MODELING IN PHYSIOLOGY
Nonlinear filter properties of the thick ascending limb
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
 |
ABSTRACT |
A
mathematical model was used to investigate the filter properties of the
thick ascending limb (TAL), that is, the response of TAL luminal NaCl
concentration to oscillations in tubular fluid flow. For the special
case of no transtubular NaCl backleak and for spatially homogeneous
transport parameters, the model predicts that NaCl concentration in
intratubular fluid at each location along the TAL depends only on the
fluid transit time up the TAL to that location. This exact mathematical
result has four important consequences: 1) when a sinusoidal
component is added to steady-state TAL flow, the NaCl concentration at
the macula densa (MD) undergoes oscillations that are bounded by a
range interval envelope with magnitude that decreases as a function of
oscillatory frequency; 2) the frequency response within the
range envelope exhibits nodes at those frequencies where the
oscillatory flow has a transit time to the MD that equals the
steady-state fluid transit time (this nodal structure arises from the
establishment of standing waves in luminal concentration, relative to
the steady-state concentration profile, along the length of the TAL);
3) for any dynamically changing but positive TAL flow rate, the
luminal TAL NaCl concentration profile along the TAL decreases
monotonically as a function of TAL length; and 4) sinusoidal
oscillations in TAL flow, except at nodal frequencies, result in
nonsinusoidal oscillations in NaCl concentration at the MD. Numerical
calculations that include NaCl backleak exhibit solutions with these
same four properties. For parameters in the physiological range, the
first few nodes in the frequency response curve are separated by
antinodes of significant amplitude, and the nodes arise at frequencies
well below the frequency of respiration in rat. Therefore, the nodal structure and nonsinusoidal oscillations should be detectable in
experiments, and they may influence the dynamic behavior of the
tubuloglomerular feedback system.
kidney; renal hemodynamics; spectral analysis; mathematical model
 |
INTRODUCTION |
EXPERIMENTS ON THE RAT RENAL vasculature have revealed
that flow and pressure, considered as functions of time, exhibit
substantial spectral structure, both in single nephrons and in whole
kidneys (3, 11, 29, 30). This spectral structure, which is manifested as the superposition of several oscillatory components distinguished by
differing characteristic frequencies, is being used to investigate renal hemodynamic regulatory mechanisms (10). Experimental measurements and theoretical considerations indicate that the spectral structure arises in part from the dynamics of the tubuloglomerular feedback (TGF)
pathway; indeed, an oscillation of 20-50 mHz has been identified as arising directly from an instability in the TGF loop (8, 9, 15, 18,
19). An intrinsic oscillation of the afferent arteriole at ~120 mHz
has been proposed as another source of spectral structure (2, 30).
Mathematical models have indicated that the frequency response of the
TGF pathway may be influenced by the transport characteristics of the
thick ascending limb (TAL) (9, 15, 22). Specifically, model simulations
showed that when oscillations in nephron flow have a period
significantly longer than the steady-state fluid transit time of the
TAL, then fluid arriving at the macula densa (MD), which has been in
the TAL for widely varying time intervals, exhibits large oscillations
in NaCl concentration. On the other hand, when flow oscillations have a
period significantly shorter than the steady-state transit time of the
TAL, then all the fluid arriving at the MD has been in the TAL for
about the same time interval, and consequently there is little
variation in NaCl concentration. These results suggest that slow
oscillations should be transmitted through the TAL, and fast
oscillations should be attenuated, i.e., the TAL operates as a low-pass
filter.
In this study, we used a simple mathematical model to investigate the
characteristics and mechanistic origins of the TAL low-pass filter. For
this model, an explicit mathematical analysis can be carried out in the
idealized case where transtubular NaCl backleak is set to zero and NaCl
active transport parameters have no spatial dependence. Under these
assumptions, luminal NaCl concentration at each location along the TAL
is a decreasing function of TAL transit time to that location.
We now summarize four important results of the analysis of this
idealized case, all of which arise directly from the luminal concentration dependence on transit time. Numerical studies demonstrate that these exact mathematical results remain valid when NaCl backleak is included.
First, when a sinusoidal component is added to steady-state TAL flow,
the model predicts that the amplitude of resulting oscillations in NaCl
concentration at the MD will be bounded by a range interval envelope
with magnitude that decreases as a function of oscillatory frequency.
For sufficiently large frequencies, the range interval magnitude is
inversely proportional to frequency. This result, consistent with
previous findings (9, 22), characterizes, in part, the low-pass filter
action of the TAL.
Second, the model predicts that the amplitude of the oscillations in
NaCl concentration at the MD, when plotted as a function of frequency,
will exhibit a series of nodes. The nodes will appear at those
frequencies where the oscillatory flow has a transit time to the MD
that equals the steady-state fluid transit time: because the fluid
transit time to the TAL is constant, a constant NaCl concentration will
result at the MD. The model further predicts that the nodes will be
associated with the formation of standing waves in luminal
concentration along the TAL, relative to the steady-state concentration
profile. These standing waves have wavelength that is inversely
proportional to the frequency of the flow oscillation, and a node in
NaCl concentration will be located at the MD when the TAL length is an
integer multiple of one-half the wavelength.
