1CellPK is a biophysical model 11–13
that can be used to analyze the transport route of CQ across an MDCK monolayer, while studying how extracellular pH, substratum porosity, and transcellular concentration gradients affect CQ’s transport behavior. As a fixed-parameter model, all input variables including compartmental volumes, areas, pH, membrane potentials and lipid fractions are held constant from the start of a simulation, and only the calculated drug concentrations vary in the different compartments. With 1CellPK, the effects of the porosity of the substratum on the permeability and transport properties of CQ were readily accounted for by considering the pores as a steric constraint on the basolateral membrane surface area (Ab
) and the effective cell cross sectional area (Aaa
) over which flux effectively occurred. Biological variability, experimental errors, and other uncertainties (such as the effective basolateral surface area over which transport takes place, and the absolute concentration and permeability of different ionic species of CQ at any given pH) were readily accounted for in Monte Carlo simulations, yielding probabilistic distributions of 1CellPK results that were reasonably accurate with respect to experimental measurements.
Geometric constraints on cell adhesions can profoundly impact cell structure and function with cell-substratum interactions potentially affecting cellular transport in many different ways. But perhaps most importantly, the non-pore area of polyester membranes is impermeant. Thus the only route for molecules to go through the membrane is the pores. Therefore, pore area, diameter and arrangements affect the number of cells per membrane area, the number and area of pores per cell, and the effective basolateral membrane area over which flux actually occurs. We found that cell density on 3 µm membranes is less than cell density on 0.4 µm membranes, and that TEER values on 3 µm membranes were lower than on 0.4 µm membranes. Whether the filter or the cell monolayer is rate limiting step, depends on the pore area available for transport, in relation to the physicochemical properties of the molecules, the permeability of the apical and basolateral membranes and the direction of transport. For example, in basolateral to apical transport studies using custom-manufactured polyester membranes, when the pore areas are small, the filter becomes the rate limiting step (N. Zheng manuscript in preparation). Because of the close relationship between simulation and experimental results, the current study suggests that, in the case of chloroquine and the commercially available 0.4 and 3 µm membranes, the cells (not the polyester) behave as the rate limiting step. However, explicit consideration of pore area vs. non-pore area makes the model applicable to other situations.
With 1CellPK, our analysis demonstrates how the logP and pKa
of ionizable functional groups of small molecules can serve as a starting point for predicting transcellular, passive transport properties. With 1CellPK, CQ is postulated to undergo very fast (instantaneous) mixing within the each subcellular compartment, with the transport of CQ across cellular membranes being the rate-limiting step determining the net rate of mass transport across the cell monolayer. The experimental results were largely consistent with this compartmental model, with CQ traversing MDCK cell monolayers via a passive, transcellular, phospholipid bilayer-limited diffusion route. The 1CellPK model captured the effects of cell biological variables (pH values in donor and receiver compartment, pore size and density of the support filter, transmembrane concentration gradients, organellar volumes and intracellular pH) on passive transport routes. However, discrepancies between simulation results and CQ uptake measured during a 4-hour exposure point to a physiological mechanism responsible for the gradual intracellular mass accumulation of CQ that is not captured by the model. Our measurements indicate that CQ accumulated intracellularly to a level higher than expected by nonspecific partitioning, ion trapping or membrane potential dependent sequestration in cytosol, lysosomes or mitochondria –all of which were included in the model or were controlled for in the experimental measurements. Other amine-containing molecules also accumulate to very high concentrations inside cells 14
Certainly, one of the limitations of a fixed-parameter model like 1CellPK is that the compartment volumes, pH, membrane potentials are held constant from the start of the simulations. Thus, the gradual biological effects of a drug on cell physiological parameters are not taken into account. CQ induced a gradual but significant increase in lysosome volume and pH in MDCK cells. By using an expanded lysosome volume (and an increased lysosome pH) as input, simulations revealed that this lysosome volume change could partly account for an increase in the total intracellular CQ mass especially in the AP → BL with pH 7.4 (). In BL → AP direction, the volume expansion cannot account for the excess CQ mass observed even during a five minute drug exposure, so an unknown factor affecting much higher-than-expected BL → AP mass sequestration remained to be identified.
CQ exists as ionic species at physiological pH and thus electrostatic interactions with negatively charged lipids and intracellular proteins and macromolecules can exert a dominant effect in terms of binding the protonated form of the base15
. Thus we considered whether such interactions might explain the discrepancy in CQ mass accumulation observed after a 4-hour incubation period. Based on results obtained with detergent-permeabilized cells, the measured partition coefficient of the ionized CQ species between the non-aqueous and aqueous cellular volume fractions were at least two orders of magnitude greater than the octanol:water partition coefficients predicted with cheminformatics software (). Therefore, by adjusting the sorption coefficients in the Monte Carlo simulations, the interaction of protonated CQ species with cellular macromolecules or anionic phospholipids, together with the measured changes in the volume of acidic organelles, could explain the higher-than-expected, gradual intracellular accumulation of CQ. Nevertheless, to confirm this hypothesis, it will be necessary to analyze the concentration of CQ in the lysosomes, the binding interaction between the protonated forms of CQ and resident cellular phospholipids and macromolecules, as well as the associated changes in organelle volumes and pH.
In summary, for monovalent weak acids and bases, the biophysical modeling approach here presented was previously tested in terms of predicting passive transcellular permeability3
. In this study we further elaborated 1CellPK to analyze the transcellular transport of dibasic molecules, to study the effects of pore size and density on transport, and more importantly, to test the model’s predictions of intracellular drug accumulation in the presence of a transcellular concentration gradient. Unlike empirical models which rely on a training data set, biophysical mechanism-based models can be used to make predictions irrespective of training data, if those predictions are aimed to guide experimental design and discover new mechanisms. When discrepancies between model predictions and observations are found, the model can point to new discoveries. Thus, although the current version of 1CellPK has been tested and validated only with CQ, it can be used to generate quantitative hypothesis and guide experimental analysis of other dibasic compounds.
To conclude, as related to drug discovery and development, permeability measurements, including in vitro
, in situ
, and in vivo
methods are low throughput and costly.16–18
Permeability assays on cell monolayers are usually done in vitro
, growing cells on semi-permeable support membranes, and monitoring the rate of mass transport across the membranes, through time.19–21
Cell permeability measurements often show huge variability between laboratories.22, 23
and many factors have been proposed to contribute to these experimental variations. Indeed, mathematical models are being increasingly used to facilitate empirical interpretation of cell-based transport mechanisms24, 25
. The ability to make predictions by using a molecule’s physicochemical properties (e.g. logP and pKa
) as input may allow biophysical modeling approaches like 1CellPK to be applied at the earliest phases of drug development, to facilitate the rational design of drug candidates with the most desirable, cell pharmacokinetic characteristics.3, 4, 13
In terms of drug mass accumulation, the fixed-parameter modeling approach here presented may be most accurate for predicting the behavior of drugs with short tissue residence times (for example, inhaled fast-acting bronchodilators or other inhaled, fast-acting medications intended to be rapidly absorbed). For predicting pharmacokinetics of drug candidates with prolonged, systemic exposures, it will be important to elucidate the mechanisms underlying gradual intracellular drug sequestration to capture the key physiological changes associated with long term exposure to drugs.