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1Department of Mathematics, Duke University, Durham, North Carolina 27708; and 2Department of Physiology and Biophysics, State University of New York, Stony Brook, New York 11794
Submitted 18 October 2002 ; accepted in final form 18 June 2003
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ABSTRACT |
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 TOP ABSTRACT METHODSRESULTSDISCUSSIONAPPENDIX AAPPENDIX BAPPENDIX CREFERENCES |
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, and on its relationship to a theoretically determined critical value of gain, c. In this study, we used that model to show how sustained perturbations in proximal tubule flow, a common experimental maneuver, can initiate or terminate LCO by changing the values of and c, thus changing the sign of - c. This result may help explain experiments in which intratubular pressure oscillations were initiated by the sustained introduction or removal of fluid from the proximal tubule (Leyssac PP and Baumbach L. Acta Physiol Scand 117: 415–419, 1983). In addition, our model predicts that, for a range of TGF sensitivities, sustained perturbations that initiate or terminate LCO can yield substantial and abrupt changes in both distal NaCl delivery and NaCl delivery compensation, changes that may play an important role in the response to physiological challenge. kidney; renal hemodynamics; autoregulation; mathematical model; nonlinear dynamics
A common and useful method of investigating TGF is to impose sustained perturbations of proximal tubule (PT) fluid flow in a freely flowing nephron where the TGF feedback loop is closed and functional (3, 5, 6, 20, 21, 26). Using this technique, Leyssac and collaborators (5, 17, 18) have demonstrated that LCO can be initiated or extinguished in halothane-anesthetized rats by insertion or removal of PT fluid, findings that indicate that PT fluid flow can have a substantial effect on the stability of the TGF system. Some insight into the basis of this phenomenon is provided by our previous investigations of the TGF-mediated LCO (12–16, 22). These modeling studies indicate that LCO will emerge for sufficiently large feedback loop gain magnitude (12), that the parameter regime that supports LCO may overlap the parameter regime of normal TGF operation, that a gain magnitude near that needed for LCO will produce maximal feedback compensation (16), and that LCO may augment NaCl delivery to the distal nephron (16). A finding of fundamental importance is that the emergence of LCO depends on key TGF system parameters and variables that, in turn, depend on the physiological state of the nephron or on the animal as a whole. For example, the parameters may be affected by an increase in systemic blood pressure, a resetting of the TGF response curve, a primary change in tubular transport, or an experimental intervention such as microperfusion of fluid into the PT. We have previously presented numerical simulations that indicate that sustained PT flow perturbations can elicit LCO or extinguish LCO (16). However, such simulations cannot determine how the feedback loop gain or the gain threshold at which LCO emerge is influenced by PT flow perturbations.
The principal objectives of this study were to analyze, in the context of our mathematical model, the effect of sustained PT flow perturbations on feedback loop gain, on the gain threshold at which LCO emerge, and on feedback compensation. By means of this analysis we provide a theoretical framework for understanding how fluctuations in physiological variables impact the stability of the TGF system and its regulatory efficacy.
Glossary
Parameters
Q/2Qref
Q
, nl/min
c
n = n when = 0, nl · min-1 · mM-1
p
Independent Variables
Specified Functions
(t)
Dependent Variables
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METHODS |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONAPPENDIX AAPPENDIX BAPPENDIX CREFERENCES |
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Concepts and Terminology
The TGF system is a negative feedback loop in which afferent arteriolar (AA) smooth muscle tension is increased in response to increased chloride concentration in TAL luminal fluid passing by the macula densa (MD) (23). AA vasoconstriction will decrease filtration pressure in glomerular capillaries and thus reduce SNGFR. Reduced SNGFR reduces both fluid flow through the TAL and chloride concentration in luminal fluid passing the MD.
SNGFR and thus TAL flow may be nearly unchanging in time and thus approximate a time-independent steady state. Associated with that steady state is an SNGFR and a TAL chloride concentration at the MD that together are called the "operating point" of the TGF response function. Frequently, that operating point is found to be near the steepest portion of the TGF response function (1); in the standard form of that function (a logistic curve that is equivalent to a scaled hyperbolic tangent function), the point of steepest slope corresponds to the point where the response function changes from concave down to concave up, i.e., to the point of inflection.
