A thermodynamically consistent approach to modeling epithelial solute and water transport in the proximal convoluted tubule

This study presents a novel approach to modeling fluid and ion transport in the proximal convoluted tubule (PCT) of the nephron using bond graphs. Bond graphs provide a robust framework for analyzing complex systems, explicitly depicting multi-domain energy exchange. Leveraging the modular nature of bond graphs, we first defined resistive modules representing membranes and capacitive modules representing solution-filled compartments, then coupled them using circuit theory. Our implementation extends beyond previous bond graph models of physiological processes by explicitly representing volumetric flow as a distinct variable within capacitive modules. In so doing, our model enables the consideration of mechanotransduction effects, where changes in fluid volume can influence membrane transporter activity, a crucial aspect of PCT function. Our bond graph model of the PCT (BG-PCT) comprises four fluid compartments bounded by five distinct membranes. The BG-PCT considers five chemical species (Na+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$<^>{+}$$\end{document}, K+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$<^>+$$\end{document}, Cl-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$<^>-$$\end{document}, HCO3-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_3<^>-$$\end{document}, and glucose) and six key membrane transporters distributed across the different membranes. Each structural subsystem comprises elementary thermodynamic processes, including dissipation, free-energy change, and power flow. This study demonstrates the advantages of bond graph modeling, particularly in its capacity to couple multiple energy domains and its modularity, which enables future extensibility. The BG-PCT provides a flexible, thermodynamically consistent platform for in silico research on epithelial transport dynamics and is available on GitHub under an open-source license to facilitate future research.