Abstract We have developed the first computational model of solute and water transport from Bowman space to the papillary tip of the nephron of a human kidney. The nephron is represented as a tubule lined by a layer of epithelial cells, with apical and basolateral transporters that vary according to cell type. The model is formulated for steady state, and consists of a large system of coupled ordinary differential equations and algebraic equations. Model solution describes luminal fluid flow, hydrostatic pressure, luminal fluid solute concentrations, cytosolic solute concentrations, epithelial membrane potential, and transcellular and paracellular fluxes. We found that if we assume that the transporter density and permeabilities are taken to be the same between the human and rat nephrons (with the exception of a glucose transporter along the proximal tubule and the H+-pump along the collecting duct), the model yields segmental deliveries and urinary excretion of volume and key solutes that are consistent with human data. The model predicted that the human nephron exhibits glomerulotubular balance, such that proximal tubular Na+ reabsorption varies proportionally to the single-nephron glomerular filtration rate. To simulate the action of a novel diabetic treatment, we inhibited the Na+-glucose cotransporter 2 (SGLT2) along the proximal convoluted tubule. Simulation results predicted that the segment’s Na+ reabsorption decreased significantly, resulting in natriuresis and osmotic diuresis.

Author summary In addition to its well-known function of waste removal from the body, the kidney is also responsible for the critical regulation of the body’s salt, potassium, acid content, and blood pressure. The kidneys perform these life-sustaining task by filtering and returning to blood stream about 200 quarts of blood every 24 hours. What isn’t returned to blood stream is excreted as urine. The production of urine involves highly complex steps of secretion and reabsorption. To study these processes without employing invasive experimental procedures, we developed the first computational model of the human nephron (which is the functional unit of a kidney). The model contains detailed representation of the transport processes that take place in the epithelial cells that form the walls of the nephron. Using that model, we conducted simulations to predict how much filtered solutes and and water is transported along each individual and functionally distinct nephron segment. We conducted these simulations under normal physiological conditions, and under pharmacological conditions. The nephron model can be used as an essential component in an integrated model of kidney function in humans.

Citation: Layton AT, Layton HE (2019) A computational model of epithelial solute and water transport along a human nephron. PLoS Comput Biol 15(2): e1006108. https://doi.org/10.1371/journal.pcbi.1006108 Editor: Daniel A. Beard, University of Michigan, UNITED STATES Received: November 23, 2017; Accepted: March 26, 2018; Published: February 25, 2019 Copyright: © 2019 Layton, Layton. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Data Availability: All relevant data are within the paper. Funding: This research was supported in part by the National Institutes of Health, National Institute of Diabetes and Digestive and Kidney Diseases, via grant DK106102 and by the National Science Foundation through grant DMS-1263995 to AT Layton. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. Competing interests: The authors have declared that no competing interests exist.

