On the role of the epithelium in a model of sodium exchange in renal tubules
Marta Marulli, Aurélie Edwards, Vuk Milišić, Nicolas Vauchelet
OOn the role of the epithelium in a model ofsodium exchange in renal tubules
Marta Marulli , Aurélie Edwards , Vuk Milišić , and NicolasVauchelet LAGA, UMR 7539, CNRS, Université Sorbonne Paris Nord, 99, avenueJean-Baptiste Clément 93430 Villetaneuse - France. University of Bologna, Department of Mathematics, Piazza di Porta S. Donato 5,40126 Bologna, Italy. Department of Biomedical Engineering, Boston University, Massachusetts.
Abstract
In this study we present a mathematical model describing the trans-port of sodium in a fluid circulating in a counter-current tubular archi-tecture, which constitutes a simplified model of Henle’s loop in a kidneynephron. The model explicitly takes into account the epithelial layer atthe interface between the tubular lumen and the surrounding interstitium.In a specific range of parameters, we show that explicitly accounting fortransport across the apical and basolateral membranes of epithelial cells,instead of assuming a single barrier, affects the axial concentration gradi-ent, an essential determinant of the urinary concentrating capacity. Wepresent the solution related to the stationary system, and we performnumerical simulations to understand the physiological behaviour of thesystem. We prove that when time grows large, our dynamic model con-verges towards the stationary system at an exponential rate. In order toprove rigorously this global asymptotic stability result, we study eigen-problems of an auxiliary linear operator and its dual.
Key words:
Counter-current, transport equation, ionic exchange, station-ary system, eigenproblem, long-time asymptotics.
One of the main functions of the kidneys is to filter metabolic wastes and toxinsfrom plasma and excrete them in urine. The kidneys also play a key role inregulating the balance of water and electrolytes, long-term blood pressure, aswell as acid-base equilibrium. The structural and functional units of the kidneyare called nephrons, which number about 1 million in each human kidney [2].Blood is first filtered by glomerular capillaries and then the compositionof the filtrate varies as it flows along different segments of the nephron : theproximal tubule, Henle’s loop (which is formed by a descending limb and anascending limb), the distal tubule, and the collecting duct. The reabsorption1 a r X i v : . [ m a t h . A P ] J a n igure 1: Simplified model of loop of Henle. q /q and u /u denote soluteconcentration in the epithelial layer and lumen of the descending/ascendinglimb, respectively.of water and solutes from the tubules into the surrounding interstitium (or se-cretion in the opposite direction) allows the kidneys to match precisely urinaryexcretion to the dietary intake [18].In the last decade, several groups have developed sophisticated models ofwater and electrolyte transport in the kidney. These models can be broadlydivided into 2 categories: (a) detailed cell-based models that incorporate cell-specific transporters and predict the function of small populations of nephronsat steady-state ([17]; [27]; [14]; [28]; [4]), and (b) macroscale models that de-scribe the integrated function of nephrons and renal blood vessels but withoutaccounting for cell-specific transport mechanisms ([24], [25]; [16]; [7]; [3]; [8]).These latter models do not consider explicitly the epithelial layer separating thetubule lumen from the surrounding interstitium, and represent the barrier as asingle membrane. We developed the model presented below to assess the impactof this set of assumptions.Specifically, in this study we present a simplified mathematical model of solutetransport in Henle’s loop. This model accounts for ion transport between thelumen and the epithelial cells, and between the cells and the interstitium. Theaim of this work is to evaluate the impact of explicitly considering the epitheliumon predicted solute concentration gradients in the loop of Henle.In our simplified approach, the loop of Henle is represented as two tubules ina counter-current arrangement, the descending and ascending limb are consid-ered to be rigid cylinders of length L lined by a layer of epithelial cells. Waterand solute reabsorption from the luminal fluid into the interstitium proceedsin two steps : water and solutes cross first the apical membrane at the lumen-cytosol interface and then the basolateral membrane at the cytosol-interstitiuminterface, [29]. A schematic representation of the model is given in Figure 1.The energy that drives tubular transport is provided by Na + /K + -ATPase,an enzyme that couples the hydrolysis of ATP to the pumping of sodium (Na + )ions out of the cell and potassium (K + ) ions into the cell, across the basolateral2embrane. The electrochemical potential gradients resulting from this activetransport mechanism in turn drive the passive transport of ions across othertransporters, via diffusion or coupled transport. We refer to diffusion as thebiological process in which a substance tends to move from an area of highconcentration to an area of low concentration [22, 23]. As described in [10], inthe absence of electrical forces, the diffusive solute flux from compartment 1 tocompartment 2 (expressed in [ mol.m − .s − ] ) is given by: J diffusion = P (cid:96) ( u − u ) , where P [ m.s − ] is the permeability of the membrane to the considered solute, (cid:96) the perimeter of the membrane, and u and u are the respective concentrationsof the solute in compartments and .We assume that the volumetric flow rate in the luminal fluid (denoted by α > ) remains constant, i.e. there is no transepithelial water transport. Thedescending limb is in fact permeable to water, but we make this simplifyingassumption in order to facilitate the mathematical analysis. Given the counter-current tubular architecture, this flow rate has a negative value in the ascendinglimb.The present model focuses on tubular Na + transport. The concentrationof Na + ( [ mol.m − ]) is denoted by u and u , respectively, in the lumen ofthe descending and ascending limb, by q and q in the epithelial cells of thedescending and ascending limbs, respectively, and by u in the interstitium. Thepermeability to Na + of the membrane separating the lumen and the epithelialcell of the descending and ascending limb is denoted by P and P , respectively. P ,e denotes the permeability to Na + of the membrane separating the epithelialcell of the descending limb and the interstitium; the Na + permeability at theinterface between the epithelial cell of the ascending limb and the interstitium istaken to be negligible. The re-absorption or secretion of ions generates electricalpotential differences across membranes. In the present model, the impact oftransmembrane potentials on Na + transport is not taken into account.The concentrations depend on the time t and the spatial position x ∈ [0 , L ] .The dynamics of Na + concentration is given by the following model on (0 , + ∞ ) × (0 , L ) a ∂u ∂t + α ∂u ∂x = J , a ∂u ∂t − α ∂u ∂x = J , (1) a ∂q ∂t = J , a ∂q ∂t = J , a ∂u ∂t = J . (2)The parameters a i , for i = 0 , , , , , denote positive constants defined as: a = πr , a = πr , a = π ( r ,e − r ) , a = π ( r ,e − r ) , a = π (cid:16) r ,e + r ,e (cid:17) . In these equations, r i , i = 1 , , denotes the inner radius of tubule i , whereas r i,e denotes the outer radius of tubule i , which includes the epithelial layer. Thefluxes J i describe the ionic exchanges between the different domains. They aremodeled in the following way: Lumen.
