Superposition properties of independent ion channels

Biophy Journal Volume 67 August1994 634-640

Superposition Properties of Interacting Ion Channels

Asbed M. Keleshian,*

Geoffrey

F.Yeo,* Robert 0.

Edeson,§

andBarryW. Madsen*

*DeparbnentofPharmacology, Unvesityof Westrn Ausa, S009;tco of I andPhysicalScdes, Uniesiy, 6150;5Deparment^ofAnaestheia,Sir CharlesGairdner 6009,WestenAusala

ABSTRACT Qunttativeanalyssofpach clam data iswidelybasedonstochastic modelsof sinl ael Mem-

bramne odte coin more than one aivechannel of a given type, and it is uually assumed these behave independentlyinordertointrettherecord and ierindividual chanel However,recent st ssu etwereare chaelintcons insomesystems.We examinea inasysten^{oftwo idnticalChannels,}

eachrodeledbyacontnuos-tme Markov chain in whichspecfid ranitionratesaredeperndet^ontheCOnduxtancestate Ofthe othercael, aingi ouslywhntheoerdhanelopensorcloses.Eachdhanel henhas,e.g.,adosed

tmedensitythat is citionalonthe other channelbeingopenorclosed,thesebeing under We rete thetwodensitesbyaconvolutio functionthatenmbiesinformationabout,andservestoquantify, in theclosed

class.Disbtrxtons^of obServabb

(su;eposition)

^soum times aregivenintermsofthese nal bies.Thebehavior

oftwocha basedontwo-and three-state Markov modesfis examnedbysimlation.Optiized fit of smulated

data usng wamesa and sampl size dicates both and gative cm be

from ^indpendenc

INTRODUCTN

Ionchannels are

specialized proteins

thatcontrolmovement of ions acrosscellmembranes. Directobservation ofsingle- channel currents with the patchclamp

technique

has pro- motedthe quantitative study of

channl

kinetics,which in turn has stimulated

development

of

app

iae models.

Thesemodels, which areusuallybasedonstochasticprocess theory,enable

proposed

kineticmechanismstobestudied by

simulation;

theyalsoprovidethe basis for inferentialanalysis ofexperimental data.

Mostionchannel modelinghas been

concered

withthe behaviorofasinglechannel acting inisolatio. However, mem e patches from which recordings are made often containmorethanoneactive channel; indeed,somechannels appeartoconsist ofanumberof identical co-channelsinthe one macromolecularcomplex (Fox, 198. In this

setting, interretation

of thepatchclamprecordreqiires anunder- stadingof how individualchannel

(or

co-channel) processes combine togive the observed

superposition

process. This

intWgration

is simplified if channls are considered to act

independently

(KijimaandKijima, 1987b;Yeo etal,1989;

Colquhon

and

Hawkes,

1990;Fredkin and Rice, 1991),an

assumption

thatwas

supported

by early studieson certain channels

(Hil

andChen,1971a, b; Neheretal., 1978; Sig- worth,

1980,

Miller, 1982). However, there is more recent evidence that

significant channl

interactionmay occur in somesystems.For

example,

observed

frequencies

of simul-

Rcdw4edforpubwica 25Februwy1994and inflina form9May1994.

Address

repmt

requeststoRobert 0.Edeson,DepartmentofAmesdhsia,

SirCharlesGardnerHospital, Nedlands6009, WestrnAnstralia. TeL:

61-9-346-4326; Fax: 61-9-3464375; E-mail: _ pharm uwa.edua

o 1994bytheBiophysilSociety

0006-3495/94Ai8/634107 $200

taneous

openinp

are highly

sggestive

of cooperafivity (Yeramianetal., 1986; Hunter andGiebisch,1987;

Matsda,

1988; Hymeletal., 1988;Schreibmayeretal.,1989),as are the

pting

kineticsofpotassiumchannelsbunits

inXenopus oocytes (Tytgat

andHess, 1992), thebehaviorofcardiac gap junctionsinchickembryos(ChenandDeHaan,1992), and the

cmpaative

p ie of

acetykicline

monomersand dimers(SchindleretaL, 1984). Nonindependencemayalso explaindeviations fromthe binomialdistnrbution of steady- state

probabilities

ofthenumber of openchannels,as com- mented uponby severalauthors

(Kiss

and

Nagy,

1985;Iwasa et aL, 1986; Krouse et

aL,

1986; Queyroy and Verdetti, 1992).

