In this chapter we discuss electrophysiological approaches to the study of renal function. The purpose is to provide an overview of the available techniques, with particular emphasis on what can be learned using the latest methods. However, the chapter is neither a technical manual nor a comprehensive review of the literature. For this, we refer the reader to other sections of the book which deal with specific nephron segments and transport mechanisms. Finally, we will not derive mathematical equations from first principles. Those equations that are essential to the text are provided in the main body of the chapter, while the more detailed formulae are described in the appendices.
Introduction
In this chapter we discuss electrophysiological approaches to the study of renal function. The purpose is to provide an overview of the available techniques, with particular emphasis on what can be learned using the latest methods. However, the chapter is neither a technical manual nor a comprehensive review of the literature. For this, we refer the reader to other sections of the book which deal with specific nephron segments and transport mechanisms. Finally, we will not derive mathematical equations from first principles. Those equations that are essential to the text are provided in the main body of the chapter, while the more detailed formulae are described in the appendices.
We have arbitrarily divided the field of epithelial electrophysiology into three major sections. The first describes measurements of transepithelial electrical properties. The second section focuses on the use of intracellular microelectrodes to discriminate apical and basolateral membrane properties. The final section deals with the technique of patch clamping to investigate the functional characteristics of individual ion channels, and in some cases their molecular identification.
The interpretation of electrical signals from epithelia is complicated by the geometry of the tissues. At least three structures within an epithelium contribute to its electrical properties: the apical plasma membrane; the basolateral plasma membrane; and the paracellular pathway. The individual cell membrane properties will, in turn, be determined by various conductive pathways, including those of passive or dissipative pathways, through which ions flow driven by their own electrochemical potential differences, and active transport, which can use metabolic energy to drive ions against these potential differences. The paracellular pathway, in turn, consists of the tight junctions connecting the epithelial cells and the lateral interspaces between the cells.
The electrical properties of this complex structure can be most easily understood in terms of equivalent circuits. A comprehensive equivalent circuit of a generic reabsorbing epithelium is illustrated in Figure 7.1 . Electrolyte diffusion across the apical membrane can be separated into its constituent ionic pathways, as shown by the expanded view of the apical membrane in Figure 7.1 . Each ionic pathway is associated with an electromotive force (EMF) or battery representing the chemical potential for each ion. Electrogenic carriers such as the Na-glucose co-transporter can also be represented by an additional resistor (R glu ) and EMF (E glu ) in parallel with the diffusional elements. All of the batteries and resistors can be lumped respectively into a single apical EMF (E ap ) and a single resistance R ap as shown in the center diagram. The basal membrane has a similar set of elements: R b and E b which represent a dissipative ion pathway in parallel with an active transport pathway, represented by a resistor (R ap ) and an EMF (E p ). Thus:
Ex=−RTziFLn[X1][X2]
Ex=−RTziFLn[X1][X2]
E x = − R T z i F L n [ X 1 ] [ X 2 ]
tx=gxgtot
tx=gxgtot
t x = g x g t o t
E=tNaENa+tKEK+tClECl=−RTF∑x=Na,K,CltxzxLn[X1][X2]
E=tNaENa+tKEK+tClECl=−RTF∑x=Na,K,CltxzxLn[X1][X2]
E = t N a E N a + t K E K + t C l E C l = − R T F ∑ x = N a , K , C l t x z x L n [ X 1 ] [ X 2 ]
The overall goal of classical electrophysiology, described in the first two sections of this chapter, is to evaluate the different elements of this equivalent circuit, to quantify the various resistances and EMFs, and to describe the extent to which they change during regulation. To this end, it has often been desirable to use reductions of the main equivalent circuit of Figure 7.1 , and to work under conditions in which these reduced circuits are applicable. Such simplifications are discussed in more detail in the sections entitled, “Transepithelial Measurements” and “Intracellular Measurements.” Finally, the section entitled, “Measurements of Individual Ion Channels” describes application of the patch clamp technique to epithelia, and permits a description of ion transport in more molecular terms.
The rationale for representing the pump by a resistance and an EMF is discussed further in the section entitled “Intracellular Measurements.” Briefly, the E p represents the maximum amount of energy that the pump derives from splitting ATP or, alternatively, the maximum electrochemical potential difference against which the pump can operate. The pump pathway must also include a non-zero internal resistance. The magnitude of this internal resistance is determined by the actual current–voltage relation for the membrane bound Na-K-ATPase. To maintain generality, all membrane resistances are shown as variable resistors to include the possibility of intrinsic or extrinsic regulation of ion channels.
Although the paracellular pathway is not a membrane barrier, it can also be modeled by resistive (R tj ) and electromotive elements (E par ). A lateral network (indicated to the right of the main circuit) is also included in the model. This network takes into consideration the finite resistance of fluid in the lateral spaces. This aspect of the circuit becomes important for calculation of individual membrane resistances from voltage deflection experiments (see Section entitled “Intracellular Measurements”).
Transepithelial Measurements
Measurements of transepithelial electrical properties are by far the simplest to perform, and the most difficult to interpret. They are easy to carry out because they are non-invasive; only extracellular electrodes are employed. They are difficult to interpret because the parameters which can be measured reflect, in most cases, a combination of many of the circuit elements shown in Figure 7.1 . In this chapter we will describe how transepithelial techniques are used to measure three basic parameters that characterize an epithelium: transepithelial voltage; transepithelial resistance; and short-circuit current. We will then discuss a number of special extracellular approaches which have been employed to gain additional insights into epithelial properties.
Measurement of Transepithelial Voltage
Methods of measuring transepithelial voltage (V te ) are conceptually simple. In principle, the potential difference between two electrodes placed on either side of the epithelium is simply determined with an appropriate electrometer. With flat epithelia that can be mounted in Ussing chambers, the transepithelial electrodes are placed in the two bathing compartments. In cylindrical epithelia such as the renal nephron, classical measurements of transepithelial potential have been performed in vivo using micropuncture techniques. In this technique, a pipette filled with electrolyte is introduced into the lumen of the tubule, and voltage is measured relative to another electrode placed in a capillary. Here, the presumption is that the voltage reflects the properties of the impaled tubule, and is not greatly influenced by those of neighboring segments. Under most conditions this should be a reasonable assumption. For a lumen of 10 μm diameter and an isoosmotic saline solution with a resistivity of around 60 Ω•cm, the axial resistance of the tubule will be about 8×10 7 Ω/cm. This is much larger than the value of the transepithelial resistance of around 100 Ω•cm 2 ( Table 7.1 ), which for the same 10 μm lumen is equivalent to 3×10 4 Ω/cm. Thus, each part of the nephron will be effectively electrically isolated from other parts of the high resistance of the luminal pathway. If the nephron segment can be isolated and perfused in vitro , the perfusion and/or collection pipette can be used to monitor the intraluminal voltage with respect to the bath potential. This is illustrated in Figure 7.2 .
| Tissue | V te (mV) | R te (Ωcm 2 ) | R par (Ωcm 2 ) | P Na /P Cl | Reference |
|---|---|---|---|---|---|
| Amphibians | |||||
| Proximal tubule ( Ambystoma ) | −10 | 70 | 70 | 0.25 | |
| Diluting segment ( Amphiuma , frog) | +10 | 290 | 306 | 4 to 5 | |
| Collecting duct ( Amphiuma ) | −24 | 160 | 200 | 0.84 | |
| Urinary bladder (Toad) | −94 | 8,900 | 50,000 | – | |
| Mammals | |||||
| Proximal tubule (rabbit) | −2 to +2 | 5 | 5 | ||
| TALH a (rabbit) | +3 to +10 | 10 to 35 | 10 to 50 | 2 to 4 | |
| CCD a (rabbit) | 0 to −60 | 110 | 160 | 0.8 | |
| OMCD a (rabbit) | −2 to −11 | 233 | |||
| IMCD a (rabbit) | −2 to 0 | 73 | |||
| Urinary bladder (rabbit) | −20 to −75 | 13,000 to 23,000 | >78,000 | – |
a TALH, thick ascending limb of Henle’s loop; CCD, cortical collecting duct; OMCD, outer medullary collecting duct; IMCD, inner medullary collecting duct.