In the lumen, we consider the diffusion of Na + towards the epithe-lium. Then, J = 2 πr P ( q − u ) , J = 2 πr P ( q − u ) . pithelium. We take into account the diffusion of Na + from the descendinglimb epithelium towards both the lumen and the interstitium, J = 2 πr P ( u − q ) + 2 πr ,e P ,e ( u − q ) . In the ascending limb (tubule 2), we also consider the active reabsorptionthat is mediated by Na + /K + -ATPase, which pumps 3 Na + ions out of thecell in exchange for 2 K + ions.The net flux into the ascending limb epithelium is given by the sum of thediffusive flux from the lumen and the export across the pump, which isdescribed using Michaelis-Menten kinetics, [10]: J = 2 πr P ( u − q ) − πr ,e G ( q ) , where G ( q ) = V m (cid:20) q K M, + q (cid:21) . The exponent of G is related to the number of exchanged sodium ions.The affinity of the pump K M, , and its maximum velocity V m , are givenreal numbers. We notice that when q → + ∞ , then G ( q ) → V m whichis in accordance with the biological observation that the pump can besaturated. Interstitium. J = 2 πr P ,e ( q − u ) + 2 πr ,e G ( q ) . We model the dynamics of a solute (here sodium) by the evolution of its con-centration in each tubule. The transport of solute and its exchange are thenmodelled by a hyperbolic PDE system at constant speed with a non-linear trans-port term and with specific boundary conditions.The dynamics of ionic concentrations is given by the following model: a ∂ t u ( t, x ) + α∂ x u ( t, x ) = J ( t, x ) a ∂ t u ( t, x ) − α∂ x u ( t, x ) = J ( t, x ) a ∂ t q ( t, x ) = J ( t, x ) a ∂ t q ( t, x ) = J ( t, x ) a ∂ t u ( t, x ) = J ( t, x ) . (3)We set the boundary conditions : u ( t,
0) = u b ( t ) , u ( t, L ) = u ( t, L ) , t > , (4)where u b is a given function in L ∞ ( R + ) ∩ L loc ( R + ) , which is such that lim t →∞ u b ( t ) =¯ u b for some positive constant ¯ u b > .Finally, the system is complemented with initial conditions u (0 , x ) = u ( x ) , u (0 , x ) = u ( x ) , u (0 , x ) = u ( x ) ,q (0 , x ) = q ( x ) , q (0 , x ) = q ( x ) .
4o simplify notations in (3), we set K := 2 πr ,e P ,e , k := 2 πr P , and k :=2 πr P . For the diffusive fluxes J and J , we include the constant πr ,e in theparameter V m and replace the parameter V m with V m, := 2 πr ,e V m , such that G ( q ) = V m, (cid:20) q K M, + q (cid:21) . (5)Moreover, the orders of magnitude of k , k are the same even if their valuesare not definitely equal, we may assume to further simplify the analysis that k = k = k . We will refer to this as the dynamic system and then (3) reads : a ∂ t u + α∂ x u = k ( q − u ) (6a) a ∂ t u − α∂ x u = k ( q − u ) (6b) a ∂ t q = k ( u − q ) + K ( u − q ) (6c) a ∂ t q = k ( u − q ) − G ( q ) (6d) a ∂ t u = K ( q − u ) + G ( q ) . (6e)The existence and uniqueness of vector solution u = ( u , u , q , q , u ) to thissystem are investigated in [15]. Several previous works have neglected the ep-ithelium region (see e.g. [25, 24]). The first goal of this work is to study theeffects of this region in the mathematical model. The main indicator quantify-ing these effects is the parameter k which accounts for the permeability betweenthe lumen and the epithelium. Then, we analyse the dependency between theconcentrations and k . In the absence of physiological perturbations, the concen-trations are very close to the steady state, thus it seems reasonable to considersolutions of (6) at equilibrium, which leads us to study the system : + α∂ x ¯ u = k (¯ q − ¯ u ) − α∂ x ¯ u = k (¯ q − ¯ u )0 = k (¯ u − ¯ q ) + K (¯ u − ¯ q )0 = k (¯ q − ¯ u ) − G (¯ q )0 = K (¯ q − ¯ u ) + G (¯ q )¯ u ( L ) = ¯ u ( L ) , ¯ u (0) = ¯ u b . (7)Section 2 concerns the analysis of solutions to stationary system (7). In par-ticular, we study their qualitative behaviour and their dependency with respectto the parameter k . Our mathematical observations are illustrated by somenumerical computations.The second aim of the paper is to study the asymptotic behaviour of thesolutions of (6). In Theorem 3.1, we show that they converge as t goes to + ∞ to the steady state solutions solving (7). Section 3 is devoted to the statementand the proof of this convergence result. Finally, an Appendix provides someuseful technical lemmas. In this section, after proving basic existence and uniqueness results, we inves-tigate how solutions of (7) depend upon the parameter k . We recall that it5ncludes also the permeability parameter as k = k i with k i := 2 πr i P i , i = 1 , .In order to study the qualitative behaviour of these solutions, we then performsome numerical simulations. We first show existence and uniqueness of solutions to the stationary system:
Lemma 2.1.
Let ¯ u b > . Let G be a C function, uniformly Lipschitz, suchthat G (cid:48)(cid:48) is uniformly bounded and G (0) = 0 (e.g. the function defined in (5) ).Then, there exists an unique vector solution to the stationary problem (7) .Moreover, if we assume that G > on R + , then we have the followingrelation ¯ q < ¯ u < ¯ q < ¯ u . Proof.
