Potential Difference Measurements of Ocular Surface Na⁺ Absorption Analyzed Using an Electrokinetic Model

Abstract

purpose. Corneal and conjunctival epithelia are capable of transcellular Na⁺ absorption and Cl⁻ secretion, which drives water movement across these tissues. A recent study demonstrated with a new open-circuit potential difference (PD) technique that Cl⁻ moves across the ocular surface in mice through Ca²⁺- and cAMP-sensitive Cl⁻ channels, the latter pathway being the cystic fibrosis (CF) transmembrane conductance regulator (CFTR). The purpose of the present study was to identify transporting mechanisms involved in Na⁺ absorption and to develop a mathematical model of ocular surface ion transport to quantify the relative magnitudes of and electrochemical coupling among transporting processes.

methods. PDs across the fluid-bathed ocular surface were measured in anesthetized wild-type and CF mice in response to Na⁺, Cl⁻, and K⁺ ion substitution and transporter agonists, inhibitors, and substrates. An electrokinetic model of the ocular surface epithelium was developed to simulate PD measurements, which involved computation of membrane potentials and cell [Na⁺], [K⁺], [Cl⁻] and volume from transporter activities and extracellular ion concentrations.

results. Na⁺ replacement produced a 6 ± 2-mV depolarization that was blocked by amiloride (K _i 0.8 μM) and benzamil (K _i 0.2 μM). The Na⁺-dependent depolarization by amiloride was significantly greater in CF mice (19 ± 3 mV). In wild-type mice, d-, but not l-glucose produced a phloridzin-sensitive, 4.1-mV hyperpolarization in the presence of Na⁺ and amiloride, with a K _m for d-glucose of 2.5 mM. Glycine and l-arginine also produced Na⁺-dependent hyperpolarizations. The epithelial transport model accurately reproduced experimental PD measurements.

conclusions. PD measurements coupled with model computations defined quantitatively the roles of Na⁺ and Cl⁻ transport processes in ocular surface ion and fluid secretion, and indicated that CFTR-dependent changes in apparent epithelial Na⁺ channel (ENaC) activity could be accounted for by electrochemical coupling, without requiring ENaC-CFTR interactions. The data and modeling also predicted significant enhancement of ocular surface fluid secretion by ENaC inhibitors and CFTR activators as possible therapies for dry eye syndromes.

The ocular surface, lined by corneal and conjunctival epithelia, can actively absorb Na⁺ from and secrete Cl⁻ into the tear film (reviewed by Dartt ¹ and Candia ² ). The relative magnitudes of net Na⁺ absorption and Cl⁻ secretion largely dictate the direction and magnitude of fluid movement across the ocular surface. A quantitative, mechanistic understanding of ocular surface ion transport is important in understanding the pathophysiology of dry eye conditions, or keratoconjunctivitis sicca (KCS), and in identifying targets for development of therapies to treat KCS. At present, INS365, a uridine triphosphate (UTP) agonist that stimulates chloride-fluid secretion through as yet unidentified channels, is in phase III clinical trials for treatment of KCS. ³ The purpose of this study was to establish, through experimental electrical potential measurements and mathematical modeling, a quantitative description of ion transport at the ocular surface. 

We recently introduced an open-circuit potential difference (PD) method to study ion transport at the ocular surface in vivo. ⁴ In this method, electrical potentials generated at the superfused ocular surface (with respect to the body) are measured by a high-impedance voltmeter in response to ion substitution, agonist-inhibitor addition, or other maneuvers. Surface PDs arise from polarized ion conductances in apical and basolateral membranes across tight epithelia and so can provide, in principle, quantitative information about electrogenic transporting systems. The corneal and conjunctival surfaces are superfused as a single compartment in this method to mimic native tear film physiology, allowing assessment of the relative contributions of individual transporters to tear film homeostasis. In our initial study, CFTR was identified as a major route for Cl⁻ secretion across the ocular surface into the tear film, and CFTR activators were proposed as a possible therapy for dry eye conditions. 

In the current study, we extend the initial experimental analysis of ocular surface Cl⁻ transport to include Na⁺-transporting mechanisms and, by mathematical modeling, to assess the contributions of individual ion transporting pathways to net solute and fluid movement across the ocular surface. Prior experimental data with regard to Na⁺ channels include short-circuit current experiments on in vitro albino rabbit cornea and conjunctiva, where amiloride-sensitive Na⁺ conductance could not be demonstrated. ^{5 6} However, short-circuit measurements on excised bovine cornea, ⁷ and more recently on primary cultures of rabbit corneal epithelium, ⁸ have provided functional evidence for amiloride-sensitive channels. Of potential relevance to human ocular disorders, the epithelial Na⁺ channel (ENaC) subunit proteins have been localized to the apical membrane of human corneal epithelium. ⁹  

In this study, ocular surface PDs were used to identify and characterize Na⁺ channel and Na⁺-coupled glucose- and amino acid-transporting pathways at ocular surface epithelia in living mice. A model of ocular surface epithelial ion transport was developed to compare quantitatively the rates of Na⁺ absorption versus Cl⁻ secretion and to analyze mechanisms of proposed coupling between CFTR and ENaC. ^{10 11} Modeling computations were applied to test the hypotheses that ENaC provides a quantitatively significant mechanism for ocular surface fluid transport that may be exploited for therapy of KCS and that enhanced Na⁺ absorption in CF mice can be accounted for by CFTR-ENaC electrochemical coupling. The mathematical model describing ocular surface epithelial transport is an extension of prior models developed by our laboratory and others for non-ocular epithelial ion transport. ^{12 13 14} Figure 1diagrams the transporting systems and pathways included in the model. The model allowed the computation of epithelial cell membrane potentials and currents, transepithelial water flow, and cellular [Na⁺], [K⁺], [Cl⁻] and volume in response to experimental maneuvers such as transporter activation-inhibition and ion substitution. Although such an epithelial transport model contains multiple parameters, constraints imposed by experimental observations allowed little freedom in selecting model parameters. The model was able to reproduce experimental findings closely and to provide quantitative information and insights not otherwise possible. 

Methods

Mice

Wild-type and CF mice (homozygous ΔF508 mutant mice) ¹⁵ on a CD1 genetic background were bred and cared for at the University of California, San Francisco, Animal Facility. Wild-type mice were fed a standard diet, and CF mice were fed a nutritional supplement (Peptamen; Nestlé, Vevey, Switzerland). Mice aged 8 to 12 weeks weighing 25 to 30 g were used. Protocols were approved by the University of California, San Francisco, Committee on Animal Research and were in compliance with the ARVO Statement for the Use of Animals in Ophthalmic and Vision Research. 

Mice were anesthetized with 125 mg/kg 2,2,2-tribromoethanol (Avertin; Sigma-Aldrich, St. Louis, MO) intraperitoneally, with supplementation during experiments to maintain deep anesthesia. Mice were immobilized, with the eye under study oriented to face upward in a custom-built stereotaxic device equipped with a rotating jaw clamp. Corneas were kept hydrated with isosmolar saline (PBS with NaCl added to give 320 mOsM) before measurements began. Core temperature was monitored with a rectal probe and maintained at 37 ± 1°C with a heating pad. 

Measurement of Ocular Surface PD

PD was measured continuously, with the ocular surface perfused serially with different solutions, as described in detail previously. ⁴ Briefly, solutions were perfused at 6 mL/min through plastic tubing using a multireservoir gravity pinch-valve system (ALA Scientific, Westbury, NY) and a variable-flow peristaltic pump (medium flow model; Fisher Scientific, Fair Lawn, NJ). An ∼50-μL fluid reservoir was maintained at the ocular surface by surface tension, with the probe catheter (PE-90 polyethylene tubing) fixed ∼1 mm from the eye surface by a micropositioner and a suction cannula positioned ∼3 mm from the orbit. Solution exchange time was generally less than 3 seconds. The measuring and reference electrodes consisted of Ag/AgCl with 1-M KCl agar bridges. The measuring electrode was located near the catheter probe and connected to a high-impedance digital voltmeter (IsoMillivolt Meter; World Precision Instruments, Sarasota, FL), having input and system electrical resistances of 10¹² and 1.1 × 10⁶ Ω, respectively. The reference electrode was connected via a continuous liquid (320 mOsM saline) column to a winged, 21-gauge needle inserted in the subcutaneous tissue at the back of the neck. PDs were recorded at 5 Hz with a 14-bit analog-to-digital converter. 

