A numerical model of acid-base transport in rat distal tubule

Meeting the multiscale challenge: representing physiology processes over ApiNATOMY circuits using bond graphs

We introduce, and provide examples of, the application of the bond graph formalism to explicitly represent biophysical processes between and within modular biological compartments in ApiNATOMY. In particular, we focus on modelling scenarios from acid-base physiology to link distinct process modalities as bond graphs over an ApiNATOMY circuit of multiscale compartments. The embedding of bond graphs onto ApiNATOMY compartments provides a semantically and mathematically explicit basis for the coherent representation, integration and visualisation of multiscale physiology processes together with the compartmental topology of those biological structures that convey these processes.

Hormonal regulation of salt and water excretion: a mathematical model of whole kidney function and pressure natriuresis

We present a lumped-nephron model that explicitly represents the main features of the underlying physiology, incorporating the major hormonal regulatory effects on both tubular and vascular function, and that accurately simulates hormonal regulation of renal salt and water excretion. This is the first model to explicitly couple glomerulovascular and medullary dynamics, and it is much more detailed in structure than existing whole organ models and renal portions of multiorgan models. In contrast to previous medullary models, which have only considered the antidiuretic state, our model is able to regulate water and sodium excretion over a variety of experimental conditions in good agreement with data from experimental studies of the rat. Since the properties of the vasculature and epithelia are explicitly represented, they can be altered to simulate pathophysiological conditions and pharmacological interventions. The model serves as an appropriate starting point for simulations of physiological, pathophysiological, and pharmacological renal conditions and for exploring the relationship between the extrarenal environment and renal excretory function in physiological and pathophysiological contexts.

Fluid dilution and efficiency of Na(+) transport in a mathematical model of a thick ascending limb cell

Thick ascending limb (TAL) cells are capable of reducing tubular fluid Na(+) concentration to as low as ~25 mM, and yet they are thought to transport Na(+) efficiently owing to passive paracellular Na(+) absorption. Transport efficiency in the TAL is of particular importance in the outer medulla where O(2) availability is limited by low blood flow. We used a mathematical model of a TAL cell to estimate the efficiency of Na(+) transport and to examine how tubular dilution and cell volume regulation influence transport efficiency. The TAL cell model represents 13 major solutes and the associated transporters and channels; model equations are based on mass conservation and electroneutrality constraints. We analyzed TAL transport in cells with conditions relevant to the inner stripe of the outer medulla, the cortico-medullary junction, and the distal cortical TAL. At each location Na(+) transport efficiency was computed as functions of changes in luminal NaCl concentration ([NaCl]), [K(+)], [NH(4)(+)], junctional Na(+) permeability, and apical K(+) permeability. Na(+) transport efficiency was calculated as the ratio of total net Na(+) transport to transcellular Na(+) transport. Transport efficiency is predicted to be highest at the cortico-medullary boundary where the transepithelial Na(+) gradient is the smallest. Transport efficiency is lowest in the cortex where luminal [NaCl] approaches static head.

Luminal flow modulates H+-ATPase activity in the cortical collecting duct (CCD)

Epithelial Na(+) channel (ENaC)-mediated Na(+) absorption and BK channel-mediated K(+) secretion in the cortical collecting duct (CCD) are modulated by flow, the latter requiring an increase in intracellular Ca(2+) concentration ([Ca(2+)](i)), microtubule integrity, and exocytic insertion of preformed channels into the apical membrane. As axial flow modulates HCO(3)(-) reabsorption in the proximal tubule due to changes in both luminal Na(+)/H(+) exchanger 3 and H(+)-ATPase activity (Du Z, Yan Q, Duan Y, Weinbaum S, Weinstein AM, Wang T. Am J Physiol Renal Physiol 290: F289-F296, 2006), we sought to test the hypothesis that flow also regulates H(+)-ATPase activity in the CCD. H(+)-ATPase activity was assayed in individually identified cells in microperfused CCDs isolated from New Zealand White rabbits, loaded with the pH-sensitive dye BCECF, and then subjected to an acute intracellular acid load (NH(4)Cl prepulse technique). H(+)-ATPase activity was defined as the initial rate of bafilomycin-inhibitable cell pH (pH(i)) recovery in the absence of luminal K(+), bilateral Na(+), and CO(2)/HCO(3)(-), from a nadir pH of ∼6.2. We found that 1) an increase in luminal flow rate from ∼1 to 5 nl·min(-1)·mm(-1) stimulated H(+)-ATPase activity, 2) flow-stimulated H(+) pumping was Ca(2+) dependent and required microtubule integrity, and 3) basal and flow-stimulated pH(i) recovery was detected in cells that labeled with the apical principal cell marker rhodamine Dolichos biflorus agglutinin as well as cells that did not. We conclude that luminal flow modulates H(+)-ATPase activity in the rabbit CCD and that H(+)-ATPases therein are present in both principal and intercalated cells.

