| HOME | HELP | FEEDBACK | SUBSCRIPTIONS | ARCHIVE | SEARCH | TABLE OF CONTENTS |
* Department of Biophysical Chemistry, Biocenter of the University of Basel, Switzerland; and 
Institut de Pharmacologie et de Biologie Structurale, Université P. Sabatier, Toulouse, France
Correspondence: Address reprint requests to Dr. Mathias Winterhalter, Institut de Pharmacologie et de Biologie Structurale, 205 Rte de Narbonne, F 31077 Toulouse, France; E-mail: Mathias.Winterhalter{at}ipbs.fr.
 |
ABSTRACT |
|---|
 TOP ABSTRACT INTRODUCTIONTHEORETICALEXPERIMENTALRESULTSDISCUSSIONCONCLUSIONSREFERENCES |
|---|
|
INTRODUCTION |
|---|
|
TOPABSTRACTINTRODUCTIONTHEORETICALEXPERIMENTALRESULTSDISCUSSIONCONCLUSIONSREFERENCES |
|---|
; Boos and Shuman, 1998). In many of these studies the respective protein channels had been reconstituted into artificial black lipid membranes (BLM) (Benz et al., 1987; Nekolla et al., 1994; Winterhalter, 1999; van Gelder et al., 2000). The penetration of substrate could be measured by means of electrical conductance experiments owing to the inherent blockade of ionic current. More recently this was even extended to noise-analysis methods applied to current fluctuations observed with single protein molecules (Bezrukov et al., 2000; Kullman et al., 2002). In such a way, stationary as well as dynamic properties of the underlying molecular mechanism have been established. The results clearly revealed a very pronounced dependence on the applied voltage, i.e., the electric membrane potential.
The available evidence calls for a theoretical basis describing the thermodynamics and kinetics of the underlying substrate-specific binding that could then be utilized to derive the translocation rate from pertinent experimental data. Naturally, the binding site should be accessible from either side of the membrane. The pertinent kinetics of those two pathways must therefore be expected to feature (at least) one individual activation barrier for each of them. Such a two-barrier kinetic model was proposed some time ago with regard to sodium channels in nerve, but was not worked out in detail for general use (Woodhall, 1973). Also, the application of a one binding site/two barrier model was briefly discussed in the case of maltoporin translocation data (Benz et al., 1987). However, this did neither actually consider different barrier heights nor the nature and quantification of the voltage dependence.
In the present article we develop a comprehensive theoretical analysis of a fundamentally general model. It gives full quantitative account of experimentally observable parameters. Then the real net translocation rate can be evaluated on the basis of measured numerical results. The voltage effect is attributed to specific reaction dipole moments occurring in the extremely large electric field of the membrane (Schwarz, 1978). How this may be utilized in practice will be tested and demonstrated with available data for maltohexaose permeation through maltoporin (i.e., LamB) channels of E. coli being reconstituted in BLM.
|
THEORETICAL |
|---|
|
TOPABSTRACTINTRODUCTIONTHEORETICALEXPERIMENTALRESULTSDISCUSSIONCONCLUSIONSREFERENCES |
|---|
The adequate reaction scheme with the respective "on" and "off" rate constants is then written
 |
 | (1a) |
 | (1b) |
General kinetics
An open channel can be observed through its ionic conductance. Since bound substrate is assumed to result in zero conductance of the channel concerned, the measured ionic current I would be proportional to the number of open channels, No = (I/Imax)NP, where Imax denotes the current in the case that all NP channels are open. Accordingly this No may be analyzed regarding its reaction kinetics. Following standard procedures, the appropriate rate equation is formulated as
 | (2) |
c''. Thus dNo/dt can be taken to be virtually zero to calculate the quantitative conditions in the prevailing intermediate stationary state (known in classical kinetics as the Lindemann-Hinshelwood approach, in biochemistry being applied to enzyme catalysis by Michaelis-Menten). This leads to a flow-equilibrium where
 | (3a) |
 | (3b,c) |
 | (4a,b) |
The stationary state probability (i.e., fraction) of open channels accordingly turns out to be
 | (5) |
subject to the relation
 | (6) |
The translocation rate
Under stationary-state conditions and c' > c'', a net transport of substrate can be expected to proceed in the negative direction, i.e., from cis to trans (due to a negative gradient of the chemical potential). It apparently becomes equal to the trans-off rate minus the reverse trans-on rate. Thus the resulting increase in the number of substrate molecules N''S on the trans side when calculated per channel becomes
 | (7) |
In the case of c' < c'', net transport naturally occurs in the opposite direction, which is described with prime instead of double prime superscripts in Eq. 7. Only for c' = c'' can no net flow of substrate take place—that is, of course, the state of the actual thermodynamic equilibrium.
