Advertisement

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

  • Anita T. LaytonEmail author
  • Leon C. Moore
  • Harold E. Layton
Original Article

Abstract

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.

Keywords

Renal hemodynamic control Spontaneously hypertensive rat Negative feedback loop Delay differential equation Nonlinear dynamics 

References

  1. Andersen, M.D., Carlsson, N., Mosekilde, E., Holstein-Rathlou, N.-H., 2002. Dynamic model of nephron–nephron interaction. In: Layton, H.E., Weinstein, A.M. (Eds.), Membrane Transport and Renal Physiology. The IMA Volumes in Mathematics and Its Applications, vol. 129, pp. 365–91. Springer, New York. Google Scholar
  2. Arendshorst, W.J., Beierwaltes, W.H., 1979. Renal tubular reabsorption in spontaneously hypertensive rats. Am. J. Physiol. (Renal Fluid Electrolyte Physiol. 6) 237, F38–F47. Google Scholar
  3. Barfred, M., Mosekilde, E., Holstein-Rathlou, N.-H., 1996. Bifurcation analysis of nephron pressure and flow regulation. Chaos 6, 280–87. CrossRefGoogle Scholar
  4. Briggs, J.P., Shubert, G., Schnermann, J., 1984. Quantitative characterization of the tubuloglomerular feedback response: effect of growth. Am. J. Physiol. (Renal Fluid Electrolyte Physiol. 16) 247, F808–F815. Google Scholar
  5. Budu-Grajdeanu, P., Moore, L.C., Layton, H.E., 2007. Effect of tubular inhomogeneities on filter properties of thick ascending limb of Henle’s loop. Math. Biosci. 209, 564–92. zbMATHCrossRefMathSciNetGoogle Scholar
  6. Carlson, N., Andersen, M.D., 1999. Mathematical modeling of nephrons. Master’s thesis, The Technical University of Denmark. Google Scholar
  7. Casellas, D., Moore, L.C., 1990. Autoregulation and tubuloglomerular feedback in juxtamedullary glomerular arterioles. Am. J. Physiol. (Renal Fluid Electrolyte Physiol. 27) 258, F660–F669. Google Scholar
  8. Chen, Y.-M., Yip, K.-P., Marsh, D.J., Holstein Rathlou, N.-H., 1995. Magnitude of TGF-initiated nephron–nephron interaction is increased in SHR. Am. J. Physiol. (Renal Fluid Electrolyte Physiol. 38) 269, F198–F204. Google Scholar
  9. Daniels, F.H., Arendshorst, W.J., 1990. Tubuloglomerular feedback kinetics in spontaneously hypertensive and Wistar–Kyoto rats. Am. J. Physiol. (Renal Fluid Electrolyte Physiol. 28) 259, F529–F534. Google Scholar
  10. Dilley, J.R., Arendshorst, W.J., 1984. Enhanced tubuloglomerular feedback activity in rats developing spontaneous hypertension. Am. J. Physiol. (Renal Fluid Electrolyte Physiol. 16) 247, F672–F679. Google Scholar
  11. Ditlevsen, S., Yip, K.-P., Holstein-Rathlou, N.-H., 2005. A stochastic model of the tubuloglomerular feedback mechanism in a rat nephron. Math. Biosci. 194, 49–9. zbMATHCrossRefMathSciNetGoogle Scholar
  12. Ditlevsen, S., Yip, K.-P., Marsh, D.J., Holstein-Rathlou, N.-H., 2007. Parameter estimation of feedback gain in a stochastic model of renal hemodynamics: differences between spontaneously hypertensive and Sprague-Dawley rats. Am. J. Physiol. Renal Physiol. 292, F607–F616. CrossRefGoogle Scholar
  13. Eaton, D.C., Pooler, J.P., 2004. Vander’s Renal Physiology, 6th edn. McGraw-Hill Medical, New York. Google Scholar
  14. Hattaway, A.L., 2004. Modelling tubuloglomerular feedback in coupled nephrons. Ph.D. diss., University of Massachusetts Amherst. Google Scholar
  15. Holstein-Rathlou, N.-H., 1987. Synchronization of proximal intratubular pressure oscillations: evidence for interaction between nephrons. Pflügers Arch. 408, 438–43. CrossRefGoogle Scholar
