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J. theor. Biol. (1999) 200, 299}305Article No. jtbi.1999.0993, available online at http://www.idealibrary.com on

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Evidence for Nonlinear Capacitance in Biomembrane Channel System

SUBHENDU GHOSH*, AMAL K. BERA AND SUDIPTO DAS

Department of Biophysics, ;niversity of Delhi South Campus, Benito Juarez Road,New Delhi 110021, India

(Received on 25 June 1998, Accepted in revised form on 24 June 1999)

The electrophysiological properties of voltage-dependent anion channels from mitochondrialmembrane have been studied in a bilayer membrane system. It was observed that theprobability of opening of the membrane channel depends on externally applied voltage and theplot is a bell-shaped curve symmetric around probability axis. A scheme of conformationalenergy levels under varying externally applied voltage was formulated. Assuming that theprobability follows Boltzmann distribution, we arrive at an expression of change in energycontaining a separate term identical to the energy of a capacitor. This fact indicates thepossibility of existence of an added capacitance due to the channel protein. Further it wasshown that the aforesaid channel capacitor could be a function of voltage leading to nonlin-earity. We have o!ered a general method of calculating nonlinear capacitance from theexperimental data on opening probability of a membrane channel. In case of voltage-dependent anion channel the voltage dependence of the capacitor has a power 0.786. Theresults have been interpreted in view of the structural organization of the channel protein inthe membrane. Our hypothesis is that the phenomenon of capacitor behaviour is a general onefor membrane channels.

( 1999 Academic Press

Introduction

The discovery of electrical behavior of biomem-branes is a milestone in understanding complexbiological phenomena. Unlike non-living objectsthe biological systems are #exible and dynamic innature and this makes the physical propertiesmore complicated. For example, the capacitanceof a bilayer membrane is a complex function ofthe solution parameters as well as other experi-mental factors (Li et al., 1994). The root cause ofthe dynamicity of a biological system lies in the#exibility of the constituent macromolecules. Asan example the transmembrane proteins, includ-

*Author to whom correspondence should be addressed.-mail: subho@del3.vsnl.net.in

022}5193/99/019299#07 $30.00/0

ing channel proteins, change shape and sizeaccording to the environmental conditions lead-ing to changes in charge distribution and hencein overall electrical properties. In addition, thecooperative interactions among the channels ina membrane play an important role in the elec-trophysiological behavior of the channels (Ghosh& Mukherjee, 1993; Ghosh, 1993). The channelstructure #uctuates around the minimum energyconformation. It is the collective e!ect of these#uctuations that leads to the emergence of newproperties, e.g. opening and closing of channels,called gating.

The process of channel gating is a highly com-plex phenomenon. Most often a channel has sev-eral substates. While changing from a closed stateto an open state (or vice versa) the channel prefers

( 1999 Academic Press

300 S. GHOSH E¹ A¸.

to attain the substates according to the orderof the conductance levels, thus making thetransition stepwise (Sakmann & Neher, 1985).A good number of channels undergo suchtransitions under the in#uence of an externallyapplied electric "eld, known as voltage sensitivechannels. It is believed that the basis of suchtransitions lies in the fast conformational changesof the channel protein (Bathori et al., 1998). Insome of the channels sensitive to externally ap-plied voltage, there lies a structural domainwhich acts as a voltage sensor and functions likea gate, i.e. the domain changes its orientation toopen or close the transmembrane passage createdby the protein (Hille, 1984). On the other hand,opening and closing of several channels areregulated by extracellular ligands (Hille, 1984).Chemical modi"cation of channel protein byphosphorylation and dephosphorylation playsa key role in regulating the activity of ion chan-nels and thereby signal transduction (Levitan,1994; Bera et al., 1995). Looking into the tem-poral behavior of the ion channel, it is observedthat the gating process is stochastic in nature;hence, one can talk about the probability of itsopening at an instant or an average probability ofopening at steady state.

In the present work we demonstrate, basedon the opening and closing probability data,that the ion channels reveal capacitance likefunctional dependence on the applied potential.This investigation was carried out on voltage-dependent anion channel (VDAC) from rat livermitochondrial membrane, known as mitochon-drial porin (Benz, 1994; Colombini, 1980a).VDAC (molecular weight of about 35 kDa, DePinto et al., 1987) with pore size of approximately1.7 nm and an exclusion limit of 2}8 kDa, showsa high single-channel conductance (Benz, 1994).It has been reported that at low membrane po-tentials the channel is open, but switches topartially closed state in response to elevated volt-ages (positive and negative) (Colombini, 1980(b);Colombini et al., 1987). Based on our experi-mental data we propose a model for the states ofVDAC under the in#uence of externally appliedelectrical potential. This model involves statist-ical mechanics and predicts channel capacitancebehavior which can also be speculated fromthe structural organization of the proteins. The

concept can be generalized to other passive-di!u-sion ion channels.