In order to analyze such multichannel systems, a theo- reic amewo is i whichhasatbasissomephysi- cally reasonable

mehanism

of

cooperativity,

yet remains

mathematically

and

compuationally

bactable. It shouldpr- vide a method of detecting and preferably quantifying in someway

channel interaction,

aswellascharacterizingit in termsofunderlying single-channel disbutional

Properties

orkinetic parameters.Ideally,a

comprehensive

theorywould enableetimation oftheseparameters froma

superpition

record.

Inthispaperwepresentkinetic modelsincorporating co-

operative

action betweentwoidentical channels. Each

chan-

nel is modeledbyacontinuous-time Markovchain inwhich specified tansitionrates aremadeinstantaneously dependent onwhethertheother channel isopen orclosed. Thesuper- positionofthetwosingle-channel models,which we refer to asthecomplete process, is also acontinuous-time Markov chain buton alargerstate space. Conditionalsojountime densitiesarederived and relatedby convolution with func- tions that embody information about and quantify the de- pendence. Analyticalsolutionsarepresentedandcompared withtheresultsfrom simulations. Other work

concerned

with 634

Properte

of

hieracilg

IonChanels modeling channel interactionscanbefound in

Queyroy

andVerdetti(1992),Luii andDilger (1993),and Draber etal.

(1993).

THEORY

Consider an isolated ion channel, where "isolated" is in- tended toguarantee that the behavior of thechannl is not influenced in any way bythe presence or behavior of neigh- boringchannels. Itisasswumedthatthechannel has two ex- perimentally disting able states, unit n

Bducting (open)

and

nonconducting

(dosed);

then,

under

equilibrium

condi- tions it willexhibitsomenatwalkineticbehavior observable asaltemating open and closed

periods.

Inthe usual way, we assume that this kinetic behavior can be modeled by a continuous-time Markov chain in which the (finite) state space isagegatedinto two

classes,

0(open) and C(closed) with n0 and

nc states, respectively.

Then successive open times are identically distnrbutedwith distbutionfunction, e.g.,

FX(t),

^mean

p,,

anddensity function

fx(t);

^likewise,

successiveclosedtimesareidenticallydistbutedwithdis- tribution function

Fy(t)

meanp., anddensity

functnfyQt).

(For any

sojoun

time T, we denote itsdis-trbutionfunction by

F,(t),

density

f1(t),

and mean i =

flF1(u)du

where F1(t)= 1-

F(t).)

Fromsandard Markovtheory, assuming distincte of

eigenvalues, fx(t)

and

fy(t)

are each linear combinations ofexponential densities, having nonnegative coefficients under detailed balance

(Kijima

^and

Kijima,

1987a).

Wenowallowasecond, identicalchannelto exist in the neighborhoodof the

first,

andconsiderthepropertiesof the superposition signal (that is, a patch clamp recording) arising fromconcurrentactivity in the two channels(see Fig. 1). Ifthe two channels are independent, the distri- butions of the various types ofobservable sojourntimes

(Zioo

^andsoon) in terms of the distributional properties of the single-channelsojourntimesaregiven byEqs. 24 and 25 of Yeo et al., (1989). Themain focus of the re- mainder of this paperis the case in which the two channels are not independent in the following sense: the closed- time density (and similarly, the open-time density) of each channel is allowed tobedependentonwhether the other channelis openorclosed.Specifiedtransition rates, and hence density parameters, are assumed to change in- stantaneously whenever the other channel opens or

Z2 Z2

z)

FIGURE 1 Idealid segmentofasperpos signalfoma patchcn-

tamingtwoacvcchannels (see also YeoetaL,1989),sowig noaom for difMre-typesofsojourtimeat4rnndaeekveQisO 1,ad2-forexample,

Z1.41

denotesasepour timeatlevel 1(onecharelopen)dot beginswith anopenmgadendswihadosure.

closes.The single-channel behavior is now characterized by conditional density functions

fy c(t) fy,O(t), fx,c(),

and

fx,O(t),

where, for example,

fy,c(t)

is the density of closed sojourn times in either channel, given that the otherchannel is closed throughout. As a way ofrelating these conditional densities to each other we define convolution functions

kc(t)

and

ko(t)

