Parameter estimation for mathematical models of a nongastric H+(Na+)-K(+)(NH4+)-ATPase.

The role of nongastric H(+)-K(+)-ATPase (HKA) in ion homeostasis of macula densa (MD) cells is an open question. To begin to explore this issue, we developed two mathematical models that describe ion fluxes through a nongastric HKA. One model assumes a 1H(+):1K(+)-per-ATP stoichiometry; the other assumes a 2H(+):2K(+)-per-ATP stoichiometry. Both models include Na+ and NH4+ competitive binding with H+ and K+, respectively, a characteristic observed in vitro and in situ. Model rate constants were obtained by minimizing the distance between model and experimental outcomes. Both 1H(+)(1Na(+)):1K(+)(1NH4 (+))-per-ATP and 2H(+)(2Na(+)):2K(+)(2NH4 (+))-per-ATP models fit the experimental data well. Using both models, we simulated ion net fluxes as a function of cytosolic or luminal ion concentrations typical for the cortical thick ascending limb and MD region. We observed that (1) K+ and NH4+ flowed in the lumen-to-cytosol direction, (2) there was competitive behavior between luminal K+ and NH4+ and between cytosolic Na+ and H+, 3) ion fluxes were highly sensitive to changes in cytosolic Na+ or H+ concentrations, and 4) the transporter does mostly Na+ / K+ exchange under physiological conditions. These results support the concept that nongastric HKA may contribute to Na+ and pH homeostasis in MD cells. Furthermore, in both models, H+ flux reversed at a luminal pH that was <5.6. Such reversal led to Na+ / H+ exchange for a luminal pH of <2 and 4 in the 1:1-per-ATP and 2:2-per-ATP models, respectively. This suggests a novel role of nongastric HKA in cell Na+ homeostasis in the more acidic regions of the renal tubules.

Transport efficiency and workload distribution in a mathematical model of the thick ascending limb.

The thick ascending limb (TAL) is a major NaCl reabsorbing site in the nephron. Efficient reabsorption along that segment is thought to be a consequence of the establishment of a strong transepithelial potential that drives paracellular Na(+) uptake. We used a multicell mathematical model of the TAL to estimate the efficiency of Na(+) transport along the TAL and to examine factors that determine transport efficiency, given the condition that TAL outflow must be adequately dilute. The TAL model consists of a series of epithelial cell models that represent all major solutes and transport pathways. Model equations describe luminal flows, based on mass conservation and electroneutrality constraints. Empirical descriptions of cell volume regulation (CVR) and pH control were implemented, together with the tubuloglomerular feedback (TGF) system. Transport efficiency was calculated as the ratio of total net Na(+) transport (i.e., paracellular and transcellular transport) to transcellular Na(+) transport. Model predictions suggest that 1) the transepithelial Na(+) concentration gradient is a major determinant of transport efficiency; 2) CVR in individual cells influences the distribution of net Na(+) transport along the TAL; 3) CVR responses in conjunction with TGF maintain luminal Na(+) concentration well above static head levels in the cortical TAL, thereby preventing large decreases in transport efficiency; and 4) under the condition that the distribution of Na(+) transport along the TAL is quasi-uniform, the tubular fluid axial Cl(-) concentration gradient near the macula densa is sufficiently steep to yield a TGF gain consistent with experimental data.

Fluid dilution and efficiency of Na(+) transport in a mathematical model of a thick ascending limb cell.

Thick ascending limb (TAL) cells are capable of reducing tubular fluid Na(+) concentration to as low as ~25 mM, and yet they are thought to transport Na(+) efficiently owing to passive paracellular Na(+) absorption. Transport efficiency in the TAL is of particular importance in the outer medulla where O(2) availability is limited by low blood flow. We used a mathematical model of a TAL cell to estimate the efficiency of Na(+) transport and to examine how tubular dilution and cell volume regulation influence transport efficiency. The TAL cell model represents 13 major solutes and the associated transporters and channels; model equations are based on mass conservation and electroneutrality constraints. We analyzed TAL transport in cells with conditions relevant to the inner stripe of the outer medulla, the cortico-medullary junction, and the distal cortical TAL. At each location Na(+) transport efficiency was computed as functions of changes in luminal NaCl concentration ([NaCl]), [K(+)], [NH(4)(+)], junctional Na(+) permeability, and apical K(+) permeability. Na(+) transport efficiency was calculated as the ratio of total net Na(+) transport to transcellular Na(+) transport. Transport efficiency is predicted to be highest at the cortico-medullary boundary where the transepithelial Na(+) gradient is the smallest. Transport efficiency is lowest in the cortex where luminal [NaCl] approaches static head.