Taking into account Eqs. 3–5, a general relation for the translocation rate can be formulated in terms of a conventional law of permeation with a permeability coefficient P. One obtains a flux J (substrate molecules per time) through one channel in the given positive direction
 | (8a,b,c) |
c = c' - c''. Thus a negative c causes a positive flux whereas a positive c results in a negative flux, i.e., one that runs in the reverse direction (i.e., cis 
trans).
Voltage dependence
According to basic principles of thermodynamics, an electric field strength E is a variable of state, just like temperature and external pressure. Equilibrium constants therefore depend on E as described by an appropriate van't Hoff equation
 | (9) |
M, i.e., the change of the overall dipole moment parallel to the field for a molar turnover of the underlying chemical reaction. Actually the effect is absolutely negligible in ordinary nonmembrane systems since M and especially E are usually much too small. The situation changes into larger orders of magnitude, however, when it comes to reactions of oriented molecules in a biological membrane (Schwarz, 1978). Here one has E = -V/d, with V being the applied voltage (positive if the electric potential is higher on cis) whereas d is the effective channel length regarding the voltage drop in the membrane. This field would readily become extremely large. Also M may take on rather substantial values. Thus a voltage effect could certainly be a significant factor in the present binding reaction. By integration of Eq. 9, one obtains an exponential dependence of K on V according to
 | (10a,b) |
m = mB - mO, i.e., the change of the channel dipole moment in the positive direction upon binding one substrate molecule.
An analogous relation applies to any elementary rate constant (i.e., one related to a single activation barrier) (Schwarz, 1978), namely
 | (11a,b) |
m
in the transition state (i.e., on top of the activation barrier). Our approach is simply empirical, being based on the Arrhenius equation with i), some preexponential factor k° subject to experimental evaluation as an adjustable parameter, and ii), an activation energy that changes proportional to the membrane potential in analogy to the thermodynamically derived equation for the equilibrium constant (Eq. 10) It stands to reason that in any case there must be an activated state to be crossed. We do not propose any structurally founded theoretical relation for the two relevant parameters. They are to be determined from the measured data. These obviously confirm quite well our predicted relationships, particularly as far as the voltage dependence is concerned.
Conformational change
So far it has been assumed that the channel itself is not affected by the applied voltage. However, the effective field could possibly induce a conformational change that alters the binding affinity. In the simplest case this may be a one step transition from a basic open state (o) to another one (•) whose ionic conductance as well as binding affinity is different. The equilibrium constant K• = N•/No would naturally be subject to an analogous field effect as expressed by Eqs. 9 and 10 (then M is the change of the molar dipole moment in the course of the structural transition). This phenomenon does of course complicate the general kinetics of the open channels if binding of substrate and structural change proceed at comparable rates. For the sake of simplicity, we do now only consider the case that the transition goes much faster than the binding reaction. Then the former process has always established an instantaneous equilibrium with N• = K• · No in the time range of binding. Also the converted state • is assumed to have completely lost its binding affinity. However, it may nevertheless still have an ionic conductance that is by a factor 
different from the basic state. Under these circumstances, the number of apparently open channels, being actually measured using the ionic conductance method, becomes equal to N = No · (1 + K•). This must be taken into account to derive the properly extended rate equation for N. As is easily seen, Eq. 2 may simply be multiplied by the factor (1 + K•) while the number of channels without bound substrate, namely No · (1 + K•), is substituted for No in the koff term. So one arrives at
 | (12) |
Now the conductance approach including noise analysis results in an apparent equilibrium constant of binding
 | (13a) |
 | (13b) |
 | (14) |
|
