838 BiophysicalJournal Volume81 August2001 838–851 2 cAMP Activation of Apical Membrane Cl Channels: Theoretical Considerations for Impedance Analysis Teodor G. Pa˘unescu and Sandy I. Helman DepartmentofMolecularandIntegrativePhysiology,UniversityofIllinoisatUrbana-Champaign,Urbana,Illinois61801USA ABSTRACT Transepithelial electrical impedance analysis provides a sensitive method to evaluate the conductances and capacitancesofapicalandbasolateralplasmamembranesofepithelialcells.Impedanceanalysisiscomplicated,duenotonly to the anatomical arrangement of the cells and their paracellular shunt pathways, but also in particular to the existence of audio frequency-dependent capacitances or dispersions. In this paper we explore implications and consequences of anatomically related Maxwell-Wagner and Cole-Cole dielectric dispersions that impose limitations, approximations, and pitfallsofimpedanceanalysiswhentissuesarestudiedunderwidelyrangingspontaneousratesoftransport,andinparticular whenapicalmembranesodiumandchloridechannelsareactivatedbyadenosine39,59-cyclicmonophosphate(cAMP)inA6 epithelia.Wedevelopthethesisthatcapacitiverelaxationprocessesofanyoriginleadnotonlytodependenceonfrequency of the impedance locus, but also to the appearance of depressed semicircles in Nyquist transepithelial impedance plots, regardlessofthetightnessorleakinessoftheparacellularshuntpathways.Frequencydependenceofcapacitanceprecludes analysisofdataintraditionalways,wherecapacitanceisassumedconstant,andisespeciallyimportantwhenapicaland/or basolateralmembranesexhibitoneormoredielectricdispersions. INTRODUCTION A variety of hormones including aldosterone and insulin 1998). In this paper we have used data derived from our selectively stimulate Na1 transport in cell-cultured A6 ep- studies of A6 epithelia (Pa˘unescu and Helman, 2001) to ithelia by increase of the density of functional amiloride- discuss and illustrate the guiding principles used in exper- sensitive epithelial Na1 channels (ENaCs) at the apical imental design, methods of analysis, and interpretation of membranes of the cells (Blazer-Yost et al., 1996; Baxen- data. Although the emphasis of our original inquiry was to dale-Cox et al., 1997; Helman et al., 1998). In contrast, understand the role of cAMP in regulation of apical mem- hormonal activation of Na1 transport mediated by the sec- brane Na1 and Cl2 channels using methods of impedance ond messenger adenosine 39,59-cyclic monophosphate analysis, the scope of this project required a rigorous ex- (cAMP) is accompanied by stimulation of Cl2 transport amination of the underlying theoretical considerations and (YanaseandHandler,1986;Chalfantetal.,1993)involving principles that have not appeared in the epithelial transport activationofapicalmembraneCl2channels(Marunakaand literature. In this manuscript we emphasize the relevant Eaton, 1990; Nakahari and Marunaka, 1995; Kokko et al., considerations and principles appropriate to epithelia. The 1997; Matsumoto et al., 1997). During the course of our experimentalfindingspresentedinacompanionpaper(Pa˘u- studiesaimedatunderstandingtheinterrelationshipsamong nescu and Helman, 2001) underscore the need for our the- hormones in regulation of Na1 transport it became neces- oreticalconsiderationsinunderstandingthemechanismsof sarytoknownotonlythemagnitudeofchangeoftheapical hormonal action that activate not only Na1, but also apical membrane cAMP-activated chloride conductance, but also membrane Cl2 channels in A6 epithelia. the consequences of activation of the chloride conductance Recognition of the existence of plasma membrane fre- in electrophysiological studies of Na1 transport, specifi- quency-dependent capacitances is critical in the design cally in the A6 epithelium and more generally in other and interpretation of studies of epithelial tissues. It is of tissues where cAMP activates Na1 and Cl2 channels. fundamentalinteresttoknowthefactorsandmechanisms To address such questions we turned to impedance anal- underlying audio frequency dielectric dispersions or re- ysis of the A6 epithelium and found, as in frog skin laxation processes in epithelial and nonepithelial plasma (Awayda et al., 1999) that the analysis of A6 epithelia was membranes in terms of the organization of the lipids and morecomplicateddueprincipallytoaudiofrequencydielec- proteins of plasma membranes and their structural/func- tric relaxation processes or dispersions (Liu and Helman, tionalrelationshipsthatgiverisetodielectricdispersions at any frequency, and in particular those at audio fre- quencies (adispersions (Schwan, 1957)) where transep- Received for publication 19 September 2000 and in final form 17 April ithelialimpedanceisusuallymeasured.Totheextentthat 2001. plasma membranes exhibit Cole-Cole-like dielectric dis- AddressreprintrequeststoDr.SandyI.Helman,Dept.ofMolecularand persions (Cole and Cole, 1941) and to the extent addi- Integrative Physiology, University of Illinois at Urbana-Champaign, 524 tionally that apical and basolateral membranes form a BurrillHall,407S.GoodwinAve.,Urbana,IL61801.Tel.:217-333-7913; Fax:217-333-1133;E-mail:[email protected]. series arrangement of leaky dielectrics leading to Max- © 2001bytheBiophysicalSociety well-Wagner dispersions, interpretation of changes of 0006-3495/01/08/838/14 $2.00 capacitanceismorecomplex,requiringconsiderationnot Frequency-DependentCapacitance 839 FIGURE 1 (A) Transepithelial lumped parameter electrical equivalent circuit of apical and basolateral plasma membranes of the cells shunted by a paracellularresistance,R .a,b,andcindicatenodeswithintheapical,basolateral,andintracellularsolutions,respectively.E andE areThe´veninemfs p a b ofapicalandbasolateralmembranes,respectively,withtheirassociatedsloperesistances,R andR,andrespectivecapacitances,C andC.