Summing up all the equations of system (7), we deduce that α ( ∂ x ¯ u − ∂ x ¯ u ) = 0 . From the boundary condition ¯ u ( L ) = ¯ u ( L ) , we obtain ¯ u = ¯ u = ¯ u .Therefore, we may simplify system (7) in α∂ x ¯ u = k (¯ u − ¯ q )2¯ u = ¯ q + ¯ q k (¯ u − ¯ q ) + K (¯ u − ¯ q )0 = k (¯ u − ¯ q ) − G (¯ q )0 = K (¯ q − ¯ u ) + G (¯ q ) . (8)By the fourth equation of (8), ¯ u = ¯ q + G (¯ q ) k , inserted into the first equation,it gives ∂ x ¯ u = G (¯ q ) α . We obtain a differential equation satisfied by ¯ q , ∂ x ¯ q = G (¯ q ) (cid:0) α + αk G (cid:48) (¯ q ) (cid:1) , (9)with α, k positive constants and provided with the initial condition ¯ q (0) thatsatisfies ¯ q (0) + G (¯ q (0)) k = ¯ u b . (10)We first remark that ¯ q (0) (cid:55)→ ¯ q (0) + G (¯ q (0)) k is a C increasing function whichtakes the value at and goes to + ∞ at + ∞ . Thus, for any ¯ u b > there existsa unique ¯ q (0) > solving (10).By assumption, G (cid:48) and G (cid:48)(cid:48) are uniformly bounded, thus we check easily thatthe right-hand side of (9) is uniformly Lipschitz. Therefore, the Cauchy problem(9)–(10) admits a unique solution, which is positive (by uniqueness since is asolution).Then, other quantities are computed thanks to the relations: ¯ u = ¯ q + G (¯ q ) k , ¯ q = ¯ q + 2 G (¯ q ) k , ¯ u = (cid:16) K + 2 k (cid:17) G (¯ q ) + ¯ q . (11)Moreover, by the fourth and fifth equations of system (8) and since G (¯ q ) > , we immediately deduce that ¯ q < ¯ u and ¯ q < ¯ u . Using the second equationof (8), we obtain the claim. 6arameters Description ValuesL Length of tubules · − [ m ] α Water flow in the tubules − [ m /s ] r i Radius of tubule i = 1 , − [ m ] r i,e Radius of epithelium layer i = 1 , . · − [ m ] K πr ,e P ,e ∼ π · − [ m /s ] k = k i πr i P i , i = 1 , changeable [ m /s ] V m, Rate of active transport ∼ πr ,e − [ mol.m − .s − ] K M, Pump affinity for sodium (
N a + ) , mol/m ]¯ u b Initial concentration in tubule mol/m ] Table 1: Frequently used parameters
We approximate numerically solutions of (8). Numerical values of the parame-ters (cf Table 1) are extracted from Table 2 in [9] and Table 1 in [13].Taking into account these quantities allow us to have the numerical ranges ofthe constants and the solution results in a biologically realistic framework. Fol-lowing the proof of Lemma 2.1, we first solve (10) thanks to a Newton method.Then, we solve (9) with a fourth order Runge-Kutta method. Finally, we deduceother concentrations u, q , u using (11).Results from Figures 2a and 2c show that in all compartments, concentra-tions increase as a function of depth ( x -axis). Physiologically, this means thatthe fluid is more concentrated towards the hairpin turn ( x = L ) than near x = 0 , because of active transport in the ascending limb. It can also be seenthat Na + concentration is higher in the central layer of interstitium and lower inthe ascending limb epithelium owing to active Na + transport from the latter tothe central compartment, described by the non-linear term G ( q ) . Furthermore,Figure 2b and Figure 2d highlight that increasing the permeability value homog-enizes the concentrations in the tubules and in the epithelium region. Taking avery large permeability value is equivalent to fusing the epithelial layer with theadjacent lumen, such that luminal and epithelial concentrations become equal.It is proved rigorously in [15] that this occurs in the dynamic system (6). Thisis derived and explained formally in Appendix A.Figures 3 and 4 depict the impact of permeability P = P = P on con-centration profiles for various pump rates V m, . Axial profiles of luminal con-centrations are shown in Figures 3a and 4a, considering different values of thepermeability between the lumen and the epithelium. The fractional increase inconcentration (FIC) is shown in Figures 3b and 4b : for each permeability value(plotted on the horizontal axis), we compute the following ratio (shown on thevertical axis): FIC(¯ u ) := 100 ¯ u ( L ) − ¯ u (0)¯ u (0) , (12)where ¯ u ( L ) is the concentration in the tubular lumen , at x = L and ¯ u (0) the concentration at x = 0 . This illustrates the impact of permeability on theaxial concentration gradient. We observe that this ratio depends also stronglyon the value of V m, . 7 a) Concentration profiles with perme-ability P i = 2 · − [m/s]. (b) Concentrations in 2D with P i = 2 · − . Length of lumen on vertical axis.(c) Concentration profiles with perme-ability P i = 2 · − [m/s]. (d) Concentrations in 2D with P i = 2 · − . Length of lumen on vertical axis. Figure 2: Concentration profiles for V m, = 2 πr ,e − and different permeabil-ity values. (a) Axial concentrations in the lumenfor different values of permeability. (b) Fractional increase in concentra-tion as a function of permeability. Figure 3: Concentration profiles for V m, = 2 πr ,e · − [ mol.m − .s − ] .8he permeability range (numerically P ∈ [10 − , − ] , equispaced 50 val-ues between these) encompasses the physiological value which should be around − m/s. As shown in Figures 3b and 4b, the FIC increases significantly with P until it reaches a plateau : indeed, as diffusion becomes more rapid than ac-tive transport (that is, pumping by Na + / K + -ATPase), the permeability ceasesto be rate-limiting. As shown by comparing Figures 3b and 4b, the FIC isstrongly determined by the pump rate V m, : if P ∈ [10 − , − ] , Na + concen-tration along the lumen increases by less than if V m, = 2 πr ,e − , andmay reach if V m, = 2 πr ,e − . This raise is expected since concentra-tion differences are generated by active