Drug Permeation against Efflux by Two Transporters

Abstract

The development of new antibiotics against Gram-negative bacteria is hampered by the powerful protective properties of their cell envelope. This envelope consists of two membranes augmented by efflux transporters, which act in synergy to restrict cellular access to a broad range of chemical compounds. Recently, a kinetic model of this system has been constructed. The model revealed a complex, nonlinear behavior of the system, complete with a bifurcation, and matched very well to experimental uptake data. Here, we expand the model to include multiple transporters and apply it to an experimental analysis of antibiotic accumulation in wild-type and efflux-deficient Escherichia coli. We show that transporters acting across the inner and outer membranes have synergistic effects with each other. In contrast, transporters acting across the same membrane are additive as a rule but can be synergistic under special circumstances owing to a bifurcation controlled by the barrier constant. With respect to ethidium bromide, the inner membrane transporter MdfA was synergistic to the TolC-dependent efflux across the outer membrane. The agreement between the model and drug accumulation was very good across a range of tested drug concentrations and strains. However, antibiotic susceptibilities related only qualitatively to the accumulation of the drugs or predictions of the model and could be fit to the model only if additional assumptions were made about the physiological consequences of prolonged cell exposure to the drugs. Thus, the constructed model correctly predicts transmembrane permeation of various compounds and potentially their intracellular activity.

Graphical Abstract

The development of new antibiotics remains a perpetual task for the biomedical community, especially in the face of spreading multidrug resistance among bacterial pathogens.^1–3 This task is especially formidable with Gram-negative bacteria, which possess a highly restrictive cell envelope.^4–6 This envelope consists of two lipid membranes fortified by numerous efflux transporters that span either the inner membrane (IM) or the outer membrane (OM) and often display a broad range of substrate specificities. This drug permeation barrier is highly efficient in the exclusion of diverse chemicals from the intracellular milieu. To ensure the accumulation of desired chemicals in this context, the cell often depends on specialized outer membrane porins and influx transporters. Understanding how this barrier can be breached is paramount for the design of antibiotics against Gram-negative pathogens.^3,5,7

Whereas biological aspects of drug permeation in Gram-negative bacteria have been largely established in the past two decades,^8–11 an adequate kinetic model of the process was only recently developed.^5,12 The model naturally explains the synergy between the low permeability of the outer membrane and active efflux and predicts an experimentally verified phase transition in drug uptake patterns. The model is based on two postulates, that both drug efflux and transmembrane diffusion are saturable, and operates with two kinetic parameters, the efflux and barrier constants. The former parameter relates the rates of active and passive drug transport across the outer membrane at low drug concentration, and the latter relates the same but when the fluxes are at saturation. The value of the barrier constant controls a bifurcation in the solution. If drug concentration can become high enough to saturate active efflux, a phase transition occurs in the behavior of the system, and the drug penetrates the cell. If, however, the barrier constant is greater than 1, such conditions cannot be met, and the drug remains excluded from the cell. This factor is especially important for lipophilic compounds, which are expected to saturate the cellular membrane at submicromolar concentrations.⁵ However, even compounds that are far from saturation would follow predictions of the model.

In this study, we sought to further examine implications of the model, extend it to cases of multiple coexisting transporters, and develop a fitting approach that maximally utilizes the information contained in the experimental data. As an experimental system, we chose the transporters of E. coli EmrA¹³ and MdfA¹⁴ from Small Multidrug Resistance¹⁵ and Major Facilitator¹⁶ superfamilies of proteins that act across the IM. These polyspecific transporters are driven by the proton-motive force and recognize a variety of compounds with some preference toward aromatic lipophilic cations. Homologues of EmrE and MdfA transporters are universally represented in genomes of Gram-negative bacteria and function in the context of two-membrane envelopes containing the efflux pumps acting across the OM. MdfA is commonly expressed in fluoroquinolone-resistant clinical isolates and contributes to resistance against these antibiotics.¹⁷ However, inactivation of neither MdfA nor EmrE separately leads to notable changes in E. coli susceptibilities to antibiotics.¹⁸ This lack of an effect of antibiotic activities is due to the expression of AcrAB-TolC, which is the major efflux pump expelling antibiotics across the OM of E. coli. The characteristic feature of this pump is its tripartite assembly spanning both the IM and OM and the periplasm. The IM component AcrB captures drugs on the periplasmic side of the IM and, with the help of the periplasmic membrane fusion protein AcrA, expels them across the OM through the TolC channel.^19–21 In E. coli and other enterobacteria, TolC is a versatile OM channel that enables activities of a variety of drug efflux systems.^22–24 Its inactivation is thought to disrupt efflux across the OM without affecting the transport activities across the IM.

Previous studies demonstrated a functional interaction between AcrAB-TolC, MdfA, and EmrE and suggested that the efflux of antibiotics across the OM by AcrAB-TolC alone is not sufficient for the protection of E. coli cells from the inhibitory activities of antibiotics.^25,26 Furthermore, the coexpression of two transporters that function across either one of the IM or OM was found to increase minimal inhibitory concentrations (MICs) of antibiotics in an additive manner, whereas the effect of the coexpression of two transporters that act across both membranes on the MICs was multiplicative.²⁶ To gain a deeper insight into this system, we measured accumulation and antibacterial activity of ethidium bromide in cells with varied OM permeability and quantitatively analyzed the relationships between MdfA and AcrAB-TolC drug efflux transporters. The results offer broad insights into the functional interaction of multiple efflux transporters as well as the relationship between the accumulation of chemical compounds inside the cell and their activity.

RESULTS

A Transporter Acting Across the IM Is Synergistic with the IM not the OM Permeation Barrier.