(B)Because a b a b impedanceismeasuredinthetimedomain,theconductancesmustbeslopeconductancesattheoperatingpointvoltagesofthemembranes.Shownare GoldmanV-I relationships(Goldman,1943)forapicalmembraneNa1andCl2channelsattheoperatingpoint(Vsc)ofshort-circuitedepitheliathatpertain a a a totheconditionsofourexperiments.NernstequilibriumpotentialdifferencesforNa1(d )andCl2(d )areindicatedatthepointsofcurrentreversal. Na Cl ThedrawingindicatesthatwhereasCl2isatorverynearelectrochemicalequilibrium,Na1transportisfiniteatanapicalmembranevoltage(Vsc)far a removedfromequilibriuminshort-circuitedtissues(Vsc2d )(Pa˘unescuandHelman,2001).AtVsc,theslopeconductanceforNa1isG 5R21and a Na a Na Na theThe´veninemfforNa1(ENa)isatorverynearzeromV.TheThe´veninemfforCl2,ECl,5d .Theconductivepropertiesofapicalmembranesof a a Cl A6 epithelia are shown in (C) for control tissues expressing amiloride-sensitive (Rbs) and -insensitive (Rbi) Na1 channels; in (D) for 100 mM Na Na amiloride-treatedtissueswithblocker-insensitiveNa1channelsandin(E)foramiloride-blockedtissuestreatedadditionallytoincreaseintracellularcAMP, whichactivatesapicalmembraneCl2channels.d isincludedin(E)toemphasizethattheoperatingpointvoltageforCl2isclosetoelectrochemical Cl equilibrium,whereasthedistributionforNa1acrosstheapicalmembraneisfarremovedfromitselectrochemicalequilibriumwithaThe´veninemf,E , Na atorverynearzero.Fromthepointofviewofimpedance,theemfs(NernstorThe´venin)areirrelevant,buttheslopeconductanceswillvarywithvoltage duetothedegreeofnonlinearityoftheV-Irelationshipsattheoperatingpointsofmembranevoltage. only of membrane areas and thicknesses but also those Thompson, 1982). It is important when dealing with non- factors that can change the frequency-dependent dielec- linear circuits to distinguish between chord and slope for- tric properties of the membrane and hence, the capaci- malismsofelectricalcircuittheory.Inournotationwehave tance measured at any frequency. used R’s and G’s to indicate slope resistances and conduc- tances (as compared to g’s in the chord formalism), and usedE’sforThe´veninemfsthatarenotnecessarilythesame THEORETICAL CONSIDERATIONS as the d’s that designate Nernst equilibrium potential dif- Electrical equivalent circuits ferences. Thus, as indicated in Fig. 1A, the epithelium is modeled as a series arrangement of apical and basolateral Inthefaceofnonlinearcurrent-voltagerelationshipsofthe membrane impedances that exists in parallel with a para- underlying channels and electrodiffusive transporters, im- cellular shunt resistance, R . The respective slope resis- pedance analysis requires electrical equivalent circuits p tances R and R and capacitances C*and C*of apical and where membranes are modeled by their slope resistances a b a b basolateral membranes give rise to the apical and basolat- (R5DV/DI).Hence,asindicatedinFig.1A,theresistances eral membrane impedances Z and Z , so that the transepi- R , R , and R are the slope resistances of the apical mem- a b a b p thelial impedance Z is: brane, basolateral membrane, and paracellular shunt, re- T spectively.Becausetheresistancesaresloperesistances,the ~Z 1Z !R EeraalanmdeEmbbarraenethse, Trehse´pveecntiivneelymf(sCohfuath,e1a9p6i9c;alHaneldmbaansoalantd- ZT5Za1a Zb1b Rpp (1) BiophysicalJournal81(2)838–851 840 Pa˘unescuandHelman and where: are relevant to those of A6 epithelia studied in our labora- tory (Liu and Helman, 1998; Pa˘unescu et al., 2000; Pa˘u- R Z 5 a (2) nescu and Helman, 2001). In amiloride-blocked states of a 11jvRaC*a Na1 transport, R 5 Rbi is near 500 kV z cm2 and R a Na b Zb511jRvbR C* (3) aNvae1ratgreasnsnpeoarrt4(P0a˘0u0nVesczucemt2a.l.R,b20m0a0y)bvuartythweriethisthperersaetentolyf b b insufficientinformationtoknowifthisissoinA6epithelia The notation of asterisk-superscripted C’s indicates that and to what extent. The R of control tissues is highly a capacitance may be complex (frequency-dependent). If the variable and depends on the rate of Na1 transport (short- C’s are complex, then the time constants, RC*, are also circuit currents) where the spontaneous amiloride-sensitive frequency-dependent.Atthelimitingfrequencybounds,ZT I can range downward to values ,1 mA/cm2 (Rbs . 