Solutions and Compounds

All perfusion solutions were isosmolar to mouse serum (320 ± 5 mOsM) as measured by freezing point-depression osmometry (Precision Systems, Inc., Natick, MA). Solution compositions are listed in Table 1 . Solution 1 (control solution) was PBS supplemented with 30 mOsM NaCl. Solution 2 (Na⁺-free solution) was made by replacing NaCl with choline chloride and Na₂HPO₄ with 8 mM phosphoric acid, titrating to pH 7.4 with choline base and then adding choline chloride to achieve 320 mOsM. In some experiments, NaCl was replaced isosmotically with 30 mM d-glucose, l-glucose, or d-mannitol (solution 3). In solution 4, choline chloride (from solution 3) was replaced isosmotically with 30 mM d-glucose or d-mannitol. Solutions 1 and 3 or 2 and 4 were mixed to give solutions of intermediate organic solute concentrations. In solution 5 (low Cl⁻ solution; 4.7 mM Cl⁻), most Cl⁻ was replaced by gluconate. Solution 6 (high-K⁺ solution; 174.2 mM K⁺) differed from solution 1 in that NaCl and Na₂HPO₄ were replaced by equimolar amounts of KCl and K₂HPO₄. Whereas Cl⁻ replacement produced a small junction potential (generally ∼ +1 mV) that was measured daily and used to correct PDs measured with low-Cl⁻ perfusates, Na⁺-free and high-K⁺ solutions did not produce junction potentials. 

Compounds were purchased from Sigma-Aldrich unless indicated otherwise. Inhibitors, activators, and substrates used in the perfusates included amiloride or benzamil (0.1–100 μM), phloridzin (100 μM), forskolin (10 μM), and l-arginine or glycine (0.1–10 mM, freshly dissolved in perfusate). CFTR_inh-172 (3-[(3-trifluoromethyl)phenyl]-5-(3-carboxyphenyl) methylene]-2-thioxo-4-thiazolidinone; 10 μM) was synthesized as described. ¹⁶ In protocols in which solution 5 (low-Cl⁻ solution) was used, all solutions contained 10 μM indomethacin to suppress CFTR activation by mechanical or other stimulation under control (unstimulated) conditions, as discussed previously. ⁴ Inhibitors-activators were prepared as 1000× stock solutions in dimethyl sulfoxide (DMSO), unless otherwise indicated. 

PD Protocols and Data Analysis

Na⁺ transport was studied by measuring PDs during continuous perfusion of the ocular surface with a series of solutions that imposed Na⁺ gradients and contained transporter substrates and/or inhibitors. Cl⁻ channel function was assessed with protocols described previously. ⁴ Data are expressed as the mean ± SE of absolute PDs or changes in PD (ΔPD), and statistical comparisons between groups were made with the two-tailed Student’s t-test. 

Model Formulation

Overview.

The model treats the ocular surface as an infinite, homogeneous monolayer of cells containing transcellular and paracellular solute transport pathways (Fig. 1) . Programming was done using Visual Fortran (Compaq; Hewlett Packard, Palo Alto, CA). Intracellular solute activities, cell membrane potentials, and cell height (reflecting volume) were allowed to vary in time. Mucosal and serosal compartments consisted of infinite well-stirred solute pools with compositions that could be altered at specified time(s). Transporting systems for Na⁺, K⁺, and Cl⁻ were included in the model. 

In ocular surface epithelia, the basolateral ouabain-sensitive 3Na⁺/2K⁺ATPase (transporter 7) establishes the electrochemical potential that drives electroneutral (bumetanide-inhibitable) Na⁺/K⁺/2Cl⁻ symport (transporter 6). Na⁺ can also enter cells through an amiloride-sensitive Na⁺ channel (transporter 1), and organic solute-coupled cotransporters (transporter 2). Distinct glucose- and amino-acid transporters, assumed to be coupled 1:1 to Na⁺ based on data from rabbit conjunctiva ^{17 18} are represented in the model as a single “Na⁺-org” transporter. A neutral exit pathway for organic solutes is included that represents cellular efflux and/or utilization (transporter 9). Cl⁻ secretion occurs at the apical membrane through transporter 4, now known to be the cAMP-regulated Cl⁻ channel CFTR, as well as a Ca²⁺-activated Cl channel (transporter 5). Apical and basolateral K⁺ conductances are included as possible exit pathways for K⁺ (transporters 3 and 8). Paracellular pathways for all charged species (transporters 10–14) are also included. Although cellular pH regulation may be important in cornea and conjunctiva, bicarbonate and proton transport pathways are not considered significant determinants of ocular surface fluid transport, as seen by the small, ∼2-mV bicarbonate effect on ocular PD. ⁴ As such, bicarbonate transport and Na⁺/H⁺ exchange pathways are not included in this model. 

Parameters.

Definitions and units of symbols are listed in Table 2 . Transporter permeability coefficients were determined from open-circuit ion flux and membrane potential data (see Tables A1and A2 in the ^Appendix) deduced from measured ocular PDs and literature values, as explained in the ^Appendix. Solute activities were related to absolute concentrations and osmotic activities according to intracellular and extracellular activity and osmotic coefficients, as listed in Table A3. 

Solute flux ( \(J_{i}^{X}\) ) through each membrane and paracellular transport pathway was defined as a function of solute concentrations on each side of the membrane and, if electrogenic, a membrane potential. Electrodiffusive flux of solute X through the ith transport pathway was assumed to be nonsaturable and nonallosteric, unless otherwise noted, as given by the Goldman-Hodgkin-Katz equation:  

\[J_{i}^{X}{=}P_{i}z_{i}U_{\mathrm{z}}\mathrm{(}X_{1}{-}X_{2}e^{{-}z_{i}U_{\mathrm{z}}})/(1{-}e^{{-}z_{i}U_{\mathrm{z}}}),\]

where \(P_{i}\) is the permeability coefficient (in centimeters per second), \(z_{i}\) is the ionic charge, and \(X_{1}\) and \(X_{2}\) are the activities of transported solute (in millimolar) in compartments to the left and right of the diffusive barrier, respectively. \(U_{\mathrm{z}}\) , the dimensionless membrane potential, is related to the appropriate potential difference (ψ_z) in Table 2 . Equations for flux through each transport pathway ( \(J_{i}^{X}\) ), and the resultant net apical, basolateral, and paracellular currents ( I_a, I_b, and I_p) are provided in the ^Appendix (equations A1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 –A17 ). 

Numerical Solution.

Cell membrane potentials at time ( \(t\ {+}\ {\Delta}t\) ) were calculated iteratively with the Newton-Raphson method based on electroneutral conditions. The open-circuit model requires equal opposing transcellular and paracellular currents ( \(I_{\mathrm{a}}\) = \(I_{\mathrm{b}}\) = − \(I_{\mathrm{p}}\) ). An option was included to model transport under short-circuit conditions, where \(I_{\mathrm{a}}\) = \(I_{\mathrm{b}}\) and \(U_{\mathrm{a}}\) = \(U_{\mathrm{b}}\) . \(U_{\mathrm{a}}\) and \(U_{\mathrm{b}}\) were computed iteratively by user-supplied initial guess. Assuming instantaneous osmotic equilibration and constant cell surface area, cell height \(h(t{+}{\Delta}t)\) (in micrometers) was computed as  

\[h(t{+}{\Delta}t){=}h(t){+}{\Delta}t{\Sigma}^{a{-}b}{\phi}_{x}J_{i}^{X}/{\psi}_{s}.\]

\({\Sigma}^{a{-}b}{\phi}_{x}J_{i}^{X}\) generically represents the net osmotic flux (in milliosmolar per square centimeter per second) into the cell and is related to individual transporter fluxes in equation A18 . \({\psi}_{\mathrm{s}}\) is mouse serum osmolarity (in milliOsmols per cubic centimeter). Intracellular solute activities were computed from fluxes and the updated cell height from equation 2according to  

\[X(t{+}{\Delta}t){=}{[}X(t)h(t){+}{\Delta}t{\Sigma}a_{x}J_{i}^{X}{]}/h(t{+}{\Delta}t).\]

