Abstract
From previous fits of drug transport kinetics across confluent MadinDarby canine kidney II cell line overexpressing human multidrug resistance 1 cell monolayers, we found that a drug's binding constant to Pglycoprotein (Pgp) was significantly smaller than its IC_{50} when that drug was used as an inhibitor against another Pgp substrate. We tested several IC_{50} candidate functions, including the standard function, the KalvassPollack function, and the efflux ratio, to determine whether any of them yielded an IC_{50} = K_{I}, as would be expected for watersoluble enzymes. For the confluent cell monolayer, the IC_{50}/K_{I} ratio is greater than 1 for all candidate functions tested. From the mass action kinetic model, we have derived a simple approximate equation that shows how the IC_{50}/K_{I} ratio depends on the elementary rate constants from our mass action model. Thus, the IC_{50} will differ between cell lines and tissues, for the same probe substrate and inhibitor, if there are different membrane concentrations of Pgp, or the probe substrate's elementary rate constants, partition coefficient, binding constant to Pgp, passive permeability, and ability to access the other transporters (if any) in the two cell lines. The mass action model and the approximate equation for the IC_{50}/K_{I} ratio derived here can be used to estimate the elementary rate constants needed to extrapolate in vitro drugdrug interactions for compounds to the in vivo environment.
The importance of membrane transporters in the metabolism and disposition of drugs is clear (Chang and Benet, 2005; Collett et al., 2005; Endres et al., 2006; Shitara et al., 2006; Bartholomé et al., 2007; Balimane et al., 2008; Glavinas et al., 2008; Kurnik et al., 2008; Nies et al., 2008). Assessing drugdrug interaction risk is an important aspect of drug development, which is often quantified by the concentration of inhibitor required to reduce probe substrate transport by 50%, reported as the IC_{50} (Gao et al., 2001; Zong and Pollack, 2003; Rautio et al., 2006). The basic function of the IC_{50} experiment is to rank order compounds with respect to inhibition of the probe substrate transport and then to use this list along with other relevant clinical information to predict in vivo activity. The IC_{50} is usually assumed to be a fairly good estimate of the true thermodynamic dissociation constant of the inhibitor, K_{I}, to the transporter.
For a onesite enzyme that competitively binds both drug and inhibitor directly from the aqueous phase, the ratio of IC_{50}/K_{I} = 1 + K_{C}[L], where [L] is the probe substrate concentration and K_{C} is the probe substrate binding constant (Cheng and Prusoff, 1973). This equation should work for membrane transporters that bind their substrate directly from the extracellular phase, e.g., glucose permeases (Hah et al., 2002).
For transporters that bind drug from the inner monolayer of the plasma membrane, such as Pgp (Loo and Clarke, 2005; Lugo and Sharom, 2005) and MRP (Borst et al., 2006), the binding site is a permeability barrier away from where the drug is added. Drug concentration at the binding site within the apical membrane depends on the partition coefficient of the drug to the inner apical membrane. At a minimum, this suggests that passive permeability through the plasma membrane and the partition coefficient should influence the IC_{50}/K_{I} ratio. We have already shown that when the steadystate MichaelisMenten equations are used to analyze Pgpmediated transport of a substrate through the confluent cell monolayer the fitted Michaelis constant K_{m} depends on the passive permeability of the substrate through the membranes (Bentz et al., 2005).
We previously fitted the elementary rate constants for substrate binding to Pgp and efflux from Pgp for amprenavir, digoxin, loperamide, and quinidine, using a MDCKIIhMDR1 cell monolayer (Table 1) (Tran et al., 2005; Acharya et al., 2006, 2008). As expected, each of these Pgp substrates inhibited the transport of the other Pgp substrates (Acharya et al., 2006). However, we discovered that the IC_{50} values we measured experimentally, which agree with those of Rautio et al. (2006), were much larger than the K_{I} predicted using our fitted dissociation constants and partition coefficients of the inhibitors, even at very low substrate concentrations.
In this work, we explain the source of the difference between an IC_{50} and a K_{I} in the confluent cell system with experiments and our mass action model. We have derived a simple approximate equation from the mass action model for the IC_{50}/K_{I} ratio in terms of the elementary rate constants for the probe substrate and the inhibitor. Our nonlinear mass action model is more difficult to solve than the standard MichaelisMenten models (Ho et al., 2000; Bartholomé et al., 2007; Sun and Pang, 2008), but we obtain a deeper understanding of how transport through the confluent cell monolayer works, which cannot be extracted from the simpler steadystate kinetic models. Thus, in vitro to in vivo extrapolations are more reliable when launched from this nonlinear mass action model.
Materials and Methods
Compounds.