Materials and Methods

MATERIALS

Diphytanoyl phosphatidyl choline (DPhPC) waspurchased from Avanti Polar Lipids (Albaster, AL).n-Decane, N-(2-hydroxy ethyl) piperazine-N@(2-ethane sulfonic acid) (HEPES) and all otherchemicals were purchased from Sigma ChemicalCo. (USA).

METHODS

Puri"cation of VDAC: VDAC was puri"edfrom rat liver miotochondria using the method ofDe Pinto et al. (1987).

RECONSTITUTION OF VDAC IN PLANAR LIPID

BILAYERS

VDAC was reconstituted into planar lipidbilayers according to the method of Roos et al.(1982). Brie#y, the apparatus constituted of apolystyrene cuvette (Warner Instrument Cor-poration, U.S.A.) with a thin wall separating twoaqueous compartments containing 1 M KCl,5 mM MgCl

2and 10 mM HEPES (pH 7.4). The

polystyrene divider had a circular aperture witha diameter of 150 lm. The aqueous compartmentswere connected to an integrating patch clampampli"er (Axopatch 200A, Axon Instruments)through a matched pair of Ag/AgCl electrodes.The cis compartment was connected to the headstage (CV-201) of the ampli"er and the trans com-partment was held at virtual ground. A 1% solu-tion of DPhPC in n-decane was painted over theaperture to form the membrane. Reconstitutionof VDAC in BLM was initiated by adding 2 ll ofVADC (2}5 ng protein dissolved in 1% TritonX-100, 10 mM HEPES, pH 7.4) to the cis cham-ber. Channel currents were recorded at full bandwidth on a video cassette recorder after digitizingthrough an analog to digital converter (VR-10B,Instrutech, NY). Data were "ltered through an8-pole Bessel "lter (Frequency Devices, U.S.A.)and sampled at a rate, greater than the cornerfrequency, using an ITC-16 interface (InstrutechCorporation, U.S.A.) connected to a Macintoshcomputer (Apple Computer, Inc., U.S.A.)

FIG. 1. Transition of VDAC from higher to lower conductance states at higher clamping potential. Immediately after theoneset of voltage, VDAC goes through di!erent substates before reaching a steady state. &&*'' at the l.h.s. indicates baseline.The point of voltage application (#60 mV) is indicated by the arrow. Conductance values (ns) of di!erent states are indicatedat the r.h.s.

FIG. 2. Voltage dependence of the open probability ofVDAC. Open probability (p) was calculated as the fractionof total time the channel spends in the open state. &&L''represents the experimental data and &&*'' is the theoreticalgraph calculated according to the eqn (1) in the text. Datashown are taken from the continuous current recording of2 min for each holding potential.

NONLINEAR CAPACITANCE IN BIOMEMBRANE CHANNEL 301

DETERMINATION OF OPEN PROBABILITY

The probability of "nding the channel in openstate at a particular voltage was determined bycalculating the fraction of total time it remainedopen at that voltage (Aldrich & Yellen, 1983).

ESTIMATION OF CONSTANT PARAMETERS OF

EQUATIONS

These were carried out by the best-"t methodsin Sigma Plot software, version 3.02 (Jandel Sci-enti"c). The method is based on minimization ofleast-squares deviations.

Experimental Results

Puri"ed rat liver mitochondrial VDAC wasidenti"ed on SDS gel as 35 kD a single band.VDAC showed channel forming activity throughelectrophysiological measurements when addedto the aqueous solution, bathing a planar lipidmembrane. Channel insertion into the membranewas indicated by the sudden jump of membranecurrent. This was not observed when only bu!er(devoid of VDAC) was added to the aqueouscompartment. It was observed that the VDACchannel is maximally open when no externalvoltage is applied. Onset of voltage induces grad-ual closing of the channel through its substates,whose existence has arleady been reported(Bathori et al., 1998). Figure 1 shows gradualtransition of VDAC from higher to lower con-ductance substates at the onset of a voltage. Fi-nally, it attains a steady state which could be

a completely closed state as shown previously(Bathori et al., 1998). The opening probability ofVDAC at di!erent clamping potentials was de-termined according to the method mentionedearlier. The plot of opening probability (steadystate) vs. voltage (Fig. 2) is a bell-shaped curvewith a maximum at a 0 mV.

