## Abstract

The mathematical model of rat proximal tubule has been extended to include calculation of microvillous torque and to incorporate torque-dependent solute transport in a compliant tubule. The torque calculation follows that of Du Z, Yan Q, Duan Y, Weinbaum S, Weinstein AM, and Wang T (*Am J Physiol* 290: F289–F296, 2006). In the model calculations, torque-dependent scaling of luminal membrane transporter density [either as an ensemble or just type 3 Na^{+}/H^{+} exchanger (NHE3) alone] had a relatively small impact on overall Na^{+} reabsorption and could produce a lethal derangement of cell volume; coordinated regulation of luminal and peritubular transporters was required to represent the overall impact of luminal flow on Na^{+} reabsorption. When the magnitude of torque-dependent Na^{+} reabsorption in the model agrees with that observed in mouse proximal tubules, the model tubule shows nearly perfect perfusion-absorption balance at high luminal perfusion rates, but enhanced sensitivity of reabsorption at low flow. With a slightly lower coefficient for torque-sensitive transporter insertion, perfusion-absorption balance in the model tubule is closer to observations in the rat over a broader range of inlet flows. In simulation of hyperglycemia, torque-dependent transport attenuated the diuretic effect and brought the model tubule into closer agreement with experimental observation in the rat. The model was also extended to represent finite rates of hydration and dehydration of CO_{2} and H_{2}CO_{3}. With carbonic anhydrase inhibition, torque-dependent transport blunted the diuretic effect and enhanced the shift from paracellular to transcellular NaCl reabsorption. The new features of this model tubule are an important step toward simulation of glomerulotubular balance.

- glomerulotubular balance
- microvilli
- sodium reabsorption
- hyperglycemia
- carbonic anhydrase

in response to variations in glomerular filtration, proximal tubule reabsorption varies proportionally (33), and this glomerulotubular balance derives from both peritubular capillary and luminal factors (14, 18, 46). Perhaps the most important luminal factor is a direct effect of axial flow velocity on reabsorption (32, 56), and along with its impact on sodium reabsorption, luminal flow has been found to influence the transport of glucose (22), bicarbonate (1, 8, 25), and chloride (15, 57). Insight into the mechanism underlying flow-dependent transport came with the demonstration that increases in axial flow velocity recruit new Na^{+}/H^{+} transporters into the luminal membrane (26, 30). With respect to the afferent signal, Guo et al. (16) proposed that the proximal tubule brush-border microvilli serve as the flow sensor and that the drag force on each microvillus produced torque on its actin filament core that was transmitted to the underlying cytoskeleton. This hypothesis received support from the experiments of Du et al. (11, 12), who studied flow-dependent transport in mouse proximal tubules perfused in vitro and found that over a fivefold variation of luminal perfusion rate, there was a predicted twofold variation in microvillous torque, which scaled identically with Na^{+} and HCO_{3}^{−} reabsorption. Moreover, luminal flow impacted H^{+} secretion via both the Na^{+}/H^{+} antiporter and the H^{+}-ATPase; there was no discernible change in paracellular parameters [tubule HCO_{3}^{−} permeability or transepithelial potential difference (PD)]. The number of different luminal membrane transporters impacted by microvillous torque has not been delineated, nor has the question of a flow effect on peritubular membrane transporters been addressed in experimental studies.

The proximal tubule of the rat has been the most intensively modeled nephron segment (48). These models have indicated that paracellular fluxes of Na^{+} and Cl^{−} constitute an important fraction of their overall reabsorption; that these fluxes assume greater significance later along the tubule (where transepithelial Cl^{−} and HCO_{3}^{−} gradients are well developed); and that both diffusive and convective components of paracellular Cl^{−} flux are substantial (45). Nevertheless, these models have supported the primacy of luminal membrane Na^{+}/H^{+} exchange in determining the rate of overall proximal Na^{+} reabsorption (47). Even with the inclusion of formate, the impact of luminal membrane Cl^{−}/ antiporter density on proximal Na^{+} and Cl^{−} fluxes was small relative to the effect that changes in NHE3 activity had on these fluxes. More recently, the proximal tubule models have been used to examine the coordination of luminal and peritubular transport pathways, required to preserve the integrity of cell volume and composition during variations on Na^{+} reabsorption (51, 52). In the present work, the proximal tubule model has been extended to allow microvillous torque to modulate transporter densities. Since microvillous torque depends on both axial flow rate and luminal diameter, proximal tubule compliance has also been represented. The model has also been extended to include CO_{2} and H_{2}CO_{3} as distinct species to permit simulation of proximal tubule diuresis using a carbonic anhydrase (CA) inhibitor. The model calculations indicate that flow-dependent reabsorption requires a coordinated impact on luminal and peritubular transporter densities. The model also predicts that inhibition of luminal membrane-bound CA can actually increase the transcellular component of Na^{+} reabsorption, due to the diuretic effect to increase late proximal microvillous torque, and due to the impact of luminal acid disequilibrium pH to increase Cl^{−}/ exchange.

## MODEL FORMULATION

This model proximal tubule extends prior versions by the inclusion of CO_{2} and H_{2}CO_{3} and their conservation equations, and thus brings the proximal convoluted tubule (PCT) model structure into conformity with the distal nephron segmental models (e.g., Ref. 53). The motivation for this extension was to be able to simulate CA inhibitors as proximal diuretics. The model is formulated both as an epithelium, with specified luminal and peritubular conditions, and as a tubule, in which luminal concentrations vary axially. Figure 1 displays both configurations, in which cellular and lateral intercellular (LIS) compartments line the tubule lumen. Within each compartment, the concentration of species *i* is designated C_{α}(*i*), where α is lumen (M), interspace (E), cell (I), or peritubular solution (S). Within the epithelium, the volume flux across membrane αβ is denoted *J*_{vαβ} (ml·s^{−1}·cm^{−2}), and the flux of solute *i* is denoted *J*_{αβ}(*i*) (mmol·s^{−1}·cm^{−2}), where αβ may refer to luminal cell membrane (MI), tight junction (ME), lateral cell membrane (IE), basal cell membrane (IS), or interspace basement membrane (ES). Along the tubule lumen, axial volume flow is designated *F*_{vM} (ml/s) and solute flows *F*_{M}(*i*) (mmol/s). There are 15 permeant model solutes, namely, Na^{+}, K^{+}, Cl^{−}, HCO_{3}^{−}, CO_{2}, H_{2}CO_{3}, HPO_{4}^{2−}, H_{2}PO_{4}^{−}, urea, NH_{3}, NH_{4}^{+}, H^{+}, , H_{2}CO_{2}, and glucose, as well as two impermeant species within the cells, a nonreactive anion and a cytosolic buffer.

