A numerical model of the renal distal tubule

Model annotation and discovery with the Physiome Model Repository

BACKGROUND

Mathematics and Phy sics-based simulation models have the potential to help interpret and encapsulate biological phenomena in a computable and reproducible form. Similarly, comprehensive descriptions of such models help to ensure that such models are accessible, discoverable, and reusable. To this end, researchers have developed tools and standards to encode mathematical models of biological systems enabling reproducibility and reuse, tools and guidelines to facilitate semantic description of mathematical models, and repositories in which to archive, share, and discover models. Scientists can leverage these resources to investigate specific questions and hypotheses in a more efficient manner.

RESULTS

We have comprehensively annotated a cohort of models with biological semantics. These annotated models are freely available in the Physiome Model Repository (PMR). To demonstrate the benefits of this approach, we have developed a web-based tool which enables users to discover models relevant to their work, with a particular focus on epithelial transport. Based on a semantic query, this tool will help users discover relevant models, suggesting similar or alternative models that the user may wish to explore or use.

CONCLUSION

The semantic annotation and the web tool we have developed is a new contribution enabling scientists to discover relevant models in the PMR as candidates for reuse in their own scientific endeavours. This approach demonstrates how semantic web technologies and methodologies can contribute to biomedical and clinical research. The source code and links to the web tool are available at https://github.com/dewancse/model-discovery-tool.

Reduced Efficacy of the KCC2 Cotransporter Promotes Epileptic Oscillations in a Subiculum Network Model

Pharmacoresistant epilepsy is a chronic neurological condition in which a basal brain hyperexcitability results in paroxysmal hypersynchronous neuronal discharges. Human temporal lobe epilepsy has been associated with dysfunction or loss of the potassium-chloride cotransporter KCC2 in a subset of pyramidal cells in the subiculum, a key structure generating epileptic activities. KCC2 regulates intraneuronal chloride and extracellular potassium levels by extruding both ions. Absence of effective KCC2 may alter the dynamics of chloride and potassium levels during repeated activation of GABAergic synapses due to interneuron activity. In turn, such GABAergic stress may itself affect Cl- regulation. Such changes in ionic homeostasis may switch GABAergic signaling from inhibitory to excitatory in affected pyramidal cells and also increase neuronal excitability. Possibly these changes contribute to periodic bursting in pyramidal cells, an essential component in the onset of ictal epileptic events. We tested this hypothesis with a computational model of a subicular network with realistic connectivity. The pyramidal cell model explicitly incorporated the cotransporter KCC2 and its effects on the internal/external chloride and potassium levels. Our network model suggested the loss of KCC2 in a critical number of pyramidal cells increased external potassium and intracellular chloride concentrations leading to seizure-like field potential oscillations. These oscillations included transient discharges leading to ictal-like field events with frequency spectra as in vitro Restoration of KCC2 function suppressed seizure activity and thus may present a useful therapeutic option. These simulations therefore suggest that reduced KCC2 cotransporter activity alone may underlie the generation of ictal discharges.

SIGNIFICANCE STATEMENT

Ion regulation in the brain is a major determinant of neural excitability. Intracellular chloride in neurons, a partial determinant of the resting potential and the inhibitory reversal potentials, is regulated together with extracellular potassium via kation chloride cotransporters. During temporal lobe epilepsy, the homeostatic regulation of intracellular chloride is impaired in pyramidal cells, yet how this dysregulation may lead to seizures has not been explored. Using a realistic neural network model describing ion mechanisms, we show that chloride homeostasis pathology provokes seizure activity analogous to recordings from epileptogenic brain tissue. We show that there is a critical percentage of pathological cells required for seizure initiation. Our model predicts that restoration of the chloride homeostasis in pyramidal cells could be a viable antiepileptic strategy.

Mathematical modeling of urea transport in the kidney

Mathematical modeling techniques have been useful in providing insights into biological systems, including the kidney. This article considers some of the mathematical models that concern urea transport in the kidney. Modeling simulations have been conducted to investigate, in the context of urea cycling and urine concentration, the effects of hypothetical active urea secretion into pars recta. Simulation results suggest that active urea secretion induces a "urea-selective" improvement in urine concentrating ability. Mathematical models have also been built to study the implications of the highly structured organization of tubules and vessels in the renal medulla on urea sequestration and cycling. The goal of this article is to show how physiological problems can be formulated and studied mathematically, and how such models may provide insights into renal functions.