Theory of the Channel System Behavior

Extensive studies on the kinetics of channelgating indicate that the process is a random oneand Markovian assumption holds good (Col-quhoun & Hawkes, 1995). According to this as-sumption the channels are characterized by fast

302 S. GHOSH E¹ A¸.

changes between various conformational statesleading to opening or closing depending on theimmediate past. The probability of openinga channel is function of time, voltage and otherenvironmental factors. The open probability cha-nges with time exponentially during relaxationon withdrawal or onset of voltage. In steady statethe open probability becomes merely a functionof voltage given other factors unchanged.

As stated in the previous section, the openingprobability (P) of the VDAC channel in steady statehas a Gaussian dependence on the externally ap-plied voltage which can be represented as follows:

P"P0

exp(!a D<iD2), (1)

where <i

is the externally applied voltage, P0

refers to the opening probability at zero externalvoltage and a is a coe$cient independent of volt-age. The conformational states of the channelprotein under externally applied electric "eldsmay be compared with the situation in electronicor molecular excitation. The inherent assumptionis that the energy levels form the ground and theexcited states designated as 0, 1, 2,2 , j,2 , ieach corresponding to a particular electric "eldand i represents the highest level. As mentionedpreviously an ion channel opens or closes ran-domly. It has also been emphasized that thisprocess is Markovian. For a system having ener-gies distributed randomly Boltzmann's statisticshold good to describe the probability. Assumingthe system of channels obey Boltzmann's statis-tics we can write the expression of probabilityunder an external voltage <

ias

Pj,i"P

0,iexp(!*E

j/k¹), (2)

where *Ej"(E

j!E

0) is the di!erence between

j-th & the ground state energies, Pj,i

and P0,i

areprobabilities of the protein channel being in thej-th and the ground states, respectively. As evi-dent from the experiment (Fig. 2) the channel ismaximally open under no external electric "eld.Hence, the opening probability under this "eldcondition is given by P

0,i. Keeping in view

that the maximum probability of opening is 1,i.e. +

jPi,j"1, we arrive at the expression

P0,i

+jexp(!*E

j/k¹)"1 (3)

Writing e P0"1 (4)

and replacing the summation by integral (assum-ing large number of states in continuum) we get(see the appendix)

P0,i"!e P

0/[k¹(1!exp(!*E

i/k¹)]. (5)

Comparing eqns (1) and (5) and recollecting thatfor external voltage <

i, P

0,istands for P, we

obtain after rearrangement

k¹(1!exp(!*Ei/k¹)"e exp(a D<

iD2). (6)

It is known that the change in energy of theprotein due to its conformational change underan electric "eld is of the order of a few kcals permole. Hence, the term *E

i/k¹ is small. If a hap-

pens to be a small quantity the exponential termsin eqn (5) can be expanded and the higher-orderterms can be neglected giving rise to the followingrelation:

*Ei"e(1#a D<

iD2). (7)

Let us rewrite eqn (7) as

*Ei"e#1/2 C D<

iD2, (8)

where

C"2ae. (9)

It may be noted here that besides its role de-"ned in eqn (4) e takes care of the correctionfactor due to approximations in eqn (7). Thephysical nature of the parameter e is discussed inthe next section of this paper. If e is not varyingsigni"cantly with voltage then C becomes a con-stant. At this juncture we may recall that theenergy of a classical capacitance is given by thesame expression as the second term on the right-hand side (r.h.s.) in eqn (8). For VDAC proteinthe value of C has been calculated from the ex-perimental probability vs. voltage plot (Fig. 2).C"10.44 lF and e"1. Let us call this termchannel protein capacitance which is distinct fromthe usual membrane capacitance.

Next, we attempt to arrive at a general expres-sion for the channel capacitance. In order to doso we rewrite eqn (1) as

P"P0

exp(!a D<iDn), (10)

NONLINEAR CAPACITANCE IN BIOMEMBRANE CHANNEL 303

where n is a real number. Correspondingly,eqns (7) and (9) are replaced by

*Ei"e(1#a D<

iDn) (11)

and

C"2ea D<iDn~2, (12)

where expression (8) remains unchanged. a andn are evaluated by the best-"t program.