To formulate the equations of mass conservation with multiple reacting solutes, consider first an expression for the generation of volume or of each solute species within each model compartment. Within a cell or interspace, volume generation [*s _{α}*(v)] and the generation of solute

*i*[

*s*(

_{α}*i*)] is equal to its net export plus its accumulation (1) (2) (3) (4) where V

_{α}is the compartment volume (cm

^{3}/cm

^{2}). Within the tubule lumen, volume or solute generation is appreciated as an increase in axial flux, as transport into the epithelium, or as local accumulation (5) (6) where

*B*

_{M}is the tubule circumference, and

*A*

_{M}is the tubule cross-sectional area. Of note,

*s*

_{I}and

*s*

_{E}have the dimension ml·s

^{−1}·cm

^{−2}or mmol·s

^{−1}·cm

^{−2}, while

*s*

_{M}has the dimension ml·s

^{−1}·cm

^{−1}or mmol·s

^{−1}·cm

^{−1}. With this notation, the equations of mass conservation for volume and the nonreacting species (Na

^{+}, K

^{+}, Cl

^{−}, urea, and glucose) are written (7) (8) where α = E, I, or M. For the phosphate and formate buffer pairs, there is conservation of total buffer (9) (10) Similar equations apply to the ammonia pair within tubule lumen and lateral interspace (α = M or E), (11) however, within the cell, there is ammoniagenesis at rate

*Q*

_{I}(NH

_{4}

^{+}) (49) (12) Although peritubular Pco

_{2}is specified, the CO

_{2}concentrations of cells, interspace, and lumen are model variables. The relevant reactions are (13) where dissociation of H

_{2}CO

_{3}is rapid, and assumed to be at equilibrium. Since HCO

_{3}

^{−}and H

_{2}CO

_{3}are interconverted, mass conservation requires (14) for α = I or E, while for the tubule lumen, (15) In each compartment (α = I, E, or M), conservation of total CO

_{2}is expressed (16) Corresponding to conservation of protons is the equation for conservation of charge for all the buffer reactions (17) where

*z*

_{i}is the valence of species

*i*. In this model, conservation of charge for the buffer reactions (

*Eq. 17*) takes the form (18) for α = E or M, while for α = I, and Buf

^{−}the impermeant cytosolic buffer (19) The solute equations are completed with the chemical equilibria of the buffer pairs: HPO

_{4}

^{2−}:H

_{2}PO

_{4}

^{−}, NH

_{3}:NH

_{4}

^{+}, HCO

_{3}

^{−}:H

_{2}CO

_{3}, and :H

_{2}CO

_{2}. Thus, if

*K*

_{α}(HPO

_{4}

^{2−}) is the equilibrium constant for the phosphate pair [with p

*K*

_{α}= log

_{10}(1/

*K*

_{α})], (20) Corresponding to the electrical potentials, ψ

_{α}, for σ = E, I, or M is the equation for electroneutrality (21) where for the cellular compartment (α = I), the sum includes the contribution of the impermeant anion (unprotonated impermeant buffer).

With respect to water flows, volume conservation equations for lumen, interspace, and cell can be used to compute the three unknowns: luminal volume flow, lateral interspace hydrostatic pressure, and cell volume. Across each cell membrane, the volume fluxes are proportional to the hydrosmotic driving forces (22) where p_{α} and *π*_{α} are the hydrostatic and oncotic pressures within compartment α, *L*_{pαβ} is the membrane water permeability, *A*_{αβ} is membrane area, and σ_{αβ}(*i*) is the reflection coefficient of membrane αβ to solute *i*. We assume that the cell is compliant in a manner that there is no hydrostatic pressure difference between the cell and lumen (23) There is a substantial oncotic force within the cell, *π*_{I}, that increases with decrements in cell volume. In the model calculations, it is assumed that cell protein content C_{I}(*π*)V_{I} is fixed and that *π*_{I} is proportional to C_{I}(*π*). In this way, the cell volume, V_{I}, replaces *π*_{I} as one of the model unknowns. With respect to the lateral interspace, its volume, V_{E}, and its basement membrane area, *A*_{ES}, are functions of interspace hydrostatic pressure, p_{E} (24) (25) where V_{E0} and *A*_{ES0} are reference values for volume and outlet area, and ξ_{AE} and ξ_{VE} are compliances. Axial variation in luminal hydrostatic pressure, p_{M}(*x*), derives from the Poiseuille flow equation (using fluid viscosity, η) (26) Solute transport is either convective (e.g., across tight junction or interspace basement membrane), electrodiffusive (e.g., via a channel), coupled to the electrochemical potential gradients of other solutes (e.g., via a cotransporter or an antiporter), or coupled to metabolic energy (via an ATPase). This is expressed in the model by the flux equation (27) In this equation, the first term indicates the role of the solute reflection coefficient in determining convective transport, and *c̄* is a (logarithmic) mean membrane solute concentration. The second term is the Goldman relation for ionic fluxes, where *h*_{αβ}(*i*) is a solute permeability, and C_{α}(*i*) and C_{β}(*i*) are the concentrations of *i* in compartments α and β. Here, (28) is a normalized electrical PD, where *z*_{i} is the valence of *i*, and ψ_{α} − ψ_{β} is the PD between compartments α and β. The third term of the solute flux equation specifies the coupled transport of species *i* and *j* according to linear nonequilibrium thermodynamics, where the electrochemical potential of *j* in compartment α is (29) For each of these transporters, the assumption of fixed stoichiometry for the coupled fluxes, allows the activity of each transporter to be specified by a single coefficient. The exception to this representation of coupled fluxes is the Na^{+}/H^{+} exchanger (NHE3) of the luminal membrane, for which a kinetic model has been developed (50). This model NHE3 represents exchange of luminal Na^{+} for cytosolic H^{+} or NH_{4}^{+}, which bind competitively, and includes a cytosolic regulatory site that confers pH dependence to transport. Although calculation of NHE3 fluxes depends on the ion affinities and carrier translocation rate coefficients, regulation of NHE3 activity in this cell model is mediated solely by changes in transporter density.

