Model of Feedback-Mediated Dynamics in Coupled Nephrons with Compliant Thick Ascending Limbs

Transcription

1 Model of Feedback-Mediated Dynamics in Coupled Nephrons with Compliant Thick Ascending Limbs Senior Thesis Amy Wen Advisor: Anita Layton Mathematics Department Duke University, Durham, NC April 27, 2010

2 Acknowledgments I would like to thank my research advisor Professor Anita Layton for all her help on this project. I am grateful for her support and guidance throughout my work on this model as well as her valuable feedback on this paper. I would also like to acknowledge Professor Elizabeth Bouzarth for serving on my committee and providing valuable suggestions. In addition, I would like to thank Professor David Kraines and Duke s PRUV program for supporting my research during Summer 2008.

3 Abstract Irregular flow and pressure have been observed in the nephrons of spontaneously hypertensive rats (SHR). Coupling of nephrons through their tubuloglomerular feedback (TGF) systems may be one of the factors involved in this phenomenon. To increase understanding of the effects due to coupling, a mathematical model was created for bifurcation analysis. This study was inspired by previous models that investigated coupled nephrons in the case of a rigid thick ascending limb (TAL) and single nephrons in the case of a compliant TAL. In the present study, a model of coupled nephrons with compliant TALs is developed. From this model, we have derived a characteristic equation for a system of coupled nephrons. Analysis of the solutions to this characteristic equation for the case of two identical coupled nephrons has revealed that TGF systems with coupling are more prone to oscillate, which could lead to the complex behavior found in SHR.

4 Contents 1 Introduction 1 2 Mathematical Model Governing Equations The Characteristic Equation Model Results: Two Identical Coupled Nephrons 16 4 Discussion 20 References 21

5 1 Introduction The kidneys play a number of vital functions in maintaining homeostasis. For example, they facilitate the regulation of water and electrolyte balance, excretion of metabolic waste, and the regulation of blood pressure [3]. The basic structural and functional unit of a kidney is the nephron. Each nephron consists of a glomerulus, which is a bundle of capillaries that performs the initial filtering, and a tubule composed of single layer of epithelial cells whose function is reabsorption and secretion of various solutes such as proteins, carbohydrates, and ions. Pressure from the glomerulus forces a filtrate of blood plasma to enter the tubule, which the tubule then transforms into urine. The process of altering the fluid composition is carried out by transepithelial transport along the tubule and is regulated by multiple mechansims, among the most important of which is the tubuloglomerular feedback (TGF) system [10]. Figure 1 displays an illustration of two coupled nephrons. The arrows denote the direction of flow, which begins at the glomerulus, enters the tubule through the proximal convoluted tubule, goes around the loop of Henle, which consists of the descending and ascending limbs, and proceeds to the distal convoluted tubule, from which the filtrate is delivered to the collecting ducts. TGF is a negative feedback mechanism that works to maintain the concentration of NaCl at a target level based on the chloride ion concentration at the macula densa (MD) of the thick ascending limb (TAL). The MD is a group of specialized cells that are in the tubular wall near the end of the TAL and close to the glomerulus. The TAL is an important section of the tubule for TGF function. The TAL is a water-impermeable portion of the loop of Henle and actively pumps NaCl from the tubular fluid into the surrounding interstitium, a process that results in diluting the tubular fluid. If the chloride concentration at the MD is high relative to the target level, TGF acts to reduce the rate of filtration. The decrease in tubular fluid flow allows more time for NaCl to be reabsorbed, hence lowering the chloride concentration. On the other hand, a low chloride 1

6 Figure 1: A schematic of two coupled nephrons. concentraion causes an increase in fluid flow, giving less time for reabsorption and resulting in an increase in chloride concentration. Thus, TGF helps to stabilize water and solute delivery to the distal portion of the nephron and allows for a balance between fluid flow rate and the tubule s capacity for absorption. Micropuncture experiments have shown that nephron flow and pressure exhibit regular oscillations in normotensive rats but may exhibit irregular oscillations in spontaneously hypertensive rats (SHR) [4]. One of the factors involved in causing irregular oscillations to arise could be coupling between nephrons that result from interactions between the TGF systems of nearby nephrons [7], as it has been found that coupling is significantly greater in SHR than in normotensive rats [12]. Coupling appears to be the result of electrotonic signal conduction along the preglomerular vasculature, with each cortical radial artery supporting approximately 10 to 25 nephrons [5]. These arteries branch off into units of two, or occasionally three, afferent arterioles which feed into the nephrons and give rise to the coupling, 2