(0 ' t < mo) satisfying

fy,o(t)

=

kc(t)

^*

fy,c(t) fx c(t)

⁼

ko(t)

^*

fx1o(t) (1)

where * denotes convolution, that is, k(t) *

J(t)

= fo kt -

u)f(u)

du. Theconditional densities are

prop-

erly normalized when

f

"

k(t)

dt=1.The functions

ke:(t)

and

ko(t)

descnrbehow the closed-time density and open- time density,

respectively,

inonechannel differs accord- ing to whether the other channel is open or closed throughout. If, say, the closed times are independent of the state of the other channel process, then

fy, c(t) fyIO(t) (-fy(t),

butnote that

fy1 c(t) fyIO(t)

does not

necessarily imply the channels behave independently) and

kc(t)

=8(t),theDirac8 function. Because of this last possibilityit ispreferabletoworkwith thecorresponding integrals

KC(t)

= fo

kc(u)

du and

Ko(t)

= fo

ko(u)

du.

AstheLaplacetransformof a convolution is aproductof Laplacetransforms and each conditionaldensity of a

sojourn

time in the closed(oropen) dass is a linear combination of exponentials, itfollows fromEq. 1 that

KC(t)

=1 - rc

wie-l'

i=l

(0 C t <00) (2) where

rc

<

2nc

-1,w1canbe lessthan,equal to, or greater thanzero, and the exponents

mi

arepositive,with ananalo- gousresult for

K(t).

Although

f= kd(u)

du= 1,

kc(t)

is not in

general

a

density

ctio

(it

may have

jumps

or take

negative

values);letKdt)=1-K(t)anddefine(byanal- ogywith

IL)

Lc =

f K-(U)d

u t =

w/m. Then

ILc = ILY1o-ILY and#L - fo K0(u)du = Ixlc^- I½x, where P

IWo 5Lo iC,

and

xIo

arethemeansoftheConditional densities above.

Theseideasare nowillustrated with a simple example.

Consider an isolated channel with one open and one closed state, having open times and closed times each exponentially

distnrbuted

with parameters a and

.,

re- spectively.Asecond,identical channel isnowpositedin the neighborhood of the first, the two interacting in suchaway that either channel has opening ratef3' during a period in which the other channel is open, and

13

otherwise. The conditional densities can be written as

fylc(t)

=

1e-O'

fx, c(t)

=

fX, O(t)

= ae-'

(3)

fy,O(t)

⁼

te-'

Using Eq. 1(and,mosteasily, aplace transforms; thus, the Laplace transform of

ko(t)

is 1 and that of

kc(t)

is

K let a. 635

Vdume67 August1994

13'(s

+

13)/((3(s

+

13')),

s

being

the

Laplace variable)

we obtain

kc(t)

⁼

('/13)8(t)

⁺

(1

-

13'/13)13'e-'t

ko(t)

= 8(t)

(4)

channel models where there may be more than one state in the open orclosedclasses, and interaction issuchthat anytransition rate may bedependentontheconductance state of the neighboring channel. For example, suppose that a channel has isolated kinetics descnibed by

KC(t)

= 1 - (1 -

13'/18)e-

'

KO(t)

= 1

with

p.c

= (1 -

1'/13)/13'

and

p.o

= 0. If the interaction between channels is such that one channel being open predisposesitsneighbortoopen(positive cooperativity), 13' >

13

andKC(t) > 1. Similarly, inthe caseof

negative

cooperativity, 13' <13 and

KC(t)

< 1; in either instance

KJ()

⁼

13'/13.

The formof

KC(t),

plottedas afunctionof time scaledinunitsof

1/13',

is shown inFig.2 forvarious valuesof

1'/1.

Of course, if the channelsareindependent,

13'

=

andlK4(t)

= 1.

We canderivetheunconditionaldensity

fi)(t)

^ofaclosed

sojourn time in the presence of the second channel

(irre- spective

of whether that secondchannel is openor

closed)

byusingthe

complete (four-state) process, obtaining

f2)(t)

=

ClAle1"1

+C2A2e-' (5) where

Al,A2½=

a+ 3'

+213+R}

C1 = 1- C2=0.5 + ^a-13' +

213'(a

+

13)/(a

+

I')

2R

qI2 fl

qn a

(Scheme 1) where 1 and 2 are closed states, and 3 is an open state.