In this model, there are two transport ATPases, the peritubular Na-K-ATPase and a luminal membrane H^{+}-ATPase. The Na-K-ATPase exchanges three cytosolic Na^{+} ions for two peritubular cations, K^{+} or NH_{4}^{+}, which compete for binding, and flux is given by (30) in which the half-maximal Na^{+} concentration, *K*_{Na}, increases linearly with internal K^{+}, and the half-maximal K^{+} concentration, *K*_{K}, increases linearly with external Na^{+} (13). The pump flux of K^{+} plus NH_{4}^{+} reflects the 3:2 stoichiometry, (31) with transport of K^{+} or NH_{4}^{+} determined by their relative affinities, *K*_{K} and , (32) Analogous expressions are written for active transport at the basal cell membranes, *J*_{IS}^{act}. Within the luminal cell membrane there is a proton ATPase, represented by an empirical expression devised by Strieter et al. (37), approximating data of Andersen et al. (2) for turtle bladder (33) where *J*_{MI}(H^{+})_{max} is the maximum proton flux and μ̄_{MI}(H^{+}) is the electrochemical potential difference of H^{+} from the cytosol to the lumen; ξ_{MI} defines the steepness of the function, and μ̄_{0} defines the point of half-maximal activity.

To represent the impact of luminal flow on proximal transport, an expression for microvillous torque is required. Heuristically, the important determinants can be roughly estimated by identifying the microvillous drag per unit tubule length, *D*_{M}, with the force on axial tubular flow (34) Assuming Poiseuille flow (*Eq. 26*) in a tubule of radius, *R*_{M}, (35) so that referring all of this force to the tips of microvilli of height, *L*_{MV}, the torque (per unit tubule length), *T*_{M}, is (36) *Equation 36* reveals the microvillous torque to vary linearly with axial volume flow, and inversely with luminal cross section. Du et al. (11) provided a more detailed assessment of the torque, in which drag along the whole microvillous, including a finite tip region of height δ_{MV} was considered. With that analysis, the microvillous torque is (37) It is *Eq. 37* that is used in the calculations below, in which transporter density is modulated by luminal flow.

Finally, the rat proximal tubule in vivo is compliant, with luminal diameter a function of luminal hydrostatic pressure. The change in diameter (per mmHg) is nearly constant over a substantial pressure range (9), although this compliance appears to be larger at lower pressures (20). In the calculations of this model tubule, a constant compliance, ν_{M}, has been assumed, and the luminal radius (relative to the reference radius, *R*_{M0}) is given by (38)

## MODEL PARAMETERS

Table 1 contains a compilation of all of the model parameters for the noncompliant tubule, and for which there is no flow dependence of transporter density. The parameters closely follow those that have been previously published (47). The major change in these parameters is that the tubule has been lengthened from a 0.5-cm segment to the full 1.0-cm tubule (Table 1 in Ref. 14). In this model of a compliant tubule, there is a greater premium on accurate representation of the axial hydrostatic pressure profile. Specifically, changes in pressure mediate changes in tubule radius, which mediate changes in microvillous torque, which mediate changes in transport. The shorter tubule segment does not capture the full development of the drop in luminal pressure. Cellular and lateral interspace dimensions have not been changed significantly. With this lengthened tubule segment, transport per unit length was halved, so as to maintain overall tubule reabsorption a comparable fraction of filtered load. Specifically, coupled transporter coefficients, H^{+}-ATPase transport maximum, cell membrane water permeabilities, and cell membrane solute permeabilities are all roughly 50% of their prior values. The tight junction water permeability, an important unknown, had previously been selected to be comparable to the transcellular water permeability, and this has also been halved to maintain this proportionality. The tight junction solute reflection coefficients and solute permeabilities have been maintained at their original values, since these yielded agreement with overall epithelial solute permeabilities. In view of the increased tubule volume, the value selected for the rate of cellular ammoniagenesis was taken as 50% that used previously (49).

The two novel solutes for this PCT model are CO_{2} and H_{2}CO_{3}, and selection of their membrane permeabilities poses a dilemma. The unit membrane permeabilities of CO_{2} and H_{2}CO_{3} for the proximal tubule had been estimated in calculations by Krahn and Weinstein (23), in a simulation of solute diffusion within proximal tubule brush border. However, if one uses their value of CO_{2} permeability for luminal and peritubular cell membranes in this model, then one obtains an epithelial CO_{2} permeability that is likely more than an order of magnitude too high, at least in comparison with the epithelial CO_{2} permeability measured by Schwartz et al. (36) in rabbit proximal tubule. This discrepancy likely derives from the fact that much of the resistance to CO_{2} transport across the epithelium derives from resistance to permeation of the cell itself; i.e., the cell is an unstirred layer. One way around this is to decrease the value assigned to membrane CO_{2} permeability uniformly until the computed overall tubule CO_{2} permeability is realistic. This has now been done: the cell membrane CO_{2} permeability has been reduced to 0.012 cm/s (8% of the value used in Ref. 23), and with this reduction, the tubule CO_{2} permeability is now 0.014 cm/s. This may be compared with the proximal tubule CO_{2} permeability measured by Schwartz et al. (36) of 10^{−4} cm^{2}/s, or for a 25-μm-diameter tubule 0.013 cm/s. Overall, the membrane CO_{2} permeability has gone from very, very large to very large. Using the proximal tubule model, the maximal luminal CO_{2} concentration has gone from 1.500 to 1.507 mM with the reduced permeability, and epithelial CO_{2} is essentially uniform. Since membrane permeability to H_{2}CO_{3} is substantially less than that to CO_{2}, the impact of cellular constraints to diffusion are also substantially less; the H_{2}CO_{3} permeability selected was half the value used by Krahn and Weinstein (23).

## MODEL CALCULATIONS

The numerical methods used in solving the model equations were those described previously (47). Only steady-state problems are considered in this paper. For the epithelial simulations, the 35 model equations were solved iteratively to double precision machine accuracy using Newton's method. For the tubule problems, there were 53 model equations, and spatial derivatives of the luminal variables were represented by first-order finite differences, backward in space. In prior proximal tubule models from this lab, a centered second-order difference scheme had been used. However, with the representation of micromolar concentrations of H_{2}CO_{3}, the model equations became a stiffer system and required the backward-differencing scheme. This backward scheme has become the default for the distal segment models. In a typical steady-state calculation, the tubule was discretized into 80 segments (compared with 40 used previously), and accuracy of the solutions was checked by refining the spatial mesh.

