Characteristics of ionic transport processes in fish intestinal epithelial cells

BioSystems 45 (1998) 123 – 140

Characteristics of ionic transport processes in fish intestinal epithelial cells Liviu Movileanu a,*, Maria Luiza Flonta b, Dan Mihailescu b, Petre T. Frangopol c a

Di6ision of Cell Biology and Biophysics, Uni6ersity of Missouri-Kansas City, School of Biological Sciences, 405 Biological Sciences Building, 5100 Rockhill Road, Kansas City, MO 64110 -2499, USA b Biophysical Laboratory, Uni6ersity of Bucharest, Faculty of Biology, Splaiul Independentei 91 -95, Bucharest R-76201, Romania c Uni6ersity of Iassy, Faculty of Physics, Department of Biophysics and Medical Physics, Iassy R-6600, Romania Received 11 March 1997; received in revised form 2 September 1997; accepted 10 September 1997

Abstract A general mathematical version of the cell model of a leaky epithelium for the NaCl absorption is presented, analysed and integrated numerically. The model consists in the adequate differential equations that describe the rate of change of the intracellular ion concentrations and are expressed in strict accordance with the law of mass conservation. The model includes many state variables representing ion concentrations, the cell volume, and membrane potentials. Ion movements are described by the Michaelis – Menten kinetics or by the constant field flux equation (Goldman–Hodgkin–Katz). In this paper, we model the intracellular ion concentrations, change in the cell volume, the transmembrane flux and membrane potentials of intestinal epithelium of both fresh water and sea water fish, and generate several simulations (in both the steady state and the transient state analysis) that appear to accord with prior experimental data in this area. For the ion movements of the sea water fish intestine, there were included a Na + /K + pump, a K + –Cl − symport system, the K + and Cl − channels in the basolateral membrane, whereas a Na + –K + –2Cl − cotransporter for NaCl absorption and K + channels are located in the apical membrane. In the fresh water fish intestinal cells, the NaCl absorption is performed by two coupled antiporters Na + /H + and Cl − /HCO3− presumably responsible for the intracellular pH regulation. In this type of cells, Na + and K + channels are located within the apical membrane, whereas Cl − channels are located within the basolateral membrane. The osmotically induced water transport across the apical and basolateral membranes has been taken into account as well. The simulations plot the steady state values for membrane potential difference, short-circuit current and intracellular ionic concentrations using the magnitude of the transmembrane flux through the Na + /K + pump and Na + –K + – 2Cl − cotransporter, or the basolateral Cl − permeability as dependent variables. The model behaves appropriately with regard to several experimental studies regarding the hyperpolarization (sea water fish intestine) and depolarization (fresh water fish intestine) of the apical membrane potential and inhibition of the short-circuit flux with reduced NaCl absorption. The model is also used to make several analytical predictions regarding the response of the membrane potential and ionic concentrations to variations in the basolateral Cl − flux. Furthermore, maintaining * Corresponding author. Tel.: +1 816 2352500; fax: +1 816 2351503. 0303-2647/98/$19.00 © 1998 Elsevier Science Ireland Ltd. All rights reserved. PII S 0 3 0 3 - 2 6 4 7 ( 9 7 ) 0 0 0 7 1 - 3

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conservation of both mass and electroneutrality and taking into account the osmolar forces is an important advantage, because it allows a rigorous analysis of the relationship between membrane potential difference, volume and flux. The model can be used in the analysis and planning of the experiments and is capable of predicting the instantaneous values of ionic fluxes and intracellular concentrations and of cell volume. © 1998 Elsevier Science Ireland Ltd. All rights reserved. Keywords: Fish intestinal cells; Ionic transport mechanisms; Computer simulation; The steady-state analysis; The NaCl absorption model

1. Introduction Computer modelling represents a powerful tool in electrophysiology providing new insights on the electrophysiological properties of different epithelial membranes. Mathematical models are formulated and analysed in order to predict the basic electrophysiological responses or to explain some strange experimental evidence on certain epithelial membranes. Nordin (1993) has proposed a mathematical model of the electrical activity of an isolated guinea pig myocyte. This model predicts with quantitative precision a wide variety of electrophysiological responses by solving a set of equations and using the experimentally measured transmembrane current and the intracellular Ca2 + and Na + concentrations. Other models are subject of a continuous improvement to resolve some contradictions between the experimental results and the computer simulations. In this respect, a re-evaluation of the three compartment model of intestinal weak-electrolyte absorption has been performed by Lucas and Whitehead (1994) to predict adequately the ratio of the forward and reverse fluxes of weak electrolyte across the intestinal cells. Computer modelling of ionic epithelial transport has been pursued very actively in the last two decades. The majority of the computer models are largely based on realistic non-linear approaches of epithelial transport (Baerentsen et al., 1982, 1983; Civan and Bookman, 1982; Duszyk and French, 1991; Fernandez and Ferreira, 1991; Fidelman and Mikulecky, 1986; Gordon and Macknight, 1991; Latta et al., 1984; Movileanu, 1995, 1996, 1997; Novotny and Jakobsson, 1996a,b; Schultz, 1979; Sohma et al., 1996; Thompson, 1986), although there are authors adopting linear non-