The opentimes remainexponentially distributedwith pa- rameter a, while the closed times are distributed as a linear combination of twoexponentials,given bytheright side of Eq. 5 with

A1L A2 =/ql2

d2[(q2+d2)241q12]5

+ 13

f-A2

I A ^-A2

(6) C2= 1-C

where

42

=

q21

+ 13.

Nowassumethatasecond,identical channelexistsinthe neighborhood of the first, thetwo

interacting

insuchaway thatwhenonechannelisopen,qu changestosome newvalue q; (notethatthe complete processdoesnot

satisfy

detailed balance). Then

fyIe(t)

=

12= 1CjA1e-Ae',

and

fyI,(t)

=

1

C'Ae- where the

respective parameters

canbe ob- tained usingEq. 6 with

q12

replacedby q. From

fyI

t).

fyIQ(t)

andEq.1,

Ke(t)

canbedeterminedby deconvolution and integration:

andR = [(a +

13,)2

⁺ 4( -

13')J1!

Again, under inde- pendence 13' =

13;

and hence, R = a + 3, A1 = , and C1= 1,thedensity

colLapsing

tomonoexponentialwith pa- rameter

13,

as

expected.

The monotonicexponential formofKCt)found in the aboveexample isnotpreserved inmorecomplex single-

(7)

x -kIt+ (C^)e -A2t I(e)_,zf

rbb+b2) ( b3)

wtmer bj = 1 +

(q,2- X)qf

^-

m)(q,2q2]),

i =

1, 2,3,

1.02

1.00

t

FIGURE 2 Interaction ⁱⁿatwo-channelsystemwhereeachchannelis modeledas atwo-stateMarkovprocesswithopeningrate |(respectively

1')when theneighboring channelis cosed(open). The functionKt(t)

relatessingle-chanelclosed timedensiiesfy (t),y (t) fndyionalonthe

othrchannelbeingcosedoropen,respectively(see Eq.3), andisplottd (fom Eq.4)forvariosvalues of '/jP ^-cae byintrceptK(0))on a

normalizedtimescale. Independenceimplies 3' =fJ andKC(t)⁼ 1.

