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.     

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.     

Feedback-mediated dynamics in a model of a compliant thick ascending limb

The tubuloglomerular feedback (TGF) system in the kidney, which is a key regulator of filtration rate, has been shown in physiologic experiments in rats to mediate oscillations in tubular fluid pressure and flow, and in NaCl concentration in the tubular fluid of the thick ascending limb (TAL). In this study, we developed a mathematical model of the TGF system that represents NaCl transport along a TAL with compliant walls. The model was used to investigate the dynamic behaviors of the TGF system. A bifurcation analysis of the TGF model equations was performed by deriving and finding roots of the characteristic equation, which arises from a linearization of the model equations. Numerical simulations of the full model equations were conducted to assist in the interpretation of the bifurcation analysis. These techniques revealed a complex parameter region that allows a variety of qualitatively different model solutions: a regime having one stable, time-independent steady-state solution; regimes having one stable oscillatory solution only; and regimes having multiple possible stable oscillatory solutions. Model results suggest that the compliance of the TAL walls increases the tendency of the model TGF system to oscillate.     

On the nature of global plasma oscillations in a tokamak. Part II: Spatial structure and propagation of perturbations observed at the T-10 tokamak

Large-scale plasma oscillations (so-called MHD oscillations) observed at the T-10 tokamak are investigated. The central electron cyclotron heating was used to enhance oscillations at the m/n = 1/1 mode with the goal of determining the internal characteristics of the process. The spatially resolved electron cyclotron emission diagnostics allowed analyzing the propagation characteristics of plasma perturbations. The experiments have revealed that excitation of oscillations in a particular mode occur simultaneously in the entire area located within the corresponding rational magnetic surface. The propagation of plasma perturbations along the torus is found to be inhomogeneous. The electron cyclotron emission diagnostics allowed finding eigen (resonance) frequencies of plasma oscillations from the parameters of their inhomogeneous propagation in the plasma core and comparing them with spectra of oscillations of the magnetic field induced by the plasma current in the edge plasma, which were recorded by magnetic probes. It is established that the frequencies of eigenmodes are independent of the electron temperature, plasma density, and auxiliary heating power. Even spatial harmonics of the principal magnetic surface are observed under strong excitation of oscillations. The rational magnetic surfaces that determine oscillation harmonics retain their position during the entire steady-state phase of the total plasma current in spite of the strong sharpening of the temperature profile due to central heating.     

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.     

Spatial instabilities and local oscillations in a lattice gas Lotka–Volterra model

 We analyze the evolution of spatially inhomogeneous perturbations in a lattice gas model for a prey-predator population. Starting with the master equation of the model, decoupled by means of a mean field approximation, spatial instabilities are seen to take place in a region of the phase diagram. This is in qualitative agreement with local oscillations already observed in computer simulations. We determine the transition line that separates the homogeneous region from the inhomogeneous region and we study the spatio-temporal self-organized structures that appear inside the inhomogeneous region. Received 3 November 1995; received in revised form 26 January 1996     

Ladder network prediction of transients in linear spatially inhomogeneous cables

A numerical method is described for finding steady state and transient responses in electrically linear, spatially inhomogeneous cables. Spatial inhomogeneities are incorporated by representing the cable by a number of finite length uniform cylindrical segments, each having the radius and electrical characteristics of a small region along the cable. Input waveforms are approximated by truncated Fourier series of sinusoidal components. Output waveforms are produced by multiplying the input Fourier series sinusoids by their respective transfer functions between input and output points on the cable and summing the resultant output point sinusoids. The transfer functions, representing attenuation and phase shift for each input sinusoid, are obtained by numerical analysis of an electrical ladder network derived from the cylindrical segment model of the cable. Results are shown for application of this method to both cylindrical and expanding radius cable geometries.     

Isosmotic volume reabsorption in rat proximal tubule

A theoretical model incorporation both active and passive forces has been developed for fluid reabsorption from split oil droplets in rat intermediate and late proximal tubule. Of necessity, simplifying assumptions have been introduced; we have assumed that the epithelium can be treated as a single membrane and that the membrane "effective" HCO3 permeability is near zero. Based on this model with its underlying assumptions, the following conclusions are drawn. Regardless of the presence or absence of active NaCl transport, fluid reabsorption from the split oil droplet is isosmotic. The reabsorbate osmolarity can be affected by changes in tubular permeability parameters and applied forces but is not readily altered from an osmolarity essentially equal to that of plasma. In a split droplet, isosmotic flow need not be a special consequence of active Na transport, is not the result of a particular set of permeability properties, and is not merely a trivial consequence of a very high hydraulic conductivity; isosmotic flow can be obtained with hydraulic conductivity nearly an order of magnitude lower than that previously measured in the rat proximal convoluted tubule. Isosmotic reabsorption is, in part, the result of the interdependence of salt and water flows, their changing in parallel, and thus their ratio, the reabsorbate concentration being relatively invariant. Active NaCl transport can cause osmotic water flow by reducing the luminal fluid osmolarity. In the presence of passive forces the luminal fluid can be hypertonic to plasma, and active NaCl transport can still exert its osmotic effect on volume flow. There are two passive forces for volume flow: the Cl gradient and the difference in effective osmotic pressure; they have an approximately equivalent effect on volume flow. Experimentally, we have measured volume changes in a droplet made hyperosmotic by the addition of 50 mM NaCl; the experimental results are predicted reasonably well by our theoretical model.     