Third, for any dynamically changing, but positive, luminal TAL fluid
flow rate, the luminal TAL concentration profile decreases monotonically as a function of TAL length, relative to the TAL entrance, since the fluid at each location along the TAL has spent progressively more time in transit through the TAL.
Finally, sinusoidal oscillations in TAL flow, except at nodal
frequencies, result in nonsinusoidal oscillations in NaCl concentration at the MD. This effect arises from the nonlinear relationship between
transit time and flow rate.
In this study, we first describe a simple mathematical model of the
TAL. Then, in RESULTS, we summarize calculations that demonstrate the dependence of luminal NaCl concentration on transit time. Using these calculations and numerical solutions of the TAL
model, we obtain bounds on MD concentration excursions as a function of
frequency, we elucidate the nodal structure of the frequency response,
and we exhibit luminal concentration profiles and associated time
records of MD concentration. In the DISCUSSION, we consider
the adequacy of the model assumptions and the physiological significance of these results.
Glossary Parameters
Co |
Chloride concentration at TAL entrance (mM)
|
f |
Frequency of flow oscillations (mHz)
|
Km |
Michaelis constant (mM)
|
L |
Length of TAL (cm)
|
p |
Period of flow oscillations (s)
|
Qop |
SNGFR (nl/min)
|
r |
Luminal radius of TAL (µm)
|
P |
TAL chloride permeability (cm/s)
|
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
|
 |
Fractional amplitude of flow oscillations
|
 |
Phase shift of flow oscillations
|
Independent Variables
x |
Axial position along TAL (cm)
|
t |
Time (s)
|
Specified Functions
Ce(x) |
Interstitial chloride concentration (mM)
|
F(t) |
TAL fluid flow rate (nl/min)
|
Dependent Variables
C(x, t) |
TAL chloride concentration (mM)
|
S(x) |
Steady-state TAL chloride concentration (mM)
|
T(x, t) |
Fluid transit time from TAL entrance (s)
|
 |
MATHEMATICAL MODEL |
Model equation.
For simplicity, we model the TAL chloride concentration only, and we
assume that sodium is absorbed in parallel with chloride. The chloride
ion is thought to be the species sensed by the MD in the TGF response
(26), and NaCl backleak is limited by the smaller epithelial
permeability of chloride, relative to sodium (24).
The principal model equation, a partial differential equation for the
chloride ion concentration C in the luminal fluid of the TAL of a
short-looped nephron (14, 15, 23), is given by
|
(1)
|
Equation 1 is in nondimensional form, i.e., all
variables and parameters have been normalized so that each is a
dimensionless quantity (see APPENDIX A of the companion
study, Ref. 17). 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). We impose a boundary condition given by C(0,
t) = 1, which means that the fluid entering the TAL has
constant chloride concentration; at time t = 0, the initial function C(x, 0) must be specified for x
(0, 1].
The rate of change of concentration in the TAL 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, taken in this study to be a specified function of time
t. The second is the transtubular efflux of chloride driven by
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 membrane chloride permeability
P.
A steady-state solution to Eq. 1 may be obtained by setting
F = 1 for 1 unit of normalized time (the transit time of the
TAL at flow speed 1). We denote the resulting steady-state TAL
concentration profile C(x, 1) by S(x).
Model parameters.
As already noted, we will establish several exact mathematical results
for the idealized case where the TAL has no transtubular NaCl backleak.
Numerical calculations and analytical results have shown that the
idealized case provides a good qualitative guide to the results
obtained when a value consistent with experimental measurements of
backleak permeability is used (15, 16). To make the results from the
two cases quantitatively comparable, the transport parameters
(Vmax and Km) for the
no-backleak case were chosen to give a steady-state concentration at
the MD nearly equal to that used for the backleak case, as well as to
closely approximate the steady-state backleak case response of MD
concentration to small flow variations relative to the steady-state
flow rate.