Also associated with steady-state flow, and thus with its corresponding operating point, is feedback gain, a measure of feedback signal amplification. Roughly speaking, gain can be measured by breaking the TGF (signal) loop (e.g., breaking the signal loop at the beginning of the thick limb), increasing the feedback loop signal by a small amount (e.g., increasing TAL flow by FAL), and measuring the resulting change in the return signal (e.g., in a short-looped nephron, measuring the change in terminal descending limb flow, FDL). Gain is then approximated by the resulting change divided by the signal increase (e.g., by FDL/FAL).
Two concepts of gain, steady-state gain (GSS) and instantaneous gain (-), have been applied to the TGF loop (13); each can be viewed as an interpretation of FDL/FAL (see APPENDIX A). Steady-state gain is the gain that has been measured in laboratory experiments; the concept of instantaneous gain arose from a theoretical analysis of a mathematical model of the TGF system (12, 13). We have previously shown that, under restricted circumstances, GSS is closely approximated by - (13).
For a given set of model parameters (see model below), there is a particular steady-state feedback loop configuration with an associated operating point and with associated values of -GSS and . In the context of our model's normalized quantities, steady-state gain magnitude -GSS corresponds to the product of the magnitude of the slope of the feedback response function (see Eq. 2) at the operating point and the derivative of chloride concentration in luminal fluid alongside the MD with respect to TAL flow, also at the operating point; corresponds to the product of the slope of the feedback response function at the operating point, the derivative of chloride concentration at the MD with respect to TAL axial position, and the steady-state TAL fluid transit time.
A model steady state may be either stable or unstable. If a steady state is stable (stable steady state, SSS), model solutions will more and more closely approximate that steady state after a transient perturbation (as, e.g., a transient addition to SNGFR). If a steady state is unstable (unstable steady state, USS), model solutions will more and more closely approximate LCO after a transient perturbation, no matter how small the perturbation.
For a sufficiently long delay in TGF signal transmission at the juxtaglomerular apparatus (JGA), the stability of a steady state is determined by the gain1 and by the critical gain c (12). If < c, then the stable state is a (time-independent) steady state; if > c, then the stable state is an LCO. The critical gain c can be determined by numerical experiments in which one varies to find the boundary between the stable and unstable steady states or by use of a characteristic equation, which can be derived from our model and which can be used to find a general expression for c in terms of model parameters.
A steady state (whether stable or not), and thus its associated operating point, may be altered by non-TGF-regulated factors that increase effective SNGFR (and thus TAL flow), e.g., an increase in systemic blood pressure or an experiment in which fluid is inserted into the PT. A steady state may also be altered by a change in the slope of the feedback response curve, if the operating point does not coincide with the point of inflection. A convenient parameter for characterizing the slope of the response curve is the feedback sensitivity , which for the purposes of this study is defined to be the magnitude of the slope of the feedback response curve at its inflection point. We define the critical sensitivity c to be the sensitivity corresponding to a case in which = c. Thus, if > c, the model solution will tend toward an LCO; if < c, the model solution will tend toward the SSS.
A sustained flow perturbation or other sustained change in model parameters has the potential to change both and c and thus change the relationship between them (i.e., change the sign of - c). Thus a stable state may be affected by a perturbation, so that an SSS may become a USS, resulting in a stable LCO, or a state characterized by LCO may become an SSS. Figure 3 (see below) summarizes the dependence of stable model behavior on gain, operating point, flow perturbations, and TGF sensitivity.
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Mathematical Model
Our model is given by the following system of coupled equations
 | (1) |
 | (2) |
 | (3) |
Each equation is in nondimensional form (see APPENDIX B). 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 MD (x = 1). Figure 1 gives a schematic representation of the model.