Table 2 contains the solution of the model equations for the open-circuited epithelium at the tubule inlet. The solutions are similar, but not identical to those of Table 2 in Ref. 47. Differences derive from inclusion of cytosolic ammoniagenesis, from the increase in cellular impermeant from 50 to 60 mM, and from the presence of a peritubular impermeant that acts to shrink interspace and cell. A peritubular impermeant was absent in early PCT models (e.g., Ref. 47) but has been included in later models (e.g., Ref. 53) to reflect the fact that these tubules are not perfused in vitro, but exchange with capillaries. The configuration of transporters leaves the cellular compartment relatively depleted of Na^{+}, Cl^{−}, and HCO_{2}^{−}, and accumulating K^{+}, phosphate, and glucose. Cytosolic ammonia accumulation derives both from ammoniagenesis and from active NH_{4}^{+} uptake on the Na-K-ATPase. There is an even balance between HCO_{3}^{−} entry and exit, so that the cell maintains a neutral pH. In these calculations, the cellular impermeant has been identified with the cellular buffer, so that in computing cytosolic osmolality, the buffer terms do not contribute to the total. The fluxes in Table 2 (nmol·s^{−1}·cm^{−2}), are depicted in Fig. 2, *left*, but with the units (pmol·mm^{−1}·min^{−1}) referred to a tubule of diameter 25 μm. In this early region of the tubule, the volume flows between transcellular and transjunctional routes are nearly evenly matched. Nearly all of the Na^{+} flux is transcellular, ∼40% with glucose, and ∼60% via NHE3. About 70% of the Cl^{−} flux is paracellular, and the bulk of this is convective, in keeping with the low tight junctional reflection coefficient; cellular Cl^{−} entry is nearly equally divided between the two luminal base exchangers. On the peritubular membrane, the bulk of the K^{+} exit is via the K^{+} channel, although most of the Cl^{−} exit is via the K-Cl cotransporter. Despite its high permeability, the Na^{+}-2HCO_{3}^{−}/Cl^{−} exchanger is poised near equilibrium, with a relatively low throughput.

The model epithelial solute permeabilities are assessed in a series of calculations using the short-circuited epithelial model. Baseline bath and lumen conditions are those of Table 2, and a series of luminal perturbations are computed: increase and decrease in hydrostatic pressure by 1 mmHg, increase and decrease in solute concentrations by 0.1 mM, and increase and decrease in transepithelial PD by 0.1 mV. Using *Eq. 23*, the change in epithelial volume flow with pressure perturbations yields the overall epithelial *L*_{p}, and the change with solute perturbations yields the “osmotic” solute reflection coefficients, σ_{o}(*i*). The solute fluxes produced by hydrostatic pressure perturbations provide an estimate of the convective solute flux, and thus the second means of determining the reflection coefficient, denoted σ_{c}(*i*). The differences between σ_{o} and σ_{c} in Table 3 are small and likely reflect changes in cell or interspace geometry with bath perturbations. The changes in transepithelial electrodiffusive solute flux with luminal solute perturbation provide the overall epithelial solute permeability, *h*_{MS}, indicated in *column 4*. A second measure of the electrodiffusive permeability is the partial ionic conductance that is obtained by dividing the change in solute flux by the perturbation of transepithelial PD, and this is indicated in *column 5*. The observation that partial ionic conductances for Na and Cl are smaller than the ionic permeabilities is due to convective ion transport across the tight junction. This was noted in prior models (47) and was the subject of an earlier work (45). For the four principal solutes shown, the overall conductance is 187 mS/cm^{2}, corresponding to a resistance of 5.3 Ω·cm^{2}. These partial conductances can be translated into solute permeabilities, and these appear in *column 6*. The apparently smaller electrical permeabilities for Na^{+} and especially Cl^{−} can be rationalized by the presence of convective tight junctional transport (45). The permeabilities shown in Table 3 are comparable to those of rat proximal tubule (47).

Figure 3 displays the axial profiles of concentration and flow along the full 1.0-cm tubule for solutes that illustrate different reabsorption patterns. The calculation is done using the baseline parameters of Table 1 and presumes a noncompliant tubule with unmodulated transport. The Na^{+} concentration is constant over distance, so that its axial flow reflects overall volume reabsorption: 80% at the tubule end. Glucose reabsorption is nearly complete by 2 mm. There is preferential HCO_{3}^{−} transport, so that by tubule end its concentration falls to 9.1 mM with lumen pH = 6.885; this corresponds to 92% reabsorption. Over the first 2 mm, there is little net K^{+} flux, so that luminal concentration increases to 5.9 mM, where it remains for the remainder of the tubule; overall, about three-fourths of filtered K^{+} is reabsorbed. The Cl^{−} profile is not shown, but its fractional changes are comparable to those for K^{+}. Both HCO_{2}^{−} and NH_{4}^{+} are initially secreted, and ultimately reabsorbed as volume reabsorption proceeds along the tubule. However, luminal acidification enhances both NH_{3} trapping and H_{2}CO_{2} reabsorption, so that NH_{4}^{+} concentration increases and HCO_{2}^{−} concentration decreases along the late tubule. For the calculation of Fig. 3, Fig. 2, *right*, displays the solute fluxes across cell and tight junction at the tubule outlet. Compared with the inlet (Fig. 2, *left*), the salient differences include the disappearance of Na^{+}-glucose uptake, the near doubling of Cl^{−}/base exchange, and a dramatic increase in tight junctional NaCl reabsorption. The junctional NaCl reabsorption is due to convective flux, driven by the transjunctional Cl^{−} and HCO_{3}^{−} concentration differences (with differing reflection coefficients for these anions). In the late proximal tubule cell, decreased Na^{+}-glucose entry leads to decreased cytosolic Na^{+} (12.8 mM compared with 19.5 mM in the early tubule cells), and thus increased peritubular flux through the Na^{+}-2HCO_{3}^{−}/Cl^{−} exchanger, which has become the dominant pathway for Cl^{−} exit.