A summary of parameters and variables, with their dimensional units as
commonly reported, is given in the Glossary. Parameter values
are given in Table 1 for both the idealized
case with no transtubular chloride backleak and the case with measured
backleak permeability. Detailed parameter selection criteria (for both cases) and supporting references can be found in Ref. 15. The chloride
backleak permeability was taken to be 1.5 × 10
5 cm/s, a value consistent with measured chloride
permeability for rabbit (1.06 ± 0.12 × 10
5 cm/s
in Ref. 25) and estimated NaCl permeability for rat (1.13 ± 0.52 × 10
5 cm/s in Ref. 20). The remaining transport
parameters for the backleak case, Vmax and
Km, were chosen to give a steady-state intratubular
chloride concentration and concentration slope at the MD consistent
with experiments. The extratubular chloride concentration
Ce is given in nondimensional form by
Ce(x) = Co(A1 exp(
A3 x) + A2), where A1 = (1
Ce(1)/Co)/(1
e
A3),
A2 = 1
A1, and
A3 = 2. The interstitial concentration at the MD,
Ce(1), corresponds to 150 mM. Graphs of Ce, the
steady-state backleak-case TAL concentration profile, and the backleak
and no-backleak steady-state MD chloride concentrations as a function of flow rate were given in figures
1 and 2 of Ref. 15.

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Fig. 1.
Range of macula densa (MD) chloride concentration as a function of
oscillatory flow frequency. The marked frequencies
n/to, for n = 1, 2, 3, 4, and 5 correspond to frequencies (in mHz) of ~63.7, 127, 191, 255, and 318, respectively [cf. figure 2 in the companion study (17)]. Dashed
curves are approximate bounds for chloride concentration excursions;
solid curves provide precise bounds for the concentration range, thus
revealing the nodal structure of the frequency response. Curves,
computed from analytic expressions, are based on the parameters for no
chloride backleak. Vertical bars give the range of chloride excursions
obtained by numerically solving Eq. 1; parameters for the
cases without and with chloride backleak were used to compute these
bars in A and B, respectively. Nodal structure for the
backleak parameters, indicated by the vertical bars in B,
follows the the same pattern as that predicted by the exact results for
the no- backleak parameters.
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|
The steady-state transit time of fluid up the TAL, from the TAL
entrance near the loop bend to the MD, is given by
to =
r2L/(
Qop), the tubular
volume divided by the steady-state flow rate. The steady-state transit
time, which corresponds to one unit of normalized time, is a key
parameter that plays a prominent role in this study and the companion
study (17). The corresponding steady-state chloride concentration at
the MD is given by Cop, which is computed numerically from
Eq. 1 and which equals S(1).
 |
RESULTS |
Chloride concentration at MD depends on TAL transit time.
In this subsection we provide mathematical expressions for TAL transit
time as a function of TAL flow speed (Eq. 2) and for MD
chloride concentration as a function of TAL transit time (Eq. 7). Except where clearly marked, dimensionless variables are used for mathematical simplicity.
If we assume that intratubular flow in the TAL is plug-flow, and if we
follow the advance of a water molecule up the (water-impermeable) TAL,
after its entry into the TAL at x = 0, then we may compute the
time of transit T(x, t) required for the
molecule to reach position x
[0, 1] at time t. We
assume that flow speed F can differ from the steady-state value
1 by no more than an amount
, with 0
< 1. Thus flow is
always positive with (1
)
F
(1 +
). With
these specifications, transit time is given implicitly by the integral
relation
|
(2)
|
which
asserts that the distance x traveled up the TAL is the integral
of speed F, taken over the interval of transit. Bounds on
T follow directly from Eq. 2 and the bounds on
F
|
(3)
|
If
flow speed F is constant, then transit time is inversely
proportionally to F, with T(x, t) = x/F.
In this study, we will frequently think of the flow speed F as
having an average of unity plus a sinusoidal component of amplitude
, 0
< 1; in that case
|
(4)
|
where
is a phase shift, and f is frequency (cycles per unit
nondimensional time). The resulting relationship between x and
T, by explicit evaluation of Eq. 2, is
|
(5)
|
Equation 5 defines an implicit function
T(x, t), which is periodic in t
with period p = 1/f.
For sufficiently large frequencies
( f >>
), the normalized distance is
approximately equal to the normalized transit time, i.e., x
T. When x = 1, this corresponds to a result previously known: for sufficiently fast oscillations, tubular fluid reaching the
MD has a transit time nearly equal to the steady-state transit time (9,
15, 22). (Because the normalized value for the steady-state flow speed
is 1, a speed relating x and T does not appear in
Eqs. 3 and 5).
For sufficiently small frequencies f, the general bound of
Eq. 3 provides a more restrictive bound on transit time
T than does Eq. 5, if one assumes only that
|cos(2
f [t
T(x,
t)] +
)
cos(2
ft +
)|
2. Thus, for
oscillatory F as in Eq. 4, Eqs. 3 and 5 taken
together, provide bounds (independent of phase shift
) on the
variation of T as a function of time at each x
|
(6)
|
When
x is set to 1, Eqs. 3, 5, and 6 provide
information about the transit time to the MD.
A particularly important case arises when x = 1 and Tf = n, n = 1, 2, 3, ... Under these conditions, the
cosine terms in Eq. 5 cancel, since cosine is 2
periodic,
and consequently T = 1, which indicates that the transit time
equals the steady-state transit time of one nondimensional unit. Since
frequency f and oscillatory period p are related by
f = 1/p, transit time T will equal the
steady-state transit time whenever the steady-state transit time is an
integer multiple of the period of an oscillation in flow speed
F, i.e., whenever F is given by Eq. 4 with
f = n.