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Equation 1 is a partial differential equation for the chloride concentration C in the intratubular fluid of the TAL of a short-looped nephron. We assume that fluid entering the TAL has constant chloride concentration; thus, we assume that C(0, t) always equals 1. At time t = 0, initial concentrations C(x, 0) (for x 
(0, 1]) and C(1, t) (for t (-
, 0)) must be specified. For t > 0, the rate of change of that concentration depends on processes represented by the three right-hand terms in Eq. 1. The first term is axial advective chloride transport at intratubular flow speed F. The second is transepithelial efflux of chloride driven by active metabolic pumps situated in the tubular walls; that efflux is assumed to be 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 effective luminal chloride concentration CMD at the MD. This feedback relationship is an empirical equation well established by steady-state experiments (23). The constant CI is the inflection point of the TGF response curve; in our model, it is also the chloride concentration at the MD when F = 1 and the chloride concentration profile in the TAL has assumed a steady state. The positive constants K1 and K2 describe, respectively, the range of the feedback response and its sensitivity to deviations from the steady state. The constant represents a flow perturbation applied into Bowman's space. Owing to the nondimensional model formulation and to our assumption that a fixed fraction of the glomerular filtrate is delivered to the TAL, can represent a fractional flow perturbation applied into Bowman's space or into the tubular lumen at the entrance of the TAL (in the nondimensional formulation, the TAL flow rate F and the SNGFR Q have the same value, because the dimensional F and Q have both been scaled by their respective base-case values).
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 mediated by AA smooth muscle. In a quasi-steady state, Eq. 2 provides a good description of the TGF response. However, dynamic experiments (2) show that a change in MD concentration does not significantly affect AA muscle tension until after a discrete (or pure) delay time p > 0, and then the effect is distributed in time so that a full response requires additional time, with greatest weight in the time interval [t - p - , t - p], where > 0 is a second delay parameter. To simulate the pure delay followed by a distributed delay, the convolution integral given in Eq. 3 is used to describe the effective signal received by the AA at time t (22). The kernel function for this integral is given by
 | (4) |
With this function, a step change in C results in a sigmoidal increase in CMD over a nondimensional time interval of .
Model parameters. A summary of parameters and variables, with their dimensional units as commonly reported, is given in the Glossary. The base-case parameters, which collectively represent the reference point for our parameter studies, are given in Table 1; the criteria for their selection and supporting references were given in Ref. 12. The extratubular chloride concentration is given in nondimensional form by Ce(x) = Co(A1e-A3x + A2), where A1 = (1 - Ce(1)/Co)/(1 - e-A3), A2 = 1 - A1,A3 = 2, and Ce(1) corresponds to a cortical interstitial concentration of 150 mM.2 A graph of Ce for F = 1 was given in Fig. 1 of Ref. 12. The steady-state operating MD chloride concentration that corresponds to = 0, which is CI, can be calculated numerically using the TAL dimensions and transport parameters, with steady flow F = 1 in Eq. 1 for the time interval equal to the washout time of the TAL at that flow, 1 dimensionless time unit.
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Model solutions. For the case of no perturbation, i.e., = 0, a steady-state solution to Eqs. 1–4 may be obtained and sustained by fixing C(1, t) = CI for t 
1, fixing F at the base-case value of 1 for the TAL transit time interval (here, t [0,1]), solving Eq. 1 for that time interval (to obtain C(1, 1) = CI), and then closing the TGF loop so that F is computed via Eqs. 2–4 for t > 1. Then F(CMD(t)) will equal 1 for all time.
The steady state for 
0 is more difficult to obtain because it corresponds to an unknown steady-state value of F, a value that in general we will call Fop (thus = 0 implies Fop = 1, whereas 0 implies Fop 1). Because we have previously established that, if p = 0 and = 0 a model steady state is stable (12), Fop can be found directly by eliminating the delay at the MD (thus, CMD(t) = C(1,t)) and solving Eqs. 1 and 2 until the solutions are sufficiently close to steady state. The long-time values of CMD(t) and F(CMD) will closely approximate Cop and Fop.
Alternatively, one can consider the inverse problem in which one specifies Fop and finds the corresponding . Thus one can fix F = Fop in Eq. 1 for an interval equal to the TAL transit time at flow speed Fop (thus for t [0,1/Fop]), find Cop = C(1, 1/Fop), assume that C(1, t) = Cop for t 1/Fop, and choose so that for t > 1/Fop, F(CMD(t)) = Fop in Eq. 2. If the feedback loop is closed at t = 1/Fop, then F(CMD(t)) = Fop for all time.
