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Am J Physiol Cell Physiol 292: C591-C602, 2007. First published September 27, 2006; doi:10.1152/ajpcell.00560.2005
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NERVOUS SYSTEM CELL BIOLOGY

Computational model of vectorial potassium transport by cochlear marginal cells and vestibular dark cells

Department of Bioengineering, Rice University, Houston, Texas

Submitted 1 November 2005 ; accepted in final form 30 August 2006

	ABSTRACT

TOP ABSTRACT MODEL RESULTS DISCUSSION APPENDIX GRANTS REFERENCES

 
Cochlear marginal cells and vestibular dark cells transport potassium into the inner ear endolymph, a potassium-rich fluid, the homeostasis of which is essential for hearing and balance. We have formulated an integrated mathematical model of ion transport across these epithelia that incorporates the biophysical properties of the major ion transporters and channels located in the apical and basolateral membranes of the constituent cells. The model is constructed for both open- and short-circuit situations to test the extremes of functional capacity of the epithelium and predicts the steady-state voltages, ion concentrations, and transepithelial currents as a function of various transporter and channel densities. We validate the model by establishing that the cells are capable of vectorial ion transport consistent with several experimental measurements. The model indicates that cochlear marginal cells do not make a significant direct contribution to the endocochlear potential and illustrates how changes to the activity of specific transport proteins lead to reduced K⁺ flux across the marginal and dark cell layers. In particular, we investigate the mechanisms of loop diuretic ototoxicity and diseases with hearing loss in which K⁺ and Cl^– transport are compromised, such as Jervell and Lange-Nielsen syndrome and Bartter syndrome, type IV, respectively. Such simulations demonstrate the utility of compartmental modeling in investigating the role of ion homeostasis in inner ear physiology and pathology.

stria vascularis; endolymph; endocochlear potential; biological modeling

Tasaki and Spyropoulos (47) carefully mapped the potential around the cochlear scala media with an electrode and found that it was highest adjacent to the lateral wall. The stria vascularis was therefore presumed to be the source of the EP. On the basis of the fact that the rates of K⁺ entry from both perilymphatic compartments (scala vestibuli and scala tympani) into the scala media are equal, it was later predicted that the stria vascularis is also the source of the endolymphatic K⁺, as it is equidistant from each of them (27). This was supported by the demonstration in transepithelial (Ussing) chamber experiments that excised stria vascularis is capable of vectorial transport of K⁺ (56). Thus any model of how the stria vascularis functions must account for the ability of the tissue to sustain a transepithelial K⁺ current and generate an electrical potential in the endolymph.

The stria vascularis is a multilaminate epithelium containing at least three cell types thought to be directly involved in its ion transport activity (Fig. 1A; Ref. 42). The outer layer, facing the scala media, is composed of marginal cells, which are connected by tight junctions and have extensive basolateral infoldings suggestive of membrane transport. An inner layer, also tight junction connected, consists of flattened cells called basal cells that appear to have little metabolic function. Between these two layers is a narrow intercellular space, the intrastrial space, with a unique fluid composition. This space is electrically isolated from the endolymph and the extracellular fluid on the opposite side of the basal cells. Within the space lie a discontinuous layer of capillaries and a collection of intermediate cells. The intermediate cells are connected by gap junctions to the basal cells below them, which in turn are connected to type I fibrocytes. These fibrocytes form one segment of a large network of gap junction-connected fibrocytes with multiple markers of ion transport activity that make up the medial portion of the spiral ligament (43, 44).

View larger version (10K): [in this window] [in a new window]   Fig. 1. A: cells of the stria vascularis. Intermediate cells, basal cells, and the fibrocytes of the spiral ligament are connected by gap junctions. The space between the marginal cells and the intermediate/basal cells is lined by tight junctions on both sides. Major transporters and conductances in marginal cells are shown. The Na+ conductance may be a leak conductance through nonspecific cation channels, if it exists. B: model variables and partial fluxes (see text for definitions). Arrows indicate the direction in which the fluxes are defined, which is not necessarily the directions in which ions flow at steady state.

Within the vestibular labyrinth, the potential difference between the scala media and the perilymphatic compartments is closer to zero, although the ion composition is similar to that within the cochlea. Both properties are believed to be provided by the nonsensory portion of the epithelium around the vestibular scala media, which is composed of dark cells (32). Vestibular dark cells have K⁺-transporting ability, channel and transporter composition, and hormonal responses similar to cochlear marginal cells (52). However, the vestibular system lacks an analog to the second tight junction-connected layer of cells of the cochlear lateral wall.