7 as seen in Figure 1 [9]. A previous model investigation of coupled nephrons has illustrated the complex behavior that arises from coupling, which supports the hypothesis that coupling is a factor of the irregular oscillations found in SHR [8]. However, in the model, the TAL was considered to be a rigid tubule and did not consider the effect of the expansion and contraction of the tubular walls as fluid flow rate changes. The model simplifies the dynamics of TGF by expressing the flow rate as a function of the single-nephron glomerular filtration rate and does not calculate the hydrodynamic pressure. The current study builds on this previous model by incorporating pressure-driven flow and compliant TAL walls, utilizing a model developed for a single nephron [6]. To examine and predict the effect of including tubular wall compliance, the resulting model was analyzed using linearization. The result of this study is an extension of an uncoupled compliant-tube TGF model to coupled nephrons, a derivation of the characteristic equation for the coupled nephron model, and the computation of solutions for the characteristic equation for the special case of two identical coupled nephrons. 2 Mathematical Model 2.1 Governing Equations We describe a model of N coupled nephrons that extends a model of TGF previously used for a single nephron with a compliant TAL [6]. We only consider the portion of the N nephrons that extends in space from the loop bend (x = 0) to the end of the collecting duct (x = L 0 ). Other structures of the nephron, such as the glomerulus, proximal tubule, and descending limb, are represented by empirical functions. Since the chloride ion concentration measured at the MD is considered to be the primary driving force for TGF, only the TAL tubular fluid chloride concentration is examined. Coupling of the nephrons is believed to be mediated by electrotonic conduction in the preglomerular vasculature. Therefore, we regard coupling as a 3

8 factor in the response of TGF, which occurs at x = 0 in our model, and we assume coupling intensifies this response. Figure 2 illustrates the overall concept of the model. Figure 2: A block diagram of the main elements of the TGF model. Pressure at the entrance of the TAL P (0, t) is modified by perturbations and coupling. These perturbations would then affect the flow Q(x, t), radius R(x, t), and chloride concentration C(x, t) along the tubule. After some delay τ, TGF response to the chloride concentration at the MD would result in an adjustment to the pressure at the loop bend. The TAL is the segment of the nephrons that extend from the loop bend (x = 0) to the MD (x = L). We model the course of the fluid within each of the nephrons as the flow of an incompressible fluid along a narrow compliant tube. Thus, the equations that describe fluid flow and pressure are 8µ P (x, t) = Q(x, t) (1) x πr(x, t) 4 ( Q(x, t) = 2πR(x, t) R ) P (x, t), (2) x P t where x is the axial position along the nephron, t is time, P (x, t) is the fluid pressure, Q(x, t) is the flow rate, and R(x, t) is the tubular radius. Eq. (1) describes how fluid flow is driven by 4

9 the axial pressure gradient. It originates from the Hagen-Poiseuille equation, which is valid for very low Reynolds number [11]. The Reynolds number gives the ratio of inertial forces to viscous forces. At low Reynolds numbers, viscosity dominates and fluid flow is laminar. Eq. (2) is derived from mass continuity for the case of an incompressible fluid. We assume that the pressure at the loop bend is determined in time through TGF and coupling with other nephrons (P 0 (t) = P (0, t)) and the outflow pressure is fixed in time (P 1 = P (L 0, t)). Solute conservation in the TAL is described by the hyperbolic partial differential equation πr(x, t) 2 t C(x, t) = 2πR(x, t)c(x, t) R(x, t) Q(x, t) C(x, t) ( t x ) Vmax C(x, t) 2πR 0 K M + C(x, t) + P M(C(x, t) C e (x, t)) (3) where C(x, t) is the concentration of chloride in the TAL, C e (x) is the interstitial chloride concentration, R 0 is the TAL radius at steady-state, V max is the maximum transport rate, K M is the Michaelis constant, and P M is the backleak permeability [6]. The last two terms acounts for active solute transport as described by Michaelis-Menten kinetics and transepithelial diffusion, respectively. We consider the chloride concentration at the bend to be fixed in time (C 0 = C(0, t)). To represent the compliant walls of the nephrons, the tubular radius varies as a function of the transmural pressure difference, which we model as R(x, t) = α(p (x, t) P e (x)) + β(x), (4) where P e (x) is the interstitial pressure, α specifies the degree of compliance, and β(x) is the tubular radius in the case where the pressure gradient or the compliance is zero. Consider modeling an arbitrary nephron indexed by i, where i = 1,..., N. The pressure at the loop bend P 0,i (t) is a function of the TGF-mediated pressure response of the nephron itself as well as coupled TGF responses in nearby nephrons. We thus model TGF-mediated 5

