Novel method for kinetic analysis applied to transport by the uniporter OCT2

Abstract

The kinetics of solute transport shed light on the roles these processes play in cellular physiology, and the absolute values of the kinetic parameters that quantitatively describe transport are increasingly used to model its impact on drug clearance. However, accurate assessment of transport kinetics is challenging. Although most carrier-mediated transport is adequately described by the Michaelis–Menten equation, its use presupposes that the rates of uptake used in the analysis of maximal rates of transport (J_max) and half-saturation constants (K_t) reflect true unidirectional rates of influx from known concentrations of substrate. Most experimental protocols estimate the initial rate of transport from net substrate accumulation determined at a single time point (typically between 0.5 and 5 min) and assume it reflects unidirectional influx. However, this approach generally results in systematic underestimates of J_max and overestimates of K_t; the former primarily due to the unaccounted impact of efflux of accumulated substrate, and the latter due to the influence of unstirred water layers. Here, we describe the bases of these time-dependent effects and introduce a computational model that analyzes the time course of net substrate uptake at several concentrations to calculate J_max and K_t for unidirectional influx, taking into account the influence of unstirred water layers and mediated efflux. This method was then applied to calculate the kinetics of transport of 1-methyl-4-phenylpryridinium and metformin by renal organic cation transporter 2 as expressed in cultured Chinese hamster ovary cells.

NEW & NOTEWORTHY Here, we describe a mathematical model that uses the time course of net substrate uptake into cells from several increasing concentrations to calculate unique kinetic parameters [maximal rates of transport (J_max) and half-saturation constants (K_t)] of the process. The method is the first to take into consideration the common complicating factors of unstirred layers and carrier-mediated efflux in the experimental determination of transport kinetics.

INTRODUCTION

The impact of carrier-mediated transmembrane transport on physiology and pharmacology is widely recognized (1, 2), and the kinetics of a transport process, i.e., the relationship between the concentration of its substrate(s) and the rate of transport into or out of a cell, provides insight into both the underlying mechanism of the process (3, 4) and the influence it may have on cellular metabolism and/or disease processes (e.g., see Ref. 5). Furthermore, many transporters directly impact the pharmacokinetics (PK) of drug elimination in humans (5). Consequently, understanding the concentration rate profile of these processes is central to efforts to use physiologically based PK (PBPK) models to describe drug clearance and predict potentially adverse drug-drug interactions and cellular toxicity in humans (6, 7). Cultured cell lines that heterologously express drug transporters are widely used for in vitro assessment of transport, frequently with the intent of deriving kinetic values for use in predicting drug clearance rates and/or unwanted drug-drug interactions in humans (5, 7–9). But while these experimental systems and the methods used for their study have provided extraordinary insights into topics of physiological, pharmacological, and clinical importance, their use can introduce factors that complicate experimental determination of accurate kinetic values (10).

The kinetics of most transport processes are adequately described by the Michaelis–Menten equation. Consequently, rates of transport generally reflect two kinetic constants: the maximal rate of substrate transport (V_max or J_max) and the substrate concentration that produces half-maximal transport (the Michaelis constant, K_m or K_t). Although determining the kinetics of transport simply requires measuring the 1) initial rate of transport as a function of 2) substrate concentration at the membrane, obtaining accurate values for these quantities is challenging. Systematic errors in those values that are often introduced when applying standard experimental protocols result in the overestimation of K_t values and underestimation of J_max values. The sources of these errors are largely technical and typically unacknowledged. A complicating factor for the measurement of K_t is the presence of unstirred water layers (UWLs); we (11) and others (12–14) have reported on this issue. The systematic underestimation of J_max reflects the small size of cells, the relatively high rates of transport into overexpressing model cell systems, and technical challenges associated with the accurate measurement of rapid substrate accumulation. Combined with the complications introduced by UWLs, these issues conspire to complicate determination of true initial rates of transport. Furthermore, the ratio of J_max to K_t is often used to estimate the intrinsic clearance (CL_int) of drugs, a parameter critical for mechanistic PBPK modeling (6), so the systematic errors inherent in the assessment of kinetic parameters generally lead to underestimates of true CL_int values.

There is increasing awareness of the complexities associated with empirical measurement of transport kinetics in cultured cells (10). Calculation of J_max and K_t generally uses estimates of the initial, unidirectional rate of uptake determined from the net accumulation of substrate into transporter-expressing cells at a single time point, typically between 30 s and 5 min, at each of several concentrations of substrate. The transporter-specific component of total substrate accumulation (i.e., the fraction that does not reflect passive diffusion and/or binding or inadequate rinsing of extracellular substrate) is usually determined by subtracting the amount of substrate measured in control cells that do not express the target transporter. Limitations of this approach, sometimes referred to as the “two-step” method, have been noted (10), including its failure to adequately account for the bidirectional nature of passive diffusion or the influence of active export pathways on estimates of the initial rate of substrate influx (15). To overcome these limitations, several analytic models have been proposed, each espousing a mechanistic approach to kinetic analysis based not on uptakes measured at a single time point but instead on the time course of uptake determined at several concentrations (10, 15–18) (see also Ref. 19).

The mechanistic models noted earlier largely focused on transporters and substrates that are particularly prone to the influence of passive diffusion [e.g., organic anion-transporting polypeptide-mediated transport of statins (15, 18)] and active efflux mediated by ATP-binding cassette transporter. These models did not, however, consider the impact on calculated K_t and J_max values of either UWLs or backflux mediated by the target transporter itself. UWLs are ubiquitous and are known to introduce a bias to experimentally determined Michaelis constants (11, 13, 20, 21), and all members of the solute carrier family of transporters (SLCs/SLCOs) are capable of supporting mediated bidirectional fluxes of their substrates (22). We have previously described an empirical method for estimating initial rates of substrate uptake into cells expressing organic cation transporter 2 (OCT2) to calculate the kinetics of transport (23). However, the method was not mechanistically based and instead involved extrapolation of measured uptakes to time 0. In the present study, we describe a robust, time course-based mechanistic model for assessment of the kinetics of SLC/SLCO-mediated substrate influx into cultured cells that takes into account the influence of UWLs and mediated substrate efflux (including the impact of intracellular partitioning of substrate) on calculated values for J_max and K_t. Novel features of the model are that it takes into account the effects of both the time-dependent development of UWLs and time-dependent rise in cytoplasmic substrate activity (and the associated increase in transporter-mediated backflux) on the time course of substrate uptake into cells. In other words, it does not make the routine assumption that the system is at a steady state (12, 21). The model was used to calculate the kinetics of transport of two prototypic substrates of human OCT2: 1-methyl-4-phenylpyridinium (MPP) and the type II diabetes drug metformin. The results confirmed the tendency of conventional analytic methods to underestimate J_max values and overestimate K_t values, resulting in ratios of these parameters that frequently underestimate CL_int by greater than threefold. The results produced by this mechanistic model were compared with those obtained using the simple empirical method described earlier (23) and validated its use for increasing the accuracy of calculated transport kinetic parameters.

MATERIALS AND METHODS

Theoretical Model

The effects of a UWL on the uptake of solutes by a monolayer of cells from an adjacent solution were analyzed using a theoretical model. The solute concentration in the solution is represented by the function C_s(x,t), where x is distance from the cell layer and t is time elapsed since the start of cellular exposure to the solute. According to Fick’s second law of diffusion, the concentration in the solution satisfies

$$\frac{\partial C_{\text{s}}}{\partial t}=\frac{\partial }{\partial x}\left(D(x)\frac{\partial C_{\text{s}}}{\partial x}\right)$$ (1)

This equation is assumed to apply within an unstirred layer of thickness L_US adjacent to the cells, where D = D_US is the molecular diffusivity of the solute and effects of convective transport due to motion of the solvent are assumed to be negligible, as shown in Fig. 1. Beyond this distance, the solution is assumed to be well stirred, with a uniform solute concentration. This condition is represented in the model by assuming a very large effective value of the diffusivity D = D_S, such that the concentration gradient is virtually zero outside the unstirred layer. Therefore,

$$D\left(x\right)=\{\begin{array}{c}{D}_{\text{US}}\text{\hspace{0.17em}if\hspace{0.17em}}0\le x\le L_{\text{US}}\\ D_{\text{S}}\text{\hspace{0.17em}if\hspace{0.17em}}{L}_{\text{US}}\le x\le L\end{array}$$

Figure 1.

A: diagrammatic representation of the dimensions of the extracellular fluid compartment, with its adjacent unstirred and well-stirred regions, and, separated by the cell membrane, the cellular compartment. B: diagrammatic representation of “virtual cell” length that accounts for the partitioning of intracellular substrate required to match calculated mediated efflux (see text for discussion).

