Modeling proximal tubule cell homeostasis: tracking changes in luminal flow

Mathematical modeling of intracellular osmolarity and cell volume stabilization: The Donnan effect and ion transport

The presence of impermeant molecules within a cell can lead to an increase in cell volume through the influx of water driven by osmosis. This phenomenon is known as the Donnan (or Gibbs-Donnan) effect. Animal cells actively transport ions to counteract the Donnan effect and regulate their volume, actively pumping Na+ out and K+ into their cytosol using the Na+/K+ ATPase (NKA) pump. The pump-leak equations (PLEs) are a system of algebraic-differential equations to model the membrane potential, ion (Na+, K+, and Cl-), and water flux across the cell membrane, which provide insight into how the combination of passive ions fluxes and active transport contribute to stabilizing cell volume. Our broad objective is to provide analytical insight into the PLEs through three lines of investigation: (1) we show that the provision of impermeant extracellular molecules can stabilize the volume of a passive cell; (2) we demonstrate that the mathematical form of the NKA pump is not as important as the stoichiometry for cell stabilization; and (3) we investigate the interaction between the NKA pump and cation-chloride co-transporters (CCCs) on cell stabilization, showing that NCC can destabilize a cell while NKCC and KCC can stabilize it. We incorporate extracellular impermeant molecules, NKA pump, and CCCs into the PLEs and derive the exact formula for the steady states in terms of all the parameters. This analytical expression enables us to easily explore the effect of each of the system parameters on the existence and stability of the steady states.

A modular and reusable model of epithelial transport in the proximal convoluted tubule

We review a collection of published renal epithelial transport models, from which we build a consistent and reusable mathematical model able to reproduce many observations and predictions from the literature. The flexible modular model we present here can be adapted to specific configurations of epithelial transport, and in this work we focus on transport in the proximal convoluted tubule of the renal nephron. Our mathematical model of the epithelial proximal convoluted tubule describes the cellular and subcellular mechanisms of the transporters, intracellular buffering, solute fluxes, and other processes. We provide free and open access to the Python implementation to ensure our multiscale proximal tubule model is accessible; enabling the reader to explore the model through setting their own simulations, reproducibility tests, and sensitivity analyses.

A model of calcium transport and regulation in the proximal tubule

The objective of this study was to examine theoretically how Ca²⁺ reabsorption in the proximal tubule (PT) is modulated by Na⁺ and water fluxes, parathyroid hormone (PTH), Na⁺-glucose cotransporter (SGLT2) inhibitors, and acetazolamide. We expanded a previously published mathematical model of water and solute transport in the rat PT (Layton AT, Vallon V, Edwards A. Am J Physiol Renal Physiol 308: F1343–F1357, 2015) that did not include Ca²⁺. Our results indicate that Ca²⁺ reabsorption in the PT is primarily driven by the transepithelial Ca²⁺ concentration gradient that stems from water reabsorption, which is itself coupled to Na⁺ reabsorption. Simulated variations in permeability or transporter activity elicit opposite changes in paracellular and transcellular Ca²⁺ fluxes, whereas a simulated decrease in filtration rate lowers both fluxes. The model predicts that PTH-mediated inhibition of the apical Na⁺/H⁺ exchanger NHE3 reduces Na⁺ and Ca²⁺ transport to a similar extent. It also suggests that acetazolamide- and SGLT2 inhibitor-induced calciuria at least partly stems from reduced Ca²⁺ reabsorption in the PT. In addition, backleak of phosphate (PO₄) across tight junctions is predicted to reduce net PO₄ reabsorption by ~20% under normal conditions. When transcellular PO₄ transport is substantially reduced by PTH, paracellular PO₄ flux is reversed and contributes significantly to PO₄ reabsorption. Furthermore, PTH is predicted to exert an indirect impact on PO₄ reabsorption via its inhibitory action on NHE3. This model thus provides greater insight into the mechanisms that modulate Ca²⁺ and PO₄ reabsorption in the PT.