Measurement of Transepithelial Resistance
For measurements of transepithelial resistance ( R te ), current must be injected across the epithelium to perturb V te . This is most easily accomplished when the epithelium can be mounted in Ussing chambers, where current flow and voltage changes are assumed to be uniform in the plane of the tissue. Resistance can then be computed from Ohms law as the ratio of the change in V te to the amount of current passed:
Rte=ΔVteΔI
Rte=ΔVteΔI
R t e = Δ V t e Δ I
Epithelia can be studied in open-circuited or voltage-clamped conditions. In open circuit the tissue is allowed to maintain its spontaneous transepithelial voltage. In this case the resistance is determined from the change in voltage produced by passing a known amount of current. Under voltage-clamp conditions a current is passed across the epithelium to maintain the transepithelial voltage at a predetermined level. In the case where this level is zero, so that the transepithelial voltage is abolished, it is called the “short-circuited” state. If the epithelium is voltage-clamped, resistance is determined from the change in current produced by a controlled voltage step. In open-circuited tubular epithelia, transepithelial current flow is not constant along the length of the tubule. In this case, cable analysis must be used to estimate the transepithelial resistance (see below).
Measurement of Transepithelial Resistance in Open Circuited Renal Tubules
The measurement of overall transepithelial resistance, R te , in renal tubules under open circuit conditions is best carried out with a double-barreled perfusion pipette system similar to the one illustrated in Figure 7.2 . In this technique, the tubule is cannulated at both the perfusion and collection ends. The double-barreled perfusion pipette, fabricated from theta glass creates separate pathways for current flow and voltage recording. An alternative technique uses the same single-barreled pipette for both current injection and voltage measurement. This is not nearly as accurate as the double-barreled technique, because the voltage deflection arising from the internal resistance of the perfusion pipette must be nulled with a bridge circuit. The microelectrodes in Figure 7.2 are for evaluation of individual cell membrane resistances. This will be discussed in the section entitled, “Intracellular Measurements.”
A thin fluid-exchange tubing (not shown) can be inserted into one barrel of the perfusion pipette of Figure 7.2 to permit rapid exchange of the perfusion solution while measuring the transepithelial potential V te ( x=0 ). Current is passed from a chlorided silver wire glued into the other barrel of the pipette. The transepithelial length constant of the tubule λ te is determined from the voltage deflections at the perfusion, ΔV te ( x=0 ), and collection, ΔV te ( x=L ), sides of the tubule, resulting from a transepithelial current pulse, I te , through the current side of the perfusion pipette. For a doubly cannulated, isolated tubule of length L , λ te is given by Eq. (7.5) from Sackin and Boulpaep :
Lλte=cosh−1[ΔVte(x=0)ΔVte(x=L)]
Lλte=cosh−1[ΔVte(x=0)ΔVte(x=L)]
L λ t e = cosh − 1 [ Δ V t e ( x = 0 ) Δ V t e ( x = L ) ]
The transepithelial resistance R te in Ωcm 2 is given by Eq. (7.6) :
Rte=2πλte3RinRitanh(L/λte)
Rte=2√πλ3teRinRi√tanh(L/λte)
R t e = 2 π λ t e 3 R i n R i t a n h ( L / λ t e )
In the above equation, R in is the input resistance measured in ohms (Ω). It is operationally defined as the voltage deflection at the perfusion end ΔV te ( x=0 ) divided by the total injected current I te . Typical injected currents for proximal tubules are 100 nA pulses of 1 to 5 seconds duration. Ideally, none of the current injected into the lumen from the perfusion pipette enters the compressed region of tubule within the holding pipette, which presumably acts like an electrical insulator when compared to the relatively low resistance of the tubule. However, artifacts may still arise from current leaks at either the perfusion or collection ends of the tubule. These can be detected by a “mismatch” between the calculated electrical radius ( r e ) of the tubule and its measured optical radius ( r o ).
re=2Riλte2Rte
re=2Riλ2teRte
r e = 2 R i λ t e 2 R t e
Use of a double-barreled perfusion pipette ( Figure 7.2 ) eliminates much of the uncertainty in R in . Double-barreled perfusion pipettes have the additional advantage that R te can be measured during changes in the perfusion solution. This is practically impossible with a single-barreled perfusion pipette, because the bridge circuit is unstable during solution changes. Finally, the term R i (Ω•cm) is the volume resistivity of the perfusion solution, as measured with a standard conductivity meter.
Measurement of Transepithelial Resistance in Voltage Clamped Renal Tubules
There have been a number of early attempts to elucidate the electrical properties of renal tubules using voltage clamp techniques similar to those originally developed for flat epithelia. The basis of these methods is to isolate a segment of tubule (usually with oil droplets) that is short enough to permit a uniform current distribution across the epithelium. To accomplish this, metallic axial electrodes are directly inserted into the lumen of the tubule. These axial electrodes can also be used for AC impedance analysis. However, one important problem with metal electrodes is the release of ions into a restricted space during continuous current flow.
An alternative technique employs segments of isolated, perfused tubules that have been shortened to such an extent that the current distribution within the lumen is virtually homogeneous. In this case R te is essentially determined from the input resistance according to Eq. (7.8) :
Rte=2πroL⋅Ri
Rte=2πroL⋅Ri
R t e = 2 π r o L ⋅ R i
Typical Results
Measurements of V te and R te in some representative epithelia are shown in Table 7.1 . The range of both these parameters is large, with values of V te ranging from ±2 mV in the proximal tubule to as much as −60 to −80 mV in the CCT. R te values vary from less than 10 Ωcm 2 in proximal tubule to more than 5000 Ωcm 2 in urinary bladder. Despite the range of values observed, the transepithelial voltages in all cases reflect two factors: the conductance of the epithelium to the major ions and the active transport of ions. In general, a high value of V te indicates that active transport is taking place across a high resistance epithelium, whereas low values of V te can reflect either a low R te or a low rate of active transport.
