We can improve our fundamental understanding of epithelial transport mechanisms and their underlying mechanisms using transport phenomena equations. Furthermore, mechanistic models generally deal with multiple variables and (non-linear) interactions, and as such, are ideal for investigating membrane transporter function and solute interactions. The following paragraph provides an overview of the various transport phenomena formulations used to model the transport across a membrane (i.e., the diffusive or reactive flux as a boundary condition).

Mechanistic transport phenomena models utilize reaction-diffusion-advection equations in the model’s spatial compartment (Eq. 1). In spatial models, transport phenomena-related processes can be diffusive [Fickian diffusion D n ( ∇ 2 C n ) ], convective [carrying of solute is flow-dependent- v n ⋅ ( ∇ C n ) ], and reactive (solute transformation favors chemical equilibrium- R n ) (Kim, 2020). The boundary conditions are prescribed at the interface between two spatial compartments and describe how the interface connects the transport of the simulated solutes at either side of the interface. Boundary conditions are typically applied as a molar flux in biological applications, as shown in Eq. 2, in which flux continuity is assumed between the diffusive flux ( D n ∇ C n ) and reactive flux ( J M ). Flux is an important term in transport phenomena models and is defined as a vector (magnitude and direction) quantity of a substance’s flow over a unit area per unit time located at a compartment’s boundary Transport Phenomena Equation :   ∂ C n ∂ t = − v n ⋅ ( ∇ C n ) ︸ Convective Term + D n ( ∇ 2 C n ) ︸ Diffusive Term + R n ︸ Reacting Term 1 General Boundary Equation : D n ∇ C n = J M 2 In the general equation, the solute is the subscript n, C_n [M] denotes the concentration of the species, D _n [m².s⁻¹] is the diffusivity, and J M [molecules.m⁻².s⁻¹] is the flux term, representing the source or sink contributions of the species. The reaction term ( J M ) models whether there is a species sink (J_M < 0) or source (J_M > 0).

For a spatial model, the flux term of the general equation (Eq. 2), captures the membrane transporter interactions with the simulated solutes. In a non-spatial setting, boundary conditions are not required and the membrane transporter solute interactions are captured with the Reacting term (Eq. 1). The reaction term ( R n ) or flux term ( J M ) , depending on a non-spatial or spatial formalism respectively, can take various kinetic forms (as depicted in Figure 1):

Flux continuity: D 2 ∇ C 2

Current mechanistic models of the ABC and SLC superfamilies are limited to lumped parameters or reduced models to describe the transport function and their influences on solute transport (Weinstein, 1986; Refoyo et al., 2018; Ito et al., 2020). For example, Weinstein (1986) developed a model to investigate the transport of solutes in the proximal tubule (PT) and distal convoluted tubule (DCT). Here, Weinstein modeled the reabsorption of Cl⁻, HCO3⁻, HPO4²⁻, H2PO4⁻¹, glucose, urea, Na⁺, and K⁺ across the epithelial membrane as a general flux dependent on solute concentration, flow rate, epithelial area, and membrane electrical conductance (see Figure 1A). The general solute flux (JB) was one of the many fluxes included, modeling the entire rat proximal tubule reabsorption. Using this model, Weinstein predicted K⁺ by a diffusive and convective flux, with a lag in the K⁺ convective flux in the early segment of the tubule, which corresponds to the previous experimental reports in rats (Weinstein, 1986).

Another lumped model was published by Refoyo et al. (2018), where they approximated indoxyl sulfate transport by the basolateral SLC22A6 transporter with Michaelis-Menten kinetics (see Figure 1B) to represent the transport mechanism in a perfused bioartificial kidney. They used a parametric study to identify particular model parameters such as the active transport kinetics through the cell monolayer as this information was not experimentally available. They concluded that the concentration of the SLC22A6 transporters (represented by the Michaelis-Menten kinetic parameters) was the most influential parameter on indoxyl sulfate clearance.

Michaelis-Menten kinetics is often applied to transporters and can also be seen in a model developed by Howe et al. (2009). Here, they developed a carboxydichlorofluroscein (CDF) transport model by the ABCC2 transporter in a hepatocyte. They used a mixture of parameter values available in the literature and experimentally determined through fitting their model to the experimental data. Their model development was similar to Refoyo et al. as they use Michaelis-Menten kinetics to lump the ABCC2 function and expression parameter values. Their simulations resulted in ABCC2 having the most significant influence on the flux of CDF.

More recent mechanistic models consider the transporter as a separate variable in the flux formulation. Layton and Layton (2019) developed a model to predict the transport of 15 solutes in the various nephron segments. They included the dependency of solute transport on solute concentration, fluid flow and transcellular, and paracellular fluxes. The model is developed with an extensive system of ordinary differential equations and algebraic expressions at steady-state. Moreover, they represented the SGLT2 transporter density as being linearly dependent on flow. As such, the number of transporters can change within the compliant tubule depending on the flow conditions. Using this model, they investigated SGLT2 and NKCC2 transporter inhibition in humans, which is impossible to measure in vivo due to ethical limitations. They used available parameters from rat renal physiological models to overcome this issue and made appropriate scaling-up assumptions for the human tubules’ geometry, transporter numbers, and flow rate. The model predictions were then successfully validated by comparing human urine clearance data to produce an accurate human proximal tubule mechanistic model. Finally, Layton et al. used the model to predict the effects of a diabetic treatment on the human proximal tubule through the inhibition of SLGT2 (Layton and Layton, 2019) and sex-difference effects on solute transporters (Hu et al., 2020, 2021).

