Stability and robustness of a generalized pump-leak model for epithelial cell and lumen volume regulation

TL;DR

The paper extends the classic pump-leak model to a two-compartment epithelial system (ABp), deriving a ten-dimensional coupled PLE framework that couples a cell to a lumen via apical, basolateral, and paracellular pathways. It provides explicit steady-state formulas for both passive and NKA-driven regimes, and proves local stability of these equilibria while examining how equilibria and dynamics depend on physiologically relevant parameters. Through global sensitivity analyses (Sobol indices) and robustness studies, the authors show a low-dimensional structure in parameter influence, with pump strength, basolateral Na$^+$ conductance, and extracellular NaCl and temperature being primary drivers of volume regulation; potassium and chloride conductances mainly affect transient behavior rather than steady states. A key finding is that basolateral NKA promotes stable homeostasis across compartments, whereas apical NKA can destabilize luminal volume beyond a threshold, highlighting how pump localization dictates epithelial transport performance and potential pathological lumen expansion. The framework and results offer a tractable, physiologically grounded tool for predicting how transporter localization and conductance changes impact epithelial volume regulation in tissues and organoids.

Abstract

Epithelial cells regulate ion concentrations and volume through coordinated membrane pumps, ion channels, and paracellular pathways which can be modeled by classical single-compartment pump-leak equations (PLEs). Many epithelial functions, however, depend on the interaction between a cell and an enclosed luminal space, a geometry that cannot be captured by classical PLEs. To address this, we develop a two-compartment model consisting of an intracellular compartment coupled to a luminal compartment through the apical membrane, with both compartments interfacing an infinite extracellular bath and connected to it through the basolateral membrane and a paracellular pathway. Building on the five-dimensional single-cell PLEs, we formulate a ten-dimensional PLE system for this geometry and derive analytical equilibria and steady-state formulas for both the passive system and the Na+/K+-ATPase (NKA) driven active system. We characterize how these equilibria depend on physiologically relevant parameters, analyze local stability across wide parameter ranges, and apply global sensitivity and robustness methods to identify the principal determinants of ion and volume homeostasis. The model reveals fundamental differences between basolateral and apical placement of the NKA, including the onset of luminal volume blow-up when apical potassium recycling is insufficient. More broadly, this framework provides a mathematically tractable and physiologically grounded foundation for studying epithelial transport and for predicting conditions under which pump localization and conductance changes lead to stable function or pathological lumen expansion.

Figures (23)

• Figure 1: (Left) Schematic of a single-cell featuring Na+, K+, Cl-, and water channels; the Na+/K+-ATPase (NKA) pump (a primary transporter); secondary transporters including KCC, NCC, and NKCC; and impermeant molecules inside (X) and outside (Y) the cell. (Middle) Schematic of the ABp model, where cell A connects to a lumen B via the apical surface and to the extracellular fluid via the basolateral membrane. The lumen B is also connected to the extracellular fluid through a paracellular pathway (p in ABp system refers to the paracellular pathway). The apical surface, basolateral membrane, and paracellular pathway contain passive ion channels and water transporters. Additionally, the basolateral membrane includes the NKA pump and other active transporters. (Right) Generalization of the ABp model to the $A_1\cdots A_NBp$ model, in which $N$ cells are connected to the common lumen B.
• Figure 2: Steady state values of ABp systems are plotted as functions of the ATP consumption rate, for the constant NKA (solid curve) and the nonlinear NKA model given in \ref{['eq:nonlinear_pump']} (circles).
• Figure 3: The admissible range of $p$ is computed numerically by plotting $f_{j,1}(p) := 4 \, \mathcal{C}_{j}(p) - \mathcal{O}_e^2$ and identifying the interval $(0, p_{\max})$ where $f_{j,1}(p) < 0$. In the zoomed-in panel on the right, the open and closed circles denote the roots of $f_{A,1}$ and $f_{B,1}$, respectively.
• Figure 4: Sodium and potassium flow across the membranes at steady state. The arrows in the left schematic of the ABp system illustrate counterclockwise sodium and clockwise potassium transport loops. Sodium moves from compartment $B$ to $A$ across the apical surface, exits $A$ through the basolateral membrane, and returns to $B$ via the paracellular pathway, whereas potassium moves from $A$ to $B$ across the apical surface, from $B$ to the ISF through the paracellular pathway, and back to $A$ across the basolateral membrane. The right panel shows that the magnitudes of the total sodium (red) and potassium (gray) fluxes across the three interfaces increase with the pump rate $p$, with Na$^+$ fluxes slightly larger than K$^+$ fluxes over the plotted range and equal across their respective interfaces. Here, total flux denotes the area-integrated flux (in $\mu$A) and is not normalized by membrane or pathway area.
• Figure 5: Steady state values of the ABp system (Equation \ref{['eq:jss_p1']}) and the largest eigenvalue of the corresponding Jacobian matrix (Equation \ref{['eq:Jacobian']}) are plotted as functions of the pump rate $p$. As shown in the right panel (and in the zoomed-in inset), all eigenvalues are negative for any given $p$, indicating that the steady states are locally stable. The vertical lines mark $p_{\min}$ and $p_{\max}$, corresponding to the pump rates at which the steady state volume reaches its minimum and to the largest value of $p$ for which steady states exist, respectively. The horizontal axis is shown on a logarithmic scale, and the vertical axis on a linear scale. All other parameters are fixed at their default values given in Tables \ref{['tab:constants']}--\ref{['tab:pump']}.

Theorems & Definitions (8)

• Lemma 1
• proof
• Proposition 1
• proof
• Lemma 2
• proof
• Proposition 2
• proof