equilibrium models (Essig and Caplan, 1979; Helman and Thompson, 1982). We would recommend, for a better understanding of these non-linear epithelial models of ionic transport processes, the paper of Latta et al. (1984). This paper shows very clearly a general computer procedure for the derivation of mathematical models of ionic epithelial transport. Their procedure is constructed on the fundamental considerations of mass conservation and physical constraints of electroneutrality and isotonicity of the bathing fluids and the intracellular medium. The environment osmolarity of the majority of euryhaline teleosts could vary in a wide range, between almost 0 mOsmol in the fresh water medium and 1250 mOsmol in the sea water medium. From this point of view, the question concerning the ion transport mechanisms and their characteristics for sea water and fresh water fish intestinal epithelia has arisen. In the model presented here, the electrodiffusional currents through ion channels, the active transport, the electrically neutral cotransport and a symport system are included. The model is based on the experimental evidence that the apical membranes of sea water fish intestinal cells possess the K + channels and the Na + –K + –2Cl − cotransporter, while the basolateral membranes contain a Na + / K + pump, a K + –Cl − symport system and the K + and Cl − channels (Ellory et al., 1984; Frizzell et al., 1984; Halm et al., 1985a; Gibson et al., 1987) (Fig. 1(A)). The NaCl absorption theory in sea water fish intestinal cells is largely based on the idea that there is a net electrolyte flux assured by the Na + and Cl − uptake through the apical membrane and the Na + and Cl − efflux through the basolateral membrane. The ion transport through the

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cellular pathway is accompanied by a Na + flux via the paracellular pathway (Fig. 1(A)). The transepithelial potential (serosa negative) is strongly dominated by the contribution of the basolateral Cl − permeability (Gibson et al., 1987). Fresh water fish intestinal cells perform the NaCl absorption using the doubly-coupled electroneutral exchanger Na + /H + and Cl − /HCO3− (Fig. 1(B)) like in the Necturus gallbladder epithelium (Baerentsen et al., 1982, 1983). The Na + and K + channels are located within the apical mem-

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brane, whereas the Cl − channels are located within the basolateral membrane. In sea water fish intestinal cells the principal adaptation changes due to the salt increasing are located in the basolateral membrane: the activity of the Na + /K + pump and the basolateral Cl − permeability are increased (Gibson et al., 1987). These characteristics are taken into account in our computations. The present method can be used in the analysis and interpretation of experiments in the steady state behaviour and it is capable to predict the instantaneous values of the intracellular ion concentrations, potential difference of the cell membranes and the cell volume. Such a theoretical approach will help to understand the concepts of transepithelial transport in sea water fish intestinal cells, will put forward to analyse qualitatively the endogenous variables when the Na + / K + pump activity, the Na + –K + –2Cl − cotransporter activity and the permeability of different ion channels are changed in the physiological range.

2. Methods

2.1. Basic model assumptions

Fig. 1. (A) Cell model of the mechanisms of ionic transport through sea water fish intestinal cells. A Na + /K + pump, a K + -Cl − symport system, the K + and Cl − channels are located within the basolateral membrane, while an electroneutral cotransport system for Cl − with stoichiometry Na + :K + :2Cl − and the K + channels are located within the apical membrane. (B) Cell model of ionic transport through fresh water fish intestinal cells. The electroneutral Na + –K + –2Cl − cotransporter in the apical membrane is replaced by doublycoupled antiporter system Na + –H + and Cl–HCO3− . There are also the Na + channels located in the apical cell membrane, whereas the basolateral membrane contain Cl − channels, but not K + channels (see for instance, Halm et al., 1985a).

The use of the electrical equivalent circuit analogy is sometimes full of pitfalls that most workers fail to avoid. To simply explain these approaches, we deal with fluxes of water and different electrolytes. An ion possesses an electric charge and a mass, so that the ionic fluxes should carry distinct charge fluxes and mass fluxes. It is immediately imposed to have two distinct constraints: mass balance and electroneutrality condition (the charge conservation). Together with the ionic flow there is the osmotically induced water transport that implies the isotonicity conservation. For more details, the reader should look over the work of Latta et al. (1984), since this paper presents a general procedure for integrating the epithelial ion transport models using the single cell approach. The bathing solutions are assumed to be infinite in volume, accordingly the transepithelial transport does not alter the bath composition. Another assumption is that the cell and the

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apical and the basolateral bathing fluids are well stirred, so the unstirred layer effects are not taken into account. Throughout this work, we normalized the membrane fluxes, the permeabilities and the hydraulic conductances to the actual surface area of the apical and the basolateral membranes. As we have seen above the present model must be constrained by three conservation relationships: mass conservation, isotonicity and electroneutrality of the intracellular compartment and the baths. We assume that the ion species that can be able to pass through the membranes (apical, basolateral or junctional) are Na + , K + and Cl − . Ca2 + transport may be present, but it would be quantitatively negligible relative to Na + , K + or Cl − . Therefore, the majority of the computer models start with law of mass balance, which is ensured by the following differential equations: dQNa = JNaa −JNab dt dQK =JKa −JKb dt dQCl =JCla − JClb dt

(1)

where QNa, QK and QCl are the total intracellular amount of Na + , K + and Cl − , respectively. Jia and Jib (i may be Na + , K + , Cl − ) are the fluxes of an ion species ‘i’ across the apical and the basolateral membranes, respectively. In the case of the fresh water fish intestinal cells, we have to take into account the H + and HCO3− production within the cell interior. The rate of the reaction CO2 +H2O = H + + HCO3− inside the cell is modulated by the presence of the enzyme carbonic anhydrase. The rate of this reaction is assumed to follow the rule: −

d[CO2] d[H + ] d[HCO3− ] = = dt dt dt



=V0 CO2 −

[H + ][HCO3− ] Ka



(1a)

with the velocity constant of this reaction V0 and Ka = 10 − 3.35 mM at the room temperature (Baerentsen et al., 1983). In this case, we have to include the additional equations to the whole system (1):

dQHCO3 =JHCO3a − JHCO3b dt dQH = JHa − JHb dt dQCO2 =JCO2a − JCO2b dt

(1b)