Distal convoluted tubule

The distal convoluted tubule (DCT) is a short nephron segment, interposed between the macula densa and collecting duct. Even though it is short, it plays a key role in regulating extracellular fluid volume and electrolyte homeostasis. DCT cells are rich in mitochondria, and possess the highest density of Na+/K+-ATPase along the nephron, where it is expressed on the highly amplified basolateral membranes. DCT cells are largely water impermeable, and reabsorb sodium and chloride across the apical membrane via electroneurtral pathways. Prominent among this is the thiazide-sensitive sodium chloride cotransporter, target of widely used diuretic drugs. These cells also play a key role in magnesium reabsorption, which occurs predominantly, via a transient receptor potential channel (TRPM6). Human genetic diseases in which DCT function is perturbed have provided critical insights into the physiological role of the DCT, and how transport is regulated. These include Familial Hyperkalemic Hypertension, the salt-wasting diseases Gitelman syndrome and EAST syndrome, and hereditary hypomagnesemias. The DCT is also established as an important target for the hormones angiotensin II and aldosterone; it also appears to respond to sympathetic-nerve stimulation and changes in plasma potassium. Here, we discuss what is currently known about DCT physiology. Early studies that determined transport rates of ions by the DCT are described, as are the channels and transporters expressed along the DCT with the advent of molecular cloning. Regulation of expression and activity of these channels and transporters is also described; particular emphasis is placed on the contribution of genetic forms of DCT dysregulation to our understanding.

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.

Mathematical modeling of kidney transport

In addition to metabolic waste and toxin excretion, the kidney also plays an indispensable role in regulating the balance of water, electrolytes, nitrogen, and acid-base. In this review, we describe representative mathematical models that have been developed to better understand kidney physiology and pathophysiology, including the regulation of glomerular filtration, the regulation of renal blood flow by means of the tubuloglomerular feedback mechanisms and of the myogenic mechanism, the urine concentrating mechanism, epithelial transport, and regulation of renal oxygen transport. We discuss the extent to which these modeling efforts have expanded our understanding of renal function in both health and disease.

A mathematical model of the diluting power of the cortical thick ascending limb of the loop of Henle

A mathematical model is presented that describes the ionic transport across the cortical thick ascending limb (cTAL) of the Henle's loop, taking into account its tubular geometry. A comprehensive description of the cTAL is given for the first time in terms of potential, ion concentrations and ion fluxes along the tubule. For given ion concentrations at the entrance of the tubule, the model simulates steady-state profiles and allows the fitting of existing experimentally measured values at its exit. Moreover, the model expands the potentialities of experiments in situ and enables testing the effect of different perturbations induced by drugs or mutation-altering transport activity. One of the main insights given by this model is the increase of the lumenal electrical potential from the entrance to the exit of the tubule with a profile determined by the transepithelial electrical potential difference and by the chemical gradients along the lumen, both reflecting transepithelial salt transport. Furthermore, model and experimental results are consistent, showing that when the TAL is perfused at high rates with a diluted NaCl solution in relation to the bath, the transepithelial electrical potential difference increases from 6.7 to 23.0 mV and the potential difference across the basolateral barrier changes very little. The model predicts that the same static head is obtained independently of the NaCl concentration at the entrance of the tubule. A final important insight concerns the lowest reported NaCl concentrations (20-30 mM) at the exit of the tubule, which is controlled by a very tight epithelium, where the back-leak is substantially reduced.

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.