Cell Volume Regulation in the Proximal Tubule of Rat Kidney : Proximal Tubule Cell Volume Regulation

We developed a dynamic model of a rat proximal convoluted tubule cell in order to investigate cell volume regulation mechanisms in this nephron segment. We examined whether regulatory volume decrease (RVD), which follows exposure to a hyposmotic peritubular solution, can be achieved solely via stimulation of basolateral K[Formula: see text] and [Formula: see text] channels and [Formula: see text]-[Formula: see text] cotransporters. We also determined whether regulatory volume increase (RVI), which follows exposure to a hyperosmotic peritubular solution under certain conditions, may be accomplished by activating basolateral [Formula: see text]/H[Formula: see text] exchangers. Model predictions were in good agreement with experimental observations in mouse proximal tubule cells assuming that a 10% increase in cell volume induces a fourfold increase in the expression of basolateral K[Formula: see text] and [Formula: see text] channels and [Formula: see text]-[Formula: see text] cotransporters. Our results also suggest that in response to a hyposmotic challenge and subsequent cell swelling, [Formula: see text]-[Formula: see text] cotransporters are more efficient than basolateral K[Formula: see text] and [Formula: see text] channels at lowering intracellular osmolality and reducing cell volume. Moreover, both RVD and RVI are predicted to stabilize net transcellular [Formula: see text] reabsorption, that is, to limit the net [Formula: see text] flux decrease during a hyposmotic challenge or the net [Formula: see text] flux increase during a hyperosmotic challenge.

Flow stimulated endocytosis in the proximal tubule

PURPOSE OF REVIEW

The proximal tubule plays a critical role in the reabsorption of ions, solutes and low molecular weight proteins from the glomerular filtrate. Although the proximal tubule has long been known to acutely modulate ion reabsorption in response to changes in flow rates of the glomerular filtrate, it has only recently been discovered that proximal tubule cells can similarly adjust endocytic capacity in response to flow. This review synthesizes our current understanding of mechanosensitive regulation of endocytic capacity in proximal tubule epithelia and highlights areas of opportunity for future investigations.

RECENT FINDINGS

Recent studies have reported that the endocytic capacity of proximal tubule cells is dramatically increased upon exposure to flow and the accompanying fluid shear stress. Modulation of this pathway is dependent on increases in intracellular calcium initiated by bending of the primary cilium, and also requires purinergic receptor activation that is mediated by release of extracellular ATP. This article summarizes what is currently known about the signaling cascade that transduces changes in flow into alterations in endocytosis. We discuss the implications of this newly described regulatory pathway with respect to our understanding of protein retrieval by the kidney under normal conditions, and in diseases that present with low molecular weight proteinuria.

SUMMARY

Primary cilia act as mechanotransducers that modulate apical endocytic capacity in proximal tubule cells in response to changes in fluid shear stress.

Regulation of glomerulotubular balance: II: impact of angiotensin II on flow-dependent transport

Underlying glomerulotubular balance (GTB) is the impact of axial flow to regulate Na(+) and HCO(3)(-) transport by modulating Na(+)-H(+) exchanger 3 (NHE3) and H-ATPase activity. It is not known whether the cascade of events following a change in flow relies on local angiotensin (ANG II) generation or receptor availability. Mouse tubules were microperfused in vitro at flows of 5 and 20 nl/min, and net fluid (J(v)) and HCO(3)(-) (J(HCO3)) absorption and cell height were measured. Na(+) (J(Na)) and Cl(-) (J(Cl)) absorption and changes in microvillous torque were estimated. Raising flow increased Na(+) and HCO(3)(-) reabsorption but did not change either Cl(-) transport or cell volume. Losartan reduced absolute Na(+) and HCO(3)(-) absorption at both low and high flows but did not affect fractional flow-stimulated transport. Compared with controls, in AT(1a) knockout (KO) mouse tubules, 53% of flow-stimulated Na(+) absorption was abolished, but flow-stimulated HCO(3)(-) absorption was retained at similar levels. The remaining flow-stimulated J(HCO3) was eliminated by the H-ATPase inhibitor bafilomycin. Inhibition of the AT(2) receptor by PD123319 increased both J(Na) and J(HCO3) but did not affect flow-mediated fractional changes. NHE3 expression at the protein level was reduced in AT(1a) KO mice kidneys. We conclude that 1) although the AT(1a) receptor is necessary for flow to impact NHE3, the effect on H(+)-ATPase is independent of AT(1a); 2) the small flow-mediated changes in cell volume suggest a coordinate flow effect on both luminal and basolateral transporters; and 3) there is no evidence of flow-dependent Cl(-) transport, and thus no evidence for convective paracellular Cl(-) transport in mouse tubules.

