A mathematical model of the urine concentrating mechanism in the rat renal medulla. I. Formulation and base-case results

A new, region-based mathematical model of the urine concentrating mechanism of the rat renal medulla was used to investigate the significance of transport and structural properties revealed in anatomic studies. The model simulates preferential interactions among tubules and vessels by representing concentric regions that are centered on a vascular bundle in the outer medulla (OM) and on a collecting duct cluster in the inner medulla (IM). Particularly noteworthy features of this model include highly urea-permeable and water-impermeable segments of the long descending limbs and highly urea-permeable ascending thin limbs. Indeed, this is the first detailed mathematical model of the rat urine concentrating mechanism that represents high long-loop urea permeabilities and that produces a substantial axial osmolality gradient in the IM. That axial osmolality gradient is attributable to the increasing urea concentration gradient. The model equations, which are based on conservation of solutes and water and on standard expressions for transmural transport, were solved to steady state. Model simulations predict that the interstitial NaCl and urea concentrations in adjoining regions differ substantially in the OM but not in the IM. In the OM, active NaCl transport from thick ascending limbs, at rates inferred from the physiological literature, resulted in a concentrating effect such that the intratubular fluid osmolality of the collecting duct increases ~2.5 times along the OM. As a result of the separation of urea from NaCl and the subsequent mixing of that urea and NaCl in the interstitium and vasculature of the IM, collecting duct fluid osmolality further increases by a factor of ~1.55 along the IM.     

A mathematical model of the urine concentrating mechanism in the rat renal medulla. II. Functional implications of three-dimensional architecture

In a companion study [Layton AT. A mathematical model of the urine concentrating mechanism in the rat renal medulla. I. Formulation and base-case results. Am J Physiol Renal Physiol. (First published November 10, 2010). 10.1152/ajprenal.00203.2010] a region-based mathematical model was formulated for the urine concentrating mechanism in the renal medulla of the rat kidney. In the present study, we investigated model sensitivity to some of the fundamental structural assumptions. An unexpected finding is that the concentrating capability of this region-based model falls short of the capability of models that have radially homogeneous interstitial fluid at each level of only the inner medulla (IM) or of both the outer medulla and IM, but are otherwise analogous to the region-based model. Nonetheless, model results reveal the functional significance of several aspects of tubular segmentation and heterogeneity: 1) the exclusion of ascending thin limbs that reach into the deep IM from the collecting duct clusters in the upper IM promotes urea cycling within the IM; 2) the high urea permeability of the lower IM thin limb segments allows their tubular fluid urea content to equilibrate with the surrounding interstitium; 3) the aquaporin-1-null terminal descending limb segments prevent water entry and maintain the transepithelial NaCl concentration gradient; 4) a higher thick ascending limb Na(+) active transport rate in the inner stripe augments concentrating capability without a corresponding increase in energy expenditure for transport; 5) active Na(+) reabsorption from the collecting duct elevates its tubular fluid urea concentration. Model calculations predict that these aspects of tubular segmentation and heterogeneity promote effective urine concentrating functions.     

Mathematical analysis of a model for the renal concentrating mechanism

A system of ordinary differential equations, designed to model the counterflow system in the renal medulla, is studied. An existence theorem for solutions of the model equations is obtained. An exact solution of the system is obtained in the limiting case of infinite water permeability. If there is diffusion in the core, evaluation of the exact solution leads to multiple stable solutions of the model equations. One solution has a large concentration ratio, which tends to a finite asymptotic limit as the pump strength tends to infinity.     

Existence and uniqueness of solutions to a mathematical model of the urine concentrating mechanism

This paper establishes some results for the existence and uniqueness of solutions to a previously published mathematical model of the mammalian urine concentrating mechanism [H.E. Layton, Distribution of Henle's loops may enhance urine concentrating capability, Biophys. J. 49:1033-1040 (1986)]. In particular, the contraction mapping principle is used to show that for sufficiently small and sufficiently large values of a positive parameter β there exist unique solutions to the model, whether it be endowed with first-order kinetics or Michaelis-Menten kinetics. Large or small β corresponds to large or small rates of active transport of NaCl from the ascending limbs. The Schauder principle is used to show that there exist solutions to the model for physiologically reasonable reabsorption kinetics, including first-order and Michaelis-Menten kinetics for all values of β.     

An inverse algorithm for a mathematical model of an avian urine concentrating mechanism

A nonlinear optimization technique, in conjunction with a single-nephron, single-solute mathematical model of the quail urine concentrating mechanism, was used to estimate parameter sets that optimize a measure of concentrating mechanism efficiency, viz., the ratio of the free-water absorption rate to the total NaCl active transport rate. The optimization algorithm, which is independent of the numerical method used to solve the model equations, runs in a few minutes on a 1000 MHz desktop computer. The parameters varied were: tubular permeabilities to water and solute; maximum active solute transport rates of the ascending limb of Henle and the collecting duct (CD); length of the prebend enlargement (PBE) of the descending limb; fractional solute delivery to the CD; solute concentration of tubular fluid entering the CD at the cortico-medullary boundary; and rate of exponential CD population decrease along the medullary cone. Using a base-case parameter set and parameter bounds suggested by physiologic experiments, the optimization algorithm identified a maximum-efficiency set of parameter values that increased efficiency by 40% above base-case efficiency; a minimum-efficiency set reduced efficiency by about 41%. When maximum-efficiency parameter values were computed as medullary length varied over the physiologic range, the PBE was found to make up 88% of a short medullary cone but only 8% of a long medullary cone.     

