Biophysal Journal Volume 67 September 1994 996-1006

Transport Properties of Single-File Pores with Two Confonnational States

Julio A. Hemandez* and Jorge FRschbargt*Departamento de Biofisica y Bioquimica, Facuttad de Ciencias, Universidad de la Rep6bica, 11200 Montevideo, Uruguay, andDeparftments of Physiology & Cellular Biophysics and Ophtalmoogy, Colege of PhysKians and Surgeons, Columbia Uniersiy,New York, New York 10032 USA

ABSTRACT Complex facilitative memfbrane bansporters of specific ligands may operate via inner channels subject to con-fornational trnsitions. To describe some properies of tese systems, we intrduce here a kinetic model of coupled transportof two species, L and w, through a two-conformational pore. The basic assumptions of the model are: a) single-file of, at most,n moecules inside the channel; b) each pore state is open to one of the compartments only; c) there is at most only one vacancyper pore; d) inside the channel, a moleule of L occupies the same positions as a molecule of w; and e) there is at most onlyone molecule of L per pore. We develop a general representation of the kinetic diagram of the model ffat is formnally similarto the one used to descnbe one-vacancy tansport through a one-conformational single-file pore. In many cases of bioogicalimportance, L could be a hydrophilic (ionic or nonionic) ligand and w could be water. The model also finds application to describesolute (w) tansport under saturation conditions. In this latter case, L would be another solute, or a tracer of w. We derivesteady-state expressions for the fluxes of L and w, and for the permeability coeffcints. The main results obtained frorn theanalysis of themodel are the following. 1) Undier the condition of equilibrium of w, the expression derived for the flux of L isfornally indistinguishable from the one obtainable from a standard four-state model of ligand bansport mediated by a two-conformatonal tansporter. 2) When L is a tracer of w, we can derive an expression for the ratio between the main isotope andtracer perneability coefficienft (PwIPd). We find that the near-equilibrium permeability ratio satsfies (n - 1) - (Pw/Pd:eqc n,a result prevxusly derived for the one-conformational, single-file pore for the case that n.-2. 3) The kinetic model studied hererepresents a generalization of the camer concept. In fact, forthe case that n = 1 (conesponding to the classical single-occupancycarrier), the near-equilibrium permeability ratio satisfies 0 - (Pw/Pdq c 1, which is characteristic of a carrier performingexchange-diffusion.

INTRODUCTION

Facilitative ransporters present in biological membranes areconsidered to be integral membrane proteins that operate viaconformational transitions between states. A classical modeldescnibes them as having the binding site for the transportedligand alternatingly facing the compartments on either sideof the membrane (Stein, 1986). Fig. 1 A shows the minimumfour-state model that describes such processes. This basicmodel or its modifications have been employed to accountfor the properties of many btasporters of ionic and nonionicligands (see, for instance, Stein, 1986; Andersen, 1989;Krupka, 1989, 1990; Daumas and Andersen, 1993; Denneret al., 1993) and of active ionic tansport systems of the typeof electrogenic enzymes (Lauger, 1980, 1984, 1991; Hansenet al., 1981).The model in Fig. 1 also represents the classical kinetic

description of carrier-mediated transport (Schultz, 1980;Stein, 1986). Thus, the term "carrier" has been adopted toaccount for the tansport processes mediated either by trans-latory molecules of the valinomycin type or by membraneproteins undergoing conformational transitions (Lauger,1980, 1984). In both cases, the main basic assumption im-plicit in the four-state scheme of Fig. 1 is that the transporter

Receivedfor publication 15 November 1993 and infinalform I June 1994.Address reprint requests to Dr. Jorge Fischbarg, Department of Physiology& Ophthalmology, Columbia University Col. of P.&S, 630 W. 168th SL,New York, NY 10032. Tel.: 212-305-9092; Fax: 212-305-2461; E-mail:fischbarg@Caccfaccc.columbiaedu.e 1994 by the Biophysical Society0006-3495/94/09/996/11 $2.00

offers a single binding site to the ligand, accessible from onlyone side of the membrane at a time (Schultz, 1980). Althoughthis scheme could constitute an accurate description of fa-cilitative tansport mediated by lipid-soluble translatory car-riers, it might not correspond with the actual mechanismstaking place in complex membrane proteins. In this case,ligand movement through an inner protein channel seemsmore realistic. Channel-like mechanisms have been sug-gested to operate, for instance, in mediated ionic tansport(Fr6hlich, 1988; Krupka, 1989; Hasegawa et al., 1992), inelectrogenic membrane systems (Andersen et al., 1985;Lagnado et al., 1988; Nakamoto et al., 1989; Hilgemannet al., 1991; Lauger, 1991; Gadsby et al., 1993; Rakowski,1993) and in tanwsporters of larger molecules, like choline(Krupka and Deves, 1988), and particularly in facilitativesugar transporters (Barnett et al., 1975; Lowe and Walmsley,1986; Walmsley, 1988; for a recent review see Baldwin,1993). In this latter example, kinetic and structural evidence(Mueckler et al., 1985; Jung et al., 1986; Alvarez et al., 1987)and data on water permeability (Fischbarg et al., 1990; Zhanget al., 1991) are suggestive of the existence of a hydrophilicchannel accessible both to sugar substrates and water mol-ecules. Similarly, urea and water appear to traverse a channelthrough the Cystic Fibrosis transmembrane conductanceregulator protein (Hasegawa et al., 1992).From the above, several transporters appear to operate via

complex inner channels, subject to conformational transi-tions and also accessible to water molecules. It seems rea-sonable to assume that this concept can be generalized to

99

Transport Properies of Sinje-File Pores

N4 - N11K 1L.