Traditionally, epithelia have been divided into the categories “tight” and “leaky,” according to their transepithelial resistances. In leaky epithelia the low value of R te is thought to largely reflect the low resistance of the tight junctions which constitute the major electrical resistance of the paracellular pathway between the epithelial cells. In tight epithelia the tight junctional resistance, and therefore the transepithelial resistance, is much higher. However, the resistance above which an epithelium is considered “tight” is not precisely defined. Even though the amphibian proximal tubule and the mammalian collecting duct have similar absolute values of paracellular resistance, the proximal tubule is considered a leaky epithelium, whereas the collecting duct is usually referred to as “tight.” Thus, a better definition of a leaky epithelium is one in which the paracellular resistance is low relative to that of the cell membranes. For example, in Table 7.1 it can be seen that in leaky proximal tubules R te is virtually equal to R par , which is small compared to the parallel transcellular resistance R c . In tight epithelia R par is significantly larger than R te . This implies that R par is of the same order of magnitude as R c or even much larger in the case of the urinary bladder. Another feature of a tight epithelium is its ability to separate two fluid compartments with very different ion compositions. High resistance tight junctions slow the “backleak” of ions and other solutes down their concentration gradients. Thus, in a tight epithelium it is harder to dissipate the ion gradients established by active transport processes.
Interpretation of Measurements of V te and R te
As discussed above, transepithelial measurements of voltage and resistance are difficult to interpret because they lump together information from many different electrical pathways arranged in parallel. To analyze such data, it is often useful to use a simplified equivalent electrical circuit.
The terms R c and E c represent the resistance and electromotive forces across the transcellular pathway, whereas R ti and E par are respectively the resistance and electromotive forces across the paracellular pathway. If the potential differences V te , E c , and E par are all defined with respect to the bath or serosal side, the overall measured transepithelial values R te and V te are related to this circuit by Eqs (7.9) and (7.10) :
Rte=RtjRcRtj+Rc
Rte=RtjRcRtj+Rc
R t e = R t j R c R t j + R c
Vte=EcRtj+EparRcRtj+Rc
Vte=EcRtj+EparRcRtj+Rc
V t e = E c R t j + E p a r R c R t j + R c
The dissection of the measured parameters into the appropriate contributions from cellular and paracellular pathways can sometimes be accomplished by using maneuvers that affect only one of the pathways or cause one pathway to dominate the other. Some of these perturbations and special conditions will be discussed below.
Contribution of Active Transport to V te
The major effect of the pump is to establish transmembrane ionic gradients, i.e., to keep cell Na low and cell K high. As was first pointed out by Koefoed-Johnsen and Ussing in their classic paper of 1958, the permeability properties of the apical and basolateral membranes of the frog skin are quite different. Since the apical membrane is selectively permeable to Na, and the driving force for this ion is inward, the entry of Na will tend to make the cell voltage positive with respect to the mucosal solution. Conversely, the basolateral membrane is selectively permeable to K. This ion will tend to flow out of the cell, making the cell voltage negative with respect to that of the serosal fluid. The EMFs E ap (= RT / F Δln[Na]) and E bl (= RT / F Δln[K]) will be in the same direction with respect to the epithelium and the transepithelial EMF, and hence V te , will reflect their sum ( Figure 7.4 ).
Although the Na-K-ATPase is ultimately responsible for the transepithelial potential in many Na-reabsorptive epithelia, the magnitude of V te does not correlate with the magnitude of active transport when different tissues are compared. In general, the effect of active ion transport will be shunted by the paracellular resistance. This shunting is least in the tight epithelia such as frog skin and toad urinary bladder, where V te can be over 100 mV. In leaky epithelia such as the proximal tubule the shunting is considerable, and the values of V te are much lower.
In leaky epithelia, E par will be much smaller than E c ( Figure 7.3 ), since ion gradients across the tight junction are relatively small. In tight epithelia where ion gradients can be significant, R tj >R c so that the term E par R c will be small compared to E c R tj , and Eq. (7.10) becomes:
Vte=Ec1+Rc/Rtj
Vte=Ec1+Rc/Rtj
V t e = E c 1 + R c / R t j
The implication of Eq. (7.11) is that if R tj >>R c , V te will approach E c , a quantity which is limited by the EMF of the Na-pump. In general, V te will be reduced according to the ratio R c /R tj . The contribution of active transport to cell membrane potential is discussed in the section entitled, “Estimation of renal Na,K pump current and electrogenic potential.”
The mammalian TALH, and its amphibian counterpart the diluting segment, have lumen-positive V te despite the fact that they are also Na-reabsorbing epithelia ( Table 7.1 , Figure 7.4 ). This turns out to be the exception that proves the rule. As discussed in detail by Greger, Na does not enter the TALH cell through a conductive mechanism, as in the frog skin and other epithelia, but through an electrically neutral co-transport system along with K and Cl. Thus, Na entry does not contribute to a lumen-negative voltage, and in fact the membrane is more permeable to K than to Na. Furthermore, the basolateral membrane has a rather high permeability to Cl. This makes the lumped EMF E ap less negative than E bl , and the potential difference between the cell and the blood side is less negative than the potential difference between the cell and the lumen. Hence, the mechanism for the lumen-positive potential in the TALH is accounted for by the different permeability properties of the two membranes, just as in the frog skin.
Contribution of Diffusion Potentials to V te
Paracellular diffusion potentials can contribute significantly to the overall transepithelial potential, especially when V te is small. For example, in the mammalian proximal tubule the early portion of the segment has a lumen-negative V te in vivo , which is thought to reflect active Na reabsorption. Farther down the nephron, however, the lumen becomes positive with respect to the blood. Preferential luminal reabsorption of HCO 3 − relative to Cl − establishes opposing gradients for Cl − and HCO 3 − across the tight junction ( Figure 7.5 ). This results in a lumen-positive potential since Cl − diffuses more rapidly across the junctions than HCO 3 − .
Diffusion potentials may also contribute to the normal lumen-positive V te in the mammalian TALH ( Table 7.1 ). In this segment NaCl is reabsorbed but water is not, leading to a dilution of the luminal fluid. Since the tight junctions of the mammalian TALH are cation-selective, Na diffuses back more rapidly than the Cl, contributing to a lumen-positive diffusion potential ( Figure 7.5 ).
Contribution of Circulating Current to V te
The different equivalent EMFs at the apical and basolateral sides of the cell produce a circulating current ( I ) which traverses both cell membranes in series, and returns via the paracellular shunt ( Figure 7.1 ). The magnitude of this current depends on the relative resistances of paracellular versus cell pathways, as well as the active transport rate for the particular epithelium. For example, in renal proximal tubules, the low shunt resistance (compared to transcellular resistance) characterizes this nephron segment as a leaky epithelia with large circulating current. On the other hand, tight epithelia like the urinary bladder or the frog skin have shunt resistances comparable to or larger than the transcellular resistance, so that the circulating current is small in comparison with proximal tubule.
The effect of circulating current on renal transepithelial potentials can be understood qualitatively by considering the electrical profiles depicted in Figure 7.6 . In this figure, the serosal or blood side of the epithelium is considered at ground and the voltage at any point is displayed as a function of distance from mucosa to serosa. For the sake of simplicity, we have assumed that the apical membrane is primarily selective to sodium, the basolateral membrane is primarily selective to potassium, and the interior of the cell is isopotential. Therefore, in the absence of circulating current, there is a “staircase” voltage profile through the epithelium determined by the respective diffusion potentials across the mucosal (or apical) membrane and across the serosal (or basolateral) membrane ( Figure 7.6a ) . Under these conditions the measured V te (mucosa minus serosa) would actually be more negative than the basolateral cell membrane potential ( E K ).
