MODELING IN PHYSIOLOGY
Spectral properties of the tubuloglomerular feedback system
H. E.
Layton1,
E. Bruce
Pitman2, and
Leon C.
Moore3
1 Department of Mathematics, Duke University,
Durham, North Carolina 27708-0320; 2 Department
of Mathematics, State University of New York, Buffalo 14214-3093; and
3 Department of Physiology and Biophysics,
State University of New York, Stony Brook, New York 11794-8661
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ABSTRACT |
A
simple mathematical model was used to investigate the spectral
properties of the tubuloglomerular feedback (TGF) system. A
perturbation, consisting of small-amplitude broad-band forcing, was
applied to simulated thick ascending limb (TAL) flow, and the resulting
spectral response of the TGF pathway was assessed by computing a power
spectrum from resulting TGF-regulated TAL flow. Power spectra were
computed for both open- and closed-feedback-loop cases.
Open-feedback-loop power spectra are consistent with a mathematical
analysis that predicts a nodal pattern in TAL frequency response, with
nodes corresponding to frequencies where oscillatory flow has a TAL
transit time that equals the steady-state fluid transit time.
Closed-feedback-loop spectra are dominated by the open-loop spectral
response, provided that
, the magnitude of feedback gain, is
less than the critical value
c required for emergence of a sustained TGF-mediated oscillation. For
exceeding
c, closed-loop spectra have peaks corresponding to the
fundamental frequency of the TGF-mediated oscillation and its
harmonics. The harmonics, expressed in a nonsinusoidal waveform for
tubular flow, are introduced by nonlinear elements of the TGF pathway,
notably TAL transit time and the TGF response curve. The effect of
transit time on the flow waveform leads to crests that are broader than troughs and to an asymmetry in the magnitudes of increasing and decreasing slopes. For feedback gain magnitude that is sufficiently large, the TGF response curve tends to give a square waveshape to the
waveform. Published waveforms and power spectra of in vivo TGF
oscillations have features consistent with the predictions of this
analysis.
kidney; renal hemodynamics; nonlinear dynamics; mathematical model
 |
INTRODUCTION |
THE POWER SPECTRUM, which quantifies the relative
prevalence of frequency components in a signal, has emerged as an
important tool in the analysis of experimental time series derived from renal hemodynamic variables. Spectral analysis helped establish that
single-nephron oscillations in intratubular pressure, flow, and distal
tubule chloride concentration, with frequency of 20-50 mHz, arise
from an intrinsic instability in the tubuloglomerular feedback (TGF)
loop (7); spectral analysis showed that significant spectral power was
distributed over a range of frequencies in tubular flow that appears to
exhibit deterministic chaos (22); and spectral analysis has been used
to distinguish oscillations arising from TGF from those of intrinsic
myogenic origin (3, 11, 23).
In this study we used a simple mathematical model to investigate the
spectral properties of the TGF system. Previously, we have
used the same model framework to elucidate the emergence of
TGF-mediated oscillations (13, 17), to distinguish steady-state gain
from instantaneous gain (14), and to characterize the nonlinear filter
properties of the thick ascending limb (TAL) (15).
We first summarize the model equations and associated physiological
parameters, and we review the TGF-mediated oscillation that may be
exhibited by the model when
, the magnitude of gain of
the TGF pathway, exceeds a critical value
c. Then, using
an open-feedback-loop configuration, we compute power spectra that characterize the low-pass filter of the model TAL and the transmission of the TGF signal from the macula densa (MD) to the afferent arteriole (AA). These results confirm the nodal structure identified in the
companion study (15) and provide additional insight into the spectral
properties of the TGF system components when operating in the absence
of feedback.
The key new results of this study, however, are obtained from the
closed-feedback-loop configuration. Power spectra, computed for
selected values of feedback-loop gain magnitude
, characterize the
evolution of the TGF system as
increases through the regime that
will not support sustained TGF-mediated oscillations,
<
c, and into the regime that does,
>
c. For
<
c, the power spectra are
dominated by the open-loop spectral characteristics, with the
fundamental resonant frequency of the TGF oscillation superimposed on
the open-loop frequency response. For
>
c, spectral power is increased, and power spectra are composed of peaks
corresponding to the fundamental resonant frequency of the TGF-mediated
oscillation and its harmonics, which are explained via the principle of
Fourier decomposition. The harmonics, which become more pronounced as
gain magnitude increases, arise from distortions in TGF oscillations
that are introduced by nonlinear properties of the TGF pathway, notably
the integrative effect of TAL transit time and the constraints of the
TGF response function. The distortions in the waveform of tubular flow
arising from nonlinear transit time consist of crests that are broader
than troughs and a slope asymmetry, in which the absolute
magnitude of the ascending slope is less than the absolute magnitude of
the descending slope. For TGF gain magnitude that is sufficiently
large, the bounds of the TGF response function tend to give a square
waveshape to the waveform. Finally, we observe that published waveforms
and power spectra from in vivo measurements of TGF-mediated
oscillations frequently exhibit features consistent with the
predictions of this theoretical analysis.
Glossary Parameters
Co |
Chloride concentration at TAL entrance (mM)
|
Cop |
Steady-state chloride concentration at MD (mM)
|
k |
Sensitivity of TGF response (mM 1)
|
Km |
Michaelis constant (mM)
|
L |
Length of TAL (cm)
|
P |
TAL chloride permeability (cm/s)
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Qop |
Steady-state SNGFR (nl/min)
|
Q |
TGF-mediated range of SNGFR (nl/min)
|
r |
Luminal radius of TAL (µm)
|
to |
Steady-state TAL transit time (s)
|
Vmax |
Maximum transport rate of chloride from TAL
(nmol · cm 2 · s 1)
|
 |
Fraction of SNGFR reaching TAL
|
 |
Distributed delay interval at the JGA (s)
|
p |
Pure delay interval at the JGA (s)
|
Independent variables
f |
Frequency of flow oscillation (mHz)
|
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 rate (nl/min)
|
Pf |
Power spectral density
|
S(x) |
Steady-state TAL chloride concentration (mM)
|
T(x, t) |
Fluid transit time from TAL entrance (s)
|
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MATHEMATICAL MODEL |
Model equations.
The mathematical model for the TGF loop (14, 17) is given by the
following system of coupled equations
|
(1)
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|
(2)
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|
(3)
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Each
equation is in nondimensional form, i.e., all variables and parameters
have been normalized so that each is a dimensionless quantity (see
APPENDIX A). Figure 1 provides a schematic representation
of the model components. The space variable x is oriented so
that it extends from the entrance of the TAL (x = 0), through
the outer medulla, and into the cortex to the site of the MD
(x = 1).
Equation 1 is a partial differential equation for the chloride
ion 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 impose the boundary condition C(0, t) = 1, which means 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. 1. The first term is axial convective chloride
transport at the intratubular flow speed F. The second is the
transtubular 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) and on chloride permeability P.
Equation 2 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 (20). 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 3 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 AA. In
quasi-steady state, Eq. 2 provides a good description of the
TGF feedback response. However, dynamic experiments (1) show that a
change in MD concentration does not significantly affect AA muscle
tension until after a pure delay time
p, and then the
effect is distributed in time so that a full response requires
additional time, with, say, 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,
we introduced the convolution integral in Eq. 3 to describe
the effective signal received by the AA at time t
(17). We require that the kernel 
satisfy