\({\Sigma}a_{x}J_{i}^{X}\) is defined as the net rate of entry of active solute X (in 10⁴ meq/cm² per second) into the cellular compartment through the various transporters. Refer to equations A19 –A22 for the specific expressions used in computing \(Na^{{+}}(t{+}{\Delta}t)\) , \(K^{{+}}(t{+}{\Delta}t)\) , \(Cl^{{-}}(t{+}{\Delta}t)\) , and \(org(t{+}{\Delta}t)\) . The model also offers the option to impose brief current or voltage spikes (one each minute) to compute total tissue resistance. The observed change in voltage or current can be used to monitor transepithelial electrical resistance (TEER, in kilo-ohms per square centimeter) under open- or short-circuit conditions, respectively (TEER = Δψ_t/ Δ I _a). Though the exact mechanism of transepithelial fluid movement remains controversial, it is agreed that it results in near-isosmolar water flux driven by solute transport. As such, net transepithelial water secretion or absorption (microliters per square centimeter per hour) was computed assuming isosmolar fluid flow across the apical membrane and paracellular tight-junctional barrier by  

\[J_{v}(t{+}{\Delta}t){=}{\Sigma}^{a{+}p}{\phi}_{x}J_{i}^{X}/{\psi}_{\mathrm{s}},\]

where \({\Sigma}^{a{+}p}{\phi}_{x}J_{i}^{X}\) is the net osmotic flux (in milliOsmols per square centimeter per second) into the apical compartment from individual transport pathway fluxes (equation A23) , and \({\psi}_{\mathrm{s}}\) is serum osmolarity (in Osmols per liter). 

Results

Amiloride-Sensitive Na⁺ Transport

Na⁺ channel function was studied by using ion replacement and transporter substrates and inhibitors. Baseline PD was first established (using solution 1) before replacement of perfusate Na⁺ by the relatively impermeant cation choline (solution 2). This exchange produced a 5- to 10-mV depolarization that was reversed by reintroduction of Na⁺ (solution 1). Representative PD recordings in Figure 2Ashow the reversible changes in PD on Na⁺ substitution, depolarization by amiloride (100 μM) in the presence of a Na⁺-containing perfusate (left), and blocking of the Na⁺-induced hyperpolarization by amiloride (right). Dose-inhibition studies were performed for amiloride and its analogue, benzamil. A representative dose–response curve for amiloride is shown in Figure 2B(left). K _i was 0.82 μM for amiloride and 0.22 μM for benzamil (right). These data indicate the involvement of amiloride- and benzamil-sensitive Na⁺ channel(s) in ocular surface Na⁺ absorption, which is probably ENaC. 

To determine the contribution of the other major cation K⁺ to apical electrogenic transport, experiments were performed using a high-K⁺ solution (solution 6). Amiloride was first added to solution 1 to minimize the influence of Na⁺ channel function on interpretation of PDs in terms of K⁺ channel function. Switching to solution 6 produced a very small 1- to 2-mV reversible hyperpolarization (data not shown, n = 5 separate experiments). Model computations supported the conclusion that apical K⁺ conductance is much lower than apical Na⁺ or Cl⁻ conductance. 

Ion Transport in CF Mice.

Ocular surface Cl⁻ transport was assayed as described previously. ⁴ Figure 3Ashows PD data from wild-type versus CF mice in which baseline PDs (of similar values) were first recorded with solution 1. Switching to an amiloride-containing solution depolarized ocular surface PDs, with CF mice showing a significantly greater response (15–20 mV in CF vs. 5–10 mV in wild-type mice). Replacing most Cl⁻ by the relatively impermeant anion gluconate (solution 5) gave a sustained hyperpolarization in wild-type but not CF mice, which is related to cAMP- but not CFTR-independent Cl⁻ secretion. The CFTR agonist forskolin produced a further hyperpolarization in wild-type mice that was reversed by CFTR_inh-172. 

The protocol used in Figure 2Awas applied to investigate amiloride-sensitive Na⁺ conductance in CF mice. Representative PD recordings in Figure 3B(left) show reversible depolarizations in wild-type and CF mice in response to Na⁺ substitution. Depolarizations of similar magnitude were produced by amiloride (100 μM) in the presence of Na⁺-containing perfusate. The CF mice consistently showed greater responses than wild-type mice to both Na⁺ replacement and amiloride addition. As summarized in Figure 3B(right), Na⁺ replacement and amiloride administration produced ΔPDs of 20 ± 3 and 19 ± 2 mV, (SE, n = 6) in CF mice, respectively. The same maneuvers produced 6 ± 2- and 7 ± 1-mV depolarizations in wild-type mice (n = 6). These results indicate the presence of an amiloride-sensitive Na⁺ channel at the ocular surface, with apparent increased activity in CF mice. 

Modeling Ocular Surface Na⁺ and Cl⁻ Transport.

The PD measurements suggest that active Na⁺ and Cl⁻ transport processes dictate net fluid movement across corneal-conjunctival epithelia, and comparative experiments performed on wild-type versus CF mice suggest a complex interaction among these pathways. As such, we developed a model to gain better qualitative understanding and to define quantitatively the key determinants of active ocular surface fluid secretion. 

According to the model in Figure 1with parameters for the ocular surface of wild-type mice selected as described in the ^Appendix, Figure 4Ashows the time course of the major cellular variables in mice after “inhibition” (by amiloride) of transporter 1 (ENaC conductance). Before inhibition, all parameters were stable to within 0.01% for a 60-minute simulation (not shown). The top graph shows the time-dependent apical, basolateral, and transepithelial potentials (ψ_a, ψ_b, and PD). Conductances were chosen to give a baseline PD of −23 mV and a ratio of active Na⁺ to Cl⁻ flux of 1:1, which produced a 5.5-mV amiloride-induced PD depolarization. The second graph shows total transepithelial current before and after amiloride. The steady-state current of 8.0 μA/cm² was reduced by 22% by ENaC inhibition. The current spikes produced by the periodic brief voltage spikes gave a baseline TEER of 5.3 kΩ/cm², which increased to 6.8 kΩ/cm² after amiloride addition. The bottom two panels in Figure 4Ashow the influence of apical Na⁺ conductance on the three major intracellular ions and cell height. There was a small decrease in cell height consequent to reduced cellular Na⁺. Secondarily enhanced transport of K⁺ into the cell produced a small 0.4% final increase in cell height. In general, each cell parameter reached a new steady state within minutes after inhibition of transporter 1. 

Figure 4Bshows the time-dependent PD response in open-circuit conditions after a series of maneuvers commonly used to elucidate transport mechanisms. As seen in the top graph, simulated Na⁺ replacement by choline yielded a depolarization similar to the experimental findings in Figures 2A and 3B . The immediate and sustained hyperpolarization of 9.5 mV predicted on replacing most apical Cl⁻ with gluconate (solution 1 switched to solution 5) also replicated experimental trends. The bottom two panels predict the time course of PD reduction toward zero after basolateral inhibition of the lone source of cellular Cl⁻, the Na⁺/K⁺/2Cl⁻ symporter (transporter 6), and the transepithelial current generator, the 3Na⁺/2K⁺ATPase (transporter 7). Because the electroneutral Na⁺/K⁺/2Cl⁻ symporter lacks a membrane-potential dependence, no immediate change in PD was present on its inhibition. 

The main findings from ocular PD experiments were modeled. In Figure 5A , the protocol for studying chloride transport was simulated, with initial ENaC inhibition, followed serially by switching to a low-Cl⁻ solution, addition of a CFTR activator, and then addition of a CFTR inhibitor. This simulation resembled experimental PD data (as in Fig. 3A , top) with one exception. Simulated instantaneous Cl⁻ channel activation yielded a significant transient hyperpolarization followed by more modest sustained hyperpolarization. Such PD behavior is not seen experimentally in response to either forskolin or direct CFTR activators, but is observed after addition of UTP. ⁴ Of note, incorporating rectification into the basolateral K⁺ channel (transporter 8) conductance (as described by Horsiberger ¹¹ ) did not alter this general behavior. To mimic the finite solution exchange time and the noninstantaneous time course of CFTR activation after addition of the agonist, an additional simulation was performed in which apical chloride conductance was increased to the same extent, but over 4 minutes (not shown). The slower channel activation blunted much of the transient hyperpolarization, but did not change steady state PD. 