In most epithelia the diffusion potential steps of Figure 7.6a would be modified by the effect of circulating current ( I ) across the resistance of the mucosal and serosal barriers. Specifically, the mucosal to cell step will be raised by an amount: I•R ap due to the circulating current crossing the mucosal membrane resistance ( Figure 7.6b ). The same current crossing the basolateral side of the cell will decrease the size of the cell-to-serosal step by I• <SPAN role=presentation tabIndex=0 id=MathJax-Element-12-Frame class=MathJax style="POSITION: relative" data-mathml='Rbl*’>?∗??R*bl
Rbl*
R*bl
R b l *
, where <SPAN role=presentation tabIndex=0 id=MathJax-Element-13-Frame class=MathJax style="POSITION: relative" data-mathml='Rbl*’>?∗??R*bl
Rbl*
R*bl
R b l *
is the effective basolateral resistance. The final values of V te and V bl can be calculated by considering the complete equivalent circuit (Appendix 7.1).
In most epithelia, the resistance of the apical membrane is larger than the resistance of the basolateral membrane, and the effect of the circulating current is to transform the staircase potential ( Figure 7.6a ) into a “well-type” potential ( Figure 7.6c ), where the intracellular region is the most negative space and V te is directly dependent on the magnitude of the current and the tightness of the epithelial cell layer. In some tight epithelia ( Necturus urinary bladder) with high paracellular resistance and low circulating current, the “staircase” potential profile is still maintained despite “IR drops” at both membranes.
Short-Circuit Current
It is also possible to measure transepithelial currents while controlling the transepithelial voltage. A special case of this voltage-clamp approach is the short-circuit current technique in which V te is maintained at zero. If the solutions on both sides of the epithelium are identical, there is no net movement of ions through the paracellular spaces, since both electrical and chemical driving forces are reduced to zero. The current across the tissue, which must also pass through the external circuit and can thus be readily measured, results only from active transport processes (defined as those which take place against an electrochemical activity gradient). Thus this current (called the short-circuit current), will equal the sum of all active ion transport processes.
The particular ion being actively transported can be identified rigorously by measuring net fluxes at the same time as the short circuit current. For some cases, such as the frog skin and toad bladder, the short-circuit current can be accounted for by the active transport of only one ion species, namely Na ( Table 7.2 ). In general, the short-circuit current will represent the sum of the net transport of Na, K, H, Cl, and HCO 3 . Another way to identify actively transported ions is to eliminate them from the bathing media and measure the resulting effects on short-circuit current. This approach is often experimentally much simpler, but it is less rigorous since the apical and basolateral solutions will not be identical. Furthermore, changing the external environment of the tissue can lead to secondary changes in cell composition and volume. In any case, once the transported species have been identified, the technique becomes a very convenient way to analyze the regulation of the active transport systems.
| Type of Epithelium | Mucosal to Serosal | Serosal to Mucosal | Net | Short-circuit Current |
|---|---|---|---|---|
| Frog skin | 24.6 | 1.5 | 23.1 | 23.6 |
| Toad urinary bladder | 35.7 | 9.5 | 26.2 | 26.8 |
An important limitation of the short-circuit current technique is that it often requires unphysiological conditions. For example, short-circuiting high resistance epithelia like frog skin will reduce the normally large V te to values near zero. This will necessarily affect the transmembrane voltage of one or both cell membranes, which may in turn affect the ionic conductances of those membranes.
The short-circuit technique also involves bathing the apical side of the tissue with a solution that has an electrolyte composition close to that of the blood. This is a highly unphysiological condition for many tight epithelia like the frog skin, which is normally in contact with pond water, and the toad bladder, which is normally in contact with dilute urine. Another important problem with short-circuit experiments is that short-circuited tissues do not have to maintain the electroneutrality of the transported species. For example, in Na-transporting epithelia, such as the frog skin, Na ions can be reabsorbed only if another cation (e.g., K, H) is secreted or if an anion (Cl) is also reabsorbed. Under physiological conditions these other ionic pathways can be rate-limiting for Na reabsorption.
Finally, the uniform current distribution required by the short-circuit technique has largely restricted its use to flat epithelia which can be mounted in Ussing chambers. However, in some cases it has been possible to voltage-clamp large-diameter amphibian tubules. Attempts have also been made to circumvent these technical problems by defining an “equivalent short-circuit current” for renal epithelia. In this method, the current at V te =0 is estimated by dividing the spontaneous value of V te by the transepithelial resistance R te . This approach assumes that R te is constant; i.e., that the current voltage relation of the epithelium is linear. Even when this condition is satisfied, it is not always possible to attribute the equivalent short-circuit current to specific ion species, since net fluxes of the ions must be measured under true short-circuited conditions.
Technical Problems
For epithelia that can be studied as flat sheets in vitro , the major technical problem with transepithelial measurements is avoiding edge damage to the tissues, particularly when these tissues are mounted in Ussing chambers. On the other hand, for renal micropuncture experiments performed in situ , the major technical problem is localization of the microelectrode tip within the tubular lumen.
The most important general problem in the measurement of transepithelial resistance is the choice of the magnitude and duration of the applied perturbations. Currents (or voltage changes) which are too large can result in changes in the electrical properties of the membranes due to voltage-dependent ion conductances. Perturbations which are either too large or too long can lead to redistribution of ions across the cell membranes, which can also alter electrical properties. For example, in toad urinary bladder modest changes in V te in the order of 10 mV under voltage-clamp conditions can result in time-dependent changes in the tissue resistance. On the other hand, if perturbations are too small they are difficult to measure accurately, and if they are applied for too short a time the capacitative, as well as the resistive, properties of the epithelium will affect the response. There are no generally accepted rules for determining the size and duration of the perturbations.
Estimation of Membrane Parameters from Transepithelial Measurements
Measurement of transepithelial electrical properties does not, in general, give any direct, quantitative information about the circuit elements of greatest interest, namely the conductances of individual membranes to specific ions. As emphasized throughout this section, R te is a lumped parameter determined by R ap , R b , R tj , and in some cases R lis (see Figure 7.1 ). V te is determined by all the R s and EMFs in the circuit. Clearly, measuring two parameters is insufficient to determine seven or eight unknowns.
However, in some cases it has been possible to either use conditions which simplify the equivalent circuit or to use experimental perturbations that selectively change only one electrical parameter. These methods have provided a good deal of information about epithelial properties from purely transepithelial measurements. Some examples are given below.
Paracellular Resistance and Selectivity
When the paracellular (tight junction) resistance is low compared to the transcellular resistance, the transepithelial resistance is dominated by the resistance of the paracellular pathway. This happens in a leaky epithelium like the proximal tubule. This condition can also be produced in some tight epithelia by blocking the major conductive pathways at the apical membrane. The most frequently used blockers are amiloride, for the Na conductance, and Ba, for the K conductance. In both of these cases the paracellular resistance can then be estimated from transepithelial measurements ( Table 7.1 ), although intracellular recordings are usually required to prove that the transcellular resistance is high.