Hypotonicity-induced increases in duodenal mucosal permeability facilitates adjustment of luminal osmolality

The integrated response to hypotonic NaCl solutions (100, 50, 25, and 0 mM NaCl) in proximal duodenum of anesthetized rats was examined. Luminal alkalinization, fluid flux, duodenal contractions, blood-to-lumen clearance of 51Cr-labeled EDTA (mucosal permeability), and perfusate osmolality were studied in the absence and presence of the cyclooxygenase inhibitor indomethacin. In response to hypotonic solutions net fluid absorption, increases in permeability and perfusate osmolality were markedly higher in indomethacin-treated animals than in controls, and these effects were diminished by the nicotinic-receptor antagonist hexamethonium. Infusion of iloprost, a stable PGI2 analog, to indomethacin-treated animals markedly reduced the hypotonicity-induced increase in mucosal permeability and diminished the rise in perfusate osmolality. Lowering the NaCl concentration in the perfusion solution but maintaining isotonicity with mannitol had no effect on mucosal permeability. Very good linear correlations were obtained between the degree of luminal hypotonicity and the increase in permeability and between increases in permeability and perfusate osmolality. It is concluded that luminal hypotonicity increases duodenal mucosal permeability. The hypotonicity-induced increase in permeability modulated by prostaglandins and nicotinic receptors fulfills the function of increasing blood-to-lumen transport of Na+ facilitating adjustment of luminal osmolality.     

Kinetics of Na(+) transport in necturus proximal tubule

The dependence of proximal tubular sodium and fluid readsorption on the Na(+) concentration of the luminal and peritubular fluid was studied in the perfused necturus kidney. Fluid droplets, separated by oil from the tubular contents and identical in composition to the vascular perfusate, were introduced into proximal tubules, reaspirated, and analyzed for Na(+) and [(14)C]mannitol. In addition, fluid transport was measured in short-circuited fluid samples by observing the rate of change in length of the split droplets in the tubular lumen. Both reabsorptive fluid and calculated Na fluxes were simple, storable functions of the perfusate Na(+) concentration (K(m) = 35-39 mM/liter, V(max) = 1.37 control value). Intracellular Na(+), determined by tissue analysis, and open-circuit transepithelial electrical potential differences were also saturable functions of extracellular Na(+). In contrast, net reabsorptive fluid and Na(+) fluxes were linearly dependent on intracellular Na(+) and showed no saturation, even at sharply elevated cellular sodium concentrations. These concentrations were achieved by addition of amphotericin B to the luminal perfusate, a maneuver which increased the rate of Na(+) entry into the tubule cells and caused a proportionate rise in net Na(+) flux. It is concluded that active peritubular sodium transport in proximal tubule cells of necturus is normally unsaturated and remains so even after amphotericin-induced enhancement of luminal Na(+) entry. Transepithelial movement of NaCl may be described by a model with a saturable luminal entry step of Na(+) or NaCl into the cell and a second, unsaturated active transport step of Na(+) across the peritubular cell boundary.     

The water permeability of toad urinary bladder. II. The value of Pf/Pd(w) for the antidiuretic hormone-induced water permeation pathway

Using the methods described in the preceding paper (Levine et al., 1984) for measuring the magnitude of the water-permeable barriers in series with the luminal membrane, we correct measured values of Pd(w) in bladders stimulated with low doses of antidiuretic hormone (ADH) or 8-bromo cyclic AMP to obtain their true values in the luminal membrane. Simultaneously, we also determine Pf. We thus are able to calculate Pf/Pd(w) for the hormone-induced water permeation pathway in the luminal membrane. Our finding is that Pf/Pd(w) approximately equal to 17. Two channel models consistent both with this value and the impermeability of the ADH-induced water permeation pathway to small nonelectrolytes are: (a) a long (approximately equal to 50 A), small- radius (approximately equal to 2 A) pore through which 17 water molecules pass in single-file array, and (b) a shower-head-like structure in which the stem is long and of large radius (approximately equal to 20 A) and the cap has numerous short, small-radius (approximately equal to 2 A) pores. A third possibility is that whereas the selective permeability to H2O results from small-radius (approximately equal to 2 A) pores, the large value of Pf/Pd(w) arises from their location in the walls of long tubular vesicles (approximately 2 micron in length and 0.1 micron in diameter) that are functionally part of the luminal membrane after having fused with it. Aggregate-containing tubular vesicles of these dimensions have been reported to fuse with the luminal membrane in response to ADH stimulation and have been implicated in the ADH-induced hydroosmotic response.     