Amprenavir and GF120918 were from GlaxoSmithKline (Uxbridge, Middlesex, UK); loperamide and quinidine were from SigmaAldrich (St. Louis, MO). [^{3}H]Amprenavir (24 Ci/mmol) was customsynthesized by GE Healthcare (Little Chalfont, Buckinghamshire, UK). [^{3}H]Quinidine (20 Ci/mmol) was from American Radiolabeled Chemicals (St. Louis, MO). Dimethyl sulfoxide was obtained from SigmaAldrich. Dulbecco's modified Eagle's medium with 25 mM HEPES buffer, high glucose (4.5 g/l), lglutamine, pyridoxine hydrochloride, without sodium pyruvate, and with phenol red was from Invitrogen (Carlsbad, CA). The same medium without phenol red was used for transport experiments. Transwell 12well plates with polycarbonate inserts (0.4μM pore size and 12 mm in diameter) were obtained from Corning Life Sciences (Lowell, MA).
Experimental Methods.
Cell line and culture conditions.
The MDCKIIhMDR1 was obtained from The Netherlands Cancer Institute (Amsterdam, The Netherlands). The cells were grown in 175cm^{2} culture flasks using Dulbecco's modified Eagle's medium with 10% fetal bovine serum, 1% lglutamine, 50 units/ml penicillin, and 50 mg/ml streptomycin at 37°C in 5% CO_{2} atmosphere. Cells were split in a ratio of 1:40 twice a week at 70 to 80% confluence after at least two washes with phosphatebuffered saline and trypsinization with 0.25% trypsin/EDTA. All transport assays were performed with cells from passages 30 to 55.
Inhibition studies.
Cell monolayers were preincubated for 1 h with inhibitorcontaining transport medium in both apical and basolateral chambers. After preincubation, fresh medium with an appropriate inhibitor concentration was added to both basolateral and apical chambers along with the radiolabeled substrate on the chosen donor side. After a 4h incubation period, samples were taken from both apical and basolateral chambers and counted using a TopCount model 9912. Passive permeability of the substrate was determined in the presence of 2 μM GF120918. Lucifer yellow was added to the donor chambers in all cases to assess monolayer integrity. Other details are described in Acharya et al. (2006, 2008).
Simulations.
For all drugs tested, there is an initial increase in the passive permeability coefficients followed by a true steady state (Tran et al., 2005; Acharya et al., 2006). For the simulations in this work, these transients will be ignored and the monolayers treated as static passive permeability barriers where passive permeability coefficients are constant in time. Including these transients would only increase the IC_{50} overestimate of the K_{I}.
In permeability studies with cell monolayers, it is only possible to measure transport across the entire monolayer, which yields the passive permeability coefficients P_{BA}, basolateral to apical chamber, and P_{AB}, apical to basolateral chamber. However, for fitting the mass action kinetics, other individual membrane passive permeability coefficients are needed, including P_{AC}, apical chamber to cytosol, and P_{BC}, basolateral chamber to cytosol. Another challenge to these simulations is that often P_{BA} does not equal P_{AB} until a true steady state occurs (Tran et al., 2005; Acharya et al., 2006). The simplest mix of experiment and theory is to set P_{AC} = P_{CA} = P_{AB} and P_{BC} = P_{CB} = P_{BA} to account for this asymmetry and to capture the basic elements of the observed kinetic process (see Tran et al., 2005, for more details). We used the stiffest integrator in MATLAB, ode23s, with absolute and relative tolerances set up to 10^{−8}.
Kinetic model of transport across a confluent cell monolayer.
Figure 1 is a cartoon of a confluent cell monolayer, featuring the polarized MDCKIIhMDR1 cells, where the basolateral membrane is attached to the polycarbonate filters and Pgp (upward arrows) is expressed on the apical surface. The apical and basolateral chambers are kept separate by the tight junctions. Active transport by Pgp occurs unidirectionally, with substrate binding to a site on Pgp within the apical membrane inner monolayer and with efflux into the apical chamber (Loo and Clarke, 2005; Lugo and Sharom, 2005). For many substrates, including those used in this study, passive permeability is a significant fraction of total transport and is quantitatively analyzed separately using the Pgp inhibitor, GF120918 (Evers et al., 2000; Acharya et al., 2008).
We measure the concentration of substrate in the apical chamber, denoted C_{A}, and in the basolateral chamber, denoted C_{B}. However, the concentration of substrate in the cytosol, denoted C_{C}, and in the inner plasma membrane in contact with the Pgp binding site, denoted C_{PC}, cannot (yet) be measured rigorously in real time. These internal concentrations are variables of the mass action model and are fitted by elementary rate constants for well defined kinetic barriers, according to the measured values of C_{B} and C_{A} over time (Tran et al., 2005; Acharya et al., 2008).