The first decision to make with respect to the representation of torque-modulated reabsorption is which of the transport pathways are affected. Evidence for an impact of flow on NHE3 is most secure, but there is also evidence that other luminal transporters are affected, at least the H^{+}-ATPase (12). There is no information available about any effect of luminal flow on peritubular transporters. In the calculations of Fig. 4, the open-circuited epithelial model is used with bath conditions as in Table 2. The *top* panel is total transepithelial Na^{+} flux, and the *bottom* panel is cell volume. Starting with the parameters of Table 1, the three sets of curves correspond to uniform scaling of model parameters: NHE3 transporter density alone, all of the luminal transporters (including solute and water permeabilities, cotransporter and pump density, and ammoniagenesis), or all of the luminal plus peritubular transporters. With respect to Na^{+} flux, it is clear that only a coordinated luminal plus peritubular scaling can yield a linear response over a broad range. From the perspective of glomerulotubular balance, the range achieved with luminal plus peritubular scaling is physiological. Indeed, increases in luminal transporter density alone produce at most a 30% increase in Na^{+} flux, falling short of observed increases in proximal Na^{+} reabsorption. An additional observation is that by itself, a broad increase in luminal transporter density produces what would be a lethal increase in cell volume. There are no significant cell volume changes with a coordinated scaling of luminal and peritubular transporter density. In the calculations that follow, all torque-modulated transporter density will affect all luminal and peritubular transporters, but no tight junction or basement membrane parameters. The calculations of Fig. 4 also suggest the magnitude for the torque-dependence of transporter density. The observation of Du et al. (11) was that Na^{+} reabsorption varied in proportion to the change in estimated microvillous torque, so that with regard to Fig. 4, a doubling of Na^{+} flux requires a 2.2-fold increase in transporter density.

Figure 5 displays the impact of torque-modulated transporter density on overall tubule function. In the panels on the *left*, the torque-modulated tubule is noncompliant; on the *right*, a compliance of ν_{M} = 0. 03/mmHg is assumed (*Eq. 38*); for each set of panels, results are compared with the curves for the nonmodulated noncompliant tubule (“Fixed”). The value for tubule compliance is motivated by data from Cortell et al. (9) and from Jensen and Steven (20). Cortell et al. (9) found a linear diameter dependence on luminal pressure of 0.45 μm/mmHg, roughly ν_{M} = 0. 02, while Jensen and Steven (20) observed a nonlinear dependence with 0.5 μm/mmHg at pressures above baseline and 1.3 μm/mmHg at reduced pressures. For these calculations, inlet and peritubular pressures are 15 and 9 mmHg, respectively (Table 2), values that have been observed in rat micropuncture (19). In each set of panels, the curves labeled “Fixed” correspond to baseline parameters in a noncompliant tubule (i.e., the calculations of Fig. 3), and the torque-modulated curves (“Tor”) have been computed by scaling luminal and peritubular transporter density by a factor of 2.2 times the relative microvillous torque (39) in which Par stands for any model parameter, and Par_{Base} is its value in Table 1. *T*_{M} is the local microvillous torque; *T*_{M0} is the torque at the tubule inlet when the inlet flow is 30 nl/min and inlet diameter is 25 μm; and *T*_{S} is the torque scaling factor. For the noncompliant tubule, the impact of torque-modulated transport is a progressive decrease in transepithelial volume flux along the tubule, so that overall volume reabsorption has been reduced from 80 to 46% of entering flow. Control and torque-dependent transport (Tor) showed a small difference in the axial pressure drop, 4 and 5 mmHg, respectively. In the compliant tubule, there is an increase in axial pressure drop to 6 mmHg, due to the 2.2-μm decrease in luminal radius (16 and 29% decreases in radius and cross-sectional area, respectively). With changes in tubule radius, luminal membrane circumference has been assumed to be constant, so that there is no loss of luminal cell membrane with tubule stretch. Since microvillous torque increases with decreasing luminal radius, the effect of tubule compliance is to blunt the axial decrease in luminal torque seen in the noncompliant tubule. Thus transepithelial volume flux does not decline as much as in the noncompliant tubule, and the overall volume reabsorption is 57%, comparable to the value reported by Schnermann et al. (33).

The principal motivation for examining torque-dependent transport is to gain insight into perfusion-absorption balance. This is examined directly in Fig. 6, in which the tubule model is used to predict proximal reabsorption over the range of inlet flows from 15 to 75 nl/min. For the higher flows, it was anticipated that the axial pressure drop would exceed the baseline inlet pressure of 15 mmHg, thus creating unrealistically negative distal pressures. To avoid this, the model was configured as a boundary value problem, in which the distal hydrostatic pressure was specified as 9 mmHg, and the initial luminal pressure was determined so as to yield this end-tubule pressure (peritubular pressure is fixed at 9 mmHg). This involved an initial guess for the inlet pressure, and iterative solution of the tubule model with Newton updates of p_{M} (0). In the case of the noncompliant tubule, without torque scaling (torque scaling = 0.0), the range in fractional proximal reabsorption (absolute reabsorption relative to inlet perfusion rate) extends from 100 to 31%, from the slowest to the highest perfusion rates. This reflects the nearly constant volume reabsorption of 24 nl/min at all inlet flows. The three other curves in the figure are computed using torque scaling coefficients of 1.0, 1.6, and 2.2, in a compliant tubule. The scaling coefficient of 1.0 corresponds to the situation in which transporter density is proportional to microvillous torque, and the value of 2.2 is based on the observation in mouse proximal tubule that Na^{+} reabsorption is proportional to microvillous torque. The intermediate value, TS = 1.6, yields a nearly constant fractional reabsorption over the fivefold range of inlet flows. At the lowest perfusion rates, the tubules with the greatest torque scaling show a progressive decline in fractional volume reabsorption. In these tubules, lower flow produces a sharp drop in transcellular fluxes; the transepithelial Cl^{−} and HCO_{3}^{−} anion gradients fail to develop, so that transjunctional convective fluxes fail to materialize; and the decrease in cellular transport is amplified. The dashed line in Fig. 6 marks the baseline perfusion rate of 30 nl/min. One measure of glomerulotubular balance has been the “gain” in proximal reabsorption, computed as one deviates from the baseline filtration rate (29). Computation of this gain for the four curves of Fig. 6 is summarized in Table 4.