112 sc Na approaches zero at infinite frequency; as frequency ap- kV z cm2) and upward to .30 mA/cm2 (Rbs , 3.7 kV z proaches zero, Z approaches the transepithelial slope re- Na T cm2).Thestaticdcorzero-frequencycapacitancesofapical sistance, R 5 [(R 1 R )R ]/(R 1 R 1 R ). If the T a b p a b p (Cdc)andbasolateralmembranes(Cdc)averagenear1.5and capacitances at the audio frequencies of impedance mea- a b 20 mF/cm2, respectively (Pa˘unescu and Helman, 2001). surements exhibit Maxwell-Wagner and/or Cole-Cole dis- Thus, for the monolayers of A6 epithelia where basolateral persions (Awayda et al., 1999) (or dielectric dispersions membraneareaisconsiderablylessthaninthefunctionally from any other source), such dispersions will give rise to coupled cells of the multicellular layers of frog skin, the power law dependence of the impedance (see below). basolateral membrane resistance of A6 epithelia is about Shown also in Fig. 1 are the electrodiffusive or conduc- fourtimesgreaterandthecapacitanceaboutthreetimesless tivecomponentsofR inthreetransportstatesofthetissues. a thanobservedinstudiesoffrogskin(Awaydaetal.,1999). In their control state (Fig. 1C), A6 epithelia, otherwise untreated by hormones or second messengers that activate Accordingly, and unlike in frog skin, where the impedance Cl2 channels, express both blocker-sensitive (bs) and to a ofthebasolateralmembranesundersomeexperimentalcon- much lesser extent blocker-insensitive (bi) Na1 currents ditions may approach negligible values, the Z of A6 epi- b thataccountcompletelyforthetotalcurrentorthenetrates thelia cannot be neglected at any frequency, as will be of Na1 entry into the cells as measured by short-circuit indicated below. currents(Baxendale-Coxetal.,1997;Helmanetal.,1998). Interpretation of transepithelial impedance data and the As indicated, R is partitioned between blocker-sensitive designofexperimentstoevaluatethetissuesiscomplicated a (Rbs)andblocker-insensitive(Rbi)sloperesistances.Nota- notonlybythejuxtaposedimpedanceofthecells(Z 1Z ) Na Na a b bly,theThe´veninemfsoftheseNa1conductances(ENa)are andtheirparacellularshunts(R ),butalsobythepossibility a p at or very near zero, because as indicated in Fig. 1B, the thattheplasmamembranesexhibitcapacitivedispersionsat slopeconductance,G ,attheoperatingpointoftheapical thefrequenciesofinterest.Asimplificationofexperimental Na membrane voltage (Vasc) intersects the voltage axis at or conditions utilizes amiloride to block apical membrane ep- very near zero when Vasc is very far removed from electro- ithelial Na1 channels (ENaCs) so that at least at very low chemicalequilibrium(dNa)(HelmanandThompson,1982). frequencies,Z ..Z andthecellularresistance(R 5R Accordingly, the blocker-sensitive and -insensitive Na1 1R )ismuchagreatebrthanR .Accordingly,withRcel.l.Ra currentsareINbsa5Va/RNbsaandINbia5Va/RNbia.Withamiloride andRb ..R ,themeasuredptransepithelialresistanace,R b, block of apical membrane ENaCs, the apical membrane R cell p T a closely approaches the values of R . At very high frequen- reduces to the Rbi, as indicated in Fig. 1D. With Ibi p Na Na cies,wheretheR andR becomenegligiblerelativetotheir averaging much less than 0.5 mA/cm2 (Helman and Liu, a b respectivecapacitivereactances,theequivalentcapacitance 1997;Baxendale-Coxetal.,1997;Blazer-Yostetal.,1999; of the cells, C*, approaches values of the series arrange- Pa˘unescu et al., 2000) and V exceeding 100 mV under eq a mentofcapacitance(C*C*)/(C*1C*),wherenotablysuch these conditions, the Rbi is expected to be far greater than a b a b 200,000 V z cm2. Na values would not be the same as (CadcCbdc)/(Cadc 1 Cbdc) if ActivationofapicalmembraneCl2channelsintroducesa either or both apical and basolateral membranes exhibit slope conductance G 5 1/R with a The´venin E . If, as dielectric dispersions. Thus, we have considered in more Cl Cl Cl indicated in Fig. 1,B and E, Cl2 is distributed across the detail the behavior of the transepithelial impedance in apical membrane at or very near the Nernst equilibrium amiloride-treatedtissuesfirst.Also,itwillbeevidentbelow potential difference, d , then E can be replaced by d . that it is critically important to know with reasonable cer- Cl Cl Cl TotheextentthatRbi ..R ,R ’R attheoperatingpoint taintythevaluesofR especiallyunderconditionswhereR Na Cl a Cl p a oftheapicalmembranevoltage,whichasindicatedelsewhere isnotmuchgreaterthanR andR ,aswouldbethecasefor b p (Pa˘unescuandHelman,2001)isveryclosetod . tissues transporting Na1 or for tissues in their amiloride or Cl For purpose of discussion below we have used values of non-amiloridepretreatedstatewherecAMPactivatesapical resistance and capacitance in the range(s) that we believe membrane Cl2 channels. BiophysicalJournal81(2)838–851 Frequency-DependentCapacitance 841 FIGURE 2 Nyquist(A)andBodeplots(BandC)areshownfortheequivalentcomplexcapacitance,C*,ofatranscellularMaxwell-Wagnerdispersion eq ifR 5‘.Maxwell-WagnerplotsofC* areshownin(D)(Nyquist)and(E)and(F)(Bode)whenR 5100kVzcm2and500kVzcm2.R 54000Vz a eq a b cm2,C 51.5mF/cm2,C 520.0mF/cm2.Thecapacitancesareassumedtobeindependentoffrequency. a b FREQUENCY-DEPENDENT CAPACITANCE plex frequency-dependent equivalent capacitance, C*, eq where: Maxwell-Wagner dispersions C* 5@jv~Z 1Z !