C H and C H = 0 for given N p .In this case, the main source of inoculum effects is the trapping of peptides in bacterial cells.(A) (right) When C B is held fixed at C B = 5 × 10 4 cells per mL, the MIC is insensitive to the value of N p used, as if bacterial cells are in the low-cell density limit (i.e., their presence creates a minimal inoculum effect).At the MIC, host cells are below the MHC.As a result, the binding of peptides to the host-cell membrane is responsible for the slow increase of the MIC with C H .The presence of a large amount of bacterial cells (C B = 10 8 cells per mL) increases the MIC about ten-fold as long as C H ( 10 8 cells per mL (the two homogenous MIC curves from the graph in the left are also included for comparison purposes.)(B) (left) In all cases, the MHC increases with increasing cell densities: either C B or C H .When C H = 5 × 10 9 cells per mL, the MHC is large and remains roughly flat as C B increases up to C B = 10 9 cells per mL.It is obviously larger for the larger N p case (squares or diamonds); it can be two orders of magnitude larger than MHC 0 .Also the MHC is larger for the competitive case C B = C H compared to the corresponding noncompetitive case C B = 0: at the MHC, the bacterial cells are above the MIC and the resulting peptide trapping in the bacterial cells raises the MHC.(B) (right) The presence of a small concentration of bacteria (i.e., C B = 5 × 10 4 cells per mL) does not alter the MHC in any significant way.Also the MHC increases faster with C H for larger N p , as expected from eqn (4b).In the presence of a large amount of bacterial cells (C B = 10 8 cells per mL), the MHC is about three times as large as in the corresponding host-cell only case, as long as C H ( 10 7 cells per mL.

Fig. 7
Fig. 7 Cell-density dependence of MHC/MIC for the noncompetitive and competitive cases, represented by solid lines with unfilled symbols and dashed lines with filled symbols, respectively.In the graph on the left, C H = C B ; in the graph on the right, the 'x' axis represents C B (C H ), when C H (C B ) is held fixed.We have chosen the same parameters as in Fig. 6: MIC 0 = 1 mm and MHC 0 = 5 mm; w B = −16.6 k B T and w H = −6.72 k B T as for melittin; the bacterial cell surface area A B = 12 mm 2 (suitable for E. coli); the host cell surface area A H = 200 mm 2 z 17 × A B (as for human red blood cells); a B = 71 Å 2 and a H = 74 Å 2 ; v p = 33 3 Å 3 and A p = 400 Å 2 .(left) In all cases shown, C H = C B .The selectivity, MHC/MIC, decreases as the cell density increases.It is larger for the competitive case (filled symbols), more so for larger C H = C B .For C H = C B ( 10 9 cells per mL, the selectivity is somewhat larger when N p is smaller; in this case, peptide trapping works in bacteria's favor by increasing the MIC.(right) The selectivity obtained with C H = 5 × 10 9 cells per mL decreases with increasing C B , more rapidly for larger N p .In this case, peptide trapping enhances the selectivity as long as C B ( 5 × 10 9 cells per mL (competitive) or C B ( 5 × 10 8 cells per mL (noncompetitive) but does not seem to have a noticeable impact outside this range.In contrast, it increases with C H , more rapidly for larger N p , when C B is held fixed at C B = 5 × 10 4 cells per mL or C B = 10 8 cells per mL.The selectivity is smaller for the latter choice of C B .With the parameter choices used, the noncompetitive selectivity can be an order of magnitude larger than the corresponding competitive one; depending on how the selectivity is measured, it can be two or three order of magnitude different; for this, compare the blue solid line with open squares with the magenta dashed curve with inverted filled triangles.

Table 1
Definitions of symbols and acronyms , the resulting C * p ; either MIC or MHC, as well as the ratio MHC/MIC are shown as a function of cell density: C B or C H .When C H (C B ) is held xed, the x axis represents C B (C H ); for the case C H = C B , it indicates both C H and C B .If the competitive cases are represented by dashed lines with lled symbols, the noncompetitive ones are described by solid lines with open symbols.