transport ; the higher the rate of activetransport, the more significant these differences. Conversely, in the absence ofpumping, concentrations would equilibrate everywhere. In regards to the ax-ial gradient, the interesting numerical results are in Figure (4a) and (4b). Weobserve that the axial gradient increases with increasing permeability when thelatter is varied within the chosen range. Therefore this indicates that takinginto account the epithelial layer in the model has a significant influence on theaxial concentration gradient.Moreover, numerical results also confirm that : ¯ u = ¯ u = u < ¯ q < ¯ u as reported in Lemma 2.1. We recall that we assume a constant water flow α which allows us to deduce ¯ u = ¯ u . As noted above, the descending limb is infact very permeable to water and α should decrease significantly in this tubule,such that ¯ u differs from ¯ u , except at the hairpin turn at x = L . On the otherhand, the last equation of system (8) implies that ¯ q < ¯ u , meaning that theconcentration of Na + is lower in the epithelial cell than in the interstitium, asobserved in vivo, [1].With the expression of G in (5), equation (10) reads ¯ q (0) + V m, k (cid:18) ¯ q (0) K M, + ¯ q (0) (cid:19) = ¯ u b . (13)In order to better understand the behaviour of the axial concentration gradientshown in Figures (3b), (4b), we compute the derivative of (13) with respect tothe parameter V m and with respect to k respectively: ∂ ¯ q (0) ∂V m + 1 k G (cid:48) (¯ q (0)) ∂ ¯ q (0) ∂V m + 1 k (cid:0) ¯ q (0) K M, + ¯ q (0) (cid:1) = 0 ,∂ ¯ q (0) ∂k + 1 k G (cid:48) (¯ q (0)) ∂ ¯ q (0) ∂k − k G (¯ q (0)) = 0 . Then, we get ∂ ¯ q (0) ∂V m = − k k G (cid:48) (¯ q (0)) (cid:0) ¯ q (0) k M + ¯ q (0) (cid:1) ≤ ,∂ ¯ q (0) ∂k = k G (¯ q (0))1 + k G (cid:48) (¯ q (0)) ≥ , because G is a monotone non-decreasing function and q (0) is positive.9e observe from numerical results (see Figures 2a, 2c, 3a, and 4a) that thegradient of u is almost constant. Thus, we may make the approximation ∂ x ¯ u ∼ ∂ x ¯ u (0) = G (¯ q (0)) α . (14)Its derivatives with respect to V m, and k are both non negative : ∂∂k [ ∂ x ¯ u ] ∼ G (cid:48) (¯ q (0)) α (cid:16) G (¯ q (0)) k k G (cid:48) (¯ q (0)) (cid:17) ≥ ,∂∂V m [ ∂ x ¯ u ] ∼ α (cid:16) ¯ q (0) K M + ¯ q (0) (cid:17) (cid:16)
11 + k G (cid:48) (¯ q (0)) (cid:17) ≥ . It means that the axial concentration gradient is an increasing function bothwith respect to the rate of active transport V m, and to the permeability k . (a) Axial concentrations in the lumenfor different values of permeability. (b) Percentage of concentration gradi-ent as a function of permeability. Figure 4: Concentration profiles for V m, = 2 πr ,e · − [ mol.m − .s − ] (a) Percentage of concentration gradi-ent 2D in tubules (b) Percentage of concentration gradi-ent projection Figure 5: Percentage of concentration gradient 2D with range V m, ∈ πr ,e · (10 − , − ) [ mol.m − .s − ] ( x − axis) and P ∈ · (10 − , − ) ( y − axis)Indeed in Fig. 5, we perform numerical simulations varying both P and V m, and observe that the ratio (12) increases monotonically with respect toboth parameters. 10 .3 Limiting cases: k → ∞ , and k → Numerical results show that above a certain high value of permeability, theepithelial concentration in tubule 2 seems to reach a plateau (see Figures 3band 4b). There are two different regimes : one for large values of permeabilities,one for small values of permeabilities, and a fast transition between them.In the large permeabilities asymptotic, we may approximate system (8) bythe limiting model k = + ∞ . In this case, (11) reduces to ¯ u = ¯ q , ¯ q = ¯ q , ¯ u = G (¯ q ) K + ¯ q . for all x ∈ (0 , L ) . This is understandable from a formal point of view, alsotaking into account computations in Appendix (A) for the stationary system(7). In this case, the gradient concentration is directly proportional to V m, : ∂ x ¯ q = G (¯ q ) α + αk G (cid:48) (¯ q ) −→ k → + ∞ G (¯ q ) α = ∂ x ¯ u. From (9), the Cauchy problem reduces to ∂ x ¯ q ( x ) = G (¯ q ) α , ¯ q (0) = ¯ u b . Additionally, it is clear that the higher pump value, the more the FIC willincrease, as observed in Figure 4b.On the other hand for small values of permeability, we obtain formally ∂ x ¯ q −→ k → , ∂ x ¯ u = G (¯ q ) α . Therefore, in a neighbourhood of the value P ∼ − , the concentration gradienttends to be constant and for this reason we notice a plateau. This section is devoted to the main mathematical result of this paper concerningthe long time asymptotics of solutions to (6) towards solutions to the station-ary system (7) as time goes to + ∞ . We first state the main result and theassumptions needed. Then, we introduce eigenelements of an auxiliary linearsystem and its dual problem. Using these auxiliary functions, we are able toshow the convergence when the time variable goes to + ∞ . A similar approachwas considered in [25] following ideas from [19]. Before stating the main result, we provide assumptions on the initial and bound-ary data.
Assumption 3.1.
We assume that the initial solute concentrations are non-negative and uniformly bounded in L ∞ (0 , L ) and in the total variation : ≤ u , u , q , q , u ∈ BV (0 , L ) ∩ L ∞ (0 , L ) . (15)11 ssumption 3.2. The boundary condition of system (6) is such that ≤ u b ∈ L ∞ ( R + ) ∩ L loc ( R + ) , lim t → + ∞ | u b − ¯ u b | = 0 , (16) for some constant ¯ u b > . BV is the space of functions with bounded variation, we notice that suchfunctions have a trace on the boundary (see e.g. [5]); hence the boundarycondition u ( t, L ) = u ( t, L ) is well-defined. Assumption 3.3.