Here, the notations are the same as in Eq. (1). The total intracellular amount of different ions is related by the cell volume (V): QNa = c iNaV QK = c iKV QCl = c iClV

(2)

where c , c and c are the intracellular Na + , K + and Cl concentrations, respectively. We have to take into account the fact that together with the ionic movements there are osmotically induced water fluxes across the membranes which influence the cell volume, especially in a leaky epithelium (Baerentsen et al., 1983): i Na

i K −

i Cl

dV = Jwa − Jwb dt

(3)

with Jwa and Jwb denoting the water fluxes through the apical and basolateral membranes, respectively. Water fluxes through the cell membranes are assumed to be proportional to the osmotic pressure across the membranes obeying to the following law (Kedem and Katchalsky, 1963):





Jwa = LpaRT % siac i − % saac a



Jwb = − LpbRT % sibc i − % sbbc b



(4) (5)

The summations are carried out over all ion types present. Here, R, T, Lpa and Lpb indicate the molar gas constant, the absolute temperature, the hydraulic conductances of the apical and basolateral membranes, respectively. sja and sjb are the solute reflection coefficients of an ion species ‘j’ across the apical and basolateral membranes, respectively. c ij, c aj and c bj denote the ionic concentrations in the intracellular medium, in the apical and basolateral baths. Using the Eqs. (1)–(3) we can

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obtain a general description of ionic movements across a leaky epithelium (which is actually the first conservation relationship). This relates the intracellular ion concentrations (for Na + , K + and Cl − ), the total ion fluxes through the cell membranes, the cell volume and the water fluxes through the apical and basolateral membranes (Schultz, 1980): dc iNa 1 = [(JNaa −JNab ) − c iNa(Jwa −Jwb )] dt V dc iK 1 = [(JKa − JKb ) −c iK(Jwa −Jwb ) dt V dc iCl 1 = [(JCla −JClb ) −c iCl(Jwa −Jwb ) dt V

(6)

The second conservation is written in such a form c iNa +c iK + c iCl + c iX + c in =p0

(7)

here, c iX, c in and p0 represent the amount of the intracellular non-diffusible anions (with the electric charge zX ), the intracellular concentration of nonelectrolyte molecules and the osmolarity of external solution, respectively. Electroneutrality (the third conservation relationship) condition can be written in the following expression: c iNa +c iK = c iCl − zXc iX

(8)

but this is included in the application of Kirchoff’s laws to the model (see equivalent electrical network). This is an usual trap that most workers do not take into account. Since the electroneutrality must be preserved, the transport of each kind of ion is related by the ionic transport performed by other ions. Accordingly, the total current through the apical membrane is always equal to the total current moving across the basolateral membrane and the total transepithelial current is at any time equal to the sum of the transcellular current with the paracellular current.

Fig. 2. An example for the equivalent electrical network of the sea water fish intestinal cells. a, i and b indicate respectively, the apical, the intracellular and the basolateral compatrment; gl, gK,a, gK,b and gCl,b are the conductance of the paracellular pathway, the conductance of the apical K + channels, the conductance of the basolateral K + channels and the conductance of the basolateral Cl − channels, respectively. EK,a, EK,b, ECl,b, and Ep denote the Nerst equilibrium potentials due to the apical K + channels, the basolateral K + channels, the basolateral Cl − channels, and the electrogenic contribution of the Na + /K + pump to the equivalent electromotive force across the basolateral membrane. El represents the equivalent electromotive force of the paracellular pathway due to the ion gradients across the tight junctions.

Fig. 2 shows the equivalent electrical circuit of the sea water fish intestinal cells, where the apical and basolateral membranes as well as the paracellular pathway are represented by electromotive forces (emf) coupled in series with resistances. It is more convenient to express the equivalent electromotive force of a membrane (Ea or Eb for the apical or basolateral side, respectively) as a function depending on the Nernst equilibrium potential for the various ions and their relative conductance and by rheogenic ion translocation mechanism (for instance, in our case the Na + /K + pump)

2.2. Equi6alent electrical circuit

Ea = EK,a

Traditionally, the analysis of the ion transport events through the epithelial membranes involves the use of the equivalent electrical circuit, which incorporates the flows and forces across the apical and basolateral membranes and the paracellular pathway.

Eb = tCl,bECl,b + tK,bEK,b + Ep

(9) (10)

where the relative conductances or transport numbers tCl,b and tK,b of the serosal membrane are given by the following relations tCl,b = gCl,b /(gCl,b + gK,b )

(11)

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tK,b = gK,b /(gCl,b + gK,b )

(12)

and EK,a, EK,b, ECl,b are the Nernst equilibrium potentials for the respective ions and membranes (see Fig. 2). gCl,b and gK,b denote the conductances of Cl − and K + channels located in the basolateral membrane, respectively. For the transport numbers and the Nernst equilibrium potentials the first and second index indicate the ionic species and membrane (apical or basolateral), respectively. Ep represents the electrogenic contribution of the Na + /K + pump to the equivalent electromotive force of the basolateral cell membrane. In the present paper, an open-circuit condition of the intestinal epithelium is assumed throughout. The intraepithelial circular current (under opencircuit condition) will be expressed by Ohm’s law formula Ic =

Eb − Ea − El Ra + Rb + Rl

(13)

with Ra, Rb and Rl as the resistances of apical, basolateral and junctional membranes, respectively. Here, El denotes the equivalent electromotive force of the cellular pathway and is given by the following expression: El = tNa,lENa,l +tK,lEK,l +tCl,lECl,l

(14)

where t’s are the transport numbers for the paracellular pathway, while E’s are the Nernst equilibrium potentials across the tight junctions due to the asymmetrical bathing fluids. The intraepithelial circular current Ic also determines voltage drops across both cell membranes (cell membrane potentials Dfai and Dfbi ) and the paracellular pathway (transepithelial potential difference Dfab ): Dfai = Ea −RaIc