Although derived with a sinusoidal perturbation, this result also holds
for any general periodic flow oscillation in F that has
average speed of 1 and a period that evenly divides the steady-state transit time, because any periodic waveform can be constructed as a
constant plus a Fourier series of sine functions with frequencies that
are integer multiples of the fundamental frequency.
In APPENDIX A we obtain a formal mathematical expression
for the TAL chloride concentration profile, under the assumption that
TAL chloride backleak permeability is zero. That expression shows that
the concentration at any location in the TAL is a function of transit
time T only. Also in APPENDIX A, we show that in
the absence of backleak or other spatially inhomogeneous influences on
transport rate, the axial TAL concentration profile is monotone decreasing in x at each time t. Numerical calculations,
such as those represented in Fig. 2, below,
suggest that concentration profiles for the backleak case are also
monotone decreasing for parameter values in the physiological range.

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Fig. 2.
Chloride concentration profiles in thick ascending limb (TAL) showing
nodal points forming and moving to the left as frequency is
increased. These profiles were computed from Eq. 1 using a
sinusoidal flow perturbation in F with the specified
frequencies f and transport parameters for the backleak case.
In A-D the wide shaded curve represents the steady-state
concentration profile. Other curves correspond to the elapsed times of
0, 1/4, 1/2, and 3/4 of the period of the oscillation. Bar to the
right in A-D gives the range of concentration
excursion of the oscillation in MD concentration for each period of the
oscillation; numbers above and below the bar give the upper and lower
bounds of the excursions in mM. B and D correspond to
the frequencies of the first two nodes illustrated in Fig. 1. A
and C correspond to intermediate frequencies, which produce
large magnitude concentration oscillations at the MD. Standing waves,
relative to the steady-state concentration profile, can be clearly
observed in B-D.
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|
For the particular case of Michaelis-Menten kinetics and no chloride
backleak, the analysis in APPENDIX A implies that the
concentration C(x , t) and the transit time
T(x, t) are related by
|
(7)
|
Because
the left side of Eq. 7 is monotone increasing in C, a decrease
in transit time T to a fixed location x will cause C to
increase, as expected.
TAL frequency response.
In this subsection and its sequel we present numerical results obtained
from the evaluation of Eq. 1 and the equations in the previous
subsection. Numerical calculations were conducted as described in
APPENDIX B.
Figure 1 shows the ranges of MD chloride excursions arising from
dimensional oscillatory flow of the form
|
(8)
|
where
Qop is the steady-state TAL flow rate (6 nl/min in our
model),
= 3/10 approximates the maximum fractional amplitude permitted by the TGF response (15), and f is frequency,
corresponding to the abscissa of the graphs.
The dashed curves in Fig. 1 are theoretical bounds on the chloride
concentration range for the parameters corresponding to no chloride
backleak; the curves were computed from Eq. 7 and the bounds on
transit time given by Eq. 6. The dashed curves in Fig. 1
provide a scaling envelope: the difference between the upper and lower
dashed curves (in mM) is about 1200/(f × mHz
1) for f > 64 mHz (64 mHz
1/to), indicating that the TAL filter exhibits 1/f scaling. This scaling arises because transit time tends to the steady-state value of 1 with 1/f scaling, i.e.,
for sufficiently large f and for x = 1, the deviation
of transit time from steady-state transit time is inversely related to
f, in the sense that |T(x,
t)
1|
/(
f ) (a
consequence of Eq. 5). The deviations in MD concentration
resulting from deviations in transit time decrease with essentially the
same scaling, relative to the steady-state MD concentration. (This
scaling appears to be unrelated to the 1/f scaling that arises
in the evolution of certain extended dissipative systems; Ref.
1.)
The solid curves, also computed for no-backleak parameters, are precise
upper and lower bounds for the chloride concentration range; they were
computed from Eqs. 5 and 7, and they take into account
the variations in transit time that are implicit in Eq. 5.
Because they are upper and lower bounds, the two solid curves do not
cross at the nodes; rather, they touch at the dimensional nodal
frequencies given by f = n/to,
n = 1, 2, 3, ... (the frequencies identified as nodal
frequencies in the previous section), and then they separate again. For
the choice of parameters in Table 1, the first five nodal frequencies
are 63.66, 127.3, 191.0, 254.6, and 318.3 mHz. The first four antinodes
are at frequencies 90.72, 156.0, 221.2, and 284.9 mHz, while the
frequencies intermediate between the nodal frequencies are 95.49, 159.2, 222.8, and 286.5 mHz. Thus the antinodes are nearly halfway
between the nodes, and antinodes are found nearer the intermediate
frequencies as frequency increases.