Direct CFTR activators and the general cAMP agonist forskolin have been found to elicit nearly identical diffusion potentials under low-Cl⁻ conditions. ⁴ To test whether isolated CFTR activation can be predicted to enhance apical Cl⁻ secretion in a manner similar to forskolin, simulations were performed in which basolateral K⁺ (transporter 8) conductance was activated to various degrees along with CFTR. Indeed, only a small augmentation of CFTR-activator–induced hyperpolarization could be achieved by concurrent stimulation of basolateral K⁺ conductance (–35.4-mV PD after threefold CFTR and K⁺ channel activation vs. −34.4 mV for CFTR activation alone, not shown). These findings have implications regarding strategies of pharmacological modulation of fluid secretion (see the Discussion section). 

The simulation in Figure 5Bfocused on the difference in amiloride effect in wild-type versus CF mice (as seen in Figs. 3A 3B ) and the possibility of CFTR-ENaC interactions. Model parameters for wild-type and CF mice were chosen to yield identical baseline PDs of –23 mV, as reported previously ⁴ and depicted in Figure 3B . Apical Cl⁻ permeability in CF mice was chosen to be 20% of that in wild-type mice, to recapitulate the observed amiloride effect in CF mice whereas Na⁺ conductance was fixed to that in wild-type mice. As seen in Figure 5B , PD depolarized by 14.5 mV upon ENaC inhibition in CF mice, similar to that measured in Figure 3B . Switching the apical compartment to a low-Cl⁻ solution (solution 5) correctly predicted a small diffusion potential of 2.1 mV in CF mice, compared with the 8.7-mV hyperpolarization in wild-type mice. Simulations were also performed in which ENaC permeability was increased twofold (using parameters from CF or wild-type mice), examining effects of ENaC inhibition and low-Cl⁻ substitution. ENaC inhibition yielded depolarizations of 19 and 10 mV with CF and wild-type parameters, respectively, and hyperpolarizations of 4.1 and 8.8 mV for low Cl⁻. In both cases, ENaC hyperactivity produced substantial low-Cl⁻ effects, which was inconsistent with experimental findings. Of importance, the model reproduced the major ocular surface electrophysiological properties in wild-type and CF mice only in the absence of CFTR-dependent ENaC conductance (see the Discussion section). 

By assuming isosmolar fluid secretion, implying fixed coupling between transepithelial solute and water transport, we also modeled the ability of transporter modulators to increase transepithelial water secretion. Computations were performed under physiological conditions (in the absence of transepithelial ionic gradients) to simulate compound action on fluid secretion into the native tear film. Figure 5Cshows that both ENaC inhibition and CFTR activation increased net fluid secretion by inhibiting Na⁺ absorption and enhancing Cl⁻ secretion, respectively, and that the effects were additive. 

Organic Solute-Coupled Na⁺ Transport.

In addition to amiloride-sensitive Na⁺ channels, electrogenic Na⁺ transport across the ocular surface, and thus fluid secretion, may also involve Na⁺-glucose and Na⁺-amino acid cotransport. Figure 6A (left) shows that isosmolar addition of d-glucose but not of d-mannitol produced a small hyperpolarization that was reversed by the Na⁺-glucose cotransporter inhibitor phloridzin. l-Glucose (5 mM) produced no significant change in PD (data not shown, n = 4 eyes). Under physiological conditions of high apical Na⁺ concentration, extracellular d-glucose saturability (K _m) was 2.5 mM as measured from PDs at increasing concentrations of d-glucose (Fig. 6A , middle and right). Hill analysis gave a d-glucose-Na⁺ coupling ratio of 0.89, consistent with 1:1 Na⁺-glucose cotransport. The PD data in Figure 6Awere modeled to determine the turnover rates of transporters 2 and 9 (see ^Appendix for explanation of parameter selection). Modeling of the experimentally measured ∼4-mV hyperpolarization under conditions of saturated cotransport and tonic ENaC inhibition (Fig. 6B)indicated a \(J_{2}\) (SGLT-1) turnover equal to ∼75% of the \(J_{1}\) (ENaC) basal activity (0.11 vs. 0.15 μeq/cm²per hour). Because SGLT-1 likely transports two solutes per turnover, this implies similar osmolar absorptive capacities of amiloride-insensitive and amiloride-sensitive pathways. The basic and neutral amino acids l-arginine and glycine also produced small, reversible hyperpolarizations in the presence of Na⁺ (Fig. 6C) . Amino acid transport was saturated only at relatively high concentration (several mM) for both amino acids studied. l-Arginine and glycine, added at 10 mM, yielded hyperpolarizations of 1.5 ± 0.9 mV (n = 4, SE) and 2.1 ± 0.4 mV (n = 5), respectively. PD analysis also revealed competitive substrate binding, where addition of l-arginine to glycine-containing solution reproducibly caused a depolarization (Fig. 6C , right). The Na⁺ dependence of these electrogenic pathways of glucose and amino acid absorption was confirmed. Figure 6D(left) shows hyperpolarizations produced by 5 mM d-glucose and 1 mM l-arginine (but not mannitol) in the presence of Na⁺ and amiloride. The hyperpolarizations were abolished after Na⁺ replacement by choline. Averaged results are summarized in Figure 6D(right). Together, these results provide evidence for at least three distinct electrogenic Na⁺ pathways at the ocular surface: amiloride-sensitive Na⁺ channels, Na⁺-glucose cotransport, and Na⁺-amino acid cotransport. 

Discussion

The goals of this study were to identify experimentally and to quantify by modeling the major Na⁺-transporting pathways at the ocular surface and to use experimental and modeling results to examine the roles of epithelial transporters in driving fluid secretion and putative CFTR-ENaC interactions. This approach provides a framework to define electrochemical coupling relevant to a variety of epithelial disorders, and for ocular surface epithelium, this approach allowed us to predict computationally the efficacy of therapies for states of tear deficiency. The PD measurement method is technically simple and permits minimally invasive in vivo measurements under physiological open-circuit conditions. The high-resistance epithelial surface, comprising cornea and conjunctiva in parallel, is responsible for generating and maintaining a large PD. The dependence of ocular PDs on specific Na⁺ and Cl⁻ transport processes, combined with transport agonist-inhibitor and ion substitution maneuvers, allows for rapid qualitative assessment of solute transport in vivo. In this study, we extended the PD measurement concept by developing a mathematical model to rigorously relate measured PDs to transporter permeabilities and transporting mechanisms. 

A major role for amiloride-sensitive apical Na⁺ absorption was found, which, according to the model, was equal in magnitude to total net Cl⁻ secretion. The inhibitory half-concentrations of amiloride (K _i, 0.82 mM) and benzamil (K _i, 0.22 mM) measured are in close agreement with those reported recently in primary cultures of pigmented rabbit corneal epithelial cells ⁸ and are consistent with the greater potency of benzamil than amiloride for ENaC. Comparable depolarizations were observed for Na⁺ replacement and amiloride application in both wild-type and CF mice, suggesting that amiloride-sensitive Na⁺ conductance provides the primary route for apical membrane Na⁺ transport under the experimental conditions (solution 1, which lacks d-glucose and amino acids). Modeling of fluid secretion in the absence of an ionic gradient predicted that Na⁺ channel inhibition or CFTR Cl⁻ channel activation would increase fluid secretion into the tear film and that both together would provide an even greater benefit than either strategy alone. 

In contrast to the large measured apical Na⁺ conductance, a weak dependence of ocular surface PD on perfusate K⁺ concentration was found in the presence of amiloride, indicating a relatively small apical surface K⁺ conductance. Similar results were reported for human nasal epithelia. ¹⁹ At steady state, basolateral K⁺ channels enhance the electrochemical driving force for fluid secretion. However, modeling of PDs in this study predicted that increased basolateral K⁺ conductance would augment apical Cl⁻ secretion little, suggesting that CFTR-specific activators would be as effective as general cAMP agonists (which activate CFTR and some basolateral K⁺ channels) in increasing apical chloride-driven fluid secretion. 