(15)

Dfbi = Eb +RbIc

(16)

Dfab = RlIc

(17)

2.3. Computing strategy In the above-present model we have dealt with three categories of quantities: (i) parameters (affinities for specialized transport mechanisms, permeabilities of different channels, maximum

rates of specialized transport mechanisms, total amount of intracellular non-diffusible anions, hydraulic conductance of the cell membranes, etc.); (ii) exogenous variables (ion concentrations within the bathing fluids, osmolarity of the bathing fluids); (iii) endogenous variables (the intracellular ion concentrations, the cell volume, the electrical potentials across the apical membrane and the basolateral membrane, the short-circuit current, the transepithelial potential etc.). Eq. (6) can be integrated numerically for the steady state condition as a function of the course time using an adequate algorithm for the numerical solution of ordinary differential equations. In our computations we used a fourth order Runge– Kutta method with quality controlled time steps. Accordingly, Eq. (6) together with the auxiliary Eqs. (7)–(16) and (A1)–(A11) have been solved for the intracellular ion concentrations, the cell volume and the potentials of the cell membranes at given values of the exogenous variables (osmolarity and the ion concentrations in the external baths) as well as the other parameters (permeability of ion channels and the pump and cotransporter activities etc.). We can obtain the conductances of ion channels by differentiating the respective voltage. The ionic mixture from the apical and basolateral compartments is assumed to contain Na + , K + , Cl − and neutral molecules at the following concentrations (Duffey et al., 1979): c aNa = c bNa = 165 mM,

c aK = c bK = 5 mM

c aCl = c bCl = 150 mM,

c aX = c bX = 20 mM

The osmolarity was p0 = 340 mOsmol l − 1 (Duffey et al., 1979; Gibson et al., 1987). Typical intracellular concentrations of Na + , K + , Cl − in sea water fish intestinal cells are: c iNa = 18 mM,

c iK = 110 mM,

c iCl = 20 mM If the Eq. (7) regarding the osmolarity condition (the osmotic equilibrium between the cell and the bath) and Eq. (8) concerning the electroneutrality of intracellular compartment are achieved and the intracellular concentration of non-electrolyte molecules (c in) is about 44 mM, then the concen-

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tration of negatively charged impermeable ions from the intracellular compartment is c iX =148 mM. Accordingly, the mean valence of the negatively charged impermeable ions is zX = −0.73. Unfortunately, at this moment there are no experimental data available for water permeabilities of individual cell membranes of sea water fish intestinal epithelium. The immersion weighing method for the continuous recording of the apparent weight of a membrane-bounded small vessel (for the intestinal epithelium as well) has been imagined to offer a cheap and reliable procedure to measure the osmotic flows (Margineanu and Flonta, 1980). This method might also be applied for the diffusional (isotopic) water flows. We used the hydraulic conductances for the apical membrane and the basolateral membrane Lpa = 0.310 − 3 cm s − 1 Osmol − 1 and Lpb =0.510 − 3 cm s − 1 Osmol − 1. These values are in the range of the leaky epithelia, for instance in the case of Necturus gallbladder Lpa =0.5Lpb =10 − 3 cm s − 1 Osmol − 1 (Baerentsen et al., 1983; Hill, 1994; Weinstein, 1987). The solute reflection coefficients are taken equal to unity (Baerentsen et al., 1983). Generally, they can be 0.95 or 1. Weinstein (1987) developed an interesting approach for the evaluation of the epithelial ionic reflection coefficients. Because our simulations were performed in the steady state condition for the symmetrical composition of the bathing fluids, the presence of the convective paracellular solute flux is not justified. Gibson et al. (1987) found that JnetNa +JnetCl is equal to the equivalent short-circuit flux (the ionic flux corresponding to the short-circuit current) in control sea water and fresh water acclimated tissues. Therefore, it is unnecessary to postulate other net transepithelial flux, although coupled fluxes due to electroneutral transport mechanisms might be involved. An immediate consequence of this equivalence is that most K + pumped into the cell to account for the Na + absorption recycles across the basolateral membrane in agreement with the reported data of Ellory et al. (1984). We introduced the parameter ‘s’ that denotes the fraction of the K + flux through the Na + /K + pump that recirculates across the basolateral K + – Cl − symport system. This type of kinetics (relation (A7)) together with the physical constraint

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for the K + flux like in the cell model of the cortical thick ascending limb of the rabbit proposed by Fernandez and Ferreira (1991) do not affect qualitatively the results (the intracellular ion concentrations, the cell volume etc.).

3. Results All the results presented here were obtained by assuming that the epithelium is in open circuit conditions. Although under open-circuit conditions the transepithelial current is zero, in leaky epithelia the paracellular pathway almost completely short-circuits the series array of the two membrane emfs causing an intraepithelial current through the cell and returning through the paracellular pathway. The maximum pump flux (J pm) and the maximum cotransporter flux (J co m ) were changed simultaneously in a balanced manner (like in other work, Movileanu and Popescu, 1995) to give an overall view of the dependence of the steady state intracellular ion concentrations or electrical responses on the electrolyte transport mechanisms through fish intestinal cells. In Fig. 3(a) we presented a 3D view of the dependence of the intracellular Na + concentration on the maximum pump flux (J pm) and the maximum cotransporter flux (J co m ). The pump and cotransport system have an opposite effect on the intracellular Na + concentration. This phenomenon seems to be obvious if we look at Fig. 1. The Na + -K + -2Cl − cotransporter plays an important role regarding the Na + uptake into the cell, while the Na + /K + pump is a rheogenic ion translocation mechanism removing Na + from the cell contributing to the decrease of the intracellular Na + concentration. Decreasing the maximum pump flux from 350 to 150 mM cm − 2 h − 1 and increasing the maximum cotransport flux from 4 to 11 mM cm − 2 h − 1 the steady state simulations indicated a 3-fold increase of the intracellular Na + concentration (from 6 to 18 mM). Both specialized mechanisms of ion movement, the pump and the cotransport systems assure the K + uptake increasing the intracellular K + concentration. Therefore, we expect to have a maximum