The diffusive solute flux must be zero at the upper surface of the solution, i.e., $D\partial C_{\text{s}}/\partial x=0$ at x = L. At the cell surface, the diffusive flux must equal the net rate of solute uptake J_net per surface area, i.e., $D\partial C_{\text{s}}/\partial x=J_{\text{net}}$ at x = 0. Within the cell, the free solute concentration C_c(t) increases in proportion to the rate of uptake, according to

$$\frac{\text{d}{C}_{\text{c}}}{\text{d}t}=\frac{J_{\text{net}}}{L_{\text{cell}}^{\text{eff}}}$$ (2)

If the cell acted as a single well-mixed compartment, without any uptake or binding of solute, the length $L_{\text{cell}}^{\text{eff}}$ would be equal to the average cell layer thickness. However, solute binding or sequestration can lead to a total intracellular concentration that is higher than the free concentration, and this is represented in the model by using an increased value of $L_{\text{cell}}^{\text{eff}}$.

The transport of solute across the cell membrane is assumed to be bidirectional, following Michaelis–Menten kinetics. In addition, a linear uptake term is included to represent nonspecific accumulation of substrate by the cell layer. As discussed in results, for the conditions relevant to the present study, this term largely reflects solute bound or trapped in regions between (or under) the cells. The net uptake rate is then given by

$$J_{\text{net}}=\frac{J_{\text{max}}^{\text{in}}{C}_{\text{s}}}{K_{\text{t}}^{\text{in}}+C_{\text{s}}}-\frac{J_{\text{max}}^{\text{out}}{C}_{\text{c}}}{K_{\text{t}}^{\text{out}}+C_{\text{c}}}+K_{\text{d}}{C}_{\text{s}}$$

where $J_{\text{max}}^{\text{in}}$ and $J_{\text{max}}^{\text{out}}$ are the maximal rates of substrate uptake and release, $K_{\text{t}}^{\text{in}}$ and $K_{\text{t}}^{\text{out}}$ are the corresponding Michaelis constants, and $K_{\text{d}}$ is the rate constant for the linear accumulation of substrate in the cell layer (reflecting the combined influence of diffusion, nonspecific binding to cell and/or plastic surfaces of the well, and incomplete rinsing of labeled substrate in the extracellular medium).

The system of Eqs. 1 and 2 consists of an ordinary differential equation for C_c(t) coupled to a partial differential equation for C_s(x,t). This system is solved numerically using the function pdepe in MATLAB (MathWorks, Natick, MA; the two resulting MatLab scripts, plus an instruction document and sample data input Excel spreadsheet, are available as Supplemental Material at https://github.com/secomb/TransportKinetics or https://doi.org/10.5281/zenodo.6621201). To allow use of this solver, ordinary differential equation (Eq. 2) for C_c(t) is replaced by a partial differential equation of the form of Eq. 1 for C_c(x,t) on the domain 0 ≤ x ≤ L, as shown schematically in Fig. 1B. The diffusivity D in this “virtual cell” domain 0 ≤ x ≤ L is set to a high value so that the concentration is uniform, representing the intracellular concentration. The boundary conditions are then $D\partial C_{\text{c}}/\partial x=0$ at x = L and $D\partial C_{\text{c}}/\partial x=f{J}_{\text{net}}$ at x = 0, where f =$L/L_{\text{cell}}^{\text{eff}}$. The factor f corrects for the larger width of this virtual domain relative to the effective intracellular domain width.

For given values of the transport and geometric parameters defined earlier, and for a given initial solute concentration in the solution that is applied to the cell layer at time t = 0, the differential equations are solved over a given time interval to yield the solute concentrations C_s(x,t) and C_c(t) in the solution and in the cells. Typically, the assumed uptake rate is such that the concentration in the bulk solution remains almost constant during the simulation period. Quantities of interest, such as the total accumulation of solute in the cells and the concentration in the solution at that cell surface, are predicted as functions of time (using the MATLAB script, tcMMsolver).

In situations of practical interest, it is often the case that net cellular solute accumulation is measured, and the aim is to deduce the values of transport parameters that describe the measured cellular uptake. In this case, the unknown parameters are varied to minimize the sum of squared deviations between model predictions and measured data on solute uptake by the cells. For this optimization procedure, the MATLAB function lsqnonlin was used. The number of fitted parameters was typically four, for example, $L_{\text{cell}}^{\text{eff}}$, $J_{\text{max}}^{\text{in}}$, $K_{\text{t}}^{\text{in}}$, and $K_{\text{d}}$. In preliminary simulations (using the MATLAB script, tcMMfitter), it was found that the results for time periods of up to 120 s were insensitive to the assumed thickness of the unstirred layer (see results for discussion of this observation), and so a fixed thickness L_US = 0.10 cm was typically assumed for the kinetic analyses. The rationale for the values used to describe the kinetics of substrate release from cells ($J_{\text{max}}^{\text{out}}$ and $K_{\text{t}}^{\text{out}}$) is presented in results.

In the presentation of simulated time courses of net substrate accumulation, unless otherwise noted, the following parameter values were assumed:

L = total depth of fluid = 0.7 cm

L_US = unstirred layer thickness = 0.10 cm

$L_{\text{cell}}^{\text{eff}}$ = the effective thickness of cell layer including the effect of solute compartmentalization [referred to in the text as the cell partition factor (CPF) = 0.0003 cm (no partitioning) or, for example, 0.0012 cm (reflecting ∼75% partitioning of intracellular substrate)]

$J_{\text{max}}^{\text{in}}$ = maximal unidirectional uptake rate = 8.33 × 10⁻⁴ nmol·cm⁻²·s⁻¹ (i.e., 50 pmol·cm⁻²·min⁻¹)

$K_{\text{t}}^{\text{in}}$ = Michaelis constant for substrate uptake = 5.0 µM

$J_{\text{max}}^{\text{out}}$ = maximal unidirectional efflux rate = 8.33 × 10⁻⁴ nmol·cm⁻²·s⁻¹

$K_{\text{t}}^{\text{out}}$ = Michaelis constant for substrate release = 25 µM (i.e., 5 × $K_{\text{t}}^{\text{in}}$)

K_d = first-order substrate accumulation constant = 3.4 × 10⁻⁷ cm·s⁻¹ (∼0.02 µL·cm⁻²·min⁻¹)

D_US = substrate diffusivity in the unstirred layer = 6 × 10⁻⁶ cm²·s⁻¹

D_S = artificial high substrate diffusivity in the stirred layer (representing effect of stirring) = 0.1 cm⁻²·s⁻¹.

Chemicals

[³H]MPP (80 Ci/mmol) was purchased from Perkin-Elmer (Waltham, MA). Nonradioactively labeled MPP was synthesized by the Department of Chemistry and Biochemistry, University of Arizona (purity > 99.5%). [¹⁴C]metformin (90 mCi/mmol) was purchased from Moravek Biochemicals (Brea, CA). Unlabeled metformin was purchased from AK Scientific (Union City, CA). Other reagents were of analytic grade and were commercially obtained.

Cell Culture

For the purpose of testing the model described here, we analyzed data collected for, and reported in, a previous study (23). For that study, Chinese hamster ovary (CHO) cells containing the Flp-In target site were purchased from Invitrogen (Carlsbad, CA) and stably transfected with OCT2 at a single Flp-In recombinase site (24). We recently showed that ∼20% of total cell OCT2 protein is expressed in the plasma membrane of these cells, with the remaining 80% diffusely expressed within the cytoplasm (25); by quantitative Western blot analysis, this represents ∼150 fmol of OCT2 protein per cm². Cells were cultured at 37°C with 5% CO₂. Culture medium consisted of Ham's F-12 Nutrient Mixture with 10% FBS (Fisher Scientific, Pittsburg, PA). Hygromyocin (100 µg/mL) and zeocin (100 µg/mL) (both purchased from Invitrogen) were added to the culture medium of CHO-OCT2 cells and wild-type (non-OCT2 expressing) CHO cells, respectively, to maintain their respective protein expression profiles. Cells were passaged every 2–4 days. When seeded into 96-well plates (Greiner, VWR, Arlington Heights, IL) for transport assays, cells were grown to confluence in antibiotic-free media.

Transport Assays

Wild-type CHO cells and cells expressing human OCT2 were typically plated at densities sufficient to reach confluence within 24 h (50,000 cells in 96-well cell culture plates), at which time they were used in transport experiments. Transport was measured using the methods described in Ref. 23. Briefly, to measure time courses of net substrate accumulation, plates containing culture media were placed in an automatic fluid aspirator/dispenser (model 406, BioTek, Winooski, VT) and automatically rinsed/aspirated three times with 300 µL of Waymouth’s buffer (WB; 135 mM NaCl, 13 mM HEPES-NaOH, 28 mM d-glucose, 5 mM KCl, 1.2 mM MgCl₂, 2.5 mM CaCl₂, and 0.8 mM MgSO₄; pH 7.4) at room temperature. Transport buffer (typically 60 μL) containing substrate and any additional required test agents was then introduced into each well at timed intervals using a VIAFLO 96-well multichannel pipet (Integra Biosciences, Hudson, NH). Following the experimental incubation, the transport reaction was stopped by the rapid addition (and simultaneous aspiration) of ∼750 μL of cold (4°C) WB. To assess the accumulation of radiolabeled substrates (MPP or metformin), following final aspiration of the cold stop, 200 μL of scintillation cocktail (Microscint 20, Perkin-Elmer) were added to each well. Plates were then sealed (Topseal-A, Perkin-Elmer) and allowed to sit for at least 2 h before radioactivity was determined with a 12-channel, multiwell scintillation counter (Wallac Trilux 1450 Microbeta, Perkin-Elmer). Accumulation of substrates is expressed as moles per centimeter squared of nominal cell surface. For the purpose of comparison with literature values that express rates of transport per milligram of cell protein, we used the conversion factor of 0.040 mg·protein·cm⁻². For each concentration of labeled substrate, extrapolation of the time course of accumulation by both OCT2-expressing and wild-type CHO cells met at a common zero-time intercept and this was taken as “plate background,” and that value was subtracted from each time point of that time course (23).