There are two natural contexts in which to consider proximal tubule diuresis, osmotic diuresis and CA inhibition. Figure 7 displays calculations using the tubule model in which luminal flow rate is constant (30 nl/min), but ambient glucose concentration (lumen inlet and peritubular bath) are varied from 2 to 80 mM. The panels on the *left* are computed using the baseline model with a noncompliant tubule and no torque scaling; on the *right*, the torque scaling coefficient is 2.2 and the tubule is compliant. The *top* panels display end-tubule glucose concentration, the *middle* panels show end-tubule volume flow, and the *bottom* panels contain the standard curves representing glucose delivery, reabsorption, and “excretion” for a clearance study. In the noncompliant tubule, glycosuria appears at an ambient glucose concentration of 30 mM. Tubule fluid reabsorption is maximal and nearly complete when ambient glucose is 24 mM, due to the effect of luminal glucose to enhance proximal Na^{+} and fluid transport, and then fluid reabsorption declines progressively with increasing glucose. This pattern has been seen previously in simulation of glucose diuresis (Fig. 3 in Ref. 44). With torque-modulated transport, glycosuria appears at a lower ambient glucose concentration (13 mM) and end-tubule flow has been substantially stabilized. With regard to the glucose reabsorption curve, it is notable that there is considerably more “splay” to this curve than in the case of nonmodulated flow. Although overall volume reabsorption is relatively constant, at higher glucose loads the pattern of luminal flow shifts more distally, thus “activating” more transporters along the tubule and enhancing the overall transport maximum for glucose reabsorption.

Figures 8 and 9 display simulations of luminal CA inhibition in the model tubule. Specifically, this denotes scaling of the rate constants for luminal CO_{2} hydration and H_{2}CO_{3} dehydration by 10^{−4}: *k*_{h} = 0.145 s^{−1} and *k*_{d} = 49.6 s^{−1}, so that the reaction is completely uncatalyzed. It is assumed that cytosolic and lateral interspace hydration remain fully catalyzed. In Fig. 8, there are two sets of panels: the *top* set uses the noncompliant tubule, with no torque-dependent transport; the *bottom* set uses the compliant tubule, with a torque scaling coefficient of 2.2. For each tubule, end-luminal HCO_{3}^{−} is ∼25 mM and Pco_{2} is 50 mmHg, providing an equilibrium pH = 7.32. Since each tubule has an end-luminal pH = 6.6, there is an acid disequilibrium pH of ∼0.7 pH units. In the noncompliant tubule, CA inhibition decreases volume reabsorption from 23.9 to 14.6 nl/min, a 39% diuretic effect; with torque-dependent fluxes, CA inhibition decreases reabsorption from 17.5 to 13.6 nl/min, a 22% effect. With CA inhibition, there is no decline in luminal HCO_{3}^{−} concentration and no increase in luminal Cl^{−}, so that the anion gradients across the tight junction that normally drive water flow are absent. Under normal circumstances, with fully catalyzed CO_{2} hydration, the model tubule predicts that transjunctional water flow is responsible for a substantial component of convective Na^{+} reabsorption across the tight junction. Moreover, as luminal flow diminishes over distance, the effect of microvillous torque on the cells will be to decrease transcellular Na^{+} flux. In sum, along the model tubule there is an axial decrease in total Na^{+} reabsorption and in the component transported via the Na-K-ATPase. This is illustrated in Fig. 9 in the panels on the *left*, which display Na^{+} fluxes for the compliant tubule with torque-scaling of transporter density. In the *bottom* panel, Na^{+} flux via the sodium pump declines from 80 to ∼30% of total Na^{+} reabsorption over the length of the tubule. In the panels on the *right*, the tubule is the same, but there is inhibition of luminal CA. In this case, there is little enhancement of convective paracellular Na^{+} flux; there is less decline in microvillous torque; and there is an acid lumen that shifts to H_{2}CO_{2}, enhances luminal Cl^{−}/ exchange, acidifies the cell, and thus sustains luminal Na^{+}/H^{+} flux. With CA inhibition, the model predicts that ∼80% of luminal Na^{+} flux is transported by the Na-K-ATPase.

## DISCUSSION

The mathematical novelty of this work has been an update of the proximal tubule model to include finite rates of catalysis of CA and to incorporate mechanisms to permit simulation of flow-dependent solute transport in a compliant tubule. The proximal tubule models from this lab have advanced in stages: the first model represented Na^{+}, K^{+}, Cl^{−}, HCO_{3}^{−}, phosphate, glucose, and urea (43), and subsequent extensions have added luminal membrane Cl^{−}/ exchange (47), ammonia transport and cytosolic ammoniagenesis (49), and a kinetic NHE3 that represented competition of H^{+} and NH_{4}^{+} in exchange for Na^{+} entry (50). In the present work, prior features have been retained, with the modification that the tubule has been lengthened from 5 to 10 mm, to obtain a more realistic estimate for the axial decline in hydrostatic pressure. To represent longer tubules, cellular transport rates were scaled back, while tight junction solute permeabilities and reflection coefficients were unchanged from earlier models. Inclusion of finite rates of hydration and dehydration of CO_{2} and H_{2}CO_{3} was straightforward and followed the approach used in all of the models of distal nephron segments (e.g., Ref. 53). Of note, however, with the representation of micromolar concentrations of H_{2}CO_{3}, the model equations became a stiffer system and required a backward-differencing scheme (first order) for solution. This was a change from the centered schemes used previously for the proximal tubule.