Regularity and boundedness of G . We assume that the non-linear function modelling active transport in the ascending limb (tube 2) is abounded and Lipschitz-continuous function on R + : ∀ x ∈ R + , ≤ G ( q ) ≤ (cid:107) G (cid:107) ∞ , ≤ G (cid:48) ( q ) ≤ (cid:107) G (cid:48) (cid:107) ∞ . (17)We notice that G defined by (5) satisfies straightforwardly (17).We now state the main result. Theorem 3.1 (Long time behaviour) . Under Assumptions 3.1, 3.2 and 3.3,the solution to the dynamical problem (6) denoted by u ( t, x ) = ( u , u , q , q , u ) converges as time t goes to + ∞ towards ¯u ( x ) , the unique solution to the sta-tionary problem (7) , in the following sense lim t → + ∞ (cid:107) u ( t ) − ¯u (cid:107) L (Φ) = 0 , with the space L (Φ) = (cid:110) u : [0 , L ] → R ; (cid:107) u (cid:107) L (Φ) := (cid:90) L | u ( x ) | · Φ( x ) dx < ∞ (cid:111) , where Φ = ( ϕ , ϕ , φ , φ , ϕ ) is defined in Proposition 3.1 below.Moreover, if we assume that there exist µ > and C such that | u b ( t ) − ¯ u b | ≤ C e − µ t for all t > , then there exist µ > and C > such that we have theconvergence with an exponential rate (cid:107) u ( t ) − ¯u (cid:107) L (Φ) ≤ Ce − µt . (18)The scalar product used in the latter claim means : (cid:90) L | u ( x ) | · Φ( x ) dx = (cid:90) L (cid:0) | u | ϕ ( x ) + | u | ϕ ( x ) + | q | φ ( x ) + | q | φ ( x ) + | u | ϕ ( x ) (cid:1) dx. The definition of the left eigenvector Φ and its role are given hereafter. In order to study the long time asymptotics of the time dependent system (6),we consider the eigen-problem associated with a specific linear system [20, 25].This system is, in some sort, a linearized version of the stationary system (7)where the derivative of the non-linearity is replaced by a constant g . When12hese eigenelements ( λ, U , Φ) exist, the asymptotic growth rate in time for asolution u of (6) is given by the first positive eigenvalue λ and the asymptoticshape is given by the corresponding eigenfunction U .Let us introduce the eigenelements of an auxiliary stationary linear system ∂ x U = λU + k ( Q − U ) − ∂ x U = λU + k ( Q − U )0 = λQ + k ( U − Q ) + K ( U − Q )0 = λQ + k ( U − Q ) − gQ λU + K ( Q − U ) + gQ , (19)where g is a positive constant which will be fixed later. This system is comple-mented with boundary and normalization conditions : U (0) = 0 , U ( L ) = U ( L ) , (cid:90) L ( U + U + Q + Q + U ) dx = 1 . (20)We also consider the related dual system : − ∂ x ϕ = λϕ + k ( φ − ϕ ) ∂ x ϕ = λϕ + k ( φ − ϕ )0 = λφ + k ( ϕ − φ ) + K ( ϕ − φ )0 = λφ + k ( ϕ − φ ) + g ( ϕ − φ )0 = λϕ + K ( φ − ϕ ) , (21)with following conditions : ϕ ( L ) = ϕ ( L ) , ϕ (0) = 0 , (cid:90) L ( U ϕ + U ϕ + Q φ + Q φ + U ϕ ) dx = 1 . (22)For a given λ , the function U := ( U , U , Q , Q , U ) is the right eigenvectorsolving (19), while Φ := ( ϕ , ϕ , φ , φ , ϕ ) is the left one, associated with theadjoint operator. The following result shows the existence of a positive eigen-value and some properties of eigenelements. We underline that in order to makethe proof easier, we consider the case k = k = k but the same result could beextended to the more general case where k (cid:54) = k . Proposition 3.1.
Let g > be a constant. There exists a unique ( λ, U , Φ) with λ ∈ (0 , λ − ) solution to the eigenproblem (19) – (22) , where λ − = (2 K + k ) − (cid:112) K + k . Moreover, we have U ( x ) > , Φ( x ) > on (0 , L ) and φ < ϕ . In order to prove this result, we will divide the proof in two steps : Lemmas3.1 and 3.2 respectively. Proposition 3.1 is a direct consequence of these twoLemmas. We start with the direct problem :
Lemma 3.1 (The direct problem) . There exists a unique λ > such that the di-rect problem (19) - (20) admits a unique positive solution U = ( U , U , Q , Q , U ) on (0 , L ) , and < λ < λ − . roof. Summing all equations in (19) we find that : U (cid:48) − U (cid:48) = λ ( U + U + Q + Q + U ) . (23)Integrating with respect to x and using condition (20), we obtain U (0) = λ .By the fourth equation in (19), we find directly: Q ( x ) = kU ( x ) k + g − λ = U ( x )1 + k ( g − λ ) . (24)Putting this expression into the second equation in (19), we find − U (cid:48) = U (cid:18) λ + λ − g k ( g − λ ) (cid:19) . Solving the latter equation, we deduce that U ( x ) = U (0) e − λx + (cid:82) x − λ + g
