| HOME | HELP | FEEDBACK | SUBSCRIPTIONS | ARCHIVE | SEARCH | TABLE OF CONTENTS |
|
|
|
|||||||
|
|||||||||
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
 |
ABSTRACT |
|---|
 TOP
 ABSTRACT
 INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES
APPENDIX A
APPENDIX B
|
|---|
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
| |
INTRODUCTION |
|---|
|
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES
APPENDIX A
APPENDIX B
|
|---|
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.
|
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.
|
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 |
 Q |
TGF-mediated range of SNGFR, nl/min |
| r | luminal radius of TAL, µm |
| Vmax | maximum transport rate of chloride from TAL,
nmol · cm 2 · s1
|
 |
fraction of SNGFR reaching TAL |
 |
distributed delay interval at JGA, s |
 p |
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 |
  (t) |
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 |
| |
METHODS |
|---|
|
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES
APPENDIX A
APPENDIX B
|
|---|
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.
|
|
(18-20, 29), which is a measure of the amplification of a signal passing through the feedback loop (see APPENDIX A for a
more detailed characterization of ). The dynamic behavior of the
model depends on whether is less than, or more than, a critical value c, the value at which LCO emerge in tubular fluid
flow and NaCl concentration at the MD. For < c, a
transient perturbation of the steady-state base case results in an
oscillatory solution with amplitude that decreases in time until the
oscillatory solution approximates the steady-state base case. After
sufficient time, that oscillatory solution, to the accuracy of a
computer simulation, is indistinguishable from the steady-state base case.
For > c, the steady-state base case remains a
solution of the model equations, but that solution is unstable. Indeed,
for > c, a transient perturbation of the
steady-state base case results in a sustained oscillatory solution
that, in time, approximates a closed trajectory known as a limit cycle.
We will use the term "limit-cycle oscillations" (LCO) to refer to
oscillations that, to the accuracy of our computer simulation, have a
trajectory that is indistinguishable from a limit cycle. An LCO
solution of the model equations is a stable solution: if an LCO
solution is transiently perturbed, then an oscillatory solution will
persist and will converge to the original LCO. The LCO increase in
amplitude with ; thus, for each > c there is a
unique LCO solution that employs the base-case parameters given in
Table 1.
The change in the nature of the stable solution, as increases
through c, is called a bifurcation. For the base-case
parameters given in Table 1, the critical gain magnitude is
c 3.24 (a method for determining the value
of c is given in Ref. 19). In model simulations, gain
magnitude was varied by changing the steepness, but not the range,
of the TGF response curve illustrated in Fig. 2A (see
APPENDIX A).
Experiments indicate that there is a time delay in TGF signal
transmission across the juxtaglomerular apparatus (JGA) (3): after a
change in tubular fluid chloride concentration at the MD, no change is
detected in SNGFR until after a discrete delay interval
(p in our model) of about 2 s, and a full effect
requires an additional distributed delay interval () of about 3 s.
The incorporation of the JGA delay into the model is a necessary
condition for a bifurcation to solutions with LCO; in the absence of a
delay, the only stable model solution, for all > 0, is the
steady-state base case (18). In some instances, we evaluated the effect
of a perturbation on an LCO solution, relative to the effect of that same perturbation on an analogous steady-state solution. To obtain a
steady-state with the same feedback gain magnitude that gives an LCO
solution, we eliminated the JGA delay from the model to provide a
hypothetical steady-state case.
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
|
(1) |
|
(2) |
Y
is the change in the system variable Y in response to a change
(i.e., a perturbation) X in another system variable
X. The denominator of Eq. 2,
(Y/ X )OL, is the ratio of
Y to X in the case where the feedback loop is
open (open-feedback-loop case, or OL). The numerator
(Y/ X )CL is the
corresponding ratio when the loop is closed (closed-feedback-loop case,
or CL). As X tends to zero, M converges to a
ratio of derivatives. However, we retain the -notation, because this
formulation is consistent with experimental studies, which necessarily
entail measurable, and therefore large, finite perturbations.