Perhaps the most important observation of this work was that scaling of luminal membrane transporter density (either as an ensemble or just NHE3 alone) had a relatively small impact on overall Na^{+} reabsorption and could produce a large derangement of cell volume (Fig. 4). Recognition of the need for coordinate regulation of luminal and peritubular transporters to preserve cellular integrity is not new. This had been articulated by Schultz and colleagues (34, 35) initially for gastrointestinal epithelia and has been studied extensively in the proximal tubule (4). An important experimental model has been the luminal application of an avidly transported organic (glucose or amino acid) for which luminal entry is Na^{+} coupled and electrogenic, with the observation that enhanced flux results in activation of peritubular K^{+} conductance (5, 27), due to an increase in the open probability of peritubular K^{+} channels (3). It is unlikely, however, that activation of peritubular K^{+} channels is the sole homeostatic mechanism. In a modeling study of the proximal tubule, it was calculated that to maintain a constant cell volume when active Na^{+} transport increased by 50%, a 10-fold increase in peritubular K^{+} permeability was required (Fig. 6 in Ref. 51). In that work, it was noted that volume activation of peritubular Na^{+}-3HCO_{3}^{−} cotransport could easily provide the desired volume homeostasis over a broad range of Na^{+} fluxes. Subsequent calculations indicated that volume activation of peritubular KCl cotransport could also serve a similar role (52). Experimental identification of a peritubular anion transporter that is activated with application of luminal organics has not been done. This issue has been studied more extensively, however, in the volume-regulatory decrease (VRD) that follows (hypotonic) osmotic shock. In mouse proximal tubules, omission of bicarbonate impairs VRD and cell swelling is associated with enhanced peritubular HCO_{3}^{−} conductance (39). In the straight proximal tubule of rabbit, however, removal of chloride impaired VRD, while omission of bicarbonate or the application of SITS had no inhibitory effect (54). In the proximal convoluted tubule of the rabbit, hypotonic swelling increases both K^{+} conductance and Cl^{−} conductance of the peritubular cell membrane (6, 55). In the absence of information identifying luminal and peritubular transporters that are activated with luminal flow, the calculations have examined a best case, in which all transporters are scaled in proportion to the luminal signal.

The luminal signal that modulates proximal epithelial fluxes has been assumed to be microvillous torque. In support of this assumption, our studies of mouse proximal tubule in vitro have shown that reabsorptive Na^{+} and HCO_{3}^{−} fluxes scale in proportion to our estimate of microvillous torque (11, 12). That this is a torque effect (rather than a flow effect) is supported by the observation that increasing tubule fluid viscosity (with constant flow) also increases Na^{+} reabsorption (11). Conversely, when flow was altered without a substantial change in estimated torque (due to increasing luminal diameter), the impact on volume reabsorption was insignificant (7). That torque impacts more than just NHE3 is supported by observations that flow dependence is seen in the presence of ethylisopropyl amiloride or in NHE3 knockout mice (12). The signal between luminal membrane events and peritubular transporter density is unknown, but in the calculations here it has been assumed that the peritubular impact of microvillous torque is equal to that on the luminal membrane. Cytochalasin experiments have implicated the actin cytoskeleton in the signal transduction from luminal flow to transporter activation (11, 12), and Weinbaum et al. (42) have emphasized that the stiff actin core of microvilli acts as a lever to amplify drag from microvillous tips to the subapical actin filaments. Whether this role for actin is only local (at the luminal membrane) or is transmitted to peritubular transporters is unknown. Other candidates for the peritubular signal must obviously include those considered in the context of flux-dependent activation, and include cytosolic ATP (e.g., Ref. 38) and cell volume per se. The calculations in this paper have all been steady state, as have the experiments defining the impact of microvillous torque. It is likely that time-dependent experiments will be needed to identify luminal-peritubular cross talk signals.

Our experimental constraint on torque scaling of tubule transporters was that reabsorptive fluxes should vary linearly with microvillous torque, with a proportionality of 1.0 (12). With the assumptions of uniform scaling of all transporters, and equal luminal and peritubular effects, this defines a single torque scaling constant, which in the context of this model was found to be 2.2. This was the local effect (as opposed to a full tubule effect), when lumen composition was similar to standard perfusates. With reference to *Eq. 39*, one might have expected torque scaling of 1.0 to achieve flux dependence of 1.0, but that equation only defines cellular parameters. In proximal tubule epithelium, transcellular fluxes are in parallel with tight junctional fluxes, and in series with the basement membrane, neither of which is torque modulated. In the absence of tubule compliance, the isolated effect of torque-modulated parameters is a substantial reduction in proximal volume reabsorption (Fig. 5), due to declining axial flow rate. Microvillous torque is directly proportional to axial flow, and (approximately) inversely proportional to luminal area (*Eqs. 36* and *37*), so that the effect of tubule compliance (a decreasing luminal diameter with the Poiseuille pressure drop) is a blunting of the flow factor on torque, and thus a less severe axial reduction in reabsorptive fluxes. The two measurements of proximal tubule compliance (9, 20) provided similar values, at least when pressures are high and there is tubular distention. The study of Jensen and Steven (20) suggested a greater compliance when luminal pressures decreased, so that with lower axial flows, the reduction in torque would not be so great, and proximal reabsorption would not so sharply diminished. While there remain uncertainties regarding transepithelial pressure differences during changes in renal perfusion, the torque-dependent modulation of transport has underscored the physiological relevance of tubular compliance. It must be acknowledged that this torque modulation gives the model tubule an axial differentiation that had been missing from prior models. In the interest of simplicity, those models had always ignored structural differences among S1, S2, and S3 segments, but the current model has now acquired a functional expression of the axial heterogeneity of cellular transport.

The primary objective of these calculations was to investigate the potential role of torque-dependent transport as a determinant of glomerulotubular balance. In broad terms, two distinct factors have been recognized that can modulate proximal tubule reabsorption in parallel with changes in glomerular filtration. One is the peritubular oncotic effect, which varies as a result of changes in filtration fraction, and which likely exerts its effect on tubular transport via changes in interstitial hydrostatic pressure. Whether the ultimate target of peritubular oncotic pressure is tight junction structure (and thus paracellular transport properties), or the cell itself remains an unresolved issue. It had been possible to represent a compliant tight junction in the context of a full proximal nephron (glomerulus, proximal tubule, peritubular capillaries, and interstitial compartment), which simulated the peritubular oncotic impact on sodium reabsorption (46). It was evident from those calculations, however, that without a luminal flow effect, there was no glomerulotubular balance. The existence of a luminal effect, perfusion-absorption balance, was inferred from the very first micropuncture observations of the mammalian proximal tubule by Walker et al. (41). Kelman (21) advanced the discussion with the insight that if perfusion-absorption balance held along the whole proximal tubule, then reabsorption must be velocity dependent, and the axial profile of luminal flow must be exponential (in a noncompliant tubule). As predicted, his plot of the logarithm of relative flow as a function of distance [measurements by Walker et al. (41)] was linear. Kelman's approach (21) can be applied to the simulated tubules of this model, and this has been done in Fig. 10, in which the *top* panels are the flow profiles from torque-modulated transport (Fig. 5). The *bottom* panels of Fig. 10 show the flows on a log scale (solid curves), along with linear regressions (dashed). If either of the solid curves had been defined by experimental data, they could pass for linear, although it is clear that the curves from the compliant tubule are a better match. In short, the model of the compliant tubule with torque-dependent transport provides an exponential flow profile, consistent with observations in the rat.