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.

A mathematical model of the myogenic response to systolic pressure in the afferent arteriole.

Elevations in systolic blood pressure are believed to be closely linked to the pathogenesis and progression of renal diseases. It has been hypothesized that the afferent arteriole (AA) protects the glomerulus from the damaging effects of hypertension by sensing increases in systolic blood pressure and responding with a compensatory vasoconstriction (Loutzenhiser R, Bidani A, Chilton L. Circ Res 90: 1316-1324, 2002). To investigate this hypothesis, we developed a mathematical model of the myogenic response of an AA wall, based on an arteriole model (Gonzalez-Fernandez JM, Ermentrout B. Math Biosci 119: 127-167, 1994). The model incorporates ionic transport, cell membrane potential, contraction of the AA smooth muscle cell, and the mechanics of a thick-walled cylinder. The model represents a myogenic response based on a pressure-induced shift in the voltage dependence of calcium channel openings: with increasing transmural pressure, model vessel diameter decreases; and with decreasing pressure, vessel diameter increases. Furthermore, the model myogenic mechanism includes a rate-sensitive component that yields constriction and dilation kinetics similar to behaviors observed in vitro. A parameter set is identified based on physical dimensions of an AA in a rat kidney. Model results suggest that the interaction of Ca(2+) and K(+) fluxes mediated by voltage-gated and voltage-calcium-gated channels, respectively, gives rise to periodicity in the transport of the two ions. This results in a time-periodic cytoplasmic calcium concentration, myosin light chain phosphorylation, and cross-bridge formation with the attending muscle stress. Furthermore, the model predicts myogenic responses that agree with experimental observations, most notably those which demonstrate that the renal AA constricts in response to increases in both steady and systolic blood pressures. The myogenic model captures these essential functions of the renal AA, and it may prove useful as a fundamental component in a multiscale model of the renal microvasculature suitable for investigations of the pathogenesis of hypertensive renal diseases.

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.

Attenuation of isoflurane-induced preconditioning and reactive oxygen species production in the senescent rat heart.

BACKGROUND: Although attenuation of anesthetic preconditioning in aged ex vivo heart models has been studied extensively, there are no comparable in vivo studies. To extend previous work and to address a possible mechanism underlying age-related differences, we investigated isoflurane-induced preconditioning and reactive oxygen species (ROS) production in the aged rat heart in vivo. METHODS: Male Fisher 344 rats were assigned from their respective age groups (young, 3-5 mo; old, 20-24 mo) to either receive 30 min of 1.0 minimum alveolar concentration isoflurane or to a control group. Rats were subjected to coronary artery occlusion for 30 min followed by 2 h of reperfusion. A fluorescent probe for superoxide anion production (dihydroethidium, 1 mg) was administered in the absence of the isoflurane or just before isoflurane exposure in four additional groups. Myocardial infarct size and superoxide anion production were assessed using triphenyltetrazolium staining and epifluorescence microscopy, respectively. RESULTS: Isoflurane decreased myocardial infarct size of young rats (26.7% +/- 3.0%) compared with young controls (50.9% +/- 1.9%; P < 0.001), whereas isoflurane did not significantly affect myocardial infarct size of old rats (39.1% +/- 0.9%) compared with old controls (46.5% +/- 2.4%; P > 0.05). Isoflurane increased ROS levels in young rats (430.5 +/- 95.9 arbitrary units [AU]) compared with young controls (162.7 +/- 25.5 AU; P < 0.01). In contrast, no significant changes in ROS levels were observed in old animals (316.4 +/- 56.3 AU isoflurane versus 233.8 +/- 59.2 AU control). CONCLUSIONS: Reduction in the cardioprotective effects of isoflurane and attenuation of isoflurane-stimulated ROS production were observed in the senescent myocardium in vivo.

A mathematical model of pH(i) regulation in central CO2- chemoreception.

In CO2 chemosensitive neurons, an increase in CO2 (hypercapnia) leads to a maintained reduction in intracellular pH (pH(i)) while in non-chemosensitive neurons pH(i) recovery is observed. The precise mechanisms for the differential regulation of pH(i) recovery between these cell populations remain to be identified; however, studies have begun to explore the role of Na+/H+ exchange (NHE). Here, we compare the results of two different formulations of a mathematical model to begin to explore pH(i) regulation in central CO2 chemoreception.

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.

Identification of transient renal autoregulatory mechanisms using time-frequency spectral techniques.