In our experimental setup with VDAC andlipid bilayer n"2.7259 and a"3.445]10~5. In-terestingly, in case of VDAC, the parameter Cis a function of < and the dependence is throughthe parameter D<

iDn~2. The latter has a

fractional index as obtained by best-"t method.This indicates that C is no more a classicalcapacitance than a voltage-dependent (nonlinear)capacitance. It may be mentioned here thatthere are number of examples of nonlinearcapacitance showing voltage dependence (Chuaet al., 1987). Here we report a similar phenom-enon in biological membrane channel system,e.g. a parameter responsible for holding electri-cal energy in the channel system. It may alsobe noted that like any other case, C is depen-dent on system constituents and the solutionconditions. The fact that in our case n'2according to the experimental results, indi-cates that the channel protein under thecircumstances has the capacity to hold morecharge or electrical energy than a classicalcapacitor.

Discussion

As mentioned in the introduction, biologicalsystems are much more complex than the non-living objects. Therefore, it is di$cult at times toapply the concepts of non-living physics to a liv-ing system straightaway. Keeping this in view, welook for the experimental basis of the nonlinearcapacitance of the channel protein. It has beenshown that VDAC closes on the onset of voltagein a stepwise fashion (Fig. 1). Averaging of thistype of data over time leads to an exponentialcurve known as relaxation (Sakmann & Neher,

1985). The latter indicates that the channel re-leases or acquires charge rather slowly on theonset or the o!set of voltage. This phenomenonoriginates from the capacitance behavior of thechannel.

What could be the physical basis of e and C?As de"ned in eqn (4), the parameter e is the ratioof maximum probability to the actual openingprobability under no external voltage. In caseof mitochondrial VDAC, e is 1. If e is greaterthan 1, the channel does not remain in a fullyopen state throughout the time period even inthe absence of an external electic "eld. Thismust be due to some structural constraint ofthe channel protein in the membrane. The con-straint depends on the internal structure ofthe channel forming protein and the lipidenvironment, hence varying from channel tochannel. The parameter may be equated tothe ratio of energies involved for maximalopening and the actual opening. In other words,e re#ects the energy constraint in full openingof a channel. The constraint can be seen asthe cumulation of various energy components,e.g. electrostatic potential, dielectric potentialdue to dielectric junctions and formation ofionic layers, conformational energy due to cha-nges in the dihedral angles and kinetic or thermalenergy of the ions leading to conformational dy-namics. It is likley that the above-mentionedenergy components overlap. For example, theelectrostatic potential and conformational strainare correlated. The thermal energy refers tohigher vibrational and rotational states of thecharges of the protein similar to the case of elec-tronic excited states. It is reasonable to state thatthe channel protein stores the energy in all theabove forms and the external electric "eld pro-vides energy required to attain the closed statecon"guration.

At the structural level, VDAC like other trans-membrane proteins consists of hydrophilic andhydrophobic domains (Colombini, 1996). Weiss(1996) argued that the capacitance current resultsfrom the net movement of surface charge onthe two membrane solution interfaces and one ofthe sources of this change in surface charge is thereorientation of dipoles for macromolecules inthe membrane. In an electric "eld, it is likleythat the protein reorients these groups in such

304 S. GHOSH E¹ A¸.

a fashion that a higher energy conformation(strained) is attained. Although it is di$cult atthe moment to quantify the contributing energycomponents it is realizable that each of thefactors mentioned earlier contributes to the param-eter e, hence C. Interestingly, in eqn (8), the en-ergy corresponding to the capacitance appears ina separate term. This indicates that the channelprotein as a whole has a capacity to hold electri-cal charges as well as energy which we identify asthe property of electrostatic capacitance. This isthe structural basis of capacitance arising out ofchannel protein. Although the validity of eqn (8)has been for smaller *E

iand a we can speculate

that even for higher order approximations in theexpansion of the r.h.s. of eqn (6) the second-orderterm in voltage will appear indicating the exist-ence of the capacitance. At this juncture, it maybe mentioned that in such a complex dynamicsystem it is likely that the capacitance is voltagedependent.

In conclusion, we may comment that agood number of membrane channels, e.g. gapjunctional protein (Mazet et al., 1992; Jongsmaet al., 1993), show bell-shaped curves at thesingle-channel level for the probability ofopening vs. voltage plot. Hence, the above-mentioned analysis can be a general methodo-logy applicable to various membrane channelswhere we expect a similar property of channelcapacitor. Our hypothesis is that the phenom-enon of capacitance is a general one in membranechannels.

