1 Department of Mathematics, Duke University, Durham, North Carolina 27708-0320; 2 Department of Mathematics, State University of New York, Buffalo, New York 14214-3093; and 3 Department of Physiology and Biophysics, State University of New York, Stony Brook, New York, 11794-8661
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ABSTRACT |
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A
mathematical model was used to evaluate the potential effects of
limit-cycle oscillations (LCO) on tubuloglomerular feedback (TGF)
regulation of fluid and sodium delivery to the distal tubule. In
accordance with linear systems theory, simulations of steady-state responses to infinitesimal perturbations in single-nephron glomerular filtration rate (SNGFR) show that TGF regulatory ability (assessed as
TGF compensation) increases with TGF gain magnitude when
is
less than the critical value
c, the value at which LCO
emerge in tubular fluid flow and NaCl concentration at the macula
densa. When
>
c and LCO are present, TGF
compensation is reduced for both infinitesimal and finite perturbations
in SNGFR, relative to the compensation that could be achieved in the
absence of LCO. Maximal TGF compensation occurs when
c. Even in the absence of perturbations, LCO increase
time-averaged sodium delivery to the distal tubule, while fluid
delivery is little changed. These effects of LCO are consequences of
nonlinear elements in the TGF system. Because increased distal sodium
delivery may increase the rate of sodium excretion, these simulations
suggest that LCO enhance sodium excretion.
kidney, renal hemodynamics, mathematical model, nonlinear dynamics
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INTRODUCTION |
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THE TUBULOGLOMERULAR FEEDBACK (TGF) system, a negative feedback loop, maintains a balance between single-nephron glomerular filtration rate (SNGFR) and absorption in predistal segments of the nephron, and it regulates the delivery of water and NaCl to the distal tubule.
Experimental studies have shown that the TGF system can exhibit
limit-cycle oscillations (LCO) in key variables, including glomerular
capillary blood pressure, fluid flow and pressure in the proximal
tubule, and fluid flow, pressure, and tubular fluid chloride
concentration in the early distal tubule (11, 22, 24). Theoretical
studies indicate that LCO emerge because of the combination of time
delays and sufficiently high system gain magnitude in the feedback loop
(12, 18, 29). In vivo recordings of the LCO of the TGF system show
that the oscillations are nonsinusoidal and thus exhibit nonlinear
features (11, 13, 15, 37, 39). The characteristic waveform of LCO
in tubular pressure is illustrated in Fig.
1B [reproduced from
Holstein-Rathlou et al. (13)]. There is a marked asymmetry between the
down slopes and up slopes and a broadening of the crests relative to
the troughs. These features are also seen in LCO produced by our
mathematical model of the TGF system, as shown in Fig. 1A.
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In a recent theoretical investigation (20, 21), we found that these
waveform features can be explained as consequences of at least three
nonlinear characteristics of the TGF system. The first is the sigmoidal
shape of the TGF response curve (Fig. 2A). This curve, which gives
SNGFR as a function of macula densa (MD) chloride concentration, shows
the limits of the range over which system flow is responsive to changes
in MD concentration. The limited range may lead to threshold and
saturation effects when the TGF system is not oscillating and to
railing of large amplitude oscillations. The second nonlinear
characteristic is the limited ability of the thick ascending limb (TAL)
to reduce luminal chloride concentration at low flow rates (Fig.
2B). This effect can lead to a dissociation in relative
amplitudes of the LCO in fluid flow and the LCO in chloride delivery
(see Fig. 5, below). The third nonlinear characteristic arises from the
dependence of TAL chloride absorption on the transit time of fluid
through the TAL. This effect distorts the waveform of chloride
concentration at the MD relative to a sinusoidal waveform in tubular
fluid flow (Fig. 2C), and it introduces a phase shift, with
extrema in flow leading extrema in concentration.