Identification of the two principal mediators of renal autoregulation from time-series data is difficult, as both the tubuloglomerular feedback (TGF) and myogenic (MYO) mechanisms interact and share a common effector, the afferent arteriole. Moreover, although both mechanisms can exhibit oscillations in well-characterized frequency bands, these systems often operate in nonoscillatory states not detectable by frequency-domain analysis. To overcome these difficulties, we have developed a new approach to the characterization of the TGF and MYO systems. A laser Doppler probe is used to measure fluctuations in local cortical blood flow (CBF) in response to spontaneous changes in blood pressure (BP) and to large imposed perturbations in BP, which elicit strong, simultaneous, transient, oscillatory blood flow responses. These transient responses are identified by high-resolution time-frequency spectral analysis of the time-series data. In this report, we compare four different time-frequency spectral techniques (the short-time Fourier transform (STFT), smoothed pseudo Wigner-Ville, and two recently developed methods: the Hilbert-Huang transform and time varying optimal parameter search (TVOPS)) to determine which of these four methods is best suited for the identification of transient oscillations in renal autoregulatory mechanisms. We found that TVOPS consistently provided the best performance in both simulation examples and identification of the two autoregulatory mechanisms in actual data. While the STFT suffers in time and frequency resolution as compared to the other three methods, it was able to identify the two autoregulatory mechanisms. Taken together, our experience suggests a two level approach to the analysis of renal blood flow (RBF) data: STFT to obtain a low-resolution time-frequency spectrogram, followed by the use of a higher resolution technique, such as the TVOPS, if even higher time-frequency resolution of the transient responses is required.

A novel quantitative method for diabetic cardiac autonomic neuropathy assessment in type 1 diabetic mice.

In this work, we used a sensitive and noninvasive computational method to assess diabetic cardiovascular autonomic neuropathy (DCAN) from pulse oximeter (photoplethysmographic; PPG) recordings from mice. The method, which could be easily applied to humans, is based on principal dynamic mode (PDM) analysis of heart rate variability (HRV). Unlike the power spectral density, PDM has been shown to be able to separately identify the activities of the parasympathetic and sympathetic nervous systems without pharmacological intervention. HRV parameters were measured by processing PPG signals from conscious 1.5- to 5-month-old C57/BL6 control mice and in Akita mice, a model of insulin-dependent type 1 diabetes, and compared with the gold-standard Western blot and immunohistochemical analyses. The PDM results indicate significant cardiac autonomic impairment in the diabetic mice in comparison to the controls. When tail-cuff PPG recordings were collected and analyzed starting from 1.5 months of age in both C57/Bl6 controls and Akita mice, onset of DCAN was seen at 3 months in the Akita mice, which persisted up to the termination of the recording at 5 months. Western blot and immunohistochemical analyses also showed a reduction in nerve density in Akita mice at 3 and 4 months as compared to the control mice, thus, corroborating our PDM data analysis of HRV records. Western blot analysis of autonomic nerve proteins corroborated the PPG-based HRV analysis via the PDM approach. In contrast, traditional HRV analysis (based on either the power spectral density or time-domain measures) failed to detect the nerve rarefaction.

The effect of size and species on lens intracellular hydrostatic pressure.

PURPOSE: Previous experiments showed that mouse lenses have an intracellular hydrostatic pressure that varied from 335 mm Hg in central fibers to 0 mm Hg in surface cells. Model calculations predicted that in larger lenses, all else equal, pressure should increase as the lens radius squared. To test this prediction, lenses of different radii from different species were studied. METHODS: All studies were done in intact lenses. Intracellular hydrostatic pressures were measured with a microelectrode-manometer-based system. Membrane conductances were measured by frequency domain impedance analysis. Intracellular Na(+) concentrations were measured by injecting the Na(+)-sensitive dye sodium-binding benzofuran isophthalate. RESULTS: Intracellular hydrostatic pressures were measured in lenses from mice, rats, rabbits, and dogs with radii (cm) 0.11, 0.22, 0.49, and 0.57, respectively. In each species, pressure varied from 335 ± 6 mm Hg in central fiber cells to 0 mm Hg in surface cells. Further characterization of transport in lenses from mice and rats showed that the density of fiber cell gap junction channels was approximately the same, intracellular Na(+) concentrations varied from 17 mM in central fiber cells to 7 mM in surface cells, and intracellular voltages varied from -45 mV in central fiber cells to -60 mV in surface cells. Fiber cell membrane conductance was a factor of 2.7 times larger in mouse than in rat lenses. CONCLUSIONS: Intracellular hydrostatic pressure is an important physiological parameter that is regulated in lenses from these different species. The most likely mechanism of regulation is to reduce the density of open Na(+)-leak channels in fiber cells of larger lenses.

