Cellular and intracellular processes are inherently complex due to the large number of components and interactions, which are often nonlinear and occur at different spatiotemporal scales. Because of this complexity, mathematical modeling is increasingly used to simulate such systems and perform experiments in silico, many orders of magnitude faster than real experiments and often at a higher spatiotemporal resolution. In this article, we will focus on the generic modeling process and illustrate it with an example model of membrane lipid turnover.

Toxin removal by the kidney is deficient in a patient suffering from end-stage kidney disease (ESKD), and current dialysis therapies are insufficient in subsidizing this loss. A bioartificial kidney (BAK) aspires to offer ESKD patients a more effective alternative to dialysis. Mathematical models are necessary to support further developments and improve designs for the BAK before clinical trials. The BAK differentiates itself from dialysis by incorporating a living proximal tubule cell monolayer to account for the active transport of protein-bound uremic toxins, namely indoxyl sulfate (IS) in this study. Optimizing such a device is far from trivial due to the non-intuitive spatiotemporal dynamics of the IS removal process. This study used mathematical models to compare two types of active transport kinetics. i.e., two-step binding and lumped parameter. The modeling results indicated that the transporter density is the most influential parameter for the IS clearance. Moreover, a uniform distribution of transporters increases the IS clearance, highlighting the need for a high-quality, functional proximal tubule monolayer in the BAK. In summary, this study contributed to an improved understanding of IS transport in the BAK, which can be used along with laboratory experiments to develop promising renal replacement therapies in the future.

Research has shown that traditional dialysis is an insufficient long-term therapy for patients suffering from end-stage kidney disease due to the high retention of uremic toxins in the blood as a result of the absence of the active transport functionality of the proximal tubule (PT). The PT’s function is defined by the epithelial membrane transporters, which have an integral role in toxin clearance. However, the intricate PT transporter–toxin interactions are not fully explored, and it is challenging to decouple their effects in toxin removal in vitro. Computational models are necessary to unravel and quantify the toxin–transporter interactions and develop an alternative therapy to dialysis. This includes the bioartificial kidney, where the hollow dialysis fibers are covered with kidney epithelial cells. In this integrated experimental–computational study, we developed a PT computational model that focuses on indoxyl sulfate (IS) transport by organic anionic transporter 1 (OAT1), capturing the transporter density in detail along the basolateral cell membrane as well as the activity of the transporter and the inward boundary flux. The unknown parameter values of the OAT1 density $\left(1.15\times {10}^7 \mathrm{transporters} \mathsf{\mu }{\mathrm{m}}^{-2}\right)$, IS uptake ($1.75\times {10}^{-5} \mathsf{\mu }{\mathrm{M}}^{-1}{ \mathrm{s}}^{-1})$, and dissociation ($4.18\times {10}^{-4}{ \mathrm{s}}^{-1})$ were fitted and validated with experimental LC-MS/MS time-series data of the IS concentration. The computational model was expanded to incorporate albumin conformational changes present in uremic patients. The results suggest that IS removal in the physiological model was influenced mainly by transporter density and IS dissociation rate from OAT1 and not by the initial albumin concentration. While in uremic conditions considering albumin conformational changes, the rate-limiting factors were the transporter density and IS uptake rate, which were followed closely by the albumin-binding rate and IS dissociation rate. In summary, the results of this study provide an exciting avenue to help understand the toxin–transporter complexities in the PT and make better-informed decisions on bioartificial kidney designs and the underlining transporter-related issues in uremic patients.

Mathematical biology has emerged as a powerful approach to describe and understand biological systems. Here, we introduce an interactive teaching tool with a practical hands-on skill session plan to introduce students to the various components of a mathematical model with 4 different mathematical approaches (i.e., ordinary differential equations, partial differential equations, stochastic differential equations, and spatial stochastic differential equations) and their advantages and disadvantages. As such, we provide a didactic summary for instructors and students interested in using VCell MathModels for mathematical modeling; this work is also valuable for mathematics-savvy users who would like to exploit fully the capabilities of the VCell software.