A negative feedback loop usually operates so that the magnification
M will fall between 0 and 1, which is to say that the closed
feedback loop will tend to reduce excursions in a controlled variable
Y arising from perturbations. In a system where there is
complete feedback control, M = 0; for weak feedback control, M is near 1. From the definition of magnification (Eq. 2), it follows that compensation (Eq. 1) is an index which
assesses the degree of control afforded to the variable Y by
the feedback system, relative to the case where no feedback is present,
when the system is presented with a specific perturbation of amplitude
X. An index value of 100% corresponds to complete
feedback compensation, whereas a value of 0% indicates no feedback
compensation. (Our definitions of magnification and compensation are
based on Refs. 27 and 30.)
Thus, compensation is a measure of how nearly a feedback loop can
return a system-regulated variable to its initial value when the system
is perturbed, relative to the case where there is no feedback. In our
context, we consider the regulated variables to be the tubular fluid
flow rate, the tubular fluid chloride concentration, and their product,
chloride delivery, all evaluated at the site of the MD. We consider the
input signal to be SNGFR, and we consider the perturbations to be
unspecified, non-TGF-induced changes in SNGFR. Alternatively, the
perturbations may represent experimental interventions, such as the
introduction of fluid into the proximal tubule by means of a
micropipette in a freely flowing nephron.
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.
X can be corrected in part by
the addition of the linear negative feedback term
Gss Y/, with negative gain Gss (the subscript will be explained below). In
this case, which is the closed-feedback-loop case, the output signal
would be Y = (Xo + X Gss
Y/). When this equation is solved for Y, one
obtains Y = X/(1 + Gss). Thus the
deviation from the base-case value Yo is reduced
through feedback by a factor of 1/(1 + Gss).
We can now express compensation in terms of gain magnitude
Gss. By using the definitions of compensation and
magnification given by Eqs. 1 and 2, we find
|
(3) |
|
(4) |
X. Moreover, as gain
magnitude Gss increases without bound, compensation
approaches 100%.
In a nonlinear system, the relationship of gain to feedback
compensation, as computed from the definition for compensation (Eqs. 1 and 2), will depend on the specific properties
of the nonlinear elements. Thus, the relationship given by Eq. 4 for linear systems may only apply as X tends to
zero, where magnification becomes a quotient of derivatives. When we
say, in the RESULTS section below, that compensation agrees
with the predictions of linear systems theory, the agreement will be
for a case where X is taken sufficiently close to zero
that the relationship obtained very nearly agrees with the relationship
for linear systems given by Eq. 4. For an experimentally
realizable perturbation, i.e., a finite perturbation, one may
reasonably expect that compensation will depend on the magnitude of the perturbation.
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 ).
> c (and
thus LCO are present), we interpret Y, in the
closed-feedback-loop term of the definition of magnification (Eq. 2), to be
(Y Yo)/Yo, where Y is the time-averaged value resulting from a sustained perturbation, and where Yo is the time-averaged
value when LCO are present but there is no sustained perturbation.
Thus, Yo is considered to be the base-case value
that is affected by the perturbation. Because LCO are never present
when the feedback loop is open (and therefore non-functional), we
interpret Y in
(Y/ X )OL to be the quotient
(Y Yo)/Yo,
where Y is the steady-state value resulting from the
perturbation and Yo is the steady-state base-case value.
The interpretation of Y when < c, and
therefore LCO are not present, is unambiguous, because the values of
Y can be taken relative to identical steady-state values.
Thus, when < c, for both the closed-feedback-loop
and open-feedback-loop terms, Y is interpreted as simply
Y Yo, where Yo is
the steady-state base-case value.
| |
RESULTS |
|---|
|
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES
APPENDIX A
APPENDIX B
|
|---|
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).
