Tubular fluid flow and distal NaCl delivery mediated by tubuloglomerular feedback in the rat kidney

The glomerular filtration rate in the kidney is controlled, in part, by the tubuloglomerular feedback (TGF) system, which is a negative feedback loop that mediates oscillations in tubular fluid flow and in fluid NaCl concentration of the loop of Henle. In this study, we developed a mathematical model of the TGF system that represents NaCl transport along a short loop of Henle with compliant walls. The proximal tubule and the outer-stripe segment of the descending limb are assumed to be highly water permeable; the thick ascending limb (TAL) is assumed to be water impermeable and have active NaCl transport. A bifurcation analysis of the TGF model equations was performed by computing parameter boundaries, as functions of TGF gain and delay, that separate differing model behaviors. The analysis revealed a complex parameter region that allows a variety of qualitatively different model equations: a regime having one stable, time-independent steady-state solution and regimes having stable oscillatory solutions of different frequencies. A comparison with a previous model, which represents only the TAL explicitly and other segments using phenomenological relations, indicates that explicit representation of the proximal tubule and descending limb of the loop of Henle lowers the stability of the TGF system. Model simulations also suggest that the onset of limit-cycle oscillations results in increases in the time-averaged distal NaCl delivery, whereas distal fluid delivery is not much affected.     

Feedback-mediated dynamics in a model of coupled nephrons with compliant thick ascending limbs

The tubuloglomerular feedback (TGF) system in the kidney, a key regulator of glomerular filtration rate, has been shown in physiologic experiments in rats to mediate oscillations in thick ascending limb (TAL) tubular fluid pressure, flow, and NaCl concentration. In spontaneously hypertensive rats, TGF-mediated flow oscillations may be highly irregular. We conducted a bifurcation analysis of a mathematical model of nephrons that are coupled through their TGF systems; the TALs of these nephrons are assumed to have compliant tubular walls. A characteristic equation was derived for a model of two coupled nephrons. Analysis of that characteristic equation has revealed a number of parameter regions having the potential for differing stable dynamic states. Numerical solutions of the full equations for two model nephrons exhibit a variety of behaviors in these regions. Also, model results suggest that the stability of the TGF system is reduced by the compliance of TAL walls and by internephron coupling; as a result, the likelihood of the emergence of sustained oscillations in tubular fluid pressure and flow is increased. Based on information provided by the characteristic equation, we identified parameters with which the model predicts irregular tubular flow oscillations that exhibit a degree of complexity that may help explain the emergence of irregular oscillations in spontaneously hypertensive rats.     

Signal transduction in a compliant thick ascending limb

In several previous studies, we used a mathematical model of the thick ascending limb (TAL) to investigate nonlinearities in the tubuloglomerular feedback (TGF) loop. That model, which represents the TAL as a rigid tube, predicts that TGF signal transduction by the TAL is a generator of nonlinearities: if a sinusoidal oscillation is added to constant intratubular fluid flow, the time interval required for an element of tubular fluid to traverse the TAL, as a function of time, is oscillatory and periodic but not sinusoidal. As a consequence, NaCl concentration in tubular fluid alongside the macula densa will be nonsinusoidal and thus contain harmonics of the original sinusoidal frequency. We hypothesized that the complexity found in power spectra based on in vivo time series of key TGF variables arises in part from those harmonics and that nonlinearities in TGF-mediated oscillations may result in increased NaCl delivery to the distal nephron. To investigate the possibility that a more realistic model of the TAL would damp the harmonics, we have conducted new studies in a model TAL that has compliant walls and thus a tubular radius that depends on transmural pressure. These studies predict that compliant TAL walls do not damp, but instead intensify, the harmonics. In addition, our results predict that mean TAL flow strongly influences the shape of the NaCl concentration waveform at the macula densa. This is a consequence of the inverse relationship between flow speed and transit time, which produces asymmetry between up- and downslopes of the oscillation, and the nonlinearity of TAL NaCl absorption at low flow rates, which broadens the trough of the oscillation relative to the peak. The dependence of waveform shape on mean TAL flow may be the source of the variable degree of distortion, relative to a sine wave, seen in experimental recordings of TGF-mediated oscillations.     