Although they provide only a small direct contribution to the EP, marginal cells play a vital role in inner ear physiology. There is strong evidence that marginal cells are directly responsible for the high endolymphatic K⁺, since they can sustain a transepithelial current and large K⁺ flux in vitro (56). Additionally, the K⁺ current they generate may sustain the potential produced by the basal and intermediate cell layer. The current would help keep the concentration of K⁺ in the intrastrial space low, which is necessary to keep a negative E for K⁺ as described above.

The importance of the stria vascularis and vestibular dark cell epithelium to cochlear and vestibular electrophysiology is underscored by the fact that multiple deafness and imbalance genetic loci are in ion transport-related genes. For example, Jervell and Lange-Nielsen syndrome, caused by loss of function in a major K⁺ channel in the apical membrane of marginal cells [as well as the slow delayed-rectifier current (I_Ks) in cardiac myocytes] includes deafness and long-QT syndrome (30). Mice lacking Na⁺-K⁺-Cl^– cotransporter (NKCC)1 found in the basolateral membrane of marginal and dark cells exhibit loss of hearing and balance (12). Moreover, hearing loss and imbalance are known side effects of some loop diuretics, such as bumetanide, which inhibit NKCC activity and reduce K⁺ transport in marginal cell epithelia (56). Mutations in genes related to the basolateral Cl^– channels in marginal cells also lead to deafness (14).

We examined the mechanism by which cochlear marginal and vestibular dark cells contribute to the unique properties of the endolymph by constructing a mathematical model of steady-state ion transport. We made use of whole cell and single-cell electrophysiological recordings in defining the flux of each ion of interest between spaces. The model is useful in understanding how and to what extent individual ion channels and transporters contribute to the EP and endolymphatic K⁺ concentration. We can also explain experimental measurements of epithelial function as well as the physiological mechanisms by which mutations and pharmacological inhibitors of channels and transporters are manifested in the inner ear. The role that ion homeostasis plays in inner ear disease is increasingly recognized; a recent review claims that "disrupted ion homeostasis processes are the final common pathway in many auditory disorders" (50). Thus the development of integrated transport models may eventually have clinical utility in guiding pharmacological therapies for inner ear disease.

	MODEL

TOP ABSTRACT MODEL RESULTS DISCUSSION APPENDIX GRANTS REFERENCES

 
In this section we develop a mathematical model with specific references to marginal cells, but the model and its results are also applicable to vestibular dark cells. The model simulates an experiment in which an epithelial layer is placed between two well-perfused baths and the effective electrical resistance between the two sides is varied to obtain different transport characteristics. The epithelium consists of a monolayer of cells connected by tight junctions. We want to account for the transmembrane voltage and current across each membrane and the concentrations of Na⁺, K⁺, and Cl^– in the cell.

The following key assumptions are made. 1) The compartments are well mixed. 2) The transmembrane voltages, intracellular concentrations, and extracellular concentrations have reached steady state. In support of this assumption, the stria vascularis has no known transient behavior at short timescales in the absence of environmental changes. 3) The extracellular compartments are well perfused or, equivalently, have volumes much larger than the cell interior. It should be noted that while this assumption is valid for the experimental case, it may not be true in vivo on the basolateral side of cochlear marginal cells, where the small volume of the intrastrial space complicates the results. 4) The intracellular space is electroneutral. We ignore the effect of capacitive charge on ion composition, because it is typically small (24).

The compartments are indicated by the labels I for intracellular space and A and B for apical and basolateral spaces adjacent to the membrane. The molar concentration c_k^I of ion k in each compartment j follows a mass balance:

(1)

J_k^a and J_k^b are the outward partial molar fluxes of ion k through the apical and basolateral membranes of the cell, respectively, and w is the volume of the cell. The fluxes are in terms of cell membrane area (A_a and A_b).

Variables v_a and v_b are the transmembrane voltages across the apical and basolateral membranes. We also define the transepithelial voltage, v_t = v_b – v_a, which represents the direct contribution of the marginal cell layer to the EP; the paracellular ion flux J_k^p; the total K⁺ flux across the epithelium, ; and the total transepithelial current density,

where z_k is the valence of k and F is Faraday's constant. J_K^p, J_K^te, and i_te are positive in the apical direction. We will occasionally refer to the K⁺ current density, i_K^te, which is simply FJ_K^te. A final parameter of interest is the equilibrium K⁺ concentration in the endolymph, c_Keq. This is calculated by setting the short-circuit transepithelial K⁺ flux equal to zero, yielding the equilibrium concentration in the absence of any other K⁺ source or sink. The value of c_Keq gives a measure of the maximum possible endolymphatic K⁺ concentration and is therefore less meaningful in the open circuit, where the current is constrained to be less than maximal.