We use the simplest competitive MichaelisMenten mass action reaction to model Pgp transport: where T_{0} is the empty transporter, C_{PC} is the substrate in the apical membrane inner monolayer, T_{C} is the transporter bound by substrate, and C_{A} is the substrate after efflux into the apical chamber (eq. 1). For the inhibitor, labeled Q, Q_{PC} is the inhibitor in the apical membrane inner monolayer, T_{Q} is the transporter bound by inhibitor, and Q_{A} is the inhibitor after efflux into the apical chamber (eq. 2). Loperamide and digoxin use additional transporters within the MDCKII cell line (Acharya et al., 2008). Both are used here as inhibitors and the additional transporters are included in the fittings.
Table 1 shows the median consensus values of the elementary parameters used to fit the transport kinetics of amprenavir, digoxin, loperamide, and quinidine (Acharya et al., 2006, 2008). Each parameter fitted gave good fits to all the data for up to 4 to 6 h of transport. The values in Table 1 make considerable sense for Pgp function and structure (Tran et al., 2005; Acharya et al., 2006, 2008). For the MDCKIIhMDR1 cells, all drugs had essentially the same rate constant for association to Pgp, k_{1}, and essentially the same fitted membrane concentration of efflux active Pgp in the apical membrane, which was a benchmark for the validity of our mass action model and kinetic analysis (Tran et al., 2005; Acharya et al., 2006, 2008). We use the term “efflux active” to denote those Pgps whose effluxed substrate can reach the apical chamber, e.g., Pgps near the tips of the microvilli, as opposed to those Pgps near the base of the microvilli, whose effluxed substrate is nearly always reabsorbed back into the membrane before reaching the apical chamber (Acharya et al., 2006).
In our simulations, the value for k_{1} was fixed at the value shown as it depends on lipid lateral diffusion coefficient and the size to the entry way into the Pgp binding site (Tran et al., 2005). We expect k_{1} to be roughly cell and tissueindependent because it depends mostly on lipid lateral diffusion in the inner plasma membrane, although this has not yet been proven. In contrast, the value of the membrane concentration of efflux active Pgp can be changed in the simulations, as this parameter can differ between cultured cells (Polli et al., 2001; Tang et al., 2002) and between tissues (Choo et al., 2006; Kurnik et al., 2008).
To be determined are three other significant parameters characterizing substrate and inhibitor interactions with the confluent cell monolayer and Pgp, all of which are probe substrate parameters:

k_{2}, the efflux rate constant of the substrate from Pgp into the apical chamber.

k_{r}, the dissociation rate constant of the substrate from Pgp back into the inner apical membrane. The binding constant of the substrate to Pgp from the inner monolayer of the apical membrane is defined by the ratio of the fitted rate constants, i.e., K_{C} = k_{1}/k_{r}. Because k_{1} is fixed at the consensus value (Table 1), here, k_{r} determines the binding constant.

K_{PC}, the partition coefficient of the substrate between the inner monolayer of the apical membrane and the cytosol. The product of K_{PC}K_{C} is binding constant to Pgp relative to the cytosolic concentration of substrate, so that the dissociation constant of the substrate to Pgp, relative to the cytosol, is K_{D} = 1/(K_{PC}K_{C}). When the drug is used as an inhibitor, K_{I} = K_{D} to keep identities straight.
The two other partition coefficients, K_{BO}, between the basolateral chamber and the outer basolateral membrane monolayer, and, K_{AO}, between the apical chamber and the outer apical membrane monolayer (Fig. 1), have also been estimated independently (Tran et al., 2005), and the values in Table 1 are used in the simulations.
Results
Candidate IC_{50} Functions for Estimating K_{I}.
The “standard” literature function, denoted SF, used to define an IC_{50} at some particular incubation time is shown in eq. 3 (Gao et al., 2001; Zong and Pollack, 2003; Rautio et al., 2006): with 〈Q〉 being the inhibitor concentration added to both compartments; nmol(〈Q〉), nmol(GF120918), and nmol(〈Q〉 = 0) refer to the nanomoles of substrate transported in the presence of 〈Q〉 in both chambers, in the presence of the potent Pgp inhibitor GF120918, and in the absence of inhibitor, respectively. If the data are without error, SF(〈Q〉) varies between 1 and 0. The inhibitor concentration required to reduce substrate transport by 50% is the IC_{50}. Throughout this article, the IC_{50} calculated by eq. 3 will be called the SF50, to identify clearly which candidate function is being tested. This candidate function can be used for B>A and A>B transport.
Other IC_{50} candidate functions were tested as well (see Balimane et al., 2008, for a current list of published candidate functions). Kalvass and Pollack (2007) proposed the candidate function in eq. 4 to replace the standard function of eq. 3 for A>B transport only. Eq. 4 is identical to Kalvass and Pollock's eqs. 12.1 and 12.4, but translated into our notation. We denote the IC_{50} obtained from eq. 4 as the KP50.