The ion selectivity of the paracellular pathway can also be evaluated under these circumstances. This involves measurement of the transference numbers for various ions across the tight junction (see Eq. (7.12) ). The most important ions in this case are Na and Cl, and their transference numbers can be estimated by reducing the concentration of NaCl on one side of the junction by diluting one of the bathing solutions. If the transcellular resistance is sufficiently high (i.e., R ap >>R tj , see Eq. (A2.3) in Appendix 2) and is unaffected by the dilution, the measured change in V te will approximately reflect the change in E par where:
ΔEpar=−RTF(tNa−tCl)Ln[NaCl]1[NaCl]2
ΔEpar=−RTF(tNa−tCl)Ln[NaCl]1[NaCl]2
Δ E p a r = − R T F ( t N a − t C l ) L n [ N a C l ] 1 [ N a C l ] 2
If sodium and chloride are the only conducting ions in the external solutions, the absolute transference numbers can be calculated from Eq. (7.12) and the requirement that t Na + t Cl =1. Some measurements of paracellular selectivity in renal epithelia are listed in Table 7.1 . This parameter is of considerable physiological interest. The results range from a significant selectivity for anions (Cl) over cations (Na) in the amphibian proximal tubule, to a cation selectivity in the thick ascending limb of Henle’s loop or its counterpart, the diluting segment, in the amphibian kidney. If ions moved through the tight junctions as if they were in free solution, a permeability ratio P Na /P Cl of 0.8 would be expected. The variations in selectivity result from differences in the expression of specific members of the tight-junction proteins claudins.
Membrane Selectivity
If the paracellular or tight junction resistance is much greater than the transcellular resistance, it is possible to determine the ion selectivity of the individual cell membranes. Specifically, if R tj is very large compared to R ap + R bl , there will be negligible current through either the paracellular pathway or the cell pathway under open-circuit conditions. Under these conditions, the circuit of Figure 7.1 predicts that changes in E ap will parallel changes in V ap which, in turn, can be estimated from the measured changes in transepithelial potential Δ V te (see Equations (A1.8) and (A1.11)–(A1.13) of Appendix 7.1). Such a situation was studied by Koefoed-Johnsen and Ussing in their classic paper on the frog skin, where pre-treatment of the skins with low concentrations of Cu +2 produced very high values of R te and V te .
This permits evaluation of individual membrane selectivities from transepithelial measurements alone if the concentration of just one ion on one side of the epithelium is replaced with an impermeant species, and the conditions associated with Eq. (A1.13) (Appendix 7.1) are satisfied. If this is the case and Na is partially replaced on the apical side, then:
ΔVte=ΔEap=−RTFtNaLn[Na]apexp[Na]apconforRtj>>Rap+Rbl*
ΔVte=ΔEap=−RTFtNaLn[Na]expap[Na]conapforRtj>>Rap+R*bl
Δ V t e = Δ E a p = − R T F t N a L n [ N a ] a p e x p [ N a ] a p c o n f o r R t j > > R a p + R b l *
where the change in potential is measured as experimental minus control. [Na] exp and [Na] con represent the concentrations of Na under experimental and control conditions, i.e., after and before the solution change.
Koefoed-Johnsen and Ussing found that changes in mucosal Na produced changes in V te close to those which would be expected if t Na =1. From this they inferred that the apical membrane was primarily conductive to Na ions. Similarly, changes in serosal K concentration produced changes consistent with the idea that the basolateral membrane conducted only K. The elegant conclusions of this study depended upon the rather unusual conditions achieved, namely a very high paracellular resistance and the absence of other “leak” pathways due to other cell types. Except for the case of the urinary bladder, such conditions are difficult to achieve in renal epithelia where paracellular pathways are usually leakier than in the frog skin.
Selectivity of the epithelial basolateral membrane has also been studied by using pore-forming polyene antibiotics to reduce apical membrane resistance so that V te becomes a reasonable estimate of the basolateral potential V bl ( see Equation A1.16 of Appendix 7.1). For example, in the turtle colon, Germann et al. were able to characterize two different conductances for K across the basolateral membrane using the amphotericin-B permeabilized epithelium. This approach has also been used mostly in flat epithelia rather than renal tubules.
Impedance Analysis
Impedance analysis permits estimation of the electrical properties of individual membranes using transepithelial measurements. In principle, transepithelial impedance can be measured from the time-course of the response to any electrical perturbation. In practice, it is usually obtained either under “current-clamp” conditions, using sine wave current perturbations at different frequencies or under voltage-clamp conditions by applying voltage perturbations. Since the apical and basolateral membranes will each have an associated complex impedance that depends on frequency, it is possible to distinguish contributions from the two membranes if they have very different time constants (τ=RC, where R is the resistance and C the capacitance).
Typically four parameters (consisting of the amplitudes and time-constants of the two membrane components) are measured to fit a circuit with five parameters (resistance and capacitance of apical and basolateral membranes and the paracellular resistance). To fully solve the system one model parameter (e.g., the paracellular resistance) usually must be determined independently.
Impedance analysis has also been used to derive detailed information about the paracellular pathway, including the distribution of resistance across tight junctions and along intercellular spaces. In this regard, it has been used to estimate individual resistances and capacitances of the apical and basolateral membranes of flat epithelial sheets obtained from frog skin, amphibian and mammalian urinary bladder, colon, and cultured epithelial cells. The elegance of the technique is that it is non-invasive, yielding information about individual membranes without the need for intracellular probes or electrodes. Because of the requirement of uniform currents or voltage fields, its application to renal tubules has been very limited.
Conclusions
Many important epithelial properties can be determined from transepithelial measurements alone. In fact, the original Koefoed-Johnsen/Ussing model for Na transport by the frog skin was based entirely on transepithelial electrical and flux measurements, and the ingenious use of special simplifying conditions. Thus, a number of important physical and thermodynamic properties of epithelia are still estimated from transepithelial measurements, particularly in renal tissues. These include the magnitude and selectivity of the paracellular shunt pathway, the ion selectivity of cell membranes (qualitatively in most instances), and the currents and EMF’s associated with active transepithelial transport. On the other hand, the quantitative description of membrane properties requires detailed intracellular measurements to specifically characterize the apical and basolateral membrane components, as well as the contribution of the paracellular pathway. These measurements will be discussed in the next section.
Intracellular Measurements
Intracellular measurements with voltage-sensitive and ion-sensitive microelectrodes permit a more detailed evaluation of individual membrane parameters than transepithelial measurements. This has been essential to our understanding of ion transport in epithelia. Three important membrane characteristics that are amenable to study with intracellular techniques are: (1) ionic selectivity; (2) membrane conductance; and (3) estimation of pump current. Although there are significant differences in the methodology of these measurements depending on the particular tissue involved, much of the underlying theory is similar in both flat epithelia and renal tubules. The emphasis of this section will be on describing the simplest and most straightforward methods for evaluation of single-membrane parameters with emphasis on renal epithelia, although much of the theory is applicable to flat epithelia as well.