Ocular surface PD measurements also provided direct evidence for transepithelial glucose- and amino acid-coupled Na⁺ absorption. The kinetics of substrate activation for the Na⁺-glucose cotransporter (SGLT-1) varies widely among species. ²⁰ Whereas tear Na⁺ concentration (>100 mM) is thought to provide a saturating concentration for Na⁺ (K _m ∼ 60 mM), the affinity of SGLT-1 for glucose must be determined experimentally for a given system. The K _m of 2.5 mM for glucose measured in this study is lower than that of 16.7 mM measured in rabbit conjunctiva. ¹⁸ The relatively low concentration of glucose in the normal human tear film (∼200 μM) is unlikely to cause significant fluid absorption. ²⁰ However, elevated tear glucose may contribute to the ocular surface disease noted in hyperglycemic diabetes (∼1 mM) by producing net absorption in the steady state. ^{21 22} This prediction is supported by data in rabbits obtained by Shiue et al., ²³ who found a ∼75% reduction in transconjunctival fluid secretion upon apical addition of saturating glucose. 

Na⁺-dependent neutral–basic amino acid cotransport has also been characterized in rabbit conjunctiva, where both high (micromolar K _m)- and low (millimolar K _m)-affinity processes have been described for l-arginine. ¹⁷ We measured qualitatively significant absorption at superphysiologic (in millimolar) amino acid concentrations, compared with the low-micromolar values reported by Puck et al. ²⁴ in native tear film. Thus, although Na⁺-coupled amino acid cotransport may not be relevant to steady-state tear fluid balance, our results support the strategy of delivering ocular therapeutics, either as amino acid analogues or conjugates, through cotransporters. ²⁵  

Although both corneal and conjunctiva epithelia are complex multilayered tissues, the ocular surface epithelial cell is modeled in the current study as a single cell layer with a parallel shunt, as was done previously to study passive solute fluxes across corneal epithelium. ²⁶ Measurements in rabbit corneal epithelium have documented intimate electrical connection among superficial and wing cell layers, in support of this assumption. ²⁷ Moreover, the apical superficial cell membrane largely dictates the electrical properties of stratified ocular surface epithelia because of its highly resistive tight junctions. ^{27 28} The magnitudes of flux chosen in this simulation yielded a TEER of 5.3 kΩ/cm², between the range of 12 to 17 kΩ/cm² measured under short-circuit conditions in rabbit cornea ^{5 29} and that of 1 to 2 kΩ/cm² reported in rabbit conjunctiva. ^{6 17 25} A potential weakness of the current model is the simplistic consideration of paracellular conductances in the context of a complicated multilayered epithelium, where possible unstirred layers could affect intercellular solute movement. In polarized epithelia that can both absorb and secrete ions to comparable extents, such as those lining the ocular surface, relative paracellular ion conductances largely determine the degree of basal fluid absorption and secretion. ²⁹ However, the treatment of paracellular conductance would only affect estimates of basal fluid secretion and not predictions regarding the utility of membrane-transport modulators. Relative impermeant-to-permeant paracellular ion permeabilities of 0.7 were selected for anions and cations, to reproduce the minimal effect of Na⁺ replacement by choline or K⁺ in the presence of amiloride. Of note, these parameters are also in accord with the measurements of Amasheh et al. ³⁰  

As has been found in nasal PD measurements in CF versus wild-type mice ^{31 32} and CF versus non-CF human subjects, ¹⁹ amiloride produced a much greater depolarization at the ocular surface in CF versus wild-type mice (seen in Fig. 3 ). We also found an enhanced depolarization in CF mice after Na⁺ replacement. There is ongoing controversy regarding the mechanism responsible for these apparent CFTR-ENaC interactions. Whereas some studies have suggested direct inhibition of ENaC function by CFTR, ^{10 33} recent modeling and careful experimentation suggest that electrochemical coupling accounts for the apparent hyperabsorption of Na⁺ across CF epithelia. ^{11 34} We investigated purported CFTR-ENaC interactions by modeling the system with parameters for CF mice chosen to test whether enhanced Na⁺ absorption across CF epithelia can be explained by electrochemical coupling between parallel-functioning transporters. PDs from the CF mouse ocular surface were accurately simulated with fivefold reduced apical Cl⁻ conductance, yet identical Na⁺ conductance, compared with wild-type mice, indicating that direct regulation of ENaC by CFTR by a mechanism other than electrochemical coupling is not necessary to explain the experimental results. Although CF epithelia are predicted to have 20% of normal function, both experiments and modeling suggest that most of the unstimulated (cAMP- and Ca²⁺-independent) Cl⁻ flux passes through CFTR-dependent channels, though not through CFTR itself. We showed previously that the low-Cl⁻ hyperpolarization in wild-type mice was reversed only to a small extent by a CFTR inhibitor. ⁴ Our model also demonstrated that reduced apical Cl⁻ conductance and unaltered Na⁺ absorptive capacity were both necessary to abolish most of the low Cl⁻ effect in CF mice (2.1-mV hyperpolarization predicted in CF as found experimentally). 

In conclusion, our results define quantitatively the principal Na⁺-transporting pathways at the ocular surface and the electrochemical coupling between Na⁺ and Cl⁻ transport in wild-type and CF ocular surface epithelia. The model predicted significant enhancement of serosal-to-mucosal fluid transport by Na⁺ channel inhibitors and Cl⁻ channel activators. Direct measurement of fluid secretion across the intact ocular surface and studies in animal models of dry eye syndrome are needed to guide and validate the modeling of fluid secretion. 

Table A1.

 

Baseline, Steady State Cell Parameters for Wild-Type Mouse Open-Circuit PDs

	Apical	Cell	Basolateral
Solute activities (mM)
Na+	119	20	112
Cl−	112	33	84
K+	3.3	75	3.2
Organic*	30	20	10
Net active ion flux (μeq/cm2		J Na+	J Cl−
per hour)		0.15	−0.15
Membrane potential (mV)	ψa	ψb	ψp
	−50	−73	−23

 * Organic (d-glucose) was absent from most apical perfusates but was included at 30 mM when calculating P ₂ and P ₉.

Table A2.

 

Transporter Permeabilities and Baseline, Steady-State Fluxes

Transporter	P i	Units	J i (μeq/cm2, per hour)
Apical membrane
1 Na+ conductance	1.63× 10−7	cm/s	0.15
2 Na+/org symporter	1.43× 10−7	cm/s · mM−2	0.11
3 K+ conductance	1.52× 10−8	cm/s	−0.001
4/5 Cl− conductance	1.21× 10−6	cm/s	−0.15
Basolateral membrane
6 Na+/K+/2Cl− symporter	1.81× 10−8	cm/s · mM−4	0.075
7 3Na+/2K+ ATPase	1.73× 10−4	μmol · cm−2 · s	0.075
8 K+ conductance	1.24× 10−5	cm/s	0.224
9 Org conductance	3.06× 10−6	cm/s	0.11
Paracellular
10 Na+ conductance	4.54× 10−7	cm/s	−0.15
11 K+ conductance	4.54× 10−7	cm/s	−0.00439
12 Cl− conductance	3.53× 10−7	cm/s	0.145
13 Choline+ conductance	3.18× 10−7	cm/s	—
14 Gluconate− conductance	2.47× 10−7	cm/s	—

 Transporter numbers correspond to those depicted in Figure 1 .

Table A3.