1+ 1 k ( g − λ ) dy = λe ( − λ + η ( λ )) x ; with η ( λ ) := − λ + g k ( g − λ ) . (25)Using the fifth equation of system (19) we recover U ( x ) = K K − λ Q ( x ) + gK − λ Q ( x ) . We inject this into the third equation to obtain Q ( x ) (cid:18) k − λ − K λK − λ (cid:19) = gK K − λ Q ( x ) + kU ( x ) . Thanks to (24) we write also: Q ( x ) (cid:18) k − λ − K λK − λ (cid:19) = K gK − λ k ( g − λ )) U ( x ) + kU ( x ) . Taking into account the first equation of system (19), we obtain : U (cid:48) ( x ) = c λ U ( x ) + k λ g k ( g − λ ) U ( x ) , (26)where we simplify notations by introducing : k λ := k K K − λ k − λ − K λK − λ , c λ := λ + k ( λ + K λK − λ ) k − λ − K λK − λ . (27)The denominator k − λ − K λK − λ vanishes for λ ± = (2 K + k ) ± (cid:112) K + k . Obviously lim λ → λ − k λ = + ∞ and we also have that < λ − < min( K , k ) .14ow we solve directly the ODE (26) with its initial condition, we get U ( x ) = λgk λ k ( g − λ ) e c λ x − e ( η ( λ ) − λ ) x c λ + λ − η ( λ ) . We are looking for a λ > such that boundary condition U ( L ) = U ( L ) issatisfied, in other words U ( L ) U ( L ) = 1 , namely F ( λ ) := gk λ k ( g − λ ) (cid:18) e ( c λ + λ − η ( λ )) L − c λ + λ − η ( λ ) (cid:19) = 1 , (28)where we recall that k λ , c λ are defined in (27) and η ( λ ) in (25). We remarkimmediately that for λ = 0 in (27), we have k = 1 , c = 0 . Then, F (0) = 1 − exp (cid:16) − gL gk (cid:17) < . We notice that for k λ , c λ > , F ( λ ) is a continuous increasing function withrespect to λ since the product of increasing and positive functions is still in-creasing (see Appendix (B.1) for more details). Moreover lim λ → λ − F ( λ ) = + ∞ .Then it exists a unique λ ∈ (0 , λ − ) such that F ( λ ) = 1 . Moreover, for < λ < λ − < min( k, K ) , the functions U , U , Q , Q , U are positive on [0 , L ] . Lemma 3.2 (The dual problem) . Let λ and U be as in Lemma (3.1) . Then,there exists Φ := ( ϕ , ϕ , φ , φ , ϕ ) , the unique solution of dual problem (21) – (22) with ϕ , ϕ , φ , φ , ϕ > . Moreover, we have φ < ϕ .Proof. By the fifth equation of system (21) we have directly : ϕ = K K − λ φ . Replacing this expression in the third equation we obtain ( k − λ − K λK − λ ) φ = kϕ . Then, ϕ ( x ) = k K K − λ k − λ − K λK − λ ϕ ( x ) = k λ ϕ ( x ) , where k λ is defined in (27). Using the first equation of (21), we have − ϕ (cid:48) = ϕ (cid:32) λ + k (cid:32) λ + K λK − λ k − λ − K λK − λ (cid:33)(cid:33) . Integrating, we obtain ϕ ( x ) = ϕ (0) e − λx e − βx , β = β λ = λk (2 K − λ ) λ − K λ − λk + K k . (29)15e easily check that β > if < λ < λ − < K .As shown in details in the Appendix B.2, for all x ∈ (0 , L ) , ( U ϕ ) (cid:48) − ( U ϕ ) (cid:48) = 0 . Integrating, we get U ( x ) ϕ ( x ) − U ( x ) ϕ ( x ) = U (0) ϕ (0) − U (0) ϕ (0) = 0 , thanks to boundary conditions U (0) = 0 and ϕ (0) = 0 .(Notice also that taking x = L in this latter relation, and using the boundarycondition U ( L ) = U ( L ) (cid:54) = 0 , we recover ϕ ( L ) = ϕ ( L ) .) Therefore, we get ϕ ( x ) = U ( x ) U ( x ) ϕ ( x ) , ∀ x ∈ [0 , L ] . (30)Using the fourth equation in (21) and thanks to (30), we obtain λφ + k U U ϕ − kφ + gϕ (cid:32) K K − λ · kk − λ − K λK − λ (cid:33) , which allows to compute φ : φ ( x ) = k U ( x ) U ( x ) ϕ ( x ) k − λ + g + gk − λ + g k λ ϕ ( x ) . Each function depends on the first component of Φ , i.e. ϕ ( x ) , and to sum up,the following relation has been obtained: ϕ ( x ) = ϕ (0) e − λx e − βx ϕ ( x ) = U ( x ) U ( x ) ϕ ( x ) φ ( x ) = ϕ ( x ) (cid:18) kk − λ − K λK − λ (cid:19) φ ( x ) = k − λ + g (cid:16) k U ( x ) U ( x ) + gk λ (cid:17) ϕ ( x ) ϕ ( x ) = k λ ϕ ( x ) , (31)where k λ , β are defined in (27) and (29). Hence the sign of Φ depends on thesign of ϕ (0) , the other quantities and constants being positive for λ ∈ (0 , λ − ) and g > by assumption. Then, we use the normalization condition (22) and(31) in order to show the positivity of ϕ (0) . It implies that ϕ (0) (cid:90) L e − λx e − βx (cid:34) U ( x ) + Q ( x ) (cid:32) kk − λ − K λK − λ (cid:33) ++ Q ( x ) k + g + λ (cid:18) k U ( x ) U ( x ) + gk λ (cid:19) + k λ U ( x ) (cid:21) dx = 1 . The integral on the left hand side is positive, thanks to properties of functionspreviously defined. Given that ϕ (0) is constant and all other quantities positive,we can conclude that ϕ (0) > .We are left to prove that the quantity φ − ϕ is negative. Using (31), werewrite : φ − ϕ = kk λ ϕ ( x ) k − λ + g (cid:18) k λ U U + λk − (cid:19) . U and U , we have φ − ϕ = k λ ϕ ( x )1 + k ( g − λ ) (cid:82) x g k ( g − λ ) e − c λ ( y − x ) e ( − λ + η ( λ )) y dye ( − λ + η ( λ )) x + λk − = k λ ϕ ( x )1 + k ( g − λ ) (cid:20) g k ( g − λ ) (cid:104) − e − c λ x − λx + η ( λ ) x c λ + λ − η (cid:105) + λk − (cid:21) , where we recall the notation η ( λ ) = − λ + g k ( g − λ ) . We set H ( x ) := g k ( g − λ ) (cid:104) − e − c λ x − λx + η ( λ ) x c λ + λ − η ( λ ) (cid:105) . (32)We have φ − ϕ < ⇐⇒ H ( x ) + λk − < . We observe that H (0) = 0 and H ( L ) = k λ thanks to (28). Moreover, H ( L ) < − λk for λ ∈ (0 , λ − ) . Indeed, we have λ < k − kk λ , which holds if and only if λ − K λ − kλK < which is true on (0 , λ − ) , since λ − < k by definition. Moreover, it is clear that H is an increasing function on [0 , L ] for λ ∈ (0 , λ − ) . Then H ( x ) ≤ H ( L ) < − λk . This concludes the proof. Now we are ready to prove Theorem 3.1. We set d i ( t, x ) := | u i ( t, x ) − ¯ u i ( x ) | i = 0 , , and δ j := | q j ( t, x ) − ¯ q j ( x ) | , j = 1 , with ¯ u i , ¯ q i satisfying (7) and u i , q i solving (6).We subtract component-wise (6) to (7). Then we multiply each of the en-tries by sign ( u i − ¯ u i ) or sign ( q j − ¯ q j ) respectively. We obtain the followinginequalities : a ∂ t d + α∂ x d ≤ k ( δ − d ) a ∂ t d − α∂ x d ≤ k ( δ − d ) a ∂ t δ ≤ k ( d − δ ) + K ( d − δ ) a ∂ t δ ≤ k ( δ − d ) − ˆ Ga ∂ t d ≤ K ( δ − d ) + ˆ G, (33)with ˆ G := | G ( q ) − G (¯ q ) | . We have used also the monotonicity of G (see (17)).We set M ( t ) := (cid:90) L ( a d ϕ + a d ϕ + a δ φ + a δ φ + a d ϕ ) dx. Multiplying each equation of (33) by the corresponding dual function ϕ i , φ i ,17dding all equations and integrating with respect to x , we obtain : ddt M ( t ) ≤ (cid:90) L (cid:16) k ( δ − d ) ϕ + k ( δ − d ) ϕ + k ( d − δ ) φ + K ( d − δ ) φ + k ( δ − d ) φ − ˆ Gφ + K ( δ − d ) ϕ + ˆ Gϕ (cid:17) dx + α (cid:90) L ( ∂ x d ϕ − ∂ x d ϕ ) dx. Integrating by parts the last integral and using the dual system (21), we cansimplify the latter inequality into ddt M ( t ) ≤ − λ (cid:90) L ( d ϕ + d ϕ + δ φ + δ φ + d ϕ ) dx + d ( L ) ϕ ( L ) − d ( L ) ϕ ( L ) − d (0) ϕ (0)+ d (0) ϕ (0)+ (cid:90) L ( gδ − ˆ G )( φ − ϕ ) dx. Using the normalization conditions in (20) and in (22), we obtain ddt M ( t ) ≤ − λ max { a , a , a , a , a } M ( t )+ d ( t, ϕ (0)+ (cid:90) L ( gδ − ˆ G )( φ − ϕ ) dx. To simplify the notation we set ¯ λ = − λ max { a ,a ,a ,a ,a } . Since G is Lipschitz-continuous and by assumption (17), ˆ G ≤ gδ with g = (cid:107) G (cid:48) (cid:107) ∞ . With this choiceof g , we apply Proposition 3.1 and deduce that the quantity ( φ − ϕ ) is negative.Then, ddt M ( t ) + ¯ λM ( t ) ≤ d ( t, ϕ (0) . Thanks to (16) and applying Gronwall’s lemma, we conclude that M ( t ) ≤ M (0) e − ¯ λt + ϕ (0) (cid:90) t d ( s, e ¯ λ ( s − t ) ds. (34)Moreover, from (16), we have d ( s,
0) = | u b ( t ) − ¯ u b | → as t → + ∞ . Then, forevery ε > , it exists ¯ t > such that d ( s, < ε for each s > ¯ t . Then for every t ≥ ¯ t , we have (cid:90) t d ( s, e ¯ λ ( s − t ) ds ≤ (cid:90) ¯ t d ( s, e ¯ λ ( s − t ) ds + ε (cid:90) t ¯ t e ¯ λ ( s − t ) ds ≤ e ¯ λ (¯ t − t ) (cid:90) ¯ t d ( s, ds + ε ¯ λ . The first term of the right hand side is arbitrarily small at t goes to + ∞ . Hence,we have proved that for any ε > there exists τ large enough such that for every t ≥ τ , M ( t ) ≤ M (0) e − ¯ λt + Cε.
Since M ( t ) = (cid:107) u ( t ) − ¯ u (cid:107) L (Φ) , it proves the convergence as stated in Theorem3.1.Finally, if we assume that there exist positive constants µ and C such that | u b ( t ) − ¯ u b | ≤ C e − µ t , then from (34) we deduce M ( t ) ≤ M (0) e − ¯ λt + C ϕ (0) e − µ t − e − ¯ λt ¯ λ − µ ≤ Ce − min { ¯ λ,µ } t . Conclusion and outlook
In this study we present a model describing the transport of sodium in a simpli-fied version of the loop of Henle in a kidney nephron. From a modelling pointof view, it seems important to take into account the epithelium in the counter-current tubular architecture since we observe that it may affect strongly thesolute concentration profiles for a particular range of permeabilities.The main limitation of the model is to not consider the re-absorption of waterin descending limb. Indeed, in Section 2, we study the steady state solutionand the assumption of a constant rate α and the boundary conditions lead to ¯ u ( x ) = ¯ u ( x ) , i.e. the luminal concentrations of sodium are the same in bothtubules for every x ∈ (0 , L ) . Conversely, in vivo, the concentrations in lumen 1and 2 are different due to the constitutive differences between the segments andpresence of membrane channel proteins, for example the aquaporins. The thindescending limb of Henle’s loop has low permeability to ions and urea, whilebeing highly permeable to water. The thick ascending limb is impermeable towater, but it is permeable to ions. For this reason, a possible extension of themodel shall assume that α is not constant but space-dependent. A first stepcould be, for instance, to take two different values of α for the first and secondequation of the model (6), α and α . From the mathematical viewpoint, thischoice slightly changes the structure of the hyperbolic system : for example,conservation of certain quantities should not be that easy to prove.Furthermore, this assumption about α has a relevant influence on other fac-tors. As already pointed out, the relation between ¯ q and ¯ u is biologicallycorrect and consistent, this means that in vivo the concentration of Na + in theepithelial cell (intracellular) is lower than in interstitium. The intracellular con-centrations (epithelium, ¯ q and ¯ q ) are usually of the order of mM whereasthe extracellular ones (therefore in the lumen and in the interstitium) are of theorder of mM, (see [6], page 692).There are also other types of source terms in the interstitium that could beadded, accounting for blood vessels and/or collecting ducts. In this case, the lastequation (6e) of the dynamic system should include a term that accounts for in-terstitium concentration storage or accumulation and for secretion-reabsorptionof water and solutes, but the impact of adding such complex mechanisms in themodel remains to be assessed.In this study, we focused our attention on the axial concentration gradientand the FIC, previously defined in Section (2), which are