Another model that displays excellent specificity in describing transport function is the research conducted by Leedale et al. (2020). In 2D and 3D spheroids, the liver transporters are modeled as microscale kinetic reactions between generic liver transporters and a generic drug. They included both passive and carrier-mediated transport as a boundary condition at the cell membrane. The carrier-mediated transport was explicitly modeled to incorporate the density of the membrane transporters, T0, and the multiscale effects of geometry on membrane transport. They investigated the spatio-temporal hepatic drug penetration dynamics by capturing varying metabolism rates in selected liver organoid zones. Their predictions can be used to inform the optimal dosages and delivery system required for in vivo experiments involving lipophilic drugs.

Modeling the transporters explicitly is instrumental when investigating the effects of disease on epithelial membrane transport. Afshar et al. (2019) developed and validated a mechanistic model of the glucose uptake by the SLGT1 and GLUT2 (both categorized in the SLC family) present in the small intestine. First, they were able to test the hypothesis that there is a varying combination of SLGT1 and GLUT2 expression along the epithelial membrane to capture the change in apical glucose concentration uptake kinetics above 10 mM. Next, they expanded the model to develop a diabetic model using a three-fold increase of SLGT1 and GLUT2 expression. The results suggested that GLUT2 expression has a more critical role in glucose uptake in a diabetic patient than SLGT1. Importantly, these hypotheses required that glucose flux across the epithelial membrane was modeled as a transporter expression function.

It is critical to select and develop an epithelial transport model that optimally aligns with the model scale, scope of research, and data availability. For example, when observing the overall input and output of the nephron, and not the individual transporter kinetics, as seen in Weinstein (1986) (Table 1), it is beneficial to use a generalized flux term for solute transport as it is part of the broader scope of the research question. However, in the Refoyo et al. (2018) and Howe et al. (2009) models (Table 1), they wanted to explore the importance of membrane transporters on solute clearance and thus use Michaelis-Menten kinetics, capturing the activity and amount of transporters (albeit in a lumped fashion). Michaelis-Menten kinetics is a standard kinetic formulation used to describe membrane transport in pharmacokinetics as it assumes the rate of transport increases non-linearly with the reacting solute concentration. It should be noted that the Michaelis-Menten equation was initially developed to describe enzyme kinetics and has a list of underlying assumptions that are sometimes neglected when applied to transport phenomena (Keener and Sneyd, 2009; Kim and Tyson, 2020).

When using Michaelis-Menten kinetics, the researcher simplifies the mechanisms affecting the transport rate and only models the overall rate while implying transporter number effects with the Michaelis-Menten parameters. More specifically, the researcher assumes that the V_Max term, representing the maximal transport rate, is a product of the transporter number and activity. In other words, the number of transporters and their activity is lumped together into one rate parameter. However, the transporter activity, in reality, depends on a combination of physical, and chemical parameters, such as the solute concentration on either side of the membrane, the particular transporter mechanism, and solute interactions.

This oversimplification concern is also applicable when using a general flux term. There is no distinction between the activities of the transporters versus their number mathematically. Additionally, transporter location is often neglected by assuming a uniform distribution over the entire transporting surface. Finally, as also discussed by Zamek-gliszczynski et al. (2003), the problem with oversimplifying the transport processes in whole organ modeling is that there is an assumption of attributing the overall kinetics to the kinetics of a single transporter, when in fact, multiple transport processes may co-exist.

There are limitations when explicitly modeling transporter density along the cell membrane. For example, in Leedale et al. (2020) the hepatic uptake was modeled as being dependent on trans (Layton and Layton, 2019) porter expression and binding kinetics on a transporter scale. They modeled the membrane transporter in general [T₀, see Eq. 2.9 in Leedale et al. (2020)] but considered the flux across the hepatocyte cell membrane as a combination of active transport and passive diffusion by altering the α_n terms. However, the multiscale model was simplified to account for only passive diffusion across the epithelium since more compound-specific data was needed accurately to parameterize the model at all scales entirely. Similar efforts were described by Layton and Layton (2019), where they assumed a linear relationship between transporter density and fluid flow (microvilli torque) due to lack of data availability.

The models of Layton and Layton, (2019); Hu et al., 2021, Leedale et al. (2020), and Afshar et al. (2019) (summarized in Table 1), are ideal examples of the importance of explicitly modeling the transport information to investigate the downstream effects on the entire system (see also Figure 1C). Layton and Layton (2019); Hu et al., 2021 were only able to investigate the influence of blood pressure, diabetes, and sex differences on solute transporters by explicitly including a transporter density in the functions. Similarly, Leedale et al. (2020) could only illustrate a non-linear increase in intracellular drug concentration with transporter protein concentration by explicitly modeling the transporter density. Afshar et al. (2019) predicted that the GLUT2 expression was the most influential parameter altered in a diabetic state instead of the SLGT1 expression. The results suggested that further research needs to be conducted on the expression of other transporters in diabetic patients and the effect thereon on solute transport. However, these investigations will only be possible if the transporter expression is an independent variable in the system of equations. Thus, it is evident that the solute-transporter relationship is intricate, and many factors may affect transporter concentration and function. However, an integrated understanding of the solute-transporter relationship is currently missing due to the following challenges:

1) Limited access to or inability to measure in vivo readouts.