10 pressure of nephron i at the loop bend by the equation P 0,i (t) = P 0,i +K 1,i tanh(k 2,i (C op C i (L, t τ i )))+ ϕ ij K 1,j tanh(k 2,j (C op C j (L, t τ j ))), j i (5) where ϕ ij quantifies the degree of coupling between nephrons i and j, K 1,i is half the range of pressure variation around P 0,i, K 2,i is a measure of the sensitivity of TGF, and C op is the time-independent steady-state chloride concentration alongside the MD when P i (0, t) = P 0,i, which represents the target concentration. The summation includes all nephrons except nephron i. However, in effect we are only summing over the nearest neighbors due to the presence of the coupling constant ϕ ij. In practice, coupling decreases exponentially as a function of distance between nephrons along the vasculature [2]. The greatest effect is observed for the pairs or triplets from the same afferent arteriole unit. Note that C(L, t τ) is the chloride concentration alongside the MD at time t τ. Parameters The base-case parameters shown in Table 1 were chosen to be consistent with experimental results in the rat kidney. The tubular radius parameter β(x) is specified (in µm) by the piecewise function β 0 0 x 1.5 L β(x) =, (6) β 1 (x) 1.5 L x L 0 where β 1 (x) is a cubic polynomial that satisfies β 1 (1.5 L) = β 0 and β 1 (L 0 ) = β 2. The values for β 0 and β 2 are selected such that the model results in a target average TAL radius and a target outflow pressure P (L 0 ) at steady state. Interstitual chloride concentration was assumed to be given by C e (x) = C 0 (A 1 e (A3x/L) +A 2 ), where A 1 = (1 C e (L))/C 0 )/(1 e A 3 ), A 2 = 1 A 1, and A 3 = 2. C e (L) corresponds to the interstitial chloride concentration at the renal cortex of 150 mm. A profile for C e (x) can be found in Figure 1 of Ref. [1]. 6

11 Table 1: Base-case parameter values. Parameter Value α cm mmhg 1 β µm β µm C mm K M 70.0 mm L cm L cm µ /(cm s) P e 5.00 mmhg P mmhg P mmhg P M cm/s Q 0 6 nl/min R µm 14.5 nmole cm 2 s 1 V max T s 2.2 The Characteristic Equation Fluctuations in blood pressure occur frequently and are triggered by sources such as respiration, heart beat, or change in posture. After a fluctuation, the blood pressure could settle down into time-independent steady state or lead to a regular, sustained, stable oscillation. Direct numerical computation of the solutions to Eqs. (1) (3) would allow for the prediction of the asymptotic behavior in response to a perturbation. However, such computation is often time-consuming. In addition, we would like to have a better understanding of the manner in which the model behavior depends on the parameter values. This motivates us to derive and analyze a characteristic equation through linearization of the model equations. First, we decouple Eqs. (1) and (2) by solving for Q(x, t) from Eq. (1) and substituting this expression into Eq. (2). This process results in an advection-diffusion equation for the 7