Cell Membrane Potentials in Epithelia
An epithelium is a sheet of polarized cells joined together to function as a selective barrier between two compartments. Epithelia not only structurally define two compartments, but also maintain the composition of those compartments via the specific transport of electrolytes, non-electrolytes, and water. The electrical voltage measured across either the apical or basolateral membrane of an epithelium is the sum of the ionic diffusion potentials across the membrane, and the voltage drops arising from current flow across the resistance of that membrane. This current flow (depicted by the thick arrow in Figure 7.1 ) arises in part from the differences in membrane ionic diffusion potentials and the sum of epithelial barrier electrical resistances (Appendix 7.1).
In practice, cell membrane potentials are determined by impaling the cell with fine tipped glass microelectrodes, filled with a highly conductive electrolyte solution (1 M or 3 M KCl). Uniform filling is often accomplished by starting with glass tubing that contains a thin glass filament, allowing solution to flow smoothly from the back to the tip of the finished electrode. The microelectrode is then mounted on a stable micromanipulator. Given the elasticity of most cell membranes, impalement usually requires a rapid forward movement of the tip that can be accomplished either mechanically or with piezoelectric headstage. Sometimes a high frequency alternating current is briefly applied to the tip to permit entry into the cell with minimum damage.
A major technical problem with microelectrode measurements is the damage to the cell that may be produced by impalement. For a discussion of this topic see Higgins et al. and Nelson et al.. Cell damage can be minimized by utilizing epithelia with large cells (e.g., Necturus ; Amphiuma ), and by recording from the basolateral rather than the apical membrane. This will result in less electrical shunting of the membrane potential, since in many cases the basolateral membrane has a much lower resistance than the apical membrane and an additional leak conductance at the basal side will have a smaller overall effect. Furthermore, epithelia like the proximal tubule, whose cells are electrically coupled, are less sensitive to impalement artifacts, since the effective cell membrane area is larger. Finally, the use of very fine-tipped micropipettes can minimize membrane damage during intracellular measurements. However, such pipettes are more likely to produce artifacts due to changes in the liquid-junction potentials (“tip potentials”) when the tip enters the cytoplasm.
Use of microelectrodes to measure cell potential may result in KCl leakage into the cell. Although these tips are extremely small (<0.2 μm), the use of concentrated KCl in the electrode to minimize liquid junction potentials can lead to KCl influx into the cell, alterations in cell composition, and cell swelling. Thus, the choice of a filling solution is a trade-off between a concentrated KCl solution, which yields low tip potentials but possible KCl leakage, versus a low salt pipette solution, which results in higher tip potentials but less salt leakage into the cytoplasm.
Intracellular potential can also be measured with relatively large diameter patch-clamp pipettes. In a “whole-cell clamp” experiment, a high resistance seal is formed between the pipette and the cell membrane. This permits direct electrical contact between the recording electrode and the cell interior with negligible impalement damage. If the amplifier is used in the voltage-clamp mode, the holding potential that reduces the membrane current to zero becomes a good measure of the cell potential. One disadvantage of using the whole-cell clamp to measure cell potential is that the cell is dialyzed with the pipette solution. The exchange of vital cell constituents with the pipette solution may alter the intracellular ion composition, as well as change the normal cell membrane permeabilities through the loss of regulatory factors. The latter effect can be minimized by using the “perforated-patch” technique, in which the patch is permeabilized by the addition of pore-forming substances such as nystatin to the pipette solution.
Evaluation of Individual Membrane Resistances from an Equivalent Circuit Analysis
The simplest technique for determining individual cell membrane resistance is to measure intracellular and transepithelial potential during an experimental maneuver that produces only a single perturbation in the parameters of the equivalent circuit. These techniques have been particularly useful in mammalian proximal tubules where multiple microelectrode impalements are difficult or in nephron segments that are not electrically coupled.
When only one microelectrode is used, the circuit of Figure 7.1 must be simplified to permit an indirect evaluation of the cell membrane resistances. This type of reduction is illustrated in Figure 7.7 , and is permissible when most of the paracellular resistance is contributed by the tight junction resistance (i.e., when R par ≈ R tj ), which is equivalent to the assumption that R tj is >> lateral interspace resistance (= R lis ). This is particularly appropriate for mammalian proximal tubule, where basal interdigitations of adjacent cells greatly reduce the lateral space resistance, and most of the paracellular resistance is contributed by the tight junction resistance.
Ebl*
E*bl
E b l *
) and an effective basolateral resistance ( <SPAN role=presentation tabIndex=0 id=MathJax-Element-17-Frame class=MathJax style="POSITION: relative" data-mathml='Rbl*’>?∗??R*bl
Rbl*
R*bl
R b l *
). “I” denotes direction of positive net circulating current under open-circuit conditions.
The circuit of Figure 7.7a can be further reduced to the simpler form of Figure 7.7b by defining an effective basolateral EMF ( <SPAN role=presentation tabIndex=0 id=MathJax-Element-18-Frame class=MathJax style="POSITION: relative" data-mathml='Ebl*’>?∗??E*bl
Ebl*
E*bl
E b l *
) and an effective basolateral resistance ( <SPAN role=presentation tabIndex=0 id=MathJax-Element-19-Frame class=MathJax style="POSITION: relative" data-mathml='Rbl*’>?∗??R*bl
Rbl*
R*bl
R b l *
), as described in Appendix 7.1. Although the electromotive forces or EMFs ( E ap , E bl , E par ) in the circuit cannot be measured directly, the potential differences across each barrier can be measured with intracellular or transepithelial electrodes. These are defined as transepithelial potential (lumen-bath)= V te , apical cell membrane potential (lumen-cell)= V ap , and basolateral potential (cell-bath)= V bl .
An important consequence of the circulating epithelial current in Figure 7.7 is that alterations in any of the electrical parameters on one side of the cell will produce changes in the measured electrical potentials on the contralateral side. This actually provides an indirect method for evaluating those resistances that remain constant during a change in loop current, I . Individual cell membrane resistances <SPAN role=presentation tabIndex=0 id=MathJax-Element-20-Frame class=MathJax style="POSITION: relative" data-mathml='Rbl*’>?∗??Rbl*
Rbl*
R b l *
and R ap can be evaluated by any experimental maneuver that changes only the parameters at one membrane. For example, rapid addition of amiloride or glucose to the apical solution presumably alters only the resistance and/or the EMF of the apical membrane by blocking Na channels or stimulating Na-glucose co-transport, respectively.
Amiloride causes a hyperpolarization of V bl by both increasing the measured apical resistance ( R ap ) and decreasing the contribution of the Na gradient to the value of E ap (see Appendix 7.1, Eq. (A1.7)) . On the other hand, addition of glucose to the luminal solution depolarizes V bl by stimulating Na-glucose co-transport, which effectively increases both the apical Na conductance and the relative contribution of the Na gradient to the apical diffusion potential. Since the primary effect of both amiloride and glucose occurs at the apical membrane, the ratio of basolateral to apical resistance is directly related to the measured ratio of basolateral to transepithelial voltage deflections according to Eq. (7.14) , which applies for addition of either apical-side amiloride or apical-side glucose:
Rbl*Rtj=−ΔVblΔVte=β
R b l * R t j = − Δ V b l Δ V t e = β
This equation implicitly assumes that neither glucose nor amiloride affect the paracellular pathway, and that the potential measurements can be performed before any changes in cell composition have occurred that would affect E ap , E bl , and R bl .