(t
s
/2) ds = 1, so that a
constant value for C(1, t
p) will be
unchanged by the distributed delay of Eq. 3. For this study
we assume that 
is given by
|
(4)
|
a
function that makes use of one oscillation of a cosine curve, centered
at the origin and scaled so that the function is nonnegative and
continuously differentiable for all u. With this function, a
step change in C results in a sigmoidal increase in CMD
over a nondimensional time interval of
(cf. Eq. 3).
A steady-state solution to Eqs. 1-4 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, the system solution will not vary in time. We denote the
resulting steady-state TAL concentration profile C(x, 1) by
S(x).
Model parameters.
A summary of parameters and variables, with their dimensional units as
commonly reported, is given in the Glossary. The values of
model parameters are given in Table 1; the
criteria for their selection and supporting references were given in
(13). The extratubular concentration is given in nondimensional form by Ce(x) = Co(A1 exp(
A3x) + A2), where A1 = (1
Ce(1)/Co)/(1
e
A3),
A2 = 1
A1, and
A3 = 2, and where Ce(1) corresponds to
a cortical interstitial concentration of 150 mM. Graphs of
Ce and the steady-state luminal profile S were given in
figure 1 of Ref. 13. The steady-state operating concentration Cop was calculated numerically
using the TAL dimensions and transport parameters, with steady flow
F = 1 in Eq. 1.
Rather than a pure delay of 4 s in signal transmission at the
juxtaglomerular apparatus (JGA), as we used in Ref. 13, a pure delay
p of 2 s was followed by a transition interval
of 3 s, providing a distributed delay in approximate agreement with experiments (1).
The steady-state TAL transit time to is a key
parameter that plays a prominent role in this study. This transit time
is the interval required for a water molecule to travel up the TAL,
from the TAL entrance to the MD, at the steady-state flow rate,
assuming plug flow; it is equal to the TAL volume divided by the
steady-state TAL flow rate, i.e., to =
r2L/(
Qop), and it
corresponds to one unit of normalized time.
Bifurcation locus.
For simplified versions of the model given by Eqs. 1-4, we
have previously shown that, for some parameter ranges, the
time-independent steady-state solution is unstable, and subsequent to a
perturbation, the solution may take the form of stable, sustained
oscillations (13, 14, 17). This parameter-dependent behavior
probably arises from a Hopf bifurcation.
A bifurcation may occur when
, the magnitude of the instantaneous
gain of the feedback response, exceeds a critical value
c (13). The instantaneous gain, investigated in detail
in Ref. 14, corresponds to the maximum reduction in single-nephron
glomerular filtration rate (SNGFR) resulting from an instantaneous
shift of the TAL flow column toward the MD, under the assumption that the SNGFR response is also instantaneous. 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 we use the phrasing "gain magnitude" when referring to
.)
The critical gain magnitude
c and the associated
critical angular frequency
c can be determined from the
model's characteristic equation, given in APPENDIX B. The
parameters from Table 1 lead to critical gain
c
3.24 and critical frequency fc =
c / (2
)
45.9 mHz, in dimensional units.
The critical frequency fc predicts the frequency of
the oscillations arising from the bifurcation.In this
study, we assume that all parameters are fixed except for the
sensitivity of the TGF feedback response, k; the gain magnitude
depends on the sensitivity through the equation K2 = kCo/2 (see APPENDIX
A). Thus, by changing sensitivity k, we change
. The
baseline value of k used previously by us in Ref. 13 is 0.24 mM
1, which results in
3.14, a value below, but
near, the critical value
c
3.24. We have previously
hypothesized that the nearness of physiologically plausible values of
to
c may explain the tendency of some short-looped
nephrons to exhibit sustained TGF-mediated oscillations while other
nephrons do not (13).

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Fig. 1.
Schematic representation of tubuloglomerular feedback (TGF) loop
model. Thick ascending limb (TAL) is modeled by Eq. 1 as a
rigid flow tube. Luminal chloride concentration C depends on the space
coordinate x and the time t; Ce,
extratubular chloride concentration. Actions of the glomerulus,
proximal tubule, and descending limb, represented by the boxes, are
modeled by phenomenological relations, given as Eqs. 2-4.
C(1, t), luminal chloride concentration at the macula densa
(MD); CMD, delayed signal; and F, TAL flow rate. In
this study, the "Input" is broad-band forcing, which is added to
the feedback response when the feedback loop is closed. "Output"
is subjected to spectral analysis.
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Power spectra.
The fast Fourier transform (FFT) was used to obtain power spectra from
numerical solutions of the model equations. A power spectrum of a
signal, roughly speaking, is a graph that shows the relative strengths
of the oscillatory components of differing frequency that combine to
make up the signal. A power spectrum is usually represented as
"power spectral density" (19); the abscissa in such a power
spectrum is frequency, and the ordinate is the sum of the squares of
the two Fourier coefficients corresponding to that frequency (see
Eqs. C2 and C3 in APPENDIX C). We will
denote the power spectral density corresponding to frequency f
by Pf . Although a power spectrum is
graphed as a continuous curve, in practice a computed spectrum is
discrete, and there tends to be leakage from frequency "bins"
corresponding to large Fourier coefficients into adjacent frequency
bins (19). APPENDIX D summarizes the methods used to
compute numerical solutions to the model equations and to compute power
spectra based on those solutions. High resolution in space and time is
required to compute solutions that are sufficiently accurate to provide
acceptable spectral resolution (see DISCUSSION and
APPENDIX D).
We will investigate the spectral properties of TGF system components by
superimposing a broad-band forcing on a variable of the system; this
variable is considered to be the input of the system. A broad-band
forcing is a signal that includes many oscillatory components, of
nearly uniform amplitude, over a range of frequencies. The effect of
the forcing on an output of the system can be used to understand the
action of the system components as a function of input signal
frequency.
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RESULTS |
Power spectra for open TGF loop.
We have previously noted that both our model for the TAL and our model
for the distributed delay contain a low-pass filter (13, 15-17),
i.e., qualitatively speaking, low-frequency oscillations pass through
these model components with little reduction in amplitude, but the
amplitudes of high frequency oscillations are attenuated. To
distinguish the contributions of the two filters to the TGF loop and to
test the adequacy of our methods, we computed power spectra for each of
the two filters separately, and then we computed the spectrum for the
filters operating in series. In all these cases, the TGF loop was open,
i.e., TGF-regulated flow did not enter the TAL.
The three power spectra arising from model components are shown in Fig.
2. In each case there was an input signal,
containing broad-band forcing, and an output signal, as specified
below; each power spectrum was computed from the output signal. The
spectra in Fig. 2 (and also Fig. 3) were normalized by dividing by the power spectrum of the input signal; the power spectrum of that input
signal, normalized by itself, is the constant value
Pf = 1.