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In a previous mathematical modeling study (21), we found that when system gain magnitude was large enough to elicit LCO, the model waveforms of oscillations in fluid flow differed from those of NaCl concentration at the MD. As a consequence of these differing waveforms and their phase differences, the amplitudes of oscillations in NaCl delivery are larger, as a percentage of the (nonoscillatory) steady-state value, than are amplitudes of oscillations in fluid delivery (see Fig. 5, below). This observation led us to hypothesize that the nonlinearities in the TGF system may have an impact, in vivo, on the important role of that system to regulate water and NaCl delivery to the distal tubule.
Here, we report the results of simulation studies designed to evaluate the influence of LCO on the ability of the TGF system to compensate for perturbations in SNGFR. The results suggest that LCO limit the regulatory ability of the TGF system and that LCO may enhance time-averaged distal sodium delivery and renal sodium excretion.
Glossary
Parameters
Co | chloride concentration at TAL entrance, mM |
Cop | steady-state chloride concentration at MD, mM |
k | sensitivity of TGF response, 1/mM |
KM | Michaelis constant, mM |
L | length of TAL, cm |
P | TAL chloride permeability, cm/s |
Qop | steady-state SNGFR, nl/min |
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TGF-mediated range of SNGFR, nl/min |
r | luminal radius of TAL, µm |
Vmax | maximum transport rate of chloride from TAL,
nmol · cm![]() ![]() |
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fraction of SNGFR reaching TAL |
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distributed delay interval at JGA, s |
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discrete (or pure) delay interval at JGA, s |
Independent variables
t | time, s |
x | axial position along TAL, cm |
Specified functions
Ce(x) | extratubular chloride concentration, mM |
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kernel function for distributed delay (dimensionless) |
Dependent variables
C(x, t) | TAL chloride concentration, mM |
CMD(t) | effective MD chloride concentration, mM |
F(CMD(t)) | TAL fluid flow, nl/min |
S(x) | steady-state TAL chloride concentration, mM |
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METHODS |
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Mathematical model and its solutions.
We used mathematical simulations to evaluate how LCO may affect
TGF-mediated regulation of fluid and NaCl delivery to the distal
tubule. The simulations were based on a model formulation that we have
used previously to study TGF system dynamics (18-21, 29). The
model is illustrated in Fig. 3; model
quantities are identified in the Glossary. For simplicity, only
the chloride concentration is explicitly represented in the model
[chloride is thought to be the principal ion sensed at the MD in the
TGF response (31)]. We assume that sodium is absorbed in parallel with
chloride. Because the model TAL is assumed to have water-impermeable rigid walls, TAL tubular fluid flow rate is a function of time only and
is equivalent to the flow rate past the MD. The mathematical formulation of the model is recapitulated in APPENDIX A; the numerical methods used to approximate solutions to the model and to
compute quantities derived from those solutions are described in
APPENDIX B.
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Perturbations. Perturbations to SNGFR were simulated by adding or subtracting stipulated amounts to the base-case SNGFR (i.e., to Qop), where it appears in the model equations; the precise role of Qop in the model is set forth in APPENDIX A, Eq. A6. Perturbations were introduced, either transiently (to initiate LCO) or continuously, with a step function. Because the model formulation assumes that a fixed fraction of SNGFR reaches the TAL, a perturbation of Qop, as a percentage of Qop, is analogous to a perturbation, in vivo, at any site before the TAL, of the same percentage of the steady-state base-case fluid flow rate at that site.
Starting from the steady-state base case, the model was perturbed as described above. After the solution reached a new steady state (or converged to an LCO), the steady (or time- averaged LCO) values of the TAL fluid flow rate, the chloride concentration at the MD, and the chloride delivery rate to the MD (and hence, into the distal tubule) were determined. Since the model includes only chloride concentration, we assumed that NaCl delivery is identical to the delivery of chloride.Feedback compensation.
The efficacy of TGF regulation was quantified by calculating feedback
compensation, an index often used in experimental investigations (see,
e.g., Refs. 14, 34). Feedback compensation is defined by
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(1) |
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(2) |
Relationship between compensation and feedback loop gain.