A computational model for emergent dynamics in the kidney

In this paper, concepts from network automata are adapted and extended to model complex biological systems. Specifically, systems of nephrons, the operational units of the kidney, are modelled and the dynamics of such systems are explored. Nephron behaviour can fluctuate widely and, under certain conditions, become chaotic. However, the behaviour of the whole kidney remains remarkably stable and blood solute levels are maintained under a wide range of conditions even when many nephrons are damaged or lost. A network model is used to investigate the stability of systems of nephrons and interactions between nephrons. More sophisticated dynamics are explored including the observed oscillations in single nephron filtration rates and the development of stable ionic and osmotic gradients in the inner medulla which contribute to the countercurrent exchange mechanism. We have used the model to explore the effects of changes in input parameters including hydrostatic and osmotic pressures and concentrations of ions, such as sodium and chloride. The intrinsic nephron control, tubuloglomerular feedback, is included and the effects of coupling between nephrons are explored in two-, eight- and 72-nephron models.

Analysis of pH gradients resulting from mass transport limitations in engineered heart tissue

Transport limitations of critical nutrients are a major obstacle in the construction of engineered heart tissues (EHTs), and the importance of oxygen in this regard is well-documented throughout the literature. An indirect effect of cellular hypoxia is the shunt to the less-efficient glycolytic metabolism, which is accompanied by a reduction in extracellular pH. Image analysis of phenol red coloration in an experimental model of EHTs demonstrated pH gradients towards the center of the construct, which were dependent on experimental variables. Based on these observations, a four-species, 2-D diffusion-reaction mathematical model was developed to predict pH in a radial-diffusion model. The mathematical model predicted lethal values of pH (<6.5) in EHTs comprised of a nominal cell density of 10(6) cells/cm(3). pH predictions were moderately dependent on O(2) concentration, and strongly dependent on cell density, CO(2) concentration, and diffusion path length. It can be concluded from this study that hypoxia-induced acidosis is an important element in the mass transport problem, and future experiments measuring pH with more sensitive methods is expected to further elucidate the extent of this effect.

The connecting tubule is the main site of the furosemide-induced urinary acidification by the vacuolar H+-ATPase

Final urinary acidification is achieved by electrogenic vacuolar H(+)-ATPases expressed in acid-secretory intercalated cells (ICs) in the connecting tubule (CNT) and the cortical (CCD) and initial medullary collecting duct (MCD), respectively. Electrogenic Na(+) reabsorption via epithelial Na(+) channels (ENaCs) in the apical membrane of the segment-specific CNT and collecting duct cells may promote H(+)-ATPases-mediated proton secretion by creating a more lumen-negative voltage. The exact localization where this supposed functional interaction takes place is unknown. We used several mouse models performing renal clearance experiments and assessed the furosemide-induced urinary acidification. Increasing Na(+) delivery to the CNT and CCD by blocking Na(+) reabsorption in the thick ascending limb with furosemide enhanced urinary acidification and net acid excretion. This effect of furosemide was abolished with amiloride or benzamil blocking ENaC action. In mice deficient for the IC-specific B1 subunit of the vacuolar H(+)-ATPase, furosemide led to only a small urinary acidification. In contrast, in mice with a kidney-specific inactivation of the alpha subunit of ENaC in the CCD and MCD, but not in the CNT, furosemide alone and in combination with hydrochlorothiazide induced normal urinary acidification. These results suggest that the B1 vacuolar H(+)-ATPase subunit is necessary for the furosemide-induced acute urinary acidification. Loss of ENaC channels in the CCD and MCD does not affect this acidification. Thus, functional expression of ENaC channels in the CNT is sufficient for furosemide-stimulated urinary acidification and identifies the CNT as a major segment in electrogenic urinary acidification.

Differences in pHi recovery in CO2-chemosensitive and non-chemosensitive cells: predictions from a mathematical model

In this paper, we present a mathematic model designed to identify potential mechanisms responsible for the observed differences in pHi recovery in CO(2)-chemosensitive versus non-chemosensitive cells. The model suggests that differences in pHi regulation may be dependent upon differences in the activation set-point of the internal modifier site of the Na(+)/H(+) exchanger (NHE).