  16. Holstein-Rathlou, N.-H., 1991. A closed-loop analysis of the tubuloglomerular feedback mechanism. Am. J. Physiol. (Renal Fluid Electrolyte Physiol. 30) 261, F880–F889. Google Scholar
  17. Holstein-Rathlou, N.-H., Leyssac, P.P., 1986. TGF-mediated oscillations in proximal intratubular pressure: Differences between spontaneously hypertensive rats and Wistar–Kyoto rats. Acta Physiol. Scand. 126, 333–39. CrossRefGoogle Scholar
  18. Holstein-Rathlou, N.-H., Leyssac, P.P., 1987. Oscillations in the proximal intratubular pressure: a mathematical model. Am. J. Physiol. (Renal Fluid Electrolyte Physiol. 21) 252, F560–F572. Google Scholar
  19. Holstein-Rathlou, N.-H., Marsh, D.J., 1989. Oscillations of tubular pressure, flow, and distal chloride concentration in rats. Am. J. Physiol. (Renal Fluid Electrolyte Physiol. 25) 256, F1007–F1014. Google Scholar
  20. Holstein-Rathlou, N.-H., Marsh, D.J., 1990. A dynamic model of the tubuloglomerular feedback mechanism. Am. J. Physiol. (Renal Fluid Electrolyte Physiol. 27) 258, F1448–F1459. Google Scholar
  21. Holstein-Rathlou, N.-H., Marsh, D.J., 1994. Renal blood flow regulation and arterial pressure fluctuations: a case study in nonlinear dynamics. Physiol. Rev. 74, 637–81. Google Scholar
  22. Holstein-Rathlou, N.-H., Yip, K.-P., Sosnotseva, O.V., Mosekilde, E., 2001. Synchronization phenomena in nephron–nephron interaction. Chaos 11(2), 417–26. CrossRefGoogle Scholar
  23. Jensen, K.S., Mosekile, E., Holstein-Rathlou, N.-H., 1986. Self-sustained oscillations and chaotic behaviour in kidney pressure regulation. Mondes Dévelop. 54–5, 91–09. Google Scholar
  24. Jensen, K.S., Holstein-Rathlou, N.-H., Leyssac, P.P., Mosekilde, E., Rasmussen, D.R., 1987. Chaos in a system of interacting nephrons. In: Degn, H., Holden, A.V., Olsen, L.F. (Eds.), Life Sciences: Chaos in Biological Systems, Plenum, New York. Google Scholar
  25. Källskog, Ö., Marsh, D.J., 1990. TGF-initiated vascular interactions between adjacent nephrons in the rat kidney. Am. J. Physiol. (Renal Fluid Electrolyte Physiol. 28) 259, F60–F64. Google Scholar
  26. Kesseler, K.J., 2004. Analysis of feedback-mediated oscillations in two coupled nephrons. Ph.D. diss., Duke University. Google Scholar
  27. Knepper, M.A., Danielson, R.A., Saidel, G.M., Post, R.S., 1977. Quantitative analysis of renal medullary anatomy in rats and rabbits. Kidney Int. 12, 313–23. CrossRefGoogle Scholar
  28. Layton, A.T., Moore, L.C., Layton, H.E., 2006. Multistability in tubuloglomerular feedback and spectral complexity in spontaneously hypertensive rats. Am. J. Physiol. Renal Physiol. 291, F79–F97. CrossRefGoogle Scholar
  29. Layton, H.E., Pitman, E.B., 1994. A dynamic numerical method for models of renal tubules. Bull. Math. Biol. 56(3), 547–65. zbMATHGoogle Scholar
  30. Layton, H.E., Pitman, E.B., Moore, L.C., 1991. Bifurcation analysis of TGF-mediated oscillations in SNGFR. Am. J. Physiol. (Renal Fluid Electrolyte Physiol. 30) 261, F904–F919. Google Scholar
  31. Layton, H.E., Pitman, E.B., Moore, L.C., 1995. Instantaneous and steady-state gains in the tubuloglomerular feedback system. Am. J. Physiol. Renal Physiol. 268, F163–F174. Google Scholar
  32. Layton, H.E., Pitman, E.B., Moore, L.C., 1997a. Nonlinear filter properties of the thick ascending limb. Am. J. Physiol. (Renal Fluid Electrolyte Physiol. 42) 273, F625–F634. Google Scholar