Additional equations are necessary to solve for the voltages. The first comes from the resistive load of the epithelium. We consider two cases. In the open circuit, v_t = 0, while in the short circuit, since capacitive currents are zero at steady state, i_te = 0. From assumption 4,

(2)

where N_f is the number of fixed charges in the cell. We choose a fixed charge equivalent to 20 mM of monovalent anions at the estimated volume of 500 fl, based on measured net concentration differences (19).

Finally, we assume that the cell is in osmolar balance. The volume changes according to

(3)

where P_w is the permeability of the membrane to water. At steady state, assuming that the osmolarity on either side of the epithelium is the same and equal to ,

(4)

Although we must include volume as a model variable in order for the cell to maintain isosmolarity, we do not fully elaborate the regulatory processes used for maintenance of normal volume, as these have not been well characterized in marginal cells or vestibular dark cells. In Eqs. 2–4, we neglect the contribution of permeant anions other than Cl^–. Therefore our calculated values of c_Cl^I should be considered overestimates, which also include some contribution from other anions, notably HCO₃^–.

We estimated the cross-sectional area of fixed marginal cells from published electron micrographs (16) to be 50 µm². This is taken to be the apical membrane area and cross-sectional area of the model cell. The basolateral area is set at 6,000 µm² based on membrane capacitance measurements (41). The large difference is due to the structural complexity and extensive infoldings of the marginal cell basolateral membrane.

Partial membrane currents in basolateral membrane. Marginal cells have a highly invaginated basolateral membrane containing an abundance of Na⁺-K⁺-ATPase (9, 36), the Na⁺-K⁺-Cl^– cotransporter NKCC1 (11), and outwardly rectified ClC-K Cl^– channels (Fig. 1).

Marginal cells and vestibular dark cells predominantly contain the ₁₂ form of Na⁺-K⁺-ATPase (48). The activity of each pump, measured in moles per second, is (17)

(5)

which assumes noncooperative binding. The intrinsic affinities (K_K,pump and K_Na,pump) are 1.16 mM for extracellular K⁺ human ₁₂ transporter expressed in Xenopus oocytes (10) and an estimated 3.3 mM for intracellular Na⁺. The density is 4,500 transporters/µm², based on immunogold labeling of vestibular dark cells (7) and v_pump,max is given a typical value of 150 s^–1 (10). To account for regulation of the cell volume by Na⁺-K⁺-ATPase, we include the arbitrary volume-dependent factor

(6)

which allows the pump to reduce activity at volumes exceeding 700 fl, ensuring that the volume remains stable over a wider range of conditions. The Na⁺ flux carried by the pump is J_Na,pump = 3v_pump and the K⁺ flux is J_K,pump = –2v_pump.

Na⁺-K⁺-Cl^– cotransporter NKCC1 is also present at high abundance in the basolateral membrane (11). Related to NKCC2 in the thick ascending limb of the renal tubule and the macula densa of the kidney, NKCC1 is found in a variety of epithelia and other tissues. The expression for the steady-state activity of each NKCC transporter, v_NKCC, follows the model of Benjamin and Johnson (4) and is given in the APPENDIX (Eq. A12). Parameters are taken from a fit of the model to data for Na⁺-K⁺-Cl^– cotransport in HeLa cells (33), as given in Table 2 of Ref. 4, because they have similar apparent affinities to transfected human NKCC1 (20). The outward flux, measured in moles per square meter per second, is J_Na,NKCC = J_K,NKCC = –_nkccv_NKCC for Na⁺ and K⁺ and J_Cl,NKCC = –2_nkccv_NKCC for Cl^–, where _NKCC is the transporter density. The value of _NKCC is estimated using the assumption that the density of the cotransporter is sufficient to remove all of the Na⁺ that enters from the Na⁺-K⁺ pump. Solving v_NKCC = –3v_pump for _NKCC with estimated values for the concentrations yields roughly 6,000 transporters/µm².