|
|
= 5. The values in Fig. 5 were normalized by the steady-state
base-case values of the fluid flow rate, chloride concentration, and
chloride delivery rate. Although oscillations in fluid flow are
relatively symmetric around the steady-state delivery rate, the
oscillations of chloride concentration have sharp crests, relative to
their troughs, and the concentration waveform is shifted upwards,
relative to that of fluid delivery. In addition, the oscillations in
chloride concentration exhibit a phase shift to the right. This phase
shift, which has been observed in experiments (11), arises because a
portion of the TAL fluid column must be expulsed before the full effect of a change in TAL fluid flow rate is observed in MD concentration (21). The chloride delivery rate, the instantaneous product of fluid
flow rate and chloride concentration, exhibits a phase shift as a
result of the phase shift in chloride concentration. Moreover, chloride
delivery has sharp crests, the sharpest among those of the exhibited
waveforms, and these sharp crests, which rise well above the crests in
normalized fluid flow rate, result in the increase in average chloride
delivery rate, as compared to the steady-state chloride delivery rate.
|
are shown, and the dots at the center of the plot give the time
averages for both fluid flow and chloride delivery.
The excursions in flow are limited by the range of the TGF response,
which in our model permits SNGFR to vary by no more than ±30% of the
base-case value, Qop. As gain magnitude increases, the
troughs in flow are limited at values of 5 and greater, while the flow maxima continue to increase, becoming limited
only at 10. This asymmetry in the flow waveform, which arises
because of the transport and transit-time nonlinearities in the TAL
(see Fig. 2, B and C), permits expansion of the
trajectories in chloride delivery to higher maxima, while the minima of
the trajectories are restricted.
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.
30% and 30%, the changes in chloride concentration were 53%
and 82%, respectively, while corresponding changes in chloride
delivery were 67% and 137%. The high sensitivity of these two
variables reflects the strong dependence of MD chloride concentration
on fluid flow (see Fig. 2B) and the fact that distal chloride
delivery is a product of both increased flow and increased concentration. In particular, for positive perturbations, increases in
MD chloride concentration were much less muted by chloride back-leak
along the TAL than were negative perturbations (see Figure 2 in Ref.
18).
Perturbations applied to the SS and LCO cases, which are both
feedback-regulated, resulted in substantially smaller deviations from
steady-state base case values than did perturbations applied to the OL
case. However, the SS case exhibited smaller deviations from the
steady-state base case than did the LCO case. This compensatory superiority was particularly marked for positive perturbations, where,
for example, a perturbation of +10% led to SS increases (%) of 1.83, 4.57, and 6.48, in fluid flow, chloride concentration, and chloride
delivery, respectively, while LCO increases (%) were 2.37, 9.77, and
10.6. Thus, while in either case the perturbation increased fluid
delivery by about 2%, the perturbation in the case of LCO increased
distal chloride delivery by 10.6%, which is 1.64 times larger than the
SS increase of 6.28%. This suggests that LCO, by reducing feedback
compensatory capability relative to the SS, increases NaCl delivery to
the distal tubule, which may lead to enhanced NaCl excretion.
Calculations for = 10, again for a +10% perturbation, yielded SS
increases (%) of 1.02, 2.52, and 3.57, in flow, chloride concentration, and chloride delivery, respectively, while LCO increases
(%) were 2.85, 13.4, and 13.8. Thus the improved feedback compensation
that was afforded to the hypothetical SS case by increased gain
magnitude resulted in an even larger disparity between chloride
delivery in the SS and LCO cases.
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.
c, the degree of feedback compensation was identical in
both cases and results agreed closely (differed by <0.01%) with the
predictions of linear systems theory (see METHODS).
Compensations for chloride concentration and delivery were identical
with compensations for fluid delivery. However, the curves in Fig. 6
diverge near the bifurcation point. In the hypothetical steady-state
case (SS), obtained when the JGA time delay was eliminated, feedback
compensation for fluid flow continued to increase with gain in
accordance with linear systems theory, and compensations for chloride
concentration and delivery were identical with those for fluid
delivery. In contrast, the onset of LCO reduced feedback compensation,
starting at gain magnitudes just above c, in comparison
to the control afforded by TGF when LCO were prevented. At a gain
magnitude of 10, approximately equal to the highest published
measurement of TGF gain (see Ref. 9), the reduction in feedback
compensation for flow was 19.1%. Compensation values for MD chloride
concentration and delivery (results not shown in Fig. 6) were similar
to those for fluid flow: the reductions in these compensations were
17.9% and 18.2%, respectively, relative to the hypothetical SS case
for = 10.
|
c, the feedback gain magnitude where LCO emerge.