Efflux transporters belong to two classes, those that act across the inner or the outer membrane of a Gram-negative bacterium. The original model analyzed transporters acting across the OM.¹² As the first expansion of the model, we applied it to transporters that act across the IM. For generality, we separately considered cases when transmembrane diffusion of the drug is saturable at the outer or inner membrane (Figure 1A). The former case applies to compounds that primarily penetrate the cell via transmembrane diffusion. The latter case would be expected for hydrophilic molecules that cross the outer membrane through an abundant porin but are delayed at the lipid bilayer. Kinetic schemes for the two cases are shown in eqs 1 and 2.

$$\mathit{0}\stackrel{k1\cdot (1-\phi )}{\underset{F\cdot \phi }{\rightleftarrows }}M\stackrel{F\cdot \phi }{\underset{k2\cdot (1-\phi )}{\rightleftarrows }}P\stackrel{k3}{\underset{k4+v1}{\rightleftarrows }}I$$ (1)

(2)

In these equations, O, M, P, and I are the external space, a saturable membrane, the periplasm, and the cytoplasm, respectively, k1 through k4 represent rate constants for transmembrane diffusion, F is the maximal flux across the membrane, φ is the fractional saturation of the barrier, and v1 is the rate of active efflux, which is postulated to follow the Michaelis-Menten kinetics, v1 = V1⋅I/(I + K_I) (Figure 1A).

Figure 1.

Kinetics of drug uptake. (A) Major drug fluxes across the Gram-negative cell envelope. Depending on the drug, the saturation of the inward flux could be happening predominantly at the inner or outer membrane of the envelope. Kinetic schemes beneath the figure describe the two cases for a system that also includes active transport across IM and OM. (B) Numeric solution for cytoplasmic drug accumulation for systems that are influx-saturated at OM and contain a transporter acting across either IM (B = 0) or OM (B = 0.5 and B = 2). The black line shows the steady-state drug accumulation in the absence of active efflux. (C, D) Drug accumulation in systems that contain one or two IM or OM efflux transporters, as indicated. Dashed lines show asymptotic behavior.

The differential rate equations for these kinetic schemes can be solved numerically. However, it is instructive to examine steady-state conditions for these systems. At steady state, compound concentrations in all compartments of the cell cease to change, and the following algebraic equations emerge for the barriers at the OM (eq 1) and IM (eq 2), respectively.

$$\{\begin{array}{ll}\phi =\frac{k1\cdot O}{k1\cdot O+F}\hfill & (3\text{a)}\hfill \\ P=\frac{k1\cdot O}{k2}\hfill & (3\text{b)}\hfill \\ k3\cdot P-k4\cdot I-\frac{V1\cdot I}{K_I+I}=0\hfill & (3\text{c)}\hfill \end{array}$$

$$\{\begin{array}{ll}P=\frac{k1\cdot O}{k2}\hfill & (4\text{a)}\hfill \\ \phi =\frac{k3\cdot P+k4\cdot I}{k3\cdot P+k4\cdot I+2F}\hfill & (4\text{b)}\hfill \\ F\cdot \phi -k4\cdot I\cdot (1-\phi )-\frac{V1\cdot I}{K_I+I}=0\hfill & (4\text{c)}\hfill \end{array}$$

With the proper substitutions, eqs 3 and 4 can be recast to the form I² + I⋅(a(1 + b) – cY) – aY =0. The sole positive root to this equation is given by the R-function, R(X,a,b,c), which describes drug accumulation in cells with a single transporter acting across the outer membrane.¹² The cytoplasmic drug concentrations can then be presented as

$$I=R\left(Y,K_I,K_E,1\right)$$ (3d)

and

$$I=R\left(Y,\frac{K_I}{1+B},K_E,\frac{1-B}{1+B}\right)$$ (4d)

Here, K_E and B are the efflux and barrier constants, K_I is the Michaelis constant of the transporter, and Y is the equilibrium concentration of the compound in the cytoplasm, Y = k1⋅k3⋅O/ (k2⋅k4). Figure 1B presents a graphic solution to eqs 3d and 4d. The solution to eq 4 has two branches for B >1 and B <1. In both cases, the intracellular compound concentration is reduced compared to its equilibrium level (shown as a black line in Figure 1B). The key distinction of the former case is the existence of a limiting internal concentration of the drug that cannot be exceeded by raising its concentration outside of the cell. Equation 3 yields only one branch, since B equals zero in this case. Thus, a single efflux transporter acts in synergy with the membrane barrier only in cases when the transporter acts across the membrane with the saturable diffusional flux.

Two Transporters Located in the Same Membrane.

We next analyzed systems where a compound is removed from the cell by two transporters. In general, these transporters can be located in the same membrane, OM or IM, or in two different membranes. For simplicity, we limited the analysis to compounds with the OM barrier. The three resulting cases are analyzed below. A numeric solution to the three systems is shown in Figure 1C,D.

Two OM Transporters:

(5)

The steady-state condition for this case leads to the following set of equations:

$$\{\begin{array}{ll}\phi =\frac{X+P}{X+P+2F/k2}\hfill & (6\text{a)}\hfill \\ F\cdot \frac{X-P}{X+P+2F/k2}-\frac{V1\cdot P}{K_{P1}+P}-\frac{V2\cdot P}{K_{p2}+P}=0\hfill & (6\text{b)}\hfill \\ I=\frac{k3\cdot P}{k4}\hfill & (6\text{c)}\hfill \end{array}$$

Here, we introduce a variable X = k1⋅O/k2, which is related to the external drug concentration with a factor k1/k2 defined by the Donnan equilibrium. To facilitate the comparison with the other systems, eqs 6b and 6c can be combined to yield:

$$\frac{Y-I}{Y+I+2F/k2\cdot k3/k4}-\frac{B1\cdot I}{K_{I1}+I}-\frac{B2\cdot I}{K_{I2}+I}=0$$ (6d)

where Y = k3⋅X/k4 is the equilibrium concentration of the drug in the cytoplasm, B1 = V1/F and B2 = V2/F are the barrier constants for the two transporters and K_I1 = K_P1⋅k3/k4 and K_I2 = K_P2⋅k3/k4 are their renormalized Michaelis constants. Equation 6d can be transformed to a third-order polynomial with respect to I and solved in an explicit form. Useful generalizations about the system can be derived from its asymptotic analysis, which is outlined in the next section.