Each ion current in the model is specified by the Goldman-Hodgkin-Katz (GHK) equation. The flux of ion k is

(7)

where p_k,max is its permeability when the channel is fully activated, P_open is the channel open probability, E_k is the equilibrium (Nernst) potential of the ion across the membrane, c_i and c_o are the intracellular and extracellular concentrations, R is the gas constant, T is temperature, and v_m is the transmembrane potential. P_open is equal to 1 unless otherwise specified.

The marginal cell requires a Cl^– conductance as an exit pathway for Cl^– ions entering the cell through the NKCC1 transporter. The conductance is composed of Cl^– channels made up of the proteins CLC-Ka and CLC-Kb (1), and the flux is termed J_ClC. The permeability was calculated from measurements of the whole cell Cl^– conductance (1).

Our final expressions for the basolateral fluxes are

(8)

(9)

(10)

where the permeability of each of the two leak fluxes is 10^–9 m/s.

Partial membrane currents in apical membrane. The apical fluxes for Na⁺, K⁺, and Cl^– (J_Na^a, J_K^a, J_Cl^a) each include only a single GHK-style expression. The apical membrane of marginal cells is dominated by an outwardly rectified K⁺ conductance (41), although a Na⁺ conductance has also been measured. The major component of the voltage-dependent current is called slowly activating K⁺ current (I_sK) and consists of four pore-forming -subunits, KCNQ1, and the modulatory subunit KCNE1 (also called minK) (37, 49). The steady-state conductance of I_sK is well modeled by a two-state Boltzmann model (41):

(11)

where _sK is v_,sK is the half-maximal activation voltage, and _sK is a slope factor. To our knowledge, the whole cell permeability has not accurately been measured, but the whole cell conductance is at least 6 nS in isolated gerbil marginal cells (41). With the ion concentrations used in those experiments, the equivalent whole cell permeability is 1.1 µm/s. However, we estimate that the maximum permeability is larger, accounting for rundown and the existence of other channels (22, 35), and use a value of p_sK = 5 µm/s. This is closer to measurements in other epithelia (cf. Ref. 25) and better accounts for measurements of the transepithelial resistance (55).

An amiloride-sensitive inwardly rectified apical Na⁺ conductance has also been recorded (34). Interestingly, the mRNA for subunits of the epithelial Na⁺ channels (ENaC) typically found in epithelial cells is not expressed (18), and Na⁺-specific channels have not been reported from single-channel patch-clamp experiments. The conductance has been proposed to be due to the numerous nonspecific cation channels found in marginal cells (46). Given the limited data, the apical Na⁺ permeability is set to be small, but a variety of values are tested.

We also incorporate the Cl^– permeability defined for the basolateral membrane into the apical membrane, which helps improve the volume stability for some extracellular concentrations.

Paracellular conductance through tight junctions. We estimate the paracellular permeability to Na⁺, K⁺, and Cl^– to be 10^–8 cm/s based on measurements in other ion-transporting epithelia (3, 21). As with the membrane currents, we use Eq. 7 to describe the current. The assumptions made in its derivation should also hold, on a larger scale, for one-dimensional transport through a narrow paracellular space.

Model simulation. The perfusion solutions were fixed at 1.3 mM Na⁺, 157 mM K⁺, and 132 mM Cl^– apically and 141 mM Na⁺, 6 mM K⁺, and 121 mM Cl^– basolaterally. The apical concentrations are measured from mammalian cochlear endolymph, with vestibular endolymph being similar, except for a higher Na⁺ concentration (53), a difference that was determined to be negligible. The basolateral concentrations are measured from cochlear perilymph. These conditions should be accurate for vestibular dark cells, but the ionic composition of the intrastrial space underlying the marginal cells is less clear, because of experimental difficulties arising from the miniscule volume of this fluid. We examine the influence of composition of the basolateral fluid below. The exact solutions for the main model variables v_a, v_b, c_Na^I, c_K^I, c_Cl^I, and w in Eqs. 1, 2, and 4 subject to v_t = 0 or i_te = 0 were obtained with a nonlinear solver in MATLAB. Solutions were obtained for the parameter values in Table 1, and with variations of each key parameter in the model. The steady state for the initial solution was estimated to be unique by multiple runs of a genetic algorithm. Simulations were done for extreme electrical load conditions to test the range of function of the epithelium. Specifically, an open circuit (infinite load resistance) tests the maximum ability of the epithelium to generate voltage and a short circuit (zero load resistance) tests the maximum ability to generate current. Although these two cases do not completely specify tissue function in an epithelium with nonlinear transport properties, they are useful functional measurements.