For a linear system, the relationship between compensation and feedback
loop gain is important for this study and is easy to derive. Let
X represent the value of an input signal and suppose that, in
response, the system produces an output signal Y = X, where
is a scalar. Suppose that at a base-case value
Xo, the system output signal has the base-case
value Yo =
Xo. Now
suppose that Xo is perturbed by an amount
X. Then the output signal, in the absence of feedback
(which is the open-feedback-loop case) would be Y =
(Xo +
X ), which differs
from the base-case value Yo by the amount
Y = Y
Yo; in this specific
linear case,
Y =
X.
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(3) |
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(4) |
Steady-state and instantaneous gain.
We have previously shown that in our model of the TGF system, a
distinction must be made between steady-state gain magnitude and
instantaneous gain magnitude; this technical point is treated in detail
in Ref. 19 (see also APPENDIX A). The gain magnitude that
determines the bifurcation of the system into LCO is the instantaneous
gain magnitude, which we designate with the symbol . The gain
magnitude Gss used in the calculation above corresponds to steady-state gain magnitude (thus the subscript). For
the parameters in this study, the instantaneous gain
exceeds the
steady-state gain Gss by ~10.3% (19). Thus, to
be precise, when we say below that a calculated compensation agrees
with the predictions of linear systems theory, we will mean that the
result obtained by a calculation of the compensation by means of the definition (Eqs. 1 and 2) very nearly approximates the
relationship in Eq. 4, because
X has been taken
sufficiently close to zero and Gss has been
interpreted to be related to
by Gss
/1.103.
Comparison and normalization of perturbed MD variables.
In the presence of LCO but in the absence of sustained perturbations,
model calculations show that the time-averaged tubular variable values
at the MD (i.e., fluid flow, chloride concentration, chloride
delivery), computed with the base-case parameters in Table 1, differ
from the corresponding steady-state base-case values. For example, the
time-averaged NaCl delivery rate differs from the steady-state NaCl
delivery rate (see RESULTS, Table 2). Thus, to obtain a
consistent and unambiguous interpretation of the definition of
magnification (and corresponding compensation), we adopted the
principle that a perturbed value should be compared to the nonperturbed
value corresponding to the stable state of the system at the given gain
magnitude . When
is less than
c, the stable state
is nonoscillatory, the base case is the steady-state base case, and the
tubular values at the MD are the steady-state values arising from the
base-case parameters (see Table 2). When
exceeds
c,
the stable state is oscillatory, the base case is a base-case LCO, and
the base-case values of the tubular variables at the MD are the
time-averaged values arising from the base-case parameters (these
time-averaged values are a function of
).
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RESULTS |
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In the absence of sustained perturbations, LCO increase distal NaCl
delivery but not distal fluid delivery.
To determine the effect of LCO on distal fluid and NaCl delivery, we
compared the rates at which fluid and chloride exited the model TAL
segment in the steady-state base case with corresponding time-averaged
rates during LCO. Results for gain magnitudes from 0 to 10 are
illustrated in Fig. 4, where the
time-averaged variables have been normalized by their corresponding
steady-state base-case values. Nonnormalized results for selected
values of
are given in Table 2. At all
gain magnitudes exceeding
c, fluid delivery was
depressed slightly by LCO, with a maximum decrease of ~0.5% at
4. This response was driven by the monotone increase in
time-averaged chloride concentration, which results in a TGF-mediated suppression of SNGFR. In contrast, time-averaged chloride delivery exhibited a biphasic relationship with increasing gain magnitude. For
small increases of
above
c, chloride delivery
decreased with time-averaged flow, but then it increased, reaching a
value of 103.7% of the steady-state base-case delivery rate for
= 10. Note that the time-averaged chloride delivery rate is the time
average of the product of the instantaneous flow rate and the
instantaneous concentration, and that, except for the steady-state case, the product of time-averaged flow and the time-averaged concentration does not equal the time-averaged chloride delivery, as a
result of phase differences in the waveforms for flow and chloride
concentration (see, e.g., Fig. 5, below).