The vertical gray bars in Fig. 1A represent MD chloride
excursions from the steady-state concentration; we computed these numerically from Eq. 1, using no-backleak parameters, with
F given by Eq. 8; these calculations provide an
independent check on the analytic results. Excursions computed at
predicted nodal frequencies had essentially zero amplitude, as
predicted; at other frequencies, the excursions reached exactly to the
solid curves in Fig. 1.
The vertical gray bars in Fig. 1B were also computed from
Eq. 1, but with transport parameters for the case with a
nonzero value for chloride backleak. These excursions from steady-state are somewhat attenuated, relative to those in Fig. 1A, except at nodal frequencies, where the excursions now have non-zero amplitude. Additional calculations showed that the amplitudes of excursions at
nodal frequencies are local minima, when amplitude is considered as a
function of frequency; thus the nodal frequencies are not displaced by
backleak. Additional evidence is provided by figure 2 and table 2 in
the companion study (17), where different methods show that the
approximate nodes associated with the backleak case are located, as a
function of frequency, where predicted by the explicit analysis, with
relative error of less than 1.3%, for f
1800 mHz.
Table 2 gives the concentration minima,
maxima, and range magnitudes, for the no-backleak and backleak cases,
corresponding to the frequencies
n/(2to), n = 1, 2, 3, ..., 10. Also shown are the analogous values computed in the limit
as f approaches zero from above; these limiting values
correspond to the maximum range magnitude permitted by the TGF
response. Table 2 shows that for the backleak case the range of the
excursions at each nodal frequency is approximately 10 times smaller
than at the immediately preceding intermediate frequency.
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Table 2.
Chloride excursions from the steady state at the macula
densa for parameter sets with and without chloride
backleak
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Chloride concentration profiles.
A natural question that emerges from this analysis is: What are the
spatial characteristics of TAL chloride concentration profiles that
lead to the nodal structure at the MD? Using Eq. 1 with flow
given by Eq. 8, we computed axial profiles for the backleak
case (Fig. 2). The amplitude of the oscillations was set to
= 9/10
so that deviations from the steady state would be sufficiently
pronounced to be easily observable in graphs. This large amplitude is
for illustrative purposes only, inasmuch as the node/antinode structure
for amplitude in the physiological range has already been determined.
However, these illustrative calculations serve to emphasize that the
nodal structure is insensitive to the amplitude of the sinusoidal
oscillation, provided that flow remains positive. Moreover, since large
amplitude flow oscillations result in large concentration deviations
from steady-state, spatial inhomogeneities in transport rate will be
magnified; nonetheless, the nodes remain at frequencies predicted by
the exact analysis for the idealized case of no chloride backleak.
In Fig. 2, A-D correspond to the frequencies f = n/(2to), n = 1, 2, 3, 4. In each
panel the wide, shaded curve is the steady-state profile corresponding
to F =
Qop. The other curves represent the
chloride profiles, after oscillatory flow has been established, at
times t = 0, p/4, p/2, and 3p/4, where
in each case p is the period of the oscillation. The bar at the
right of each panel in Fig. 2 gives the chloride excursion at the MD
during each flow oscillation.
In Fig. 2A, for f = 0.5/to, which
corresponds to a period twice that of the steady-state transit time
to, there are large excursions of the concentration
from the steady-state curve, except at the point of entrance
(x = 0), which is a fixed boundary value. In Fig. 2B,
for f = 1/to, which corresponds
to a period that equals the steady-state transit time, there are large
excursions from the steady-state curve, except at the TAL entrance and
the MD (at normalized length x = 1), where there is an
approximate node. The dynamic concentration profiles in Fig. 2B
show that a standing wave has been established, relative to the
steady-state chloride concentration profile. The standing wave has an
internodal distance equal to the length of the TAL.
In Fig. 2C, for f = 1.5/to, we see
the effect of increasing frequency. The approximate node moves from the
normalized length 1 to 2/3, and the excursions at the MD are again
large, as in Fig. 2A. As frequency increases further to
f = 2/to in Fig. 2D, the node moves
to normalized length 1/2, and a new node appears at length 1. A full
standing wave with wavelength equal to the length of the TAL has been
established.
This pattern of node formation and displacement to the left, as a
function of increasing frequency, is a property of the standing waves
that are established as a consequence of the sustained flow oscillations. When Tf = n, with n = 1, 2, 3, ..., Eq. 5 implies that transit time is time
independent at locations along the TAL where normalized length
x equals n/f. This time independence, combined
with concentration dependence on transit time, implies that these TAL
locations correspond to nodes (or approximate nodes in the presence of
chloride backleak). If we let xn designate these
nodal locations and write the relationship between nodal location and
frequency in terms of dimensional variables, we obtain xn = n(L/to)/f, where L
is TAL length, to is steady-state transit time,
and, consequently, L/to is steady-state
flow speed. According to this relationship, an oscillatory component
with frequency f will produce standing waves with nodes at
lengths xn, n = 1, 2, 3, ..., along the
TAL. (The wavelength of a standing wave is equal to twice the
internodal distance, i.e., 2xn). In addition, the
relationship xn = n(L/to)/f implies that when
xn = L, the frequencies that will produce
nodes at the MD are given by fn = n/to.