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Fig. 3. A 3D view of the intracellular Na + (a), K + (b) and Cl − (c) concentrations (in mM) as a function of the maximum pump co −2 flux (J m h − 1. The range p ) and the maximum cotransporter flux (J m ). The range of the maximum pump flux is 150 – 350 mM cm of the maximum cotransporter flux is 4–18 mM cm − 2 h − 1.

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Fig. 3. (Continued)

intracellular K + concentration, when the maximum pump flux and the maximum cotransport flux are higher like in Fig. 3(b). In other words, for the same value of the cotransport rate the K + uptake is enhanced monotonously by increasing the Na + /K + pump rate and vice-versa. The present model does not include the constant cell volume condition or other regulatory constraint. Therefore, the intracellular Cl − concentration is not invariable or given by the intracellular concentration of the impermeable negatively charged molecules. The Na + –K + –2Cl − cotransporter also carried Cl − into the cell and this fact suggests us an increase of the Cl − uptake when the cotransport coefficient is raised (Fig. 3(c)). Since Cl − is not involved in the ion transport achieved by the Na + /K + pump, obviously we might obtain a slight dependence of the intracellular Cl − concentration on J pm (as an influence of electroneutrality and isotonicity condition). The short-circuit current is strongly dependent on the maximum activity of the Na + – K + – 2Cl −

cotransporter. Decreasing the activity of the Na + –K + –2Cl − cotransporter in a range 4–18 mM cm − 2 h − 1 we can see the steady state level of the short-circuit current is reduced from − 110 to − 10 A cm − 2 (Fig. 4). The short-circuit current reaches the steady state level after about 200 s. Generally, the intracellular ion concentrations and the cell volume attain the steady state level after about 100–150 s. Frizzell et al. (1979) carried out experiments with the furosemide addition to the mucosal side of the flounder intestinal cells. They found that furosemide rapidly inhibited the transepithelial potential (Dfab ) and the short-circuit current (Ic ), whereas the transepithelial conductance remained practically unchanged. Therefore, it is natural to think that the short-circuit current is strongly dependent on the Na + – K + –2Cl − cotransporter flux. Accordingly, our simulations are consistent with the experimental evidence regarding the furosemide addition to the apical side of the flounder intestinal cells. The NaCl absorption in fish intestinal cells can also be reduced by the Na + and Cl − replacement

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Fig. 4. Changes in the short-circuit current as a function of the maximum cotransporter flux and the course time. The maximum pump flux is 200 mM cm − 2 h − 1.

experiments performed to the mucosal bath with the condition that the replacement ions do not pass rapidly through the tight junctions. Such ion replacement experiments have been performed by Halm et al. (1985b). Interestingly enough, by reducing the activity of the Na + – K + – 2Cl − cotransporter using ion replacement approaches, Halm et al. determined a hyperpolarization of the apical membrane potential (Halm et al., 1985b). In Fig. 5 we showed in a 3D view the influence of the pump and cotranspoter toward the apical membrane potential. Our model solved for the steady state condition offers the same strange response of the apical membrane potential as a function of the maximum cotransporter flux. We think that this odd electrophysiological response can be explained as a consequence of the strong influence of the maximum cotransporter flux on the short-circuit current. Another of the possibilities to test the validity of the present ion transport model is to check up the changes of different electrical parameters determined by the changes of the absolute ion permeabilities. In the present model we have the following sequence:

P aK increases [ Ki decreases; Isc increases; g aK increases; Vac hype rpolarizes P bK increases [ Ki decreases; Isc decreases; g bK increases; Vac hyp erpolarizes P bCl increases [ Cli decreases; Isc increases; g bCl increases; Vac dep olarizes Decreasing the apical K + permeability from 3× 10 − 5 to 5× 10 − 6 cm s − 1 caused a slight decrease of the intracellular Na + (around 18 mM) and Cl − (around 20 mM) concentrations, while the intracellular K + concentration increased from 102 to 112 mM. Intracellular K + concentration is almost linearly related to the apical K + permeability. Therefore, a decrease in the apical K + permeability as occurred in experiments with mucosal Ba2 + determined a net K + absorption against its electrochemical potential difference (Frizzell et al., 1984). The intraepithelial circular current has no significant changes produced by the variation of the conductivity of the K + chan-

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Fig. 5. Changes in the apical membrane potential D8ai (mM) as a function of the maximum pump flux and the maximum cotransporter flux. The ranges of the maximum pump and the cotransport coefficient are given in Fig. 3.

nels located in the apical membrane, while the cell volume has small changes (B 0.4%). Intracellular Na + concentration was not affected significantly by the changes in the basolateral Cl − permeability. By increasing the basolateral Cl − permeability an enhancement of the intracellular K + concentration and reduction of the intracellular Cl − concentration can be seen (Fig. 6). This observation suggests us a decrease of the cell volume due to the increasing number of the Cl − channels located in the basolateral membrane (according to the electroneutrality condition in the cell compartment, Eq. (8)). Fig. 7 shows the influence of the basolateral Cl − permeability on the short-circuit current (Isc ) and the basolateral membrane potential (Dfbi ). If the conductivity of Cl − channels located within the basolateral membrane is increased, then the Cl − efflux is higher and the basolateral membrane potential (Dfbi ) will be lower and, accordingly, the short-circuit current is increased in the absolute value. The computed transepithelial potential difference