Another candidate function used to calculate an IC_{50} is the efflux ratio (ER) in eq. 5: ER50 denotes the IC_{50} from eq. 5. This is a very good numerical approximation to the equation currently recommended in the Food and Drug Administration Guidance on Drug Interactions for measuring the IC_{50} for Pgp (Balimane et al., 2008), and it is simpler.
We also tested the simplest plausible candidate functions, i.e., just the A>B transport versus inhibition, termed absorption inhibition (AI) and the B>A transport versus inhibition termed secretion inhibition (SI), AI50 denotes the IC_{50} from eq. 6, and SI50 denotes the IC_{50} from eq. 7.
Experimental IC_{50} Curves.
The error bars in Figs. 2 and 3 are S.D.s calculated using the equations derived in Supplemental Data Appendix A. These are the appropriate equations to use for functions with products and quotients of variables with error, i.e., the candidate functions (Taylor, 1997). In all cases, our original triplicate data for nanomoles transported had <10% S.D.
Figure 2 shows the inhibition of 3 μM quinidine A>B transport across the MDCKIIhMDR1 confluent cell monolayer by different Pgp competitive inhibitors/substrates. We chose quinidine as the probe substrate because Pgp is its only kinetically relevant transporter in the MDCKIIhMDR1 cell line, like amprenavir (Acharya et al., 2008). Figure 2A shows amprenavir as inhibitor. The fitted K_{I} for amprenavir is shown by the arrow at ∼5 μM. It is clear that the KP50 ∼10 μM from eq. 4 makes a better estimate of K_{I} than the SF50 ∼30 to 40 μM from eq. 3 or the AI50 ∼20 μM from eq. 6. The inhibition curves have the shape of the onesite competitive inhibition curve denoted HH, for HendersonHasselbalch, reasonably well.
Figure 2B shows loperamide as inhibitor. The KP50 ∼0.4 μM from eq. 4 makes a better estimate of the fitted K_{I} ≅ 0.1 μM than the SF50 ∼1 to 2 μM from eq. 3 or the AI50 ∼1 μM from eq. 6. There is also a “bounce” in the KP curve starting at a concentration approximately 1 order of magnitude below the K_{I}. This experiment has been done three times with basically the same result: the fraction transported goes down to 0.8 to 0.9 around 0.01 μM loperamide and rises to approximately 1.1 around 0.01 to 0.05 μM loperamide. For all three experiments the KP50 is in the range 0.3 to 0.5.
This bounce became less variable in magnitude when the cells were preincubated with the inhibitor for 1 h and the measurement was made 2 h later, as we have done here. Shorter preincubation times (0.5 h) or longer measurement times (4 h) typically yielded greater variability. Our simplest hypothesis involves changes in microvilli morphology, e.g., longer or shorter, which can strongly alter passive and active efflux via drug reabsorption by microvilli (Tran et al., 2005; Acharya et al., 2006, 2008).
Figure 2C shows quinidine itself as the inhibitor; i.e., excess cold quinidine was added. The KP50 ∼1 to 2 μM from eq. 4 makes a better estimate of the K_{I} ≅ 0.1 μM than the SF50 ∼4 to 5 μM from eq. 3 or the AI50 ∼3 μM from eq. 6. So, for all three cases in Fig. 2, the Kalvass and Pollock (2007) KP50 was closer to the K_{I}.
Figure 3, shows the inhibition of digoxin transport in both directions. Figure 3, A and B, shows 0.03 μM digoxin transport inhibited by quinidine in the A>B and B>A directions, respectively. The fitted K_{I} values are shown by the arrow and the dotted line represents 50% inhibition of Pgp activity. Figure 3, C and D, shows 0.1 μM digoxin transport inhibited by verapamil in the A>B and B>A directions, respectively. The ER candidate function (eq. 5), is closer to the K_{I} for quinidine, but there is not much difference between the candidate functions. The K_{I} for verapamil has not been fitted yet.
Simulations.
We have observed that the IC_{50} measured for a number of inhibitors using several commonly used candidate functions overestimate the K_{I} by 2 to 100fold, depending on the probe substrate and the inhibitor. The mass action kinetic model was analyzed to determine what causes IC_{50}/K_{I} > 1.
The first step is to determine whether simulations show the same overestimate for IC_{50}/K_{I}, which may not be expected because the model fitted the K_{I} in the first place. Figure 4 shows the simulation of the nanomoles transported over time for a 2 μM quinidinelike probe substrate in both directions as a function of concentration for a quinidinelike inhibitor whose K_{I} = 0.1 μM. Complete inhibition of Pgp is shown by the thick black line, denoted as GF120918 here, in the middle and yields the passive permeability of the probe substrate through the bilayer, because there is no transporter for quinidine in the mass action model. Below the GF120918 line is the A>B transport and above the GF120918 line is the B>A transport, as a function of inhibitor concentration (0.3 and 1 μM, respectively). The dashed lines at top and bottom show inhibitorfree transport in both directions; i.e., when 〈Q〉 = 0, and Pgp efflux is fully functioning (e.g., the B>A rate is larger than the A<B rate). According to eq. 3, the SF50 in either direction would be halfway between the dashed line (without inhibitor, 0 μM) and the black solid line (for completely inhibited Pgp, GF120918).