Introduction The parenchyma of a kidney is divided two major structures: the medulla and the outer renal cortex. In the multi-lobed human kidney, these structures take the shape of 8–18 cone-shaped renal lobes, with each resembling a uni-lobed rodent kidney, The outer region is the cortex, in which are clusters of capillaries, and convoluted segments of renal tubules. The inner region is the medulla, which further divides into the outer and inner medulla. Within the medulla one finds almost parallel arrangement of tubules and vessels [1]. Each human kidney is populated by about a million nephrons. Each nephron consists of an initial filtering component called the glomerulus and a renal tubule specialized for reabsorption and secretion. The renal tubule is the portion of the nephron in which the glomerular filtrate circulates before being excreted as urine. The functional role of the nephron is to adjust the composition of the urine so that wastes are excreted and that daily intake roughly equals urinary excretion. The renal tubule consists of a number of segments. Given in an order consistent with fluid flow direction, the segments are: the proximal tubule, which consists of two segments, the proximal convoluted tubule (or, the S1-S2 segments) and the S3 segment; the loop of Henle, which in turn consists of a descending limb and an ascending limb; the distal convoluted tubule, the connecting tubule, and the collecting duct. Each tubular segment is lined by a single layer of epithelial cells. The ultrastructure and transport properties of the epithelial cells vary widely among different tubular segments, so that different tubular segments specialize in different roles in renal water and solute transport. Generally, the proximal tubule reabsorbs the largest fraction of the glomerular filtrate, including about two-thirds of the water and NaCl, in addition to filtered nutrients like glucose and amino acids. The thick ascending limb of the loop of Henle that follows actively pumps NaCl into the interstitium of the medulla, without water following. As a result, the fluid that reaches the distal tubule is dilute relative to blood plasma. Depending on the hydration status of the body, the collecting duct exploits this hypotonicity by either allowing (anti-diuresis) or not allowing (diuresis) water to return to general circulation via osmosis [1]. To represent physiological processes and function changes of the kidney in diseases, one may employ a useful and non-invasive approach: computational modeling. Detailed models of solute and transport have been developed for renal epithelial cells [2, 3], tubular segments [4–6], and populations of nephrons [7, 8]. All these models, and other published ones, are formulated for the rat, due to the relatively plentiful anatomic, micropuncture, and electrophysiologic data available in rodents. It goes without saying that significant differences exist between the rat kidney and the human kidney, in terms of anatomy and hemodynamics. Consequently, while results obtained using a rat kidney model may shed insights into human kidney function, those results don’t always or entirely translate. To investigate human kidney function under physiological, pathophysiological, and pharmacological conditions, we have developed the first computational model of epithelial solute and water transport of the human nephron. Starting with our published computational model of the rat nephron [9], we incorporated anatomic and hemodynamic data from the human kidney, and we adjusted key transporter data so that the predicted urine output is consistent with known human values. Due to the relative sparsity of data on the renal transporter expression levels in humans, we identified a set of compatible transport parameters that yielded model predictions consistent with human urine and lithium clearance data. Using the resulting model, we then explored the effects of two renal transporter inhibitors on kidney function. First, we considered an inhibitor of the sodium-glucose cotransporter 2 (SGLT2) cotransporter, which is expressed on the apical membrane of the proximal convoluted tubule and is a novel target of diabetes drugs [10]. Under normoglycemic conditions, how does the drug impact segmental Na+ transport and urine excretion? We also simulated inhibition of the Na+-K+-Cl− cotransporter (NKCC2), which is expressed on the apical membrane of the thick ascending limbs of the loops of Henle and aids in the active transport of Na+, K+, and Cl− into the cell. How substantial are the compound’s diuretic, natriuretic, and kaliuretic effects?

Discussion Our understanding of the kidney has been vastly improved by studies employing techniques that evaluate renal function at the single nephron level. Particularly instrumental and indispensable is the micropuncture technique [37], which has facilitated studies of glomerular filtration and hemodynamics, and tubular epithelial activity in animal models. However, such experiments are considered too invasive to be conducted in the human kidney. To reveal microscopic processes and physiological function in the human kidney, one may utilize functional MRI [38], a non-invasive technique that could facilitate translation of many studies performed in controlled animal models using technologies that are invasive to humans. Another alternative is the computational modeling technique. One notable application of computational models is the simulation of “clean” knockout experiments. Because unlike an animal, a computational knockout model does not need to live, it will not attempt to compensate by adjusting other transport mechanisms. In this study, we developed the very first computational model of solute and water transport along a human nephron. A number of studies have suggested that a similar set of transporters are expressed along the human and rodent nephrons (e.g., [39–41]). Thus, we based the human nephron model on our published rat model [9, 11]. We first incorporated anatomic and hemodynamic data from humans, and then determined what additional transport parameters need to be adjusted to ensure that model predictions are consistent with known human data. Even though we did not expect a human to be a big rat, a rat model using human anatomic and hemodynamic data, without any change in transport expression levels, generate predictions that are largely (albeit not perfectly) consistent with human data. See Table 3, column “GFR & dimensions.” This result suggests general similarity between the electrophysiology of mammals that survive in similar living environment. With additional adjustments in transporter expression (see below), the model predicts key tubular transport and urine output that are consistent with human values (Table 3, column “Baseline”). Model simulations were validated by comparing predicted urine output, and possibly other predictions, with measurements in humans. However, it is important to note that the model does not merely recapitulate experimental observations. The model predicts, at every single point along the nephron, luminal fluid flow, luminal solute concentrations, cytosolic solute concentrations, epithelial membrane potential, transcellular solute and water fluxes, and paracellular solute and water fluxes—most of which are virtually impossible to determine in humans in vivo under current ethical guidelines. Hence, the model suggests, under various physiological or pharmacological conditions, what transport processes might be taking place within the nephron in order to produce the urine that we observe. Key model predictions are summarized below: Model simulations predict two major differences in transport activity between human and rat: (i) lower SGLT2 density along the human proximal convoluted tubule, and (ii) lower H + -ATPase and H + -K + -ATPase densities along the human collecting duct.