(Dfab ) averaged around − 2 mV, in good agreement with the experimental results of Smith et al. (1975). A basolateral positive orientation of Eb would involve a higher permeability of the K + channels located in the basolateral membrane (Fig. 2). Accordingly, the basolateral negative transepithelial potential (Dfab ) across the sea water fish intestine is mainly a consequence of the lower basolateral K + permeability. A very interesting feature of this model is that although ‘s’ denotes the fraction of the K + flux through the Na + /K + pump, which recirculates across the K + –Cl − symport, the kinetics of this mechanism depending on ‘s’ does not produce any important change between 0.01 and 0.8. However, the intracellular Na + concentration increased with 3.1%, while the intracellular Cl − concentration followed a reduction from 20.1 to 18 mM. It is a good idea to extend and develop the applications of this model on other goals like the concrete replacement experiments in intestinal

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cells in transition-state condition (Movileanu, 1994), the action of different channel blockers, the modulation of ionic conductances by different anaesthetic compounds (Flonta et al., 1987, 1988). We have seen in the section ‘Introduction’ that the fresh water fish intestinal cells perform the NaCl absorption through two coupled neutral antiporters (Na + /H + and Cl − /HCO3− ), which are presumably responsible for the intracellular pH (pHi ). We assumed that a certain ratio of the apical K + channels are pH-sensitive and that there is a saturation kinetic curve relating the increase of the blocked pH-sensitive K + channels and the increase of the intracellular pH. We pursued the membrane potential behaviour in the transient state simulations in the case of the Na + replacement with the impermeable cation NMDG + .

Fig. 7. Changes of electrical parameters as function of the basolateral Cl − permeability. The range of the basolateral Cl − permeability is between 3 ×10 − 5 – 13 × 10 − 5 cm s − 1. Basolateral membrane potential is given in mV and short-circuit current in mA cm − 2. The maximum pump flux and the maximum cotransporter flux are 200 and 16 mMcm − 2h − 1, respectively.

Fig. 6. Dependence of intracellular ion concentrations (Nai, Ki, Cli are given in mM) on the permeability of Cl − channels located within basolateral membrane (P bCl). The rage of the basolateral Cl − permeability is between 3 ×10 − 5 –13×10 − 5 cm s − 1. The maximum pump flux and the maximum cotransporter flux are the same as in Fig. 3.

Fig. 8 shows a large, but slow depolarization of the apical membrane potential with decrease of the apical Na + concentration. This result is in contrast with the sea water fish intestinal epithelia, where reducing NaCl absorption induces a hyperpolarization of the apical membrane potential (Halm et al., 1985b; Movileanu, 1995). This finding could be adequately explained by looking at Fig. 9. Reducing Na + concentration in the apical bath, the rate of the Na/H + antiporter is decreased and the H + efflux across the apical membrane becomes lower. This promotes an increase of the intracellular H + concentration, and finally the intracellular pH (pHi ) is decreased. On the other hand, the apical membrane contains pH-sensitive K + channels and so, the apical membrane potential will depolarize as a result of the intracellular medium acidification.

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Fig. 8. A typical simulation showing an isomolar replacement of Na + with the impermeable cation NMDG + in the apical side of the fresh water intestinal epithelium. Each simulation proved a very slow, but large depolarization of the apical cell potential. Apical Na + concentrations in these ion-replacements were indicated on each curve.

4. Discussion We formulated and integrated numerically an ion transport model for fish intestinal cells. The modelling is an extension of the several prior papers by other authors regarding the epithelial transport and the results appear to be congruent with several prior experimental studies on both fresh water and sea water fish intestinal cells. We evaluated the model by comparing its predictions (steady state and transient state behaviour) with a set of experimental observations of both sea water and fresh water fish intestinal cells. Obviously, the method is based on the principles of mass conservation, and maintenance of electroneutrality and isotonicity within intracellular, apical and basolateral compartments. We have attempted to analyse the steady state condition for ion transport in this type of cells changing both the transport rate of the Na + /K + pump and the Na + – K + – 2Cl − cotransporter in a balanced, coordinated manner. As we have seen above, the transepithelial transport of one substance (for instance, Na + ) will always be related by the transport of other sub-

stances (K + and Cl − ) either directly due to specific electrogenic, cotransport or symport mechanisms, or indirectly owing to the changes in electrochemical potential difference or osmotic driving forces. In flounder intestine the activity of the Na + – + K –2Cl − cotransport system could be reduced by cyclic nucleotides such as cGMP and cAMP (Rao et al., 1984), while the K + conductance of the apical membrane is suppressed by decreased cellular pH. Rao et al. reported that cAMP modulates the tight junction perm-selectivity in the intestine of the winter flounder. Musch et al. reported a coupling ratio of 1.3 9 0.4 Na + per Cl − suggesting that in flounder intestine the brush border cotransport process could be an electrogenic one and that it translocates more cations than anions (Musch et al., 1982). Simulations of the steady state condition in the presence of bumetanide have led to a large reduction of the intracellular Na + and Cl − concentrations from 21 to 6 mM and from 22 to 8 mM, respectively (Fig. 3(a) and (c)). Accordingly, a 3.5-fold decrease of the intracellular Na + concen-

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Fig. 9. Computer simulation of the transient-state condition for the Na + replacement with an impermeable cation NMDG + . Effect on the intracellular pH (pHi ). Apical Na + concentrations in these ion-replacements were indicated on each curve.