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In the presence of sustained perturbations, LCO result in larger
deviations in distal NaCl delivery than in distal fluid delivery.
For gain magnitude = 5, we computed the responses of MD variables
to perturbations of up to ±30% of the steady-state flow rate. The
percent deviations from corresponding steady-state base-case values are
given in Table 3, which reports the unregulated responses for the
open-feedback-loop case (OL), the hypothetical steady-state feedback-controlled responses (SS) obtained by removing the time delay
at the JGA (see METHODS), and the LCO responses, based on time-averaged values.
In the presence of sustained infinitesimal perturbations in fluid
flow, the regulatory ability of TGF is reduced by LCO.
We next sought to quantify, by evaluating feedback compensation (the
index defined in METHODS), how LCO may influence the regulatory function of the TGF system. We first examined the effect of
LCO on TGF compensation for an infinitesimal perturbation in SNGFR, as
a function of feedback gain magnitude . Two cases were examined. In
the first, LCO were prevented by eliminating the time delay at the JGA;
in the second, the base-case time delay was used, which led to the
emergence of LCO when gain magnitude exceeded
c
3.24.
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In the presence of sustained finite perturbations in fluid flow, the regulatory ability of TGF is reduced by LCO. As discussed in METHODS, the effects of finite perturbations may differ substantially from those elicited by infinitesimal perturbations, because of TGF system nonlinearities. Therefore, we also used the index of feedback compensation to quantify the impact of sustained, finite perturbations having physiologically relevant magnitudes. Such perturbations simulate a typical micropuncture protocol in which tubular fluid is added to, or removed from, the proximal tubule to permit estimation of feedback compensation (see, e.g., Refs. 27 and 33).
Figure 7 illustrates the responses in fluid flow, chloride concentration, and chloride delivery to perturbations of up to ±30% of steady-state flow rate. Three cases are shown for gain magnitude
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DISCUSSION |
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Adequacy of the model. This investigation has shown that a simple model of TGF, a model that describes a seemingly straightforward proportional negative feedback control system, can exhibit complex behavior when its nonlinear elements are engaged by perturbations of physiologically relevant magnitude. However, the applicability of the model results and of the model predictions discussed below depends upon whether our model of the TGF system provides an adequate representation of the key features of the TGF system in vivo. The model is based on conservation of mass and on experimental data from volume-replete rats (18, 29); the model's assumptions have been previously examined in substantial detail in Refs. (18-21, 29). The model's properties and previous predictions have agreed well with published experimental measurements, including feedback system gain magnitude (19), the approximate feedback gain value needed to support LCO (18), and the temporal characteristics of the LCO waveform (21). The nonlinearities in the model are responsible for these temporal characteristics, which distort the LCO waveform in distinctive ways and which underlie the results of this study. Thus the similarity of in vivo recordings of LCO, which generally exhibit these temporal characteristics (21), lends substantive support to the adequacy of the model.
Model predictions. This investigation of the impact of LCO on the regulatory role of the TGF system has yielded a number of predictions that are of potential physiological importance. First, the model predicts that, if LCO are suppressed, then the TGF system will be particularly effective in stabilizing distal NaCl delivery when SNGFR is increased above its steady-state base-case value by a perturbation (Fig. 7F ). Indeed, because of system nonlinearities, the model feedback compensation in this case exceeds that of a linear system with equivalent feedback gain magnitude. However, the model's ability to maintain distal NaCl delivery in the case of a reduction in SNGFR is less than that of a linear control system. In contrast to this asymmetrical compensation for NaCl, feedback compensation for distal fluid delivery is symmetrical (Fig. 7D) and is similar to that reported in experimental studies (9, 33, 34, 35).