12 pressure P t P (x, t) R(x, t)2 4µ R P R(x, t) x x P (x, t) = R(x, t)3 16µ R P 2 P (x, t), (7) x2 where the boundary conditions are P (0, t) = P 0 (t) and P (L, t) = P 1, as stated previously. We then simplify Eqs. (3) and (7) by using nondimensionalization. Let x =x/l t =t/t 0 τ =τ/t 0 c A0 =c A0 /c A0 = 1() C =C/C 0 C e =C e /C 0 Q =Q/Q 0 Ṽ max =V max /(C 0 Q 0 /(c A0 L)) K M =K M /C 0 P M =P M /(Q 0 /(c A0 L)) P =P/P 0 R =R/R 0 µ =µ/(πp 0 R 4 0/(Q 0 L)), where c A0 = 2πR 0 and T 0 = πr 2 0L/Q 0. Using these nondimensional variables in Eqs. (3) 8

13 and (7), the equations become R 2 C t P t R2 4µα ( ) R = 2RC t Q C x Vmax C K M + C + P M(C C e ) R x P x = R3 16µα (8) 2 P x 2, (9) where the tilde symbols have been dropped. We assume that the tubular fluid pressure is linear and the parameter β(x) results in a constant tubular radius at steady state. We then apply these assumptions to linearize Eqs. (8) and (9) by assuming infinitesima. perturbbtions in C, P, and R: C(x, t) = C SS (x) + ϵd(x, t) (10) P (x, t) = P SS (x) + ϵg(x, t) (11) R(x, t) = 1 + ϵr(x, t), (12) where ϵ 1, C SS (x) and P SS (x) are the steady-state chloride concentration and pressure, respectively, and steady-state tubular radius is assumed to be normalized to 1. Pressure Perturbation We first linearize Eq. (9) by substituting in Eqs. (11) and (12), from which we obtain ( t (P (1 + ϵr)2 SS + ϵg) ϵ r ) 4µα x x (P SS + ϵg) = (1 + ϵr)3 16µα 2 x 2 (P SS + ϵg). (13) The pressure-radius relationship from Eq. (4) implies that ϵr(x, t) = α(p (x, t) P SS ) = ϵαg(x, t) (14) 9

14 since the pressure gradient is zero at steady state. In other words, P (x, t) = P e (x) and β(x) = 1. With steady-state tubular flow rate and tubular radius normalized to 1, and keeping in mind the normalization for µ, we find from Eq. (1) that P SS x = 8µ. (15) Disregarding terms above O(ϵ) and applying Eqs. (15) and (14), we find that Eq. (13) simplifies to an advection-diffusion equation for G. We further consider the case of coupling, and let the subscript i represent the ith model nephron. We then obtain the equation G i t + 2 G i x = 1 16µα i 2 G i x 2 (16) with the boundary conditions G i (0, t) = P i (C op )D i (1, t τ i ) + ϕ ij G j (0, t), (17) j i G i (L 0 /L, t) = 0. (18) The boundary condition at x = 0 specified by Eq. (17) gives the change in P 0,i (t) (the pressure at the loop bend) in response to a change in the chloride concentration at the MD, which has a delay of τ to allow time for signal transduction. It also includes terms that represent the dependence of the the TGF-mediated response on nearby nephrons at the loop bend. To illustrate the strengthening effect of coupled TGF responses, the perturbation from steady-state pressure for the model nephron indexed by i is increased by j i ϕ ijg j (0, t), where ϕ ij is the coupling coefficient that quantifies the influence of nephron j on nephron i. The other boundary condition at x = L 0 /L fixes the pressure at the end of the nephron. Assuming that D i (x, t) can be written as D i (x, t) = f i (x)e λit [8], where f i (x) is a differentiable real function for x and λ i is an element of the field of complex numbers, and further 10