NS3 - N2

A BFIGURE 1 The four-state model of ligand btansport mediated by a two-conformational transporter. (A)The trasporter exhibits a binding site eitherto compartment e (form T,) or to compartment i (form T1). T,L, TEL: theligand-binding states, with the binding sites facing compartments e and i,respectively. (B) The equivalent description used to derive a steady-stateexpression for the ligand flux in Appendix 1.

most tansporters. We analyze here a kinetic model of fa-cilitative transport of two species (w and L) through a com-

mon two-conformational channel. We elect to treat thesimple case of a single-file channel. Species w is present innearly saturating activities, whereas species L is present insmall activities in the two compartments. In many biologicalcases, w could be water, whereas L would be a specific ionicor nonionic ligand. Because our purpose is to understandsome basic kinetic properties, we do not consider here theeffects of transmembrane electric fields.

In the first part, we introduce a procedure to lump tran-sitional steps into overall diagram components, thus simpli-fying the diagrammatic analysis of the model. We thenshow the general steady-state expressions derived for thespecies fluxes and suggest possible applications of themodel. To perform the analysis, we employ the King-Altman method (King and Altman, 1956) as modified byHill (1977).

In the second part, we discuss the conditions under whichthe present single-file channel model behaves as a four-statemodel for the transport ofL (like the one in Fig. 1). Crucially,we show that when w has equal activities in the twocompartments (equilibrium condition), the expression forthe steady-state flux of L is identical to that derived fromthe four-state model of Fig. 1 (Appendix 1; see also Stein,1986).

In the third part, we derive explicit expressions of thepermeability coefficients for w and L In particular, when Lis a tracer of w, we show that the near-equihibrium perme-

ability ratio (P,,IPd, (equivalent to the Ussing coefficient(Hille and Schwartz, 1978; Kohler and Heckmann, 1979))has lower and upper limits of (n - 1) and n, respectively, forany value of n (n - 1). These limits are analogous to thosearising from the analysis of the one-conformational one-

vacancy single-file pore (Kohler and Heckmann, 1979;Finkelstein, 1987, pp. 52-55; Hernandez and Fischbarg,1992), valid for n - 2. When n = 1, the single-file channelmodel analyzed here becomes the classical carrier perform-ing exchange-diffusion, characterized by a near-equilibriumpermeability ratio obeying 0 c (P|/PdOq c 1.

THE KINETIC MODEL

We assume that the transport ofw across a membrane sepa-rating compartments e and i takes place through a number Nof identical single-file pores that exist in two conformationalstates. When saturated, the pores contain a single file of nmolecules of w. The molecules occupy specific positionsinside the pore, numbered 1, 2, -, n. Each conformationalstate exposes a binding site to one of the compartments, whilesimultaneously closing the access to a binding site on theother compartment. To account for the transport process, weassume that vacancies are generated at the extreme positionsof the pore (positions 1 and n), as a consequence of therelease of molecules to the compartments. We consider herethe case that, at most, only one vacancy exists per pore. Forthe case of water, this assumption may be justified by thelarge water activity of biological compartments (Kohler andHeckmann, 1980). The one-vacancy mechanism has beenpreviously adopted to account for the single-file transportprocess in near-saturation conditions in ionic and waterchannels (Kohler and Heckmann, 1979; Schumaker andMacKinnon, 1990; Hernandez and Fischbarg, 1992).The pore is also capable of binding another ligand L re-

versibly to any of the inner positions of the pore. For thepurpose of this paper, we assume that a molecule of L oc-cupies the same positions inside the pore as a molecule of w.This assumption implies that, similarly to the molecules ofw, L is also restricted by the "no-pass" condition. No furtherstructural restriction is imposed by this assumption, becausethe binding of L to a particular position is, in general, de-termined by different rate constants than the ones corre-sponding to the binding of w to the same position. At aparticular position, w and L could bind to different chemicalgroups present at the channel walls. We also assume thatthere is at most only one molecule of L per pore. This as-sumption is valid for sufficiently small activities of L in thecompartments. As can be seen (Fig. 4), when n = 1 thesingle-file channel model becomes the classical six-state ki-netic model of countertransport (Stein, 1986).