The simulated SF50 is shown by the large black dots at different time points and is approximately 0.7 μM, i.e., 7fold larger than the K_{I} used to calculate these simulations in the first place. The simulated nanomoles transported starting with either 0.1 or 5 μM probe substrate gave essentially the same results (data not shown), proving that the difference between SF50 and K_{I} is not due to probe substrate concentration. Likewise, because there were no transporters in this simulation, aside from Pgp, the overestimate cannot be due to another transporter, but rather is due to the IC_{50} candidate function.
The IC_{50}/K_{I} Overestimation Equation.
Because our data and the mass action kinetic model predict that the IC_{50} will overestimate the K_{I} of the inhibitor, we can use simulations to understand why. Figure 5 shows the simulated concentrations of a 2 μM quinidinelike probe substrate over time. Figure 5A shows that for B>A transport, the concentration in the basolateral chamber, C_{B}, starts at 2 μM and decreases continuously, whereas the concentration in the apical chamber, C_{A}, increases continuously to greater than 3 μM, which exceeds the initial concentration in the donor chamber because the volume of the basolateral chamber is 3fold greater than that of the apical chamber. The cytosolic concentration, C_{C}, rises slowly and converges with C_{B} at steady state. After 6 h, the system has reached its true steady state, at which the flux from the apical chamber by passive permeability into the cytosol equals the ATPdriven active transport by Pgp into the apical chamber, plus the passive permeation from the cytosol to the apical chamber.
The usual assumption of an IC_{50} experiment is that there is a steadystate period, wherein the rate of product formation is approximately constant over some period of time. For B>A transport, there is no clear steady state for C_{A} until the true steady state, which occurs sometime after 6 h (Fig. 5A). Note that the “jump” of the concentration in the cytosol, C_{C}, at the first time point in the simulation (6 min), is not a computational artifact. This is the first time point taken in our experiments (Tran et al., 2004, 2005; Acharya et al., 2006). This jump is due only to connecting a straight line between t = 0 and t = 6 min and would become smooth if we had plotted earlier time points from the simulation.
Figure 5B shows for A>B transport that there is a striking difference in the shapes of the concentration curves. The concentration in the apical chamber, C_{A}, starts at 2 μM, decreases slowly, and reaches a true steady state at approximately 1.5 μM. The concentration in the cytosol, C_{C}, jumps rapidly to approximately 0.16 μM, a quasisteady state, and then decreases slightly over time to the true steady state of approximately 0.13 μM. We use quasisteady state to denote a relatively flat concentration curve in the cytoplasm that eventually reaches the true steady state. The term cannot be defined rigorously because there is a continuum of more or less relatively flat concentration curves observed in these simulations. However, the quasisteady state in conjunction with the mass action kinetic equations can be used to provide guidance for understanding the overestimate of K_{I} by the IC_{50} values of the candidate functions.
The conditions met during the quasisteady state for A>B transport is that the cytosolic C_{C} is essentially constant over time, yielding a timeindependent IC_{50} and that C_{C} ≪ C_{A}. We use these conditions on the mass action kinetic model of Pgpmediated transport through a confluent cell monolayer to derive an approximate solution for SF50/K_{I}, which is shown in Appendix B in the Supplementary Data. This equation (eq. 8) is All terms on the righthand side, both numerator and denominator, are for the probe substrate and are defined in Table 1 using the same units. The numerator shows all of the parameters that control Pgpmediated efflux from the cells, including the membrane concentration of efflux active Pgp, denoted [Pgp]. Increasing the numerator leads to a greater overestimate. The denominator shows all of the parameters that control influx into the cells, i.e., the +GF120918 lipid bilayer/tight junction permeation, denoted P_{BC} and P_{AC}, and the transport by other transporters, denoted k_{B} and k_{A}. The bilayer thickness is denoted d, expressed in nanometers, and we use d = 4 nm for our simulations; i.e., 4/d = 1 nm^{−1}. Thus, all of these terms have the units of second^{−1}. Increasing the denominator leads to a smaller overestimate. The value of SF50/K_{I} is always greater than 1 when the approximations used in the derivation are valid. Eq. 8 is the molecular expression of the schematic diagram proposed in Litman et al. (2003).