15 that D i = D j, f i = f j, and λ i = λ j, Eq. (17) becomes G i (0, t) = P i (C op )f(1)e λ(t τi) + ϕ ij G j (0, t), (19) j i where f f i and λ λ i. For the case of two nephrons, the matrix representation of Eq. (19) is 1 ϕ 12 G 1(0, t) = P 1(C op )f(1)e λ(t τ 1). (20) ϕ 21 1 G 2 (0, t) P 2(C op )f(1)e λ(t τ 2) Thus, we can uncouple the boundary conditions to get: G 1(0, t) = G 2 (0, t) = 1 1 ϕ 12 P 1(C op )f(1)e λ(t τ 1) 1 ϕ 12 ϕ 21 ϕ 21 1 P 2(C op )f(1)e λ(t τ 2) 1 P 1(C op )f(1)e λ(t τ1) + ϕ 12 P 2(C op )f(1)e λ(t τ 2). (21) 1 ϕ 12 ϕ 21 ϕ 21 P 1(C op )f(1)e λ(t τ1) + P 2(C op )f(1)e λ(t τ 2) In other words, for two nephrons, we have G i (0, t) = 1 1 ϕ ij ϕ ji [P i (C op )f(1)e λ(t τ i) + ϕ ij P j(c op )f(1)e λ(t τ j) ], (22) where i j. Now, assume that the solution takes the form G i (x, t) = G i (0, t)g i (x) = 1 1 ϕ ij ϕ ji [P i (C op )f(1)e λ(t τ i) + ϕ ij P j(c op )f(1)e λ(t τ j) ]g i (x), (23) 11

16 where g i (0) = 1 and g i (L 0 /L) = 0. Substituting into Eq. (16), we find that ( ) 1 g i (x) 2g 16µα i(x) λg i (x) (P i (C op )e λτ i + ϕ ij P j(c op )e λτ j ) = 0, (24) i which implies that 1 16µα i g i (x) 2g i(x) λg i (x) = 0 (25) if we assume a nonzero pressure response at the loop bend. Using the boundary conditions g i (0) = 1 and g i (L 0 /L) = 0, we then find that g i (x) = c + i ek+ i x + c i ek i x, (26) where k ± i = 16µα i ± (16µα i ) µα i λ. (27) and e k i L 0/L c ± i = ± e k i L 0/L e. (28) k+ i L 0/L Now consider more than two nephrons. We can still represent the boundary condition in Eq. (19) for n nephrons in matrix form as follows: 1 ϕ 12 ϕ 1n G 1 (0, t) P 1(C op )f(1)e λ(t τ 1) ϕ 21 1 ϕ 23 G 2 (0, t) P 2(C op )f(1)e λ(t τ 2) =. (29) ϕ n1 1 G n (0, t) P n(c op )f(1)e λ(t τ n) Then, we can uncouple the boundary conditions G 1 (0, t), G 2 (0, t),..., G n (0, t) by mutliplying 12

17 both sides by the inverse of the matrix of ϕ s, as long as the matrix is nonsingular. G 1 (0, t) 1 ϕ 12 ϕ 1n G 2 (0, t) ϕ 21 1 ϕ 23 = G n (0, t) ϕ n1 1 1 P 1(C op )f(1)e λ(t τ 1) P 2(C op )f(1)e λ(t τ 2). (30). P n(c op )f(1)e λ(t τ n) Again, assuming the equation takes the form in Eq. (23) and substituting into Eq. (16), we find that 1 ϕ 12 ϕ 1n ϕ 21 1 ϕ ϕ n1 1 1 ( P 1(C op )f(1)e λ(t τ 1) ( P 2(C op )f(1)e λ(t τ 2) ( P n(c op )f(1)e λ(t τ n) 1 16µα 1 g 1 16µα 2 g 1 16µα n g ) 1(x) 2g 1(x) λg 1 (x) ) 2(x) 2g 2(x) λg 2 (x) = 0.. ) n(x) 2g n(x) λg n (x)) Still assuming nonsingularity of the matrix of ϕ s, we again arrive at Eq. (25) for g i (x) with solution as found in Eq. (26). (31) Concentration Perturbation We now want to linearize Eq. (8), the solute conservation equation. This is accomplished by substituting in Eqs. (10)-(12) and Eq. (1) for Q to obtain (1 + ϵr(x, t)) 2 (C SS(x) + ϵd(x, t)) t = 2(1 + ϵr(x, t))(c SS (x) + ϵd(x, t)) (1 + ϵr(x, t)) t (1 + ϵr(x, t))4 + 8µ x (P SS(x) + ϵg(x, t)) x (C SS(x) + ϵd(x, t)) ( ) Vmax (C SS (x) + ϵd(x, t)) K M + C SS (x) + ϵd(x, t) + P M(C SS (x) + ϵd(x, t) C e ) (32) 13