Two IM Transporters:

$$\mathit{0}\stackrel{k1\cdot (1-\phi )}{\underset{F\cdot \phi }{\rightleftarrows }}M\stackrel{F\cdot \phi }{\underset{k2\cdot (1-\phi )}{\rightleftarrows }}P\stackrel{k3}{\underset{k4+v1+v2}{\rightleftarrows }}I$$ (7)

The steady-state condition in this kinetic scheme leads to two equations identical to eqs3a and 3b and the third one shown below in eq 8.

$$Y-I-\frac{V1/k4\cdot I}{K_{I1}+I}-\frac{V2/k4\cdot I}{K_{I2}+I}=0$$ (8)

IM and OM Transporters:

(9)

Under steady-state conditions, the two cellular compartments behave independent from each other, and the solution can be expressed via two R-functions:

$$\{\begin{array}{l}P=R\left(X,\frac{K_{P1}}{1+B1},K_{E1},\frac{1-B1}{1+B1}\right)\hfill \\ I=R\left(\frac{K3}{K4}p,K_{I2},K_{E2},1\right)\hfill \end{array}$$ (10)

It can be shown that the cytoplasmic drug concentration, I, can be explicitly expressed as a fourth-order polynomial of the external drug concentration, O, which has a single positive root on the defined domain. A prediction can also be made for more complicated systems with more transporters. Given the structure of eqs 6d and 8, each additional transporter would raise the power of the underlying polynomial by one. Thus, a numeric or asymptotic analysis of such systems becomes imperative.

Asymptotic Analysis of Drug Accumulation.

The asymptotic behavior of the system is illustrated in Figure 1C,D. Two limits are of particular interest, the very low and high drug concentrations. For systems with the transporters in the same membrane, this can be readily achieved through the analysis of the inverse function, Y(I), which can be derived in an analytical form (eqs 6d and 8). In both cases, a simple expression emerges for the limit of low drug concentrations, when I ≪ K_I1, K_I2:

$$I=\frac{Y}{1+K_{E1}+K_{E2}}$$ (11a)

When multiple transporters contribute to drug efflux, the equation must include efflux constants of all contributing transporters:

$$I=\frac{Y}{1+\sum K_{Ei}}$$ (11b)

When transporters are located in two different membranes, however, the efflux constants contribute in a multiplicative manner. Indeed, in the limit of low concentrations, the R-function is approximated as a straight line with the slope equal to Y/(1 + K_E).¹² Given that, eq 10 leads to the following estimate for the steady-state cytoplasmic drug concentration:

$$I=\frac{Y}{\left(1+K_{E1}\right)\left(1+K_{E2}\right)}$$ (11c)

As illustrated in Figure 1C, this results in a greater reduction of drug permeation compared to the case when two transporters with the same characteristics are located in the same membrane.

A similar analysis for the limit of high drug concentrations leads to the following set of equations. For two OM transporters,

$${\text{lim}}_{Y\to \infty }I(Y)\{\begin{array}{lll}Y\cdot \frac{1-B1-B2}{1+B1+B2}-\frac{k3}{0.5k2\cdot k4}\cdot \frac{V1+V2}{1+B1+B2},\hfill & B\text{1}\text{+}B\text{2}<\text{1}\hfill & (12\text{a})\hfill \\ \mathit{\text{root}}\left(\frac{B1\cdot I}{K_{I1}+I}+\frac{B2\cdot I}{K_{I2}+I}=1\right),\hfill & B\text{1}\text{+}B\text{2}<1\hfill & (12\text{b})\hfill \end{array}$$

where the function root() represents the positive root of the enclosed equation. Because each term in the equation is a monotonic function of I, this expression always has a single positive root when B1 + B2 > 1. The solution for two IM transporters is similar to eq 12a but has only one branch since B1 = B2 = 0 in this case:

$${\text{lim}}_{Y\to \infty }I(Y)=Y-\frac{V1+V2}{k4}$$ (12c)

The case of transporters distributed between the two membranes is somewhat more convoluted arithmetically than the two systems described above but has a familiar form:

$${\text{lim}}_{Y\to \infty }I(Y)=\{\begin{array}{lll}Y\cdot \frac{1-B1}{1+B1}-A1,\hfill & B\text{1 <1}\hfill & (12\text{d})\hfill \\ R\left(\frac{K_{I1}}{B-1},K_{I2},K_{E2},1\right),\hfill & B\text{1 >1}\hfill & (12e)\hfill \end{array}$$

where A1 is a constant that can be expressed through the kinetic parameters of the system.

For all three cases, eqs 12a–d can be readily generalized to include multiple efflux transporters. On the basis of that, several heuristics can be derived for qualitative analysis of empirical data. Specifically, multiple efflux transporters located in the same membrane cooperate in an additive manner so that the net efflux constant of the system equals the sum of individual efflux constants whereas the net barrier constant equals the sum of all the barrier constants. In contrast, transporters located in different membranes of the cell envelope interact in a multiplicative manner. Efflux transporters are also synergistic with the saturation of the transmembrane diffusion but only if the barrier and the transporters act on the same membrane.

Minimal Inhibitory Concentrations.

The above analysis establishes a quantitative relationship between external and internal drug concentrations. In principle, this should suffice to predict the MIC of the drug provided that its biochemical mechanism is known. In the simplest view, an antibiotic arrests bacterial growth when it inhibits the activity of its target to a certain, target specific degree. The intracellular drug concentration that makes it happen should be traceable to the inhibition constant of the drug, K_i. Given the K_i, the inhibitory external drug concentration would then be defined by eqs 6d, 8, and 10. In contrast to the direct equations, the expressions for MIC as a function of K_i can be written explicitly. For the sake of simplicity, K_i in the following analysis refers to the cytoplasmic drug concentration that arrests cell growth, which is related with a constant factor to its biochemical inhibition constant.

For the system with two IM transporters, the expression for MIC can be immediately derived from eq 8:

$$\text{MIC}=A\cdot \left(K_{\text{i}}+\frac{V1/k4\cdot K_{\text{i}}}{K_{I\text{1}}+K_{\text{i}}}+\frac{V2/k4\cdot K_{\text{i}}}{K_{I2}+K_{\text{i}}}\right)$$ (13)

where $A=\frac{k2\cdot k4}{k1\cdot k3}$ is the factor that describes the equilibrium between the cytoplasm and the external space. For two OM transporters, the inverse function to eq 6d yields

$$\{\begin{array}{c}\text{MIC}=A\cdot \frac{K_{\text{i}}\cdot (1+\text{EFL})+\frac{F}{0.5k2}\cdot \frac{k3}{k4}\cdot \text{EFL}}{1-\text{EFL}},\text{ where }\\ \text{EFL}=\frac{B1\cdot K_{\text{i}}}{K_{I1}+K_{\text{i}}}+\frac{B2\cdot K_i}{K_{I2}+K_{\text{i}}}\end{array}$$ (14)

In the absence of the barrier, i.e., when B1 = B2 = 0, eq 14 is reduced to the form of eq 13. The defined domain for this function is limited to EFL < 1 because the MIC must remain positive. At EFL = 1, the function displays a singularity, with MIC reaching the infinity. This condition can be achieved when B1 + B2 > 1. In such cases, certain intracellular drug levels cannot be reached by simply increasing their external concentration.

