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.
 |
INTRODUCTION |
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.
View larger version (17K):
[in this window]
[in a new window]
|
Fig. 1.
Comparison of model waveform of single-nephron glomerular filtration
rate (SNGFR) (A) with experimental waveform of proximal
tubule pressure (B) measured by Holstein-Rathlou et al. (13).
Model waveform (A) was computed for gain magnitude = 4 and scaled temporally to have the same period as the experimental
waveform. Vertical dashed lines mark times that correspond to maximum
slope magnitudes in the model waveform. Dashed curve in A,
given for interval extending from 100 to 180 s, is a sine wave,
provided for comparison only. Experimental waveform in B
exhibits typical tubuloglomerular feedback (TGF)-mediated oscillations
in proximal tubule pressure, which closely corresponds to proximal
tubule flow (11). Waveforms in A and B both exhibit
wide crests, relative to troughs, and small ascending slope magnitudes,
relative to descending slope magnitudes. (Figure is reprinted from Ref.
21).
|
|
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.
View larger version (22K):
[in this window]
[in a new window]
|
Fig. 2.
Elements of TGF pathway that introduce nonlinearities. A: TGF
response function used in model, based on extensive experimental
records (31). SNGFR Q is given as a function of steady-state macula
densa (MD) chloride concentration CMD. Values labeled along
gray bars correspond to base-case steady-state values used in model.
B: chloride concentration at MD, CMD, predicted by
model for steady values of SNGFR Q. Values labeled along gray bars
correspond to the base-case steady-state values. C: effect of
transit time and other nonlinear elements of thick ascending limb (TAL)
transport on prescribed sinusoidal SNGFR waveform (left; the
TGF response has been suspended). Resulting waveform (right)
in MD chloride concentration (CMD) has wide troughs,
relative to crests, and large ascending slope magnitudes, relative to
descending slope magnitudes. (In this figure, CMD indicates
chloride ion concentration in tubular fluid near MD; it should not be
identified with the delayed, effective concentration CMD
used in Fig. 3 and in Eqs. A1-A3, A5, A6.)
|
|
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 |
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.
View larger version (16K):
[in this window]
[in a new window]
|
Fig. 3.
Model configuration. Model represents three essential elements of the
TGF pathway: thick ascending limb (cylinder), delay at the MD (box at
right), and TGF response function (box at left).
The dimensionless composite parameters K1 and
K2 are identified in APPENDIX A; other
symbols are identified in the Glossary. Perturbations were
introduced by adding or subtracting stipulated percentages of the
steady-state base-case flow rate.
|
|
Model parameter values, given in Table 1,
are the same as those used in our previous studies; these parameter
values serve as our "base-case" parameter set. Model results
obtained with the base-case parameters are generally consistent with
experimental measurements of nephron and TGF system function. In
particular, the key system characteristics shown in Fig. 2, A
and B, were obtained using the base-case parameter set. In this
study, we use the term "steady state" to designate a model
state in which fluid flow and chloride concentration at each
spatial location is not varying as a function of time; i.e., a
"steady state" corresponds to a time-independent solution of the
model equations given in APPENDIX A. For the base-case
parameters, there is a unique steady-state solution, which we call the
"steady-state base case."
We have previously shown that the stable state of our model of the TGF
system depends on the value of feedback gain magnitude (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)
|
where M, the magnification, is defined by
|
(2)
|
In the definition for magnification, Y is a system
variable that is regulated by means of the feedback loop; 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.
However, if a linear feedback loop is operative, then the input signal
Xo + 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)
|
which simplifies to
|
(4)
|
Thus in a linear negative feedback system, compensation and
negative feedback gain magnitude Gss are simply
related through Eq. 4, and the relationship is independent of
the size of the perturbation 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 ).
Following this principle, for cases where > 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 |
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).
View larger version (18K):
[in this window]
[in a new window]
|
Fig. 4.
Time-averaged MD chloride concentration and chloride and fluid delivery
rates as functions of gain magnitude . These variables are expressed
as percent of their steady-state base-case values. Limit-cycle
oscillations (LCO) emerge at the critical gain magnitude
c 3.24. Time-averaged fluid delivery differs from
the steady-state base case by less than 0.5% and closely approximates
steady-state base-case flow for large , but time-averaged chloride
delivery increases by nearly 4% as gain magnitude increases through
the physiological range. Because of phase differences in LCO, chloride
delivery does not track the increase in chloride concentration to
nearly 107%. Gray bar marks 100% of steady-state base-case values.
|
|
The reason for the increase in time-averaged chloride delivery and the
relative stability in fluid delivery can be seen in Fig.
5A, which shows the fluid flow,
chloride concentration, and chloride delivery LCO waveforms at the MD,
for = 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.
View larger version (32K):
[in this window]
[in a new window]
|
Fig. 5.