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Fig. 2.
Open-feedback-loop power spectra for distributed delay only, TAL only,
and TAL with distributed delay, demonstrating the accuracy of the
methods employed, through agreement with analytical predictions (see
text). The low-pass filter of the distributed delay does not much
affect the spectral properties of the model TAL for frequencies of 300 mHz and below. Nodes marked with asterisks correspond to values marked
with asterisks in Table 2.
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For the separate TAL filter, Eq. 1 was solved with input given
as steady-state flow F =
Qop plus
small-amplitude broad-band forcing. The output was given by Eq. 2, but as F(C(1, t)), i.e., the flow was
determined by the TGF response function, but without the delays that
would be introduced by Eq. 3. The sensitivity k was
adjusted to provide a gain magnitude of
= 1. In this case, we
obtained the spectrum in Fig. 2 marked "TAL only"; the general trend of decreasing amplitude as a function of frequency indicates that
the TAL operates as a low-pass filter. However, the spectrum exhibits
local minima, which correspond to nodes, and local maxima, which
correspond to antinodes. The nodal structure is explained in the
companion study (15), which shows that the range of NaCl excursions at
the MD depends on the fluid transit time through the TAL, with nodes
corresponding to frequencies where the oscillatory flow has a transit
time to the MD that equals the steady-state TAL fluid transit time. In
Table 2 we list selected nodes predicted by
analytical techniques in the companion study (15) and the corresponding
nodes found in this study through numerical calculation of model
solutions and subsequent spectral analysis. There is excellent
agreement up to 500 mHz, but there is increasing divergence as
frequency increases, arising from the approximate nature of the
numerical calculations. Nonetheless, there is agreement with error less
than 1.3% from 0 mHz through at least 1800 mHz.
In APPENDIX D we explain that the power spectrum value for
"TAL only" at f = 0, given by P0
0.8190, indicates a steady-state gain of the TAL that is within 0.3%
of a value computed previously by other means (14). This close
agreement provides confirmation that the power spectrum has been
correctly computed and scaled.
The curve marked "Distributed delay only" in Fig. 2 shows the
spectral response of the distributed delay of Eq. 3; in this case, the input C(1, t
p) was replaced by
broad-band forcing with mean value zero, and CMD was taken
as the output. This spectrum also indicates a low-pass filter, and this
spectrum also exhibits nodes, at frequencies of (2 + n)/3 per
second, for n = 0, 1, 2, ... These nodal frequencies arise
because the integral in Eq. 3 vanishes when the integrand has
the form 
(t
s
/2) × sin(2
(2 + n)s/3 +
), with 
given by Eq. 4, for any phase shift
. A different kernel
function 
would, of course, yield a different
spectral structure, since it would be composed of different Fourier
components, but physiologically reasonable choices of

are likely to have little qualitative effect below
500 mHz (see APPENDIX C). In this spectrum for
"Distributed delay only," the response at f = 0 is
P0 = 1, because a constant signal is transmitted
undiminished.
The thick shaded curve marked "TAL with distributed delay" in
Fig. 2 is the power spectrum of the TAL and distributed delay acting in
series. The input was the same as for the case of "TAL only," and
the output was obtained from Eq. 2, via the distributed delay
of Eq. 3. Thus, this spectrum is the power spectrum for the
open-feedback-loop configuration of the TGF model. We see from this
spectrum that the only effect of the distributed delay below 500 mHz is
to attenuate the spectral power of the TAL spectrum, especially above
300 mHz.
In the companion study (15) we found that the TAL low-pass filter, in
the absence of chloride backleak, exhibited 1/f scaling for
frequencies larger than about 64 mHz, i.e., the amplitude of chloride
excursions at the TAL decreased inversely with frequency. When the
spectra for "TAL only" and "Distributed delay only" are graphed on a log-log plot (not reproduced here), the resulting plots
are linear for sufficiently large frequencies, which indicates that
both spectra exhibit 1/f scaling. The TAL exhibits 1/f
scaling, with chloride backleak present, in about the same range as in the absence of backleak. However, the distributed delay exhibits 1/f scaling only above 300 mHz; consequently, the spectrum
produced by the two filters in series exhibits 1/f scaling only
above 300 mHz. Thus the combined action of the two filters does not
exhibit 1/f scaling in the range that includes much of the
frequency domain of TGF and myogenic autoregulation. In particular, the
scaling predicted by the model cannot be the source of 1/f
scaling observed in experimental records of blood pressure in rat for
frequencies ranging from 0.01 mHz up to 3 mHz (10).
Power spectra for closed TGF loop.
Figure 3 gives power spectra for the
closed-feedback-loop configuration illustrated in Fig. 1, corresponding
to increasing values of instantaneous gain magnitude
, which is a
measure of feedback strength. For each spectrum, the input was
small-amplitude broad-band forcing added to flow entering the TAL; the
output, which was subjected to spectral analysis, was the TAL flow
predicted by the feedback response (see Fig. 1). In the model, the
spectral characteristics of TAL flow are the same as those of SNGFR,
since by assumption TAL flow is a fixed fraction of SNGFR (see
APPENDIX A and Ref. 13).

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Fig. 3.
Power spectra for increasing magnitudes of instantaneous gain magnitude
. Shaded curves, power spectral density (PSD) for the
open-feedback-loop case; solid curves, PSD for closed-feedback-loop
case. Dashed line, PSD of the small-amplitude input perturbation.
Numbers beside the curves represent frequencies of local extrema in PSD
for the closed-feedback-loop case (except in F, where extrema
for both cases are shown). Open-feedback-loop curve in A is a
portion of the shaded curve in Fig. 2. Note that for gain less than
the critical gain magnitude, c 3.24, power spectra
are dominated by the spectral structure of TAL NaCl transport. For > c, the spectra are dominated by the fundamental
frequency of the stable TGF-mediated oscillation and its harmonics.
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In Fig. 3, dashed lines are the (normalized) power spectrum of the
input; solid curves are the closed feedback loop power spectra; the
shaded curves, for comparison, are the spectra for open feedback loops
(exhibited as "TAL with distributed delay" in Fig. 2). In Fig. 3,
A-E, the labels along the curves are frequencies corresponding to nodes or antinodes of the closed-loop spectrum; in
Fig. 3F, the extrema of both spectra are labeled.
Figure 3, A-D, shows the development of a resonant
frequency, emerging from the open-loop spectrum, and increasing from
~34 to ~46 mHz as the gain magnitude
is increased from 1 to
3.24. A second resonant frequency emerges from the open-loop spectrum at ~89 mHz; this frequency can be identified by detailed analysis of
the characteristic equation (Eq. B1 in APPENDIX B). The antinodes in Fig. 3 are slightly to the left of the frequency midpoints between nodes.
As noted in the section describing the MATHEMATICAL MODEL,
a bifurcation may occur at the critical value of gain magnitude,
c
3.24. Consequently, for
exceeding
c, the small applied perturbation elicits sustained TGF
oscillations at a frequency near the critical frequency
fc
45.9 mHz. This has dramatic effects on the
power spectrum, as shown in Fig. 3, E and F. Two
features are particularly noteworthy. First, the power at all
frequencies is greatly increased, as indicated by the upward shift in
the power spectral density curve. This additional power does not arise from the small-amplitude broad-band forcing; rather, it arises from
sustained flow oscillations of large amplitude. Indeed, additional numerical studies employing a transient perturbation, but no broad-band forcing, produced spectra that are almost identical to those in Fig. 3,
E and F. Additional studies also showed that the
transition to spectra qualitatively like Fig. 3, E and
F, occurred for
< 3.26, confirming, in
conjunction with the result of Fig. 3D, an abrupt transition to
sustained oscillatory flow localized within ~0.6% of
c.
The second noteworthy feature is the emergence of a series of harmonics
of the fundamental resonant frequency of the TGF system, ~46 mHz. As
increases, the higher frequency harmonics become stronger, as can
be seen through comparison of E with F of Fig. 3. The
harmonics in Fig. 3, E and F, arise from the action of the nonlinear elements in the TGF system in shaping the large-amplitude oscillations.
Waveshape distortion in open TGF loop.
If all elements of the TGF system were linear, then
oscillations in key variables would be pure (i.e., single-frequency)
sine waves, and the high-frequency components that represent distortion from a pure sine wave (i.e., the harmonics) would not be present in
Fig. 3, E and F. The nonlinear elements in our model
include the filter and transport characteristics of the TAL (Eq. 1) and the TGF feedback relationship (Eq. 2). The pure and
distributed delays in the feedback pathway (which enter through
Eqs. 3 and 4) are linear elements; indeed, the effect
of the distributed delay on a sinusoidal component is to attenuate its
amplitude without changing its frequency.
Examples of distortion by nonlinear elements of the TGF pathway are
illustrated in Figs. 4 and 5. Columns
A and B in Fig. 4 show open-feedback-loop responses to
specified sinusoidal input flows. Columns C and D in
Fig. 4 exhibit waveforms arising in the closed-feedback-loop case for
= 5 and
= 10, which both exceed
c.
[Experimental studies indicate that steady-state, in vivo gain
magnitude, which slightly underestimates instantaneous gain magnitude
(14), ranges from 1.5 to 9.9 (6).]