 

Activity and Osmotic Coefficients

Solute	a x	φ x	a x e	φ x e
Na+, K+, Cl−	0.75	0.7	0.75	0.92
org	1	1	1	1

Appendix 1

Model Flux Equations

For electrogenic fluxes, individual transport pathways were governed by simple electrodiffusion unless otherwise specified, with fluxes defined by the Goldman-Hodgkin-Katz equation (equation 1in the main text). Neutral transport pathways lacked a potential dependence. Each equation is provided below, along with any necessary explanation.  

\[J_{1}{=}P_{1}U_{\mathrm{a}}{[}\mathrm{Na}_{\mathrm{a}}^{{+}}{-}\mathrm{Na}^{{+}}(t)e^{{-}U_{\mathrm{a}}}{]}/(1{-}e^{{-}U_{\mathrm{a}}}).\]

 

\[J_{2}{=}P_{2}U_{\mathrm{a}}\ \frac{\mathrm{Na}_{\mathrm{a}}^{{+}}\mathrm{org}_{\mathrm{a}}{-}\mathrm{Na}^{{+}}(t)\mathrm{org}(t)e^{{-}U_{\mathrm{a}}}}{\left(\ \frac{\mathrm{Na}_{\mathrm{a}}^{{+}}}{K_{2}^{\mathrm{Na}}}{+}1\right)\ \left(\ \frac{\mathrm{org}_{\mathrm{a}}}{K_{2}^{\mathrm{org}}}{+}1\right)\ (1{-}e^{{-}U_{\mathrm{a}}})}.\]

Transporter 2, when modeled as the Na⁺-glucose cotransporter, was assigned extracellular saturability values for both Na⁺ ( \(K_{2}^{\mathrm{Na}}\) ) and glucose ( \(K_{2}^{\mathrm{org}}\) ). \(K_{2}^{\mathrm{Na}}\) was assumed to be 60 mM based on data of Horibe et al. ¹⁸ in rabbit conjunctiva. \(K_{2}^{\mathrm{org}}\) was determined experimentally to be 2.5 mM at the mouse ocular surface by dose–response experiments under conditions of saturating apical Na⁺.  

\[J_{3}{=}P_{3}U_{\mathrm{a}}{[}\mathrm{K}_{\mathrm{a}}^{{+}}{-}\mathrm{K}^{{+}}(t)e^{{-}U_{\mathrm{a}}}{]}/(1{-}e^{{-}U_{\mathrm{a}}}),\]

 

\[J_{4}{=}{-}P_{4}U_{\mathrm{a}}{[}\mathrm{Cl}_{\mathrm{a}}^{{-}}{-}\mathrm{Cl}^{{-}}(t)e^{U_{\mathrm{a}}}{]}/(1{-}e^{U_{\mathrm{a}}}),\]

 

\[J_{5}{=}{-}P_{5}U_{\mathrm{a}}{[}\mathrm{Cl}_{\mathrm{a}}^{{-}}{-}\mathrm{Cl}^{{-}}(t)e^{U_{\mathrm{a}}}{]}/(1{-}e^{U_{\mathrm{a}}}),\]

 

\[J_{6}{=}P_{6}\ \frac{\mathrm{Na}^{{+}}(t)\mathrm{K}^{{+}}(t)\mathrm{Cl}^{{-}}(t)^{2}{-}\mathrm{Na}_{\mathrm{b}}^{{+}}\mathrm{K}_{\mathrm{b}}^{{-}}(\mathrm{Cl}_{\mathrm{b}}^{{-}})^{2}}{\left[1{+}\ \frac{\mathrm{Na}_{\mathrm{b}}^{{+}}}{K_{6}^{\mathrm{Na}}}\right]\ \left[1{+}\ \frac{\mathrm{K}_{\mathrm{b}}^{{+}}}{K_{6}^{K}}\right]\ \left[1{+}\ \frac{\mathrm{Cl}_{\mathrm{b}}^{{-}}}{K_{6}^{\mathrm{Cl}}}\right]\ ^{2}},\]

 

\[J_{7}{=}P_{7}\left[\ \frac{\mathrm{Na}^{{+}}(t)}{\mathrm{Na}^{{+}}(t){+}K_{7}^{\mathrm{Na}}}\right]\ ^{3}\left[\ \frac{\mathrm{K}^{{+}}(t)}{\mathrm{K}_{\mathrm{b}}^{{+}}{+}K_{7}^{K}}\right]\ ^{2}(a\ {\cdot}\ U_{\mathrm{b}}{+}b).\]

Flux equations for transporters 6 and 7 (the Na⁺/K⁺/2Cl⁻ symporter and 3Na⁺/2K⁺ATPase) were identical with those used in the model of tracheal epithelia by Hartmann and Verkman. ¹⁴ Saturability (in millimolar) was assigned accordingly ( \(K_{6}^{\mathrm{Na}}\) = 3.8; \(K_{6}^{K}\) = 7.5; \(K_{6}^{\mathrm{Cl}}\) = 26; \(K_{7}^{\mathrm{Na}}\) = 11.8; and \(K_{7}^{K}\) = 1.4), as were the constants in equation A7that define the weak basolateral membrane potential dependence of 3Na⁺/2K⁺ATPase ( \(\mathit{a}\) = 0.006; \(\mathit{b}\) = \(1{-}a\ {\cdot}\ U_{b}\) ).  

\[J_{8}{=}P_{8}U_{\mathrm{b}}{[}\mathrm{K}^{{+}}(t){-}\mathrm{K}_{\mathrm{b}}^{{+}}e^{{-}U_{b}}{]}/(1{-}e^{{-}U_{b}}),\]

 

\[J_{9}{=}P_{9}{[}\mathrm{org}(t){-}\mathrm{org}_{\mathrm{b}}{]},\]

 

\[J_{10}{=}P_{10}U_{\mathrm{p}}{[}\mathrm{Na}_{\mathrm{a}}^{{+}}{-}Na_{\mathrm{b}}^{{+}}e^{{-}U_{\mathrm{p}}}{]}/(1{-}e^{{-}U_{\mathrm{p}}}),\]

 

\[J_{11}{=}P_{11}U_{\mathrm{p}}{[}\mathrm{K}_{\mathrm{a}}^{{+}}{-}\mathrm{K}_{\mathrm{b}}^{{+}}e^{{-}U_{\mathrm{p}}}{]}/(1{-}e^{{-}U_{\mathrm{p}}}),\]

 

\[J_{12}{=}{-}P_{12}U_{\mathrm{p}}{[}\mathrm{Cl}_{\mathrm{a}}^{{-}}{-}\mathrm{Cl}_{\mathrm{b}}^{{-}}e^{U_{\mathrm{p}}}{]}/(1{-}e^{U_{\mathrm{p}}}),\]

 

\[J_{13}{=}P_{13}U_{\mathrm{p}}{[}\mathrm{chol}_{\mathrm{a}}^{{+}}{-}\mathrm{chol}_{\mathrm{b}}^{{+}}e^{{-}U_{\mathrm{p}}}{]}/(1{-}e^{{-}U_{\mathrm{p}}}),\]

 

\[J_{14}{=}{-}P_{14}U_{\mathrm{p}}{[}\mathrm{gluc}_{\mathrm{a}}^{{-}}{-}\mathrm{gluc}_{\mathrm{b}}^{{-}}e^{U_{\mathrm{p}}}{]}/(1{-}e^{U_{\mathrm{p}}})\]

 

Model Parameter Selection

Transporter permeability coefficients were calculated from the flux equations using estimated ion activities, membrane potentials, and net transcellular Na⁺ and Cl⁻ fluxes under open-circuit conditions (see Table A1 for a summary of parameters for simulations in wild-type mice). Apical ion activities were chosen from the composition of solution 1. Basolateral ion activities were assigned based on serum concentrations from CD1 wild-type mice (measured by the University of California, San Francisco, Moffitt Hospital Clinical Laboratory; n = 5). Cellular ion activities and apical and basolateral membrane potentials were estimated from intracellular microelectrode measurements made in frog and rabbit corneal epithelial in vitro systems. ^{35 36} Baseline cellular Cl⁻ activity was set at 33 mM, 1.9-fold greater than predicted for passive distribution across the apical membrane. ³⁵ Values of −50 and −73 mV were chosen for ψ_a and ψ_b, respectively, to yield the experimentally measured steady state transepithelial potential (ψ_a − ψ_b = −23 mV). 