A: two periods of model oscillations in TAL fluid flow rate
(dotted curve), chloride concentration at the MD (dashed curve), and
chloride delivery rate to the MD (solid curve), for gain magnitude = 5. Variables are expressed as percent of steady-state base-case
values. Chloride delivery rate is the instantaneous product of fluid
flow rate and chloride concentration. Each oscillation has a period of
~21.75 s. B: phase plots, showing delivery rate of chloride
to MD as a function of TAL fluid flow rate, for labeled values of gain
magnitude . Arrows indicate direction of time evolution. Dots
indicate time-averaged values corresponding to gain magnitudes. Gray
cross bars indicate intersection of 100% of steady-state base-case
chloride delivery and fluid flow. Figure is drawn so that ranges of
chloride delivery rate, for = 5, have the same magnitude in each
panel; extrema of those ranges are connected by gray horizontal lines.
Maxima of fluid flow and chloride delivery rates increase significantly
as increases while minima are restricted.
|
|
Figure 5B provides further insight into how the nonlinear
features of the TGF system contribute to the dissociation between the
waveforms of fluid flow and chloride delivery. Figure 5B is a
phase plot, where the instantaneous values for chloride delivery and
fluid flow are plotted against each other. The result is a depiction of
the trajectory of the oscillating system, which evolves in time in the
direction indicated by the arrows. Trajectories for four values of 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.
The absence of feedback in the OL case led to large deviations from the
steady-state base-case values. While the OL percent change in flow was
equal, under model assumptions, to the percentage of the applied
perturbation, the unregulated percentage changes of both chloride
concentration and delivery were much larger. For flow perturbations of
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.
Feedback compensation for fluid delivery to the MD is shown in Fig.
6. For gain magnitudes less than
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.
View larger version (17K):
[in this window]
[in a new window]
|
Fig. 6.
Percent feedback compensation for fluid delivery to MD as a function of
instantaneous gain magnitude , for infinitesimal perturbations. LCO
arise for gain magnitudes exceeding 3.24. To obtain
hypothetical steady-state data for dashed curve (SS), LCO were
suppressed by eliminating juxtaglomerular apparatus (JGA) delay. Solid
curve for ranging from 0 to 10 represents compensation for the
stable solution, steady state or LCO, corresponding to base-case
parameters. These results show that for these stable solutions, optimal
compensation is achieved just above the critical gain magnitude
c. [For < c, and for the SS
case, compensations for chloride concentration and chloride delivery to
the MD were identical to compensations for fluid delivery; for LCO,
compensations for chloride concentration and chloride delivery (not
shown) closely approximated compensations for fluid delivery.]
|
|
These results for infinitesimal perturbations suggest that LCO limit
the ability of TGF to stabilize distal delivery of fluid and NaCl.
Further, maximal regulatory efficacy is predicted to be achieved near
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).
View larger version (38K):
[in this window]
[in a new window]
|
Fig. 7.
Effects of finite perturbations of SNGFR on MD fluid flow rates,
chloride concentrations, and chloride delivery rates, and associated
feedback compensations, for TGF gain magnitude = 5. Effects of
perturbations are computed for three cases: no TGF control of SNGFR
(open loop), TGF control, but with LCO suppressed by eliminating MD
delay (closed-loop, steady), and TGF control with LCO (closed-loop,
oscillations). For perturbations of ±25 and ±30%, the two
closed-loop cases agree because LCO were suppressed by offsets from the
center of the TGF response curve. A: flow rate at MD, given as
a percentage of respective base-case, in response to flow perturbations
in SNGFR, showing that TGF limits the effects of perturbations on fluid
flow and that, in the model, LCO do not impair this effective TGF
regulation. B: chloride concentration at MD, as a percentage of
respective base case. C: delivery rate of chloride ion by TAL
flow to MD, as a percentage of respective base case. B and
C show that chloride concentration and delivery are much more
sensitive to flow perturbations than is fluid flow, both in presence
and absence of TGF regulation. Moreover, when TGF loop is closed, there
is increased sensitivity to perturbations when LCO are
present, compared to the absence of oscillations (i.e., steady
flow). D-F: feedback compensation for fluid flow, chloride
concentration, and chloride delivery, respectively. Horizontal bar is
compensation predicted by linear systems theory, for = 5. For all
three regulated variables, compensation when LCO are present is
significantly less than compensation for steady flow. For chloride
concentration and chloride delivery, compensation is more effective for
positive perturbations than for negative perturbations, for both LCO
and steady flow.
|
|
The responses to the perturbations, illustrated in Fig. 7,
A-C, were normalized with respect to the corresponding
base-case values at zero perturbation for either steady state or LCO,
as appropriate (values given in Table 2 for < 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).
The open-feedback loop responses (Fig. 7, A-C,
squares) exhibit the inherent sensitivity of the
unregulated system, described above in conjunction with Table 3. The
regulatory action of TGF can be seen in the smaller magnitudes of the
closed-feedback-loop responses (both steady-state and LCO), relative to
the corresponding open-loop responses. Furthermore, the magnitudes of
the closed-loop steady-state responses were less than those obtained in
the presence of LCO, except for the perturbations of ±25% and
±30%, because the curves become coincident as LCO are sup- pressed
by the large changes in average flow, which result in large offsets
from the center of the TGF response curve. Thus Fig. 7,
A-C, show that the feedback-regulated system was much less
affected by perturbations than was the unregulated system, and superior
regulation around the natural base-case was provided by the
hypothetical steady-state case, in which LCO were prevented.