#21 (4) Althoughthemeasurementofimpedanceisstraightforward, eq a b the analysis and interpretation of data is complicated when NyquistandBodeplotsofC* areplottedinFig.2.Forthe eq capacitances are frequency-dependent. Frequency-depen- casewhereR isinfiniteandwhereCdc51.5mF/cm2,Cdc a a b dent capacitances may arise from Maxwell-Wagner disper- 520mF/cm2,andR 54000Vzcm2,C* decreasesfrom b eq sions (Daniel, 1967; Jonscher, 1983) due to the juxtaposi- the Cdc of 1.5 mF/cm2 to 1.395 mF/cm2 as frequency in- a tion of leaky dielectrics that exist in series. From the creases toward ;100–200 Hz. Despite the fact that C and a transepithelial point of view, the apical and basolateral C areconstantatallfrequencies,theseriesarrangementof b membranes of the cells behave as a series arrangement of membranes gives rise to a Maxwell-Wagner dispersion at leaky dielectrics with capacitances C and C paralleled by lowaudiofrequenciessothatthecellsbehaveasafrequen- a b resistancesR andR ,respectively,asindicatedinFig.1A. cy-dependentcapacitancethatexistsinparallelwiththeR . a b p Accordingly,thetranscellularimpedancebehavesasacom- The dispersion appears as an ideal semicircle in Nyquist BiophysicalJournal81(2)838–851 842 Pa˘unescuandHelman plots with a characteristic frequency at the apex of the referredtoasbdispersions.Ourconsiderationsinthispaper semicircle of 1.79 Hz, as indicated in Fig. 2A. When R is will be limited to adispersions. Plasma membranes may a not infinite and in the range of 100–500 kV z cm2 as exhibit multiple relaxation processes (Eq. 5) (Cole and illustrated in Fig. 2D, the C* due to Maxwell-Wagner Cole, 1941; Awayda et al., 1999) where the complex ca- eq dispersions are quantitatively more complex, reflecting the pacitance C* at any frequency will vary with the magni- fact that a finite value of resistance behaves in the equiva- tudes of the dielectric increments (C), the time constants i lent sense as a frequency-dependent capacitance (C(v) 5 (tr), or characteristic frequencies (fr) of the relaxation pro- i i (jvR)21). Accordingly, as illustrated in Fig. 2,D–F, the cesses and power law coefficients (a) that lead to obser- i imaginarycomponentsofC* (ImC*)becomeenormousin vation of depressed semicircles in Nyquist plots of C*. eq eq value relative to those of the real components of C* C‘ represents the limiting static capacitance at frequen- eq a (ReC*) as frequency decreases toward zero. Thus, despite cies far greater than the highest relaxation frequency eq relatively high values of Rbi in amiloride-blocked tissues, process (fr 5 (2ptr)21) that exists in the audio range of a i i such values of R will markedly affect the magnitudes and frequencies. a phases of the C* at low frequencies ,;100 Hz for the Resolving plasma membrane dielectric relaxation pro- eq parametersofthecircuitusedhereinourcalculations(Fig. cesses in intact epithelial tissues at high frequencies is 2,EandF).Ifcapacitancesdonotexhibitdielectricdisper- limited by the uncertainties in measurement of the exact sions at higher frequencies than ;100 Hz, then with C 5 magnitude of the series solution resistance of the cell cyto- a 1.5 mF/cm2 and C 5 20 mF/cm2, C would be 1.395 plasm and the solutions bathing the apical and basolateral b eq mF/cm2athigherfrequenciesbeyondthoseexpectedofthe bordersofthetissues.Withhighvaluesofcapacitance(1.5 Maxwell-Wagner dispersions. Indeed, because C is con- mF/cm2), the capacitive reactance of C at 10 kHz is 10.6 b a siderably larger than C , the Maxwell-Wagner frequency- V z cm2 and only 1.06 V z cm2 at 100 kHz, which a dependent decrement of C* is relatively small (7%), but approaches the value of the cytoplasmic resistance of the eq neverthelessimportantattheloweraudiofrequencieswhere cells. Thus, detection of dispersions that exist in the attempts are made to determine the dc capacitances. Such range of radio frequencies (bdispersions) or higher fre- considerationsareespeciallyimportantinthosetissuesthat quencies is essentially if not completely precluded be- spontaneously transport Na1 at low rates, in tissues where causetheerrorsindeterminingZ fromsmalldifferences T Na1 transport is suppressed by drugs, hormones, or other betweenthemeasuredimpedancesandthoseoftheseries experimentalmaneuversthataffectRbs,andinparticularin resistances of the cells and bathing solutions cannot be Na those tissues treated with amiloride to completely inhibit adequatelyassessed.Thus,ourexperimentsandourconsider- blocker-sensitive apical membrane Na1 channels. With ationsarelimitedtoaudiofrequencyadispersionsofthetype shunt resistances in the range of 10 kV z cm2, the useful describedbyEq.5. frequency range of interest (see below) includes those fre- quenciesdownto;0.1Hz,whereMaxwell-Wagnerdisper- O n C sions will affect the magnitudes and phases of the transep- C*5 i 1C‘ (5) ithelial impedance. It must be stressed, owing to the series i5111~jvtir!ai a arrangementofapicalandbasolateralmembranesofepithe- lia