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 = 5 in Fig. 7: open-feedback-loop responses where TGF was
nonfunctional (squares), responses for steady flows where LCO were
prevented by eliminating the time delay at the JGA (open circles), and
time-averaged responses with LCO present (solid circles).
|
< c,
or for = 5, as appropriate). This different normalization
convention, relative to Table 3, affects
only the LCO case; the different convention was adopted because
emphasis, in this context, was placed on regulatory capability around
the base case in a given situation, either steady-state or LCO (see
METHODS, Comparison and normalization of perturbed MD
variables).
|
= 5. In the hypothetical steady-state (open circles in Fig. 7), the shapes of the compensation curves for the three variables differ markedly from
the linear system compensation. The fluid delivery compensation curve
is relatively symmetric, in that the deviations from the predictions of
linear systems theory are similar for both positive and negative
perturbations. The shape of this compensation curve is in general
agreement with experimental studies (9, 33, 34, 35).
In contrast to compensation for fluid delivery, the steady-state
compensation curves for chloride concentration and delivery rapidly
decline for negative perturbations, while they equal or exceed
performance of a linear system at most positive perturbations. The
asymmetry in the chloride compensation curves is primarily a
consequence of the nonlinearity of the relationship between TAL flow
and MD chloride concentration (Fig. 2B), which results in
large concentration increases when positive perturbations are applied
in the absence of feedback, relative to the magnitude of concentration
decreases obtained when negative perturbations are applied. The
comparison of TGF-regulated excursions, which are more nearly linear as
a function of perturbations (Fig. 7, B and C), with
the nonregulated excursions results in the asymmetrical compensation
(see Eqs. 1 and 2). In contrast, by model assumptions, TAL flow is a linear function of SNGFR when there is no feedback, and
the comparison with nearly linear TGF-regulated fluid delivery as a
function of perturbations (Fig. 7A) results in nearly
symmetrical compensation.
Together, these results indicate that, in the absence of oscillations,
the nonlinear features of the TGF system can both limit and enhance
regulation and that the system may be especially efficacious at
stabilizing distal chloride delivery in the face of increasing fluid
input into the nephron.
In the presence of LCO (closed circles in Fig. 7,
D-F ), feedback compensation was reduced in comparison
to the steady-state case, except at perturbations of ±25% and
±30%, where LCO are suppressed. As in the steady-state case, the
compensation curve for average fluid flow is relatively symmetrical,
whereas those for average chloride concentration and delivery are
skewed, thereby indicating that the system is better able to compensate
for positive perturbations than for negative perturbations. The
asymmetrical curves reflect the higher open-feedback loop sensitivities
of MD chloride concentration and distal chloride delivery to increases in TAL flow rate, relative to decreases in flow rate (already noted
above for the steady-state case). As a result of those sensitivities, the LCO compensation values for chloride concentration and chloride delivery, corresponding to positive perturbations, exceeded those for
fluid flow. However, as a consequence of the high inherent sensitivity
of chloride delivery to flow perturbations, the increments in average
distal chloride delivery during LCO were larger, as a percentage of
base-case delivery, than the increments in average fluid flow. These
results illustrate yet again that the regulated variable most affected
by LCO is distal chloride delivery.
The basis for the complex responses to finite perturbations can be seen
in Fig. 8, where LCO waveforms for TAL
flow, MD chloride concentration, and MD chloride delivery are shown for
three cases: sustained flow perturbations of 20%, 0%, and +20%.
The flow waveforms (Fig. 8A) have been phase-shifted to all
begin at a common point in time; the corresponding chloride
concentration and delivery waveforms (Fig. 8, B and
C) have been adjusted by the same phase shifts, thereby
preserving the phase relationships for each set of three curves
obtained with the same perturbation.
|
| |
DISCUSSION |
|---|
|
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES
APPENDIX A
APPENDIX B
|
|---|
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 for = 10).