When Fop is known, Cop may be obtained by solving the time-independent form of Eq. 1
 | (5) |
A dynamic solution to Eqs. 1–4 can be found by taking the corresponding steady-state solution as initial data and perturbing transiently. The long-time solution, depending on model parameters, will tend either to a steady state or to an LCO.
Steady-state gain. The steady-state gain, which has been measured in experiments (1, 3), is expressed (13) in terms of dimensionless model variables as
 | (6) |
Characteristic equation and instantaneous gain. Information about the stability of a steady-state solution of the model TGF system (i.e., how it will respond to a transient perturbation) can be obtained from the model's characteristic equation,
 | (7) |
, in terms of normalized variables, is given by
 | (8) |
This characteristic equation generalizes previous versions (12, 15) by including the effect of Fop 1; previously, we assumed Fop = 1 so that Fop did not appear. The method of derivation of the characteristic equation was explained previously (12, 13, 22).
There are two ways in which the characteristic equation may be used. First, given a particular set of model parameters, one can find the corresponding steady-state solution S(x,Fop), from that solution obtain via Eq. 8, substitute that in the characteristic equation, and then solve the characteristic equation (by techniques of numerical analysis) for the corresponding real and complex parts of the eigenvalues 
n (n = 1, 2, 3,...) that satisfy the characteristic equation when = n. If all the real parts of the n's are negative, then the model solution is an SSS. If the real part of any one of the eigenvalues is positive, then the steady state is a USS.
Conversely, given a set of model parameters, and the corresponding steady-state solution S(x, Fop), one can use the characteristic equation to solve for a particular value of , the critical gain c, which corresponds to the transition between the SSS and the USS: c is the smallest such that the real part of an eigenvalue n equals 0. Thus, if one is given an external concentration Ce, a steady-state profile S with associated Fop (a steady state based on the feedback response parameters), delay parameters p and , and backleak permeability P, then one can determine the critical gain c, and compare it with the feedback gain (computed by Eq. 8) to determine whether a steady-state solution is stable; this is how the characteristic equation was used in this study. Because a steady-state solution and its corresponding operating values Cop and Fop are altered by a sustained flow perturbation introduced through , both and c depend on (see Fig. 3).
Changing the physiological state: sustained flow perturbation and TGF sensitivity. In this study, we assume that the model physiological state is determined by two parameters. The first parameter, the sustained flow perturbation (which appears in Eq. 2), quantifies all non-TGF-related factors affecting SNGFR. Such factors include sustained microperfusion into the PT, changes in mean systemic arterial pressure, and changes in extracellular volume (ECV). The second parameter, the TGF sensitivity , is defined to be the magnitude of the slope of the TGF response curve at its inflection point, which has been shown to depend on the physiological state (1, 23). By maximizing the slope of the TGF response given by Eq. 2 as a function of CMD, one finds that the dependence of model TGF sensitivity on model parameters is given by
 | (9) |
was varied by changing K2.
Feedback compensation. The efficacy of model TGF regulation can be quantified by calculating feedback compensation, an index used in experimental investigations (10, 20, 26). Feedback compensation is defined by
 | (10) |
 | (11) |
In the definition for magnification, Y is a system variable that is regulated by means of the feedback loop (e.g., distal chloride delivery); Y is the change in the system variable Y in response to a change X (i.e., a perturbation) in another system variable X (e.g., PT flow). The denominator of Eq. 11, (Y/X)OL, is the ratio of Y to X in the case where the feedback loop is open (open-feedback-loop case, or OL). The numerator (Y/X)CL is the corresponding ratio when the loop is closed (closed-feedback-loop case, or CL). If X tends to zero, M converges to a ratio of derivatives; however, the -notation is retained because it is consistent with experimental studies, which necessarily entail measurable perturbations.
Because the stable state of the model TGF system may be an LCO or a stable steady state, and the type of stable state (LCO or steady state) may differ for X = 0 and X 0, "Y" requires further interpretation to be well defined. We thus adopt two conventions with respect to computing Y: 1) specific values of Y are time averages that correspond to the stable state of the system under the given perturbation; and 2) changes in Y are normalized with respect to the stable, nonperturbed state of the system. We thus set Y = (Y - Y0)/Y0, where Y0 is the time-averaged value of Y obtained from the stable state when X = 0, and Y is the time-averaged value of Y obtained from the stable state when X 0 (if the stable state of the system is an LCO for zero perturbation, Y is compared to the zero perturbation stable-state average, even if the perturbation X has resulted in an SSS). We normalize with respect to Y0 because Y0, the time-averaged value of Y obtained from the stable state when X = 0, may differ for the OL and CL model TGF systems.