The compensation values corresponding to Fig. 7, A-C, are
plotted in Fig. 7, D-F. In each panel of Fig. 7, the gray
horizontal line corresponds to a feedback compensation of 81.9%, the
compensation for a linear feedback system with gain magnitude = 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.
View larger version (18K):
[in this window]
[in a new window]
|
Fig. 8.
Waveforms arising from perturbations in SNGFR of 20%, 0%, and
+20%. For each perturbation, the waveforms of TAL fluid flow rate,
chloride concentration at MD, and chloride delivery rate to MD are in
the appropriate relative phase relationship. Horizontal dashed line in
each panel corresponds to steady-state base-case value. Curve labeled
0% in each panel represents the LCO for = 5, for no
perturbation; upper and lower solid curves correspond to perturbations
of +20% or 20%, respectively. A: fluid flow waveforms,
phase-adjusted so that all waveforms begin at the steady-state
base-case value. B: waveforms of chloride concentration.
C: waveforms of chloride delivery. At each instant of time,
each value of each waveform curve for chloride delivery equals the
product of the corresponding values for fluid flow and chloride
concentration in A and B.
|
|
The marked nonlinearity of the TGF system is evident in the distortion
of the waveforms obtained in the perturbed cases compared with the case
of no perturbation. Not only were LCO in both flow and chloride
concentration translated to generally higher or lower levels, but also
the waveforms underwent fundamental shape changes as the excursions
approached the limits of the TGF feedback response curve and as the
flow decreased to values where the NaCl concentration at the MD
approached its minimum level. Chloride delivery, which is the product
of fluid flow and concentration, was shaped by the phase difference
between flow and chloride concentration, and the phase lag was
influenced by the sign and magnitude of the perturbation. For example,
note that the chloride delivery waveform for zero perturbation has a
shoulder region on the rising portion of the curve. For the negative
perturbation, this inflection is more pronounced and is shifted to the
right on the waveform.
| |
DISCUSSION |
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).
Figure 9 situates the 13 measurements of
TGF gain or compensation on the graph showing model compensation
arising from infinitesimal perturbations (previously Fig. 6). It is
noteworthy that the measured values are distributed below the critical
gain magnitude, 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.
View larger version (23K):
[in this window]
[in a new window]
|
Fig. 9.
Measured values of compensation and gain, indicated along the
compensation curve exhibited in the previous Fig. 6. Experimental
values tend to group below the range of gain magnitudes that correspond
to LCO, suggesting that under normal circumstances feedback gains take
on values that support maximal feedback compensation.
|
|
Of course, TGF gain magnitude in some individual nephrons will be
greater than the means reported in Table 4 and illustrated in Fig. 9.
Furthermore, barbiturate anesthetics, which were used in the majority
of the studies cited in Table 4, may attenuate the sensitivity and
magnitude of the TGF response (23). Thus LCO are likely to be present
in a substantial subset of nephrons, as shown by Holstein-Rathlou and
coworkers (8, 9, 11, 13, 15, 37, 39). The model predicts that
regulation of the delivery of fluid and NaCl to the distal nephron will
be less effectively regulated in these oscillating nephrons, and as the proportion of oscillating nephrons increases, renal sodium excretion may also increase (see below). This potential natriuretic effect of LCO
may be relevant to hypertension: several studies in genetically hypertensive rat strains have revealed both high TGF gains and spontaneous oscillations in a substantial fraction, or in a majority, of nephrons (4, 10, 14, 15, 37, 38). Indeed, in these hypertensive
animals, the oscillations, which usually appear to be chaotic but can
also take the form of LCO, may act to enhance sodium excretion and
thereby limit the degree of hypertension. However, it should be noted
that the effect of chaotic oscillations in tubular fluid flow on distal
sodium delivery is unknown and is not addressed in this study.
The natural evolution of the operating point of a system to a region
near a bifurcation is an organizational characteristic of some types of
complex systems; it is called "self-organizing criticality" (1).
In this context, the term "criticality" refers to a system poised
near a boundary where the stability of the system changes, and
"self-organizing" indicates that the system spontaneously moves
to a critical state. The function of a system exhibiting
self-organizing criticality may change abruptly when the phase
transition is crossed, as in this study when LCO emerge. Because small
changes in key parameters can lead to marked functional changes in
systems exhibiting self-organized criticality, they are thought to be
particularly effective and adaptable regulatory systems.
The pattern shown in Fig. 9, where the TGF system in hydropenia or
volume depletion appears to operate near the boundary of a phase
transition, suggests that this system may exhibit self-organizing criticality. Moreover, our model suggests an advantageous consequence of operating near the bifurcation boundary: TGF compensation is maximal
there. Although the questions of whether nephrons exhibit self-organizing behavior and what mechanisms might mediate such behavior are intriguing, the answers are unknown.
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 flu
TOP
ABSTRACT
INTRODUCTION