Second, the model predicts that the onset of LCO, in the absence of a sustained perturbation, results in increases in time-averaged distal NaCl delivery, while time-averaged distal fluid delivery is little affected (Fig. 4). Although the magnitude of the increment in distal NaCl delivery is modest, this behavior illustrates yet again that the nonlinear elements in the TGF system can result in a dissociation of the regulation of fluid and electrolyte delivery to the distal nephron. Third, the model predicts that LCO markedly reduce the ability of the TGF system to compensate for perturbations in SNGFR, both infinitesimal and finite (Figs. 6 and 7). In vivo, the kidney is continually perturbed by substantial fluctuations in blood pressure, and one important role of the TGF system is its participation in the autoregulation of renal blood flow and GFR (26). Any decrease in the efficacy of renal autoregulation, as a consequence of the development of LCO, would result in increased fluctuations in the baseline level of SNGFR and distal delivery of water and solutes. Indeed, the model predicts that a sustained perturbation in SNGFR would in some cases result in nearly double the increments in time-averaged fluid and/or NaCl delivery into the distal nephron, when compared with an otherwise similar case where LCO are absent (Table 3 and related results in text forModel predictions and measured gains.
Thirteen steady-state measurements of TGF gain or compensation are
collected in Table 4. The measured values
aregroup means for open-feedback-loop gain magnitude or regulatory
compensation in adult rats obtained by four different laboratories
using differing methodologies. Data from hypertensive animals or rats
treated with drugs that alter the function of the TGF system, e.g.,
angiotensin II and nitric oxide synthase inhibitors, have not been
included. The measured gain magnitudes have not been adjusted for the
5-10% underestimation of instantaneous gain magnitude by
steady-state gain magnitude (19).
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Potential effects on sodium excretion. The model predicts that LCO tend to increase time-averaged NaCl delivery to the distal nephron, attributable to both waveform distortion and reduced TGF regulatory ability. This prediction leads to an important question: Would an LCO-mediated increase in delivery of NaCl to the distal nephron also increase renal sodium excretion? This is a complex issue that involves several considerations.
The first concerns the linkage between distal sodium delivery and renal sodium excretion. The distal nephron exhibits some degree of short-term load adaptation, driven by increased luminal sodium concentration (16, 36), which would tend to attenuate a perturbation in distal sodium delivery. On the other hand, a rise in average tubular fluid NaCl concentration at the MD, subsequent to the emergence of LCO, will suppress renin secretion (31) and thereby tend to enhance sodium excretion. In addition, it is well established that perturbations in tubular flow driven by fluctuations in blood pressure are associated with acute changes in renal sodium excretion, a phenomenon called pressure natriuresis (6, 7). Moreover, renal sodium excretion is a process that exhibits integral characteristics, in that the effects of small increases in sodium excretion accumulate over time until the losses are sufficient to reduce arterial blood pressure and/or alter renal sodium handling to reestablish long-term sodium balance (6, 7). Hence, it is reasonable to expect that the decrease in TGF regulatory ability associated with LCO will result in parallel changes in distal sodium delivery and renal sodium excretion that are physiologically significant. Experiments have shown that LCO in tubular fluid chloride concentration persist well into the early segment of the distal tubule in the rat (11, 24). Thus, a second consideration concerning the effect of LCO on renal sodium excretion is the response of the transporting cells in the distal nephron to oscillations in tubular fluid NaCl concentration and tubular fluid flow. Although we could not find any experimental data concerning the influence of such oscillations on sodium transport in the distal nephron, some evidence suggests that there may be significant effects. In many types of sodium-transporting tight epithelial cells, apical sodium entry does not passively follow changes in mucosal sodium concentration. Rather, apical membrane sodium permeability varies inversely with mucosal sodium concentration (5, 25, 32). Several mechanisms may be involved in this phenomenon, which appears to