RESULTS

Simulated Effect of UWLs and Mediated Backflux on the Time Course of Substrate Concentration at the Membrane

Figure 2A shows the concentration of substrate in the bulk solution (in this case, 0.1 µM, depicted by the black dashed line) and at the membrane (solid lines) during a 120-s exposure to OCT2-expressing cells, as determined using tcMMsolver. In this simulation, the rate of substrate uptake into the cells was calculated using a J_max of 50 pmol·cm⁻²·min⁻¹ and a K_t of 5 µM [the former reflects the high end and the latter the average of values we recently reported for MPP transport in a CHO cell line that stably expresses OCT2 (23, 26)]. In addition, all accumulated substrate was assumed to remain within the cell; in other words, there was no efflux of accumulated substrate. If mixing of the medium were perfect (i.e., if the UWL were absent), then the substrate concentration at the membrane (depicted by the solid black line, Fig. 2A) would remain effectively equal to that in the bulk medium (dashed black line, Fig. 2A), despite removal of substrate from the medium into the cells. However, UWLs are invariably present above static surfaces (13). The solid red line in Fig. 2A shows the predicted time course of substrate concentration at the membrane in the presence of a UWL of 150 µm. Whereas the concentration at time 0 was 0.1 µM (equal to that in the well-mixed bulk medium), transport activity rapidly reduced the concentration at the membrane, dropping it to a near steady-state level of ∼0.07 µM within 20–30 s. At a steady state (i.e., the value calculated by allowing the simulation to run to a time point of ∼3,000 s), the concentration at the membrane was reduced to a value that was maintained by diffusion from the bulk medium that, in this case, was 150 µm away. When the UWL was increased to 300 µm, it took ∼90 s for the concentration at the membrane to be reduced to a near steady-state value of ∼0.06 µM, reflecting the increased length of time required for solute particles to diffuse 300 µm. However, increasing the depth of the UWL beyond ∼300 µm, to values of 500 µm, 1,500 µm, or more, had effectively no impact on the profile of substrate concentration depletion at the membrane over a time course of 120 s; the influence of UWLs of >300 µm in depth required extended time courses to be realized. Indeed, in the case of UWL values of >1,000 µm, the true steady-state condition that was the underlying assumption in previous analyses of UWL effects (12, 20, 21, 27, 28) required tens of minutes to be achieved. This rather counterintuitive observation is a consequence of the influence of distance on the length of time required for particle diffusion through a UWL of depth δ (29):

$$t=\frac{{\text{δ}}^2}{2D}$$ (3)

Figure 2.

Time-dependent parameters calculated using tcMMsolver. In all cases shown, the maximal rates of transport (J_max) for mediated influx into the cells was set at 50 pmol·cm⁻²·min⁻¹, with a half-saturation constant (K_t) of 5 µM; the concentration of substrate in the well-stirred bulk medium was set at 0.1 µM (dashed black line). A: influence of unstirred water layers (UWLs) of increasing depth (with units of µm) on the change in concentration of substrate at the extracellular face of the membrane due to mediated transport activity. B: effect of UWLs of increasing depth on the net accumulation of substrate in the cells. For A and B, it was assumed that no efflux of accumulated substrate occurred during the depicted time course. C: influence of mediated efflux of accumulated substrate on the concentration of substrate at the extracellular face of the membrane. The solid red and green lines show the profiles of substrate concentration when all intracellular substrate is freely available [i.e., no intracellular partitioning; cell partition factor (CPF) of 3]; the dashed red and green lines show these profiles if 75% of accumulated substrate is assumed to be unavailable for mediated efflux; CPF of 12. D: influence of mediated efflux on the net accumulation of substrate.

Over a time course of 120 s or less, substrate molecules in the test solution that are more than 350–400 µm from the cell surface at the start of the experiment cannot reach the membrane (by diffusion) to replace the molecules removed by transport. Consequently, despite the expected presence of UWLs between 1,000 and 2,000 µm in depth above cells in unstirred/unshaken culture dishes (14), rates of transport during incubations that are ≤2–3 min in duration and involve acute exposures to substrate-containing media are effectively diffusion limited.

The effect of the gradual depletion of substrate at the membrane on unidirectional rates of uptake that occurs due to the presence of a UWL is shown in Fig. 2B. The kinetic parameters (J_max and K_t) were the same as those used to generate Fig. 2A, and it is worth emphasizing that, again, accumulated substrate was not permitted to leave the cell (i.e., no efflux). The black straight line in Fig. 2B reflects the unidirectional rate of transport arising from a constant exposure to 0.1 µM substrate; in this case, that rate was 0.98 pmol·cm⁻²·min⁻¹. The colored lines in Fig. 2B show the predicted accumulation of substrate under the influence of UWLs of 150, 300, 500, or 1,500 µm. The deviation of these lines from the control slope reflects the decline in the rate of transport arising from the gradual decrease in substrate concentration at the membrane. The vertical dotted line in Fig. 2B marks the levels of accumulation occurring at 60 s. It is evident that had 60-s uptakes been used to estimate the initial rate of transport, the presence of UWLs would result in underestimating by 25–30% the true unidirectional rate produced by exposure to 0.1 µM substrate; longer time points exacerbated the problem. As discussed earlier (see Eq. 3), the presence of a 500- to 1,500-µm UWL exerted no greater effect on the apparent rate of transport than did a UWL of 300 µm.

Unlike the situation modeled in Fig. 2, A, and B, accumulated substrates are likely to exit the cell as their free cytoplasmic concentration rises. Passive (diffusive) flux is a quantitatively significant mechanism by which certain substrates can both enter and exit cells. This issue, and the impact it has on kinetic measurements, has been rigorously examined (15). The present discussion is, however, limited to OCT2-mediated transport of MPP and metformin (and similar small, hydrophilic compounds). With logP values of −2.53 and −1.8, respectively (ALOGPS 2.1), these substrates are sufficiently polar that trans-membrane diffusive fluxes are effectively zero, at least for the timeframe and extracellular concentrations of these experiments (unpublished observations). Nevertheless, as discussed later, the influence of nonspecific binding of substrates (particularly charged substrates) to cell surfaces and the exposed plastic of culture wells/dishes as well as inadequate rinsing of substrate-containing media (which can occupy residual, i.e., unrinsed, volumes between and under cells), both of which can produce a first-order, i.e., nonsaturable, component to apparent net cellular accumulation of substrate, should be considered in any kinetic analysis.

Although a transmembrane diffusive flux may not, for many substrates, be a complicating factor in the assessment of transport kinetics, OCT2, a uniporter, can itself mediate efflux as well as uptake (30, 31), and this fact can complicate interpretation of the time course of net substrate accumulation. The rate of mediated efflux is defined by the free concentration of substrate in the cytoplasm and the kinetics of its interaction with the cytoplasmic face of OCT2. Direct measurement of the kinetics of efflux is challenging (31, 32), but for the current purpose we can estimate mediated efflux based on kinetic and thermodynamic constraints that exist for uniporters (4), namely, 1) that the ratio of J_max to K_t for influx and efflux must, in the absence of a net driving force, be equal and 2) that the presence of a driving force, which for an electrogenic facilitated diffusion process like OCT2 reflects the substrate’s electrochemical gradient, will shift that equality in direct proportion to the driving force. The electrogenic nature of OCT2 (30) permits it to support passive accumulation of its positively charged substrates, with the maximum cytoplasmic-free concentration of a monovalent substrate at 25°C given by the following relationship (4):

$$\frac{{[\text{O}{\text{C}}^{+}]}_{\text{cyto}}}{{[\text{O}{\text{C}}^{+}]}_{\text{out}}}={10}^{-(\mathrm{\phi }/59)}$$ (4)

where φ is the electrical potential difference (in mV) across the CHO cell membrane. Reported values for the CHO membrane potential vary from as small as −5 to −10 mV (33, 34) to as large as −70 mV (35), but the median of the reported values we found is approximately −40 mV (36, 37). Consequently, at equilibrium, the free cytoplasmic concentration of OC⁺ that can be supported by OCT2 is approximately five times that in the extracellular solution. Therefore, to simulate OCT2-mediated efflux, the model multiplies the ratio of J_max to K_t for influx by a factor of 0.2 (thereby accounting for the development of a fivefold outwardly directed gradient of free substrate at steady state).