If a steady state is the stable state, distal chloride delivery is computed as the product of the steady-state MD chloride concentration and the steady-state TAL fluid flow rate at the MD. If an LCO is the stable state, distal chloride delivery is computed as the time average of the product of the chloride concentration at the MD and the TAL fluid flow rate at the MD.
Numerical methods. Details of the numerical methods used to obtain results are given in APPENDIX C. The base-case parameter values (excepting those for and ) were used in all calculations.
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RESULTS |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONAPPENDIX AAPPENDIX BAPPENDIX CREFERENCES |
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Because we aim to make predictions that are related to experiments, it is incumbent on us to provide evidence that the instantaneous gain that arises in our theoretical analysis can be estimated from experiments that measure steady-state gain.
Previously, we have calculated, for the case of no sustained flow perturbation and for base-case parameters, that the instantaneous gain magnitude exceeds steady-state gain magnitude -GSS by 
10% (13). Here we report further calculations to demonstrate good agreement between instantaneous gain and steady-state gain in steady states corresponding to sustained flow perturbations over a range of feedback sensitivities. Some results of those calculations are illustrated in Fig. 2, which shows instantaneous and steady-state gain magnitudes and -GSS for TGF sensitivities 4 and 8 (sensitivities that yield = 4 and = 8 at zero perturbation, = 0). The value of used in this figure and subsequently in this section is the dimensional value of a sustained flow perturbation in early PT.
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The relative difference between the two measures of gain, at zero perturbation, is 10.3%, for both 4 and 8. The maximum relative difference between the two measures, for 4, is 13.4% at = -12 nl/min; for 8, that difference is 12.5%, also at = -12 nl/min. The maximum absolute difference between the two measures, for 4, is 0.374, at = -0.060; for 8, that difference is 0.748 at = -0.027. We computed the two measures of gain over the range of = -12 to 12 nl/min (in increments) and for sensitivities ranging from 0.25 to 10 (in increments). As sensitivities increase, we found that the relative differences tend to stabilize near 10%. Also, we found that, although both measures of gain tend to zero as sensitivity tends to zero, the relative difference increases as sensitivity tends to zero and perturbations tend to -12 nl/min. Indeed, for sensitivities greater than 2 and perturbations greater than -8 nl/min, the maximum relative difference was 12.4%. However, in the "corner" region marked off by sensitivities such that 1 2 and by perturbations such that -10 nl/min -8 nl/min, the maximum relative difference was 16.3%. The difference between the two gain values in this corner is not likely to be physiologically significant, however, because in that region both values are less than 1.2 and their difference is never larger than 0.14.
By means of an analysis like that in Ref. 13, we found that exceeds -GSS for all perturbations at a given sensitivity.
Shape of the -Curve
The dependence of on flow perturbations and feedback sensitivity can be understood by considering the two steps involved in calculating (Fig. 3). Given a flow perturbation and a feedback sensitivity , one determines the operating steady-state TAL flow Fop, and then one evaluates the three factors comprising (given by Eq. 8): F'(Cop), 1/Fop, and 
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The model's steady-state operating point is represented by Qop and Cop, where Qop is the effective SNGFR (which is defined to be the sum Q + ), and Cop is the MD chloride concentration that corresponds to the steady-state TAL flow Fop. The calculation of the steady-state operating point is represented graphically in Fig. 4; the operating point corresponds to the intersection of the MD chloride concentration curve and the TGF response curve. Figure 4, A and B, illustrates the variation of the intersection point with sustained perturbation for two different feedback sensitivities . A sustained flow perturbation affects the location of the intersection point by translating the TGF response curve vertically (see Eq. 2). The range of operating points is smaller for the case of higher feedback sensitivity, because the higher sensitivity represents a stronger TGF response and thus a response with greater feedback compensation. Figure 4C explicitly shows the dependence of Qop and Fop on for the feedback sensitivities used in Fig. 4, A and B. For the higher sensitivity 8, Fop deviates less from the base-case TAL operating flow rate of 6 nl/min, compared with the lower sensitivity 4.