stabilize cytosolic sodium activity and which has been referred to as "feedback inhibition" (25, 32). The decrease in sodium permeability initiated by increased luminal sodium concentration has a time constant on the order of a few seconds (5). Because this time interval is much shorter than the period of the LCO (20 to 50 s; Ref. 10), apical sodium permeability may vary in time and decrease, in its time-averaged value, when time-averaged luminal sodium concentration increases. Thus, apical sodium uptake may not directly track the oscillations in luminal sodium concentration; consequently, the ability of the tubular epithelial cells to adapt to an LCO in sodium load may be limited, in comparison to the ability to adapt to an increased steady load. A third consideration is that the parameters that determine whether LCO emerge in our model are not fixed; rather, they are strongly influenced by the physiological state of the nephron. These parameters may be reset by alterations in a number of physiological variables, including the factors that influence renal sodium transport. For example, alterations in dietary sodium intake, effective circulating volume, and arterial blood pressure influence renal hemodynamics and sodium transport via changes in levels of renal nerve activity, angiotensin II, and atrial natriuretic peptide. These factors alter nephron flow and can shift the operating point and sensitivity of the TGF system (31). Such functional changes can affect the key parameters that determine whether a nephron will exhibit steady flow or LCO. These key parameters are the system time delays, including delays related to tubular fluid flow rate (18, 29), and the gain magnitude, which is a function of the steepness of the TGF response curve and the slope of the chloride concentration profile in TAL flow along the MD in the steady state (18, 19). Hence, the physiological state of an animal can affect both the propensity for LCO to emerge and the rate of sodium transport in the distal nephron.Summary. The findings of this study support the accepted view that TGF plays a key role in the regulation of the delivery of fluid and electrolytes to the distal nephron. However, these findings also predict a rich ensemble of behaviors that may mediate a differential regulation of fluid and electrolyte delivery to the distal nephron and differential compensatory responses to positive and negative perturbations in SNGFR. A notable prediction is that the emergence of LCO in vivo will enhance the delivery of NaCl into the distal nephron and thereby tend to enhance sodium excretion. Both experimental tests and additional modeling studies are warranted to further elucidate the functional significance of the behaviors of this complex, nonlinear negative-feedback control system.
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APPENDIX A |
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Mathematical Model
Model equations.The model is formulated as a system of coupled equations
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(A1) |
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(A2) |
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(A3) |
Equation A1 is a partial differential equation for the chloride
concentration C in the intratubular fluid of the TAL of a short-looped nephron. At time t = 0, initial concentrations C(x, 0)
(for x [0, 1]) and C(1, t) (for
t
(
, 0)) must be specified. We assume the boundary
condition C(0, t) = 1, meaning that fluid entering the TAL
has constant chloride concentration. The rate of change of that
concentration at x
(0,1 ] depends on
processes represented by the three right-hand terms in Eq. A1.
The first term is axial convective chloride transport at intratubular
flow speed F. The second is transepithelial efflux of chloride driven by active metabolic pumps situated in the tubular walls; that efflux is
approximated by Michaelis-Menten kinetics, with maximum transport rate
Vmax and Michaelis constant KM.
The third term is transtubular chloride backleak, which depends on a
specified fixed extratubular chloride concentration profile
Ce(x) (see below) and on chloride permeability P.
Equation A2 describes fluid speed in the TAL as a function of
the effective luminal chloride concentration CMD at the MD
(see below). This feedback relation is an empirical equation
well-established by steady-state experiments (31). The constant
Cop is the steady-state chloride concentration obtained at
the MD when F 1. The positive constants K1 and
K2 describe, respectively, the range of the
feedback response and its sensitivity to deviations from the steady state.