Carrier-mediated efflux of accumulated substrate can exert a large effect on the time course of both substrate concentration at the membrane (Fig. 2C) and net substrate accumulation by cells (Fig. 2D). Even in the absence of a UWL, and despite the substrate concentration at the membrane remaining equal to that in the bulk medium, the rate of net uptake can deviate markedly from unidirectional influx when accumulated substrate is permitted to efflux back out of cells at a rate defined by the kinetics of that process (as defined earlier) and the rise in free concentration of substrate in the cytoplasm (as influenced by mediated influx). For a uniporter like OCT2, of particular importance is the rate of rise of free substrate in the cytoplasm, which is influenced by 1) cell size/volume and 2) the extent to which cytoplasmic substrate becomes sequestered by compartmentalization or binding/partitioning to cytoplasmic constituents.

Cytoplasmic volume of cultured cells (including CHO cells) is ∼0.3 µL·cm⁻² of a confluent monolayer (32, 35, 38–44)¹, which, for a confluent monolayer, results in an average “cell height” of 3 µm·cm⁻². When that was used as the volume into which accumulated substrate is distributed (obtained by using a CPF of 3 in tcMMsolver), and all that substrate was considered to be freely available for interaction with the OCT2-mediated efflux pathway, the time course of net substrate uptake (solid red line in Fig. 2D, which assumed a UWL of 0) rapidly deviated from the true initial rate (solid black line) and approached steady state (where influx = efflux) within ∼30 s. The basis for the rapid achievement of steady-state substrate accumulation is evident in Fig. 3, which shows the calculated time course of free cytoplasmic substrate concentration arising from the combined influence of mediated uptake and efflux of substrate distributed freely in the volume of cytoplasm (in this case, 0.3 µL·cm⁻²). With a J_max of 50 pmol·cm⁻²·min⁻¹ and a K_t of 5 µM, unidirectional influx (straight black line, Fig. 3) would produce a cytoplasmic concentration equal to that outside (0.1 µM; horizontal dashed black line, Fig. 3) in <2 s and surpass the equilibrium concentration (0.5 µM; dotted black line, Fig. 3) in <10 s. This rapid rise in availability of cytoplasmic substrate fed the efflux pathway, resulting in efflux rates that rapidly approached the unidirectional rate of influx supported by 0.1 µM extracellular substrate and producing the profile of cytoplasmic substrate concentration shown by the solid red line in Fig. 3. The presence of a 500 µM UWL resulted in a transient decrease in the concentration of substrate at the membrane (Fig. 2C, solid green line), but the comparatively rapid achievement of a steady-state condition eliminated the net removal of substrate from the medium thereby permitting diffusion to raise the concentration at the membrane back to the level in the bulk medium. The result, then, of a UWL was a modest right shift in the time course of both substrate uptake and a decrease in cytoplasmic substrate concentration (solid green lines in Fig. 2D and Fig. 3, respectively).

Figure 3.

Effect of an unstirred water layer (500 µm) and intracellular partitioning of substrate [75%; cell partition factor (CPF) of 12] on the intracellular concentration of free substrate resulting from mediated uptake (kinetic parameters as in Fig. 2). The solid black line shows the profile if no accumulated substrate is allowed to leave the cell; the colored lines assume that mediated efflux can occur. See text for a discussion of the effect of membrane potential on the boundary conditions.

The simulations described earlier assumed that all accumulated substrate was freely dissolved in the cytoplasm, and that assumption is often not warranted. Substantial fractions of accumulated cationic molecules, including small polar OCT2 substrates like metformin, MPP, and tetraethylammonium (TEA), can behave as if either sequestered within membrane-bound intracellular compartments or loosely bound to or intercalated within cytoplasmic constituents (e.g., see Refs. 31, 38, and 45). There have been efforts to model the time course of sequestration and relative distribution of sequestered compounds (e.g., see Refs. 38 and 46), but the underlying assumptions are substantial. For the present purpose, we used a nonmechanistic approach that simply provided a larger cell volume [modeled as an increase in effective cell height (i.e., “cell length” $L_{\text{cell}}^{\text{eff}}$)] into which accumulated substrate was distributed, thereby reducing the rate of rise of cytoplasmic substrate concentration and thus mimicking the effect of sequestration of a fraction of this material; in the model, $L_{\text{cell}}^{\text{eff}}$ is output as CPF. Our own work with MPP suggests that ∼75% of accumulated MPP at steady state “behaves” as if it is not freely available (31), and this was modeled by assuming a nominal cell volume of 1.2 µL·cm⁻² (a CPF of 12), rather than the measured 0.3 µL·cm⁻² (CPF of 3)². Using a CPF of 12 increased the simulated net rate of OC⁺ accumulation in the absence and presence of a UWL (dashed red and green lines in Fig. 2D), although the decrease in mediated efflux that follows the delayed rise in cytoplasmic concentration of free substrate still resulted in net uptake rates that were lower than the unidirectional rate of influx (∼60% lower at 60 s; Fig. 2D). The lower rate of rise of free cytoplasmic substrate under the CPF12 condition (Fig. 3, dotted lines) also exacerbated the effect of a UWL on the profile of substrate concentration at the membrane (dashed green line in Fig. 2C) and, therefore, further reduced the rate of substrate uptake (green vs. red dashed lines in Fig. 2D).

Effect of UWLs and Mediated Backflux on the Apparent Kinetics of Substrate Transport

Figure 4 shows the kinetic profiles predicted to occur for a process with a K_t of 5 µM when J_max was increased from 3 to 30 to 300 pmol·cm⁻²·min⁻¹, with 3 reflecting the low range of reported OCT2 J_max values and 300 near the high end of reported values (26, 47). Each plot also shows the calculated kinetic profiles resulting from estimates of initial rates of transport based on net fluxes measured at 30 s, 60 s, 2 min, 10 min, 100 min, or under the UWL condition expected to occur at true steady state (the latter calculated according to Ref. 21). The UWL was presumed to be 1,500 µm, reflecting the average of values determined for unstirred/unshaken cell culture plates (14), although as noted earlier, the effective UWL over time courses of up to 2–3 min is likely <500 µm (and the simulation results were effectively the same using either of these UWL values). The bold dashed black lines in Fig. 4 show the kinetic profiles that reflected the listed J_max values and true K_t of 5 µM with no UWL and no backflux of accumulated substrate; the dotted black lines in Fig. 4 show the profiles predicted to occur at steady state, i.e., when the impact of the UWL on bias introduced to the assessment of the “apparent” K_t (K_tapp) was fully achieved. First, note that the presence of a UWL (in the absence of backflux) had virtually no effect on calculated J_max values (Fig. 4, A, C, and E). This is consistent with predictions of earlier, steady state-based models (20, 21). The influence of J_max on the impact of the UWL is evident; increasing J_max from 3 to 300 pmol·cm⁻²·min⁻¹ increased the bias added to the true K_t (the right shift in kinetic profiles) from a low of only 0.6 µM to more than 63 µM, resulting in K_tapp values that overestimated the true K_t of 5 µM by only ∼10% at the low value of J_max, to >13-fold at the high extreme. The time point used to estimate the initial rate of transport was also important. When J_max was low, the time point used to estimate the initial rate of transport had virtually no effect on assessment of K_t; the concentration at the membrane remained virtually equal to that in the bulk medium (despite the presence of the UWL) and, so, in the absence of efflux, net uptake was effectively equal to unidirectional influx. But as J_max increased, the time-dependent decrease of substrate concentration at the membrane (see Fig. 2A) exerted a marked effect on the estimate of K_tapp. When J_max was 300 pmol·cm⁻²·min⁻¹, the use of 30-s uptakes resulted in a kinetic profile that produced a K_tapp of 9.7 µM (a bias of 4.7 µM), but this increased to 11.8 or 14.7 µM when initial rates were based on 60- or 120-s uptakes, respectively. Further increases in the time point used to estimate the initial rate converged on the profile reflecting steady state.

Figure 4.

Predicted impact of unstirred water layers (UWLs) and mediated efflux on the experimentally determined kinetic profile for mediated substrate uptake. For maximal rate of transport (J_max), units are pmol cm^-2 min^-1; for half-saturation constant (K_t), units are µM. A, C, and E show the influence on the kinetic profile of increasing J_max values (from 3, to 30, to 300, respectively), but assume no efflux of substrate during the several time points allowed for substrate accumulation. B, D, and F introduce the influence of carrier-mediated efflux of accumulated substrate [using an assumed cell partition factor (CPF) value of 12 to account for modest intracellular substrate sequestration]. See text for a complete description.