For the system with transporters in both the inner and outer membranes, the expression for MIC is similar to eq 14 but has a more complex structure, which reveals the multiplicative contribution of the two transporters to the reduction of the drug susceptibility.

$$\{\begin{array}{c}\text{MIC}=A\cdot \frac{\left(K_{\text{i}}+v2\right)\cdot (1+\text{EFL})+\frac{F}{0.5k2}\cdot \frac{k3}{k4}\cdot \text{EFL}}{1-\text{EFL}},\text{ where }\\ \text{EFL}=\frac{B1\cdot \left(K_{\text{i}}+v2\right)}{K_{I1}+K_{\text{i}}+v2},\\ v2=\frac{V2/k4\cdot K_{\text{i}}}{K_{I2}+K_{\text{i}}}\end{array}$$ (15)

A deeper insight into the relationship between key parameters of the system and the MIC is available from the analysis of a single transporter. The general case here is described by eq 14 with B2 set to zero. Further rearranging the equation and recalling that B⋅F = V and V/(0.5k2⋅K_P) = V/ (0.5K2⋅K_I)⋅K4/K3 = K_E, one can obtain

$$\text{MIC}=A\cdot K_{\text{i}}\cdot \frac{1+K_E+\raisebox{1ex}{$K_i$}\!\left/ \!\raisebox{-1ex}{$K_I$}\right.\cdot (1+B)}{1+\raisebox{1ex}{$K_i$}\!\left/ \!\raisebox{-1ex}{$K_I$}\right.\cdot (1-B)}$$ (16)

The term A⋅K_i on the right side of eq 16 represents the external drug concentration needed to inhibit bacterial growth in the absence of any efflux whereas the fraction defines the correction due to efflux. The denominator in the fraction can turn to zero at certain combinations of the system parameters defining the point of singularity. Such conditions can occur when the affinities of a drug to the target and the efflux transporter are about equal. When K_i/K_I ≪ 1, the correction due to efflux approaches 1 + K_E. Figure 2 illustrates the landscape of the correction factor as a function of K_i/K_I and B for compounds with a large efflux constant, K_E ≫ 1.

Figure 2.

Predicted minimal inhibitory concentrations for compounds with large efflux constants, K_E ≫ 1.

Contributions of MdfA and EmrE Efflux Pumps to the Intrinsic Resistance of E. coli to Lipophilic Cations.

To analyze the interplay between efflux transporters coexisting in the two-membrane cell envelope of E. coli, we constructed a series of strains lacking either MdfA or EmrE in the presence and absence of TolC (Table 1). Since in enterobacteria the OM channel TolC is required for the activities of all efflux pumps acting across the outer membrane,^23,27 ΔTolC cells lack drug efflux across the OM, whereas efflux across the IM remains active. In addition, the constructed strains were further modified to control the permeability of the OM by expressing an inducer-controlled large pore, which increases the influx of various compounds across the outer membrane.²⁸ Therefore, both modifications, the hyperporination and the deletion of tolC, are expected to increase the intracellular concentrations of lipophilic cations and hence the substrate load on MdfA and EmrE transporters, albeit by different mechanisms.

Table 1.

Strains and Plasmids Used in This Study

strains/plasmids	description	source
M6394 (WT)	wild-type GC4468 Δ(argF-lac)169 λ−IN(rrnD-rrnE)1 rpsL179(Strr)	42
M6394ΔtolC (ΔTolC)	M6394 ΔtolC	42
GKCW105 (WT-pore)	M6394 attTn7::mini-Tn7T Tpr araC ParaBAD fhuAΔC/Δ4L	28
GKCW109 (ΔTolC-pore)	M6394ΔtolC attTn7::mini-Tn7T Tpr araC ParaBAD fhuAΔC/Δ4L	28
GKSL1 (WT-pore ΔemrE)	M6394 ΔemrE attTn7::mini-Tn7T Tpr araC ParaBADfhuAΔC/Δ4L	this study
GKSL2 (WT ΔemrE)	M6394 ΔemrE attTn7::mini-Tn7T Tpr ara ParaBADMCS	this study
GKSL3 (WT-pore ΔmdfA)	M6394 ΔmdfA attTn7::mini-Tn7T Tpr araCParaBADfhuAΔC/Δ4L	this study
GKSL4 (WT ΔmdfA)	M6394 ΔmdfA attTn7::mini-Tn7T Tpr araC ParaBADMCS	this study
GKUS1 (ΔTolC-pore ΔemrE)	M6394 ΔtolC ΔemrE attTn7::mini-Tn7T TpraraC ParaBADfhuAΔC/Δ4L	this study
GKUS2 (ΔTolC ΔemrE)	M6394 ΔtolC ΔemrE attTn7::mini-Tn7T Tpr araC ParaBADMCS	this study
GKUS3 (ΔTolC-pore ΔmdfA)	M6394 ΔtolC ΔmdfA attTn7::mini-Tn7T Tpr araC ParaBADfhuAΔC/Δ4L	this study
GKUS4 (ΔTolCΔmdfA)	M6394 ΔtolC ΔmdfA attTn7::mini-Tn7T Tpr araC ParaBAD MCS	this study
pUC18T-R6K-mini-Tn7T	a suicide delivery vector	38

The inactivation of either one of the transporters in the WT background did not change the MICs of the tested compounds (Table 2). However, the IC₅₀’s modestly decreased for acriflavine and ethidium bromide (EtBr) in both mutants. This result suggests that in WT cells the intracellular concentrations of these compounds are very low and the contribution of efflux across the IM is not significant. As expected, the inactivation of TolC reduced the MICs of all four tested compounds since they are the substrates of the AcrAB-TolC efflux pump. Surprisingly, in ΔtolC cells, the inactivation of MdfA but not EmrE significantly decreased both the MICs and IC₅₀’s of ethidium bromide and acriflavine (Table 2). Hence, in these E. coli cells, only MdfA is efficient enough to affect the intracellular accumulation of lipophilic cations.