in general, that Maxwell-Wagner dispersions are un- IllustratedinFig.3,A–CareNyquistandBodeplotsofa avoidableregardlessofthemagnitudesofconductancesand singlerelaxationprocess.Wehaveassumed,baseduponour capacitances in tight and leaky epithelia and regardless of experimental findings (Pa˘unescu and Helman, 2001) that additionaldispersionsthatmayarisefromtheplasmamem- theapicalmembraneischaracterizedbyaCole-Colerelax- brane dielectrics. ation process, so that the complex apical membrane capac- Thus, independent of the values of R, interpretation of itance,C*,canbecalculatedwithEq.5.Thedccapacitance, a a measurements of impedance can be problematic at very low Cdc, is 1.5 mF/cm2 (Pa˘unescu and Helman, 2001). The a frequenciesduetounavoidableMaxwell-Wagnerdispersions. dielectricincrement,C 5Cdc2C‘,is0.6mF/cm2witha 1a a a relaxation frequency of 110 Hz. The complex capacitance approaches C‘ 5 0.9 mF/cm2 as frequency far exceeds 10 Cole-Cole dielectric dispersions a kHz approaching values near 1 MHz. The semicircle is In contrast to Maxwell-Wagner dispersions, frequency-de- depressed due to a power law dependence of the capaci- pendent capacitances can also arise from dielectric relax- tance,a 50.6,whichisconsistentwiththeideathatthere a ation processes as described originally by Cole and Cole, is a distribution of time constants (Cole and Cole, 1941; Schwan, and others (Cole and Cole, 1941; Schwan, 1957; Cole,1968;Gabler,1978;Pethig,1979;Jonscher,1983)of Kell and Harris, 1985; Takashima, 1989) and can be asso- the relaxation processes centered at a relaxation frequency ciatedwithC and/orC .Whenobservedataudiofrequen- of 110 Hz. For such dispersions the complex capacitance a b cies these relaxation processes are refereed to as adisper- vectors vary in magnitude (uC*u) and phase angle (f) as a sions, while those at higher radio frequencies have been indicated in the Nyquist plots (Fig. 3A) or in rectangular BiophysicalJournal81(2)838–851 Frequency-DependentCapacitance 843 FIGURE 3 Apical membrane Nyquist (A) and Bode plots (B and C) are shown for the complex capacitance C*of a Cole-Cole audio frequency a a dispersionwitharelaxationfrequencyof110Hzandpowerlawdependence,a 50.6.Thetranscellularcomplexequivalentcapacitance,C*,isshown a eq inNyquist(D)andBodeplots(EandF)whentheapicalmembranecapacitanceexhibitsanadispersionasin(A).R 5500kVzcm2;R 54000Vz a b cm2,andC 520.0mF/cm2.Thedashedlinesin(D)–(F)indicatetheC* ifR 5R 5‘. b eq a b coordinates as the real (ReC*) and imaginary (ImC*) com- Hz is markedly influenced by the Maxwell-Wagner disper- a a ponentsofC* asindicatedinFig.3,BandC,respectively. sion.If,additionally,C exhibitsadispersions,thebehavior eq b The absolute magnitude of ImC*is maximal at the relax- of C* would be more complex especially at the lower a eq ationfrequencyof110Hz(Fig.3C).TheReC*isdecreased frequencies.Consequently,fromthetransepithelialpointof a from Cdc by one-half of the magnitude of the dielectric view, capacitance of the cells would appear frequency- a increment(C 5Cdc2C‘)at110Hz(Fig.3B).Itshould dependentovertheentirerangeofaudiofrequencies,where 1a a a be noted that a 5 22ImC*/C where 0 # a # 1. at higher frequencies the changes of capacitance would be a a 1a a dominated by the a dispersions but with more complex behavior at the lower frequencies due to the Maxwell- Combined Maxwell-Wagner and Wagner dispersion. Assuming, as indicated above, that C Cole-Cole dispersions b (20 mF/cm2) is devoid of adispersions, the C‘ is 0.861 eq With Maxwell-Wagner dispersions dominant at lower fre- mF/cm2 (Fig. 3,D and E) and the Cdc would be 1.395 eq quenciesandadispersionsathigherfrequencies,resolving mF/cm2 if R and R were infinitely large, as indicated by a b the adispersions at lower frequencies is complicated, as the dashed lines in these figures. The contribution of the illustratedinFig.3,D–F.RegardlessofthevaluesofR and Maxwell-Wagner dispersion to C* can be assessed by a eq assuming for the sake of simplicity that C is devoid of a examination of the differences of C* when R and R are b eq a b dispersions,thebehaviorofC* atfrequencies,;100–200 either finite or infinite in value. eq BiophysicalJournal81(2)838–851 844 Pa˘unescuandHelman Determination of the complex equivalent capacitance, C* eq C* isbestdeterminedunderconditionswhereideallyR is eq a infinitely high so that the transepithelial resistance is due solelytotheshuntresistance,R .Undersuchconditionsthe p transepithelialimpedanceisdeterminedbytheparallelcom- bination of R and C* so that: p eq R Z 5 p (6) T 11jvR C* p eq Rearranging Eq. 6 allows calculation of the C* from the eq measuredvaluesofZ .