All transcellular permeability coefficients (with the exceptions of P₂ and P₉) were determined from estimated steady state net active Na⁺ and Cl⁻ fluxes. The ratio of active Na⁺ absorption to Cl⁻ secretion (1:1 in wild-type mice and 5:1 in CF mice) was deduced experimentally from the magnitude of the amiloride effect, as described in the Results and Discussion sections. Because no levels of ion fluxes across mouse ocular surface epithelia have been determined, the absolute magnitudes of active Na⁺ and Cl⁻ flux for wild-type mice were chosen as intermediate values between those for cornea and conjunctiva, which were reported from rabbit and frog under both open- and short-circuit conditions. ^{6 8 17 18 29}  

We had some freedom in selecting the relative apical and basolateral K⁺ fluxes (J₃ and J₈, respectively). However, modeling of the very weak experimentally determined dependence of PD on apical K⁺ confirmed that most of the K⁺ flux occurred through transporter 8 ( \(J_{3}\) : \(J_{8}\) = −1:224 in wild-type mice). Paracellular permeability values were selected based on the small effect of Na ⁺ replacement on PD (see Fig. 2A . Equal and opposing transcellular and paracellular Na ⁺ fluxes were assumed. ²⁹ A combination of higher assumed baseline absolute transcellular Na ⁺ and Cl ⁻ and relative paracellular Na ⁺ flux values would increase the predicted baseline ocular surface fluid secretion rate. Based on constraints imposed by our own experimental observations and in accordance with flux measurements through tight junctions, ³⁰ Na⁺ and K⁺ permeabilities ( \(P_{10}\) and \(P_{11}\) ) were assumed to be equal, with relative selectivity for choline ⁺ versus Na⁺ and K⁺ as \(P_{13}\) : \(P_{10}\) : \(P_{11}\) = 0.7:1:1, and for gluconate versus chloride as \(P_{14}\) : \(P_{12}\) = 0.7:1 (Table A2). 

Simulations involving Na⁺-organic cotransport (Fig. 6B)focused on Na⁺-coupled glucose permeation. \(P_{2}\) was chosen to produce in simulations the 4- to 5-mV hyperpolarization measured experimentally in the presence of amiloride. \(P_{9}\) was then selected to give equal apical and basolateral membrane glucose fluxes in the steady state ( \(J_{2}\) = \(J_{9}\) ). Parameters for these simulations were selected based on the intracellular Na⁺ activity (12.1 mM) and ψ_a (−59.9 mV) achieved after amiloride addition and on the apical Na⁺ and Cl⁻ contents of solution 3. 

Iterative Procedure

Computations in this study were performed under open-circuit conditions. Fluxes through each transport conduit were computed using the guesses for ψ_a and ψ_b over time intervals of Δt = 1 second. Computations using a smaller step size (Δt = 0.1 second) gave similar results, confirming the adequacy of the 1-second step size. Total currents across each major barrier were then calculated at the end of each time interval:  

\[I_{\mathrm{a}}{=}96500\ {\cdot}\ (J_{1}{+}J_{2}{+}J_{3}{-}J_{4}{-}J_{5}),\]

 

\[I_{\mathrm{b}}{=}96500\ {\cdot}\ (J_{7}{+}J_{8}),\]

 

\[I_{\mathrm{p}}{=}96500\ {\cdot}\ (J_{10}{+}J_{11}{-}J_{12}{+}J_{13}{-}J_{14}),\]

where 96,500 represents Faraday’s constant, which converts flux into current. The threshold for acceptable relative deviation from open-circuit electroneutrality was set at 0.01%. If boundary conditions were not met, both ψ_a and ψ_b were modified by 1 mV and the two-dimensional Newton-Raphson method was used to update guesses for both ψ_a and ψ_b for the next iteration. 

Once electroneutrality was established ( \(I_{\mathrm{a}}\) = \(I_{\mathrm{b}}\) = − \(I_{\mathrm{p}}\) ), changes in cell volume (expressed as height, assuming a constant surface area) were computed based on net isosmolar water fluxes. The equation for computing net osmotically active solute flux into the cellular compartment is  

\[{\Sigma}^{a{-}b}{\phi}_{X}J_{i}^{X}{=}{\phi}_{\mathrm{Na}}J_{1}{+}({\phi}_{\mathrm{Na}}{+}{\phi}_{\mathrm{org}})J_{2}{+}{\phi}_{K}J_{3}{+}{\phi}_{\mathrm{Cl}}J_{4}{+}{\phi}_{\mathrm{Cl}}J_{5}{-}({\phi}_{\mathrm{Na}}{+}{\phi}_{\mathrm{K}}{+}2{\phi}_{\mathrm{Cl}})J_{6}{-}(3{\phi}_{\mathrm{Na}}{-}2{\phi}_{\mathrm{K}})J_{7}{-}{\phi}_{\mathrm{K}}J_{8}{-}{\phi}_{\mathrm{org}}J_{9}.\]

Steady state cell height was assumed to be 5 μm, similar to the height of the superficial ocular surface cell layer. The exact choice of cell height was important only for pre–steady-state kinetics after perturbation. Employing the updated cell height as well as solute fluxes, new intracellular solute activities at time (t + Δt) were computed:  

\[\mathrm{Na}^{{+}}(t{+}{\Delta}t){=}{[}\mathrm{Na}^{{+}}(t)\ {\cdot}\ h(t){+}{\Delta}t\ {\cdot}\ a_{\mathrm{Na}}\ {\cdot}\ 10^{4}\ {\cdot}\ (J_{1}{+}J_{2}{-}J_{6}{-}3J_{7}){]}/h(t{+}{\Delta}t),\]

 

\[\mathrm{K}^{{+}}(t{+}{\Delta}t){=}{[}\mathrm{K}^{{+}}(t)\ {\cdot}\ h(t){+}{\Delta}t\ {\cdot}\ a_{K}\ {\cdot}\ 10^{4}\ {\cdot}\ (J_{3}{+}2J_{7}{-}J_{6}{-}J_{8}){]}/h(t{+}{\Delta}t),\]

 

\[\mathrm{Cl}^{{-}}(t{+}{\Delta}t){=}{[}\mathrm{Cl}^{{-}}(t)\ {\cdot}\ h(t){+}{\Delta}t\ {\cdot}\ a_{\mathrm{Cl}}\ {\cdot}\ 10^{4}\ {\cdot}\ (J_{4}{+}J_{5}{-}2J_{6}){]}/h(t{+}{\Delta}t),\]

 

\[\mathrm{org}(t{+}{\Delta}t){=}{[}\mathrm{org}(t)\ {\cdot}\ h(t){+}{\Delta}t\ {\cdot}\ a_{\mathrm{org}}\ {\cdot}\ 10^{4}\ {\cdot}\ (J_{2}{-}J_{9}){]}/h(t{+}{\Delta}t).\]

Net isosmolar water movement across the apical cell membrane and intercellular space into the apical compartment was calculated (equation 4in the Methods section; Table A3 ) using the summed osmotically active solute movement:  

\[{\Sigma}^{a{+}p}{\phi}_{X}J_{i}^{X}{=}{-}{[}{\phi}_{\mathrm{Na,e}}J_{1}{+}({\phi}_{\mathrm{Na,e}}{+}{\phi}_{\mathrm{org,e}})J_{2}{+}{\phi}_{\mathrm{K,e}}J_{3}{+}{\phi}_{\mathrm{Cl,e}}J_{4}{+}{\phi}_{\mathrm{Cl,e}}J_{5}{+}{\phi}_{\mathrm{Na,e}}J_{10}{+}{\phi}_{\mathrm{K,e}}J_{11}{+}{\phi}_{\mathrm{Cl,e}}J_{12}{+}{\phi}_{\mathrm{cho,e}}J_{13}{+}{\phi}_{\mathrm{gluc,e}}J_{14}{]}.\]

 

³  Contributed equally to the work and therefore should be considered equivalent authors.

 Supported by Grants EY13574 from the National Eye Institute; DK72517 and DK35124 from the National Institute of Diabetes and Digestive and Kidney Diseases; HL73856 and HL59198 from the National Heart, Lung, and Blood Institute; EB00415 from the National Institutes of Biomedical Imaging and Bioengineering; and Drug Discovery and Research Development Program grants from the Cystic Fibrosis Foundation.

 Submitted for publication August 16, 2005; revised September 19, 2005; accepted November 22, 2005.

 Disclosure: M. H. Levin, None; J . K. Kim, None; J . Hu, None; A.S. Verkman, None

 The publication costs of this article were defrayed in part by page charge payment. This article must therefore be marked “advertisement” in accordance with 18 U.S.C. §1734 solely to indicate this fact.

 

Figure 1.