Several features predicted by the explicit analysis for the idealized
case with no chloride backleak can be observed in the concentration
profiles shown in Fig. 2. First, despite large excursions in flow, the
profiles are monotone decreasing along the TAL (see text near Eq. 7 and near Eq. A7 in APPENDIX A). Second, since
increasing frequency results in transit times that more nearly
approximate steady-state transit times (as indicated by Eq. 5),
maximal excursions from the steady-state concentration profile decrease
in amplitude with increasing frequency.
Finally, the asymmetry of concentration profiles around the
steady-state profile in Fig. 2, A-D, indicates that the
oscillatory time course of chloride concentration at each nonnodal site
along the TAL is nonsinusoidal. Figure 3
illustrates the oscillations at the MD corresponding to Fig. 2,
A-D. The oscillations at frequencies 0.5/to and 1.5/to (cf. Fig. 2,
A and C) are clearly nonsinusoidal and exhibit an
asymmetry in which concentration increases more rapidly than it
decreases. This asymmetry is a consequence of the implicit nonlinear
relationship, between time t and transit time T in
Eq. 5, for fixed x and for Tf
n. A
second nonlinear feature is a flattening of the trough of the
oscillation with largest amplitude
( f = 0.5/to). This is attributable,
in large part, to the approach of luminal chloride concentration to the limiting minimal value attainable by transepithelial chloride transport
when solute backleak is present (cf. figure 2 in Ref. 15). These
distorted waveforms are considered in detail in the companion study
(17), where the effects of the TAL low-pass filter on the dynamics of
the TGF system are evaluated.

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Fig. 3.
Oscillations in MD chloride concentration associated with the cases of
Fig. 2, as a function of time. Steady-state MD concentration is
~32 mM. At MD nodal frequencies, concentration excursions have
amplitudes contained within the shaded bar (cf. Fig. 2, B and
D); at intermediate frequencies, concentration excursions are
much larger (cf. Fig. 2, A and C) and are
nonsinusoidal as a result of the nonlinear action of the TAL filter. As
a result of periodicity, the pattern here is repeated in each time
interval of 2to.
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Figure 3 also contrasts MD chloride concentration oscillations at nodal
frequencies (cf. Fig. 2, B and D). The shaded bar in
Fig. 3, which includes the steady-state concentration value of ~32
mM, contains the approximate nodes, which have oscillations in
concentration between 30 and 35 mM.
 |
DISCUSSION |
This study provides quantitative predictions of the degree by which the
amplitude of oscillations in MD chloride concentration is reduced as a
function of frequency of oscillatory fluid flow, and it predicts that
the frequency response of NaCl concentration at the MD will have a
nodal structure, which arises from the establishment of standing waves
in luminal concentration. These results explain and extend earlier
model simulations that indicated that the TAL acts as a low-pass filter
(9, 15, 22).
In this study we have considered two basic cases, an idealized case
where the transepithelial chloride transport rate was assumed to depend
only on local luminal chloride concentration, with no spatial
inhomogeneity arising from chloride backleak, and a more realistic case
with chloride backleak determined by a computed luminal concentration
and a time-independent interstitial concentration formulated to
approximate the cortical and outer medullary concentration profile (see
Ref. 15). The idealized case is important because exact mathematical
equations can be obtained in which the effects of all parameters can be
readily ascertained. In particular, the equations obtained in the
RESULTS section demonstrate that the nodal response pattern
emerges from the structural assumptions of the model and are not
sensitive to the particular parameter choices given in Table 1: for the assumptions of the idealized case, luminal chloride concentration depends only on transit time, and nodes will appear at spatial locations where transit time is constant.
In an actual TAL, however, the transepithelial chloride transport rate
will depend, to some extent, on spatial inhomogeneities such as those
introduced by backleak, which will likely be more significant in the
cortex than in the outer medulla as a result of a larger
transepithelial concentration difference. By finding numerical
solutions, we have evaluated the effect of a chloride backleak
permeability with magnitude in the upper range of two experimental
measurements (see section MATHEMATICAL MODEL, Model parameters). The numerical solutions exhibit approximate nodes rather than the perfect nodes of the exact analysis, but, nonetheless, the nodes arise in the locations predicted by the exact analysis, both
as a function of frequency and of TAL length. That is to say, the
approximate nodes appear at the sites that would be predicted if
luminal chloride concentration were a strict function of transit time
only.