In Fig. 2, we show the kinetic diagram of the model forthe case that n = 3. Because the number of states equals2(1 + n + n2), the steady-state kinetic analysis becomes verymuch involved for larger n s. However, the diagram of Fig. 2is suggestive of a general form of representation of the pro-cess, which permits us to derive some results without per-forming a detailed kinetic analysis. This general represen-tation is shown in Fig. 3 A. The parts of the diagramcontaining the vacancy states have been collected into "com-ponents," called Cw for the case that only molecules ofw arepresent, and CL (with i = 1, 2, , n-1) for the case thata molecule of L is also present. The superindex "i, indicatesthe position from which L is moved in the e to i directionwhen the corresponding component is traversed. Notice that,except for the case that n = 1, each component may betraversed through several different trajectories. In general,the component's state having the vacancy exposed to the icompartment is called state "E," independently of theparticular component. Analogously, the opposite state is

T* -4 -TjIL 1 I

To L _ - T L

Hemandez and Fischbarg 997

Volume 67 Septenrer 1994

' O 01_ -10 0 0oo

KAb ____,

ox 01o -- ox o

00 x 100 x

L_________ K_

0

1-00 00

*ox -_*ox xo Fo-xo

_z 1,K1 _1,xo -exo ,ox.-l Fox-

_ 1,bKl1 1<xeo_-Ix*o o-x15o10xOVe

00 0 1,F 00

FIGURE 2 The ompldete kinetic diyagm of the model for the cse thatn = 3. (0) Molecls of w-, (0) Molcules ofL; (X) Vaancie. Compo-nents C, and CL are eclosed by dashed recagles.

called "r. In the e to i direction, components are traversedfrom state E towards state I.The general diagram of Fig. 3 A resembles the kinetic

description of the one-vacancy, single-file transport processthrough a one-conformational pore (Kohler and Heckmann,1979; Hern&ndez and Fischbarg, 1992). The difference re-lates to the presence of the conformational transiton steps.Hence, the "components," containing the conformationaltransitions between vacancy states, substitute the linear se-quences of steps of ljumping" between sequential positionsof the one-conformational pore model. Also, conformationaltransitions between saturated states appear in the linear por-tions ofthe diagram. However, the steps conesponding to therelease and binding of L and w maintain their relative po-sitions. As a consequence of the similarity between the gen-eral kinetic diagrams involved, some of the results obtainedhere are analogous to those previously obtained by us fromthe kinetic analysis of a one-conformational pore model ofwater transport (Hernandez and Fischbarg, 1992).The diagram of Fig. 3 A can be used with advantage to

perform the steady-state kinetic analysis of the process. Weuse the di ammtic formalism developed by Hill (1977) toderive steady-state expressions for the fluxes ofL and w. Toperform the analysis, we disinguish the three "general"

cycles contained in the diagram ofFig. 3A. These three cyclesare called "a," "b," and "c" and are shown in Fig. 3B. InAppendix 2, we derive the general steady-state expressionsof the model.From (A5), we express the molar flux ofw for L4 = L =

0, taken positive in the e to i direction, by (A6) as

E(N)XE'P-J. = (We~- wj) (1)

Therefore, the permeability coefficient P, ofw is given by

= (N)Y.E'PPl E'I (2)

For the case that w is water, and considering the assump-tions of the model, we can identify P,, with the osmoticpermeability coefficient Pf of the membrane (or at least, withthe contnrbution of the system analyzed here to the totalpermeability):

Pw = Pf. (3)Because E' is afunction of the activities ofw in thecom-

partments, in this case P,, depends upon these activities. Asdiscssed previously (Hemnindez and FLschbarg, 1992), thisis a consequence of the vacancy mechanism responsible ofthe transport process, and it constitutes a typical property ofdiscontinuous difusion (Schultz, 1980).The molar flux of L is given by (A9):

JL = (N)Ra Ekp (WC-1 WiLc- WcW-14)+ -wcEjPj(wF-1iIE (4)

When we = wi = w, we obtain from (A12) the permeabilitycoefficient of L as a function ofw and of the actities of L:

=(N) (wnY.Ekpk +w '.cE j

(Er)w (5)From (4) we notice that the model implies that a nonzero

flux JL may be obtained for equilibrium activities of L, thatis for L4 =4, provided that wC and w; are different This isa consequence of the flux coupling between both speciesimplied in the model. In particular, this property provides uswith the possibility of experimentally testing models wherew (e.g., water) is also a ligand of the solute (L) hansporter.

There are two main situations of biological importancewhere the model analyzed here may find application: 1) fa-clitative hansport of a solute (L) through a path also ac-cessible to water molecules (w), and 2) tranort of an ionicor nonionic ligand (w) present in saturating activities. In thislatter case, L could be another ligand transported by the sys-tem and present in sufficiently small activities or a tracer ofw. In both situations, the accomplishment of the "no-pass"(or single-file) condition is required.

In the section "Validity of the Four-State Model" we ana-lyze the conditions under which the general model shown inFig. 3A exhibits the macroscopic behavior of a four-statemodel used to represent the tansport of L, similar to the

998 Bop Jua

Transport Poperties of Ske-FDe Pores

A

0.

*0.

FIGURE 3 (A) The general kinetic diagram of themodel The small blick circes represent pore states C,is a "c onaining vacancy states having wa-ter moculesonly;Cly , n-l e "components"c aining vancy states having the ligad mokacle

(see te for fuhr details) (solid lns) Actual indvidual t si (brok line) several mediatesteps, conssing of a rpeated sequence of componentsCL and trasiio. (B) The three genral cycles com-

prising the generI diagram of A- The symbols em-ployed here have the same me n s in that figurSee the text for the meaning of the geanral cycles

(a)

0.

B

(b) (c)

one shown in Fig. 1. Later, in the section "PermeabilityCoefficients," we use the general expressions shown aboveto analyze a more particular situation and also to discuss thecase where L is a tracer of w.