Using simulations, we assessed the accuracy of eq. 8. Changing each parameter 10fold, up or down, produced appropriate 7 to 12fold changes in the predicted values of SF50/K_{I} (data not shown). The decrease in SF50/K_{I} as the passive permeability coefficients of the probe substrates increase is well predicted by eq. 8, R^{2} > 0.95, for both A>B and B>A transport. This result shows that the direction of transport does not matter for eq. 8, despite the differences in the shapes of the kinetic curves. Thus, the quasisteadystate condition for A>B transport was not necessary to derive eq. 8, but it was sufficient. This directional independence is not shared by all other membranespecific candidate functions for IC_{50} calculations; e.g., the Kalvass and Pollack (2007) equation in the B>A direction, appropriately transformed, has a very different IC_{50} than that for A>B transport.
Experimental validation of eq. 8 is examined in Table 2 wherein the measured SF50 values in this work are “corrected” by eq. 8 and compared with the K_{I} values shown in Table 1. The kinetically fitted K_{I} and the K_{I} estimated using eq. 8 are in good agreement, except for the case of digoxin as the probe substrate and quinidine as the inhibitor. Because the simulations showed no obvious discrepancy, it appears that there remain other unknown factor(s) in the IC_{50} experiment. Further work involving a broader database will be required to understand these factors.
What is the relationship between a K_{I}, an IC_{50}, and the fraction of Pgp bound by the substrate? Figure 6 shows a simulation of the inhibition of 0.03 μM digoxinlike A>B transport by a quinidinelike inhibitor using the standard function (eq. 3), the Kalvass and Pollack equation (eq. 4), and the decrease in the fraction of Pgp bound by the digoxinlike probe substrate relative to that bound without inhibitor. It is interesting to note that the fraction of Pgp bound by digoxin is nearly identical to the inhibition curve of the standard equation. The KP50 ∼2 μM underestimates the concentration of 50 reduction in digoxin binding to Pgp, ∼4 μM.
Compared with the experimental data for the same case (Fig. 3, A and B), the simulation shows a greater separation between KP50 and SF50 than the experimental data. For the data, KP50 ∼6 μM and SF50 ∼10 μM. The larger IC_{50} values for the experiments may be due to the bounce seen in Fig. 2, B and C, or it may suggest that digoxin and quinidine do not compete as well as the other pairs tested. The simulations assume pure competitive binding using fitted binding constants obtained with drugs alone.
Physical Mechanism for the Overestimate Equation.
The physical mechanism for the overestimate of SF50 compared with that of K_{I} is explained by the basic shape of a singlesite competitive binding curve. We can vary any of the kinetic parameters in eq. 8 to show this and have chosen to use the Pgp concentration in the membrane. Figure 7 is a simulation of the concentration of Pgp bound to the digoxinlike probe substrate as a function of concentration of a quinidinelike inhibitor. There is a 0.03 μM digoxinlike probe substrate, and the elementary parameters from Table 1 were used in the simulations. The solid line is for 200 μM as the membrane concentration of efflux active Pgp, as in Table 1. Without inhibitor, approximately 0.12 μM Pgp is bound by substrate after a 2h incubation. To reduce the substratebound Pgp by 50% required approximately 1.2 μM inhibitor. This value simulates the data in Fig. 3, A and B. On the other hand, the dotted line simulation has half the membrane concentration of effluxactive Pgp (100 μM), and the inhibitorfree concentration of digoxinbound Pgp was 0.1 μM. To reduce this value by 50% requires approximately 0.7 μM inhibitor. A onesite competitive binding curve flattens out as the fraction bound decreases, and more inhibitor is required to reduce the substratebound Pgp by 50%. In both cases, K_{I} = 0.1 μM is the same, and thus the overestimate is 7 and 12fold, respectively.
InsideOut Vesicles.
Because of the difficulty in getting the confluent cell monolayer system to yield a K_{I} simply, simulations of insideout plasma membrane vesicles (Glavinas et al., 2008) were undertaken. These vesicles have been proposed as a simpler system for fitting a K_{I} or a Michaelis constant K_{m}, depending on the experiment, because the binding site is directly exposed to the incubation medium. The drug would partition into the membrane, but there is no known permeability barrier from that membrane monolayer to the Pgp binding site, unlike the drug flipflop across the plasma membrane that is required when the drug binds to the outer basolateral monolayer.
Starting from the appropriate mass action kinetic reactions, shown in Supplemental Data Appendix C, we calculated that the small volume of the vesicles, 1 to 10 μm diameter, would allow the true steady state to be achieved within seconds (eq. D2 in Supplemental Data Appendix D). This allowed approximations that yielded the eq. 9: i.e., Pgp in insideout vesicles should behave like a soluble enzyme (Cheng and Prusoff, 1973). It is important to note that these simulations assumed that the vesicles were unilamellar and that the larger concentration of probe substrate and inhibitor within the vesicle due to Pgp had no impact on the Pgp binding site within the outside monolayer of the vesicle. These assumptions remain to be tested by experiment.