The addition of mediated efflux to the conditions modeled in Fig. 4, A, C, and E, resulted in marked changes in the predicted kinetic profiles of OCT2-mediated uptake (Fig. 4, B, D, and F). In all cases, introduction of mediated efflux resulted in rates of net transport that underestimated unidirectional influx, resulting in systematic underestimation of J_max values that became progressively exaggerated as a function of both the true J_max and the time point used to estimate the initial rate of transport. The intermediate value of J_max (30 pmol·cm⁻²·min⁻¹), which is near the median value of J_max reported for OCT2-mediated transport in heterologously expressing cultured cells (23, 26, 47), was underestimated by 38% using 30-s uptakes and by 50% using 120-s uptakes, whereas K_tapp values were overestimated by 30% and 60%, respectively. The systematic and opposing inaccuracies introduced to the assessment of J_max and K_t values have a synergistic effect on estimation of CL_int, which is generally determined from the ratio of J_max and K_t (48); values based on 30-s uptakes produced a twofold underestimate of CL_int, whereas values based on 120-s uptakes produced a threefold underestimate of CL_int.

Mechanistic Modeling of the Kinetics of Transport Derived from Time Courses of Net Substrate Uptake

We developed a computational method (tcMMfitter) for determining the kinetic parameters for OCT2-mediated unidirectional influx into cultured cells that assumes that the time course of transporter-mediated net uptake reflects the combined influence of the 1) rate of unidirectional influx and 2) rate of unidirectional efflux. The former is a function of the kinetics of uptake and the concentration of substrate at the extracellular face of the membrane (and, consequently, is influenced by the UWL); the latter is a function of the kinetics of efflux and the concentration of (free) substrate at the cytoplasmic face of the membrane (and, consequently, is influenced by the rate of substrate entry, cytoplasmic volume, and the partitioning of accumulated substrate via binding and/or sequestration within intracellular compartments). As noted earlier, the kinetics of efflux have a definable relationship that is correlated with the kinetics of influx (4). Consequently, J_max and K_t for unidirectional influx can be calculated from a series of time courses of net substrate accumulation determined for a range of extracellular (i.e., bulk medium) substrate concentrations, through the simultaneous assessment of values for the UWL and CPF required to account for the time-dependent profile of net accumulation at each concentration of substrate.

Sample data.

To test the method, we first calculated the predicted time courses of net substrate uptake (using tcMMsolver) for a series of concentrations as supported by a process with a J_max for unidirectional influx of 30 pmol·cm⁻²·min⁻¹ and K_t of 5 µM (approximating the kinetics of OCT2-mediated MPP transport) plus a first-order component of nonsaturable substrate accumulation (K_d). As noted earlier, the J_max-to-K_t ratio for OCT2-mediated efflux from CHO cells can be assumed to be ∼20% of that for uptake (4). Therefore, for these calculations, we simply assumed J_max for efflux was equal to that for uptake, whereas K_t for efflux was five times larger (i.e., 25 µM). We also assumed the presence of a fixed extracellular UWL of 1,500 µm [the average of values reported for UWLs above cultured cells in multiwell plates (14)], acknowledging that the effective UWL for a time course of 3 min will be <500 µm (indeed, the calculated time courses were virtually unchanged when a value for the UWL of 500 µm was used rather than 1,500 µm). We assumed that 75% of accumulated substrate is partitioned within the cell (CPF of 12) rather than free [again, reflecting our experience with steady-state accumulation of MPP (31); also see Ref. 38]. We ran simulations for three different values of K_d, to represent the influence of a broad range of nonsaturable processes (that include diffusion, nonspecific binding, and residual extracellular substrate). At the low extreme, we used a K_d of 0 (no first-order component of substrate accumulation). We also modeled the influence of a K_d of 0.02 µL·cm⁻²·min⁻¹ (3.4 × 10⁻⁷ cm⁻¹·s⁻¹), which reflects our experimental observations with OCT2-mediated transport of both MPP and metformin (23), and 0.2 µL·cm⁻²·min⁻¹ (3.4 × 10⁻⁶ cm⁻¹·s⁻¹), to provide a test of the anticipated upper end of first-order accumulation of comparatively permeant cationic substrates.

Figure 5, A, C, and E, shows the family of curves representing the simulated 3-min time courses of net substrate uptake calculated, using tcMMsolver, from the above listed input parameters. As expected, substrate accumulation increased as a function of concentration; at the low end of the concentration range (0.3–3.0 µM, less than the K_t of 5 µM), substrate accumulation increased nearly in proportion to increases in substrate concentration, regardless of the K_d value. It was at the high end of substrate concentration (10–100 µM, greater than the K_t) that the influence of a nonsaturable component of accumulation became apparent. When K_d was 0 or 0.02 µL·cm⁻²·min⁻¹, increases in substrate accumulation changed modestly (as saturation of the mediated pathway was approached) despite sequential threefold increases in concentration. However, when K_d was increased another 10-fold, to 0.2 µL·cm⁻²·min⁻¹, net substrate accumulation continued to increase substantially as substrate concentration increased (e.g., from 30 to 100 µM), despite the effective saturation of the mediated transport pathway.

Figure 5.

Calculation of kinetic parameters [maximal rate of transport (J_max), half-saturation constant (K_t), and dissociation constant (K_d)] using tcMMfitter and simulated time courses of net uptake from increasing concentrations of substrate. A, C, and E: net uptakes from six bulk medium concentrations of substrate were calculated using tcMMsolver with the following as input parameters: J_max = 30 pmol·cm⁻²·min⁻¹, K_t = 5 µM, cell partition factor (CPF) = 12, and K_d = 0 (A), 0.02 (C), or 0.2 µL·cm⁻²·min⁻¹ (E). The kinetic values returned by tcMMfitter are shown in Table 2. Insets in A, C, and E show Michaelis–Menten plots generated from estimates of initial rates of transport calculated from net uptakes at 0.9, 28.8, 54, 126, and 180 s (refer to colored vertical lines in the time courses). Kinetic parameters from these plots (calculated using Prism, GraphPad) are shown in Table 1. B, D, and F: averages (±SD) of five separate data sets generated by introducing 15% random error into the data sets plotted in A, C, and E. The associated analysis used the 11 time points shown in the panels. The kinetic values returned by tcMMfitter are shown (and in Table 2). MPP, 1-methyl-4-phenylpyridinium.

Also evident is that none of these predicted time courses were linear over any range of time points. The impact of that fact is evident in the systematic shift in kinetic values obtained using substrate accumulations at single time points as estimates of the initial rate of unidirectional influx. Table 1 shows J_max, K_tapp, and K_d values estimated from rates of transport determined from net substrate accumulations predicted to exist at 0.9-, 28.8-, 54-, 126-, or 180-s points within these three time course profiles (these time points are indicated as vertical colored dashed lines in the simulated time courses). The insets in Fig. 5, A, C, and E, show Michaelis–Menten plots derived from these initial rate estimates. J_max and K_tapp values determined from net accumulation at ∼1 s (29.7 pmol·cm⁻²·min⁻¹ and 5.1 µM) deviated from the input values by <2% (regardless of the K_d value), reflecting the fact that at such a short period of time neither UWLs nor mediated backflux will have had time to develop (Table 1). However, these values were markedly dependent on the time point used to estimate the initial rate of transport. The calculated J_max decreased to ∼25 and then to ∼17 pmol·cm⁻²·min⁻¹ (at 28.8 and 180 s, respectively), and K_t increased to 5.7 and then to ∼9 µM compared with the input values of 30 pmol·cm⁻²·min⁻¹ and 5 µM. Accordingly, the J_max-to-K_t ratio (i.e., CL_int) decreased to 110 and then to <50 µL·mg⁻¹·min⁻¹ (using the conversion factor of 0.04 mg protein·cm⁻²) compared with the input value of 150.

Table 1.

Kinetic parameters based on net substrate accumulation at single time points of the calculated time courses of transport shown in Fig. 4A

Time Point for Initial Rate Estimation	Jmax, pmol·cm−2·min−1	Kt, µM	Kd, µL·mg−1·min−1	CLint, µL·mg−1·min−1
0.9 s	29.7	5.1	0	145
28.8 s	25.1	5.7	0	110
54 s	22.7	6.2	0	92
126 s	18.6	7.6	0	61
180 s	16.9	8.6	0	49
0.9 s	29.8	5.1	0.001	146
28.8 s	25.3	5.8	<<0.001	109
54 s	22.9	6.4	<<0.001	89
126 s	18.8	7.7	<<0.001	61
180 s	17.0	8.7	<<0.001	49
0.9 s	29.7	5.1	0.200	146
28.8 s	25.2	5.9	0.175	107
54 s	22.7	6.5	0.167	87
126 s	18.7	8.2	0.159	57
180 s	16.9	9.6	0.157	44

Intrinsic clearance (CL_int) values were calculated from the ratio of maximal rate of transport (J_max) to half-saturation constant (K_t) following the conversion of uptake expressed per cm² to uptake per mg cell protein, using the conversion factor of 0.04 mg·cm⁻². K_d, dissociation constant.