Table 2.

Antibacterial Susceptibilities of E. coli Strains Lacking MdfA and EmrE in Different Genetic Backgrounds

	nalidixic acid (μg/mL)		ethidium bromide (μg/mL)		acriflavine (μg/mL)
strains	MIC	IC50	MIC	IC50	MIC	IC50
WT	16	5.4 ± 0.96	256–512	124.0 ± 63.8	32	14.5 ± 3.4
WT ΔmdfA	16	4.27 ± 0.46	256–512	95.6 ± 27.4	32	13.0 ± 1.4
WT ΔemrE	16	3.9 ± 0.5	256–512	79.9 ± 38.1	32	9.7 ± 1.4
ΔTolC	2	0.46 ± 0.05	8	4.3 ± 0.6	4	2.3 ± 0.2
ΔTolC ΔmdfA	2	0.46 ± 0.04	2–4a	0.9 ± 0.1	1	0.62 ± 0.4
ΔTolC ΔemrE	2	0.46 ± 0.06	8	4.0 ± 1.0	4	2.0 ± 0.2
WT-pore	8–16	1.08 ± 0.06	128–256	30.3 ± 4.8	32	7.7 ± 0.7
WT-pore ΔmdfA	4–8	1.09 ± 0.16	128–256	16.8 ± 0.8	32	5.8 ± 0.3
WT-pore ΔemrE	4–8	1.06 ± 0.12	128–256	24.7 ± 0.6	32	5.5 ± 0.9
ΔTolC-pore	2	0.53 ± 0.07	8	3.4 ± 1.2	4	1.8 ± 0.4
ΔTolC-pore ΔmdfA	2	0.52 ± 0.02	2	1.2 ± 0.4	2	1.1 ± 0.3
ΔTolC-pore ΔemrE	2	0.55 ± 0.05	8	3.6 ± 1.5	4	1.8 ± 0.2

^aValues significantly different from a parental strain are shown in bold and underlined.

We next constructed hyperporinated variants of WT and efflux-deficient strains. In WT-pore cells, the MICs of all four tested compounds did not change or, as in the case of EtBr decreased only by 2-fold (Table 2). However, the IC₅₀’s of all compounds decreased by at least 4-fold, demonstrating that hyperporination increased influx of compounds across the outer membrane and reduced efflux efficiency. As expected, hyperporination of the OM in ΔTolC-pore cells affected neither the MICs nor IC₅₀’s of compounds. In contrast, the inactivation of either mdfA or emrE in WT-pore and ΔTolC-pore cells reproduced changes in cells with the intact OM. Only the inactivation of mdfA in ΔTolC-pore cells significantly decreased the MICs and IC₅₀’s of EtBr and acriflavine.

Taken together, these results suggest that only MdfA is efficient in reducing the susceptibility of E. coli to lipophilic cations in the absence of TolC-dependent efflux. The activity of neither MdfA nor EmrE is affected by the increased influx of compounds across the OM.

Inactivation of MdfA Increases the Intracellular Accumulation Level of Ethidium Bromide in E.coli Cells.

To analyze the kinetic contributions of efflux pumps in the intracellular accumulation of lipophilic cations, we next developed a quantitative intracellular accumulation assay using EtBr as a fluorescent probe (Figures 3 and 4). In agreement with the antibacterial activity measurements (Table 2), we found no differences in the intracellular accumulation levels of EtBr in cells with different genetic backgrounds and with or without emrE (Figure 3). Hence, the detailed kinetic analyses were focused on the effect of MdfA. In WT cells, the addition of increasing concentrations of EtBr led to a rapid increase in fluorescence without further changes of fluorescence in the time course of the experiment (Figure 4). The levels of accumulation of EtBr only slightly increased in WT ΔmdfA, suggesting that in the presence of the intact OM barrier the contribution of MdfA is low (Figure 3A).

Figure 3.

Accumulation of EtBr in strains with varied efflux capacities across the inner and outer membranes and varied permeation across the outer membrane. (A, B) Cells with intact efflux across the outer membrane and variable efflux across the inner membrane with the intact (A) and hyperporinated (B) outer membrane. (C, D) Cells lacking efflux across the outer membrane and varied efflux across the inner membrane with the intact (C) and hyperporinated (D) outer membrane. The time-dependent changes in fluorescence of ethidium bromide (see Figure 4) were converted into intracellular concentrations and fitted into a two-exponential function to determine the steady-state accumulation levels of the probe. Plots are the steady-state state levels of intracellular ethidium bromide as a function of the extracellular concentration of the probe. Error bars are SD (n = 3).

Figure 4.

Measured (symbols) and fitted (lines) time courses of ethidium bromide accumulation in cells with different genetic backgrounds. The measured values are averages of three independent experiments. The fitting parameters are shown in Table 3.

The kinetics of the EtBr fluorescence changes was different in ΔTolC cells. The fluorescence intensity of EtBr continuously increased over time without reaching steady-state levels in the course of the experiment (Figure 4). Hence, as expected, the inactivation of TolC-dependent efflux breached the permeability barrier of E. coli cells. As with WT cells, only a small increase in the intracellular levels of EtBr could be seen in ΔTolC ΔmdfA (Figures 3C and 4).

Hyperporination reduced the MICs of EtBr in WT-pore and ΔTolC-pore cells only moderately by 2–4-fold (Table 2) but significantly changed the kinetics of EtBr accumulation in the cells (Figure 4). The rapid partition of EtBr in WT-pore cell envelopes was followed by an increase in EtBr fluorescence, reaching the steady-state levels within 5 min. These steady-state levels were about 2.5 times higher in the WT-pore cells than in WT and further increased in WT-pore ΔmdfA (Figure 3B). Similar changes in the kinetics of EtBr accumulation were seen in ΔTolC-pore, but the steady-state levels of intracellular EtBr further increased. The highest levels of accumulation were detected in ΔTolC-pore ΔmdfA cells (Figure 3D). At an external EtBr concentration of 8 μM, these levels were at least 8-fold higher than in WT cells.