A mathematical model of rat distal convoluted tubule. II. Potassium secretion along the connecting segment

A simulation of the rat distal convoluted tubule (DCT) is completed with a model of the late portion, or connecting tubule (CNT). This CNT model is developed by relying on a prior cortical collecting duct (CCD) model (Weinstein AM. Am J Physiol Renal Physiol 280: F1072–F1092, 2001), and scaling up transport activity of the three cell types to a level appropriate for DCT. The major difference between the two tubule segments is the lower CNT water permeability. In early CNT the luminal solution is hypotonic, with a K+ concentration less than that of plasma, and it is predicted that osmotic equilibration requires the whole length of CNT, to end with a nearly isotonic fluid, whose K+ concentration is severalfold greater than plasma. With respect to potassium secretion, early CNT conditions are conducive to maximal fluxes, whereas late conditions require the capacity to transport against a steep electrochemical gradient. The parameter dependence for K+ secretion under each condition is different: maximal secretion depends on luminal membrane K+ permeability, but the limiting luminal K+ concentration does not. However, maximal secretion and the limiting gradient are both enhanced by greater Na+ reabsorption. While higher CNT water permeability depresses K+ secretion, it favors Na+ reabsorption. Thus in antidiuresis there is a trade-off between enhanced Na+-dependent K+ secretion and the attenuation of K+ secretion by slow flow. When the CNT model is configured in series with the early DCT, thiazide diuretics promote renal K+ wasting by shifting Na+ reabsorption from early DCT to CNT; they promote alkalosis by shifting the remaining early DCT Na+ reabsorption to Na+/H+ exchange. This full DCT is suitable for simulating the defects of hyperkalemic hypertension, but the model offers no suggestion of a tight junction abnormality that might contribute to the phenotype.

A mathematical model of rat distal convoluted tubule. I. Cotransporter function in early DCT

A model of rat early distal convoluted tubule (DCT) is developed in conjunction with a kinetic representation of the thiazide-sensitive NaCl cotransporter (TSC). Realistic constraints on cell membrane electrical conductance require that most of the peritubular Cl(-) reabsorption proceeds via a KCl cotransporter,along with most of the K(+) recycled from the Na-K-ATPase. The model tubule reproduces the saturable Cl(-) reabsorption of DCT but not the micropuncture finding of linear Na(+) flux in response to load, more likely a feature of late DCT (CNT). As in proximal tubule, early DCT HCO(3)(-) reabsorption is mediated by a luminal Na(+)/H(+) exchanger (NHE), but in contrast to proximal tubule, the DCT exchanger is operating closer to equilibrium. In the model DCT, two consequences of the lesser driving force for NHE exchange are an acidic cytosol and wider swings in NHE flux with perturbations of luminal composition. Variations in luminal NaCl provide a challenge to cell volume, which can be blunted by volume dependence of the KCl cotransporter. Cell swelling can also be induced by increases in peritubular K(+) concentration. In this case, volume-dependent inhibition of TSC could provide volume homeostasis that also enhances distal Na(+) delivery, and ultimately enhances renal K(+) excretion. In the model DCT, proton secretion is blunted by peritubular HCO(3)(-), so that there is little contribution by this segment to the maintenance of metabolic alkalosis. During alkalosis, the model predicts that increasing luminal NaCl concentration enhances NHE flux, so that these calculations provide no support for a role of early DCT in recovery from Cl(-) depletion alkalosis.

Mathematical models of renal fluid and electrolyte transport: acknowledging our uncertainty

Mathematical models of renal tubular function, with detail at the cellular level, have been developed for most nephron segments, and these have generally been successful at capturing the overall bookkeeping of solute and water transport. Nevertheless, considerable uncertainty remains about important transport events along the nephron. The examples presented include the role of proximal tubule tight junctions in water transport and in regulation of Na(+) transport, the mechanism by which axial flow in proximal tubule modulates solute reabsorption, the effect of formate on proximal Cl(-) transport, the assessment of potassium transport along collecting duct segments inaccessible to micropuncture, the assignment of pathways for peritubular Cl(-) exit in outer medullary collecting duct, and the interaction of carbonic anhydrase-sensitive and -insensitive pathways for base exit from inner medullary collecting duct. Some of these uncertainties have had intense experimental interest well before they were cast as modeling problems. Indeed, many of the renal tubular models have been developed based on data acquired over two or three decades. Nevertheless, some uncertainties have been delineated as the result of model exploration and represent communications from the modelers back to the experimental community that certain issues should not be considered closed. With respect to model refinement, incorporating more biophysical detail about individual transporters will certainly enhance model reliability, but ultimate confidence in tubular models will still be contingent on experimental development of critical information at the tubular level.