Equation A3 represents time delays in the feedback pathway
between the luminal fluid chloride concentration at the MD,
C(1, t), and an effective MD concentration,
CMD(t), which is used to calculate the flow
response that is modulated by smooth muscle of the afferent arteriole
(AA). In quasi-steady state, Eq. A2 provides a good description
of the TGF response. However, dynamic experiments (3) show that a
change in MD concentration does not significantly affect AA muscle
tension until after a discrete (or pure) delay time p,
and then the effect is distributed in time so that a full response
requires additional time, with greatest weight in the time interval
[t
p
, t
p],
where
is a second delay parameter. To simulate the pure delay
followed by a distributed delay, the convolution integral given in
Eq. A3 is used to describe the effective signal received by the
AA at time t (29). The kernel function
for
this integral is given by
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(A4) |
A steady-state solution to Eqs. A1-A4 may be obtained by
setting F = 1 for 1 unit of normalized time (the transit time of the TAL at flow speed 1), starting at t = 0, to give the
steady-state operating concentration Cop = C(1, 1) at the
MD. If one specifies that C(1, t) = Cop for
t (
, 1), then the input flow to the TAL, F, is fixed
at 1 for all previous time. The feedback loop can then be closed at
t = 1. If the system remains unperturbed, then the system
solution will not vary in time. The steady-state TAL concentration
profile C is denoted by S(x).
Normalization of equations.
The dimensional forms of Eqs. A1 and A2 are given
by
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(A5) |
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(A6) |
Model parameters.
A summary of parameters and variables, with their dimensional units as
commonly reported, is given in the Glossary. The values of
model base-case parameters are given in Table 1; the criteria for their
selection and supporting references were given in Ref. 18. The
extratubular concentration is given in nondimensional form by
Ce(x) = Co(A1eA3
x + A2), where
A1 = (1
Ce(1)/Co)/(1
e
A
3), A2 = 1
A1, A3 = 2, and
Ce(1) corresponds to a cortical interstitial concentration of 150 mM. Graphs of Ce and the steady-state
luminal profile S(x) were given in Fig. 1 of Ref. 18. The
steady-state operating concentration Cop was calculated
numerically using the TAL dimensions and transport parameters, with
steady flow F = 1 in Eq. A1.
Gain magnitude.
A bifurcation in model solution can occur when the magnitude of the
instantaneous gain of the feedback response exceeds a critical
value,
c (18). The instantaneous gain is given by
= K1K2S'(1), where
K1K2 is a measure of the
strength of the feedback response and S'(1) (a negative quantity) is
the slope of the steady-state chloride concentration profile at the MD. (In a negative feedback loop, the feedback gain is negative by convention; thus the phrase "gain magnitude" is used when
referring to
.) The instantaneous gain, investigated in Ref. 19,
corresponds to the maximum reduction in SNGFR resulting from an
instantaneous shift of the TAL flow column toward the MD, under the
assumption that the response in SNGFR is also instantaneous.
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APPENDIX B |
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Numerical Methods
Methods are identified by corresponding figure numbers.Figure 1A.
This curve was computed from the model equations given in
APPENDIX A. Equation A1 was solved using a
second-order essentially nonoscillatory (ENO) scheme, coupled with
Heun's method for the time advance. This algorithm yields solutions
that exhibit second-order convergence in both space and time (17). The
integral of Eq. A3 was evaluated by the trapezoidal rule. The
numerical time and space steps in normalized units were
X = 1/640 and
t = (320 × to)
1, where
to is the steady-state TAL transit time in seconds
(see APPENDIX A, Normalization of equations). These
time steps, which correspond to dimensional values of
X = 7.8125 × 10
4 cm and
t = 3.125 × 10
3 s, were used for all dynamic calculations required
for Figs. 1, 2, 4-9, and for Tables 2 and 3. The high
degree of numerical grid refinement was required, both to faithfully
represent the nonlinearities that are embodied in the model equations
(21) and to compute with sufficient accuracy the time-averaged fluid flow rates, chloride concentrations, and chloride delivery rates. Oscillations in Fig. 1A were initiated by a brief transient
perturbation of F. The waveform was recorded only after the oscillation
had converged to a LCO.
Figure 2A.
This standard curve was obtained by evaluating Eq. A6, with Q = F/.
Figure 2B.