Fourth, the model predicts that maximal regulatory efficacy will be
attained at the phase transition boundary between steady and
oscillatory flows, that is, when the feedback gain magnitude nearly
equals the critical gain magnitude c (Fig. 6). This implies that the relatively low feedback gain of the TGF system, as
typically measured experimentally (see below) is sufficient to result
in near optimal TGF regulation of distal delivery of fluid and NaCl.
Increasing the feedback gain above c does not further
increase TGF compensation, because LCO emerge.
These four predictions, taken together, suggest that LCO have the
capacity to increase time-averaged NaCl delivery to the distal nephron
while time-averaged distal fluid delivery is not increased or is
increased fractionally less than NaCl. The potential for LCO to
differentially increase time-averaged distal NaCl delivery and thus
enhance sodium excretion is considered in a separate subsection (see below).
Finally, a general, overarching prediction is that TGF system
nonlinearities play a significant role in the regulatory function of
TGF. Even if the results and specific predictions arising from the
model simulations are not confirmed in full detail in subsequent, more
comprehensive, simulations, or by in vivo experiments, new insights
should nonetheless emerge in such investigations, for evidence of the
potential of nonlinearities to distort TGF waveforms is manifest in
experimental records (21).
Model 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).
|
c, where the model predicts that LCO
emerge. Five of these measurements cluster just below c
(two compensations of 70% superimpose). These five measurements were
obtained in hydropenia or extracellular volume depletion, states
associated with enhanced renal sodium retention. The nearness of the
measurements to c suggests that the TGF system in
hydropenia or volume depletion may operate, on average, at or near
maximal regulatory efficiency, at least in terms of the stabilization
of distal fluid and sodium delivery.
|
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.
| |
APPENDIX A |
|---|
|
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES
APPENDIX A
APPENDIX B
|
|---|
Mathematical Model
Model equations.The model is formulated as a system of coupled equations
|
|
(A1) |
|
(A2) |
|
(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
|
(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
|
|
(A5) |
|
(A6) |
is the
(dimensionless) fraction of SNGFR reaching the TAL, Qop is
the steady-state (operating) SNGFR, Q is the TGF-mediated range of
SNGFR, and k is the sensitivity of the TGF response (18). To
express these equations in nondimensional form, let 
= x/L, 
= t/to, 
= 
(, ) = C(x,
t)/Co,
e() = Ce(x)/Co,
MD() = CMD(t)/Co, 
(MD()) = F(CMD(t))/Fo,
max = Vmax/(Vmax)o,
M = KM/Co, 
= P/Po,
K1 = Q/2Qo,
K2 = kCo/2,
op = Cop/Co,
= s/to, 
p = p/to,
= /to, and
() = (s)/(1/to),
where L is TAL length and the quantities subscripted with an
"o" are conveniently chosen reference values:
Ao = 
r2,
to = AoL/Fo, Co = C(0), Fo = Fop = Qop,
(Vmax)o = FoCo/(2rL), Po = Fo/(2rL), and Qo = Qop. With these conventions, to is the filling time (and thus the transit time) of the TAL at flow rate Fo, and (Vmax)o is the
rate of solute convection into the inlet of the TAL at flow rate
Fo, divided by the area of the sides of the TAL. When
Eqs. A5 and A6 are rewritten in dimensionless terms and
the tilde symbols are dropped, Eqs. A1 and A2 follow
directly. The dimensional form of Eqs. A3 and A4 are
the same as their nondimensional forms.
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 eA
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.
c can be determined from the
model's characteristic equation, given in Ref. 21. In this study, we
assume that all parameters affecting the gain are fixed except for the
sensitivity of the TGF feedback response k, which is used to
vary through the equation K2 = kCo/2. Thus, by increasing k, is increased.
| |
APPENDIX B |
|---|
|
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES
APPENDIX A
APPENDIX B
|
|---|
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 × 104 cm and t = 3.125 × 103 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 (2t/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.
| |
ACKNOWLEDGEMENTS |
|---|
We thank Paul P. Leyssac for insightful comments, which led to improvements in this article.
| |
FOOTNOTES |
|---|
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.
| |
REFERENCES |
|---|
|
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES
APPENDIX A
APPENDIX B
|
|---|
1.