Similarly, measurement of the ratio of apical to transepithelial potential change following addition of barium to the basolateral solution permits estimation of the apical membrane resistance via Eq. (7.15) :
RapRtj=−ΔVapΔVte=ΔVblΔVte−1=α
R a p R t j = − Δ V a p Δ V t e = Δ V b l Δ V t e − 1 = α
Again, it has been assumed that basolateral application of barium has no effect on the paracellular pathway, and that the measurement can be performed rapidly enough to avoid changes in cell composition.
Finally, in cases where it is feasible to make changes on only one side of the epithelium, the voltage divider ratio can be used instead of either Eq. (7.14) or Eq. (7.15) . When current is injected into the tubule lumen via the perfusion pipette, a certain fraction of that current will cross the apical and basolateral cell membrane in series, producing voltage deflections ΔV ap and ΔV bl . If the lateral and basal resistances of Figure 7.1 are combined into a single effective resistance <SPAN role=presentation tabIndex=0 id=MathJax-Element-23-Frame class=MathJax style="POSITION: relative" data-mathml='Rbl*’>?∗??Rbl*
Rbl*
R b l *
(defined in Appendix 7.1, Eq. (A1.4) ) the ratio of apical to basolateral resistance during transepithelial injection of current is given by Eq. (7.16) :
γ=RapRbl*=ΔVapΔVbl=ΔVteΔVbl−1
γ = R a p R b l * = Δ V a p Δ V b l = Δ V t e Δ V b l − 1
The term “γ” is sometimes referred to as the voltage divider ratio. An alternative to Eq. (7.16 ) is to define the “fractional resistance” of either the apical ( fR ap ) or basolateral membranes ( fR bl ), according to Eqs. (7.17a), (7.17b) :
fRap=RapRap+Rbl*=1−ΔVblΔVte
f R a p = R a p R a p + R b l * = 1 − Δ V b l Δ V t e
fRbl*=Rbl*Rap+Rbl*=ΔVblΔVte
f R b l * = R b l * R a p + R b l * = Δ V b l Δ V t e
Since γ, α, β, and R te are all measured quantities, the individual resistances: <SPAN role=presentation tabIndex=0 id=MathJax-Element-27-Frame class=MathJax style="POSITION: relative" data-mathml='Rbl*’>?∗??Rbl*
Rbl*
R b l *
, R ap , R tj , can be evaluated by using any three of the four equations Eqs. (7.14), (7.15), (7.16), and Eq. (A1.5) .
An example of these methods is illustrated by the experiment depicted in Figure 7.8 . In this experiment, changes in transepithelial ( V te ) and basolateral potential ( V bl ) are shown during addition of 1 mM barium to the bath ( Figure 7.8a ), followed by addition of 8 mM glucose to the lumen ( Figure 7.8b ). The superimposed smaller deflections are due to periodic current injection for evaluating “ γ ” from Eq. (7.16) .
In both these experiments, ΔV bl and ΔV te were taken as the initial changes in voltage resulting from a particular maneuver. For example, in the barium experiment, only changes in V bl and <SPAN role=presentation tabIndex=0 id=MathJax-Element-28-Frame class=MathJax style="POSITION: relative" data-mathml='Rbl*’>?∗??Rbl*
Rbl*
R b l *
were presumed to occur, and ΔV bl was taken as the difference between the baseline V bl and the V bl at the inflection point (asterisk). The origin of the slow secondary depolarization of V bl in Figure 7.8a is not known.
The above methods have been used to determine individual cell membrane resistance in mammalian proximal tubule. Some of these results are summarized in Table 7.3 . Although there are some differences in the absolute values of cell resistances, there is general agreement that in proximal tubule the resistance of the cellular pathway is between 20 and 30 times higher than the resistance of the shunt pathway. This factor is even higher in amphibian proximal tubule.
| Segment | Apical Resistance | Basolateral Resistance | Shunt Resistance | Method | Species | Reference |
|---|---|---|---|---|---|---|
| PCT | 238 | 68 | 16 | Apical glucose | Rabbit (PCT) | |
| Basal Ba 21 | ||||||
| Voltage divider | ||||||
| PCT | 118 | 39 | 8 | Apical glucose | Rabbit (PCT) | |
| Voltage divider | ||||||
| PCT | 255 | 92 | 5 | Apical glucose | Rat | |
| Voltage divider | ||||||
| TALH | 88 | 47 | 47 | Apical high K + , Ba 2+ | Rabbit | |
| Voltage divider | ||||||
| TALH | 57 | 21 | 37 | Apical Ba 2+ | Mouse | |
| Voltage divider | ||||||
| CCD | 149 | 123 | 166 | Apical Ba 2+ | Rabbit | |
| Voltage divider | ||||||
| CCD | 57 | 80 | 230 | Apical amiloride, Ba 2+ + | Rabbit | |
| Voltage divider | ||||||
| OMCD | 707 | 176 | 393 | Apical glucose | Rabbit | |
| Voltage divider | ||||||
| Urinary bladder | 3700 to 154,000 | 5000 to 10,300 | 6500 to 38,000 | Apical amiloride | ||
| Voltage divider | Rabbit | |||||
| Cell cable |
Determination of individual membrane parameters from Eqs. (7.14)–(7.16) involves certain practical problems. These methods require measurement of ΔV te and ΔV bl in the same tubule at the same axial distance along its length. As indicated in Figure 7.8 , the small magnitude of the change in transepithelial potential renders the two ratios α and β in Eqs. (7.14) and (7.15) particularly susceptible to errors in ΔV te . Measurements of ΔV te obtained by advancing a microelectrode into the lumen are unreliable, because damage to the epithelium can produce artificially low values of ΔV te . Since transepithelial potential is measured only at the perfusion or collection ends of the tubule, the value of ΔV te must be calculated from the electrotonic voltage spread along the tubule using a terminated cable analysis (see section entitled, “Transepithelial Measurements”).
Evaluation of Individual Membrane Resistances Using Multiple Intracellular Recordings
In the relatively large cells of amphibian proximal tubule (30 μm diameter), direct electrical measurement of cell membrane resistance is possible. Exploiting the property of electrical coupling between adjacent proximal cells, it is possible to pass current from an intracellular microelectrode through an annular syncytium, with an outer specific resistance of <SPAN role=presentation tabIndex=0 id=MathJax-Element-29-Frame class=MathJax style="POSITION: relative" data-mathml='Rbl*’>?∗??Rbl*
Rbl*
R b l *
and an inner specific resistance of R ap . This cannot be done in nephron segments like the collecting duct, where adjacent cells are not electrically coupled.
The use of cellular cable analysis to evaluate the individual resistances <SPAN role=presentation tabIndex=0 id=MathJax-Element-30-Frame class=MathJax style="POSITION: relative" data-mathml='Rbl*’>?∗??Rbl*
Rbl*
R b l *
, R ap , and R tj still depends on the definition of transepithelial resistance ( Eq. (A1.5) ) and the “voltage-divider” ratio ( Eq. (7.16) ) that were discussed in the section entitled, “Cell Membrane Potentials in Epithelia.” However, instead of relying on the ratio ΔV bl / ΔV te , obtained during application of barium or amiloride, the parallel resistance of the cell layer ( R z ) is computed directly from the electrotonic voltage spread along the double core cable, where R z is defined by Eq. (7.18) .