Tubuloglomerular Feedback Signal Transduction in a Short Loop of Henle

In previous studies, we used a mathematical model of the thick ascending limb (TAL) to investigate nonlinearities in the tubuloglomerular feedback (TGF) loop. That model does not represent other segments of the nephron, the water, and NaCl transport along which may impact fluid flow rate and NaCl transport along the TAL. To investigate the extent to which those transport processes affect TGF mediation, we have developed a mathematical model for TGF signal transduction in a short loop nephron. The model combines a simple representation of the renal cortex with a highly-detailed representation of the outer medulla (OM). The OM portion of the model is based on an OM urine concentrating mechanism model previously developed by Layton and Layton (Am. J. Renal 289:F1346–F1366, 2005a). When perturbations are applied to intratubular fluid flow at the proximal straight tubule entrance, the present model predicts oscillations in fluid flow and solute concentrations in the cortical TAL and interstitium, and in all tubules, vessels, and interstitium in the OM. Model results suggest that TGF signal transduction by the TAL is a generator of nonlinearities: if a sinusoidal oscillation is added to constant intratubular fluid flow, the time required for an element of tubular fluid to traverse the TAL is oscillatory, but nonsinusoidal; those results are consistent with our previous studies. As a consequence, oscillations in NaCl concentration in tubular fluid alongside the macula densa (MD) will be nonsinusoidal and contain harmonics of the original sinusoidal frequency. Also, the model predicts that the oscillations in NaCl concentration at the loop-bend fluid are smaller in amplitude than those at the MD, a result that further highlights the crucial role of TAL in the nonlinear transduction of TGF signal from SNGFR to MD NaCl concentration.     

Multistable Dynamics Mediated by Tubuloglomerular Feedback in a Model of Coupled Nephrons

To help elucidate the causes of irregular tubular flow oscillations found in the nephrons of spontaneously hypertensive rats (SHR), we have conducted a bifurcation analysis of a mathematical model of two nephrons that are coupled through their tubuloglomerular feedback (TGF) systems. This analysis was motivated by a previous modeling study which predicts that NaCl backleak from a nephron’s thick ascending limb permits multiple stable oscillatory states that are mediated by TGF (Layton et al. in Am. J. Physiol. Renal Physiol. 291:F79–F97, 2006); that prediction served as the basis for a comprehensive, multifaceted hypothesis for the emergence of irregular flow oscillations in SHR. However, in that study, we used a characteristic equation obtained via linearization from a single-nephron model, in conjunction with numerical solutions of the full, nonlinear model equations for two and three coupled nephrons. In the present study, we have derived a characteristic equation for a model of any finite number of mutually coupled nephrons having NaCl backleak. Analysis of that characteristic equation for the case of two coupled nephrons has revealed a number of parameter regions having the potential for differing stable dynamic states. Numerical solutions of the full equations for two model nephrons exhibit a variety of behaviors in these regions. Some behaviors exhibit a degree of complexity that is consistent with our hypothesis for the emergence of irregular oscillations in SHR.     

Effect of tubular inhomogeneities on filter properties of thick ascending limb of Henle's loop

We used a simple mathematical model of rat thick ascending limb (TAL) of the loop of Henle to predict the impact of spatially inhomogeneous NaCl permeability, spatially inhomogeneous NaCl active transport, and spatially inhomogeneous tubular radius on luminal NaCl concentration when sustained, sinusoidal perturbations were superimposed on steady-state TAL flow. A mathematical model previously devised by us that used homogeneous TAL transport and fixed TAL radius predicted that such perturbations result in TAL luminal fluid NaCl concentration profiles that are standing waves. That study also predicted that nodes in NaCl concentration occur at the end of the TAL when the tubular fluid transit time equals the period of a periodic perturbation, and that, for non-nodal periods, sinusoidal perturbations generate non-sinusoidal oscillations (and thus a series of harmonics) in NaCl concentration at the TAL end. In the present study we find that the inhomogeneities transform the standing waves and their associated nodes into approximate standing waves and approximate nodes. The impact of inhomogeneous NaCl permeability is small. However, for inhomogeneous active transport or inhomogeneous radius, the oscillations for non-nodal periods tend to be less sinusoidal and more distorted than in the homogeneous case and to thus have stronger harmonics. Both the homogeneous and non-homogeneous cases predict that the TAL, in its transduction of flow oscillations into concentration oscillations, acts as a low-pass filter, but the inhomogeneities result in a less effective filter that has accentuated non-linearities.     