When these three simulated time courses were analyzed using our iterative model, tcMMfitter, it returned J_max, K_t, K_d, and CPF values that were generally within ∼5% of the original input values independently used to generate the simulated accumulation profiles (Table 2). The robustness of the model was subsequently assessed by determining the influence on the output values of introducing increasing amounts of error to the time points while also reducing the number of data points used to establish the time courses. Figure 5, B, D, and F, shows time courses that reflect the introduction of 15% error to each of 11 time points (n = 5 at each point, displayed as means ± SD) distributed over the 3-min time course [using the “NORMINV(RAND)” function of Excel]. The lines show the calculated time courses arising from the computed values for J_max, and K_t, K_d, and CPF from analysis of each simulated data set. Despite the substantial error introduced to the individual points of the several time courses, the computed parameters were within 15% of the input values (and usually <10%; Table 2). Importantly, the CL_int values calculated from these estimates were within 20% of the input value (123, 153, and 127 vs. 150 µL·mg⁻¹·min⁻¹), despite the introduction of substantial error to the input data.

Table 2.

Kinetic parameters determined from simulated time courses (Fig. 5) generated using tcMMfitter and three different values for K_d (0, 0.02 and 0.20 µL·mg⁻¹·min⁻¹)

Input Kd	Jmax, pmol·cm−2·min−1	Kt, µM	Kd, µL·mg−1·min−1	CPF (Effective Cell Height), µm
MPP
0	30.0	5.0	<0.001	12.0
15% error, n = 5	31.0 ± 1.9	6.3 ± 1.4	<0.001	11.0 ± 2.3
0.02	29.5	4.8	0.026	12.1
15% error, n = 5	29.3 ± 2.9	4.8 ± 0.5	0.020 ± 0.015	12.9 ± 1.6
0.2	28.3	4.9	0.207	13.2
15% error, n = 5	26.0 ± 2.4	5.1 ± 1.2	0.223 ± 0.034	12.6 ± 2.1
Metformin
0	302	304	<0.001	11.4
15% error, n = 5	285 ± 21	309 ± 7.8	<0.001	14.0 ± 5.5
0.02	300	300	0.020	12.0
15% error, n = 5	297 ± 6.4	301 ± 8.4	0.020 ± 0.002	12.5 ± 0.7
0.2	300	300	0.204	12.0
15% error, n = 5	300 ± 9.3	299 ± 3.0	0.200 ± 0.012	12.0 ± 0.4

Input values for 1-methyl-4-phenylpyridinium (MPP) were as follows: maximal rate of transport (J_max), 30 pmol·cm⁻²·min⁻¹; half-saturation constant (K_t), 5 µM; and cell partition factor (CPF), 12. Input values for metformin were as follows: J_max, 300 pmol·cm⁻²·min⁻¹; K_t, 300 µM, and CPF, 12. For each input, dissociation constant (K_d) value kinetic parameters were determined from single simulated time courses derived directly from the listed input values (for six substrate concentrations) and from five separate time courses that had random error inserted into the calculated value for each time point] using the “NORMINV(RAND)” function of Excel]; these latter values are shown as means ± SD.

We also examined simulated time courses reflecting the kinetic characteristics of OCT2-mediated metformin transport, which displays both lower affinity of the transporter for this substrate and a higher capacity for maximum transport than those for MPP transport (23, 26, 49). For the metformin simulations, the input values were J_max of 300 pmol·cm⁻²·min⁻¹, K_t of 300 µM, CPF of 12, and the three K_d values noted previously. The resulting time course profiles (Fig. 6, A, C, and E) were qualitatively similar to those for the MPP simulations, although the impact of the first-order component was substantially larger (refer to the insets in Fig. 6, C and E), reflecting the 100-fold higher substrate concentration used to saturate OCT2-mediated metformin transport (10 mM for MPP vs. 0.1 mM for MPP). As with analysis of the simulated MPP time courses, tcMMfitter returned kinetic characteristics that were within ∼1% of the input values when using the “errorless” time course values and generally well within 5% of the input values when using the data sets that included 15% random error (Table 2).

Figure 6.

Calculation of kinetic parameters [maximal rate of transport (J_max), half-saturation constant (K_t), and dissociation constant (K_d)] using tcMMfitter and simulated time courses of net uptake from increasing concentrations of substrate. A, C, and E: net uptakes from six bulk medium concentrations of substrate were calculated using tcMMsolver with the following as input parameters: J_max = 300 pmol·cm⁻²·min⁻¹, K_t = 300 µM, cell partition factor = 12, and K_d = 0 (A), 0.02 (C), or 0.2 µL·cm⁻²·min⁻¹ (E). The kinetic values returned by tcMMfitter are shown in Table 2. Insets in A, C, and E show Michaelis–Menten plots generated from estimates of initial rates of transport calculated from net uptakes at 0.9, 28.8, 54, 126, and 180 s (refer to colored vertical lines in the time courses). Kinetic parameters from these plots (calculated using Prism, GraphPad) are shown in Table 1. B, D, and F: averages (±SD) of five separate data sets generated by introducing 15% random error into the data sets plotted in A, C, and E. The associated analysis used the 11 time points shown in the panels. The kinetic values returned by tcMMfitter are shown (and in Table 2).

Experimental data.

To provide a “real world” test of the model, we analyzed the time courses of OCT2-mediated uptake of two commonly used (radioactively labeled) OCT2 substrates: MPP and metformin [using data sets we previously reported in Ref. 45). Figure 7A shows the results of a typical experiment that measured the 3-min time courses of total (net) MPP accumulation into OCT2-expressing cells. When these data, from a single experiment performed in triplicate, were analyzed using tcMMfitter, calculated J_max and K_t values were 19.3 pmol·cm⁻²·min⁻¹ and 6.4 µM, respectively. Figure 7B shows the time course profiles averaged (±SD) from four separate experiments that resulted in tcMMfitter-generated values for J_max, K_t, and K_d of 22.5 pmol·cm⁻²·min⁻¹, 6.1 µM, and 0.02 µL·cm⁻²·min⁻¹, respectively. Table 3 shows the individual results from these four experiments. Figure 8 shows the tcMMfitter results applied to data from seven experiments that measured time courses for OCT2-mediate metformin transport (Table 3).

Figure 7.

A: 3-min time courses of organic cation transporter 2 (OCT2)-dependent net uptake for six concentrations of 1-methyl-4-phenylpyridinium (MPP). Each point is the mean of three wells (±SD) from a single experiment [data reported previously (23)]. Individual data points from each time course were corrected for time 0 background, based on first-order extrapolation to time 0 of that time course (23). The dashed lines show time courses calculated using tcMMsolver using the values for maximal rate of transport (J_max), half-saturation constant (K_t), dissociation constant (K_d), and cell partition factor (CPF) determined by analyzing these data with tcMMfitter. B: 3-min time courses of OCT2-dependent net uptake for six concentrations of MPP. Each point is mean uptake determined in four separate experiments (each performed in triplicate) (±SD) (corrected for time 0 background). The dashed lines show time courses calculated using tcMMsolver using the values for J_max, K_t, K_d, and CPF determined by analyzing these data with tcMMfitter.

Table 3.

Transport parameters (J_max, K_t, CPF, and unstirred water layer) calculated using tcMMsolver for four individual experiments [data reported previously (23)] that measured the time courses of organic cation transporter 2-mediated 1-methyl-4-phenylpyridinium transport (concentrations from 0.3 to 100 µM; see Fig. 6)

	Jmax, pmol·cm−2·min−1	Kt, µM	CLint, µL·mg−1·min−1	CPF, µm	Kd, µL·cm−2·min−1
Experiment 1	23.7	5.5	108.2	18.2	0.023
Experiment 2	19.3	6.4	75.3	8.4	0.022
Experiment 3	39.5	5.3	186.9	21.6	0.029
Experiment 4	21.7	5.7	95.9	11.7	0.009
Average	24.3 ± 7.6	7.5 ± 2.0	86.7 ± 39.2	15.0 ± 6.0	0.014 ± 0.010
Fit of average	22.5	6.1	92.6	21.6	0.020

Also shown are the values obtained by fitting the average time course uptakes from all four experiments. Intrinsic clearance (CL_int) was calculated by converting maximal rate of transport (J_max) expressed per cm² to uptake per mg cell protein using the factor of 0.04 mg·cm⁻². CPF, cell partition factor; K_d, dissociation constant; K_t, half-saturation constant.

Figure 8.

Three-minute time courses of organic cation transporter 2-dependent net uptake for six concentrations of metformin. Each point [data reported previously (23)] is mean uptake determined in seven separate experiments, each performed in triplicate (±SD) (corrected for time 0 background). The dashed lines show time courses calculated using tcMMsolver using the values for maximal rate of transport (J_max), half-saturation constant (K_t), dissociation constant (K_d), and cell partition factor (CPF) determined by analyzing these data with tcMMfitter.