	RESULTS

TOP ABSTRACT MODEL RESULTS DISCUSSION APPENDIX GRANTS REFERENCES

We validated the model by comparing it to several published experimental results, adjusting model parameters to match different experimental situations. Ikeda and Morizono (19) measured ion concentrations and electrical potentials in situ throughout chinchilla stria vascularis and found values of Na⁺ concentration ([Na⁺]) = 1.7 mM, [K⁺] = 141.7 mM, [Cl^–] = 121 mM, v_a = 0.2 mV, and v_b = 17.9 mV for marginal cells. A simulation with the extracellular concentrations set to the values measured in that study, assuming that the load is close to an open circuit, yields v_a = 0.582 mV, v_b = 20.7 mV, c_Na^I= 2.17 mM, c_K^I= 148 mM, and c_Cl^I= 128 mM. The predicted value of open-circuit v_t is slightly more than the value measured by Ikeda and Morizono, which is consistent with the fact that the open circuit usually sets the upper limit of v_t.

Wangemann (52) directly measured open-circuit voltage and short-circuit currents across both vestibular dark cell and marginal cell layers in vitro with fluids of equal composition (150 mM Na⁺, 3.6 mM K⁺, and 153.4 mM Cl^–) on both sides. The measured open-circuit voltages were 8 and 11 mV and the short-circuit currents were 712 and 849 µA/cm² in vestibular dark cells and cochlear marginal cells, respectively. When run with the conditions used in those experiments, the model epithelium generates v_t = 16.3 mV in the open circuit and i_te = 681 µA/cm² in the short circuit.

That study also measured the effects of ouabain and bumetanide administration, which can be simulated in the model. Experimentally ouabain caused the open-circuit v_t to drop from 6.5 mV to 0.46 mV and the short-circuit current to drop from 564 µA/cm² to 44 µA/cm². In the model, we achieve similar results with a 60% reduction in the Na⁺-K⁺-ATPase density, which yields values of 0.38 mV and 26.8 µA/cm², compared with the control values above. Bumetanide caused the experimental open-circuit v_t to change from 7.3 mV to 0.5 mV and the short-circuit current to change from 824 µA/cm² to 81 µA/cm². With a 96% reduction in _NKCC, the model predicts corresponding values of 1.29 mV and 24.4 µA/cm². The response of the short-circuit K⁺ flux to bumetanide was also measured directly by an ion-selective vibrating electrode; this value should be similar to J_k^te in the model. The study noted a decrease in the local voltage gradient as measured by a K⁺-selective electrode to 18 ± 14% of the control value. The model epithelium predicts a change in J_K^te, which should be approximately proportional to the measured voltage gradient, from 1,372 µA/cm² to 726 µA/cm² after bumetanide administration.

Together, these results are in good agreement and validate the model as a physiologically realistic description of the epithelium, despite the approximate estimates used for some of the parameters.

The effective density of Na⁺-K⁺-ATPase, _pump, affects the open-circuit voltage, short-circuit current, K⁺ flow across the epithelium, and endolymphatic K⁺ concentration. In the open circuit (Fig. 2, top), a minimal pump activity is required to maintain a positive v_t (which becomes negative for very low pump densities). The pump's effect on v_t occurs primarily through changes in v_b, which becomes more positive with increased pump activity, following the Nernst potential for Cl^– (E_Cl^b), the most permeable ion across the basolateral membrane, as it increases in concentration. This Cl^– increase is largely accounted for by a heightened contribution by NKCC in response to the Na⁺ concentration change induced by the Na⁺-K⁺ pump. When the Na⁺-K⁺ pump activity drops considerably below normal, NKCC activity drops sharply and eventually reverses direction. In the short circuit, the values of v_a and v_b are necessarily equal and take on a value intermediate to E_K^a and E_Cl^b, with E_Cl^b having a somewhat stronger effect because of its high net permeability. As a result, the change to the membrane potential is minimal in the short circuit. The K⁺ current i_K^te and net current i_te are strongly dependent on _pump, driven by both the direct contribution to the K⁺ flow and the secondary contribution through NKCC. Under various conditions we found that both transporters reach much higher levels of activity in the short circuit than in the open circuit as they are not limited by the reverse flow of Na⁺ and other leaks in the circuit.