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Fig. 4.
Shaping of waveforms by components of the TGF pathway. In each column,
QIN is an input single-nephron glomerular filtration rate
(SNGFR), T is the TAL fluid transit time to the MD (in units of
steady-state transit time to), C is the luminal
chloride concentration at the MD, and Q is the output SNGFR (for the
closed-loop cases, columns C and D,
QIN = Q). Vertical dashed lines in rows 1-3
coincide with local extrema of transit time; horizontal shaded bars in
row 1, which span the trough and the crest of the SNGFR
waveform, correspond to maximum and minimum transit times,
respectively. Wide-line shaded waveforms in rows 4-6 are
sine waves, for comparison. See text for detailed explanation.
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In each column of Fig. 4, row 1 shows the input SNGFR
(QIN) as a function of elapsed time (the time-scale is at
the base of Fig. 4). In the model, QIN is related to input
TAL flow by a constant factor, FIN =
QIN. Row 2 of Fig. 4 gives TAL fluid transit
time T from the TAL entrance to the MD, which is computed from
Eq. 2 in the companion study (15) and which is expressed in
units of the steady-state transit time to
15.7 s. Transit time is an important quantity since theoretical
considerations indicate that chloride concentration at the MD depends
largely on TAL transit time (15). Row 3 of Fig. 4 gives luminal
chloride concentration at the MD. The vertical dashed lines in rows
1-3 of Fig. 4 coincide with a local maximum and a local
minimum of the transit time; the shaded bars in row 1 of Fig. 4
indicate the corresponding transit-time intervals.
The thin solid curves in rows 4-6 of Fig. 4 give the TGF
responses, expressed as SNGFR, arising from Eq. 2
(Q = F/
), for the indicated magnitudes of gain
(C4, C6, D4, and D5 of Fig. 4 are
intentionally left blank). The wide shaded curves in Fig. 4, rows
4-6, are sine functions, adjusted to match the frequency of
the solid curves and to have a similar amplitude. They are provided so
that distortion relative to sine curves will be apparent.
The frequency of the sinusoidal waveforms in A1 and B1
in Fig. 4 was specified at 45.9 mHz, the value of the critical
frequency fc, to permit comparison with the closed
loop results of columns C and D. Column A of
Fig. 4 has SNGFR excursions of amplitude
QIN = 6 nl/min,
so that SNGFR has lower and upper bounds of 27 and 33 nl/min,
respectively. For an input oscillation of this small amplitude, the
response of the system, at every stage, and for values of
up to 10, appears to be essentially linear, i.e., the resulting waveforms in
rows 2-6 of column A of Fig. 4 all appear to be
nearly sinusoidal. Nonetheless, there is a small degree of distortion,
which is apparent in Fig. 4, A4-A6, and in the
corresponding power spectra (see, e.g., Fig. 5A).
In Fig. 4A1 we observe that the maximum transit-time interval,
indicated by the top left shaded bar, spans a trough in
QIN, and the minimum transit-time interval, indicated by
the other shaded bar, spans a crest. These transit-time intervals
correspond to the maximum and minimum transit times marked by the
dashed lines on Fig. 4A2. In Fig. 4A3, maximum and
minimum transit times correspond to minimum and maximum MD chloride
concentrations, respectively, consistent with the analysis of the
companion study (15). For the special choice of the critical frequency
fc, the response waveform Q in Fig. 4A4 is
in phase with the input QIN.
In column B of Fig. 4,
QIN = 9, and oscillations
in QIN are therefore bounded by minimum and maximum flow
rates of 21 and 39 nl/min, respectively. As a consequence of these
larger amplitude oscillations, two nonlinear features emerge. First,
transit time T increases slowly relative to its rate of
decrease, an asymmetry arising from the larger time interval
corresponding to the maximum T, relative to the minimum
T. This asymmetry, which may be seen by comparing the shaded
bars in Fig. 4A1 to those in Fig. 4B1, and which is
apparent in Fig. 4B2, is reflected in asymmetry in the
concentration record in Fig. 4B3, which leads to the output waveform for Q in Fig. 4B4. This waveform has two particular
features, arising from TAL transit, that distinguish it from the shaded sine wave: the wide crest of the wave, relative to the trough, and a
rise to the crest that is slower than the fall from the crest. We call
this slope asymmetry "slow up/fast down."
A second nonlinear element arising through the TGF response given by
Eq. 2 is apparent in Fig. 4, B5 and B6. Because
the TGF response has maximum amplitude of
Q = 18 nl, the response is bounded below and above by 21 and 39 nl/min, respectively (see APPENDIX A); consequently, a "railing" effect occurs
for values of MD concentrations that lead to extreme values of
effective concentration CMD through Eq. 3. In Fig.
4B5 we observe railing at the lower bound, corresponding to
large MD concentration. In Fig. 4B6, for
= 10, we observe
railing at both extremes, which tends to produce a square waveform.
Waveshape distortion in closed TGF loop.
When the feedback loop is closed, as in columns C and D
of Fig. 4, the nonlinear behavior of the system is compounded by the nonlinear feedback, leading to a broader crest and generally squarer waveshape in the waveform for TAL flow. (With the closure of the loop,
the waveforms in Fig. 4, C1 and C5, coincide, as do the waveforms in D1 and D6.) However, with the increasing
nonlinearity, the action of the TAL low-pass filter becomes apparent in
the waveforms for transit time in Fig. 4, C2 and D2:
because the filter integrates flow, the large slopes in QIN
are reduced.
In Fig. 4C5, with the closure of the loop, the slow up/fast
down characteristic is enhanced, relative to Fig. 4B5, although the distributed delay of Eq. 3 tends to reduce the degree of
the effect that could be expected, given the pronounced fast up/slow down waveform in Fig. 4C3. The flattening in the crest of the waveform in Fig. 4C5 arises from the broad trough in Fig.
4C3; and small negative slope within the crest of Fig.
4C5 arises from the small positive slope in the trough of Fig.
4C3. Also, by comparison with the peaked maxima in Fig.
4C3, one sees that that railing has clipped the lower range of
SNGFR flow in Fig. 4C5. Thus the case illustrated in column
C of Fig. 4 represents the mixed effects of the nonlinear TAL and
TGF responses.
In column D of Fig. 4, where the gain magnitude corresponds to
the extreme physiological range, the clipping effect of railing dominates the transformation of MD concentration, in Fig. 4D3, to flow, in Fig. 4D6. Transit time in Fig. 4D2 has the
largest maximum and the smallest minimum of the cases examined, leading to the most pronounced extrema in MD concentration, in Fig.
4D3. Compared to the other waveforms for flow in Fig. 4, the
square waveform in Fig. 4D6 has the most pronounced deviation
from the reference sine wave.
Effect of nonlinear elements on power spectrum.
Figure 5 provides power spectra
corresponding to some of the waveforms of Fig. 4. The thin solid curves
in Fig. 5, A and B, are the spectra for the sinusoidal,
and thus single-frequency, flow inputs QIN illustrated in
Fig. 4, A1 and B1. The thick shaded curves in Fig. 5,
A and B, are the power spectra of the TGF response, corresponding to the solid-line curves in Fig. 4, A4 and
B4. Although the response illustrated in Fig. 4A4
appears to be substantially linear, the power spectrum for the response
shows that the elements of the TGF pathway have introduced substantial
spectral structure: the peaks in the gray curve in Fig. 5A
correspond to the fundamental frequency of the input, plus a series of
harmonics. When the amplitude of the input is increased, as in Fig.
4B1, the clearly nonlinear response observed in Fig.
4B4 corresponds to the more pronounced excitation of harmonics
shown in Fig. 5B.