Stabilization through spatial pattern formation in metapopulations with long-range dispersal

Many studies of metapopulation models assume that spatially extended populations occupy a network of identical habitat patches, each coupled to its nearest neighbouring patches by density-independent dispersal. Much previous work has focused on the temporal stability of spatially homogeneous equilibrium states of the metapopulation, and one of the main predictions of such models is that the stability of equilibrium states in the local patches in the absence of migration determines the stability of spatially homogeneous equilibrium states of the whole metapopulation when migration is added. Here, we present classes of examples in which deviations from the usual assumptions lead to different predictions. In particular, heterogeneity in local habitat quality in combination with long-range dispersal can induce a stable equilibrium for the metapopulation dynamics, even when within-patch processes would produce very complex behaviour in each patch in the absence of migration. Thus, when spatially homogeneous equilibria become unstable, the system can often shift to a different, spatially inhomogeneous steady state. This new global equilibrium is characterized by a standing spatial wave of population abundances. Such standing spatial waves can also be observed in metapopulations consisting of identical habitat patches, i.e. without heterogeneity in patch quality, provided that dispersal is density dependent. Spatial pattern formation after destabilization of spatially homogeneous equilibrium states is well known in reaction–diffusion systems and has been observed in various ecological models. However, these models typically require the presence of at least two species, e.g. a predator and a prey. Our results imply that stabilization through spatial pattern formation can also occur in single-species models. However, the opposite effect of destabilization can also occur: if dispersal is short range, and if there is heterogeneity in patch quality, then the metapopulation dynamics can be chaotic despite the patches having stable equilibrium dynamics when isolated. We conclude that more general metapopulation models than those commonly studied are necessary to fully understand how spatial structure can affect spatial and temporal variation in population abundance.     

Temporal oscillations in neuronal nets

Summary A model for the interactions of cortical neurons is derived and analyzed. It is shown that small amplitude spatially inhomogeneous standing oscillations can bifurcate from the rest state. In a periodic domain, traveling wave trains exist. Stability of these patterns is discussed in terms of biological parameters. Homoclinic and heteroclinic orbits are demonstrated for the space-clamped system.The research reported in this paper was supported in part by NIH GM2037     

Analysis of standing droplets in rat proximal tubules

Volume, osmolality, and concentrations for Na, Cl, and raffinose have been measured as a function of time in standing droplets within rat intermediate and late proximal tubules. Standing droplet reabsorption proceeds without the development of a measurable osmotic difference across the epithelium. After 140 s of tubular exposure, droplet-to- plasma concentration differences are observed for raffinose, Na, and Cl with the observed Na concentration difference, usually referred to as limiting gradient, being approximately 9 mM. It is possible that a smaller or even no limiting difference would be attained with longer exposure times. Previous values measured for the limiting Na concentration in the rat proximal tubule were determined before the attainment of constant concentrations. Assuming that the Na concentration we measured is the limiting value, we estimate that active NaCl transport accounts for a very small fraction, less than 6%, of the volume reabsorption; using an alternative approach of fitting a theoretical model to our experimental data, active NaCl transport is again estimated to account for only 6% of the total reabsorbate. The previous interpretation that a limiting Na concentration gradient constitutes the most direct evidence for active Na transport may be in error; the gradient we measure can be modeled without incorporating active NaCl transport.     

Dynamics of cellular homeostasis: Recovery time for a perturbation from equilibrium

In the collecting ductin vivo, the principal cell encounters a wide range in luminal flow rate and luminal concentration of NaCl. As a consequence, there are substantial variations in the transcellular fluxes of Na⁺ and Cl⁻, conditions which would be expected to perturb cell volume and cytosolic concentrations. Several control mechanisms have been identified which can potentially blunt these perturbations, and these entail cellular regulation of the luminal membrane Na⁺ channel and peritubular membrane K⁺ and Cl⁻ channels. To illustrate the impact of these regulated channels, a mathematical model of the principal cell of the rat cortical collecting duct has been developed, in which ion channel permeabilities are either constant or regulated. In comparison to the model with fixed permeabilities, the model with regulated channels demonstrates enhanced cellular homeostasis following steady-state variation in luminal NaCl. However, in the transient response to a cytosolic perturbation, the difference in recovery time between the models is small. An approximate analysis is presented which casts these models as dynamical systems with constant coefficients. Despite the presence of regulated ion channels, concordance of each model with its linear approximation is verified for experimentally meaningful perturbations from the reference condition. Solution of a Lyapunov equation for each linear system yields a matrix whose application to a perturbation permits explicit estimation of the time to recovery. Comparison of these solution matrices for regulated and non-regulated cells confirms the similarity of the dynamic response of the two models. These calculations suggest that enhanced homeostasis by regulated channels may be protective, without necessarily hastening recovery from cellular perturbations.     

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.