Feedback-mediated dynamics in two coupled nephrons

Previously, we developed a dynamic model for the tubuloglomerular feedback (TGF) system in a single, short-looped nephron of the mammalian kidney. In that model, a semi-linear hyperbolic partial differential equation was used to represent two fundamental processes of solute transport in the nephron’s thick ascending limb (TAL): chloride advection by fluid flow along the TAL lumen and transepithelial chloride transport from the lumen to the interstitium. An empirical function and a time delay were used to relate glomerular filtration rate to the chloride concentration at the macula densa of the TAL. Analysis of the model equations indicated that stable limit-cycle oscillations (LCO) in nephron fluid flow and chloride concentration can emerge for sufficiently large feedback gain magnitude and time delay. In this study, the single-nephron model was extended to two nephrons, which were coupled through their filtration rates. Explicit analytical conditions were obtained for bifurcation loci corresponding to two special cases: (1) identical time delays but differing feedback gains, and (2) identical gains but differing delays. Similar to the case of a single nephron, our analysis indicates that stable LCO can emerge in coupled nephrons for sufficiently large gains and delays. However, these LCO may emerge at lower values of the feedback gain, relative to a single (i.e., uncoupled) nephron, or at shorter delays, provided the delays are sufficiently close. These results suggest that, in vivo, if two nephrons are sufficiently similar, then coupling will tend to increase the likelihood of LCO.     

Swelling of the Transverse Tubular System in Frog Sartorius

Electron microscopy shows that the transverse tubular system of frog sartorius swells in Ringer fluid in which NaCl is partially replaced by sucrose (sucrose isotonic solutions). At constant tonicity, the degree of swelling is roughly proportional to the decrease in ionic strength and to the sucrose concentration of the bathing solution. Swelling is time-dependent and reversible within 2 hr. The late after potential which follows a train of impulses is prolonged with swelling, but not to the extent expected from the model of Adrian and Freygang. This discrepancy remains unexplained, as does the mechanism of swelling of the transverse tubular system, although some suggestions are offered. One is that the transverse tubular system contains fixed charges and swells like a fixed charge gel.     

Nonlinear interactions in renal blood flow regulation

We have developed a model of tubuloglomerular feedback (TGF) and the myogenic mechanism in afferent arterioles to understand how the two mechanisms are coupled. This paper presents the model. The tubular model predicts pressure, flow, and NaCl concentration as functions of time and tubular length in a compliant tubule that reabsorbs NaCl and water; boundary conditions are glomerular filtration rate (GFR), a nonlinear outflow resistance, and initial NaCl concentration. The glomerular model calculates GFR from a change in protein concentration using estimates of capillary hydrostatic pressure, tubular hydrostatic pressure, and plasma flow rate. The arteriolar model predicts fraction of open K channels, intracellular Ca concentration (Ca(i)), potential difference, rate of actin-myosin cross bridge formation, force of contraction, and length of elastic elements, and was solved for two arteriolar segments, identical except for the strength of TGF input, with a third, fixed resistance segment representing prearteriolar vessels. The two arteriolar segments are electrically coupled. The arteriolar, glomerular, and tubular models are linked; TGF modulates arteriolar circumference, which determines vascular resistance and glomerular capillary pressure. The model couples TGF input to voltage-gated Ca channels. It predicts autoregulation of GFR and renal blood flow, matches experimental measures of tubular pressure and macula densa NaCl concentration, and predicts TGF-induced oscillations and a faster smaller vasomotor oscillation. There are nonlinear interactions between TGF and the myogenic mechanism, which include the modulation of the frequency and amplitude of the myogenic oscillation by TGF. The prediction of modulation is confirmed in a companion study (28).     

One-dimensional computer analysis of oscillatory flow in rigid tubes.