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and has a substantial effect on F'(Cop) (the slope of the TGF response curve at the operating MD chloride concentration Cop) because of their impact on the steady-state operating point. As the perturbation increases over its full range, the operating point moves along the TGF response curve from a region of small slope to a region of large slope and then to another region of small slope (see Fig. 4, A and B). This is the primary reason that depends on in the way shown in Fig. 5A1: is largest for near zero, and decreases as the magnitude of increases. Figure 5A1 also shows that, as long as the magnitude of the perturbation is not too large, increases in magnitude as the TGF sensitivity increases. This results from the way that affects F'(Cop). As increases, the maximal slope of the response curve increases, which increases the slope of the response curve at the operating point. Thus F'(Cop), and hence , increases in magnitude as the TGF sensitivity increases, for values of that are not too large. The factors 1/Fop and do not change as much as F'(Cop) does when or changes; thus the shape of the -curve primarily reflects the variation of the factor F'(Cop). However, the effects of the factors 1/Fop and can be seen in Fig. 5A1, which shows that the graph of is not exactly symmetrical with respect to = 0.
Shape of the c-curve. Figure 5A2 illustrates the dependence of critical gain c on for sensitivities 4 and 8, a dependence computed by means of the characteristic equation, Eq. 7 ( and play a role in determining c through their influence on the steady-state TAL flow Fop; see Fig. 3). The decrease in c as a function of increasing perturbation largely arises from the accompanying increase in Fop, which increases the ratio of the JGA delay times (p and ) to TAL fluid transit time, 1/Fop (based on an analysis similar to that given in Ref. 12). Because of the smaller variation of c, relative to , as a function of (note the different scales on the vertical axes of Fig. 5, A1 and A2), variation of primarily determines the sign of - c and thus the stable state of the model system (LCO or SSS).
Dependence of the Sign of - c on
In Fig. 5, B1 and B2, the gain curves from Fig. 5, A1 and A2, are matched according to the sensitivities 4 or 8. For sustained flow perturbations between a and b or between c and d, the stable state is an LCO ( - c > 0), and for outside those regions the model solution tends toward an SSS ( - c < 0). As TGF sensitivity increases, the gain changes more rapidly than critical gain c, and therefore the range of values for which an LCO is the stable state increases as increases. These results provide a theoretical explanation for a well-known observation: if the TGF response curve is sufficiently steep at its inflection point, and if the administered PT microperfusion is sufficiently small, then TGF mediates regular, sustained oscillations in tubular flow.
Critical Sensitivity Curve
The curve in Fig. 6A gives critical sensitivity as a function of perturbation (recall that, given a sensitivity , if there is a sustained perturbation such that the instantaneous gain is equal to the critical gain c, then that sensitivity is defined to be the critical sensitivity c for that ; in Fig. 5B1, 4 equals the critical sensitivity for = a and = b, and in Fig. 5B2, 8 is the critical sensitivity for = c and = d). Because both and c depend on , the curve in Fig. 6A can be calculated directly only by means of exhaustive numerical simulations based on Eqs. 1–4. Therefore, we used the following implicit procedure to calculate points on the curve in Fig. 6A. We began by selecting a value of Fop; next, we calculated the corresponding c via the characteristic equation (Eq. 7); finally, Eqs. 8 and 2 were used to determine the sensitivity and the sustained perturbation , respectively, needed to achieve the selected value Fop.
The points a, b, c, and d on the perturbation axis in Fig. 6A are the same as those shown in Fig. 5, B1 and B2. The critical sensitivity corresponding to points a and b in Fig. 6A, 4, corresponds to the sensitivity 4 in Fig. 5, and the critical sensitivity 8 corresponding to points c and d in Fig. 6A is the sensitivity 8 used in Fig. 5. Thus, Fig. 6A can be viewed as a summary of the results of many experiments like those shown in Fig. 5, B1 and B2, although the curve given in Fig. 6A was calculated by means of the efficient and accurate technique described in the previous paragraph. Above the curve, the stable solution is an LCO, whereas below and outside the curve the stable solution is a steady state.