VAUDITY OF THE FOUR-STATE MODELIf n = 1 and only one ligand is present (either w or L), thefour-state model of Fig. 1 applies. For larger ns, and whenonly w is present, a four-state model is still valid as an ap-proximation, under conditions of the parameters that permitto collapse the transitions conecting the states having thevacancy in positions 1 and n, into a single transitional step.We now analyze whether the model may exibit a four-state

diagram behavior for the tansport of I, for the case thatn . 1 and when both w and L are presentUnder certain necessary conditions, a kinetic diagram may

be reduced to a diagam consiting of a lower number ofstates (HIll, 1977). For the case analyzed here, one possibilityis to consider that the vacancy states are transient interme-diates and, therefore, can be eliminated from the diagram.The reduced iagram now includes some of the original tran-sition steps with their original rate constants and also thereduced transitional steps corrsonding to the "reducedrate constants. These reduced rate constants are functions ofthe oiginal ones. In general, under this hypothesis, thenumber of states of the reduced diagram equals 2(n + 1).Therefore, only in the case that n = 1 we obtain a four-state

Hommidez and Fmd*arg 999

Volurne 67 Septenber 1994

reduced diagram (Fig. 4). In Appendix 3, we perform themodel reduction for the case that n = 1 and assuming thatthe vacancy states are transient intermediates. As it is shownthere, the necessary condition for reduction ((A17)) may beaccomplished for large activities ofw in the compartments.

Inspection of Fig. 4 B shows that the reduced model forn = 1 becomes similar to the four-state model of Fig. 1 if thereduced rate constants r14, r41, rT, and r63 are nil (or negli-gible). From (A19) we see that if t, = ti, = 0, then r14 =r4l = r,6 = r6l = r34 = r43 = r36 = r63 = 0. However, thiscondition does not make the reduced rate constants r,3, r3l,r46, and r64 nil. Hence, in this parficular example, because theoriginal rate constants uc,, u,c, s6, and se are not affected bythe reduction procedure, the model in Fig. 4 B becomes simi-lar to the model in Fig. 1. Therefore, for the hypothesis as-sumed, only for the case that n = 1, and under a severelyrestrictive condition, the model introduced here may be re-duced to the four-state model of solute transport. We mayconclude that this situation is extremely improbable, andthat the four-state reduced model does not represent arelevant particular case of the model of a complex trans-porter analyzed here.

Although, from the above, we cannot justify the standardfour-state model as a particular case of a general model oftransport ofw and L in complex membrane systems, we mayask ourselves whether the model introduced in this article isable to exhibit kinetic properties not distinguishable fromthose of a four-state model under some circumstances. In-deed, when w, = wi = w, the flux of L is given by (A12):

W (N) (WaEkk ± cLTh) (L-eI) (6)(ET)W

In this case (wc = wi = w = constant), every binding stepinvolving molecules ofw will be characterized by a pseudo-first-order rate constant j, given by

(Bj = bjw, (7)

where bj is the second-order rate constant ofbinding of stepj.

Under these conditions, the directional diagrams of themodel in Fig. 4 fall into four groups, those having ascommon factors L4L, L4, and Li, and those independent ofthe activities of L (Fig. 5). Therefore, the denominator(ET)W is of the form

(ET)s = R1L Li + R2Le + R3Li + R4, (8)

where the coefficients R ,., R4 are functions of w andof the rate constants.

Comparison of (6) and (8) with (A1)-(A3) shows that,under the condition of equilibrium of w between the com-partments, the flux (JL) is given by an expression formallyanalogous to the one derived for the four-state model of Fig.1. If the model analyzed here represents an actual hansportprocess, the coefficients of expressions of the form of (8) or(A3) should be functions of the equilibrium activity w. Inparticular, we may conclude that, for a transporter descnrbedby a solute-water (L-w) model having similar properties tothe one introduced here, kinetic experiments of ligand trans-port performed at a zero or very small water activity differ-ence between compartments can be interpreted in terms of asimple four-state solute model. However, this four-statemodel will not provide with an accurate mechanistic descrip-tion of the trnsport process.

Other particular situations simulating somehow a four-state model behavior may occur (for instance, involving par-ticular values of the intermediate rate constants), but they arenot analyzed here.

PERMEABILITY COEFFICIENTS

From (A5), we notice that the flux ofw is prtally coupledto the ransport of L If the activities of L are sufficientlysmall, the contribution of the coupling terms to the flux ofw may be negligible. Hence, we assume that the permeabilitycoefficient of w is, for all practical purposes, given by ex-pression (2), which was obtained under the condition thatL4 = L= 0. When we w, = w, we obtain the near-equilibrium permeability coefficient (P|),q

(6) j (1)

l l(5) X - X (2)

l1 l(4)ixi-~ 'L (3

(N)Y\E'PiW.)eq E'E,1\4

(6) j_ (1).1 t,rs

I I.'

