Bifurcation analysis of TGF-mediated oscillations in SNGFR

Abstract

Recent micropuncture studies in rats have demonstrated the existence of oscillatory states in nephron filtration mediated by tubuloglomerular feedback (TGF). We develop a minimal mathematical model of the TGF system, consisting of a first-order hyperbolic partial differential equation describing thick ascending limb (TAL) NaCl reabsorption and an empirical feedback relation. An analytic bifurcation analysis of this model provides fundamental insight into how oscillatory states depend on the physiological parameters of the model. In the special case of no solute backleak in the TAL, the emergence of oscillations explicitly depends on two nondimensional parameters. The first corresponds to the delay time of the TGF response across the juxtaglomerular apparatus, and the second corresponds to the product of the slope of the TGF response curve at the steady-state operating point and the space derivative of the steady-state NaCl concentration profile in the TAL at the macula densa. Numerical calculations for the case without TAL backleak are consistent with this result. Numerical simulation of the more general case with TAL backleak shows that the bifurcation analysis still provides useful predictions concerning nephron dynamics. With typical parameter values, the analysis predicts that the TGF system will be in oscillatory state. However, the system is near enough to the boundary of the nonoscillatory region so that small changes in parameter values could result in nonoscillatory behavior.

Effect of sustained flow perturbations on stability and compensation of tubuloglomerular feedback

“…Steady-state gain is the gain that has been measured in laboratory experiments; the concept of instantaneous gain arose from a theoretical analysis of a mathematical model of the TGF system (12,13). We have previously shown that, under restricted circumstances, G SS is closely approximated by Ϫ␥ (13).…”

“…For a sufficiently long delay in TGF signal transmission at the juxtaglomerular apparatus (JGA), the stability of a steady state is determined by the gain 1 ␥ and by the critical gain ␥ c (12). If ␥ Ͻ ␥ c , then the stable state is a (time-independent) steady state; if ␥ Ͼ ␥ c , then the stable state is an LCO.…”

“…Using this technique, Leyssac and collaborators (5,17,18) have demonstrated that LCO can be initiated or extinguished in halothane-anesthetized rats by insertion or removal of PT fluid, findings that indicate that PT fluid flow can have a substantial effect on the stability of the TGF system. Some insight into the basis of this phenomenon is provided by our previous investigations of the TGF-mediated LCO (12)(13)(14)(15)(16)22). These modeling studies indicate that LCO will emerge for sufficiently large feedback loop gain magnitude (12), that the parameter regime that supports LCO may overlap the parameter regime of normal TGF operation, that a gain magnitude near that needed for LCO will produce maximal feedback compensation (16), and that LCO may augment NaCl delivery to the distal nephron (16).…”

“…Some insight into the basis of this phenomenon is provided by our previous investigations of the TGF-mediated LCO (12)(13)(14)(15)(16)22). These modeling studies indicate that LCO will emerge for sufficiently large feedback loop gain magnitude (12), that the parameter regime that supports LCO may overlap the parameter regime of normal TGF operation, that a gain magnitude near that needed for LCO will produce maximal feedback compensation (16), and that LCO may augment NaCl delivery to the distal nephron (16). A finding of fundamental importance is that the emergence of LCO depends on key TGF system parameters and variables that, in turn, depend on the physiological state of the nephron or on the animal as a whole.…”

“…kidney; renal hemodynamics; autoregulation; mathematical model; nonlinear dynamics THE TUBULOGLOMERULAR FEEDBACK (TGF) SYSTEM is a key moment-to-moment regulator of the single-nephron glomerular filtration rate (SNGFR). A large body of experimental and theoretical evidence indicates that TGF can mediate sustained, regular oscillations of 20-47 mHz in tubular flow, pressure, and intratubular thick ascending limb (TAL) NaCl concentration (4,6,7,12,18). These regular oscillations are called limit-cycle oscillations (LCO) because, after initiation, they tend to more and more closely approximate a fixed cycle, provided that system parameters do not change.…”