This point is most forcefully made by results shown in Fig. 2,
C and D, where approximate nodes appear within the
model TAL. The exact analysis predicts that the first nodes will appear
at normalized lengths 2/3 and 1/2. Despite unrealistically large oscillations in flow, which tend to magnify the spatial inhomogeneity arising from chloride backleak, careful examination of the numerical results showed that the approximate nodes are within 0.2% of the predicted lengths, which is on the order of the error introduced by the
numerical methods. In the companion study (figure 2 of Ref. 17), we
find for this case with chloride backleak that the nodes differ by less
than 1.3% from the predicted locations, as a function of oscillatory
frequency, for small-amplitude oscillations below 1800 mHz.
It must be acknowledged that other forms of spatial inhomogeneity in
transport may affect these results, e.g., inhomogeneities in luminal
diameter and transport capacity. However, in view of incomplete
experimental knowledge of these factors and the good agreement between
experiments and model predictions described in the companion study
(17), simulation studies based on a more detailed mathematical model
may not now be warranted.
Other nonideal factors that may affect the application of model results
to the physiological setting include the elastic compliance of the TAL
and axial diffusion of NaCl within TAL luminal flow. However,
experimental evidence suggests that tubular compliance becomes a
significant effect at frequencies of 1 Hz (8); thus spectral
characteristics below 500 mHz are not likely to be significantly affected [see APPENDIX C in the companion study (17)].
Standard estimates indicate that diffusion of NaCl along the axis of
the lumen will not be a significant factor (14).
A final nonideal factor is the use of a simple, single-barrier,
pump-leak representation of TAL NaCl transport that ignores the
response dynamics of the TAL cells. This approach implicitly assumes an
instantaneous response to changes in luminal NaCl concentration, whereas the response of epithelial cells should exhibit at least two
time constants. The first represents the initial response of NaCl
uptake through the apical membrane. This response should be very rapid,
as it is only limited by the molecular dynamics of the cotransport
systems and ion channels in the apical membrane. Most epithelial cell
models assume that this step has an instantaneous response (see, e.g.,
Refs. 5 and 13). The second component of the cellular response
represents the transition of cytosolic ion concentrations and cell
volume to new steady states in response to the altered apical membrane
ionic fluxes. This response is slower (ca. 2-3 s; Ref. 7) and is
determined by the relationship between the magnitude of the ionic
fluxes and cell volume. However, these slower changes in cytosolic ion
concentrations alter the driving force for apical membrane NaCl uptake.
These effects should have only minor, if any, effects on our analysis
for two reasons. First, the disappearance of NaCl from the tubular
lumen relies on apical uptake of Na+ and Cl
,
and this response is very rapid. In addition, cytosolic Na+
levels are well regulated by Na-K-ATPase in the basolateral membrane, so that the slower secondary adjustments in apical ionic flux will be
small relative to the magnitude of the initial change in apical uptake.
The second reason is that processes with a 2- to 3-s time constant are
likely to have little effect on spectral structure below 300-500
mHz, which is above the frequencies corresponding to the first few
nodes (see APPENDIX C in Ref. 17).
If, as we expect, the nonideal factors considered above have only a
minor impact on the results of our model analysis, then the predicted
nodal structure should be observable in open-feedback-loop experiments
employing oscillatory perturbations in TAL flow that are ~30% of
steady-state flow. The first two approximate nodes shown in Fig.
1B have oscillatory amplitudes of 1.6 and 0.8 mM (see Table 2),
whereas the near-antinodal amplitudes at the surrounding intermediate
frequencies are 30.1, 9.6, and 5.7 mM. Differences of this magnitude
should be measurable with existing methods (8). The detection of these
nodes would validate the simple equation for mass conservation in the
TAL used in this study (Eq. 1), a formulation that has been
widely used to obtain both steady-state and dynamic simulation results
(see, e.g., Refs. 6, 8, 12, 15, 21, 27).
In summary, the nodal structure of the TAL frequency response curve,
which arises from the establishment of standing waves in luminal
concentration, illustrates the nonlinear character of the TAL filter.
This unusual filter not only attenuates high-frequency perturbations,
but it also distorts the waveform. Simulations in the companion study
(17) indicate that TGF-mediated oscillations in nephron flow will be
affected by the TAL filter, and, in some cases, the spectral structure
of the TAL filter will be imposed on that of the entire TGF system.
Considerations in the DISCUSSION of the companion study
(17) suggest that the nodal structure predicted in this study may have
already been observed in some experimental records.
 |
APPENDIX A |
TAL Concentration is a Function of Transit Time
We use the method of characteristics (4) to show that, under
appropriate hypotheses, TAL concentration is a function of transit
time. The method of characteristics has been used previously to compute
profiles of key variables in proximal tubule flow (28).