Bak, P.,
C. Tang,
and
K. Wiesenfield.
Self-organized criticality.
Phys. Rev. A
38:
364-374,
1988[Medline].
2.
Briggs, J. P.,
G. Schubert,
and
J. Schnermann.
Quantitative characterization of the tubuloglomerular feedback response: effect of growth.
Am. J. Physiol. Renal Fluid Electrolyte Physiol.
247:
F808-F815,
1984.
3.
Casellas, C.,
and
L. C. Moore.
Autoregulation and tubuloglomerular feedback in juxtamedullary glomerular arterioles.
Am. J. Physiol. Renal Fluid Electrolyte Physiol.
258:
F660-F669,
1990
4.
Chen, Y.-M.,
K.-P. Yip,
D. J. Marsh,
and
N.-H. Holstein-Rathlou.
Magnitude of TGF-initiated nephron-nephron interactions is increased in SHR.
Am. J. Physiol. Renal Fluid Electrolyte Physiol.
269:
F198-F204,
1995
5.
Fuchs, W.,
E. H. Larsen,
and
B. Lindemann.
Current-voltage curve of sodium channels and concentration dependence of sodium permeability in frog skin.
J. Physiol.
267:
137-166,
1977
6.
Guyton, A. C.
Arterial Pressure and Hypertension. Philadelphia: Saunders, 1980, p. 32-34.
7.
Guyton, A. C.,
T. G. Coleman,
A. W. Cowley, Jr.,
R. D. Manning, Jr.,
R. A. Norman, Jr.,
and
J. D. Ferguson.
A systems analysis approach to understanding long-range arterial blood pressure control and hypertension.
Circ. Res.
35:
159-176,
1974
8.
Holstein-Rathlou, N.-H.
Synchronization of proximal intratubular pressure oscillations: evidence for interaction between nephrons.
Pfluegers Arch.
408:
438-443,
1987[Web of Science][Medline].
9.
Holstein-Rathlou, N.-H.
A closed-loop analysis of the tubuloglomerular feedback mechanism.
Am. J. Physiol. Renal Fluid Electrolyte Physiol.
261:
F880-F889,
1991
10.
Holstein-Rathlou, N.-H.,
and
P. P. Leyssac.
TGF-mediated oscillations in the proximal intratubular pressure: differences between spontaneously hypertensive rats and Wistar-Kyoto rats.
Acta Physiol. Scand.
126:
333-339,
1986[Web of Science][Medline].
11.
Holstein-Rathlou, N.-H.,
and
D. J. Marsh.
Oscillations of tubular pressure, flow, and distal chloride concentration in rats.
Am. J. Physiol. Renal Fluid Electrolyte Physiol.
256:
F1007-F1014,
1989
12.
Holstein-Rathlou, N.-H.,
and
D. J. Marsh.
A dynamic model of the tubuloglomerular feedback mechanism.
Am. J. Physiol. Renal Fluid Electrolyte Physiol.
258:
F1448-F1459,
1990
13.
Holstein-Rathlou, N.-H.,
A. J. Wagner,
and
D. J. Marsh.
Tubuloglomerular feedback dynamics and renal blood flow autoregulation in rats.
Am. J. Physiol. Renal Fluid Electrolyte Physiol.
260:
F53-F68,
1991
14.
Karlsen, F. M.,
C. B. Andersen,
P. P. Leyssac,
and
N.-H. Holstein-Rathlou.
Dynamic autoregulation and renal injury in Dahl rats.
Hypertension
30:
975-983,
1997
15.
Karlsen, F. M.,
P. P. Leyssac,
and
N.-H. Holstein-Rathlou.
Tubuloglomerular feedback in Dahl rats.
Am. J. Physiol. Regulatory Integrative Comp. Physiol.
274:
R1561-R1569,
1998
16.
Kunau, R. T., Jr.,
H. L. Webb,
and
S. C. Borman.
Characteristics of sodium reabsorption in the loop of Henle and distal tubule.
Am. J. Physiol.
227:
1181-1191,
1974.
17.
Layton, H. E.,
and
E. B. Pitman.
A dynamic numerical method for models of renal tubules.