1/Rz=1/Rap+1/Rbl*
1 / R z = 1 / R a p + 1 / R b l *
In practice, R z is evaluated by injecting current I o into the cell layer at x=0 via a microelectrode and measuring the voltage deflection ΔV x at two or more locations “ x ,” downstream from the injection site. The arrangement of microelectrodes is illustrated in Figure 7.2 . If x is at least twice the diameter of the tubule, the electrotonic voltage spread along the cable will be given by Eq. (7.19) , where λ c is the cellular length constant of the tubule :
Ln[ΔVx]=Ln[RzIo4πrλc]−1λcx
L n [ Δ V x ] = L n [ R z I o 4 π r λ c ] − 1 λ c x
The best fit for the two unknown parameters λ c and R z is determined by evaluating Eq. (7.19) at a number of locations along the tubule. The radius of the tubule, r o , is measured with an optical micrometer. Combining Eqs. (7.16), (7.18), (7.19), and (A1.5) , the individual membrane resistances R ap , <SPAN role=presentation tabIndex=0 id=MathJax-Element-33-Frame class=MathJax style="POSITION: relative" data-mathml='Rbl*’>?∗??Rbl*
Rbl*
R b l *
, and R tj are uniquely determined by the two parameters R z , γ, and the measured value of R te according to Eqs. (7.20)–(7.22):
Rap=Rz[1+γ]
R a p = R z [ 1 + γ ]
Rbl*=Rz[1+1γ]
R b l * = R z [ 1 + 1 γ ]
Rti=Rte[Rbl*+Rap]Rap+Rbl*−Rte
R t i = R t e [ R b l * + R a p ] R a p + R b l * − R t e
The transepithelial resistance of the tubule, R te , is determined by passing current and measuring voltage through a doubled-barreled perfusion pipette according to the methods described in the section entitled, “Transepithelial Measurements” and Figure 7.2 .
In amphibian proximal tubule, most measurements of individual cell resistances have been performed using a cellular cable analysis with two or more intracellular microelectrodes. On the other hand in mammalian tubules, the method of relative voltage deflections ( Eqs. (7.14) and (7.15) ) have been used exclusively. However, in Ambystoma proximal tubule, a direct comparison of the two methods has been made under similar conditions. These results are summarized in the first two rows of Table 7.4 . As indicated, the absolute values of resistance are quite close considering the propagation of errors that occur with both types of measurements. The remaining rows of Table 7.4 summarize additional resistance measurements in amphibian renal epithelia.
| Segment | Apical Resistance | Basolateral Resistance | Shunt Resistance | Method | Species | Reference |
|---|---|---|---|---|---|---|
| Proximal tubule | 2509 | 683 | 71 | Apical glucose | Ambystoma | |
| Basal barium | ||||||
| Voltage divider | ||||||
| Proximal tubule | 2305 | 591 | 53 | 2 electrode | Ambystoma | |
| Cable analysis | ||||||
| Proximal tubule | 6957 | 2399 | 267 | 2 electrode | Necturus | |
| Cable analysis | ||||||
| Proximal tubule | 2700 | 1900 | — | 2 electrode | Frog | |
| Cable analysis | ||||||
| Proximal tubule | 1350 | 2100 | 166 | 2 electrode | Triturus | |
| Cable analysis | ||||||
| Dilute segment a | 550 | 219 | 306 | 2 electrode | Amphiuma | |
| Cable analysis | ||||||
| Collecting tubule | 154 | 192 b | 454 | Voltage clamp | Amphiuma | |
| Voltage divider | ||||||
| Urinary bladder | 9000–65,000 | 1000–7000 | 100,000 | Apical amiloride | Necturus | |
| Voltage divider |
a These measurements assumed that the Amphiuma diluting segment has only one cell type.
b The basolateral conductance of this segment is strongly inward rectifying and quoted value applies only at a membrane potential of −60 mV.
As with the mammalian proximal tubule, the ratio of shunt to cell resistance clearly establishes the amphibian proximal tubule as a “leaky” epithelium. In contrast, the Necturus urinary bladder (a “tight” epithelium) possesses a shunt resistance that is several times larger than the cell resistance pathway. Neither the diluting segment nor the collecting tubule of the Amphiuma fall neatly into the “tight” or “leaky” category, since both of these segments have cellular and paracellular resistances that are in the same order of magnitude. Interestingly, both the diluting segment and collecting duct have higher shunt resistances and lower cellular resistances than the corresponding membrane of the proximal tubule. Mammalian collecting ducts and diluting segments also have higher shunt resistances than mammalian proximal tubules, although differences in cell resistance are less dramatic along the mammalian nephron.
The resistances quoted for the Amphiuma diluting segment in Table 7.4 assumed a single cell type throughout the cable analysis. This greatly simplified the calculation of membrane resistance. Unfortunately, subsequent experiments indicated that this nephron segment actually consists of two different cell types with dissimilar conductive properties. One cell type has a high basal K and Cl conductance (HBC), whereas the other cell type (LBC) has a low basolateral conductance for both ions. There is also some evidence that mammalian TALH may exhibit a certain amount of cell heterogeneity as well. Since it is unlikely that different cell types in the same nephron segment are directly coupled to each other, a unique value of cell membrane resistance cannot be determined from cable analysis on these nephron segments unless all recordings are made from the same cell type.
The cellular cable equations ( Eqs. (7.18) and (7.19) ) were originally derived for in situ proximal tubules, in which the cables are effectively infinite in length. However, they should be reasonably valid for isolated perfused tubules as long as the voltage recording microelectrodes are several tubule diameters from either end of the tubule. Under these conditions the electrotonic voltage spread would be effectively the same as for an infinite cable. The problem of “cross-talk” or interactions between the transepithelial and cellular cables has been suggested as a source of error in these measurements. The complication of cable–cable interactions would only be significant if current injected into the cell layer leaks into the lumen and then re-enters the cell layer at the some point downstream. A thorough analysis by Guggino et al. for Necturus proximal tubule suggests that cross-talk will be negligible if intracellular voltage deflections ΔV x are recorded at locations x>λ c .
Evaluation of cell membrane resistance via Eqs. (7.14)–(7.16) is only as accurate as the equivalent circuit of Figure 7.7 . An important aspect of this electrical model is the assumption that the tight junction constitutes the principal resistance of the paracellular pathway, or R par ≈ R tj . This is probably true in tight epithelia like the urinary bladder, where the lateral intercellular space has a negligible resistance compared to that of the tight junction. However, the low transepithelial resistance of the proximal tubule raises the possibility that the resistivity of free solution in the lateral space contributes significantly to overall shunt resistance. In this case, evaluation of the divider ratio is complicated by the lateral resistive network shown in Figure 7.1 . Current flow through a distributed network of this kind will cause the measured value of ΔV ap / ΔV bl to underestimate the actual value of <SPAN role=presentation tabIndex=0 id=MathJax-Element-37-Frame class=MathJax style="POSITION: relative" data-mathml='Rap/Rbl*’>???/?∗??Rap/Rbl*
Rap/Rbl*
R a p / R b l *
by an amount that depends on the ratio of lateral space resistance, R lis , to paracellular resistance R par (see Figure 7.9 ).