I

~~~~~~0.2

0.96

0.0 20.0 t(Ms) 40-0

FIGURE 3 Interaction in atwo-dannelsystemwhereeachchanndelis modeledasathre-stateMarkovprocess basedonScheme 1,and thetran- siin rateq12isdependentontheeigoin chad being dksed (when it is

qo

oropen (q. Pkots weregenerated Eq.7with paraeter

valusq12=0.82,

q2,

=0.023,1l=0.115,a=0.06ms-',andarange of valuesfor 41(idictedbytheratio

qLjq2

wrtten besideeachcuve).

B kic Jounma 636

Kc(t)

⁼ I I q.

qu

I--_i

PrperesofInteractn IonChanels andm =A, m2 = A,andm3=

q12.

^Itcanbe shown that

I;bi

+C/b2 = 1/b3;

thus,

inthisinstance

KJO)

= 1.Also Lc=

(1-q;Jq)q21/(q;J3).

Clearly, under independence

q;

= q12 and

KC(t)

= 1. These results are

illustrated

in Fig. 3 where

KC(t)

for this model is plotted for various values ofq;/q12.Regardlessof independence,K((t) = 1.

Suppose now that it is theopeningrate,Bthatchanges(e.g., to ,B') when the other channel is open (in which case the complete system satisfies detailedbalance).Then

KO(t)

= 1 and, usingmethods similar to thosegivingEq. 7,

KC(t)

= 1 - (1 -

B'I(3)[C'e-Ai'

+C

e-v'1

(8) where

A',

A, C;, and C' are obtained from Eq. 6 with ,3 replaced by (3' and d2 replaced by

d2

=

q21

+ 3'. Note thatKR(O) = (3'/3and thehomologywithEq. 4; also

Pc

=

(1

-

13'/.3Xqn

+

q2JY(q%3')'

We nowconsider what isexperimentally observable in such a two-channel system, thatis, sojourn times of the typesillustrated inFig. 1. The distributions of these so- journtimescanbe obtainedusing properties of thetran- sitionratematrix of thecomplete process and conditional independence, allowing us to generalize Eqs. 24 and 25 of Yeo et al. (1989) as

F4(t)

=

Fy c(t)Hy c(t) FZ2(t)

=

Fx 0(t)Hx

o(t) (9)

zj>t)

^{= v}

OFxI c(t){-d[HyIO(t)]dt}

fZC:(t)

⁼

VcFy1O(t)-d[Hx,c(t)]dt}

⁽¹⁰⁾

(t) =

v1

fx

C(t)Hy

o(t)

fzlco(t)

= vco

fy 0(t)Hx c(t)

where thenormalizingconstantsv aresuch thatthe func- tions integrate to unity, and, for example,

HyIo(t)

is the probability that a channel remains closed for at leasttime t,giventhat the other channel hasjust opened and remains open for at least time t. In the case of in- dependent channels, thisis the tail distribution function of a closed sojourn from a random time point ("a re- maining sojourn time"), which would satisfy

HyO(t)

=

i¼Y lo Fy o(u)

du

(Hy c(t)),

consistent with standard renewal theory; furthermore,

fz,.(t)

⁼

fzcc(t),

t . 0.

Theseresultsforindependentchannelsapplyvirtuallyun- changedin certaindependentchannelmodels, andwerestrict consideration here and in the nextsection to such special cases(themoregeneral theorywith

applications

will be dealt withelsewhere).Anexampleis the two-state channelsystem above,in whichany closed(or open)sojourn,orremaining sojourn, hasa single exponential

distnrbution.

Another ex- ampleis the three-state model ofScheme 1, provided that only the channelopening rate can changewhen the other channel chanes conductance level (thesituinis different if

q2

can change).Forthese

special

cases,

VOO=

VtL10 4vc

= LyIc voc= (KLY10 -

nl)/I4ioc

where 7 =

foF1 O(u)Fx,

(u) du;

hene,fz,(t)

=

fz(t)

=

FxIC(0PYIOtY1q'

Steady-state probabilities

po, p1

and

p2

that the two- channelsystem is in each of the three conductance levels 0, 1, and 2,

respectively,

can be derived using the transition rate matrix of the complete process. For Scheme 1 this can be reduced to six states (Colquhoun and Hawkes, 1990) cor- responding to one channel being in state i and the other in

state

j (i j

⁼

1, 2, 3), namely (1, 1), (1, 2), (2, 2), (1, 3),

(2, 3), and (3, 3), the first three for level 0, the next two for level 1, and the last for level 2.Afull discussion will be given elsewhere. This six-state model may beapproximated, in the case where (3changes to (3' when the other channel opens, by a three-state continuous-time Markov chain on the con- ductance levels 0, 1, and 2, with

2, fr Oa 1±I2

a 2a

(Scheme 2) Choosing (3 and ,' so thatthisapproximate process has the same steady-state probabilities as the complete pro- cess wefind

(3

=

(3q12/(ql2

⁺

q2J) O'

⁼

(3'qd(ql2

⁺

q21)

and

po

=

or/h, p1

=

2ac/h

and

P2

=

f3'/h,

where h = & +

2a$3

+ ,BB'.Notethata3= l/C, ,' = and a =

l/;LxIc

=

l/Lx,o.

SIMULATIONS

Two-channel superposition records were simulatedfor a se- lection ofsingle-channel Markovmodels,with andwithout interaction of the formdescribed above. The distributions

peraining