In Fig. 9, the data shown in Fig. 7B and Fig. 8 were used to calculate J_max and K_tapp values and the resulting CL_int values (in the insets) using the “two-step” analytic approach (i.e., initial rates estimated from single time points). For comparison, the kinetic profiles calculated from the full time courses using tcMMfitter are shown as dashed lines in Fig. 9. Estimates of the initial rate of transport at each concentration of MPP (Fig. 9A) or metformin (Fig. 9B) were based on net uptakes at 30 s, 1 min, 2 min, and 3 min, and the resulting substrate concentration-rate plots were analyzed by fitting the data to the Michaelis–Menten equation. As anticipated from the issues raised earlier, the two-step method resulted in a systematic decrease in apparent J_max as the time point for the estimate of the initial rate increased, reflecting the deviation between measured net flux and the rate of unidirectional influx produced by mediated backflux of accumulated substrate. There were also increases in K_tapp associated with increasing the time point for the estimation of the initial rate, reflecting the combined influence of the developing UWL and the backflux of substrate from the cells.

Figure 9.

“Two-step” determination of the kinetics of organic cation transporter 2 (OCT2)-mediated transport of 1-methyl-4-phenylpyridinium (MPP; A) and metformin (B). Initial rates of transport were estimated from net uptakes determined at either 30 s (red), 60 s (blue), 120 s (green), or 180 s (orange), from the data presented in Fig. 7 (for MPP) and Fig. 8 (for metformin). The black dashed lines are the kinetic profiles calculated using the maximal rate of transport (J_max) and half-saturation constant (K_t) values obtained using tcMMfitter. Inset: intrinsic clearance (CL_int; i.e., J_max/K_t) values determined from the parameters obtained from each kinetic profile (note: the calculation of CL_int values used the conversion factor of 0.04 mg of cell protein per cm⁻² of confluent cells).

The analytic method described here estimates the kinetics of transport based on the influence of known mechanisms of transport and selected physical phenomena (UWLs). We previously determined the kinetics of OCT2-mediated MPP transport that were also based on the estimates of initial rates of transport derived from time courses of transport (23). That method, however, was not mechanistically based and, instead, fit the time course profile to a rectangular hyperbola as a means to estimate the slope of the relationship at time 0. The intent was to minimize the impact of UWLs and mediated backflux using calculated values for what, we suggested, were valid estimates of initial rates of uptake. J_max and K_t values obtained for OCT2-mediated transport of MPP (Fig. 10, A and B) and metformin (Fig. 10, C and D) were compared using mechanistically based tcMMfitter analysis with those obtained using the empirical approach. There was no difference in values calculated using the two methods: in four separate experiments, J_max and K_t values for MPP transport obtained using the mechanistic model were 24.3 ± 7.6 (SD) pmol·cm⁻²·min⁻¹ and 7.5 ± 2.0 µM compared with 21.7 ± 6.3 pmol·cm⁻²·min⁻¹ and 5.6 ± 0.7 µM for the empirical protocol; J_max and K_t values for metformin transport obtained using the mechanistic model were 356 ± 118 (SD) pmol·cm⁻²·min⁻¹ and 351 ± 59.0 µM compared with 339 ± 159 pmol·cm⁻²·min⁻¹ and 278 ± 90 µM for the empirical protocol.

Figure 10.

Comparison of kinetic constants [maximal rate of transport (J_max; A) and half-saturation constant (K_t; B)] for organic cation transporter 2-mediated 1-methyl-4-phenylpyridinium (MPP) transport calculated using the MATLAB model tcMMfitter (see Table 3) and an empirical (nonmechanistic) method (to estimate initial rates of transport). The vertical bars show the mean values for J_max and K_t determined using the two methods; the values did not differ significantly. The data points connected by dashed lines show the values derived from the same individual data sets [all four of which were taken from a previous study (23)].

DISCUSSION

The kinetic characteristics of a transport process provide insights into the influence it may have on cellular and organismic physiology. In addition, they provide insights into matters of pharmacological, clinical, and regulatory importance. For example, PBPK models are increasingly used to quantify the influence of metabolism and transport on the clearance of drugs from the body (e.g., Refs. 6 and 50–53) and for predicting potential unwanted drug-drug interactions; these predictions are being codified into regulatory recommendations (8, 9). However, the accuracy of such predictions relies on the accuracy of the data used for their calculation, including the kinetics of the transport processes that influence drug assimilation, distribution, metabolism, and excretion. Although the predictive potential of metabolic PBPK is generally acknowledged (54), transport PBPK has, to date, not proven to be reliable predictor of clearance (51) and, instead, has largely been restricted to so-called “retrospective” data analyses (e.g., see Refs. 6 and 7). The failure of PBPK to accurately model the influence of transport on assimilation, distribution, metabolism, and excretion is due, at least in part, to the questionable accuracy of the in vitro assessment of the kinetics of drug transport (6, 51). In the present study, we used computational methods to quantify the influence of several generally unacknowledged technical issues, namely, UWLs and carrier-mediated efflux of accumulated substrate, both of which can compromise the accurate assessment of transport kinetics. We then introduced a mathematical model that analyzes the time course of net substrate accumulation to determine kinetic parameters of transport that are corrected for the influence of these complicating issues.

That technical issues can influence the accurate experimental assessment of transport kinetics has long been acknowledged (e.g., see Refs. 13, 19, and 55). One such complication reflects the ubiquitous association of UWLs with cell surfaces, be they in isolated tissues (e.g., strips of intestine) or in layers of cultured cells. An UWL is composed of the static water layer immediately adjacent to cell membranes and the overlying layers of slow laminar flow that extend to the bulk (experimental) solution and through which movement of solutes is limited to diffusion (56). As noted by Barry and Diamond (13), UWLs can result in concentrations at the membrane that differ from that in the bulk solution by 10-fold or more. For transport processes that support net substrate entry across the plasma membrane, the inevitable consequence of an UWL is a substrate concentration at the membrane that is less than that in the overlying, well-stirred bulk solution. Consequently, experimentally determined K_t values routinely overestimate the true K_t of a transport process. Winne (21) first quantified the theoretical impact of UWLs on experimentally determined Michaelis constants, and Wilson and Dietschy (57) were the first to confirm that manipulation of UWLs above tissue (intestinal) surfaces influences measurement of K_t values. Our own work extended this to show that experimental manipulation of either UWL thickness or maximal rates of transport influences K_tapp values in cultured cell models (11), as predicted by Winne’s theoretical treatment of the issue. However, previous studies, including our own, routinely assumed that the system was at steady state, thereby ignoring the influence of the time course of a typical transport experiment on the impact of UWLs on the resulting kinetic profile of the process. The results presented here, obtained using the tcMMsolver model, provide the first view of the influence of time on the development of the “UWL effect” and revealed two practical insights into the impact of UWLs on measurement of transport kinetics. First, while transport activity can rapidly decrease the concentration of substrate at the membrane (Fig. 2, A and C) (with a concomitant decrease in the rate of transport into the cell; Fig. 2, B and D), it takes substantially longer than the 30 s to several minute time course of a typical experiment to approach the steady-state conditions assumed in previous assessments of UWL effects (e.g., see Refs. 12, and 2021). Consequently, the actual influence of UWLs on the determination of kinetics of transport is generally substantially less than that predicted to exist at steady state. Second, while the UWL over the cell surface in unstirred culture wells typically exceeds 1,000 µm (14), during the course of a ∼2- to 3-min transport experiment the rate of diffusion is such that substrate molecules within the UWL, but more than 300–500 µm from the membrane at the start of the experiment cannot reach the membrane surface (Fig. 2A). Overall, the model suggests that the ubiquitous presence of UWLs can be expected to have a moderate effect on K_t estimates, with the magnitude of the effect most markedly influenced by the maximal rate of mediated transport (Fig. 4).

The second technical issue that influences the accurate experimental determination of transport kinetics reflects the challenge of measuring the initial rate of mediated substrate influx into cells. One of the underlying assumptions of Michaelis–Menten kinetics is that the values of J_max and K_t reflect unidirectional rates of transport, i.e., that there is no backflux of substrate over the course of the measurement (3). Efforts to satisfy this requirement typically involve showing that the time course of net substrate accumulation (the measured variable) is linear with time and extrapolates back to zero accumulation at time 0 (e.g., see Ref. 58). However, achieving this using standard cultured cell models is generally challenging and practical compromises are typically required. A review of the literature reveals that most studies of (for example) OCT2-mediated MPP transport into cells grown in multiwall plates measured net substrate accumulation following incubation times of 1–4 min to provide an estimate of the initial rate of transport (e.g., see Refs. 59–63). However, inspection of these published time courses of MPP uptake reveals that, as predicted by the model (Figs. 2 and 4) and shown in our own experimental times courses (Fig. 6), the net accumulation of MPP is not linear over this time course. This general observation is, arguably, predictable from the comparatively small size of cultured cells. As shown in Fig. 3, even modest rates of substrate entry will result in increasing intracellular substrate concentrations that, in turn, will drive increasing the rates of efflux over the course of experiments. In theory, if one could measure net uptake within the first seconds of exposure to substrate, the model predicts that neither backflux nor UWLs would introduce major errors to the estimation of the initial, unidirectional rate of uptake (Figs. 2 and 3). However, achieving adequate precision in the measurement requires sufficient substrate accumulation to result in an experimental signal well above the background noise of the system, and this is generally the criterion that dictates reliance on longer incubation times.