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Fig. 5.
Power spectra of selected waveforms from Fig. 4. A-D
correspond to columns A-D in Fig. 4. A and
B: open feedback loop. C and D: closed feedback
loop. A: even in the case of small-amplitude sinusoidal input
QIN, significant harmonics may be generated in resulting
SNGFR Q by the nonlinearities of the TGF pathway. B: harmonics
become more pronounced for larger-amplitude input. C: power
spectra of luminal chloride concentration C at the MD and SNGFR Q for a
gain magnitude near the middle of the experimentally measured
range. Harmonics apparent in Q are reduced in concentration C with each
passage through the model TAL and then rebuilt by the TGF response.
D: power spectra for a gain magnitude at the high end of the
physiological range. Harmonics are more pronounced in this case, but
still the model TAL tends to reduce their power. Labels above peaks are
the fundamental frequency and its harmonics.
|
|
Figure 5, C and D, give power spectra of the MD
chloride concentration C (thin solid curves), corresponding to Fig. 4,
C3 and D3, and power spectra of SNGFR Q (thick shaded
curves), corresponding to Fig. 4, C5 and D6. These
spectra show the substantial increase in the power of the harmonics as
the distortion from the sine waveform becomes more pronounced with
increasing gain magnitude. Also, these spectra for the MD concentration
show the action of the TAL low-pass filter in reducing the strength of
the harmonics that are present in TAL flow. These harmonics are then
reconstructed in the flow by the TGF response function, largely through
the effect of railing.
The pattern of harmonic frequencies in Fig. 5 can be understood in
terms of the Fourier components of a periodic wave. A superposition of
sine curves, with varying frequencies and amplitudes, is required to
represent a nonsinusoidal periodic oscillation, and if that oscillation
has frequency f, then the sine curves must have frequencies of
nf, n = 1, 2, 3, ..., since the oscillation is
periodic. As
increases, the waveforms of the TGF pathway become
more distorted, with the curve segments connecting extrema becoming
steeper; consequently, the high-frequency Fourier components make
larger contributions to the waveform representation.
 |
DISCUSSION |
We have used a mathematical model to investigate the spectral
properties of the TGF pathway. For an open-feedback-loop configuration, the results of this study are consistent with the nodal TAL response pattern predicted in the companion study (15). For the
closed-feedback-loop configuration, this study predicts that the
spectral properties of TGF-regulated flow depend largely on whether the
gain magnitude exceeds the critical gain required for the emergence of
sustained TGF-mediated oscillations. The nodal structure of power
spectra for subcritical gains is a consequence of the filter properties of the TAL; for sustained oscillations, power spectra exhibit a
harmonic structure that arises from nonlinear properties of the model
TGF pathway.
Although the results presented here are based on numerical calculations
that employ a fixed parameter set (with the exception of TGF
sensitivity, which is used to vary feedback gain), the study's
qualitative conclusions are independent of the particular parameter
choices in Table 1; indeed, the results and conclusions depend only on
the structural characteristics of the model, through its dependence on
steady-state transit time to, and on the
combinations of parameters that generate the critical gain magnitude
and critical frequency through the characteristic equation (Eq. B1). Thus the nodal patterns and harmonic frequency structure
observed in Fig. 3 should arise for any choice of parameters in the
physiological range, through a rescaling of the frequency axis.
Effects of idealizations in model formulation.
Factors not included in the model formulation may affect the in vivo
spectral characteristics of the TGF system. Several of these factors
were considered in the companion study (15), including the elastic
compliance of the tubular walls, axial diffusion of NaCl within luminal
flow, spatial inhomogeneities in TAL luminal diameter and transport
capacity, and the effect of time-varying luminal concentrations on
transepithelial transport rate. Other factors, particular to the
applicability of this study, include oscillations introduced by
respiration, the dynamic properties of absorption by the proximal
tubule and descending limb, and spectral characteristics contributed by
spontaneous vasomotion of the renal vasculature. However,
because rat respiration has a frequency of ~1 Hz, this factor is not
likely to significantly affect tubular flow spectral characteristics at
frequencies below 500 mHz (see APPENDIX C). Little research
has been conducted on the dynamic properties of glomerulotubular
balance (GTB), but existing experimental studies suggest that GTB is
robust for flow variations within the physiological range (21).
The effect of spontaneous vasomotion awaits further investigation.
A final factor that may impact the applicability of the results is the
phenomenological characterization of the delay in feedback response
given by Eqs. 3 and 4. However, theoretical
considerations, developed in APPENDIX C, indicate that the
spectral structure introduced by the delay in feedback, in the range of
applicability of the model, will be insensitive to the precise
mathematical characterization of the delay, provided that the
characterization has certain general features. The insensitivity is a
consequence of the short time scale of the delay, relative to the
steady-state TAL transit time.
Numerical methodology.
In the course of this investigation, we found that great care must be
exercised to obtain power spectra that faithfully represent the
nonlinear features predicted by the mathematical model. Numerical solutions to model equations must be computed with sufficient accuracy
to preserve the structure inherent in the model equations, and good
frequency resolution must be attained in the power spectra that are
computed from the numerical solutions. The computation of power
spectra, based on given data, has been heavily studied (19); the
methodology used in this study is summarized in APPENDIX D.
We considered the computation of accurate numerical solutions for
dynamic TAL flow in Ref. 18, where we reviewed research which shows
that numerical methods may produce approximate solutions to model
equations that exhibit artifactual diffusion (which redistributes spectral power) and/or artifactual dispersion of Fourier
components (which displaces propagation speed of spectral components as
a function of frequency). In this study we used a low-diffusion, low-dispersion method, combined with high resolution in space and time,
to faithfully represent the high-frequency Fourier components in the
concentration profiles of the TAL and thus preserve spectral structure
(see APPENDIX D). The results for test cases were confirmed
by comparison with the analytical results in the companion study (15),
which predict a regular nodal pattern (cf. figure 1 in Ref. 15 and Fig.
2 in this study).
Three previous model studies that used similar formulations for the TAL
(8, 9, 11) appear to have not detected the regular pattern of harmonics
predicted by this study, and the waveforms exhibited in Ref. 8 do not
exhibit marked nonsinusoidal features. The discrepancy between these
studies and our results may be due, at least in part, to a highly
dispersive numerical method that was used in the previous studies,
coupled with low spatial and temporal resolution.
Comparison with published experimental data: waveforms. This
model study predicts specific patterns of waveform distortion in
tubular flow, which are associated with specific spectral
characteristics. Are the salient characteristics of these waveforms and
associated power spectra observable in vivo? The answer to this
question speaks directly to the adequacy of our model to represent
essential features of the TGF system. Moreover, if waveform distortion
is observable in vivo, the results of this model study have important implications for the interpretation of power spectra derived from experimental records.
To determine whether the features of waveform distortion shown in Fig.
4 were present in vivo, we examined published tracings of TGF
oscillations. The first feature we sought to identify is a broadening
of the crest of the flow waveform, relative to the trough, which
retains a more pointed shape, as seen in Fig. 4C5. This feature
is most obvious at gain magnitudes of
~ 4-5, before the
waveform is large enough to be significantly constrained by the limits
of the TGF response function. The second feature was a difference in
the magnitude of ascending and descending slopes of the flow waveform,
which was also seen in Fig. 4C5. These features, which arise
from the inverse relationship between fluid speed and TAL transit time,
are particularly evident in the MD chloride concentration waveforms in
Fig. 4 of this study and in figure 3 of the companion study (15),
before the distributed delay of AA response has acted to smooth the
curves. Note that when examining concentrations, the trough, rather
than the crest, is broadened, and the ascending slope magnitude is
larger than descending magnitude. (In figure 3 of the companion study
(15), the flatness of the trough of the curve corresponding to
f = 0.5/to may be accentuated by solute
backleak, which impairs the capability of the TAL to reduce chloride
concentration at low flow rates.)
Eight experimental time records for single-nephron pressure, flow, or
MD concentration were examined; the results are summarized in Table
3. Every record showed differences, in the
majority of the displayed periods, in ascending and descending slope
magnitudes, with the fall in the recorded variable being discernibly
more rapid than the rise (except for the reverse case in distal
chloride concentration). Four of the eight records exhibited broadening of the crests of the waveform, whereas the excursions through the
minima were sharp. Because the broadening of the crests of the
oscillatory flow waveform may be accentuated by increasing TGF gain
(see Fig. 4), its appearance in some records but not others may be
attributable to internephron heterogeneity.
A typical measured TGF oscillatory waveform of late proximal tubule
pressure, from a study by Holstein-Rathlou et al. (11) (their figure 1, bottom right), is reproduced in Fig.
6B; an earlier study has shown that
the waveforms of flow and pressure have similar shapes (7). In Fig.
6A we exhibit model SNGFR, for
= 4; the waveform has been
scaled temporally to have the same period as the experimentally
measured waveform, 28.4 s, which corresponds to a frequency of 35.2 mHz. In Fig. 6A, the dashed curve, appearing in the fourth
through the sixth oscillation, is a sine wave, provided for comparison
with the nonsinusoidal model waveform. In Fig. 6, the vertical dashed
lines, appearing in the seventh through the tenth oscillations in both
A and B, are drawn to coincide with the points on the
model waveform with local maximum slope magnitude. Comparison of the
waveforms in Fig. 6, A and B, indicates a close
correspondence between the wave shape predictions of the model and the
in vivo measurement: both waveforms exhibit broad crests and a smaller
ascending slope magnitude relative to descending magnitude.