Phospholipid metabolism in the rat renal inner medulla

In view of the importance of phospholipids as a source of precursor fatty acids for the high prostaglandin synthesis in the renal inner medulla, we studied pathways of phospholipid esterification and degradation in the rat inner medulla. De novo acylation of [14C]arachidonate occurred predominantly in position 2 of phosphatidylcholine in the microsomal fraction. This newly esterified [14C]arachidonate was accessible to deacylation by a microsomal phospholipase A2 (EC 3.1.1.4) with alkaline optimum which was Ca2+-dependent and resistant to 0.1% deoxycholate. No phospholipase A1 (EC 3.1.1.32) activity against endogenous labeled phosphatidylcholine could be demonstrated in the microsomal fraction. When exogenous phosphatidylcholine labeled at position 2 was deacylated by renomedullary homogenates, labeled free fatty acid but no labeled lysophosphatidylcholine was recovered in the reaction products. This could be attributed to further degradation of generated lysophosphatidylcholine by a cytosolic lysophospholipase (EC 3.1.1.5). Sodium deoxycholate at a concentration of 0.1% or higher inhibited the lysophospholipase and allowed the demonstration of both A2 and A1 alkaline phospholipase activities in the homogenate. The major in vitro pathway of lysophosphatidylcholine disposition is further degradation by a cytosolic lysophospholipase, while reutilization for phosphatidylcholine synthesis through the action of a predominantly microsomal acyltransferase appears to be a minor pathway. In the presence of several acyl-CoAs, reutilization of lysophosphatidylcholine is significantly increased by an acyl-CoA:lysophosphatidylcholine acyltransferase (EC 2.3.1.23) but there is no preferential transfer of arachidonyl-CoA compared to other acyl-CoAs.     

On a simplified model of the renal medulla.

A set of equations that approximate the central core model of the human kidney are solved. It is noted that the solution indicates a limit to the increase in solute concentration in the descending limb of Henle for a given metabolic pump term. Comparison is made with the results of an alternative theory, and numerical results are presented for the increase in solute concentration in the descending limb of Henle.     

Countercurrent exchange in the inner renal medulla: Vasa recta-descending limb system

A model of countercurrent exchange has been developed to simulate transport of salt, urea and water among vasa recta and descending limbs of the loop of Henle in the inner medulla. These vessels are abstracted as three concentric cylinders: the inne one represents descending vasa recta, the middle one represents ascending vasa recta and the outer one represents descending limbs. The capillary plexus, which connects the ascending and descending vasa recta, is modeled as a series of well-mixe compartments. Multicomponent transport equations for the sytem are derived from steady state mass balances and simple passive flux relations. The resulting set of nonlinear equations are solved numerically by an iterative Gauss-Seidel algorithm with under-relaxation. Simulations yield the salt and urea concentrations as well as volume flow rates in all tubes and compartments. The simulations indicate that solute concentrations can increase monotonically toward the papillae even if all transport processes within the exchanger are passive and source fluxes decrease monotonically toward the papillae.     

Reaction of alpha-actinin from renal inner medulla on prolonged dehydration

A western-blot analysis using monoclonal antibodies revealed that a reduction of alpha-actinin occurred in the renal inner medulla under the long-lasting dehydration. The ratio between protein content measured in rats of WAG line being hydrated or after 3-days water deprivation consisted of 52.7+/-6.2 against 23.9+/-3.3 as evaluated in relative units. The alpha-actinin level changes similarly in mutant rats of Brattleboro line not capable of synthesizing vasopressin. It was 57.5+/-4.6 in hydrated animals, and statistically lower in rats being under 3-day water deprivation--26.4+/-5.7 in relative units of protein.     

A mathematical model of C4 photosynthesis: The mechanism of concentrating CO2 in NADP-malic enzyme type species

A computer model comprising light reactions in PS II and PS I, electron-proton transport reactions in mesophyll and bundle sheath chloroplasts, all enzymatic reactions and most of the known regulatory functions of NADP-ME type C₄ photosynthesis has been developed as a system of differential budget equations for intermediate compounds. Rate-equations were designed on principles of multisubstrate-multiproduct enzyme kinetics. Some of the 275 constants needed (ΔG₀′ and K _m values) were available from literature and others (V _m) were estimated from reported rates and pool sizes. The model provided good simulations for rates of photosynthesis and pool sizes of intermediates under varying light, CO₂ and O₂. A basic novelty of the model is coupling of NADPH production via NADP-ME with ATP production and regulation of the C₃ cycle in bundle sheath chloroplasts. The functional range of the ATP/NADPH ratio in bundle sheath chloroplasts extends from 1.5 to 2.1, being energetically most efficient around 2. In the presence of such stoichiometry, the CO₂ concentrating function can be explained on the basis of two processes: (a) extra ATP consumption for starch and protein synthesis in bundle sheath leads to a faster NADPH and CO₂ import compared with CO₂ fixation in bundle sheath, and (b) the residual photorespiratory activity consumes RuBP by oxygenation, NADPH and ATP and causes the imported CO₂ to accumulate in bundle sheath cells. As a wider application, the model may be used for predicting results of genetic engineering of plants. This revised version was published online in June 2006 with corrections to the Cover Date.     