Regulation of K-Cl cotransport: from function to genes

This review intends to summarize the vast literature on K-Cl cotransport (COT) regulation from a functional and genetic viewpoint. Special attention has been given to the signaling pathways involved in the transporter's regulation found in several tissues and cell types, and more specifically, in vascular smooth muscle cells (VSMCs). The number of publications on K-Cl COT has been steadily increasing since its discovery at the beginning of the 1980s, with red blood cells (RBCs) from different species (human, sheep, dog, rabbit, guinea pig, turkey, duck, frog, rat, mouse, fish, and lamprey) being the most studied model. Other tissues/cell types under study are brain, kidney, epithelia, muscle/smooth muscle, tumor cells, heart, liver, insect cells, endothelial cells, bone, platelets, thymocytes and Leishmania donovani. One of the salient properties of K-Cl-COT is its activation by cell swelling and its participation in the recovery of cell volume, a process known as regulatory volume decrease (RVD). Activation by thiol modification with N-ethylmaleimide (NEM) has spawned investigations on the redox dependence of K-Cl COT, and is used as a positive control for the operation of the system in many tissues and cells. The most accepted model of K-Cl COT regulation proposes protein kinases and phosphatases linked in a chain of phosphorylation/dephosphorylation events. More recent studies include regulatory pathways involving the phosphatidyl inositol/protein kinase C (PKC)-mediated pathway for regulation by lithium (Li) in low-K sheep red blood cells (LK SRBCs), and the nitric oxide (NO)/cGMP/protein kinase G (PKG) pathway as well as the platelet-derived growth factor (PDGF)-mediated mechanism in VSMCs. Studies on VSM transfected cells containing the PKG catalytic domain demonstrated the participation of this enzyme in K-Cl COT regulation. Commonly used vasodilators activate K-Cl COT in a dose-dependent manner through the NO/cGMP/PKG pathway. Interaction between the cotransporter and the cytoskeleton appears to depend on the cellular origin and experimental conditions. Pathophysiologically, K-Cl COT is altered in sickle cell anemia and neuropathies, and it has also been proposed to play a role in blood pressure control. Four closely related human genes code for KCCs (KCC1-4). Although considerable information is accumulating on tissue distribution, function and pathologies associated with the different isoforms, little is known about the genetic regulation of the KCC genes in terms of transcriptional and post-transcriptional regulation. A few reports indicate that the NO/cGMP/PKG signaling pathway regulates KCC1 and KCC3 mRNA expression in VSMCs at the post-transcriptional level. However, the detailed mechanisms of post-transcriptional regulation of KCC genes and of regulation of KCC2 and KCC4 mRNA expression are unknown. The K-Cl COT field is expected to expand further over the next decades, as new isoforms and/or regulatory pathways are discovered and its implication in health and disease is revealed.

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.

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

The purpose of this study is to develop a numerical model that simulates acid-base transport in rat distal tubule. We have previously reported a model that deals with transport of Na(+), K(+), Cl(-), and water in this nephron segment (Chang H and Fujita T. Am J Physiol Renal Physiol 276: F931-F951, 1999). In this study, we extend our previous model by incorporating buffer systems, new cell types, and new transport mechanisms. Specifically, the model incorporates bicarbonate, ammonium, and phosphate buffer systems; has cell types corresponding to intercalated cells; and includes the Na/H exchanger, H-ATPase, and anion exchanger. Incorporation of buffer systems has required the following modifications of model equations: new model equations are introduced to represent chemical equilibria of buffer partners [e.g., pH = pK(a) + log(10) (NH(3)/NH(4))], and the formulation of mass conservation is extended to take into account interconversion of buffer partners. Furthermore, finite rates of H(2)CO(3)-CO(2) interconversion (i.e., H(2)CO(3) &rlharr; CO(2) + H(2)O) are taken into account in modeling the bicarbonate buffer system. Owing to this treatment, the model can simulate the development of disequilibrium pH in the distal tubular fluid. For each new transporter, a state diagram has been constructed to simulate its transport kinetics. With appropriate assignment of maximal transport rates for individual transporters, the model predictions are in agreement with free-flow micropuncture experiments in terms of HCO reabsorption rate in the normal state as well as under the high bicarbonate load. Although the model cannot simulate all of the microperfusion experiments, especially those that showed a flow-dependent increase in HCO reabsorption, the model is consistent with those microperfusion experiments that showed HCO reabsorption rates similar to those in the free-flow micropuncture experiments. We conclude that it is possible to develop a numerical model of the rat distal tubule that simulates acid-base transport, as well as basic solute and water transport, on the basis of tubular geometry, physical principles, and transporter kinetics. Such a model would provide a useful means of integrating detailed kinetic properties of transporters and predicting macroscopic transport characteristics of this nephron segment under physiological and pathophysiological settings.