!- I/ "' !,(4) 3 "-I (3'

__% _ _

(9)

where E' is a fimction of w.From (2), (3), and (9) we see that the effective ratio

P,,/(P)rq is a function of w, w1, and of the particular equi-librium activity w,

(10)

A BFIGURE 4 Kinetic diagram of the model for the case that n = 1. (A) Thekinetic diagram incding all the states The numbering of the states is usedto perform the model reduction in Appendix 3. (B) The corresponding re-duced model, under the n that the vacancy sates are transientintermediats. The sud lines represent the acua oiginA trntions, thebroke lines are "reduced ait a rizd by "reduced" rateconstants (see text).

and can be smaller or larger than one.The general expression for the flux of L in the model is

given by (A9) (or Eq. 4). If the activities of L are sufficientlysmall, to account for the following condition:

bLc Le < rL and b.4 <rLT,then Er is given by

(11)

ET = EEL, (12)

Bio~ Jotumal1 000

P,./(Pw),q = E'IEF 31

Herniez and Fucbarg Transpout Properfies ot SkRe-FIe Pores

A,.-i F -I

p~~~~~

I I

II

N f

,

I I

-A,8s W

N~~~~~

FIGURE 5 General repr of the diec-

tional digrams for the case ofequal water acivities mthe twoAMMtnts Tfhe symbols h;nre thesm

as in Fig. 3. The symbol ([) repesents a com-

plee in of the diagram- In the linear seg-ments, such asiuption occs when a sngle tran-

sitioal step is absent, in the ceanxnents it ocams byinteruIg all the possibe tajectoies between statesE and I by ex ng the Immmimmnumberof traiioa steps (A) GenerAl schemes of the di-

rcional diagrns id ofthe lgand actities.

(B) Anaogs schemes for diretional diagrms co-tainin L,4L,,,and Lascommonfactrs

N~~~~

IfI

N~~~~~I I

I I

1F1

,_\ _\I

I I

I I

III,~~~~

N /1

(L )N1-

,_i F-N

f

II I

I

)~ ~ ~ ~ ~ ~~1

(L-

I

p

N-

, F -11N

I

i

x~~~~~~~~

BN

I I

I I

\N

"I .1

t o,II I

I I

I \

I I

IN /~~~~~~~~~

,-I t-NII

I I

Nle II I

I II

N

,-I

I I

I I,~~~~ I- I

N, /

(N e i)

L -)I

where EL is the sum of all the parts of the flux diagrams thatfeed into cycle b and is a function of the activities of w. Forthe casethatw = w. = w,

(Er)w = E', (EL)wUnder condition (13), expression (5) becomes

(N) (WEY kPk + -Ejpj)E (EL)w

From (9) and (14), we express

(WpaEkpk+W EcEEPj)LPO ( w Cq (EL)wEb EPjP

(13)

In this case, the following identities between the rate con-stants and component terms occur (see Appendix 2):

bwe = bLx; bwi = bw-; r = rL.;

r,- = ; Sci = Uci; sie uiu;

Cw,jO= C4,. (for alli=1, 2, -,n -1);

Cw,,> = C"Li, (for alli =1, 2,- ,n-1);(17)(14)

and

(15) D, =DL (foralli= 1,2, ,n-1).

If L is a tracer of w having diffusive properties identicalto those of common w, we can identify (PJ) with the dif-fusive permeability of w, Pd:

The terms DL represent sums analogous to DW (seeAppendix 2) provided by the particular componens C

Under (17), (EJ. is given by

(16) (EL) = n (Hw)W-I Q + (n- 1) ,)w2w',

1001

Z,-l F-"'

I II IIx

N

"I"

(POW = Pd - (18)

Volme 67 Septerrer 1994

where

Q =sicrjW +sirWi + r.Cr.,iand (19)

S= bw,eb,si sra, r,,,D,.,and where H,, is defined by (A7).

Using expressions (A74 (A8) (A), (All), (18), and(19) and conditions (16) and (17), we obtain from (15) thenear-equilibrium ratio of the permeability coefficients of w

(P)q HnQ + (n-1)wS (20)Pd HWQ + Ws

For the case of water, this expression gives the ratio be-tween the osmotic and difsive permeability coefficients,(P|)./Pd.From the inspection of Eq. 20, we see that (n - 1) '

(P,).Pd < n, a result encountered for the one-conformational single-file pore (Kohler and Heckmann,1979; Hern&ndez and Fischbarg, 1992) for the case that n '2 (for the case that n = 1, the one-conformational single-filepore has a permeability ratio that always equals one). Asalrady commented, this result is a consequence ofthe formalsimilarit between the diagram of Fig. 3A and the diagamsused by those authors to descibe single-file htanportthrough the one-conformational pore. From (10) and (20), werealize that, depending on the activities of w, the ratio PW/Pdcan be larger than n, a property attnrbutable to the discretenature of the kinetic process and already found for the caseof the one-conformational single-file pore (Henrmndez andFischbarg, 1992).As already mentioned (see "Validity of the Four-State

Model"), for the case that n = 1, the model analyzed herebecomes the classical description of a single-occupancy car-rier able to perform countrtraport oftwo different ligans,w and L (see Fig. 4A). Therefore, in this case, ifL is a traof w, the near-equilibrium permeability ratio (Eq. 20) sat-isfies 0 ' (P,w)q/Pd ' 1, a result already derived for carriersperforming "exchange diffusion" (Liuger, 1980; Schultz,1980). Hence, the dassical carrier constitue a particularkinetic case (the case that n = 1) of the two-conformationalmulti-occupancy channel analyzed here. As previously dis-cussed by Liuger (1980, 1984), "carrier-lIke" behaviors oc-cur as limiting cases of single-occupancy channels withmany conformational states, in the presence of high activa-tion energy bafriers for the binding steps, that determinethe condition of being "cdosed" on one side of the pore at agiven time.