An online tool for calculation of free-energy balance for the renal inner medulla

Concentrating models of the renal inner medulla can be classified according to external free-energy balance into passive models (positive values) and models that require an external energy source (negative values). Here we introduce an online computational tool that implements the equations of Stephenson and colleagues (Stephenson JL, Tewarson RP, Mejia R. Proc Natl Acad Sci USA 71: 1618-1622, 1974) to calculate external free-energy balance at steady state for the inner medulla (http://helixweb.nih.gov/ESBL/FreeEnergy). Here "external free-energy balance" means the sum of free-energy flows in all streams entering and leaving the inner medulla. The program first assures steady-state mass balance for all components and then tallies net external free-energy balance for the selected flow conditions. Its use is illustrated by calculating external free-energy balance for an example of the passive concentrating model taken from the original paper by Kokko and Rector (Kokko JP, Rector FC Jr. Kidney Int 2: 214-223, 1972).     

Effect of varying salt and urea permeabilities along descending limbs of Henle in a model of the renal medullary urine concentrating mechanism

Modelling studies have played an important role in research on the mechanism of urine concentration and dilution by the medulla of the kidney ever since Hargitay and Kuhn (1951,Z. Elektrochem. 55, 539–558) first proposed that the parallel tubular structures in the kidney medulla must function as a “countercurrent multiplication” system. Present-day models, in keeping with our considerably improved understanding of most aspects of medullary structure-function relationships, have evolved into rather sophisticated systems of parallel tubes. In spite of this increasing complexity, it has remained the case that “model medullas” do not concentrate as well as the real kidney, especially in the inner medulla where only passive, diffusional transport occurs. Inasmuch as these models take into account the majority of contemporary ideas making up our global hypothesis about the functioning of this system, their failure to behave physiologically indicates that our understanding remains incomplete. The purpose of the present modelling study was to evaluate the implications of some recent measurements showing that permeabilities of NaCl (P _s ) and urea (P _u ) vary along the length of the descending thin limbs of Henle (Imaiet al., 1988,Am. J. Physiol. 254, F323–F328), rather than being constant throughout this segment as had been assumed earlier. It was hoped that these newly measured values might explain, by a passive, diffusional process, the net solute addition at the bend of Henle’s loop observed under some circumstances and heretofore attributed (though without any supporting experimental evidence) to active transport into the descending limb. The results of the present study show that whereas incorporation of the new values forP _s andP _u in the descending limbs of short nephrons does indeed improve the concentrating power of the model, these new values are nonetheless not sufficient to allow the model to build an osmolarity gradient that increases all the way through the inner medulla. This failing, which is common to virtually all modelling studies to date using measured values from rat kidneys, probably points to a key role for preferential exchange supposed by some to exist among certain tubule segments within vascular bundles in species whose kidneys have the highest concentrating power.     

Properties of soluble cyclic AMP-dependent protein kinase activity of renal inner medulla

Studies of the chromatographic distribution of soluble protein kinase in rat kidney demonstrated that the type I isoenzyme predominates in cortex, whereas activity in outer and inner medulla is almost exclusively the type II form. The type II isoenzyme also predominates (95% or greater) in human, canine, bovine, porcine and rabbit inner medulla. Compared to soluble type I activities from rat renal cortex or medulla, type II activity of inner medulla demonstrates a marked resistance to activation by NaCl and/or urea in subcellular preparations. However, with respect to solute activation, the resistance of the type II enzyme of inner medulla does not differ from that of type II activities from other tissues. In contrast to the effects on basal activity, NaCl and urea potentiated inner medullary type II activation by cyclic AMP and also delayed the rate of subunit reassociation after chromatographic removal of cyclic AMP. Incubation of inner medullary slices in high osmolality buffer (NaCl and urea) did not alone activate soluble protein kinase, an observation which implied that the enzyme was also resistant to solute activation in the intact cell system. Moreover, at 1650 mosM, vasopressin activation of soluble protein kinase was enhanced compared to responses at 750 mosM despite comparabel levels of cyclic AMP accumulation at the two osmolalities. However, a cyclic AMP-independent action of high osmolality to reduce the rate of inactivation of arginine vasopressin-stimulated protein kinase was not demonstrable in inner medullary slices.The present data suggest the possibility that the resistance of inner medullary protein kinase to solute activation could be related to the isomeric form of enzyme (type II) present in this tissue. The high concentrations of NaCl and urea routinely found in inner medulla during hydropenia also influenced protein kinase responses to arginine vasopressin, and may do so in part by directly potentiating the action of cyclic AMP on subunit dissociation.