significant factors inthe urinary concentration mechanism, [11, 12]. The axial gradient is an im-portant determinant of urinary concentration capacity. When water intake islimited, mammals can conserve water in body fluids by excreting solutes in areduced volume of water, that is, by producing a concentrated urine. The thickascending limb plays an essential role in urine concentration and dilution, [21]:the active reabsorption of sodium without parallel reabsorption of water gen-erates an interstitial concentration gradient in the outer medulla that in turndrives water reabsorption by the collecting ducts, thereby regulating the con-centration of final urine.In summary, our model confirms that the active trans-epithelial transport of Nafrom the ascending limbs into the surrounding environment is able to generatean osmolality gradient. Our model indicates that explicitly accounting for the2-step transport across the epithelium significantly impacts the axial concen-19ration gradient within the physiological range of parameters values consideredhere. Thus, representing the epithelial layer as two membrane in series, asopposed to a single-barrier representation, may provide a more accurate under-standing of the forces that contribute to the urinary concentrating mechanism. A Large permeability asymptotic
In this section we consider the case where the permeability between the lumenand the epithelium is large, i.e. when P i → ∞ , with i = 1 , in the definition ofconstants k and k . For this purpose, we set k = k = k = ε and we let ε goto . Physically, this means fusing the epithelial layer with the lumen.Rewriting (6) in this perspective gives ∂ t u ε + α∂ x u ε = 1 ε ( q ε − u ε ) (35a) ∂ t u ε − α∂ x u ε = 1 ε ( q ε − u ε ) (35b) ∂ t q ε = 1 ε ( u ε − q ε ) + K ( u ε − q ε ) (35c) ∂ t q ε = 1 ε ( u ε − q ε ) − G ( q ε ) (35d) ∂ t u ε = K ( q ε − u ε ) + G ( q ε ) . (35e)We expect the concentrations u ε and q ε to converge to the same quantity. Thesame happens for u ε → u and q ε → u . We denote u , respectively u , thelimit of u ε and q ε , respectively u ε and q ε . Adding (35a) to (35c) and (35b) to(35d), we obtain ∂ t u ε + ∂ t q ε + α∂ x u ε = K ( u ε − q ε ) ∂ t u ε + ∂ t q ε − α∂ x u ε = − G ( q ε ) . Passing formally to the limit ε → , we arrive at ∂ t u + α∂ x u = K ( u − u ) (36) ∂ t u − α∂ x u = − G ( u ) , (37)coupled to the equation for the concentration in the interstitium obtained bypassing into the limit in equation (35e) ∂ t u = K ( u − u ) + G ( u ) . (38)The equations (36), (37), (38) describe the same concentration dynamics ina system without epithelium, previously studied in [24] and [25]. The formalcomputation above shows that this × system may be considered as a goodapproximation of the larger system (6) for large permeabilities.Such a convergence result may be proved rigorously and it is investigatedin [15]. It relies on specific a priori estimates and the introduction of an initiallayer. 20 Technical results
B.1 Function F ( λ ) In this subsection we prove the monotonicity of the function F ( λ ) which appearsin the proof of Lemma 3.1. First let’s recall it F ( λ ) := gk λ k ( g − λ ) (cid:18) e ( c λ + λ − η ( λ )) L − c λ + λ − η ( λ ) (cid:19) . (39) Lemma B.1.
The function F defined by (39) is monotonically increasing on (0 , λ − ) .Proof. The product of positive increasing functions is increasing.• λ (cid:55)→ k λ = K kλ − K λ − kλ + kK is a positive and increasing function if λ ∈ (0 , λ − ) . Indeed ∂k λ ∂λ = − λkK +2 kK + k K ( λ − K λ − kλ + kK ) is positive for < λ < K + k and λ − < K by definition.• We set f ( λ ) := g k ( g − λ ) ; if λ < g + k the function f is positive since g > by hypothesis and it is also increasing since ∂∂ λ f ( λ ) = gk (1+ k ( g ( y ) − λ )) > , and λ − ≤ k .• The function x (cid:55)→ e x − x is increasing on R + and the function λ (cid:55)→ c λ + λ − η ( λ ) is increasing on (0 , λ ) . Indeed, we have straightforwardly c λ + λ − η ( λ ) = 2 λ + 2 k + k k − λ − K λK − λ + k k + g − λ . B.2 Relation between direct and dual system
We recall the eigenelements problem written as below: ∂ x U ( x ) − ∂ x U ( x )000 = λ U ( x ) + A U ( x ); U ( x ) = U U Q Q U (40) − ∂ x ϕ ( x ) ∂ x ϕ ( x )000 = λ Φ( x ) + t A Φ( x ); Φ( x ) = ϕ ϕ φ φ ϕ (41)with related matrix defined by A = − k k − k k k − k − K K k − k − g
00 0 K g − K . t Φ , we deduce ϕ ∂ x U − ϕ ∂ x U = λ t Φ U + t Φ A U . Taking the transpose of (41) and multiplying on the right by U , we also have − ∂ x ϕ U + ∂ x ϕ U = λ t Φ U + t Φ A U . As a consequence, we deduce the relation ( U ϕ ) (cid:48) − ( U ϕ ) (cid:48) = 0 , ∀ x ∈ [0 , L ] . (42)Since U ( L ) = U ( L ) in (20) and by initial conditions U (0) = 0 , ϕ (0) = 0 ,then also ϕ ( L ) = ϕ ( L ) , as set in (22). It means that ( U ϕ ) = ( U ϕ ) ∀ x ∈ [0 , L ] . Thanks to this relation, we can consider in our previous computation: ϕ ( x ) = U ( x ) U ( x ) ϕ ( x ) , ∀ x ∈ [0 , L ] . References [1] J.C. Atherton, R. Green, S. Thomas, J.A. Wood,
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