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Fig. 6.
Comparison of model waveform of SNGFR (A) with experimental
waveform of proximal tubule pressure (B). Model waveform,
computed for gain magnitude = 4, has been scaled temporally to have
the same period as the experimental waveform. Dashed curve in
A, given for the interval extending from 100 to 180 s, is a
sine wave; vertical dashed lines mark times that correspond to maximum
slope magnitudes in the model waveform. B: typical TGF-mediated
oscillations in proximal tubule pressure. Both waveforms exhibit wide
crests, relative to troughs, and small ascending slope
magnitudes, relative to descending slope magnitudes. [B is
reprinted from Holstein-Rathlou et al. (11).]
|
|
From this examination of experimental results, we conclude that in vivo
TGF oscillations exhibit features consistent with the predictions of
our model. However, the mechanisms that operate in the model may not be
the sole causes of waveform distortion in vivo. For example, the slope
asymmetry could also result from differences in the time constants of
the turn-on and turn-off transitions of the TGF mechanism acting across
the JGA, a feature not represented in the model; indeed, experiments
suggest such a hysteresis in TGF responses to manipulations of Henle's
loop flow between zero and high flows (figure 13 of Ref. 20). However, these results are not strictly comparable to our simulation studies, since the TGF on-transition will be principally determined by the
washout of the TAL, while the off-transient will be dominated by the
kinetics of TAL NaCl absorption, as the epithelium dilutes the
stationary fluid column within the TAL. To a lesser extent, all
commonly used experimental techniques will include, and may be
influenced by, the dynamics of TAL absorption. Hence, there is no
definitive evidence, at present, regarding asymmetry in the
transmission of the TGF signal across the JGA; but if such asymmetry
exists, it would reinforce the asymmetry arising from the transit-time
dependence of TAL NaCl absorption.
Another alternative cause for waveform distortion is a displacement of
the TGF response operating point to a site near the top of the response
curve, which could result in the broadening of the waveform crest, but
not the trough, by imposing a constraint on the flow increase allowed
by the feedback response (in Eq. 2 we assume that the operating
point is at the midpoint of the response curve). However, experimental
evidence indicates that the TGF operating point is usually near the
center of the TGF curve in normal, extracellular volume-replete rats
(20), consistent with the model assumptions (13).
Comparison with published experimental results: power spectra.
Regardless of the mechanism of waveform distortion, the TGF waveform in
vivo has marked similarities to those shown in Figs. 4 and 6A,
which suggests that the harmonics of the TGF fundamental should be
present in power spectra computed from experimental data. In
particular, power spectra from in vivo data should have some of the
features of the spectra shown in Fig. 3, E and F, especially for low frequencies (for instance, <200 mHz), where confounding factors may be reduced.
The prediction that power spectra of TGF oscillations will be
characterized by maxima at the fundamental frequency and its harmonics
is supported by spectra in the experimental literature. Studies by
Holstein-Rathlou and Marsh (7) and Yip et al. (22) appear to show a
resonant TGF oscillation in rat proximal tubular pressure and its first
harmonic, similar in this respect to our Fig. 3E. In figure
1B in Ref. 7, an oscillation in pressure of 35-44 mHz
appears to have a harmonic in the range of 74-88 mHz. [The higher
frequency components at 133-163 mHz in figure 1B of Ref. 7
may arise from intrinsic vascular oscillations of the AA (1, 23); also
note that the plot in figure 1B of Ref. 7 uses a linear, rather
than a logarithmic, ordinate, which may account for the absence of
discernible harmonic structure in the power spectrum of distal chloride
concentration given in the same figure.] In figure 1D in Ref.
22, an oscillation of 33-35 mHz appears to have a harmonic in the
range of 66-70 mHz. In figure 1E of Ref. 22, a resonant
frequency of ~23 mHz appears to have a harmonic at ~47 mHz.
Another study by Yip et al. (23), however, contains power spectra of
rat efferent arteriolar flow that appear to exhibit a nodal pattern.
For example, figure 5B (dashed curve) of Ref. 23 exhibits local
minima at ~50, 97, 153, and 204 mHz and local maxima at 24, 66, 123, and 172 mHz, patterns that are very similar to the nodal structure
observed in our Fig. 3, A-D. However, it appears that
these experimental flow measurements already exhibit a resonant TGF
oscillation of modest power (see figure 5A of Ref. 23), in
which case comparison should be made with our Fig. 3, E-F,
where there is better qualitative agreement in shape but somewhat less
agreement in nodal pattern. A plausible explanation for the apparent
discrepancy is that the nephron examined may have a subcritical gain
magnitude, and its oscillations may be driven through coupling with a
neighboring nephron that is spontaneously oscillating (5). Indeed, when
we perturbed our model configuration with
= 3, using a
large-amplitude signal with a sweeping frequency, from 0 to 1 mHz, we
obtained a power spectrum that was in good qualitative agreement with
the nodal structure in the closed-loop spectrum of our Fig. 3C
and similar to the dashed-line spectrum reported by Yip et al. (23) in
their figure 5B.
Finally, studies of whole kidney renal blood flow and arterial pressure
by Cupples et al. (3) yielded complex power spectra that exhibit
elements suggestive of both nodal patterns and TGF harmonics (see
figure 1 in Ref. 3).
Based on these comparisons with experimental data, we conclude that
sustained TGF-mediated oscillations in vivo can be sufficiently nonsinusoidal to exhibit substantial power in harmonics that lie above
the fundamental frequency. The same is true for oscillations in
nephrons with subcritical gain magnitude, since the TGF system may
express the nodal structure of the TAL filter in response to
perturbations. The harmonics and nodal structure may be confounding factors in studies of in vivo power spectra, where detailed analysis has been used to identify and quantify other oscillatory elements, e.g., the intrinsic myogenic oscillation and interactions between TGF
and the myogenic oscillation (2, 23). High numerical resolution in
computer simulations and careful consideration of experimental design
and data analysis are needed in studies of TGF dynamics, because
nonlinear characteristics of the system can be easily lost or obscured.
 |
APPENDIX A |
Normalization of Equations
The dimensional forms of Eqs. 1 and 2 are given
by
|
(A1)
|
and
|
(A2)
|
where
r is the tubular radius,
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 (13). To express these equations in a
nondimensional form, let
= x/L,
= t/to,
= r/
,
(
,
) = 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,
=
/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/(2
rL), Po = Fo/(2
rL), 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. A1 and A2 are rewritten in dimensionless
terms and the tilde symbols are dropped, Eqs. 1 and 2
follow directly. The dimensional form of Eqs. 3 and 4
are the same as their nondimensional forms.
 |
APPENDIX B |
The Characteristic Equation
The characteristic equation for Eqs. 1-4, obtained by
procedures described in Refs. 13 and 17, is given by
|
(B1)
|
where all variables have been nondimensionalized as
described in APPENDIX A. As described previously (13, 17), the bifurcation locus corresponds to a value of
that is pure imaginary, i.e.,
= i
, where i =
, and
is a real
number that represents the angular frequency of oscillatory solutions.
When
= i
, the specification for