tration and a 2.7-fold reduction of the intracellular Cl − concentration owing to the bumetanide effect account for the prominent role of the Na + – K + –2Cl − cotransporter in NaCl absorption through sea water fish intestinal cells. The changes in the basolateral K + permeability due to the external factors suggest us the capability of the K + channels to limit the changes in the cell volume. In the present approach we used the Na + /K + pump flux in such a way no interaction exists between the inner and the outer sites of the pump. One considers the Na + /K + pump be a current source whose output is explicitly independent of the basolateral membrane potential. Yet, there is some experimental evidence indicating a voltage sensitivity of the Na + /K + pump (De Weer et al., 1988; Goldshlegger et al., 1987; Rakowski et al., 1997). Because, this dependence is not very strong in out case, the above mentioned effect was neglected. However, the epithelial modelling is a very useful tool due to a high degree of flexibility. Epithe-

lial models allow including a newly proposed transport mechanism in order to investigate its potential consequences on the epithelial transport. Different processes with regulatory functions relating the Na + –K + –2Cl − cotransporter and ion channels or changes in apical area resulting from fusion of cytoplasmic vesicles were not included in the present model. As concluding remarks concerning the specificity of the electrolyte transport mechanisms through fish intestinal cells we can say that: (i) the Na + /K + pump and the Na + –K + –2Cl − cotransporter have a cumulative effect on the intracellular Na + concentration; (ii) the maximum transport rate of the Na + /K + pump has little effect on the cell volume or the intracellular Cl − concentration and a weak effect on the transcellular current (Ic ) or the apical membrane potential (Vac ) when changed in the physiological range; (iii) the present model confirms the experimental evidence for hyperpolarization of the apical membrane potential and inhibition of the short-circuit current as result of decreasing the NaCl absorp-

L. Mo6ileanu et al. / BioSystems 45 (1998) 123–140

tion according to the studies of Frizzell et al. (1979) and Halm et al. (1985b), respectively; (iv) with the present approach we can also explain a depolarization of the apical membrane potential following a Na + replacement with the impermeable cation NMDG + , that means a decrease of the NaCl absorption; (v) an inhibition of the NaCl absorption across the apical membrane (by the bumetanide-sensitive Na + – K + – 2Cl − cotransporter from 18 to 4 mM cm − 2 h − 1) will affect significantly the intracellular Na + and Cl − concentrations; (vi) an inhibition of the apical K + permeability will determine a net K + absorption against its electrochemical potential difference like in the experiments with Ba2 + (Frizzell et al., 1984; Musch et al., 1982); (vii) the changes of the basolateral Cl − permeability have an opposite effect on the intracellular concentrations of K + and Cl − (Fig. 6). This finding could be explained by the following reasoning: an increase of the basolateral Cl − permeability might cause a decrease of the intracellular Cl − concentration. Consequently, the Cl − gradient across the apical membrane will be higher and this fact leads to an enhancement of the activity of the Na + – K + – 2Cl − cotransporter. Therefore, the intracellular K + concentration will tend to increase. As you see in Fig. 6, the intracellular Na + concentration will change very slightly. This behaviour is a response to the presence of the Na + /K + pump into the basolateral membrane. Increasing effect for the intracellular Na + concentration performed by the Na + – K + – 2Cl − cotransporter is partially compensated by the decreasing effect carried out by the Na + /K + pump; (viii) although the apical K + permeability is at least 5-fold higher than the basolateral K + permeability the cell volume and the intracellular K + concentration are very sensitive to the changes of the basolateral K + permeability. Therefore, the K + channels located within the basolateral membrane might play an osmoregulatory role. The main feature of the ion transport through the sea water fish intestinal cells is that both the Na + /K + pump and the basolateral membrane permeability for Cl − are increased compared with the ones for the fresh water fish intestinal cells.

137

5. Conclusions A general model of the fish intestinal epithelium consisting of a neutral entry of NaCl into the cell, an electrogenic entry of Na + and K + in a ratio of 3:2, a neutral exit of Na + and K + (the symport system), and electrodiffusive fluxes of Na + , Cl − and K + across both cell membranes and tight junctions, can explain most measurements carried out on the tissue. There is no need to involve other transport mechanisms for K + (or Cl − ) across the apical or the basolateral membrane. The model also explains both the depolarization (fresh water fish intestine) and hyperpolarization (sea water fish intestine) of the apical membrane potential as a result of the NaCl absorption decrease. This procedure of the analysis of epithelial transport is largely based on fundamental considerations of mass balance, electroneutrality conservation and the isotonicity constraints. The method is applicable to many different epithelial cell membranes, because each differs only in the form of the auxiliary expressions containing the membrane solute fluxes (diffusive movements, electrogenic active transport, electroneutral transporters) as well as the water fluxes.

Acknowledgements The present work was supported by a grant of the National Council for Advanced Scientific Research of Romania (Nr. 5009/A6/1997).

Appendix A. Analytical expressions for the ion transport mechanisms in fish intestinal cells The system of differential Eq. (6) can be solved numerically if we have all the auxiliary equations, which describe the ion fluxes through the apical, basolateral and junctional membranes, and the water fluxes through the apical and basolateral membranes. As we mentioned above, the present model includes four ways for the ions to enter or exit the cell: through the conductive pathways (K + channels in the apical membrane and K +