DSCUSIONThe model introduced in this article represents an exampleof the kinetic descrption ofcoupled tansport of two speciestirough a two-ofmational btansporter. As already men-tioned, the purpose of this article has been to provide withsome basic kinetic aspects ofa plausible mechanism of trans-

port through complex membrane proteins and not to mod-elize a particula case of mediated tansport As it is, themodel analyzed here could constitute, for instance, a basis toreresent single-occupancy transport of inorganic ionsthrough an aqueous channel undergoing conformational tran-sitions. In this case, the application of the model requires theintrduction of the electrical effects on the rate constants, aproblem under discussion (see, for instance, Cooper et al.,1988; Dani and Levitt, 1990). As already suggested here, themodel could also be employed to reprsent ionic ttansportunder near-saturation conditions. In this case,wwould be thesaturating ion, and L would be another ion present in lowactivities, or a trac of w. As in the previous case, the elec-trical effects should be introduced. Conceming the possibleapplicio of the model to btanpoters of hydrophili li-gands of larger size, such as amino acids or sugars, we offerthe following eculations. Of all the mptions of themodel, the fat that each conformational state is open at anygiven time to only one ofthe compartments seems a plausiblep rty oftis type of transporters. Altgh it is possible,for instance, that the glucose transporters operate by meansof an inner hydrophilic channel, there is no evidence avail-able to support the idea of a single-file of water moleculesinside it. In fact, considering the dimensions of the glucosemolecule obtainable either from a graphic simulation (ap-prox. 9.2 x 6.0 X 5.2 A in Van der Waals size) or fromdiffusion coefficient measurements (ILngsworth, 1953;Stokes-Einstein radius: -3.2 A), compared with the dimen-sions of the water molecule (--1.4 A Van der Waals radius),one would expect more than one file of water moleculesinside such a channel. In this type of situation, schemes stillmore complicated than the one analyzed here should be de-veloped to represent the transport processes.One ofthe main conclusions ofthe analysis presented here

is that, if a given solute tansporter is also able to bind watermolecules in the same path available for the solute, the mod-els developed to describe the process need to inchde wateras a second ligand. In this connection, under the condition ofwater equilibrium, thek i expression derived here for theflux of ligand (L) (Eqs. 6-8) is formally stin lefrom the one obtainable for the minimum four-state modelof solute tanport twough a two-conformational htansporter(Eqs. A1-A3). If this standard four-state model descriptionis adopted, and the tlsporter is actually performing trans-port of both solute and water, the rate constants of the four-state model will be misinterpreted as true fist-order rateconstants independent of the water actities, whereas theycould actually correspond to expressions of the rate constantsof a more complicated diagram involving water as a ligand.It is intesting to note that, because of the similarity of dia-grams commented below, an analogous conclusion could bereached from the analysis of the tansport process of a ligandL mediated by a one-conformational, single-file pore underequilibrium saturatig acivities of w (Hernindez andFischbarg, 1990). These result suggest that the analysis of

1002 RBnp Aoua

Transport Properties of Sixle-Fie Pores

kinetic data about species fluxes is not sufficient to dis-tinguish between a simple four-state-carrier mediated trans-port and more complicated mechanisms, such as one-conformational or two-conformational single file pores.The general scheme developed here to represent the trans-

port process through the two-conformational single-file poreis formally similar to the one employed in the case of theone-conformational single-file pore. As a result of thissimilarity, the kinetic expressions derived here for the per-meability coefficients are formally analogous to those ob-tainable from the analysis of the one-conformational single-file pore under near-saturation conditions (Kohler andHeckmann, 1979; Hernandez and Fischbarg, 1992). Corre-spondingly, we also derive here the noteworthy result that,analogously to the one-conformational single-file pore withn -2, the near-equilibrium ratio between the main and tracerpermeability coefficients of the saturating speciesw satisfies(n - 1) ' (P)eq/Pd ' n.An important conclusion of the present study is that the

model analyzed here represents a generalization of a carriersystem, and that the classical carrier able to perform coun-terransport oftwo different species, and to exhibit exchange-diffusion of the main and tracer isotopes of a single species,constitutes a particular case of the two-conformational pore.This result contnrbutes to the concept that classical carriersand channels are particular cases of more general types oftansport systems, an idea previously advanced by Lauger(1980, 1984, 1987) and confirmed for the case of ionic trans-port by using physical models (Berry and Edmonds, 1992,1993) and by a theory combining the Poisson and Nermst-Planck equations (Chen and Eisenberg, 1993).