( y) given in Eq. 4 yields
|
(B2)
|
With
this evaluation, the real and complex parts of Eq. B1 can be
solved numerically (as described in APPENDIX C of Ref. 14), to find the critical gain magnitude
c and the associated
critical angular frequency
c.
 |
APPENDIX C |
Spectral Properties of the Distributed Delay
The distributed delay is characterized by the choice of the kernel
function 
appearing in Eq. 3 and specified in
Eq. 4. We have previously shown that the bifurcation locus is
unlikely to be much affected by the specific form of the kernel
function (17). Here we reason from general considerations that the
spectral properties of the TGF pathway, for frequencies below 500 mHz, do not much depend on the form chosen for the kernel function, in the
sense that a physiologically reasonable choice for the function is
unlikely to introduce significant structure (i.e., pronounced local
extrema) in spectral power at frequencies below 500 mHz.
The basis of these general considerations is that the time scale of the
distributed MD delay (2-4 s; Ref. 1) is much shorter than that of
the transit time of the TAL (15-20 s) or of the sustained oscillations that may be mediated by the TGF pathway (with period 20-33 s; Ref. 7). These time scales correspond to characteristic frequencies of 250-500 mHz for the MD delay, 50-60 mHz for
the TAL transit time, and 30-50 mHz for the sustained
oscillations. Thus spectral structure below 500 mHz (and particularly
below 250 mHz) will be dominated by contributions from TAL transit time or TGF-mediated oscillations. It follows as a corollary that tubular compliance, estimated in Ref. 7 to have a characteristic time of 1 s
(corresponding to 1000 mHz), will not significantly affect spectral
structure below 500 mHz.
The kernel function 
represents the time course of
the signal that modulates AA diameter. Experiments show that a step increase in MD chloride concentration produces a sigmoidal decrease in
AA diameter like that shown in figure 6 of Ref. 1. To represent this
sigmoidal transition, we make the following mathematical assumptions
about the kernel function: the function is continuous on the real line;
the function is nonnegative in an interval W = [
/2,
/2] of duration
and equal to zero outside the interval; and in
W, the function increases monotonically from a value of zero to
a maximum amplitude, near the center of the interval W, and
then decreases to a value of zero (thus 
is
symmetric, or nearly so, about the center of W). The form of
the kernel specified in Eq. 4 is consistent with these
assumptions.
A general kernel function, meeting these assumptions for

on W, can be expressed as a Fourier series
(4),
|
(C1)
|
where
cos(2
nu/
) and sin(2
nu/
) are the basis
functions, and where an and bn
are the Fourier coefficients, which are given by
|
(C2)
|
|
(C3)
|
Outside
W, 
0. Note that for n
1, the
sine and cosine functions in Eq. C1 have periods
/n, which evenly divide the interval for which

may be nonzero.
Since we require that a constant input concentration C in Eq. 3
pass through the distributed delay without a change in value (i.e.,
CMD must equal C if C is unchanging in time), the kernel function must have weight one on the interval W (i.e.,

(u)du = 1), which implies that
ao = 2/
for all choices of the kernel function
(the definite integrals of all other terms of Eq. C1 vanish). A
hypothetical choice of 
using only this constant term
(i.e., 
= 1/
, in W) is a worst case,
because Fourier components of the input function of the form
sin(2
nu/
+
), for arbitrary phase shift
and for n = 1, 2, 3, ..., will be orthogonal to

, and will therefore produce pronounced minima in the power spectrum at the frequencies n/
. If
= 3 s, as in
this study, these minima will fall at frequencies (in mHz) of 333, 666, 1000, 1333, ...
The choice of a constant function 
produces a linear
transition in response to a step change in input C, but the choice of

given by Eq. 4 produces a sigmoidal
transition. The corresponding power spectrum will have minima with the
same spacing, but the minima will start at 666 mHz, since only input
components of the form sin(2
nu/
+
) for n = 2, 3, 4, ..., will be orthogonal to 
. These
minima occur in the curve labeled "Distributed delay only" in
Fig. 2.
For the general kernel, taken to possess the assumed properties of the
physiological kernel function, we expect that the cosine terms, which
are even functions, will dominate the sine terms, which are odd
functions, and that lower frequency terms will dominate high frequency
terms. Thus, the largest Fourier coefficients will be
ao and a1, as in Eq. 4,
and consequently the most significant orthogonal cancellations will
occur for input frequencies of n/
, n = 2, 3, ..., and orthogonal cancellations arising from the input frequency
1/
will be small. If the distributed delay is spread on an interval
of duration 4 s or less, then 2/
will be no smaller than 500 mHz,
and the less significant frequency 1/
will be no smaller than 250 mHz. It follows that the conclusions of this study are unlikely to be
affected by the choice of 
.
Now consider a more general formulation of the MD delay in which the
formal distinction between the pure and distributed delays is removed.
Let Eq. 3 be replaced by CMD(t) = 