18 Letting K(C) = V max C/(K M + C), we find that at steady state (ϵ = 0), 1 8µ x P SS x C SS = K(C SS ) + P M (C SS C e ). (33) Discarding terms above O(ϵ) and using using Eqs. (33) and (14), Eq. (32) simplifies to D t = 2α G i ic SS t + 1 ( P SS C SS 4r i 8µ x x + P SS x D x + G i x ) C SS (K (C SS ) + P M )D. x (34) Notice from Eq. (30) that G i t (0, t) = λg i(0, t) since the only time dependence arises from e λt. Using this fact and substituting D(x, t) = f(x)e λt, G i (x, t) = G i (0, t)g i (x), and r i (x, t) = α i G i (x, t) into Eq. (34), we find λe λt f(x) = 1 8µ (C SSG i (0, t)g i(x) + P SSe λt f (x) + 4α i C SSP SSG i (0, t)g i (x)) 2α i C SS λg i (0, t)g i (x) (K (C SS ) + P M )f(x)e λt. (35) From Eq. (30), we can express G i (0, t) as G i (0, t) = n a ij P j(c op )f(1)e λ(t τj), (36) j=1 where a ij are constants that come from the inverse of the matrix of ϕ s. Applying Eq. (36) and Eq. (15) to Eq. (35) and rearranging, we find f (x)+(λ+k (C SS )+P M )f(x) = n j=1 ( ) a ij P j(c op )e λτ j 1 f(1) 8µ C SSg i(x) 2α i g i (x)(λc SS + 2C SS). (37) Using integrating factors, we can solve for f(x), using the boundary condition f(0) = 0. 14

19 Thus, ( f(s) = exp s 0 ) (λ + K (C SS ) + P M ) dy f(1) s n ( ) a ij P j(c op )e λτ j 1 0 8µ C SSg i(x) 2α i g i (x)(λc SS + 2C SS) j=1 ( x ) exp (λ + K (C SS ) + P M ) dy dx. (38) 0 Letting s be 1, this simplifies to 1 1 = 0 exp n ( ) a ij P j(c op )e λτ j 1 8µ C SSg i(x) 2α i g i (x)(λc SS + 2C SS) j=1 ( 1 ) (λ + K (C SS ) + P M ) dy x dx. (39) By combining Eqs. (15) and (33), we know that d dx C SS(x) = K(C SS ) P M (C SS (x) C e (x)), (40) which leads to K (C SS ) + P M = P MC e(x) C SS SS (x). (41) C C SS Substituting Eq. (41) into Eq. (39), we find 1 = C SS(1) ( exp n a ij P j(c op )e λτ j j=1 1 x P M C e C SS 1 0 ( ( 1 e λ(1 x) 8µ g i(x) 2α i g i (x) λ C SS C SS )) + 2 (42) ) dy dx. (43) 15

20 We define the TGF gain γ i as C SS (1)P i (C op ). Then, Eq. (42) can be re-written as: 1 = n a ij γ j e λτ j j=1 1 0 ( ( 1 e λ(1 x) 8µ g i(x) 2α i g i (x) λ C SS C SS )) ( 1 ) P M C e + 2 exp dy dx. x C SS (44) Taking the partial derivative of the pressure response function given in Eq. (5) with respect to C i then setting the concentration C i to C op, we find that the gain can also be described as γ i = K 1,i K 2,i C SS. (45) In other words, the gain for a given nephron is related to the product of the range of pressure variation, the sensitivity of the TGF system, and the steady-state concentration gradient. In the limit where there is no coupling, the matrix of a ij values becomes the identity matrix and we recover 1 = γ i e λτ i 1 0 ( ( 1 e λ(1 x) 8µ g i(x) 2α i g i (x) λ C SS C SS )) ( 1 ) P M C e + 2 exp dy dx, x C SS (46) the equation found in Ref. [6] for a single nephron with compliant TAL walls. It was shown in this single nephron case that the gain factors γ for the rigid and complaint TAL models are related by γ = 8µL 0 γ rigid [6]. (47) 3 Model Results: Two Identical Coupled Nephrons A bifurcation analysis was performed to elucidate the dynamics of coupled nephrons with compliant TAL walls. We focused on the the special case of two identical nephrons that possess the same parameters. Using the model s characteristic equation Eq. (44), the parameter boundaries that divide regions with differing behaviors were found. Throughout this anlysis, we assume that coupling is symmetric (ϕ ϕ ij = ϕ ji ). 16