Thus, the inactivation of MdfA leads to significant changes in the intracellular accumulation of EtBr in both WT and ΔTolC backgrounds, as could be measured by both the cell growth inhibition and fluorescence-based accumulation assays.

Fitting Intracellular Accumulation of EtBr into Kinetic Models.

The model has too many parameters to determine them from fitting to a single time course. The use of multiple strains that differ only in efflux or hyperporination alleviates the problem. Many of the rate constants are conserved among these strains. However, the remaining transporters in efflux-attenuated strains were set to have their own K_m and V_max values, whereas the pore was postulated to support a maximal flux of ω⋅F and interact with the compound with its own Michaelis constant, K_d2.

To maximize the amount of utilized experimental information, we used time courses (Figure 4) rather than steady-state concentrations (Figure 3) for fitting. Diffusion across the pore and the OM bilayer was modeled explicitly as two parallel processes. Accumulation kinetics was approximated using the following set of equations.

$$\{\begin{array}{l}\frac{\text{d}\phi 1}{\text{d}t}=\frac{2F}{K_{\text{d}}}\cdot (X+P)(1-\phi 1)-2F\cdot \phi 1\hfill \\ \frac{\text{d}\phi 2}{\text{d}t}=\frac{2\omega F}{K_{\text{d}2}}\cdot (X+P)(1-\phi 2)-2\omega F\cdot \phi 2\hfill \\ \frac{\text{d}P}{\text{d}t}=F\cdot \phi 1+\omega F\cdot \phi 2-\frac{2F}{K_{\text{d}}}\cdot P\cdot (1-\phi 1)-\frac{2\omega F}{K_{\text{d}2}}\cdot P\cdot (1-\phi 2)-\frac{V1\cdot P}{K_p+P}-\frac{\text{d}I}{\text{d}t}\hfill \\ \frac{\text{d}I}{\text{d}t}=k3\cdot P-k4\cdot I-\frac{V2\cdot I}{K_{I2}+I}\hfill \end{array}$$ (17)

Here, the microscopic rate constants for diffusion across the OM and the pore are expressed via their respective Michaelis constants $k2=\frac{2F}{K_{\text{d}}}$ and $k{2}_{\text{pore }}=\frac{2\omega F}{K_{\text{d}2}}$. We then assumed that the detected increase in fluorescence of ethidium bromide upon its mixing with the cells (Figure 4) arises primarily due to its interaction with DNA (no significant fluorescence enhancement was found for the other cellular components; see Materials and Methods) and represents, therefore, its cytoplasmic concentration, I, and fit eq 17 to the experimental data.

The quality of fit was reasonably good with an R-squared value of 0.99. The best fit parameters are summarized in Table 3, whereas the fitted time courses for each strain are shown in Figure 4. The estimated 95% confidence intervals appeared reasonably tight, suggesting that the approach can yield the values of most if not all kinetic parameters of the model to within an order of magnitude or better.

Table 3.

Best-Fit Parameters for Ethidium Bromide Uptake Rates

parameter	mean	95% confidence interval
B	1.5	0.7	2.3
B2	0.05	0.009	0.109
F, μM/s	0.021	0.01	0.032
KE·KI, μM	49	4.1	93
KE2·KI, μM	18	−5	42
Kd, μM	30	7.6	53
Kd2, μM	3.1	2.6	3.6
KI, μM	37	19	55
KP, μM	0.43	0.35	0.51
k3/k4	1.25	0.9	1.6
k4, s−1	1.7	1.2	2.2
w	3.4	1.6	5.2

The analysis revealed that the barrier constant for EtBr declines from 1.5 ± 0.8 in WT cells to 0.05 ± 0.06 in tolC mutants. This result is in agreement with previous experimental findings that most, if not all, active efflux of EtBr across the OM is accomplished via TolC-dependent transport. The efflux constant for this transport was estimated as K_e = B⋅K_d/K_p = 110. For comparison, the efflux constant for trans-IM transport was much smaller, 1.3, and declined to 0.5 upon inactivation of mdfA. Thus, MdfA is responsible for 60% of EtBr efflux across the inner membrane. The Michaelis constant for MdfA, 40 μM, ensures that the transporter operated below saturation at all tested concentrations of EtBr. This value is in good agreement with the experimentally estimated equilibrium dissociation constant for EtBr and the purified MdfA protein, which was found to be ~10 μM.²⁹ For TolC-dependent efflux, the Michaelis constant was much lower, 0.4 μM. The maximal flux of ethidium bromide through the pore was 3.4-fold faster than across the intact OM. At low concentrations of the dye, however, the effect of hyperporination on transmembrane diffusion was much greater, about 34-fold, owing to the higher affinity of the molecule to the pore. Given the depth of the periplasm, δ, of 20 nm, the permeability coefficient of the cellular membrane for EtBr was calculated as Pm = δ⋅F/K_d = 1.4 × 10⁻⁹ cm/s, which increased in hyperporinated cells to 5 × 10⁻⁸ cm/s.

Relationship between the Intracellular Accumulation and Bacterial Growth Inhibition.

The set of best-fit parameters determined from the uptake studies were then used to predict the susceptibilities of the eight tested strains to ethidium bromide. Solving eq 17 for X under steady-state conditions resulted in an explicit expression that linked the expected inhibitory concentration of the drug, IC₅₀, to its effective cytoplasmic inhibitory concentration, K_i. K_i was the only additional fit parameter needed to predict the IC₅₀.

We found only a qualitative match between the model and the experiment when we adopted the parameter values best fitted to EtBr accumulation (Figure 5). Whereas the overall trend was well recapitulated by the model, numeric discrepancies could be as high as 15-fold, especially for the hyperporinated strains. This prompted us to investigate the possibility that the pore might contribute differently to accumulation of the drug and its biological activity. To this end, we treated both K_i and w as adjustable parameters and indeed found a good agreement between the model and experiment, with the largest mismatch of about 3-fold observed for the mdfA strain (Figure 5).

Figure 5.

Measured and predicted inhibitory concentrations of ethidium bromide, IC₅₀, in the permeability-variant strains. The data set “ω = 3.4” was obtained using the uptake-optimized parameters listed in Table 3 and using K_i as the sole adjustable parameter. When both K_i and m were varied, the data set “ω = 0.4” was obtained.