Concerning the possible application of some of these re-sults to the problem of water tranort, it has been demon-strated that part of the water movement across biologicalmembranes takes place through diverse protein channels ortransporters. Most of such movement takes place acrosschannels specific for water (e.g., the CHIP28 water channelprotein (Preston et al., 1992); the WCH-CD water chan-nel protein (Fushimi et al., 1992); the CHIP28k waterchannel protein (Zhang et al., 1993); the tonoplast intrinsicprotein (Maurel et al., 1993), but some movement also takesplace across other membrane proteins (the nicotinic acetyl-choline receptor channel (Dani, 1989); glucose tansporters(Fischbarg et al., 1990; Zhang et al., 1991); the CFTR (Ha-segawa et al., 1992)). It might be necessary to employ kineticanalysis to interpret the permeability properties of such com-plex tansporters. A detailed kinetic analysis of the processesmediated by these systems can become rather involved. Onthe other hand, it is possible that the use ofgeneral techniquessuch as the diagrammatic method employed here will resultsufficient to understand some basic aspects of solute andwater transport. Our use in this article of the algorithm de-veloped by Hill (1977) to obtain conclusions from the analy-sis of the simplified scheme of Fig. 3 constitutes a successfulexample of this type of approach.

Supported by National stitutes of Health Grants EY06178 and EY08918,and in part by the Juvenile Diabts Fonai, and by Research to PreventBlindness J. A. Hernindez held an International Research Scholaship fromResearch to Prevent Blindness. We thank Dr. Carlos de los Santos for hisassistance with molecular graphics simulao

APPENDIX 1

Steady-state solution of the four-state model

We derive here a steady-state expression of the ligmd flux for the modelof Fig. 1 usine. as in the rest of the article, the diagrammatic method de-veloped by Hil (19T7). Our purpose is to compare this exp n with thatobtined in the section "Validity of the Four-State ModeL" Other analysisof this model have aleady been perforM (see, for instance, Stein, 1986).

For the analysis, we refer to Fig. 1 B, where N1, -, N4 represent thesttes of Fig. 1 A. We assume the consevaonm dition, that

is, that the total number of tranotes (or mols) per umit area (N) remam sa nt (N) = (N1) + +(N4) = constant, where (N) design thenumber of tansporters (or moles) per unit area in state N, The rate constantk determnes the transiton between states N1 and N in the i toj direction,and ,- determines the transition in the opposite, j to , decion (i,j = 1,2, 3, 4). Because there is only one cycle, only transitions between sequentialstates in the cycle are taking place. k2 and k, are second-order rate con-stants; the rest are fit-order rate a t The lignd ativities in thecompatments are desiged by Le and L;. Tbe steady-sate ligand fluxJL isthen given by the cycle flux:

JL= [(N)HIA] (L. - 4), (Al)

where, from the condition of detailed balance,

tI =k12k3 k4 k4l = k14 k43 ka kM I (A2)

and where A is the sum of all the dirctional diagrams of al the states ofthe modeL Takig the ligand acvities as common facis, we expressA by

A = A1L,L + A2L, + A34 + A4,

A1 = k2k43(k3+2)A2 =k2[14(kn+ k32 +k C+,) +kAklA3 =k [2[k4l (k23 + k32 +k2 ) +kB3a4]

(A3)

(A4)

and

A4 = k4 [k4 (k23 + k) + k32k2k] + k14 [k21 (k32 + k34) + k23k41

APPENDIX 2

Steady-state sution of the two-conformaftnlsingle-file pore model

We derive here the general epression of the stedy-sate fluxes of L andw for the model of Fig. 4. The folbowing symbols r esent b,,, b-,: rateconstants of binding w from compartments e and i; rW ,, r,i: rate cnstantsof release ofw to comprtments e and i; bl,, b,.: rate constants of bindingL from compartments e and i; rL,, r,: rate ontants of release of L tocompartments e and i; sa, s,,: rate constants ofthe cformaional tanionof pores comletely filled with molecules ofw, "transporting the open sitein the e to i and i to e directions, respectively-, ua, u,,: rate constants of theconformaional transitions of pores without vacancies and having the ligandL in any position, "ansporting" the open site in the e to i and i to e di-recions, respectively; we, w,: activities ofw in compartments e and i; L,,4:acdvities of L in compartments e and i; (N): total number of pores per unitarea of the memlbrae.

Hemandez and Fischbarg 1003

Volume 67 SepWnter 1994

The plicaion of the diarammic method to the scheme of Fig. 4allows one to derive general expressins for the fluxes ofL and w. The fluxof w, J, is given by

J. = -Nw,)bEiP (we-wj)

the ones defined for component C,, differing between them as a con-sequence of the different position of the ligand L inside each componenLThe term D. is the sum of all the appendages generated inside C,,feeding into the general cycle a via states E and I.

When w. = w; = w, JLbe1mes

JL=(NXw P. + w EE4

2L,

+ (n - 2)aE.kPk(w -w,L. - w,wF' 4L) (A12)(A5)

+ (n -l1)XE,P,1wC'- -w LI)l

where Er is the sum of all the directional diagrams of all the statesof the model, and where the terms IEPh represent sums, taken overthe general cycles a, b, and c of Fig. 6, of all the cycle fluxes EWP.(h = i, j, k) determined by each general cycle. In expression A5, thecondition of detailed balance has been applied to all the cycle fluxes.Hence, P. represents the product of the rate constants in any of thedirections, for the particular cycle h. E is the sum of all the parts ofthe flux diaams that feed into cycle h.