Discussion
The K_{I} and IC_{50} of an inhibitor are the two basic ways of rank ordering inhibitors with respect to their activity against an enzyme or transporter, but they measure two quite different physicochemical properties. The K_{I} = 1/(K_{QPC}K_{Q}) is the dissociation constant for the inhibitor from Pgp to the cytosol, i.e., the inverse of product of the binding constant of the inhibitor from the bilayer to Pgp, K_{Q}, and the partition coefficient of the inhibitor into the bilayer from the cytosol, K_{QPC}. This is an equilibrium thermodynamic parameter defined by the sum of two equilibrium thermodynamic free energies. The IC_{50} is the concentration of inhibitor required to reduce the transport of the probe substrate by 50% from a chosen probe substrate concentration and at a chosen incubation time. This is a profoundly kinetic parameter that depends on the K_{I} and the kinetic parameters required to model the evolution of the system. Here, we have discovered which kinetic parameters make this definition.
We previously showed that the measured IC_{50} was much greater than the fitted K_{I} for Pgp to amprenavir, loperamide, and quinidine (Acharya et al., 2006). Starting from the standard literature equation for the SF50 (eq. 3), we have derived a very simple approximate equation that defines the ratio of SF50/K_{I} (eq. 8), shown in Supplemental Data Appendix A. The SF50/K_{I} ratio increases as the parameters driving substrate efflux increase, e.g., membrane concentration of efflux active Pgp, substrate binding constant, substrate partition coefficient, and substrate efflux rate constant. Increases in these parameters reduce the cytosolic concentration of the substrate and the amount of substrate bound Pgp and increase the concentration of inhibitor needed to reduce the substratebound Pgp by 50% (Fig. 7). Eq. 8 also predicts that the SF50/K_{I} ratio decreases as the passive permeability increases, either by bilayer permeation or the presence of other transporters. It is simply the math of a onesite binding reaction with inhibition by competitive binding happening within a small volume.
If it were the case that the MDCKIIhMDR1 cell line had another active transporter in the apical membrane that shared the substrate range of Pgp and was inhibited by GF120918, then the efflux active surface density of Pgp shown in Table 1 would be an average of the efflux active surface densities of Pgp and this other transporter, weighted by their respective kinetic parameters. Although loperamide does use another transporter in the basolateral membrane of this cell line (Table 1) (Acharya et al., 2008), our fits show that it is bidirectional, i.e., not an active transporter like MRP2 and breast cancer resistance protein. It appears that the MDCKII cell line shows no functional expression of MRP2 or breast cancer resistance protein (Lalloo et al., 2004; Wang et al., 2007; Weiss et al., 2007; Solazzo et al., 2009).
Equation 8 can be used to correct the SF50 and yield a good estimate for the K_{I} in most cases (Table 2). The exception was digoxin inhibited by quinidine, for which the correction reduced the overestimate from approximately 90fold to approximately 20fold, which is still a large overestimate. Taub et al. (2005) found several Pgp substrates that did not inhibit other Pgp substrates very well and noted that Pgp can bind more than one substrate (Litman et al., 1997; Shapiro and Ling, 1997; Allers et al., 2009). One hypothesis is that digoxin binds predominantly to one site and quinidine binds predominantly to the other site, yielding weaker inhibition compared with that of drugs that bind predominantly to the same site. If so, then the two sites have cooperativity. We are unaware of any study showing that both substrates are transported from the same Pgp simultaneously or synchronously.
A second hypothesis starts with the finding that digoxin transport across the confluent MDCKIIhMDR1 cell monolayer requires one or more transporters other than Pgp, in both the apical and basolateral membranes (Table 1) (Acharya et al., 2008). Our analysis of quinidine transport showed that only Pgp is required for its transport (Acharya et al., 2008). However, this finding does not mean that quinidine cannot interact with these other transporters, it is just that the transporters do not provide a kinetically significant transport pathway for quinidine, whose bilayer permeability is much larger than that of digoxin (Table 1). This hypothesis is consistent with the idea that there are two binding sites for these drugs, but one is on Pgp and the other is on the other digoxin transporter.
Using standard steadystate MichaelisMenten equations, Kalvass and Pollack (2007) predicted that the standard equation (eq. 3), would overestimate the K_{I}. Our experiments showed that the KP50 and the ER50 values were usually closer to the K_{I} than the IC_{50} values for the other candidate functions. However, none of the IC_{50} values of the candidate functions estimated the K_{I} very well (Figs. 2 and 3). It is interesting to note that when the analysis given to the SF candidate function (eq. 3) to derive the overestimation equation (eq. 8) in Supplemental Data Appendix A was performed on the KP candidate function, the result was KP50/K_{I} = 1 (data not shown), even though Figs. 2 and 3 show that was not the case.