-ATPase and H -K -ATPase densities along the human collecting duct. Under baseline conditions, the model proximal tubule and the thick ascending limb reabsorb most of the filtered Na + , K + , and Cl − ; most of the urinary K + is secreted by the distal tubular segments; and the thick ascending limb and, to a lesser extent, the inner-medullary collecting duct acidify the urine.

, K , and Cl ; most of the urinary K is secreted by the distal tubular segments; and the thick ascending limb and, to a lesser extent, the inner-medullary collecting duct acidify the urine. The model human proximal tubule exhibits glomerulotubular balance (Fig 5), so that tubular reabsorption of filtered volume and Na + varies with SNGFR, although that balance is not perfect.

varies with SNGFR, although that balance is not perfect. Under physiological conditions the SGLT2 of the model S1-S2 segments and the SGLT1 of the model S3 segment mediate ∼90 and 10% of the filtered glucose, respectively (Fig 6). When SGLT2 is inhibited, the transport capacity of SGLT1 more than doubles so that >40% of the filtered glucose is reabsorbed.

Inhibition of SGLT2 significantly reduces Na + reabsorption by the model proximal tubule and increases Na + reabsorption by the thick ascending limb (Fig 7).

reabsorption by the model proximal tubule and increases Na reabsorption by the thick ascending limb (Fig 7). Under euglycemic conditions, inhibition of SGLT2 induces significant osmotic diuresis, resulting in natriuresis and kaliuresis; see Table 3, column “SGLT2 inhibition,” and Fig 7.

NKCC2, which is expressed along the apical membrane of the thick ascending limb, plays an essential role in renal Na+ transport. Even an incomplete inhibition of NKCC2 (80%) induces substantial diuresis, natriuresis, and kaliuresis, with urine output elevated several folds (Table 3, column “NKCC2 inhibition,” and Fig 7). The present model simulates solute and water transport along the nephron of a healthy adult. The model can be used to simulate the nephron of a diabetic patient, if model parameters are appropriately adjusted to capture pathophysiological changes in hemodynamics (to represent diabetes-induced glomerular hyperfiltration), anatomic (tubular hypertrophy), transport and other relevant model parameters (see Table 2 in Ref. [9] or Na+ transport expression changes in a diabetic rat). It must be acknowledged that quantifying these pathophysiological changes is a challenging task. Nonetheless, such a model. if successfully formulated, can be used to assess the actions of SGLT2 inhibitors in a diabetic kidney, as was done in a rat [9, 42]. By further adjusting model parameters to simulate a remnant kidney (using our previous approach for the rat [12, 43]), we can simulate the administration of SGLT2 inhibitors to a kidney with diabetic nephropathy. To the extent that renal fluid and Na+ excretion can determine blood pressure and heart failure, model results can be used to assess the degree to which cardiovascular benefits SGLT2 inhibitors persist in patients with reduced GFR, and why. To directly predict blood pressure, however, one would need a more comprehensive model such as Ref. [44]. By adjusting SNGFR (to represent changes in renal blood flow regulation) and key transport activity levels (e.g., NHE3 [45]), one can simulate a hypertensive kidney and assess the relative effectiveness of diuretics, and why. A major motivation for developing a computational model of the human nephron is to to provide a platform for pharmacokinetics and pharmacodynamics simulations; that platform can be applied to predict the effects of a new drug, or to explain the underlying mechanisms of observed effects. In this regard, it is noteworthy that important species differences have been reported in relation to the expression of various membrane transporters that mediate transport of organic anions and cations in the mammalian kidneys and other organs [46]. (Organic anions and cations are not represented in the present model.) A computational model that includes the correct expression of the relevant organic anions/cations (in humans) would prove useful in translating drug test results in rodents to humans. The present nephron model predicts cellular solute concentrations, tubular flow, and luminal fluid solute concentrations. Except for the proximal tubule, nephron segments are represented as rigid tubules, and cell volume regulation [3] is not represented. Also, SNGFR is assumed known a priori. To model autoregulation, SNGFR can be set as a function of downstream tubular flow composition [47–51]. Interstitial fluid composition is assumed known a priori (Table 2). Additionally, the model does not represent the vasculature. As a result, the model does not represent the interactions among the nephron segments, or the interactions between nephrons and the renal vasculature. To properly simulate renal handling of a given drug compound, a model that represents the interactions among renal tubule and vessels is needed. Such a model can be formulated by embedding the nephron model into a human renal medullary model (e.g., Ref. [52–54]), similar to the recent rat kidney model by Weinstein [55]. Indeed, the present model can be used as an essential component in an integrated model of kidney function in humans for studying clinically relevant questions such as K+-induced natriuresis [55].