21 We examined the effect of coupling for the case of two nephrons with identical γ and τ. We computed the γ τ pairs within the physiological range that form a transition between different long-term solution behaviors. These boundaries are the regions where the real part of λ is zero, as the roots occur where there is a solution bifurcation or a transition between stable solution behaviors. Defining λ as ρ+iω, we forced ρ to be zero to locate the boundaries. We considered cases of coupled and uncoupled nephrons with rigid and compliant walls. Since the nephrons considered have identical γ, τ, and ϕ, and we assumed λ for the coupled nephrons are also the same, the characteristic equations for the two nephrons are the same. Therefore, any coordinates (γ, τ) in parameter space that satisfy one equation will concurrently satisfy the other. For this reason, we simply have a system of two nonlinear equations from the real and imaginary parts of the characteristic equation for one of the nephrons. To find the bifurcation curves, the parameter γ (or τ) was fixed at a particular value, and Newton s iteration method for solving systems of nonlinear equations was used to determine what values of τ (or γ) and ω (since ρ=0), if any, satisfy the two equations simultaneously. The noncompliant TAL case occurs in the limit where α 0. We simulated the rigid tubule case in our model by choosing a compliance value α three orders of magnitudes lower than the base-case value of α = cm mmhg 1. We considered cases where ϕ is 0, 0.1, and 0.2 (Fig. 3). We see that these curves resemble those found in Figure 2 (panels A1 and C1) of Ref. [8]. The points (γ, τ) that are located below all curves with ρ n = 0 indicate points where transient perturbations lead to the time-independent steady-state solution. For all other points (γ, τ), limit-cycle oscillations result. The effect of coupling is to bring down all the bifurcations between the various regions. We see the same effect for the compliant TAL case, where α = cm mmhg 1 (Fig. 4). Again, we considered situations where ϕ is 0, 0.1, and 0.2. When there is no coupling (ϕ = 0), we obtain curves that match those found in Figure 2 (panel B1) of Ref. [6]. As was noted in Ref. [6], compliance lowers the root curves. This indicates TAL compliance 17

22 increases the tendency of the TGF system to oscillate. Coupling appears to further increase this tendency to oscillate, as was found in the rigid tubule case. Figure 3: Root loci for a rigid TAL as a function of gain γ and delay τ for the case of identical nephrons. 18

23 Figure 4: Root loci for a compliant TAL as a function of gain γ and delay τ for the case of identical nephrons. 19

24 4 Discussion We have developed a mathematical model of coupled nephrons with compliant TALs. The main accomplishment of this study was the derivation of a modified characteristic equation to account for the effects of coupling. The characteristic equation was obtained similarly to the derivation of the characteristic equation for a TGF model representing a single compliant TAL from a previous study [6], specifically from a linearization of the model equations around the steady state values. The difference arises from additional coupling terms in the boundary condition of the pressure perturbation at x = 0 that represent electrotonic signal conduction along the preglomerular vasculature. The characteristic equation is useful as a method for determining the location of bifurcations in parameter space and characterizing those solutions. The characteristic equation allows us to distinguish regions of stable time-independent steady-state solutions as well as the regions of various limit-cycle oscillations. Utilizing the characteristic equation, we computed bifurcation curves for two identical coupled nephrons. Analysis of the validity of the characteristic equations was performed by comparing the bifurcation curves of coupled nephrons with rigid TALs with those found in Ref. [8] as well as the bifurcation curves of nephrons with compliant TALs with no coupling with those found in Ref. [6]. The bifurcation curves agreed with the results found from these earlier studies of TGF systems. Coupling for identical nephrons with compliant TALs result in a similar trend as found for coupling for identical nephrons with rigid TALs in that the bifurcation curves are shifted downwards. Thus, our model predicts that coupling increases the region for limit-cycle oscillations and decreases the region for steady-state solutions. 20

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