When k = 4 =0, the flux of w is determined by the singe generlcycle b

(A6)

where F is the sum of all the diectional diagrams of all the staes corre-sponding to the gneral cycle b only, and where 4EYPV is the sum of all thecycle fluxes determined by the general cycle b. For the deom_itor E, theappendages to the cyces only cmonect the states contained in the generalcycle b (E1). In relation with this, it can be demonstated f the kineticanalysis that E,' = E,/EL for every cycle i in the general cycle b (seeexpression 12)We define

H,, = E-P'P- (A7)

where (Er) is Er under the conditon of equal activity of w m the com-par

The expessions derived here are general, except for the fact that theIrabnal constants Ma and u, have been idered indennt of theposiion of L inside the pore.

APPENDIX 3

Reduction of the model for the case that n = 1

We perform here the reduction of the model in Fig. 3A, under thecondition that the vacancy states are trasient intermediates, and forthe case that n = 1. The methodology used here follows the one dis-cussed in Hill (19T7). For the analysis, we refer to Fig. 4. The inter-mediate pore states are designed as N,, N2, - - -, N. xacoding to thenumbering of the states shown in that figure. The rate constants t,, andt,, correspond to the conformational transitions of the vacancy-containing pores "transporting" the open site in the e to i and i to e

directions, respectively.The necessAry cAndition for redution is that, under any shuation,

P(N2) < P(N1), P(N3) and P(N3)<P(N3), P(N6), (A13)From the detaled balance condiion,

H., = b.,s.c r,,,C,xc = b.s,.S r.,.C.,a, (A8)

where C,,. and C, are the sums of all the prodts ofunint di-i trajectries inside C,, and their cArre dg ap es feeding

into them from all the states inside the componen, from the E to I and I toE states, repectively.

TIhe flux of 1 JL, is given by

L= (NH[aEP4k(Wr'WiL. - w.w7'4)

+ I. EjPj (w -L,-wF'4)YLEV,where, from the detailed baance condition,

where P(X) repesents the p (fequency) of state X in the poreensembk.

From condiion A13, we umethat states N2 and N5 always approxi-mately ac -bmpL

d(N2)/dt = d(Ns)/dt = 0. (A14)

From thece snons obtainable ford(N2dtand d(NYdtfmthe modelin Fg 4 A, and after some alg4eb, condion A14 lends to

[r-.(N,) + rL,(Ns))ta + bL,,L, + b,,1w,) + t4jrL(N) +(NO = D)t )+ ..N)

and

YaEIPk (AIO)

= hLj (Ui,)" (r,, r 1(b,," )^' rLCC .. CZ2C*Lt1bs,s. r,.D,

= bL,(a )" (r.,j1' (b,,)' rLiC,C,,C----C-L2 C':bs,s,,r.,D.ad

ICEJPj = bL,(u .A(r,,)-'(b,,r -rLC' C2

... C2Ca-L,Cjjr +S,ir,,.] (All)

=L ("it;r(,c o LCC

'.' 'CLaeeC .[r r,rj +s6r.,j +sir.,,;

The terms IA£EPk and I,EJPJ are sums, taken over the general cyclesa and c, similar to the sum defined above for the general cycle b. Theterms CL,,, --, Ca,,, and CL,,--- C.,CI represent sunms analogous to

[rL.,,N4) + r,, (NQ()j(ti + b,4,j + b,jw,) + t,jr,,j(N1) + rLW3)]D

(A15)

D = tea (b,,w, + bo) + tfr(bw,,wC + bL,,) + (b,,w, + bL.,liXbwc + bL,4).

(A16)

From (A15) and (A16), we recognize that condition A13 is achieved if,for intance,

(A17)

We proceed through successive substitutions to obtain, from(A15), expressions for d(N6)/dt, d(N3ydt, d(N4ydt, and d(NVdt asfunctions of (N,), (N3), (N4), (NJ, and the rate constants. We show

1004 R BpmJua

iw (N)Y-bE.pi (W.-

Wi)9E'

b.,w., b..iwi > r.,., r.,i, rL,, rLj

Hen and Fisdbarg Transport roperties of SWVe-FIe Pores 1005

the expressions obrained:

d(Nt) = [r3lwi(N3) - rt34(N1)] + [r4lwi(N4) - r14L,(N1)]

+ [(rtlwi + siXN6) - (rl,,w + sk)XNI)l

dt)=-r 4(N,) - wr3w(N3)I + [r6L1(N*) -r3w,(N3)]+ [(rL434 + Ui)(N4) - (r34L + uie)(N3)1

d(N4)= [r14L(N1)-r4lwi(N4)] +-rL 4(N6)-r,w,(N4)]

(A18)+ [(r344 + ui)(3)(r43L + uXN4)

and

dt - [r= w r(N3)-r.3L(N,)] + [r,we(N4) - r.L4(N6)]

+ [(ri*w, + s)(N1)N-(r61wi +

In these expressi, the riis represent "reduced" rate constants, deter-miig the tnsitions shown by broken lines in Fig. 4 B. The meaning ofsubindex "ij" is the same as in Appendix 1. The reduced rate constants areexpressed in terms of the original rate constants:

r = rw.b,(b.,w, + boL, + t.) _ rbjb...w. + bLL, +r,+ )D D

=r,_

. rlb

-r_

t

=

rbt=r6,14- D r41- D ' 16- D D

______ _______ =rUjb.,t. r.r,wsb,tfD r43 = =

D D

r4= rL'b.(b.w.+ +r4 )ft ~D (A19)

and

rWc bL,(b., w, + bw,4L + t.)D

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