[ItshouldbenotedaccordingtoEq. T 6andthedefinitionofthecomplexcapacitance,C*,inEq. eq 4 that the transepithelial impedance of an epithelium can always be modeled by an R paralleled by the cellular p impedance(Z 1Z )regardlessofthetightnessorleakiness a b of the junctional complexes in the paracellular shunt path- ways of the epithelium. To the extent that R is practically a infinitely large, as in amiloride-blocked tissues with un- stimulated intracellular levels of cAMP, R can be equated p directly with the R and determined from the measured T values of Z as frequency approaches zero, thereby simpli- T fying analysis, as R can be measured directly. When R p a cannotbeneglected,R mustbedeterminedbyanindepen- p dent method because R . R .] To the extent that R can p T a vary over wide ranges depending on the rates of Na1 transport, it is always possible to use amiloride to block Na1 channels (Rbs 3 ‘) thereby elevating R to values of Na a Rbi that will exceed 200 kV z cm2 (see above) when the Na amiloride-insensitive Na1 currents are ,0.5 mA/cm2. Un- dertheseamiloride-blockedconditionswhereR ..R ,the a b transepithelial resistance, R , approaches values of the par- T allelcombinationofR andR orvaluesofR whenR .. a p p a R . Thus, with R averaging near 10 kV z cm2 and cell p p resistancesmuchgreaterthan200kVzcm2,C* determined FIGURE 4 Shown are the frequency-dependent absolute magnitudes eq withEq.6wouldtoaverygoodapproximationprovidedata (uC*u)(A)andphaseanglesindegrees(f)(B)ofatranscellularMaxwell- eq thatwouldallowdiscriminationbetweendispersionsarising WagnerdispersionwhenRa5‘,100kVzcm2,or500kVzcm2.Rb5 4000Vzcm2,C 51.5mF/cm2,C 520.0mF/cm2.Notethatthevalues from Maxwell-Wagner behavior and dielectric relaxation a b ofuC*ucanexceedC (1.5mF/cm2)asfrequencydecreases(seetext). processes. Plotted in Fig. 4 are the magnitudes, uC*u, and eq a eq phase angles, f, of C* for R between 100 kV z cm2 and eq a infinity attributable alone to the Maxwell-Wagner disper- R is100kVzcm2(Fig.4B).Itisimportanttorealizethat a sion.Atfrequencies.;10Hz,uC*uisconstant.However, with R ’s in the range of 10 kV z cm2 the characteristic eq p at frequencies ,10 Hz, uC*u increases precipitously as frequencies of impedance (fz 5 (2pR uC*u)21) are in the eq p eq frequency decreases at finite values of R . Accordingly, rangeof10Hz,sothatimpedancemeasuredattheselower a despite block of the amiloride-sensitive Na1 channels, the frequenciesisespeciallysensitivetochangesofcapacitance Maxwell-Wagner effect makes it appear as though capaci- associated with Maxwell-Wagner dispersions. tanceexceedstheactualCdcof1.5mF/cm2.Thisisapitfall Although the Maxwell-Wagner effect on uC*u is domi- a eq of the analysis where what seemingly may be a reasonable nantatlowfrequencies,thedielectricdispersionsarereadily assumption,namelyaveryhighand“negligible”valueofR distinguishedfromthoseofMaxwell-Wagnerathigherfre- a relative to R and R , is in fact a poor assumption when quencies,asindicatedinFig.5A.Fortherelaxationprocess b p analysisofdataiscarriedoutatlowfrequencies(seebelow) centered at 110 Hz, which we have used in our consider- in the range of the Maxwell-Wagner dispersions. Not only ations of A6 epithelia, the uC*u are virtually the same at eq aretheuC*uoverestimatedatfrequencies,10Hz,thephase frequencies .;30–40 Hz irrespective of the values of eq angles of C* also vary with frequency up to 1 MHz when R $100kVzcm2.Ananalysissuchasthiscanandshould eq a BiophysicalJournal81(2)838–851 Frequency-DependentCapacitance 845 would be relatively small compared to those arising from the apical membrane if the dispersions at both membranes areproportionallythesame.Thus,itismorelikelythatifa dispersions are observed under the conditions considered here, they arise from the apical membranes of the cells (however, see also below). It should be recognized that A6 epithelia are capable of widerangingratesofspontaneousNa1transportwithlower limits near 1 mA/cm2. In such tissues the values of R will a benear100kVzcm2.Consequently,fromthepointofview oftransepithelialimpedance,suchlowtransportratetissues will behave more like amiloride-blocked tissues. However, when apical membrane conductance to Na1 is spontane- ouslyhighorwhenCl2conductanceisactivatedbycAMP inthepresenceorabsenceofamiloride,itwouldnolonger bepossibletoassume,asabove,thatvaluesofR approach T valuesofR .R mayalsobeeitherlargerorsmallerthanR . p a b Such conditions impose special limitations in evaluation of thefrequency-dependentcharacteristicsoftheplasmamem- branes (see below). Consequently, our discussion to follow focusesonexperimentalconditionswhereR ..R ,andin a p particular on the influence of the R on the analysis of p experimental data. IMPEDANCE Depressed impedance loci (semicircles) in Nyquist plots (R >> R ) a b It has been a consistent and inviolate observation in our laboratory (Liu and Helman, 1998; Awayda et al., 1999; Pa˘unescu and Helman, 2001) and others (Schifferdecker and Fromter, 1978; Gordon et al., 1989) that Nyquist im- FIGURE 5 Shown are the frequency-dependent absolute magnitudes pedanceplotsinvariablyexhibitdepressedsemicircularbe- (uC*u)(A)andphaseanglesindegrees(f)(B)ofatranscellularMaxwell- havior regardless of the state of transport of the epithelium eq Wagnerdispersionwheretheapicalmembranecapacitance,C*a,iscomplex studied(R .R ;R ,R ),requiringfittingequationswith (Cdc51.5mF/cm2,C‘50.9mF/cm2,fr5110Hz,a 50.6).R 