 

Ocular surface epithelial cell ion transport. Transport pathways are numbered and time-dependent intracellular solute activities, cell height, and membrane potentials are shown. Solute fluxes are defined as positive in the left (apical) to right (basolateral) direction. Arrowheads: direction of solute movement under open-circuit conditions. See Table A2 for specific transporter parameters under baseline, steady state conditions.

Table 1.

 

Composition of Perfusate Solutions

Solution	Name	Na+	Cl−	K+	Gluconate−	Choline+	Organic
1	Control	170	160	4.2	0	0	0
2	Na+-free	0	160	4.2	0	170	0
3	Organic	154	144	4.2	0	0	30
4	Organic (−Na+)	0	144	4.2	0	154	30
5	Low Cl−	170	4.7	4.2	155	0	0
6	High K+	0	160	174.2	0	0	0

 Values are expressed in millimolar. All solutions contained (in mM): 9.5 phosphate, 1 Ca²⁺, 0.5 Mg²⁺ (pH 7.40). Organic is d-glucose, l-glucose, or d-mannitol.

Table 2.

 

Model Symbols and Definitions

X(t)	Activity of solute X in cytoplasm (mM)
X a	Activity of solute X in apical bathing solution (mM)
X b	Activity of solute X in basolateral bathing solution (mM)
a x	Cellular activity coefficient of solute X
φ x	Cellular osmotic coefficient of solute X
a x e	Extracellular activity coefficient of solute X
φ x e	Extracellular osmotic coefficient of solute X
ψs	Serum osmolarity (units vary and are specified for each equation)
h(t)	Cell height (μm)
J i	Turnover rate of ith transporter (Fig. 1) . Solute flux for X is the product of J i and the number of X transported per turnover event (J i x , μeq/cm2per second), defined as positive from left to right
I a	Apical-to-cell membrane current (μA/cm2)
I b	Cell-to-basolateral membrane current (μA/cm2)
I p	Apical-to-basolateral paracellular current (μA/cm2)
P i	Permeability coefficient for ith transporter (units provided in Table A2)
K i x	Apparent binding constant for solute X to transporter i (mM), used for saturable transporters as described in the Appendix.
ψa	Apical membrane potential (cell with respect to apical solution); (mV)
ψb	Basolateral membrane potential (cell with respect to basolateral solution); (mV)
ψt	Transepithelial potential (PD; ψb − ψa); (mV)
U z	Dimensionless potential (U z = ψz F/RT, where F = 23 cal/mV per mole, R = 1.9872 cal/K per mole, and T = 310 K)

Figure 2.

 

Amiloride-sensitive Na⁺ transport at the ocular surface. (A) PD recordings showing effects of switching between sodium-containing (+Na⁺) and sodium-free (−Na⁺, sodium replaced by choline) solutions. Horizontal line: presence of amiloride, added to either the +Na⁺ (left) or −Na⁺ (right) solutions. Representative of six experiments. Solutions used were left: 1 alone, 2 alone, 1 alone, 1+amiloride, and 2+amiloride; right: 1 alone, 2 alone, 1 alone, 2 alone, and 2+amiloride. (B) Dose-dependent inhibition of sodium absorption by amiloride and benzamil. Left: Representative time course of PD inhibition, first by Na⁺ replacement, and then by adding amiloride to the perfusate at increasing concentrations. Solutions used: 1 alone, 2 alone, 1 alone, and 1+amiloride. Right: dose response of benzamil and amiloride for depolarizing PD, determined from experiments as shown on the left (mean ± SE, n = 5 for benzamil; n = 6 for amiloride).

Figure 3.

 

Ion transport at the ocular surface in CF mice. (A) PD recordings in CF (top) and wild-type (bottom) mice in response to amiloride, low Cl⁻, forskolin (10 μM), and CFTR_inh-172 (10 μM). Solutions used: 1 alone, 1+amiloride, 5+amiloride, 5+amiloride+forskolin, and 5+amiloride+forskolin+CFTR_inh-172. (B) Elevated amiloride-sensitive Na⁺ absorption in CF mice. Left: PD tracings in wild-type versus CF mice (curves overlaid) in response to Na⁺ replacement and amiloride. Solutions used: 1 alone, 2 alone, 1 alone, and 1+amiloride. Right: Summary of ΔPD for wild-type and CF mice for indicated maneuvers (SE, n = 6 eyes per genotype). *P < 0.01 comparing wild-type versus CF mice.

Figure 4.

 

Ocular surface epithelial ion transport model. For each maneuver, individual transporter conductances or apical compartment solute compositions were changed instantaneously. See the Appendix for model parameters. (A) Influence of amiloride on time-dependent model parameters in open-circuit conditions. Transporter 1 (apical Na⁺) conductance was fully inhibited (at dashed line). Changes in (from top to bottom) membrane potentials, currents, intracellular ion activities, and cell height are shown. Brief voltage spikes were applied every minute (top tracings) by an imposed current spike (second trace from top) with a magnitude that was a fixed proportion of the instantaneous current. (B) Effects of different maneuvers on open-circuit PD. From top to bottom, apical Na⁺ replaced with choline, apical Cl⁻ replaced by gluconate, basolateral Na⁺/K⁺/2Cl⁻ symporter inhibited, and basolateral 3Na⁺/2K⁺ ATPase inhibited.

Figure 5.

 

Modeling PD protocols and fluid secretion. (A) Simulation of protocol for studying chloride transport (as in the experiment in Fig. 3A ). CFTR activation was produced by a threefold increase in apical Cl⁻ permeability. (B) Simulation of protocol for studying sodium transport. Comparison of wild-type versus CF mouse ocular PDs in response to amiloride and then low Cl⁻. Solutions simulated: 1 alone, 1+amiloride, and 5+amiloride. CF mouse apical Cl⁻ permeability was 20% of that in wild-type mouse, whereas apical Na⁺ permeability was not changed in the CF mouse parameter set. (C) Incremental steady state water secretion (above baseline), modeled in response to addition to solution 1 of amiloride, a CFTR activator (increasing apical Cl⁻ conductance threefold), or both together.

Figure 6.

 

Solute-coupled Na⁺ absorption. (A) Left: PD response to additions of d-mannitol (30 mOsM, replacing NaCl) and d-glucose (30 mOsM, replacing d-mannitol), followed by the Na⁺-glucose cotransporter inhibitor phloridzin (100 μM), all in solutions containing amiloride (100 μM). Solutions used: 1 alone, 1+amiloride, 3(d-mannitol)+ amiloride, 3(d-glucose)+amiloride, 3(d-glucose)+amiloride+phloridzin, 3(d-mannitol)+amiloride+phloridzin, and 1+amiloride+phloridzin. Middle: Concentration-dependent effect of d-glucose on PD in the presence of amiloride (100 μM). Solutions used: 1 alone, 1+amiloride, and a series of mixtures of 1 and 3(d-glucose)+amiloride. Right: Corresponding concentration–response curve (mean ± SE, n = 5). (B) Simulated PD tracing of maneuvers identical with those performed experimentally and depicted in (A, left). (C) PD responses to amino acids, in the presence of amiloride. Left: l-arginine (1 mM) was added to and then removed from solution 1. Representative of six experiments. Solutions used: 1 alone, 1+amiloride, 1+amiloride+l-arginine, and 1+amiloride. Right: addition to solution 1 of glycine (5 mM) followed by l-arginine (5 mM) before Na⁺ replacement by choline. Representative of four experiments. Solutions used: 1+amiloride, 1+amiloride+glycine, 1+amiloride+glycine+l-arginine, and 2+amiloride+glycine+l-arginine. (D) Left: representative PD response to serial isosmolar additions of 30 mM d-mannitol, 30 mM d-glucose, and 1 mM l-arginine in the presence of amiloride. Solutions used: 1 alone, 1+amiloride, 3(d-mannitol)+amiloride, 3(d-glucose)+amiloride, 3(d-glucose)+amiloride+l-arginine, 4(d-glucose)+amiloride+l-arginine, 4(d-glucose)+amiloride, 4(d-mannitol)+amiloride, and 2+amiloride. Right: Paired analysis of ΔPD measured in +Na⁺ or –Na⁺ solutions. SE, n = 3–5 mice; *P < 0.01, comparing PD responses to zero.

The authors thank Liman Qian for mouse breeding and genotype analysis and Oscar Candia for advice on model parameter selection.