To obtain data for this curve, Eq. A1 was solved for specified
constant values of fluid flow, F. For each value of F, a steady-state concentration profile was obtained for C. The curve was constructed by
plotting the concentration values at the MD as a function of the values
of SNGFR, Q, which are given by Q = F/.
Figure 2C.
To obtain data for Fig. 2C, Eq. A1 was solved for a
sustained sinusoidal flow given by F = Q =
Qo(1 + 0.30 sin (2
t/22)), where t is in seconds. The resulting concentration waveform at the MD was recorded after the initial concentration profile had passed
out of the model TAL.
Figure 4.
LCO were computed for integer values of (dots on curves) and for
additional values between 3 and 4. Oscillations were initiated by a
brief transient perturbation (+10% of steady-state base-case flow).
Waveforms were recorded for analysis only after oscillations were
indistinguishable from LCO; to ensure this convergence, simulations were conducted for 968,640 time steps, corresponding to about 50.5 min
of simulated oscillations, before waveforms were recorded. Slightly
more than two periods of each waveform were recorded (periods differ
slightly as a function of
). The period of each waveform was
determined, and two periods of each were used to compute the time
averages of MD variables. Simpson's rule was used to approximate the
integrals that represent the time averages.
Figure 5. Selected waveforms computed for Fig. 4 were used for Fig. 5.
Figure 6.
Compensation was evaluated at integer values of and at selected
other values to produce sufficiently smooth curves. The hypothetical
steady-state case for
>
c was computed by
replacing Eq. A3 with the nondelay relation
CMD(t) = C(1, t). For the cases where the stable solution to the model equations is a steady state, i.e., for
c and for cases where both
>
c but the MD delay was eliminated (marked "SS" in
the figure), compensation was evaluated from the defining equations
(Eqs. 1 and 2) as follows: sustained perturbations of
0.01 and +0.01 of the value for Qop were added to
Qop; solutions were computed until new steady-states were
achieved;
Y was identified with changes in the MD variables
F, C, or FC; centered difference quotients
Y/
X, with
X = 0.02, were
computed for both open and closed feedback loop (open loop corresponds to
= 0); compensation was then computed using Eqs. 1 and 2. For the oscillatory cases, sustained perturbations of
0.01 and +0.01 of the value for Qop were also added to
Qop; solutions were computed until LCO had been attained,
and averages were computed (as for Fig. 4);
Y was identified
with changes in the time-averages of the MD variables F, C, or the
product FC; centered difference quotients
Y/
X
were computed for the closed-feedback-loop cases (open-loop cases had
already been computed for the steady-state compensations), using the
normalization conventions described in the METHODS;
compensation was then computed using Eqs. 1 and 2.
Figure 7. The values represented were computed by the methods already described for Figs. 4 and 6.
Figure 8. Waveforms computed for Fig. 7 were used in Fig. 8.
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ACKNOWLEDGEMENTS |
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We thank Paul P. Leyssac for insightful comments, which led to improvements in this article.
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FOOTNOTES |
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We thank John M. Davies and Kayne M. Arthurs for assistance in preparation of Figs. 1-9, which was supported by National Science Foundation Group Infrastructure Grant DMS-9709608 (to M. C. Reed, H. E. Layton, and J. J. Blum).
Portions of this work were completed while H. E. Layton was on sabbatical leave at the Institute for Mathematics and Its Applications at the University of Minnesota, Minneapolis, MN.
This work was principally supported by National Institute of Diabetes and Digestive and Kidney Diseases Grant DK-42091 (to H. E. Layton).
This work was presented in poster format at Experimental Biology '98 (Abstract 630, FASEB J. 12: 108, 1998).
The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. §1734 solely to indicate this fact.
Address for reprint requests and other correspondence: H. E. Layton, Department of Mathematics, Duke University, Box 90320, Durham, NC 27708-0320 (E-mail: layton{at}math.duke.edu).
Received 20 May 1999; accepted in final form 13 September 1999.
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