Perfusion-absorption balance in the compliant model tubule with torque-dependent transport was displayed in the calculations of Fig. 6. That figure contained curves using several values of the torque-scaling factor, corresponding to torque-proportional fluxes (TS = 2.2), torque-proportional transporter density (TS = 1.0), and an intermediate value (TS = 1.6). At high perfusion rates, all three values blunted changes in fractional reabsorption and yielded predictions that would be difficult to distinguish experimentally. At low perfusion rates, model predictions diverged. Experimentally, perfusion-absorption balance in the rat has been studied in micropuncture protocols in which luminal flow was acutely varied by activating tubuloglomerular feedback (TGF), or by addition or subtraction of luminal fluid from the early tubule. In such experiments, Haberle et al. (17) computed a linear regression for proximal reabsorption as a function of single-nephron glomerular filtration rate and observed a proportionality coefficient of 0.92 when TGF was manipulated, and a coefficient of 0.76 when tubule fluid was removed by a suction pump. Similarly, Peterson et al. (29) computed the gain of the single-nephron glomerular filtration rate signal and found it to be 0.44 with TGF manipulations, and 0.68 when tubule fluid flow was decreased; they observed no gain when luminal fluid was added. Thus, with regard to the gains computed in Table 4, the rat data would suggest torque scaling in the range 1.0–1.6, rather than 2.2. It may be the case that the mouse proximal tubule is different from rat, perhaps with a tighter tight junction, so that the relative changes in cellular and epithelial transport are closer. It might also be possible that in the micropuncture protocols, fluid collections were made before full development of transporter insertion or retrieval was complete, or there were changes in luminal diameter that influenced microvillous torque.

Two derangements of proximal tubule function were examined in these simulations, hyperglycemia and CA inhibition. As might be expected, in each case, the impact of torque-dependent transport was to attenuate the diuretic effect. In the simulation of hyperglycemia (as previously; see Ref. 44), the tubule without torque modulation showed a decrease in proximal volume reabsorption concomitant with the appearance of end-luminal glucose. With torque modulation, end-luminal volume flow was found to be nearly constant over the range of ambient glucose (20–80 mM), and end-luminal glucose concentration was found to approximate the perfusate concentration. Both of these predictions bring the model tubule into agreement with micropuncture observations of hyperglycemia in the rat (40). In addition, torque-dependent transport gives the model tubule a less sharply defined maximal glucose reabsorptive flux. This splay in the glucose reabsorption curve had been an object of interest in the early study of renal glucose transport (reviewed in Ref. 28). Explanations considered have included nephron heterogeneity or alteration of tubular transport kinetics by extracellular volume expansion. What the present model adds to the discussion is an additional rationalization for the observed splay, namely, that torque-dependent transporter density recruits more glucose carriers to the luminal membrane as increasing luminal glucose load delivers higher flow rates to later portions of the proximal tubule.

This is the first model to simulate CA inhibition in the proximal tubule. With complete inhibition of luminal membrane CA, calculations predicted a luminal acid disequilibrium pH ∼0.7 units, comparable to measurements in the rat (0.83 ± 0.26) by Rector et al. (31). In the compliant tubule, with torque-dependent fluxes, the decrease in proximal volume reabsorption produced by luminal CA inhibition was 22%. There is considerable disparity in measurements of the proximal diuretic effect of CA inhibition, but in a careful study in the rat, Kunau (24) found proximal sodium inhibition of 34% after administration of benzolamide. In the calculations of Fig. 8, the diuretic effect of luminal CA inhibition was 39% in the tubule without torque dependence, so that one possible explanation for the greater diuretic effect in vivo might have been interference with flow sensing by microvilli. In calculations not shown here, inhibition restricted to cytosolic CA had a greater diuretic effect than luminal membrane inhibition. Thus, an alternative resolution of the difference between model and experiment might be combined inhibition of luminal membrane and cytosolic CA by benzolamide. Of note, one effect of luminal CA inhibition was obliteration of the lumen-to-blood anion gradients for Cl^{−} and HCO_{3}^{−} along the tubule, and elimination of much of the convective paracellular Na^{+} flux. There was also enhancement of transcellular NaCl transport, driven by increased axial flow (and torque-dependent transporter recruitment), plus an acid lumen that accelerated luminal membrane formic acid cycling. These model results are concordant with the recent observations of Deng et al. (10), who found that benzolamide increased absolute renal oxygen consumption while inhibiting proximal Na^{+} transport. In their experiments, they were able to localize the increase in oxygen consumption to the proximal tubule, and they suspected that both the decrease in paracellular Na^{+} flux and the enhancement of luminal Cl^{−} uptake were responsible for the increased metabolic cost of Na^{+} reabsorption.

In sum, the mathematical model of rat proximal tubule has been extended with computation of luminal microvillous torque, and with dependence of tubule diameter on transtubular hydrostatic pressure. Both features are necessary for representation of torque-dependent membrane transporter density. We conclude that to represent torque-dependent changes in transepithelial Na^{+} reabsorption (of the observed magnitude and without lethal cellular consequences), there must be modulation of both luminal and peritubular transporter density. When the magnitude of torque-dependent Na^{+} reabsorption in the model agrees with that observed in mouse proximal tubules, the model tubule shows nearly perfect perfusion-absorption balance at high luminal perfusion rates, but enhanced sensitivity of reabsorption at low flow. With a slightly lower coefficient for torque-sensitive transporter insertion, perfusion-absorption balance in the model tubule is closer to observations in the rat over a broader range of inlet flows. Another consequence of torque-dependent transport is attenuation of the diuretic effect of hyperglycemia, and brings the model tubule into closer agreement with experimental observation in the rat. The model was also extended to represent the finite rates of hydration and dehydration of CO_{2} and H_{2}CO_{3}. With CA inhibition, torque-dependent transport was predicted to blunt the diuretic effect, and to enhance the shift from paracellular to transcellular NaCl reabsorption.

## GRANTS

This investigation was supported by Public Health Service Grants R01-DK-29857 (A. M. Weinstein) and RO1-DK-62289 (T. Wang) from the National Institute of Arthritis, Diabetes, and Digestive and Kidney Diseases.

## Footnotes

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- Copyright © 2007 the American Physiological Society