Critical Sensitivity Curve: Tests and Examples
Figure 6B gives the results of two numerical experiments conducted to test the critical sensitivity curve in Fig. 6A. Values of the sustained perturbation and the TGF sensitivity were selected to place the model TGF system just inside (at point S) or just outside (at point T) the critical sensitivity curve. For each case, the model system was initialized at the time-independent steady-state operating point determined by those values (see Fig. 3). Then, a transient flow perturbation was applied at an elapsed time of 6 s. For the state just inside the curve, an LCO developed; however, for the state just outside the curve, the oscillations initiated by the transient flow perturbation rapidly decreased in amplitude and the solution approached an SSS. These tests confirm the explicit analytical results summarized in Figs. 5 and 6A.
Figure 6C1 depicts the effects of two different sustained flow perturbations, = 5.0 and = 9.5 nl/min, on the TGF response curve and on the corresponding operating points, indicated by the open and closed circles. The critical sensitivity curve in Fig. 6A and the points U and V corresponding to these perturbations predict that the stable state for = 9.5 is a steady state (marked by the closed circle in Fig. 6C1) and that the stable state for = 5.0 is an LCO (the associated unstable steady-state operating point is marked by the open circle in Fig. 6C1). The steady-state operating point for the smaller perturbation is closer to the steepest portion of the TGF response curve, compared with the operating point for the larger perturbation. Figure 6C2 illustrates the effect of suddenly changing (at time 4.0 min) the applied perturbation from = 9.5 to = 5.0 and then suddenly restoring the original perturbation (at time 8.0 min). Oscillations emerge and are damped out as predicted by the locations of points U and V in Fig. 6A. Note that the flow range of the oscillation, indicated by the thick gray curve in Fig. 6C1, occupies a significant portion of the sloped part of the TGF response function. This model simulation gives results similar to those obtained in experiments by Leyssac and Baumbach (18): in halothane-anesthetized rats, PT pressure oscillations can be initiated or terminated by adding fluid to (or removing fluid from) the late PT via a micropipette.
Figure 6D1, adapted from a study by Holstein-Rathlou and Leyssac (5), illustrates PT pressure from an experiment in a halothane-anesthetized Sprague-Dawley rat. The stable state approximated an LCO, but the removal of PT fluid (a perturbation of -10 nl/min) resulted in an SSS (albeit with a superimposed high-frequency oscillation from respiration). The removal of the perturbation resulted in a return to an LCO, and the response to the perturbation was repeatable. Figure 6D2 contains analogous results from our model. We began in a state, corresponding to X in Fig. 6A, in which the steady-state operating point is at the inflection point of the TGF response curve; moreover, because > c, the stable state is an LCO. The introduction of a -10 nl/min flow perturbation3 resulted in the relocation of the steady-state operating point to the point corresponding to W in Fig. 6A, and that steady state was stable. The removal of the perturbation resulted in a return to an LCO. The flow pattern in Fig. 6D2 is very similar to the pressure pattern in Fig. 6D1, except that the model oscillation has a higher frequency. However, by a temporal rescaling of our base-case parameters, that aspect of the experimental record could be matched also.
Figure 6E1, also adapted from Ref. 5, illustrates PT pressure from an experiment in which a 7.5 nl/min flow perturbation increased the amplitude of oscillations in a preexisting LCO. Figure 6E2 shows results from our model that predict this pattern. If one initializes the model at a steady-state operating point that corresponds to a sustained negative perturbation and that is inside the LCO regime, but is near the bifurcation boundary (e.g., an operating point corresponding to point Y in Fig. 6A), and then one introduces a positive perturbation that results in a steady-state operating point nearer the point of inflection (e.g., corresponding to point Z in Fig. 6A); then the increased gain results in a stronger feedback response and an increased oscillation amplitude.
Relationship to Previous Work
In our previous model investigations (12, 13, 15, 22), Fop was always set to the (nondimensional) base-case value of 1. Thus Fop did not appear in the earlier versions of the characteristic equation, and the steady-state concentration profile S (which depends on Fop) was a fixed function. In Ref. 12 (with Fop equal to 1), we used the characteristic equation to determine the dependence of c on the JGA delay .