In the present study, we moved away from the traditional reliance on initial rate estimates based on net accumulations at fixed time points and instead used numerical analysis of full time courses of substrate uptake to solve for J_max and K_t values that best described the observed transport profiles. This required an experimental protocol capable of obtaining net uptake measurements at a number (generally 6–12) of time points and several (typically 6) substrate concentrations. To achieve this, we used cells grown in 96-well culture plates in conjunction with semiautomated delivery of substrate-containing solutions and automated aspiration and rinsing of experimental media. The resulting data, once corrected for background substrate accumulation [Fig. 6A (23)], were analyzed using a model that took into account the time-dependent influence of UWLs and the impact of the mediated backflux of accumulated substrate from the cells. The kinetics of efflux, required for the analysis, were themselves based on the iteratively derived values for J_max and K_t for influx, using the constraints on efflux parameters described by Stein (4). For the present analyses, we assumed that K_t for OCT2-mediated efflux is five times the value of the computed K_t for influx [resulting in a J_max-to-K_t ratio for efflux that is 20% of that for influx, as dictated by the net driving force associated with the −40-mV membrane potential of CHO cells (4)]. In addition, the model solved for the extent of intracellular sequestration of accumulated substrate (CPF) required to account for the observed deviation from linearity of the time courses of uptake arising from mediated backflux. The MATLAB tcMMfitter script used for this analysis proved quite robust and was capable of converging on the parameters used to generate sample data sets (Fig. 5) despite the introduction of substantial random error (15%) and the use of 100+-fold ranges for the upper and lower bounds for J_max, K_t, and CPF (the UWL was restricted to be between 500 and 2,000 µm). When experimentally determined time courses for OCT2-mediated MPP uptake were analyzed, the model predicted J_max, K_t, and CPF (and UWL) values (Figs. 6 and 7 and Table 2) well within the range expected for these values from our previous work (discussed in the context of Fig. 11). Also, as expected, when single points within the time courses were used as the basis for estimating initial rates of MPP transport, the resulting calculated values for J_max were lower, and K_tapp values were higher, than those calculated using the model (Fig. 9). This reflected the fact that the model deduced the rate of net uptake at time 0, uncomplicated by either UWLs or mediated backflux of substrate, which provides the most accurate estimate of the unidirectional rate of influx.

Figure 11.

Relationship between maximal rate of transport (J_max) and half-saturation constant (K_t) values for organic cation transporter 2 (OCT2)-mediated 1-methyl-4-phenylpyridinium (MPP) transport reported in the literature. Open circles represent values reported by other laboratories (all of which were conducted at 37°C); squares represent values reported in previous papers from our laboratory (11, 23, 26, 65,66) (all of which were conducted at ∼24°C). The green square is from Sandoval et al. (23) and was based on the time course of MPP transport; the red triangle represents the numbers generated in the present study. K_tapp, apparent K_t.

We have recently reported that time courses of net uptake can be used to estimate the rate of transport at time 0 by analyzing the shape of the time course as a means to calculate the time-dependent slope of the relationship at time 0 as a means to estimate the unidirectional initial rate of substrate influx. This analytic approach is capable of producing kinetic parameters that are effectively identical to those produced by the proposed mathematical model (Fig. 10). However, while the empirical approach is distinctly nonmechanistic, the parameters produced by the current model reflect the assumption that time courses of net uptake quantitatively reflect the kinetics (for both influx and efflux) of a mediated transport processes working within the physical constraints of a common experimental system (i.e., limited by diffusion through UWLs).

One intent of this study was to test if a model capable of correcting for the influence of UWLs and mediated backflux on measurement of transport kinetics could increase the accuracy of estimates of the kinetic parameters J_max, K_t, and CL_int. But how is the accuracy of these values to be assessed? This is a difficult question to answer. A review of the literature (e.g., see Ref. 34) reveals a substantial degree of variability in reported kinetic values for OCT2 (J_max and K_t for transported substrates and K_i/IC₅₀ values for inhibitors of transport). Indeed, the “accuracy” of transport kinetics is recognized as a major issue in the field of drug transport (e.g., see Refs. 51 and 64). But what are the “real” values? J_max of transport is the product of an individual transporter’s “turnover number” and the number of functional transporters (functional expression) in the cells in question (3, 7). Because of differences in functional expression between the cell lines used by different laboratories and between those lines and the tissue(s) where these transporters are natively expressed, the critical parameter required for physiological modeling is, in fact, not “J_max” but the turnover number (7, 25). However, as secure knowledge of this parameter is generally lacking, it is, arguably, a fool’s errand to make quantitative comparisons of literature J_max values with the intent of establishing their “accuracy.”

Evaluating the accuracy of Michaelis constants is, perhaps, more achievable; whereas J_max is an extensive property, K_t is an intensive property (i.e., independent of amount). It is reasonably presumed that there is, in fact, a unique (true) K_t value for any transporter (operating under a defined set of physiological conditions) that reflects the characteristics of substrate/ligand interaction with a unique protein structure. Consequently, experimental methods that take into account factors that complicate calculation of K_t (see Fig. 4) should result in a value that reflects the characteristics of the transporter. Importantly, the factors that influence empirical determination of K_t (including UWLs and substrate backflux) result in overestimates of its value. Thus, given a series of K_tapp values for transport of a test substrate, those at the lower end, given suitable confidence in the precision of the underlying calculations, are likely a more accurate reflection of the true value for that substrate.

With these ideas in mind, we compared 11 sets of J_max and K_t values (ours and others) for OCT2-mediated MPP transport. Figure 11 shows K_tapp values versus associated J_max values (expressed here as pmol·mg⁻¹·min⁻¹ using, when needed, a factor of 0.04 mg·cm⁻²) for these studies (including the values determined from our model; Table 3). If our hypothesis is correct, then accurate kinetic parameters for OCT2-mediated MPP transport are likely those in which comparatively large values for J_max are associated with comparatively low values for K_tapp. Figure 11 (arbitrarily) divides the distribution of K_tapp versus J_max values into four quadrants. The top left quadrant displays values that combine comparatively high K_t estimates with comparatively low J_max values; it is, perhaps, telling that the values in that quadrant used initial rate estimates that were based on 10-min exposures of cells to substrate (67, 68). The comparatively high J_max estimates of the data sets in the top right quadrant (62, 63, 69, 70) were associated with comparatively high estimates for K_t, which could reflect a failure to adequately correct for parallel first-order compartments (which seldom can be quantitatively mimicked using nonexpressing cells) and/or the influence of UWLs (which is exacerbated by high rates of transport; Fig. 4). The comparatively low K_t estimate in the bottom left quadrant was associated with a comparatively low J_max value. In fact, we have previously shown that low rates of transport can minimize the influence of both UWLs and mediated backflux on the assessment of transport kinetics (11). However, low rates of transport can compromise obtaining the precision needed for studies on, for example, kinetic mechanisms of transport, or for large-scale screening required in drug development studies.

The seven sets of data in the bottom right quadrant of Fig. 11 combined comparatively high J_max values with comparatively low K_t values. All these values were generated by our laboratory and were either published previously [Fig. 11, squares (11, 23, 26, 49, 65, 66)] or reported in the present study (Fig. 11, red diamond; see Table 3). Although the methods used in our previous studies were not identical to those used in the present work, they reflected the gradual development of the methods we now use. In fact, the study by Sandoval et al. (23) (Fig. 11, green square) used the time course-based empirical method discussed in results (Fig. 10). Considered as a cohort, the experimental and analytic methods we have used combined the (relatively) high J_max values with the (relatively) lower K_t values that we suggest will be the hallmark of accurate transport kinetics. The model introduced here provides a more robust, mechanistically based assessment of kinetic values than the previous empirical method (23) while also validating the use of that analytically simpler approach.

In summary, the calculation of J_max and K_t values for transport based on analysis of time courses of net substrate uptake increases confidence in the accuracy of these values because it permits mechanistically driven correction for common technical artifacts that complicate traditional kinetic experiments. The methods used are comparatively simple, both experimentally and computationally. The expected increase in accuracy of in vitro estimates of J_max and K_t should increase confidence in the accuracy of CL_int values that are increasingly used in PBPK assessments of the impact of transport processes on drug clearance.

SUPPLEMENTAL DATA

Supplemental Material: https://doi.org/10.5281/zenodo.6621201.

GRANTS

This work was supported by the National Institutes of Health Grants 1R01GM129777 and 5P30ES006694.

DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the authors.

AUTHOR CONTRIBUTIONS

S.H.W. conceived and designed research; S.H.W. and T.W.S. performed experiments; S.H.W. and T.W.S. analyzed data; S.H.W. and T.W.S. interpreted results of experiments; S.H.W. and T.W.S. prepared figures; S.H.W. drafted manuscript; S.H.W. and T.W.S. edited and revised manuscript; S.H.W. and T.W.S. approved final version of manuscript.