  33. Layton, H.E., Pitman, E.B., Moore, L.C., 1997b. Spectral properties of the tubuloglomerular feedback system. Am. J. Physiol. (Renal Fluid Electrolyte Physiol. 42) 273, F635–F649. Google Scholar
  34. Layton, H.E., Pitman, E.B., Moore, L.C., 2000. Limit-cycle oscillations and tubuloglomerular feedback regulation of distal sodium delivery. Am. J. Physiol. Renal Physiol. 278, F287–F301. Google Scholar
  35. Leyssac, P.P., Baumbach, L., 1983. An oscillating intratubular pressure response to alterations in Henle loop flow in the rat kidney. Acta Physiol. Scand. 117, 415–19. CrossRefGoogle Scholar
  36. Leyssac, P.P., Holstein-Rathlou, N.-H., 1989. Tubulo-glomerular feedback response: enhancement in adult spontaneously hypertensive rats and effects of anaesthetics. Pflügers Arch. 413, 267–72. CrossRefGoogle Scholar
  37. Marsh, D.J., Sosnovtseva, O.V., Chon, K.H., Holstein-Rathlou, N.-H., 2005a. Nonlinear interactions in renal blood flow regulation. Am. J. Physiol. Regul. Integr. Comp. Physiol. 288, R1143–R1159. Google Scholar
  38. Marsh, D.J., Sosnovtseva, O.V., Pavlov, A.N., Yip, K.-P., Holstein-Rathlou, N.-H., 2005b. Frequency encoding in renal blood flow regulation. Am. J. Physiol. Regul. Integr. Comp. Physiol. 288, R1160–R1167. Google Scholar
  39. Marsh, D.J., Sosnovtseva, O.V., Mosekilde, E., Holstein-Rathlou, N.-H., 2007. Vascular coupling induces synchronization, quasiperiodicity, and chaos in a nephron tree. Chaos 17, 015114-1-015114-10. CrossRefGoogle Scholar
  40. Mason, J., Gutsche, H.U., Moore, L.C., Müller-Suur, R., 1979. The early phase of experimental acute renal failure. IV. The diluting ability of the short loops of Henle. Pflügers Arch. 379, 11–8. CrossRefGoogle Scholar
  41. Mosekilde, E., Sosnovtseva, O.V., Holstein-Rathlou, N.-H., 2005. Collective phenomena in kidney autoregulation. Transplantationsmedizin 17, 115–30. Google Scholar
  42. Nyengaard, J.R., Bendtsen, T.F., 1992. Glomerular number and size in relation to age, kidney weight, and body surface in normal man. Anat. Rec. 232, 194–01. CrossRefGoogle Scholar
  43. Oldson, D.R., 2003. Flow perturbations in a mathematical model of the tubuloglomerular feedback system. Ph.D. diss., Duke University. Google Scholar
  44. Oldson, D.R., Layton, H.E., Moore, L.C., 2003. Effect of sustained flow perturbations on stability and compensation of tubuloglomerular feedback. Am. J. Physiol. Renal Physiol. 285, F972–F989. Google Scholar
  45. Perko, L., 2001. Differential Equations and Dynamical Systems, 3rd edn. Springer, New York. zbMATHGoogle Scholar
  46. Pitman, E.B., Layton, H.E., Moore, L.C., 1993. Dynamic flow in the nephron: filtered delay in the TGF pathway. Contemp. Math. 114, 317–36. Google Scholar
  47. Pitman, E.B., Layton, H.E., Moore, L.C., 1994. Numerical simulation of propagating concentration profiles in renal tubules. Bull. Math. Biol. 56(3), 567–86. zbMATHCrossRefGoogle Scholar
  48. Pitman, E.B., Zaritski, R.M., Kesseler, K.J., Moore, L.C., Layton, H.E., 2004. Feedback-mediated dynamics in two coupled nephrons. Bull. Math. Biol. 66(6), 1463–492. CrossRefMathSciNetGoogle Scholar
  49. Reeves, W.B., Andreoli, T.E., 2000. Sodium chloride transport in the loop of Henle, distal convoluted tubule, and collecting duct. In: Seldin, D.W., Giebisch, G. (Eds.), The Kidney: Physiology and Pathophysiology, 3rd edn. pp. 1333–349. Lippincott Williams & Wilkins, Philadelphia. Google Scholar