(t
s
/2) C(1, s) ds, with the interval W of duration
lengthened to include the duration of the pure delay,
p,
and with 
(u) modified to be nearly zero for
some portion of the left side of the interval W, for instance,
[
/2, 
/2 +
p]. Again, the kernel function can be represented as a Fourier series, with the same generic form as
given in Eq. C1, but the basis functions of the Fourier series
will not have periods that are multiples of the interval during which

is significantly different from zero. Nonetheless, significant cancellations will only occur for the input functions that
have periods that evenly divide the interval for which

is significantly different from zero, and the
consequences for power spectra will not differ from those already
noted. Thus, the conclusions of this study are unlikely to be affected
by the formal separation of the MD delay into pure and distributed
components.
 |
APPENDIX D |
Numerical Methods
Numerical methods are identified by corresponding figure numbers.
Figures 2 and 3. The power spectra in Figs. 2 and 3 arise from
an imposed perturbation of the form I(t) =
(t)/Io, where
(t) =
cos
(2
fnt + (
1)n
/2),
with fn = tonM/(N × 1 s), and where
Io = N × max{
(t)|t
[0,
p]}, with period p specified below. For Fig. 2,
M = 2 and N = 2048; in this case, I(t)
has dimensional period p = to/f1 = 1024 s (~17 min) and
a maximum amplitude of ~0.973 × 10
4. For Fig. 3,
M = 1 and N = 1024; in this case, I(t) has
the same period, but a maximum amplitude of ~1.94 × 10
3. The function I(t) is equivalent to a sum
of sine functions with alternating phase shifts of 0 and
, a
construction which allows the perturbation to evolve gradually from
I(0) = 0, thus avoiding the excitation of high-frequency modes that may
lead to aliasing. Division of
by Io yields a maximum
amplitude of ~10
3; this scaling ensures that we
observe the linear response properties of the system, at least for gain
magnitude below the critical bifurcation value
c.
In Fig. 2, for the case designated "Distributed delay only," C(1,
s
p) in the integrand of Eq. 3 was
replaced by I(t), and CMD(t) was
considered to be the output O(t). For the case "TAL
only," F was taken to be 1 + I(t), and
O(t) was taken to be the dimensionless SNGFR, computed
without any delay, i.e., O(t) = 1 + K1
tanh[K2(Cop
C(1,
t))]. For "TAL with distributed delay," F was
taken to be 1 + I(t), and O(t) = F(CMD(t)), computed from Eqs.
1-4.
Equation 1 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 (12). The integral of Eq. 3
was evaluated by the trapezoidal rule. The numerical time and space
steps were
x = 1/640 and
t = 1/320 = 3.125 × 10
3 s, where
t, here and below, is given in
dimensional units. This high degree of numerical grid refinement is
required for sufficiently accurate resolution of oscillations up to 1 Hz, and it provides good qualitative results up to 2 Hz (see Fig. 2 and
Table 2). As shown in the companion study (15), flow oscillations
produce standing waves in luminal chloride concentration along the TAL; consequently, each frequency component of the broad-band forcing having
frequency greater than 1/to will produce one or
more nodes, relative to the steady-state concentration profile, at
sites along the length of TAL. To obtain valid information about the
spectral properties of the model, the standing-wave components must be resolved by the numerical methods. For oscillations of 2 Hz, the nodes
will be separated (see Ref. 15) by dimensional length (L/to)/(2 Hz)
0.0159 cm, or
nondimensional length 0.0318, and the associated wavelength will be
twice this length. Thus the 640 subintervals used for the TAL resolved
the wavelength of this highest frequency component on a numerical grid
of ~10 points.
Sampling of model output O(t) for Fig. 2 and for Fig. 3,
A-D, began after one period of I(t). In Fig. 3,
E -F, the perturbation I(t) allowed sustained
oscillatory solutions to develop; in these cases, sampling began after
two conditions were met: 1) the oscillations in F
reached maximum amplitude, and 2) an integral number of periods of I(t) had elapsed. The output O(t) for all cases
was sampled at 5 Hz (i.e., every 64
t = 0.2 s) for 5 × 2048 points, corresponding to a real time interval of
2048 s, exactly twice the period of the perturbation
I(t). The mean of the output O(t) was computed and
subtracted from O(t) to prepare the data for spectral
analysis.
The spectra displayed in Figs. 2-3 are estimates of power spectral
density, called periodograms, which are computed from the discrete
Fourier transform of the demeaned output O(t). In our implementation, the periodograms were computed
via a FFT and a supplementary algorithm from Ref. 19, both adapted to
double precision arithmetic. The supplementary algorithm minimizes
spectral variance per data point by averaging periodograms
obtained from overlapping data sets. We used four overlapping sets of
4096 points, and we chose the Welch window for the FFT. The
periodograms obtained from O(t) were normalized through
division by the periodogram of I.
The domain of the periodogram values corresponds to the frequencies
fn = 2fN n/4096, where n = 0, 1, ..., 2048, and where the Nyquist frequency
fN is given by
(2 × 64
t)
1. Thus, domain values are
spaced at intervals of about 1.221 mHz, from 0 to 2.5 Hz.
For n > 0, the ordinate values of the periodogram,
Pn, approximate the sum of the squares of the
Fourier coefficients (corresponding to the fn) of
the response to the input signal I(t). However, P0 corresponds to the square of the constant term
of the Fourier series, and
should therefore equal the absolute value of the open-feedback-loop
steady-state gain Gss. For the parameters used in
the back-leak case, we showed in Ref. 14 that
|Gss|/
0.9069. For the
case "TAL only" and in each case shown in Fig. 3, we find that
/
0.9050, which is excellent agreement, given that different
numerical methods were used in the two studies. [Although a frequency
component for zero frequency is not introduced by the perturbation
I(t), a value of P0 emerges in the
spectra as a consequence of spectral leakage in the limit as frequency
tends to zero.]
Figure 4. The steady-state TAL profile S(x)
corresponding to flow F = 1 was computed from Eq. 1 via
the ENO scheme. Normalized versions of the oscillations in Fig. 4,
A1 and B1, were then introduced through F. The
waveforms in rows 2-6 were recorded after at least one
period of the flow oscillation to ensure that the initial profile
S(x) had been expelled. The transit-time integral was evaluated
by the trapezoidal rule. To elicit the oscillations in Fig. 4,
columns C and D, TAL flow was initially perturbed by a
square pulse (10% of steady-state flow) lasting for one transit time
interval to. Waveforms were recorded after
oscillations reached full amplitude.
Figure 5. The signals in Fig. 5 were processed as described for
Figs. 2-3, including normalization by the spectrum of the
broad-band perturbation used in Figs. 2-3, to allow magnitude
comparisons among the figures. The spectra for MD concentration C (1, t), which corresponds to the signal that is magnified by
TGF, were multiplied by 102 to permit comparison with the
spectra for SNGFR.
Figure 6. The waveform in Fig. 6A, for
= 4, was
computed like the waveforms in columns C and D of Fig.
4. The waveform in Fig. 6B was digitally scanned at
300 dots per inch from a reprint of Ref. 11 and stored as
a PostScript file. To clearly exhibit the shape of the waveform, the
image was stretched horizontally by a factor of about 2.5 (by scaling
within the PostScript file), and the axes and legends were
redrawn.
 |
ACKNOWLEDGEMENTS |
We thank Chris Clausen for helpful discussions on the methodology
of spectral analysis. We thank John M. Davies for assistance in
preparation of Figs. 1-6.
 |
FOOTNOTES |
This work was supported in part by National Institute of Diabetes and
Digestive and Kidney Diseases Grant DK-42091.
A brief report on this work appeared in Proc. Int. Congr. Industr.
Appl. Math. 3rd (Zeitschrift für Angewandte Mathematik und Mechanik, 76: 33-35, 1996).
Address for reprint requests: H. E. Layton, Department of Mathematics,
Duke University, Box 90320, Durham, NC 27708-0320 (E-mail:
layton{at}math.duke.edu)
Received 1 April 1996; accepted in final form 19 June 1997.
 |
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