The dynamic characteristics of catheter-transducer systems using rigid tubes with compliance lumped in the transducer and oscillatory flow of fluid in rigid tubes were analyzed. A digital computer model based on one dimensional laminar oscillatory flow was developed and verified by exact solution of the Navier-Stokes Equation. Experimental results indicated that the damping ratio and resistance is much higher at higher frequencies of oscillation than predicted by the one dimensional model. An empirical correction factor was developed and incorporated into the computer model to correct the model to the experimental data. Amplitude of oscillation was found to have no effect on damping ratio so it was concluded that the increased damping ratio and resistance at higher frequencies was not due to turbulence but to two dimensional flow effects. Graphs and equations were developed to calculate damping ratio and undamped natural frequency of a catheter-transducer system from system parameters. Graphs and equations were also developed to calculate resistance and inertance for oscillatory flow in rigid tubes from system parameters and frequency of oscillation.     

Oscillatory isozymes as the simplest model for coupled biochemical oscillators

We analyze a simple model for two autocatalytic reactions catalyzed by two distinct isozymes transforming, with different kinetic properties, a given substrate into the same product. This two-variable system can be viewed as the simplest model of chemically coupled biochemical oscillators. Phase-plane analysis indicates how the kinetic differences between the two enzymes give rise to complex oscillatory phenomena such as the coexistence of a stable steady state and a stable limit cycle, or the co-existence of two simultaneously stable oscillatory regimes (birhythmicity). The model allows one to verify a previously proposed conjecture for the origin of birhythmicity. In other conditions, the system admits multiple oscillatory domains as a function of a control parameter whose variation gives rise to markedly different types of oscillations. The latter behavior provides an explanation for the occurrence of multiple modes of oscillations in thalamic neurons.     

Dynamic reduction with applications to mathematical biology and other areas

In a difference or differential equation one is usually interested in finding solutions having certain properties, either intrinsic properties (e.g. bounded, periodic, almost periodic) or extrinsic properties (e.g. stable, asymptotically stable, globally asymptotically stable). In certain instances it may happen that the dependence of these equations on the state variable is such that one may (1) alter that dependency by replacing part of the state variable by a function from a class having some of the above properties and (2) solve the 'reduced' equation for a solution having the remaining properties and lying in the same class. This then sets up a mapping Τ of the class into itself, thus reducing the original problem to one of finding a fixed point of the mapping. The procedure is applied to obtain a globally asymptotically stable periodic solution for a system of difference equations modeling the interaction of wild and genetically altered mosquitoes in an environment yielding periodic parameters. It is also shown that certain coupled periodic systems of difference equations may be completely decoupled so that the mapping Τ is established by solving a set of scalar equations. Periodic difference equations of extended Ricker type and also rational difference equations with a finite number of delays are also considered by reducing them to equations without delays but with a larger period. Conditions are given guaranteeing the existence and global asymptotic stability of periodic solutions.     

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.     

Dynamic Flight Stability of a Model Hoverfly in Inclined-Stroke-Plane Hovering

Most hovering insects flap their wings in a horizontal plane,called ‘normal hovering'.But some of the best hoverers,e.g.true hoverflies,hover with an inclined stroke plane.In the present paper,the longitudinal dynamic flight stability of a model hoverfly in inclined-stroke-plane hovering was studied.Computational fluid dynamics was used to compute the aerodynamic derivatives and the eigenvalue and eigenvector analysis was used to solve the equations of motion.The primary findings are as follows.(1) For inclined-stroke-plane hovering,the same three natural modes of motion as those for normal hovering were identified:one unstable oscillatory mode,one stable fast subsidence mode,and one stable slow subsidence mode.The unstable oscillatory mode and the fast subsidence mode mainly have horizontal translation and pitch rotation,and the slow subsidence mode mainly has vertical translation.(2) Because of the existence of the unstable oscillatory mode,inclined-stroke-plane hovering flight is not stable.(3) Although there are large differences in stroke plane and body orientations between the inclined-stroke-plane hovering and normal hovering,the relative position between the mean center of pressure and center of mass for these two cases is not very different,resulting in similar stability derivatives,hence similar dynamic stability properties for these two types of hovering.     

On the infusion of a therapeutic agent into a solid tumor modeled as a poroelastic medium

The direct infusion of an agent into a solid tumor, modeled as a spherical poroelastic material with anisotropic dependence of the tumor hydraulic conductivity upon the tissue deformation, is treated both by solving the coupled fluid/elastic equations, and by expressing the solution as an asymptotic expansion in terms of a small parameter, ratio between the driving pressure force in the fluid system, and the elastic properties of the solid. Results at order one match almost perfectly the solutions of the full system over a large range of infusion pressures. Comparison with experimental results is acceptable after the hydraulic conductivity of the medium is properly calibrated. Given the uncertain estimates of some model constants, the order zero solution of the expansion, for which fluid and porous matrix are decoupled, yields acceptable values and trends for all the physical fields of interest, rendering the coupled analysis (in the limit of small displacements) of little use. When the deformation of the tissue becomes large nonlinear elasticity theory must be resorted to.     