DISCUSSION

The creation of a quantitative model of drug accumulation and activity in Gram-negative bacteria remains a challenge in the biomedical community. The main difficulty here is the sheer complexity of this multicomponent system, which is further aggravated by the physiological response of bacteria to a chemical exposure. In this respect, it is critical to identify the key parameters that define compound penetration into bacteria and construct a minimal model that meaningfully approximates the complexity of the process. Recently, such a model was created and applied to a Gram-negative bacterium with a varied permeability of the cell envelope.¹² Here, we applied the model to E. coli strains with varied genetic backgrounds as well as the hyperporination of the OM state.

We again observed a good performance of the model. For a set of eight genetically distinct strains whose antibiotic susceptibility varied over 2 orders of magnitude, the time courses of drug accumulation were matched with the R-squared factor of 0.99 (Figure 4), whereas the MICs could be predicted to within a 3-fold accuracy given an additional assumption (Figure 5). This suggests that the model indeed quantitatively recapitulates the key aspects of drug accumulation and activity. Moreover, many of the model parameters were determined through fitting to a reasonable precision (Table 3). Therefore, the approach is suitable for para-metrization of compound permeation into bacteria and various ensuing applications in structure–activity analysis. Notably, a similar pattern of accumulation has been previously observed for radioactive erythromycin²⁸ and prospective drugs quantified via mass spectrometry.³⁰

The analysis presented in eqs 1 through 17 expands the model to bacteria with multiple efflux transporters acting across the inner or outer membrane. All such systems were predicted to display the same qualitative behavior and obey the same kinetic formalism as developed for a system with a single transporter.¹² In particular, the steady-state drug accumulation levels were predicted to follow a bifurcation pattern controlled by the value of the barrier constant. If B is greater than 1, an increase in the external drug concentration is not expected to produce a concomitant increase in intracellular drug levels, which would be limited by the Michaelis constant of the dominating efflux transporter for the given drug (Figure 1).

The analysis further provided useful guidelines on how kinetic parameters of the system cooperate in a multi-transporter system. The keys of the guidelines are as follows. First, a transporter acts in synergy only with the same membrane barrier across which it transports. For example, an IM transporter would not be synergistic with an OM barrier, and the barrier constant would equal one in this case. Second, the efflux constants of transporters located in the same membrane, be it IM or OM, should be added together (eq 11a). Third, the efflux constants of transporters acting across different membranes are multiplied (eq 11c). In this sense, the transporters located in different membranes act in synergy with each other. Fourth, the barrier constants of the transporters in the same membrane should be added together. It is the sum of all barrier constants that defines the asymptotic behavior of the system at high drug concentrations. Fifth, the barrier constants of transporters located in different membranes are not combined or multiplied. Only the transporters acting across the flux-limiting membrane matter in defining the barrier constant for the system. These heuristics reveal that the efflux and barrier constants behave in an intuitive manner and that their use could indeed simplify quantitative analysis of drug accumulation.

Despite the overall good performance of the model, the predicted fit was not perfect (Figure 4). How can the quality of the fit be improved? A number of simplifying assumptions have been made in the model to reduce the number of adjustable parameters and allow the convergence of the fit to a single solution. Some of them are difficult to assess. Among these is the expectation that all elementary fluxes in eqs 1 and 2 are not diffusion-controlled and have reached their steady state. Likewise, the cell compartments were assumed to have a uniform, averaged structure. Accounting for these factors may or may not improve the quality of the fit but will certainly increase the number of model parameters beyond the hope of practical utility.

Other factors can be addressed within the basic formalism of the model but require additional parametrization. One of the tenets of the aforementioned reaction schemes is the absence of other fluxes. This, however, is not entirely true since diffusing drug molecules could be subjected to enzymatic or nonenzymatic degradation. Indeed, drug modification is a well-recognized mechanism of antibiotic resistance. Mathematically, this channel of compound disappearance would be indistinguishable from its efflux and would only be detected in efflux-deficient strains. For EtBr, this process was not a major factor, since the B-factor (which is proportional to the maximal efflux rate, B = V/F) declined from 1.5 in the wild-type cells to virtually being undetectable at 0.05 ± 0.04 in tolC mutants (Table 3).

Another factor that is worth considering is the change in cell physiology caused by genetic manipulations. In principle, the inactivation of efflux transporters could result in other alterations to the cell envelope, necessitating appropriate adjustments to the model parameters.^31,32 We decided against such adjustments in this study to avoid overparametrization of the model. Such adjustments would be advised if independent estimates on the model parameters become available.

Notably, cell growth inhibitory concentrations, IC₅₀’s, of EtBr could not be fit to the model using the same parameters as found by fitting its intracellular accumulation. This was not, however, due to presumed deficiencies of the model or fitting algorithm. There was little correlation between experimentally determined EtBr levels and the IC₅₀’s. For example, there was a significant increase in EtBr accumulation upon hyperporination of ΔTolC-pore and ΔTolC-pore ΔmdfA cells (Figure 3), which was not accompanied by any notable decline in IC₅₀’s (Table 2 and Figure 5). A similar lack of correlation between the intracellular accumulation of chemicals and the cell susceptibility to them has been found in other studies.^33–35 Clearly, additional processes must be introduced into the model to correctly predict drug susceptibilities.

The most prominent difference between the two types of measurements is in the duration of the experiment. Uptake measurements were done in nongrowing cells over a short period of time, 15 min. The cell is unlikely to mount any significant response under such conditions. In contrast, drug susceptibility studies involve continuous incubation with an antibiotic, when physiological changes are inevitable. These changes are unlikely to affect the gross architecture of the cell, such as the Donnan equilibrium between its compartments, but can be expected to alter gene expression or lipid modifications on the membrane. We indeed found a decent match between measured and simulated IC₅₀’s when we allowed variations in the expression level of the pore (Figure 5). This indicates that a compound induces greater damage to the cell during a prolonged incubation than would be expected on the basis of its short-term permeability. In order to correctly model this phenomenon, its mechanism needs to be elucidated. With these reservations in mind, however, we find the match between theory and experiment highly encouraging and posit that the model analyzed in this study offers a solid base for the simulation of drug permeation into the cell and its activity.