View larger version (10K): [in this window] [in a new window]   Fig. 2. Effect of maximum Na+-K+-ATPase activity on epithelial function in the open-circuit potentials (top), short-circuit currents (per marginal cell) (middle), and equilibrium luminal K+ (bottom). The maximum activity is the product of the effective pump density and the turnover number of an individual pump. Dashed lines represent equilibrium potentials (E) of the major conductance in each membrane. K+ currents through the individual transporters are shown as apically directed (negative, in terms of the definitions in the text). The physiological value of effective density of Na+-K+-ATPase (pump)is 4,500 transporters/µm2 (vertical dotted line).

View larger version (10K): [in this window] [in a new window]   Fig. 3. Effect of Na+-K+-Cl– cotransporter (NKCC)1 density (functional transporters/cell) on epithelial function in the open-circuit potentials (top), short-circuit currents (per marginal cell) (middle), and equilibrium luminal K+ (bottom). Key equilibrium potentials are shown as dashed lines. The estimated density is 6,000 transporters/µm2 (vertical dotted line).

View larger version (10K): [in this window] [in a new window]   Fig. 4. Effect of maximal K+ permeability (psK,max) in the apical membrane on epithelial function on open-circuit potentials (top), short-circuit currents (per marginal cell) (middle), and equilibrium luminal K+ (bottom). Equilibrium potentials for the major conductances of each ion are shown as dashed lines. The estimated value of the permeability is 5 µm/s (vertical dotted line).

View larger version (9K): [in this window] [in a new window]   Fig. 5. Effect of Na+ permeability in the apical membrane (pNa,ap) on epithelial function on open-circuit potentials (top), short-circuit currents (per marginal cell) (middle), and equilibrium luminal K+ (bottom). Dashed lines are equilibrium potentials of the major conductance in each membrane. The estimated value of the Na+ permeability is 0.3 µm/s (vertical dotted line).

View larger version (10K): [in this window] [in a new window]   Fig. 6. Effect of basolateral Cl– conductance on open-circuit potentials (top), short-circuit currents (per marginal cell) (middle), and equilibrium luminal K+ (bottom) Equilibrium potentials for ions with the largest conductance are shown as dashed lines. The physiological value of the permeability (pClb) is 0.09 µm/s (vertical dotted line).

View larger version (17K): [in this window] [in a new window]   Fig. 7. Influence of extracellular K+ on short-circuit transepithelial K+ current per marginal cell. Labels represent the current in pA/cell at different concentrations of K+ on the apical (cKA) and basal (cKB) sides of the epithelium. The current reverses direction for high apical K+. The zero locus corresponds to equilibrium K+ concentrations, as in the calculations of the apical concentrations, cKeq.

View larger version (17K): [in this window] [in a new window]   Fig. 8. Influence of extracellular Cl– on the short-circuit transepithelial K+ current per marginal cell. Labels represent the current in pA/cell at different concentrations of Cl– on the apical (cClA) and basal (cClB) sides of the epithelium. The current reverses direction for high basolateral Cl–.

	DISCUSSION

TOP ABSTRACT MODEL RESULTS DISCUSSION APPENDIX GRANTS REFERENCES

For example, the model clearly shows how changes to p_K^a have an important effect on transepithelial K⁺ transport (Fig. 4, middle). The potassium channel subunits KCNE1 and KCNQ1 are most well known for their involvement in long QT syndrome (LQT-1 and LQT-5) since they are responsible for the I_Ks current, which helps prolong the cardiac action potential (23). Interestingly, patients with heterozygous mutations in KCNQ1 have LQT-1, because of a dominant-negative effect of the mutant gene, but maintain normal hearing (Romano-Ward syndrome). This differs from the rarer homozygous form, which consists of both LQT-1 and congenital deafness (Jervell and Lange-Nielsen syndrome). This difference is also seen in KCNQ1-knockout mice (8). Heterozygous knockouts have normal hearing and balance, while homozygous knockouts are deaf and display classic Shaker-Waltzer behavior indicative of vestibular dysfunction, presumably as a result of complete loss of I_Ks from the marginal and dark cells. The continued function of these cells in the presence of only one functional gene might include other subunit combinations not found in cardiac myocytes or the presence of additional K⁺-conductive channels in the apical membrane. These effects may be understood with respect to the saturation of i_K^te seen in Fig. 4. If p_K^a is in fact larger then we have estimated, then a decrease in the number of channels due to a heterozygous mutation could have a relatively small (<50%) effect on K⁺ transport, while a homozygous mutation would have a large effect. We expect this may be the case, due to either an underestimate of I_sK or the existence of other channels capable of conducting K⁺, such as known two-pore K⁺ channels (22, 35), which would raise the total K⁺ permeability into the saturating region of the curve. Regulatory activity may also be important in maintaining a high conductance in heterozygotes; I_sK exhibits -adrenergic and muscarinic responses consistent with those seen in cardiac myocytes (54). If the K⁺ permeability is actually close to our estimate, its position at the steepest part of the curves suggests that the K⁺ permeability would be a powerful variable for regulation.