5‘, a b a b a a a a powerlawdependencetoaccountforthedepressedlocusof 100kVzcm2,or500kVzcm2.R 54000Vzcm2,C 520.0mF/cm2. b b the impedance vectors. Complex frequency-dependent ca- pacitances give rise to power law dependencies of imped- be done for any set of parameters. Our calculations pre- ance as indicated below. Eq. 6 can be rewritten as: sented here apply only to those values that we have reason to believe are representative of A6 epithelia studied in our R own laboratory. ZT511jvR ~ReC*p 2jzImC*! (7) p eq eq We emphasize that our analysis for the purpose of dis- cussionpresumesthatanadielectricdispersionoccursonly whichcanbeexpandedinrectangularcoordinates(ReZ 2 at the apical membranes of the cell. Quite generally, there T j z ImZ ) to: maybemorethanoneadispersionatapicalmembranesand T it is likely that basolateral membranes also exhibit adis- 11~vR ImC*! persions.Presently,wehavenospecificinformationonthe Z 5 p eq dielectric behavior of any basolateral membrane. It is nev- T ~11vRpImC*eq!21~vRpReC*eq!2 ertheless clear that with values of Cdc much greater than b ~vR2ReC*! tlhatoesrealomfeCmadcb(r2an0evdsi.e1le.5ctmricF/dcimsp2e),rstihoenscotonttrhibeuvtiaoluneosfobfaCso*eq- 2 j~11vRpImC*epq!21e~qvRpReC*eq!2 (8) BiophysicalJournal81(2)838–851 846 Pa˘unescuandHelman At the frequency v5 2pfz 5 (R uC*u)21: p eq 1 fz5 ˛ (9) 2pR ReC*21ImC*2 p eq eq and it is readily shown that the ReZ 5 R /2 when f 5 fz, T p regardless of the value of the ImZ (or regardless of the T degreeofdepressionoftheimpedancelocus).Definingthe degree of depression of the Nyquist impedance semicircle, g, as ImZ /ReZ 5 2 z ImZ /R at f 5 fz, gis: T T T p U ReC* g5 ˛ eq (10) ReC*21ImC*21ImC* eq eq eq f5fz Consequently, depression of the impedance semicircles in NyquistplotsisindependentofthemagnitudeofR .Itshould FIGURE 6 Relationship between power law dependence of Cole-Cole p be noted and emphasized that depression of the impedance dispersion (a) (depression of complex capacitance) and depression of a locusrequiresuImC*u.0,likethosethatarisewithMaxwell- impedancesemicirclesinNyquistplots(g)(seetext). eq WagnerandCole-Coleadispersions.Itshouldbeappreciated thatifImC* iszero,thenchangealoneofReC* doesnotgive 5 R /2, the small deviations from ideal depressed semicir- eq eq p risetoadepressedimpedancelocus. cles exist because of the greater contribution of the Max- To illustrate the relative importance of the ImC* and well-Wagner dispersion to the impedance at the lower fre- eq ReC* on depression of the impedance locus we calculated quencies leading to an asymmetry in the impedance locus. eq gfor a simple parallel combination of an R and an apical The exact impedance locus will depend not only on the p membranewithasingleCole-Colerelaxationprocesswhere capacitivedispersionsbutalsoonthevaluesofR ,because p Cdc was 1.5 mF/cm2 and where C‘ was 0.5, 0.9, or 1.2 R is a major component of transepithelial impedance that a a p mF/cm2, thereby giving dielectric increments of 1.0, 0.6, willdeterminethefrequenciesatReZ 5R /2.ForR ’sof T p p and 0.3 mF/cm2, respectively, at characteristic frequencies 1, 10, and 50 kV z cm2 (Fig. 8,A–C), the corresponding of 110 Hz, as above. Depression of capacitance, a, was frequencies are 142.2, 12.09, and 2.45 Hz. Thus, in tissues a varied between extremes of 0 and 1. with very high R 5 50 kV z cm2 (Fig. 8B), the principal p The relationship between a and gis nonlinear and in- changesofZ occuroveraverysmallbandwidth(0.1to10 a T verseasillustratedinFig.6.Thedegreeofdepressionofthe Hz)thatisprimarilyinthefrequencyrangeoftheMaxwell- impedance locus will vary not only with the magnitude of Wagner dispersion. When R 5 1 kV z cm2 (Fig. 8A), the p the dielectric increment but also requires, as indicated majorchangesofimpedanceoccuroverarangeoffrequen- above, that a . 0 (or ImC* (cid:222) 0). It is also apparent that cies (10 Hz–1 kHz) in the range of the adispersion. At an a eq the maximal influence of the capacitive dispersion occurs intermediate R 5 10 kV z cm2 (Fig. 8C), the major p when a approaches unity. Although we have dealt here changes of Z occur between 0.1 Hz and ;50 Hz, thereby a T withspecificvaluestoillustrateapoint,thisdiscussioncan encompassing the Maxwell-Wagner dispersion and a rela- begeneralizedtoanyfrequency-dependentdielectricbehav- tivelysmallportionoftheadispersion.Thus,itshouldnot ior(andtoMaxwell-Wagnerbehavior)which,accordingto be surprising, especially at these intermediate values of R , p thegeneralizedthesispresentedabove(Eqs.7–10),maybe that the impedance loci cannot be described exactly by a used to argue for the existence of frequency-dependent simple equation of a single depressed semicircle. capacitances when the impedance locus is depressed. For experimental data that appear to conform by inspec- DepressedNyquistimpedancelociareplottedinFig.7A tion to an ideal (non-skewed) depressed semicircle, it is for epithelia with apical and basolateral membrane charac- tempting to fit the data to a general and empirical equation teristicsdescribedabove.Thea