Bull. Math. Biol.
56:
547-565,
1994[Web of Science][Medline].
18.
Layton, H. E.,
E. B. Pitman,
and
L. C. Moore.
Bifurcation Analysis of TGF-mediated oscillations in SNGFR.
Am. J. Physiol. Renal Fluid Electrolyte Physiol.
261:
F904-F919,
1991
19.
Layton, H. E.,
E. B. Pitman,
and
L. C. Moore.
Instantaneous and steady-state gains in the tubuloglomerular feedback system.
Am. J. Physiol. Renal Fluid Electrolyte Physiol.
268:
F163-F174,
1995
20.
Layton, H. E.,
E. B. Pitman,
and
L. C. Moore.
Nonlinear filter properties of the thick ascending limb.
Am. J. Physiol. Renal Fluid Electrolyte Physiol.
273:
F625-F634,
1997
21.
Layton, H. E.,
E. B. Pitman,
and
L. C. Moore.
Spectral properties of the tubuloglomerular feedback system.
Am. J. Physiol. Renal Fluid Electrolyte Physiol.
273:
F635-F649,
1997
22.
Leyssac, P. P.,
and
L. Baumbach.
An oscillating intratubular pressure response to alterations in Henle loop flow in the rat kidney.
Acta Physiol. Scand.
117:
415-419,
1983[Web of Science][Medline].
23.
Leyssac, P. P.,
and
N.-H. Holstein-Rathlou.
Tubulo-glomerular feedback response: enhancement in adult spontaneously hypertensive rats and effects of anaesthetics.
Pflügers Arch.
413:
267-272,
1989[Web of Science][Medline].
24.
Leyssac, P. P.,
F. M. Karlsen,
and
O. Skøtt.
Dynamics of intrarenal pressures and glomerular filtration rate after acetazolamide.
Am. J. Physiol. Renal Fluid Electrolyte Physiol.
261:
F169-F178,
1991
25.
Ling, B.,
and
D. C. Eaton.
Effects of luminal Na+ on single Na+ channels in A6 cells, a regulatory role for protein kinase C.
Am. J. Physiol. Renal Fluid Electrolyte Physiol.
256:
F1094-F1103,
1989
26.
Moore, L. C.,
and
D. Casellas.
Tubuloglomerular feedback dependence of autoregulation in rat juxtamedullary afferent arterioles.
Kidney Int.
37:
1402-1408,
1990[Web of Science][Medline].
27.
Moore, L. C.,
and
J. Mason.
Perturbation analysis of tubuloglomerular feedback in hydropenic and hemorrhaged rats.
Am. J. Physiol. Renal Fluid Electrolyte Physiol.
245:
F554-563,
1983.
28.
Moore, L. C.,
and
J. Mason.
Tubuloglomerular feedback control of distal fluid delivery: effect of extracellular volume.
Am. J. Physiol. Renal Fluid Electrolyte Physiol.
250:
F1024-F1032,
1986.
29.
Pitman, E. B.,
H. E. Layton,
and
L. C. Moore.
Dynamic flow in the nephron: filtered delay in the TGF pathway.
Contemp. Math.
141:
317-336,
1993.
30.
Riggs, D. S.
Control Theory and Physiological Feedback Mechanisms. Huntington, NY: Krieger, 1976.
31.
Schnermann, J.,
and
J. Briggs.
Function of the juxtaglomerular apparatus: control of glomerular hemodynamics and renin secretion.
In: The Kidney: Physiology and Pathophysiology (2nd ed.), edited by D. W. Seldin,
and G. Giebisch. New York: Raven, 1992, p. 1249-1289.
32.
Schultz, S. G.
Membrane cross-talk in sodium-absorbing epithelial cells.
In: The Kidney: Physiology and Pathophysiology (2nd ed.), edited by D. W. Seldin,
and G. Giebisch. New York: Raven, 1992, p. 287-299.
33.
Thomson, S. C.,
and
R. C. Blantz.
Homeostatic efficiency of tubuloglomerular feedback in hydropenia, euvolemia, and acute volume expansion.