Clinicians have long known that loop diuretics, which inhibit NKCC, cause reversible ototoxicity. These drugs, which include ethacrynic acid, furosemide, and bumetanide, reduce the EP and alter the composition of the endolymph to varying degrees (6, 28, 29). Their action on the function of cochlear marginal cells can be inferred from Fig. 3, which shows that the direct contribution of the marginal cells to the EP (v_t) and, more significantly, to i_K^te, decreases sharply with NKCC inhibition. The effect on the endolymphatic K⁺ concentration would be especially large because of the secondary modulation of the Na⁺-K⁺ pump. Loop diuretics may also have effects on the ability of the second layer of the stria to produce a potential caused by the secondary rise in intrastrial K⁺ or by inhibition of other transporters within the cochlea. The prominent synergistic relationship between Na⁺-K⁺-ATPase and NKCC that we have demonstrated is consistent with histology showing that the expression of the two transporters is closely matched throughout development and during age-related changes (38). It also explains why experimental application of bumetanide to marginal cell layers causes a near-complete loss of the open-circuit voltage, short-circuit current, and transepithelial K⁺ flux (56), rather than only a partial loss.

Histopathology of the stria vascularis shows that bumetanide causes enlargement of the intrastrial space, which is prevented by simultaneous administration of ouabain (2). Our results suggest that the difference is not due to ouabain's effect on the marginal cell's Na⁺-K⁺-ATPase, since bumetanide would have already indirectly inactivated it. Instead, it is likely that ouabain is preventing K⁺ from reaching the intrastrial space by blocking other components of the proposed K⁺-recycling pathway (54). In this situation, inactivation of marginal cell activity would have little effect on intrastrial volume.

The effect of the density of NKCC on transepithelial voltage and current reaches a plateau well below our estimated value of 6,000 transporters/µm². In the open circuit the plateau is reached well below that value, and in the short circuit the plateau is reached right at that value. At physiological values of the load current, we expect that the function plateaus between those two extremes, which would be just below the estimated density. This would indicate that small concentrations of loop diuretics would have no effect on marginal and vestibular dark cell function. Mice with mutations in or lacking the gene for NKCC1 are deaf, display a Shaker-Waltzer phenotype, and have collapsed semicircular canals and Reissner's membranes (12, 13, 15). These changes are all consistent with the effects we have attributed to NKCC. Interestingly, heterozygotes had normal auditory brain stem responses and showed no abnormal pathology. This finding is unusual given the crucial role of NKCC but, as with the KCNQ1-knockout mice, may be explained by the saturation in its effects. Realizing that Fig. 3 is plotted on a logarithmic scale, it is clear that if the wild-type density of NKCC is indeed slightly above the plateau, a 50% decrease would cause little or no change to any of the voltages and ion currents considered.

Cl^– channel diseases provide another clinical example of defective ion transport in marginal cells and dark cells. The basolateral membrane of marginal cells contains two Cl^– channel proteins, ClC-Ka and ClC-Kb (1). However, mutations in either of the genes encoding these channels do not cause auditory or vestibular symptoms. Bartter syndrome, type IV, which includes salt wasting and deafness, is caused by mutations in the gene encoding Barttin (5). Barttin associates with ClC-Ka and ClC-Kb and is now known to be an essential subunit for both (14). Since it affects both Cl^– channels, its effect is more potent than a mutation in either individual channel. Similarly, simultaneous mutations in the genes encoding both Cl^– channels have been associated with sensorineural deafness (40). In light of our results, it appears that the complete lack of effect with loss of half of the channels may once again be related to saturation in the effects of the protein's expression level. Since the cell's Cl^– permeability greatly outweighs the permeability to any other ion, nearly complete loss is necessary for the open-circuit transcellular potential v_t to decrease. Transepithelial K⁺ flow and other measures of function in the short-circuit case are much more sensitive, but i_K^te is still positive (though small) after a 50% drop in the permeability, and would be further resistant if the measured value is a slight underestimate.