For simplicity and generality, let K(C) denote the
transtubular active transport of chloride, and assume that K is
Lipschitz continuous in C, that K > 0 for C > 0, and that
K decreases monotonically to 0 as C decreases to 0. In the
absence of backleak, the rate of transepithelial chloride transport at
position x and time t depends only on C(x,
t), the local intratubular concentration at time t,
and Eq. 1 becomes
|
(A1)
|
The
fluid at location x and time t entered the TAL at time
so = t
T(x,
t). At time s
[so,
t], the fluid passed through the location
|
(A2)
|
where
0
y(s)
x, and y(t) = x. If g(s) is defined to be the chloride concentration
along the path determined by Eq. A2, then g(s) = C( y(s), s), and
g(so) = C(0, so) = 1. If we
differentiate the expression for g(s) and employ Eq. A2 with so considered as a fixed (constant)
time, then we obtain
|
(A3)
|
Comparing
Eqs. A1 and A3, we obtain an ordinary differential
equation (indeed, an initial value problem) for the concentration along
the TAL transit path
|
(A4)
|
Because
of the properties assumed for K, g is positive and monotone
decreasing in s.
For any so, define a function h by
|
(A5)
|
Then
h(0) = g(so) = 1, and h is a uniquely determined
function that is independent of the choice of
so. Since for each so, t
so = T, it
follows that
|
(A6)
|
We
conclude from Eq. A6 that C(x, t) depends
only on the transit time T and not on the distribution of time
that solute-laden fluid has spent at the locations between 0 and
x. This result does not depend on a specific form for K
but, rather, only on the general properties of K set forth
before Eq. A1.
Now we show that, under our hypotheses, the profile of C at each time
t is monotone decreasing. If we differentiate Eq. A6 with respect to x, and evaluate
T/
x by
differentiating Eq. 2 with respect to x, we obtain
|
(A7)
|
Since
h > 0, we have K > 0, and since we assume that F > 0, it follows that
C/
x < 0.
In the case were K is Michaelis-Menten kinetics, Eq. A4
has an implicit solution, analogous to Eq. A6, given by Eq. 7.
 |
APPENDIX B |
Numerical Methods
Numerical methods are identified by corresponding figure numbers.
Figure 1.
Dashed lines providing the outer envelope for the MD chloride
concentration range were computed for discrete values of f, incremented by 1/(40to). Dimensionless lower and
upper bounds of T(1, t) (Tm and
TM, respectively) corresponding to each value of
f were computed from Eq. 6; the resulting range of C(1,
t) was computed from Eq. 7, using the dimensionless
iteration map Cn+1 = exp((1
VmaxTCn)/K
m), where C1 = 0.1 and T = Tm or TM. The iteration was
terminated when successive iterates of Cn differed
by
10
8. The plotted values were obtained by
multiplying dimensionless values of Cn by the
reference concentration Co.
In Fig. 1 the solid curves for the precise bounds were computed from
the same discrete values of f used above. Bounds on
T(1, t) were computed from Eq. 5, for each
f, by finding extremal values of T as t varied
in increments of to/100 from 0 to
40to, the period of one oscillation at the lowest
frequency evaluated. Bounds on C(1, t) were then computed by
the iteration map described above.
The vertical bars in Fig. 1 were obtained from the numerical solution
of Eq. 1, which was computed in double-precision arithmetic 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
(14). The numerical space and time steps, in nondimensional units, were
x = 1/640 and
t = 1/214,
respectively, corresponding to dimensional values of 7.8125 × 10
4 cm and 9.58738 × 10
4 s. The initial
data for each numerical run was the steady-state profile S(x).
Sampling of values of C(1, t) began after a computational time interval that was an integer multiple of both the transit time
to of the TAL and the period of the oscillation in
F. This served both to expel the initial profile S(x)
and to initiate the sampling interval at onset of an oscillation. The
range of the oscillation in C(1, t) was then determined over
a time interval corresponding to the period of the oscillation.
Figures 2 and 3.
The profiles in Fig. 2 and the MD concentrations in Fig. 3 were
generated from the numerical solution of Eq. 1 using the ENO scheme. The space and time steps, in nondimensional units, were
x = 1/5120 and
t = 1/214. High
resolution in space and time was required because axial profiles
contain segments where
C/
x is very large. In each case, preceding the time marked t = 0, the numerical solution was
computed for an interval that was an integer multiple of both the
transit time to of the TAL and the period of the
oscillation in F. For Fig. 2, sampling was conducted for one
period of an oscillation in F; for Fig. 3, sampling was
conducted for an interval 2to, which is an integer
multiple of the periods of the four oscillations: 2to, to,
to/3, and
to/2.
 |
ACKNOWLEDGEMENTS |
We thank Niels-Henrik Holstein-Rathlou for the suggestion that we
examine the simulated TAL chloride concentration profiles. We thank
John M. Davies for assistance in preparation of Figs. 1-3.
 |
FOOTNOTES |
This work was supported in part by National Institute of Diabetes and
Digestive and Kidney Diseases Grant DK-42091.
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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