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and Cl − channels in the basolateral membrane), via the Na + /K + pump, via the K + – Cl − symport, or via the Na + – K + – 2Cl − cotransporter. The ionic flux generated by the active transport of Na + and K + due to Na + /K + pump with noncooperative sites is given by the following equation according to the saturation kinetic scheme, which is a Michaelis – Menten type expression with a pump ratio of three sodium ions per two potassium ions (Baerentsen et al., 1982, 1983; Civan and Bookman, 1982; Duszyk and French, 1991; Fernandez and Ferreira, 1991; Fidelman and Mikulecky, 1986; Latta et al., 1984): J p = J pm



c iNa c iNa + k pNa

 3

c bK c bK +k pK

n 2

(A1)

where k’s and J pm are the affinity constants for Na + and K + and the maximum pump flux, respectively. Superscripts ‘b’ and ‘i’ indicate the basolateral and intracellular compartments, respectively. The affinity constants k pNa and k pK can be expressed as function depending on concentration (Civan and Bookman, 1982):



k pNa =k bNa 1+



k pK =k cK 1+

 

c iK k iK

c bNa k iNa

(A2) (A3)

with k bNa = 0.2 mM, k iK =8.3 mM, k cK =0.1 mM and k iNa =18.5 mM. Apart the description of the Na + /K + pump with non-cooperative sites (Civan and Bookman, 1982), some authors use a cooperative site pump model, in which the Na + /K + pump flux is a Michaelis–Menten– Hill equation with the Hill coefficients equal to 3 and 2 for Na + and K + , respectively (see for instance, Fidelman and Mikulecky, 1986). There are some situations where the Na + /K + pump flux expression is given by a Michaelis–Menten factor and another factor linearly depending on transmembrane potential (Novotny and Jakobsson, 1996a,b; Sohma et al., 1996). Pursuing the factors that contain the membrane potential, we can see in Novotny and Jakobsson (1996a) a very week influence of the membrane potential (its coefficient is equal to 0.005), whereas we observe in Sohma et al. (1996)

a strong dependence of the Na + /K + pump flux on the membrane potential factor (its coefficient is equal to 1). However, in all the cases, the Na + / K + pump flux should contain at least a Michaelis–Menten factor, but the whole expression strongly depends on the respective case (Apell, 1989; Levitt, 1980). A detailed study regarding the effect of the membrane potential on the mammalian Na + /K + pump has been carried out by Goldshlegger et al. (1987). The Na + flux through the Na + /K + pump (J pNa) coincides with Jp, while the K + flux through the Na + /K + pump is related to the Na + flux by the following relation: J pK = − 23 J pNa

(A4)

The NaCl absorption in sea water fish intestinal cells is performed by a Na + –K + –2Cl − cotransporter (Geck and Heinz, 1986; O’Grady et al., 1987a,b). Musch et al. (1982) have investigated the characteristics of the Na + –K + –2Cl − cotransporter in the intestine of a marine teleost, the winter flounder. The ionic fluxes via the Na + – K + –2Cl − cotransporter can be estimated seeing as we have no electrochemical potential under the equilibrium conditions (Baerentsen et al., 1983): J co = J co m −





 n

c aNa c aK c aCl a a a c + k Na c K + kK c Cl + kCl a Na



c iK c iCl c iNa c iNa + kNa c iK + kK c iCl + kCl

2

2

(A5)

where kNa (5 mM), kK (4.5 mM), kCl (20 mM) denote the affinity constants of the cotransporter. J co m is the maximum cotransporter flux (cotransport coefficient) and the subscript ‘a’ indicates the apical solution. The influxes of K + and Cl − via the cotransporter are related to the Na + influx by stoichiometry of the cotransporter that can be expressed by the following equality: 1 co co J co = J co Na = J K = − 2 J Cl

(A6)

As far as the K + –Cl − symport system is concerned, we have no information available about the affinity constants of this type of cells, but we can introduce the simplest kinetic description (Fernandez and Ferreira, 1991): J sym = A sym(c iKc iCl − c bKc bCl)

(A7)

L. Mo6ileanu et al. / BioSystems 45 (1998) 123–140

where A sym is a phenomenological constant, and, c iK, c bK, c iCl, c bCl, are the ion concentrations of K + and Cl − in the intracellular medium and basolateral bath, respectively. J sym is the K + or Cl − flux across the K + –Cl − symport system that is located in the basolateral membrane (Gibson et al., 1987; Halm et al., 1985a). The ionic fluxes through the ion channels within both cell membranes or in the parracellular pathway are given by the Goldman–Hodgkin – Katz equation (Hille, 1984): n cm i − c i exp( − zi umn ) mn (A8) J mn i =P i zi umn 1 − exp( − zi umn ) with FDfmn umn = (A9) RT mn m n where J mn i , Dfmn, P i , zi, c i and c i and F denote the diffusional flux of an ionic species ‘i’ through the cell membrane located between the compartments ‘m’ and ‘n’, the potential difference between the compartments ‘m’ and ‘n’, the permeability of ion channel ‘i’ at the membrane located between the compartments ‘m’ and ‘n’, the charge valence of ion species ‘i’, the concentrations of ion species ‘i’ in the compartments ‘m’ and ‘n’ and the Faraday constant, respectively. Solute permeabilities are treated as invariant parameters. However, the membrane conductances are strongly dependent on both the membrane potentials and ionic concentrations. The analysis of the equivalent electrical network provides us with the relationships between the cell membrane potentials, equivalent electromotive forces (Eqs. (9) and (10)) and the cell membrane conductances (Eqs. (14) and (15)). We could obtain the analytical form of the electrical conductances of the ion channels by differentiating the Goldman–Hodgkin – Katz formula (A6) as a function of the respective voltage (Civan and Bookman, 1982): 2 P mn i F g mn i = RT[1− exp(−zi umn )]2



139

if umn " 0 and g mn i =

(A10)





2 n cm P mn i F i +ci RT 2

if umn = 0

(A11)



n × {(c m i −c i exp( −zi umn ))

[1−(1+ zi umn ) exp( − zi umn )] +zi umnc ni (1− exp( −zi umn )) exp( −zi umn )},

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