tothesix types ofsojourntime (Fig. 1)were es- timated from binned data Steady-state probabilities were computed as the ratio of the total duration of a particular sojourn type to the duration of the whole record. From all theseestimates, using Eqs. 9 and 10 and the predictedsojourn typeprobabilities, the decay rates and weights of the

expo-

nentials of thefour conditional density functions of Eq. 1 wereestimatedbynonlinearleast-squaresregression(Nelder andMead, 1965). Thefunctions

KC(t)

andK<(t)were then obtained from Eq. 1 by deconvolution andintegration.

For thetwo-statesingle-channelmodel(see Fig.2for ana- lyticalresults)witha=1.0 and,3=2.0

ms-1,

the simulated steady-stateprobabilitiesof zero, one, or twochannels being open are0.1105, 0.4442, and 0.4453 if the channels are in- dependent

((3'

= ,3, ^see Eq. 3). As expected, these prob- abilitiesshift in the presence ofcooperativity; for example, when(3' =3

ms-1

they are 0.0911, 0.3635, and 0.5454 (very close tothe theoretical values of

¼ii,

4/11, and6Yl, respec- tively).Asegment of the simulatedsuperpositionprocess for the abovetwo casesisillustratedinFig.4.Estimates ofKC(t) andK<,(t)obtained from thesimulated record areplotted in Fig. 5 (dashed lines) for the cases (3' = 3.0, 2.0, and 1.75

ms'l.

Inallcasestheyareessentially

indistingishable

from thefunctionscomputedanalytically usingEq. 4 from knowl- edge ofthe underlying single-channel parameters.

Wereturn inFig. 6tothe two-channel system wherean isolated single-channel model is based on Scheme 1 with

Kektsiaet]. 637

'Pco^.(ILX c IWAX c

Vodume 67 August1994

2.0

0.0 5.0 10.0

t(ms)

FIGURE 4 Actinuo segment(64 mu)of thesmulated ion

proces for thetwo-channeLtwo-statesystemofFig.2 witha=1.0,B1=

2.0 ms-'.(a)independentchannels(ie,13' = 2.0ms-') (b)positivey

cooeativechannels(13' =3.0 ms-u'

0.0 0.5 1.0

t (mIs) 1.5

FIGURE 5 FuntionsKc(t)andmK(t) (dentedaclectivelyKRt) here and

infolowing figures) est simuated data 50,000

sojo isfor eachof thesIxtypes,dawd lnes)and derivedanalytically

(fromEq.4,soiid lines)forthetw two-staesystemofFig.2with

a = 1.0,13 =2.0 ms-'.Curves fromabove downwardsrepresentKQ(t)

forI= 3.0ms- (posivecaopeivtyI =2'0s-2 (indepenlence) and13'=1.75 ms- (negative , respectivel.KO(t)=1for all

cases

parametervaluesgiveninFig.3, and with interaction via the

openingratefrom,B (= 0.115

ms-1)

tosome newvalue (' while theneighboring channel isopen. When (3' = 0375 ms-' (upper traces Fig. 6) the estimate for

Kc(t)

(dashed curve)isveryclosetotheanalytical solution (solid curve).

With negative cooperativity ((3' = 0.0575 ms-l, lower traces)

Kc(t)

is slightlyunderestimated atlowvalues oft,

reflecting difficulty in fittingfastexponentials and a rela- tivelysmal samplesize(collected byconsidering^sufficient ofas ladrecordsegment togiveatleast100sojourns in each of the six types). Asimilar diffculty is evident in

FIGURE 6 Interaction in atwo-channelsystemwhere eachchannel is modeledacacordng to 1,and the l openingrateis dependent entheneighboringchannelbeing cosed(whenit is13)oropen

(13').Usingparamete valoes giveninFig.3,Kc(t)andKt)were ted fr simulated data cosistingofaminimum of 100sojournspertype (dasd and doe lines)andbysotion ofEq.8(solid lies)forthecases

13' =0.375ms-u1[qper trce:KR(t)J 13' =0.0575ms-1 [Lower traer.

K&t)J

and13' ⁼0.115ms-1 [dotedtrace:Kc(t)]. KO(t)isgoupedaround K(t)= 1inall case&

Ke(t)

under independence (dotted curve), but in all ces

KO(t)

is well estimated (dashedlinesclosetoK(t) = 1).

Fig. 7presentssimulation results and analytical solutions for

Kjt)

andK0t) inaspecialcaseofthetwo-statemodel (seeFig. 5)wherethesymmetryof therate constantsunder closed(a = 1.0, (3 = 2.0ms-1) andopen(a' = 2.0, 3' =

4.0ms-')conditions renderssuperpositiondatainsensitive

to testsfor

nonindependence

basedonsteady-state

probabili-

tiesandbinomialtheory.However, estimation of

KJQ)

and

KO(t)

from simulated data (again with aminimum of100

sojourns)

resultedinwelldefinedfunctions^veryclosetothe analytical solutions.

2.0-

1.5

0.0 1.0 2.0

t (ms)

FIGURE 7 Interactio ⁱⁿa two-channel two-state systemwhere both

openingandclo;ingratesaredependentontheneigb channelbeing cosed(whentheyare 13anda,respectively)oropen(1' md a',seetext) Estimatesfo i teddata(iimof100sojmns,dahedlins)and

analyical soutions(solidlines)aregivenforKAt)(apertaces)andKO(t)

(kwertrces)

(a)

0-

(b)

2-

15.0

638 Boma

Kekehianeta. Prpertie ofInteracfingIon Channels 639

DISCUSSION