CONCLUSION

In conclusion, the kinetics of drug accumulation in Gram-negative bacteria is well described by a model that postulates the saturability of transmembrane diffusion and active drug efflux. The model appears universal and applies to conditions when the working concentration of the drug is below or above its saturation levels, when saturation occurs at the inner or outer membrane and regardless of the number of efflux transporters. The kinetic behavior of the system is derived from the Michaelis-Menten kinetics but is more complex and includes a bifurcation. The kinetic behavior is defined by the values of two kinetic parameters, the efflux and barrier constants. The two constants behave in an intuitive manner in multiefflux systems, and their use can simplify a quantitative analysis of drug accumulation. Transporters acting across the same membrane are additive to each other but are synergistic with transporters in the other membrane. Active efflux is also synergistic with the limitations on transmembrane diffusion but only if the two fluxes occur across the same membrane. Accumulation of a drug is a poor predictor of its biological activity but can be if additional assumptions about the drug are made.

MATERIALS AND METHODS

Strains and Growth Conditions.

Strains and plasmids used for this study are tabulated in Table 1. Luria-Bertani (LB) broth (10 g/L Bacto Tryptone, 5 g/L yeast extract, and 5 g/L NaCl at pH 7.2) or LB agar (LB broth containing 15 g/L agar) were used for growing cells at 37 °C. Ampicillin (100 μg/ mL) was used as a selection marker, as required.

Cells were inoculated into LB media supplemented with ampicillin (100 μg/mL) for overnight growth at 37 °C with constant shaking at 200 rpm. A subculture was done by diluting the overnight culture at a 1:100 ratio into fresh LB broth. When required, at an optical density (OD₆₀₀) of ~0.3, cells were induced with 0.1% l-arabinose to induce the expression of the pore at a copy number of 140 per cell.²⁸

Construction of Mutants.

The one step inactivation method was used to create E. coli ΔemrE and ΔmdfA mutants.³⁶ This was done by amplification of a chloramphenicol resistance marker that contains flanking sequences homologous to the targeted gene(s) to be deleted. The temperature-sensitive pDK46 plasmid and PCR products were transformed into the WT and ΔTolC strains. Cells were then plated onto LB agar plates containing 7.5 μg/mL chloramphenicol and incubated overnight at 30 °C. The deletions were confirmed by PCR.

A mini-Tn7T-based protocol was used to insert fhuAΔCΔ4L to encode the pore onto the E. coli chromosome.^28,37,38 For this purpose, the suicide vector pUC18T-R6K-mini-Tn7T was used. The suicide delivery vector was electroporated along with the pTNS2 helper plasmid into corresponding E. coli strains, and the cells were grown for 1 h in LB medium containing 20 mM glucose. The cells were then plated onto LB agar containing kanamycin (20 μg/mL) and incubated overnight at 37 °C. Colonies that appeared on the plate were screened for insertion of the pore using specific primers by PCR.

Determination of Minimum Inhibitory Concentration (MIC) of Antibiotics.

We used the 2-fold broth dilution method³⁹ to detect the antibiotic susceptibility of different strains of E. coli, following an earlier protocol.²⁸ Cells were grown in LB broth to an OD₆₀₀ of ~0.3 at 37 °C with shaking at 200 rpm and induced with 0.1% l-arabinose to express the pore. These were further grown until the OD₆₀₀ reached 1.0. These cells were used to determine the MIC in a 96-well microtiter plate containing 2-fold increasing concentrations of different antibiotics. The inducer concentration of 0.1% l-arabinose was maintained in each microtiter plate. Approximately 10 000 cells were added to each well of a microtiter plate and incubated at 37 °C for 18 h. The OD₆₀₀ of these cells was measured in a Spark 10M microplate reader (Tecan) to determine the MIC and IC₅₀ values for the antibiotics studied. The MIC was determined as the drug concentration that inhibits >90% of cell growth. The IC₅₀ was determined by fitting the measured OD’s to the Hill equation.

Ethidium Bromide Accumulation Assay.

Fluorescence of free- and DNA-bound EtBr was measured for calibration using a black F-bottom nonbinding 96-well plate (Greiner Bio-One, Inc.), as described in an earlier protocol.¹² EtBr solutions at different concentrations (0.5, 1, 1.5, 2, 3, 4, 5, 6, 7, and 8 μM) were mixed with 17 μg of salmon sperm DNA (Invitrogen Inc.) in HMG buffer containing 50 mM HEPES-KOH (pH 7.0), 1 mM MgSO₄, and 0.4% glucose. Fluorescence emission of free and DNA-bound EtBr was measured for excitation at 480 nm with a 25 000 Z-value and gain 65 in the Spark 10M microplate reader (Tecan). The emission maxima for free EtBr was found to be at 620 nm and that of its DNA-bound form was at 610 nm. We observed a linear dependence of fluorescence emission intensity for the above-mentioned concentration range of EtBr. Similarly, fluorescence measurements were performed for EtBr in the presence of E. coli lipids, 27 μg (a polar fraction, Avanti Lipids Inc.), lipopolysaccharide, 1.33 μg (Sigma), and bovine serum albumin, 20 μg (Sigma).⁴⁰ However, we did not observe a substantial change in EtBr fluorescence in the presence of these biomolecules. To analyze the kinetics of the intracellular accumulation of EtBr, cells were induced and grown to an OD₆₀₀ of 1.0 as described above. These cells were harvested by centrifugation at 4000 rpm for 30 min at room temperature, and the pellets were washed with 30 mL of HMG buffer. Washed cell pellets after centrifugation were again resuspended in HMG buffer at room temperature, and the OD₆₀₀ was adjusted to 2.0. Drug uptake experiments were performed using a temperature-controlled Spark 10M microplate reader (Teca n) that had a sample injector, in fluorescence mode. The fluorescence emission of HT is collected for excitation at 355 nm and emission at 450 nm,⁴¹ whereas that of EtBr is collected at 480 nm excitation and 610 nm emission. All experiments were performed in duplicates at gain 65 and averaged over three sets of measurements. Data were plotted in Microsoft excel and fitted in Matlab using the exponential expression F = A₁ + A₂ (1 – e^−kt), as reported earlier before.¹²

ABBREVIATIONS

IM

inner membrane

OM

outer membrane

MIC

minimal inhibitory concentration

IC₅₀

inhibitory concentration that reduces cell growth by 50%

HT

Hoechst 33342

EtBr

ethidium bromide