Abstract
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.
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Acknowledgments
This research was supported by the National Institutes of Health: National Institute of Diabetes and Digestive and Kidney Diseases, Grant DK089066 to A.T. Layton, and by the National Science Foundation, through Grants DMS-0701412 to A.T. Layton and Research Training Grant DMS-0943760 to the Department of Mathematics at Duke University. Portions of this work were presented at Experimental Biology 2012 (FASEB J 26:690.1, 2012)
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Ryu, H., Layton, A.T. Tubular fluid flow and distal NaCl delivery mediated by tubuloglomerular feedback in the rat kidney. J. Math. Biol. 68, 1023–1049 (2014). https://doi.org/10.1007/s00285-013-0667-5
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DOI: https://doi.org/10.1007/s00285-013-0667-5