The possibility of interaction between ion channels is im- portant for several reasons. Independence or nonindepen- dencein this senseisrelevant in anyintegrativeapproach to understandingmacroscopic membrane phenomena based on single-channelbehavior. Conversely, this is equally impor- tant instatistical analysis of macroscopic signals for eluci- dation of underlying single molecular events; examples of this include interpretation of concurrent openings in patch clamp records, and membrane noiseanalysis. Inanyevent, thequestion ofcooperativity betweenchannels has consid- erableintinsicinterest inrelation to macromolecular prop- erties and transmembrane signaling ingeneral. Moreover, if cooperativity does exist, it might potentially be exploited pharmacologically.

Thereappears to beaccumulating evidence that interaction does occur in some systems. TIhisisusually inferred from some discrepancy between observed behavior and the pre- dictions of anindependenceassumption. Although this is an appropriateinitial step inanalysis, andsomeof these tests are quite sophisticated (Dabrowski et al., 1990), they are not necessarily conclusive orquantitative (Uteshev, 1993), nor baseduponpossiblemechanisms ofcooperativity,and there- fore cannot serve in modeltesting. The approach in this paper hasbeen to proposeasimple mechanism for interaction and to examine consequential properties of the observable process.

The mechanism studied here postulates that the isomer- ization involved in changingbetweenconducting and non- conductingstatesinonechannelcanresult inanessentially instantaneouschange inparticularrate constantsinaneigh- boring channel. If the two channels exist within a single macromolecularcomplex(that is,they areco-channels), this couldoccurthroughallostericcoupling.In thecaseof sepa- ratebutcontiguouschannels theeffect could be mediatedby electrical field alterations associated with conformational change. Theassumptionofinstantaneityisnotunreasonable, given the known time scale ofgatingisomerization in these molecules. It should be noted that the model need not be restricted to conditioning on one channel being open or

closed;

intheory,occupationof anystatecould alter behavior in the other channel, althoughrelating model results to ex- perimental datamay bemore difficult.

In the Markov framework used to model the single- channel process, it is natural toexpressinteraction in terms ofan alteration in specified transition rates, which them- selves relatetoidentifiable chemical kineticprocesses such asagonistbindingorchannelgating.If,as inexamples pre- sentedhere, these are transition rates within or outof the closed class, their alteration will affect the distribution of closed times.

Thus,

thedistributions of closed times condi- tionalon an eventof interest(here,theneighboringchannel beingopenorclosed)canbecompared,with their difference being some indication and measure of the interaction. It is convenient tomakethis

comparison

usingaconvolutionre- lation which provides a function

KC(t)

that quantifies this

difference; in the two- and three-state models examined where only the opening rate constant may change,

KC(t)

is lessthan, equal to, or greater thanunity in the case of nega- tive, zero(independence), and positive cooperativity, respec- tively.(Mhe function K<(t) is similarly derived and has analo- gousbehavior.) Using simulated data,

KC(t)

andK,,(t)have beenestimated fromsuperpositionrecordsbyoptimizedfit- ting. For the modelsexamined, these estimates have been almost exactly those predicted analytically in the case of large sample size (more than 50,000 sojoums/type), with only small deviations when using samples consisting of a miniimum of 100sojourns/type.

Although we have only considered some simple two- channel models ofinteraction, variousextensions wouldnot seeminsurmountable inprinciple. Allosteric couplingacross several co-channelswithinthe same macromolecularcom- plex is quite feasible. In the case ofmultiple spatially sepa- rated(but electrostatically interacting) channels, the possi- bility of time dependenceorofdistance-dependent delayor attenuation ofcooperativeresponsesneedstobeconsidered.

A further extension concerns L-type calcium channels, where it appears to be thelocalizedaccumulation ofcalcium ions following activation (rather than the conductance change per se) that mediates interchannel dependence (DeFelice, 1993).

We arecurrently developinga general theory for aggre- gated,interacting Markovchains applicableto morecomplex modelshaving larger state space and channel number than in theexamples presented here.

We thank ourcolleague Robin Milne for critical comments on the manuscript.

Financialsupport wasprovided bythe Australian Research Council.

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