The correction factor given by eq. 8 has no effect on the rank order of Pgp inhibitors when a single probe substrate with the same in vitro cell line was used, because it would be the same correction for all inhibitors. Changing the probe substrate with the same in vitro cell line would change the IC_{50} values but would not affect the rank order.
What is not clear is whether inhibitor rank order would be maintained from one in vitro cell line to another in vitro cell line or when data were extrapolated in vivo. Recall for the Pgp substrate used as a competitive inhibitor that K_{I} = 1/(K_{QPC}K_{Q}). If the binding constant of the inhibitor, K_{Q}, were about the same between two cell types, which is not known but is a reasonable speculation insofar as the acyl chains in plasma membranes are similar, then their rank ordering would depend largely on their relative partition coefficients to the inner apical membrane, K_{QPC}. Whichever drug had the larger partition coefficient would have the smaller K_{I} and possibly a smaller IC_{50}. We found that partition coefficients had a lot of sensitivity between the drugs (Table 1) (Tran et al., 2005). In a different cell line or in vivo, it would be very hard to predict whether the partition coefficients would remain in the same rank order. More measurements of the partition coefficients as a function of liposome composition would clarify this part of the extrapolation problem.
Equation 8 has an immediate application to the in vivo experiments. Choo et al. (2006) found that the tariquidar dose in mice needed to increase the tissue penetration of [^{11}C]Ndesmethylloperamide to 50% of complete inhibition of Pgp was much higher for the brain than for the testes. They speculated on reasons to explain this, including the possibility that there was a higher Pgp surface density in the bloodbrain barrier, following the analysis in Litman et al. (2003). Kurnik et al. (2008) found in humans that tariquidar could fully inhibit Pgp in lymphocytes but not in the bloodbrain barrier.
Equation 8 gives us a testable hypothesis to explain these findings. If the ratio of membrane concentration of efflux active Pgp to probe substrate passive permeability coefficients across the apical membrane is greater in the brain than in the testes and the lymphocytes, which seems reasonable, then this equation can explain these observations and allow us to estimate other essential kinetic parameters. If the ratios are opposite, then a very different mechanism than commonly believed must dominate the kinetics of Pgpmediated transport in these organs.
Equation 8 can also clarify another in vivo issue. Kannan et al. (2009) used an analysis in Kalvass and Pollack (2007) to speculate that the difference between the KP50, identified by them as the K_{I} of the inhibitor, and the SF50, implied that more than 80% of the Pgp in the bloodbrain barrier must be bound to achieve 50% inhibition of transport into the brain. They proposed that there could be “spare transporters” in the bloodbrain barrier to account for this effect. We see from eq. 8 that the overestimate cited can be explained by our model, without invoking spare transporters. The Kalvass and Pollack (2007) candidate function does not equal the K_{I} for the drugs we have tested, and the SF candidate function predicts the fraction of substrate bound Pgp quite well (Fig. 6).
In summary, we have found that all of the IC_{50} candidate functions tested in this work overestimate the K_{I}. The SF50/K_{I} overestimate depends on the membrane concentration of efflux active Pgp in the apical membrane and the probe substrate kinetic parameters (eq. 8). This equation can be used to refine the estimate for the K_{I} (Table 2). Thus, our kinetic model yields a simple and accurate equation that can serve as a tool for in vitroin vivo extrapolation.
Footnotes

↵^{} The online version of this article (available at http://dmd.aspetjournals.org) contains supplemental material.

Article, publication date, and citation information can be found at http://dmd.aspetjournals.org.
doi:10.1124/dmd.109.029843.

 K_{I}
 K_{D} of a Pgp substrate when it is used as an inhibitor
 MDCKIIhMDR1
 MadinDarby canine kidney II cell line overexpressing human multidrug resistance 1
 K_{C}
 the substrate binding constant to Pgp relative to the lipid bilayer
 Pgp
 the Pglycoprotein product of the hMDR1 gene
 MRP
 multidrug resistanceassociated protein
 GF120918
 N(4[2(1,2,3,4tetrahydro6,7dimethoxy2isoquinolinyl)ethyl]phenyl)9,10dihydro5methoxy9oxo4acridine carboxamide
 A>B (or B>A)
 transport across the confluent cell monolayer when the donor chamber is apical (or basolateral) and the receiver chamber is basolateral (or apical), respectively
 K_{D} = 1/(K_{C}∗K_{PC})
 substrate dissociation constant from Pgp relative to the aqueous concentration in the cytosol, where K_{PC} is the substrate’
 Received August 20, 2009.
 Accepted November 2, 2009.
 Copyright © 2010 by The American Society for Pharmacology and Experimental Therapeutics