The authors thank Dr Amitabha Mukerjee andDr Shobhit Mahajan of the Department of Physics and DrParamita Ghosh of the Department of Biophysics,Univeristy of Delhi for critical reading of the manuscript.Financial support of the Council of Scienti"c andIndustrial Research, Government of India is thankfullyacknowledged.

REFERENCES

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BENZ, R. (1994). Permeation of hydrophilic solutes throughmitochondrial outer membranes: review on mitochondrialporins. Biochim. Biophys. Acta 1197, 167}196.

BERA, A. K., GHOSH, S. & DAS, S. (1995). MitochondrialVDAC can be phosphorylated by cyclic AMP-dependentprotein kinase. Biochim. Biophys. Res. Commun. 209,213}217.

BATHORI, G., SZABO, I., SCHMEHL, I., TOMBOLA, F., DE

PINTO, V. & ZORATTI, M. (1998). Novel aspects of theElectrophysiology of Mitochondrial Porin. Biochim. Bio-phys. Res. Commun. 243, 258}263.

CHUA, L. V., DESOER, C. A. & KUH, E. S. (1987). ¸inear andNonlinear Circuits, pp. 295}362. McGraw Hill, Singapore.

COLOMBINI, M. (1980a). Pore size and properties of chan-nels from mitochondria isolated from Neurospora crassa.J. Membr. Biol. 53, 79}84.

COLOMBINI, M. (1980b). Structure and mode of action ofa voltage-dependent anion selective channel (VDAC)located in the outer mitochondrial membrane. Ann. N>Acad. Science 341, 552}563.

COLOMBINI, M., YEUNG, C. L., TUNG, J. &, KOG NIG, T.(1987). The mitochondrial outer membrane channel,VDAC is regulated by a synthetic polyanion. Biochim.Biophys. Acta 905, 279}286.

COLOMBINI, M., BALCHLY-DYSON, E. & FORTE, M. (1996).VDAC, a channel in the outer mitochondrial membrane.In: Ion Channels (Narahashi, T., ed.) Vol. 4, pp. 169}202.New York: Plenum Press.

COLQUHOUN, D. & HAWKES, A. G. (1995). In: Single Chan-nel Recording (Sakmann, B. & Neher, E., ed.), pp. 457}460.N.Y. and London: Plenum Press.

DE PINTO, V., PREZIOSO, G. & PALMIERI, F. (1987).A simple and rapid method for the puri"cation of themitochondrial porin from mammalian tissues. Biochem.Biophys. Acta 905, 499}502.

GHOSH, S. & MUKHERJEE, A. (1993). Statisticalmechanics of membrane channels. J. theor. Biol. 160(2),151}157.

GHOSH, S. (1993). Relaxation of membrane channels: astatistical mechanical approach. J. theor. Biol. 165(2),171}176.

HILLE, B. (1984). Ionic Channels of Excitable Membranes.Sunderland, MA: Sinauer Associates.

JONGSMA, H. J., WILDERS, R., TAKENS-KWAK, B. R.& ROOK, M. B. (1993). Progress in Cell Research (Hall,H. E. Zampighi, G. A. & Davis, R. M. eds), Vol. 3, pp.187}192. Amsterdam: Elsevier.

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LI, J., DOWNER, N. W. & SMITH, H. G. (1994). Evaluation ofSurface-bound Membranes. In: Biomembrane Electrochem-istry (Blank, M. & Vodyanoy, I., eds), p. 505. Washington,DC: American Chemical Society.

MAZET, J.-L., JARRY, T., GROS, D. & MAZET, F. (1992).Voltage dependence of liver gap-junction channels recon-stituted into liposomes & incorporated into planar bi-layers. Eur. J. Biochem. 210, 248}255.

ROOS, N., BENZ, R. & BRDICZKA, D. (1982). Identi"cationand characterization of the pore-forming protein in theouter membrane of rat liver mitochondria. Biochim. Bio-phys. Acta 686, 204}214.

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NONLINEAR CAPACITANCE IN BIOMEMBRANE CHANNEL 305

APPENDIX

Details of the Steps from Eqns (A.1)}(A.2)

We rewrite,

P0,i

&jexp (!*E

j/k¹)"1. (A.1)

Assuming large number of states in continuumand writing *E

j/k¹"x, we replace the summa-

tion by integration, thus obtaining

1/P0,j"k¹

0:z exp(!x) dx"k¹ [1!exp(!z)];

where z"*Ei/k¹. (A.1@)

Hence, combining eqns (4) and (3@)

P0,j

"e P0/k¹ [1!exp(!*E

i/k¹)]. (A.2)