are0.25,0.5,0.75,and1.0 of the form (Van Driessche, 1986): a and R 5 1000 V z cm2 (Cdc 5 1.5 mF/cm2; C‘ 5 0.9 p a a Rfit mF/cm2; fr 5 110 Hz; R 5 500 kV z cm2; R 5 4000 V z Z 5 (11) cm2; C 5 20 mF/cm2).aDepression of ImZ ibs maximal at T 11~jvRfitCefiqt!gfit b T a 5 1.0 (Fig. 7,A and C) at frequencies near 142 Hz for Such an equation assumes that the impedance locus is a a 50.25,0.50,and0.75andnear148Hzfora 51.0(Fig. symmetrical at frequencies above and below the frequency a a 7C).Thecorrespondingvaluesofganda aregiveninFig. at the apex of the semicircle. The capacitance, Cfit, is the a eq 7C; granges between 0.785 and 0.953 for a between 1.0 capacitanceatv5(RfitCfit)21.However,dependingonthe a eq and 0.25. Although the maximal absolute values of ImZ degree of asymmetry of the semicircle, the values of Rfit T existatfrequenciesveryclosetothoseexpectedwhenReZ maybelargerorsmallerthanthoseofR ,dependingonthe T T BiophysicalJournal81(2)838–851 Frequency-DependentCapacitance 847 cm2), the values of Cfit are remarkably close to the actual eq values of uC*u at the frequencies, fz. eq Itisalsoimportanttoknowthatinthefaceoffrequency- dependent capacitances there are experimental circum- stances where impedance loci cannot be approximated by Eq. 11. For example, as illustrated in Fig. 9, we have calculated the impedance at values of R of 200, 600, and p 1000Vzcm2assuming,ideally,thatR 5‘andthata 5 a a 1.0.InspectionoftheplotsinFig.9Aindicatesclearlythat it would be inappropriate to attempt to fit such data to Eq. 11, as the behavior of the impedance loci is certainly more complicated. This behavior, although influenced by the basolateral membrane R and C , is not due to a Maxwell- b b Wagnereffectbutrathertotheapicalmembranerelaxation processasillustrated inFig.9B,wheretheimpedanceloci werecalculatedwithR 50.Accordingly,theexaggerated b asymmetryoftheimpedancelociinFig.9Bisduealoneto the apical membrane dispersion that would be observed prevalently in “leaky” epithelia. If fitting of data at the higher frequencies is attempted with Eq. 11, the capaci- tancescalculatedwouldreflectthoseoftheadispersionsat frequencies near 963 Hz when R 5 200 V z cm2. p Ifitisassumedthatcapacitanceisconstantatallfrequen- cies,thenitwouldbetemptingtobelievethatdatalikethose in Fig. 9 are best fit with a mathematical model consisting of an apical membrane R C and basolateral membrane a a R C that exist in parallel with R (see below). However, b b p this would clearly be inappropriate here and would repre- sent a serious pitfall when plasma membranes exhibit a dispersions. Thus, to avoid this pitfall, it would be prudent toevaluateepitheliafortheexistenceofdispersionsbyany method appropriate to their transport physiology. For A6 epithelia and other tight epithelia where apical membranes express only Na1 channels, the amiloride-blocked state of FIGURE 7 InfluenceofanapicalmembraneCole-Coledispersiononthe the tissues serves as such a method. Similarly, for very transepithelialimpedance(Z ).R 51000Vzcm2,R 54000Vzcm2, leakytissueswhereR ,,(R 1R ),frequencydependence T p b p a b Cb520.0mF/cm2,Cadc51.5mF/cm2,Ca‘50.9mF/cm2,fr5110Hz. due to Maxwell-Wagner or adispersions can be assessed Calculationsweredonewitha 50.25,0.5,0.75,and1.00.Corresponding a when the conditions of the experiments conform to Eq. 6. valuesofdepressionoftheimpedanceloci(g)areindicatedin(C). Activation of apical membrane conductance range of Z included in fitting data to Eq. 11. Thus, for In response to second messengers like cAMP, the apical T example,iftheZ between0.1Hzand50HzinFig.8Care membrane conductances to Na1 and Cl2 are activated in T fittoEq.11,thefitteddashedlinedeviatesfromtheactual tissues like A6 epithelia resulting in decrease of R , and a dataatfrequencies.50Hz(Fig.8D)andthevalueofCfit hence decrease of the fractional transcellular resistance fR eq a approximates the value of uC*u at 12.09 Hz. Such fitting 5R /(R 1R ).DependingontherelativevaluesofR and eq a a b a provides no information on the Maxwell-Wagner or adis- R andtheirassociatedvaluesofC*andC*,thetranscellular b a b persions. Capacitance is calculated at a singular value of impedancelociwill,accordingtotheirtimeconstantsR C* a a frequency at the apex of the fitted semicircle, where the andR C*,reflecttheexistenceoftwoplasmamembranesin b b frequency is determined not only by the capacitance, but series by partially overlapping semicircles in Nyquist im- also in particular by the value of R . Consequently, if R pedance plots dictated by the magnitudes of the impedance p p changes, then capacitance will appear to change due to the vectorsofapicalandbasolateralmembranesandtheR ofthe p change of R and not due to change of the dielectric prop- paracellular shunt. Under absolutely ideal conditions when p erties or area changes of the plasma membranes. For the R 5 ‘ and C and C are constant at all frequencies, the p a b conditions illustrated in Fig. 8 (R 5 1, 10, and 50 kV z transcellular impedance locus would appear as illustrated in p BiophysicalJournal81(2)838–851