Am. J. Physiol. Renal Fluid Electrolyte Physiol.
264:
F930-F936,
1993
34.
Thomson, S. C.,
R. C. Blantz,
and
V. Vallon.
Increased tubular flow induces resetting of tubuloglomerular feedback in euvolemic rats.
Am. J. Physiol. Renal Fluid Electrolyte Physiol.
270:
F461-F468,
1996
35.
Thomson, S. C.,
V. Vallon,
and
R. C. Blantz.
Reduced proximal reabsorption resets tubuloglomerular feedback in euvolemic rats.
Am. J. Physiol. Regulatory Integrative Comp. Physiol.
273:
R1414-R1420,
1997
36.
Wright, F. S.
Flow-dependent transport processes: filtration, absorption, secretion.
Am. J. Physiol. Renal Fluid Electrolyte Physiol.
243:
F1-F11,
1982
37.
Yip, K.-P.,
N.-H. Holstein Rathlou,
and
D. J. Marsh.
Chaos in blood flow control in genetic and renovascular hypertensive rats.
Am. J. Physiol. Renal Fluid Electrolyte Physiol.
261:
F400-F408,
1991
38.
Yip, K.-P.,
N.-H. Holstein-Rathlou,
and
D. J. Marsh.
Dynamics of TGF-initiated nephron-nephron interactions in normotensive rats and SHR.
Am. J. Physiol. Renal Fluid Electrolyte Physiol.
262:
F980-F988,
1992
39.
Yip, K.-P.,
N.-H. Holstein Rathlou,
and
D. J. Marsh.
Mechanisms of temporal variation in single-nephron blood flow in rats.
Am. J. Physiol. Renal Fluid Electrolyte Physiol.
264:
F427-F434,
1993
This article has been cited by other articles:
|
|
 |
|
  O. V. Sosnovtseva, A. N. Pavlov, E. Mosekilde, K.-P. Yip, N.-H. Holstein-Rathlou, and D. J. Marsh Synchronization among mechanisms of renal autoregulation is reduced in hypertensive rats Am J Physiol Renal Physiol, November 1, 2007; 293(5): F1545 - F1555. [Abstract] [Full Text] [PDF]
|
|
|
|
|
|
W. A. Cupples and B. Braam Assessment of renal autoregulation Am J Physiol Renal Physiol, April 1, 2007; 292(4): F1105 - F1123. [Abstract] [Full Text] [PDF]
|
|
|
|
 |
|
 V. Vallon, B. Muhlbauer, and H. Osswald Adenosine and kidney function. Physiol Rev, July 1, 2006; 86(3): 901 - 940. [Abstract] [Full Text] [PDF]
|
|
|
|
|
|
A. T. Layton, L. C. Moore, and H. E. Layton Multistability in tubuloglomerular feedback and spectral complexity in spontaneously hypertensive rats Am J Physiol Renal Physiol, July 1, 2006; 291(1): F79 - F97. [Abstract] [Full Text] [PDF]
|
|
|
|
|
|
A. T. Layton and H. E. Layton A region-based mathematical model of the urine concentrating mechanism in the rat outer medulla. I. Formulation and base-case results Am J Physiol Renal Physiol, December 1, 2005; 289(6): F1346 - F1366. [Abstract] [Full Text] [PDF]
|
|
|
|
 |
|
 C. Walstead and K.-P. Yip Acute arterial hypertension inhibits proximal tubular fluid reabsorption in normotensive rat but not in SHR Am J Physiol Regulatory Integrative Comp Physiol, April 1, 2004; 286(4): R726 - R733. [Abstract] [Full Text] [PDF]
|
|
|
|
|
|
D. R. Oldson, L. C. Moore, and H. E. Layton Effect of sustained flow perturbations on stability and compensation of tubuloglomerular feedback Am J Physiol Renal Physiol, November 1, 2003; 285(5): F972 - F989. [Abstract] [Full Text] [PDF]
|
|
| ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| HOME | HELP | FEEDBACK | SUBSCRIPTIONS | ARCHIVE | SEARCH | TABLE OF CONTENTS |
| Visit Other APS Journals Online |