The results of experiments and models of strial marginal cells and vestibular dark cells should be interpreted in the context of the transporter and channel interactions demonstrated here. How the cells would behave in a larger system also needs to be considered, because the behavior of the monolayer can change significantly when placed into a closed-loop configuration. We partially address this concern by incorporating simple electrical feedback of the net current on the system through a resistor, which is implicit in the open (infinite resistance)- and short (zero resistance)-circuit analyses. These cases are the two extremes of electrical feedback and indicate the maximum ability of the epithelium to generate current or voltage. However, because of the nonlinear nature of transport across the epithelium, these two cases may not be sufficient to fully describe the functional possibilities. Additionally, the extracellular concentrations, which were constrained in our model, connect the epithelium to another closed loop, as epithelial transport is dependent on concentration changes produced by itself or by other cells and tissues. Proper solution of the model equations in the normal state and with parameter changes requires knowledge of the extracellular concentrations in each condition. In particular, we demonstrated that transport is highly sensitive to the basolateral K⁺ concentration, so this value needs to be accurately measured. If the intrastrial K⁺ concentration is indeed 1–2 mM, as has been suggested (45), it would need to be carefully regulated because of the marginal cells' extreme sensitivity to basolateral K⁺ in that range.

Our results suggest several other places where new experiments would help more clearly delineate the function of the epithelium. For example, it would be desirable to have more accurate measurements of the Na⁺ and K⁺ permeability of the apical membrane of the marginal cell in situ or even in vivo. In particular, the presence of additional K⁺ channels besides KCNQ1/KCNE1 should be systematically investigated. Finally, further characterization of hormonal effects on marginal and dark cell function would help in building a more complete model that also incorporates endocrine regulation (51).

	APPENDIX

TOP ABSTRACT MODEL RESULTS DISCUSSION APPENDIX GRANTS REFERENCES

 
The kinetics of the NKCC transporter follow glide symmetry (31), a first-in, first-out scheme in which the ions bind in a specific order on one side of the membrane and dissociate in the same order on the opposite side. Various forms of NKCC from different species have been shown to fit the model well (4). We derived an explicit form for the steady-state activity based on this model for a cell with the number of transporters equal to S_NKCC. We assume that the ion binding constants are the same on either side of the membrane and that both Cl^– binding steps have the same rate constant. The subscripts 1 and 2 refer to the extracellular and intracellular states of the transporter, respectively. The total amount of transporters is conserved:

(A1)

In the above equation, E designates the unbound transporter, and EN, ENC, ENCK, and ENCKC are the various steps of ion binding. We define the following for simplicity:

(A2)

in which K_Na, K_K, and K_Cl are intrinsic binding constants for Na⁺, K⁺, and Cl^– binding to the enzyme. The individual reaction equations are

(A3)

Inserting these individual reaction equations into Eq. A1 yields

(A4)

where

(A5)

(A6)

We also know from thermodynamics that there is no net flux of transporters from one side of the membrane to the other:

(A7)

where k_F^e and k_F^e are the forward and backward rate constants for E₁ E₂ and k_F^f and k_B^f are the forward and backward rate constants for ENCKC₁ ENCKC₂. The turnover rate per unit volume is

(A8)

We want to express this in terms of [E] and the ion concentrations. Sequential substitutions of Eqs. A3 result in

(A9)

where

(A10)

Applying Eqs. A4 and A7 to Eq. A9,

(A11)

If we define _NKCC as the density of transporters per cell, then this is equivalent to a total turnover density of

(A12)

	GRANTS

TOP ABSTRACT MODEL RESULTS DISCUSSION APPENDIX GRANTS REFERENCES

 
This work was supported by a grant from the Whitaker Foundation (RG-02-0024) and a training fellowship from Keck Center for Computational Biology of the Gulf Coast Consortia (National Library of Medicine Grant No. 5T15-LM-07093).

	ACKNOWLEDGMENTS

 
We thank Dr. Ruth Anne Eatock for helpful comments on the manuscript.

	FOOTNOTES

 

Address for reprint requests and other correspondence: R. M. Raphael, Dept. of Bioengineering, M. S. 142, PO Box 1892, Rice Univ., Houston, TX 77251-1892 (rraphael{at}rice.edu)

The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

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