In the present study, the delays p and were fixed, and we studied the dependence of c on Fop (more precisely, we studied how c depends on and , which determine Fop). For each value of Fop, there is a curve, analogous to the n = 1 bifurcation curve in Ref. 12, that gives the dependence of c on = p + /2, as illustrated in Fig. 7. For = 0 and any sensitivity, Fop equals 1, and therefore the middle curves in Fig. 7, A and B, are the same (these two middle curves are analogous to the curve labeled n = 1 in Fig. 4 of Ref. 12).
Along the Critical Sensitivity Curve, Chloride Compensation and Distal Chloride Delivery Change Abruptly as One Passes from Stable Steady State to Stable LCO
For the ranges of and used in Fig. 6A, feedback compensation and delivery of chloride to the MD were computed as described in METHODS and APPENDIX C. Column B in Fig. 8 illustrates results for sensitivity 8. Figure 8B1 is a reproduction of Fig. 5B2, Fig. 8B2 is feedback compensation for chloride delivery to the MD corresponding to sensitivity 8, and Fig. 8B3 is chloride delivery to the MD for sensitivity 8. In Fig. 8, B2 and B3, the solid curve corresponds to the stable-state case (LCO or SSS); the dashed curve corresponds to the steady-state case where LCO have been suppressed in the USS regime by setting p and equal to 0.
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Column A in Fig. 8, which illustrates model results for the stable-state case, is analogous to results in column B but also represents the role of variable , as shown by the axes accompanying Fig. 8A4. Column C illustrates analogous results for the case in which LCO are suppressed.
Figure 8A1 is analogous to Fig. 5, B1 and B2. The (mostly) upper surface is the gain as a function of sustained perturbation and sensitivity ; the lower surface, largely hidden by the upper surface, is the critical gain c. The bold line that appears on each surface corresponds to sensitivity 8. The intersection of these surfaces corresponds to the critical sensitivity curve exhibited in Fig 6A; specifically, the projection of that intersection onto the - plane, shown in Fig. 8A4, is the critical sensitivity curve. Figure 8C1 is the same figure as Fig. 8A1, but here the intersection of these surfaces is interpreted to mark the boundary between SSS and USS. In a physiological context, the unstable steady state is unrealizable, because frequent transient perturbations, always present in a living system, will result in LCO. However, the unrealizable USS is useful for evaluating the effects of stable oscillations, relative to steady-state operation.
Figure 8A2 illustrates model predictions for TGF compensation, assessed in terms of chloride delivery to the MD, assuming stable-state operation: compensation decreases abruptly when a change in or moves the model physiological state from just outside to just inside the critical sensitivity curve. In contrast, steady-state operation yields a smooth relationship between compensation and the independent variables and , as shown in Fig. 8, B2 and C2. Thus, in the USS parameter regime, the effect of LCO, relative to steady-state operation, is to reduce the system's capacity to keep chloride delivery to the MD within a narrow range around the zero-perturbation chloride delivery.
Figure 8A3 illustrates model predictions for chloride delivery to the site of the MD. Comparison with Fig. 8, B3 and C3, indicates that, in the presence of LCO, perturbations result in an asymmetric delivery change. Positive perturbations result in increased chloride delivery relative to the USS, whereas negative perturbations result in decreased delivery relative to the USS.
Figure 9 shows that water delivery to the MD, as a function of perturbation, varies much less than chloride delivery to the MD, and that the variation of water compensation with respect to perturbation is more symmetrical than that of chloride compensation. For = 8, Fig. 9, A and B, shows water compensation and delivery as a function of perturbation, both for stable-state and for steady state, along with the stable-state result for chloride. Figure 9C shows the maximum difference, over sustained perturbations of -12 to 12 nl/min, between percent stable-state and percent steady-state delivery as a function of sensitivity, for both chloride and water: the model predicts that LCO affect MD chloride delivery much more than they affect water delivery.
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The results in Figs. 8 and 9 generalize results in Ref. 16 by showing that the pattern reported there holds for a substantial range of physiologically plausible sensitivities and by relating the stability thresholds observed in Ref. 16 to the conceptual framework developed in Refs. 12 and 13.4
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DISCUSSION |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONAPPENDIX AAPPENDIX BAPPENDIX CREFERENCES |
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