Lens intracellular hydrostatic pressure is generated by the circulation of sodium and modulated by gap junction coupling.

We recently modeled fluid flow through gap junction channels coupling the pigmented and nonpigmented layers of the ciliary body. The model suggested the channels could transport the secretion of aqueous humor, but flow would be driven by hydrostatic pressure rather than osmosis. The pressure required to drive fluid through a single layer of gap junctions might be just a few mmHg and difficult to measure. In the lens, however, there is a circulation of Na(+) that may be coupled to intracellular fluid flow. Based on this hypothesis, the fluid would cross hundreds of layers of gap junctions, and this might require a large hydrostatic gradient. Therefore, we measured hydrostatic pressure as a function of distance from the center of the lens using an intracellular microelectrode-based pressure-sensing system. In wild-type mouse lenses, intracellular pressure varied from ∼330 mmHg at the center to zero at the surface. We have several knockout/knock-in mouse models with differing levels of expression of gap junction channels coupling lens fiber cells. Intracellular hydrostatic pressure in lenses from these mouse models varied inversely with the number of channels. When the lens' circulation of Na(+) was either blocked or reduced, intracellular hydrostatic pressure in central fiber cells was either eliminated or reduced proportionally. These data are consistent with our hypotheses: fluid circulates through the lens; the intracellular leg of fluid circulation is through gap junction channels and is driven by hydrostatic pressure; and the fluid flow is generated by membrane transport of sodium.

Parameter estimation for mathematical models of NKCC2 cotransporter isoforms.

An optimization problem, formulated using a nonlinear least-squares approach, was used to estimate parameters for kinetic models of the three isoforms of the kidney-specific Na-K-2Cl (NKCC2) cotransporter. Specifically, the optimization problem estimates the magnitude of model parameters (i.e., off-binding and translocation rate constants) by minimizing the distance between model unidirectional fluxes and published unidirectional (86)Rb(+) uptake curves for the A, B, and F isoforms of the NKCC2 cotransporter obtained in transfected Xenopus oocytes. By using different symmetry assumptions, NKCC2 models with five, six, seven, or eight parameters were evaluated. The optimization method identified parameter sets that yielded computed unidirectional fluxes consistent with the uptake data. However, the parameter values were not unique, in that systematic exploration of the parameter space revealed alternative parameter sets that fit the data with similar accuracy. Finally, we demonstrate that the optimization method can identify parameter sets for the three transporter isoforms that differ only in ion binding affinities, a result that is consistent with a published mutagenesis analysis of the molecular and structural bases for the differences in (86)Rb(+) uptake among the A, B, and F isoforms. These NKCC2 cotransporter models will facilitate the development of larger scale models of ion transport by thick ascending limb cells.

Analysis of nonstationarity in renal autoregulation mechanisms using time-varying transfer and coherence functions.

The extent to which renal blood flow dynamics vary in time and whether such variation contributes substantively to dynamic complexity have emerged as important questions. Data from Sprague-Dawley rats (SDR) and spontaneously hypertensive rats (SHR) were analyzed by time-varying transfer functions (TVTF) and time-varying coherence functions (TVCF). Both TVTF and TVCF allow quantification of nonstationarity in the frequency ranges associated with the autoregulatory mechanisms. TVTF analysis shows that autoregulatory gain in SDR and SHR varies in time and that SHR exhibit significantly more nonstationarity than SDR. TVTF gain in the frequency range associated with the myogenic mechanism was significantly higher in SDR than in SHR, but no statistical difference was found with tubuloglomerular (TGF) gain. Furthermore, TVCF analysis revealed that the coherence in both strains is significantly nonstationary and that low-frequency coherence was negatively correlated with autoregulatory gain. TVCF in the frequency range from 0.1 to 0.3 Hz was significantly higher in SDR (7 out of 7, >0.5) than in SHR (5 out of 6, <0.5), and consistent for all time points. For TGF frequency range (0.03-0.05 Hz), coherence exhibited substantial nonstationarity in both strains. Five of six SHR had mean coherence (<0.5), while four of seven SDR exhibited coherence (<0.5). Together, these results demonstrate substantial nonstationarity in autoregulatory dynamics in both SHR and SDR. Furthermore, they indicate that the nonstationarity accounts for most of the dynamic complexity in SDR, but that it accounts for only a part of the dynamic complexity in SHR.

Multistability in tubuloglomerular feedback and spectral complexity in spontaneously hypertensive rats.