  50. Schnermann, J., Briggs, J.P., 2000. Function of the juxtaglomerular apparatus: Control of glomerular hemodynamics and renin secretion. In: The Kidney: Physiology and Pathophysiology, 3rd edn. pp. 945–80. Lippincott Williams & Wilkins, Philadelphia. Google Scholar
  51. Skeldon, A.C., Purvey, I., 2005. The effect of different forms for the delay in a model of the nephron. Math. Biosci. Eng. 2(1), 97–09. zbMATHMathSciNetGoogle Scholar
  52. Sosnovtseva, O.V., Pavlov, A.N., Mosekilde, E., Holstein-Rathlou, N.-H., 2002a. Bimodal oscillations in nephron autoregulation. Phys. Rev. E Stat. Nonlinear Soft Matter. Phys. 66, 061909-1–61909-7. Google Scholar
  53. Sosnovtseva, O.V., Postnov, D.E., Nekrasov, A.M., Mosekilde, E., Holstein-Rathlou, N.-H., 2002b. Phase multistability of self-modulated oscillations. Phys. Rev. E Stat. Nonlinear Soft Matter. Phys. 66, 036224-1–36224-9. Google Scholar
  54. Sosnovtseva, O.V., Postnov, D.E., Mosekilde, E., Holstein-Rathlou, N.-H., 2003. Synchronization of tubular pressure oscillations in interacting nephrons. Chaos Solitons Fractals 15, 343–69. CrossRefMathSciNetGoogle Scholar
  55. Sosnovtseva, O.V., Pavlov, A.N., Mosekilde, E., Holstein-Rathlou, N.-H., Marsh, D.J., 2004. Double-wavelet approach to study frequency and amplitude modulation in renal autoregulation. Phys. Rev. E Stat. Nonlinear Soft Matter. Phys. 70, 031915-1–31915-8. Google Scholar
  56. Wagner, A.J., Holstein-Rathou, N.-H., Marsh, D.J., 1997. Internephron coupling by conducted vasomotor responses in normotensive and spontaneously hypertensive rats. Am. J. Physiol. (Renal Physiol. 41) 272, F372–F379. Google Scholar
  57. Wang, H., Siu, K., Ju, K., Chon, K.H., 2006. A high resolution approach to estimating time-frequency spectra and their amplitudes. Ann. Biomed. Eng. 34(2), 326–38. CrossRefGoogle Scholar
  58. Wittner, M., Di Stefano, A., Wangemann, P., Nitschke, R., Greger, R., Bailly, C., Amiel, C., Roinel, N., de Roufignac, C., 1988. Differential effects of ADH on sodium, chloride, potassium, calcium and magnesium transport in cortical and medullary thick ascending limbs of mouse nephron. Pflügers Arch. 412, 516–23. CrossRefGoogle Scholar
  59. Yip, K.-P., Holstein-Rathlou, N.-H., Marsh, D.J., 1991. Chaos in blood flow control in genetic and renovascular hypertensive rats. Am. J. Physiol. (Renal Fluid Electrolyte Physiol. 30) 261, F400–F408. Google Scholar
  60. Yip, K.-P., Holstein-Rathlou, N.-H., Marsh, D.J., 1992. Dynamics of TGF-initiated nephron–nephron interactions in normotensive rats and SHR. Am. J. Physiol. (Renal Fluid Electrolyte Physiol. 31) 262, F980–F988. Google Scholar
  61. Yip, K.-P., Marsh, D.J., Holstein-Rathlou, N.-H., 1995. Evidence of low dimensional chaos in renal blood flow control in genetic and experimental hypertension. Physica D 80, 95–04. zbMATHCrossRefGoogle Scholar
  62. Zhai, X.-Y., Thomsen, J.S., Birn, H., Kristoffersen, I.B., Andreasen, A., Christensen, E.I., 2006. Three-dimensional reconstruction of the mouse nephron. J. Am. Soc. Nephrol. 17, 77–8. CrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 2008

Authors and Affiliations

  • Anita T. Layton
    • 1
    Email author
  • Leon C. Moore
    • 2
  • Harold E. Layton
    • 1
  1. 1.Department of MathematicsDuke UniversityDurhamUSA
  2. 2.Department of Physiology and BiophysicsState University of New YorkStony BrookUSA

Personalised recommendations