The effect of solution non-ideality on membrane transport in three-dimensional models of the renal concentrating mechanism

Previous models of the renal concentrating mechanism employ ideal approximations of solution thermodynamics for membrane transport calculation. In three-dimensional models of the renal medulla, predicted urine concentrations reach levels where there idealized approximations begin to break down. In this paper we derive equations that govern membrane transport for non-dilute solutions and use these equations in a three-dimensional model of the concentrating mechanism. New numerical methods were employed that are more stable than those employed previously. Compared to ideal solution models, the urea non-ideality tends to increase predicted osmolarities, whereas NaCl non-ideality decreases predictions.     

The Relation Between the Late After-Potential and the Size of the Transverse Tubular System of Frog Muscle

This is an investigation of the effects on the late after-potential of immersing frog sartorius muscles in three kinds of modified Ringer's fluid; hypertonic, low chloride, and potassium-free. The late after-potential has been attributed to the depolarizing effect of an accumulation of potassium, during a preceding train of impulses, in the intermediary space of the model of a muscle fiber proposed by Adrian and Freygang. Both the hypertonic and low chloride solutions prolonged the late after-potential reversibly and the potassium-free solution shortened it. The effect of the low potassium solution fitted those data calculated from the model, but the effect of the hypertonic and low chloride solutions required an increase in size of the intermediary space of the model in order to fit the calculated data. An electron microscopic study of the muscles showed that the size of the transverse tubular system changed reversibly in the hypertonic and low chloride solutions in almost the amount necessary to fit the experimental data to the calculated data. This agreement between the change in size of the transverse tubular system and that of the intermediary space indicates that the intermediary space may be the transverse tubular system.     

Dynamics of a biochemical system with multiple oscillatory domains as a clue for multiple modes of neuronal oscillations

We analyze the behavior of a two-variable biochemical model in conditions where it admits multiple oscillatory domains in parameter space. The model represents an autocatalytic enzyme reaction with input of substrate both from a constant source and from non-linear recycling of product into substrate. This system was previously studied for birhythmicity, i.e. the coexistence between two stable periodic regimes (Moran and Goldbeter 1984), and for multithreshold excitability (Moran and Goldbeter 1985). When two distinct oscillatory domains obtain as a function of the substrate injection rate, the system is capable of exhibiting two markedly different modes of oscillations for slightly different values of this control parameter. Phase plane analysis shows how the multiplicity of oscillatory domains depends on the parameters that govern the underlying biochemical mechanism of product recycling. We analyze the response of the model to various kinds of transient perturbations and to periodic changes in the substrate input that bring the system through the two ranges of oscillatory behavior. The results provide a qualitative explanation for experimental observations (Jahnsen and Llinas 1984b) related to the occurrence of two different modes of oscillations in thalamic neurones.     

A SPAK isoform switch modulates renal salt transport and blood pressure

The renal thick ascending limb (TAL) and distal convoluted tubule (DCT) play central roles in salt homeostasis and blood pressure regulation. An emerging model suggests that bumetanide- and thiazide-sensitive NaCl transporters (NKCC2 and NCC) along these segments are phosphorylated and activated by WNK kinases, via SPAK and OSR1. Here, we show that a kidney-specific SPAK isoform, which lacks the kinase domain, inhibits phosphorylation of NCC and NKCC2 by full-length SPAK in?vitro. Kidney-specific SPAK is highly expressed along the TAL, whereas full-length SPAK is more highly expressed along the DCT. As predicted from the differential expression, SPAK knockout in animals has divergent effects along TAL and DCT, with increased phosphorylated NKCC2 along TAL and decreased phosphorylated NCC along DCT. In mice, extracellular fluid volume depletion shifts SPAK isoform abundance to favor NaCl retention along both segments, indicating that a SPAK isoform switch modulates sodium avidity along the distal nephron.