Single-nephron proximal tubule pressure in spontaneously hypertensive rats (SHR) can exhibit highly irregular oscillations similar to deterministic chaos. We used a mathematical model of tubuloglomerular feedback (TGF) to investigate potential sources of the irregular oscillations and the corresponding complex power spectra in SHR. A bifurcation analysis of the TGF model equations, for nonzero thick ascending limb (TAL) NaCl permeability, was performed by finding roots of the characteristic equation, and numerical simulations of model solutions were conducted to assist in the interpretation of the analysis. These techniques revealed four parameter regions, consistent with TGF gain and delays in SHR, where multiple stable model solutions are possible: 1) a region having one stable, time-independent steady-state solution; 2) a region having one stable oscillatory solution only, of frequency f1; 3) a region having one stable oscillatory solution only, of frequency f2, which is approximately equal to 2f1; and 4) a region having two possible stable oscillatory solutions, of frequencies f1 and f2. In addition, we conducted simulations in which TAL volume was assumed to vary as a function of time and simulations in which two or three nephrons were assumed to have coupled TGF systems. Four potential sources of spectral complexity in SHR were identified: 1) bifurcations that permit switching between different stable oscillatory modes, leading to multiple spectral peaks and their respective harmonic peaks; 2) sustained lability in delay parameters, leading to broadening of peaks and of their harmonics; 3) episodic, but abrupt, lability in delay parameters, leading to multiple peaks and their harmonics; and 4) coupling of small numbers of nephrons, leading to multiple peaks and their harmonics. We conclude that the TGF system in SHR may exhibit multistability and that the complex power spectra of the irregular TGF fluctuations in this strain may be explained by switching between multiple dynamic modes, temporal variation in TGF parameters, and nephron coupling.

A robust method for detection of linear and nonlinear interactions: application to renal blood flow dynamics.

We have developed a method that can identify switching dynamics in time series, termed the improved annealed competition of experts (IACE) algorithm. In this paper, we extend the approach and use it for detection of linear and nonlinear interactions, by employing histograms showing the frequency of switching modes obtained from the IACE, then examining time-frequency spectra. This extended approach is termed Histogram of improved annealed competition of experts-time frequency (HIACE-TF). The hypothesis is that frequent switching dynamics in HIACE-TF results are due to interactions between different dynamic components. To validate this assertion, we used both simulation examples as well as application to renal blood flow data. We compared simulation results to a time-phase bispectrum (TPB) approach, which can also be used to detect time-varying quadratic phase coupling between various components. We found that the HIACE-TF approach is more accurate than the TPB in detecting interactions, and remains accurate for signal-to-noise ratios as low as 15 dB. With all 10 data sets, comprised of volumetric renal blood flow data, we also validated the feasibility of the HIACE-TF approach in detecting nonlinear interactions between the two mechanisms responsible for renal autoregulation. Further validation of the HIACE-TF approach was achieved by comparing it to a realistic mathematical model that has the capability to generate either the presence or the absence of nonlinear interactions between two renal autoregulatory mechanisms.

Differences in pHi recovery in CO2-chemosensitive and non-chemosensitive cells: predictions from a mathematical model.

In this paper, we present a mathematic model designed to identify potential mechanisms responsible for the observed differences in pHi recovery in CO(2)-chemosensitive versus non-chemosensitive cells. The model suggests that differences in pHi regulation may be dependent upon differences in the activation set-point of the internal modifier site of the Na(+)/H(+) exchanger (NHE).

Very low frequency modulation in renal autoregulation.

This study aims to examine the presence of a possible third renal autoregulatory mechanism in the very low frequency (VLF) band (approximately 10 mHz) using a high-resolution time- frequency spectral method. Blood pressure and renal blood flow data were measured from conscious and anesthetized Sprague-Dawley and spontaneously hypertensive rats, at the level of the whole kidney (via ultrasound flow probe) and local cortical tissue of a kidney (via laser Doppler flow probe). In addition, N-nitro-L-arginine (LNAME) was used in order to assess the effect of nitric oxide on the third mechanism. Using a complex demodulation method with high time and frequency resolution, a VLF band was often observed, as well as amplitude modulation at the VLF of the two other autoregulation mechanisms. The presence of amplitude modulation is an indication of a particular form of nonlinear interaction between the autoregulatory mechanisms. Physically, such interactions may arise from the fact that all three mechanisms share a common effector, the afferent arteriole. In addition, the magnitude of amplitude modulation of the VLF on the other autoregulatory mechanisms was enhanced by the addition of LNAME, suggesting an important role of nitric oxide in the autoregulatory process.