Mathematical properties of pump-leak models of cell volume control and electrolyte balance

Homeostatic control of cell volume and intracellular electrolyte content is a fundamental problem in physiology and is central to the functioning of epithelial systems. These physiological processes are modeled using pump-leak models, a system of differential algebraic equations that describes the balance of ions and water flowing across the cell membrane. Despite their widespread use, very little is known about their mathematical properties. Here, we establish analytical results on the existence and stability of steady states for a general class of pump-leak models. We treat two cases. When the ion channel currents have a linear current-voltage relationship, we show that there is at most one steady state, and that the steady state is globally asymptotically stable. If there are no steady states, the cell volume tends to infinity with time. When minimal assumptions are placed on the properties of ion channel currents, we show that there is an asymptotically stable steady state so long as the pump current is not too large. The key analytical tool is a free energy relation satisfied by a general class of pump-leak models, which can be used as a Lyapunov function to study stability.

Mechanotransduction in the renal tubule

The role of mechanical forces in the regulation of glomerulotubular balance in the proximal tubule (PT) and Ca(2+) signaling in the distal nephron was first recognized a decade ago, when it was proposed that the microvilli in the PT and the primary cilium in the cortical collecting duct (CCD) acted as sensors of local tubular flow. In this review, we present a summary of the theoretical models and experiments that have been conducted to elucidate the structure and function of these unique apical structures in the modulation of Na(+), HCO(3)(-), and water reabsorption in the PT and Ca(2+) signaling in the CCD. We also contrast the mechanotransduction mechanisms in renal epithelium with those in other cells in which fluid shear stresses have been recognized to play a key role in initiating intracellular signaling, most notably endothelial cells, hair cells in the inner ear, and bone cells. In each case, small hydrodynamic forces need to be greatly amplified before they can be sensed by the cell's intracellular cytoskeleton to enable the cell to regulate its membrane transporters or stretch-activated ion channels in maintaining homeostasis in response to changing flow conditions.

Modeling transport in the kidney: investigating function and dysfunction

Mathematical models of water and solute transport in the kidney have significantly expanded our understanding of renal function in both health and disease. This review describes recent theoretical developments and emphasizes the relevance of model findings to major unresolved questions and controversies. These include the fundamental processes by which urine is concentrated in the inner medulla, the ultrastructural basis of proteinuria, irregular flow oscillation patterns in spontaneously hypertensive rats, and the mechanisms underlying the hypotensive effects of thiazides. Macroscopic models of water, NaCl, and urea transport in populations of nephrons have served to test, confirm, or refute a number of hypotheses related to the urine concentrating mechanism. Other macroscopic models focus on the mechanisms, role, and irregularities of renal hemodynamic control and on the regulation of renal oxygenation. At the mesoscale, models of glomerular filtration have yielded significant insight into the ultrastructural basis underlying a number of disorders. At the cellular scale, models of epithelial solute transport and pericyte Ca2+ signaling are being used to elucidate transport pathways and the effects of hormones and drugs. Areas where further theoretical progress is conditional on experimental advances are also identified.