EXPERIMENTAL |
|---|
|
TOPABSTRACTINTRODUCTIONTHEORETICALEXPERIMENTALRESULTSDISCUSSIONCONCLUSIONSREFERENCES |
|---|
Materials and Methods
BLM experiments
Planar lipid bilayers have been prepared of diphytanoyl phosphatidylcholine (Avanti Polar Lipids, Alabaster, AL) according to the technique of Montal and Mueller (1972). They are formed across a 60-µm diameter hole in a 25-µm thick Teflon film (Goodfellow, Cambridge, UK) being sandwiched between two Delrine chambers, each containing 2 ml of a freshly purified aqueous solution (1 M KCl, 1 mM CaCl2, 10 mM Tris buffered to pH = 7.4). A small magnetic stirrer was placed in each compartment to quickly homogenize the solution after adding a new substance. The whole setup was shielded from external electromagnetic fields as well as from vibrations to minimize irrelevant membrane current fluctuations. The Delrine cell was enclosed in a double isolated Faraday cage connected with the signal ground and also with a homemade acoustically isolating closet placed on a piezo-electric vibration isolating table (model Elite 3, Newport Corporation, Irvine, CA). The quality of the bilayer membranes was checked by capacitance and residual conductance measurements. The capacitance of the whole system proved to be 
150 pF. The residual membrane conductance (<7 pS) was subtracted from the overall conductance.
The apparatus had been connected with the external circuit through a pair of homemade Ag/AgCl electrodes encased in 200-µl pipette tips filled with 5% agarose soaked with 1 M KCl during the fabrication process. The electrode on the cis side of the measuring cell was grounded whereas the other (on trans) was connected with the headstage of an Axopatch 200B amplifier (Axon Instruments, Foster City, CA) in the voltage clamp mode.
The purification of maltoporin has previously been described in detail (Dumas et al., 2000). Small amounts of wild-type maltoporin from a 0.1 µg/ml buffer solution with 1% octyl-POE detergent (Alexis, Lauchringen, Switzerland) were injected into the cis-side compartment. Incorporation of maltoporin into the bilayer was promoted by applying a transmembrane voltage of 100–200 mV and by stirring for a few seconds after addition. A single maltoporin molecule in the bilayer could be kept stable for several hours without any significant change of its physical properties. The concentration of maltohexaose (Senn Chemicals, Basel, Switzerland) was adjusted by adding appropriate small amounts of a concentrated stock solution. The exact sugar concentration was then determined by means of optical polarization measurements (PerkinElmer 241) (PerkinElmer, Wellesley, MA). This revealed a nonnegligible surplus of the commercially quoted amount (up to 25%). After sugar addition, the aqueous solution was homogenized by stirring during a few seconds. Then signals were recorded 20 min later. Titrations for the whole range of applied voltages were carried out with the same maltoporin molecule to avoid possible divergences between different individual single protein molecules. All the measurements were performed at room temperature. The applied transmembrane voltage refers to the potential on the cis side relative to the trans side.
Data acquisition and analysis
The data were filtered with the low pass Bessel filter of the amplifier at 2–5 kHz and then monitored with a LeCroy LT342 digital storage oscilloscope (LeCroy, Geneva, Switzerland). The resulting standard deviation of noise after amplification was 1 pA. The entire experiment was recorded on videotape with a digital-type recorder (DTR-1204, Bio-Logic, Claix, France). The averaged power spectrum of the current noise was recorded using the fast Fourier transform module of the oscilloscope. The data were sampled at a rate reflecting a Nyquist frequency of 12.5 kHz. With these parameters, the frequency interval between two successive points in the power spectrum becomes 2 Hz. To overcome the additional noise sources mentioned above, the background spectrum of the membrane without sugar was subtracted from each individual spectrum. The curve fitting was carried out using the Marquardt-Levenberg method. The total current recording was transferred to a personal computer via a GPIB card using the graphical program LabVIEW 4.01 from National Instruments (Austin, TX). Statistical analysis of the blockade events was performed by means of Bio-Patch analysis (Bio-Logic, Grenoble, France) and further homemade software.
Sugar-induced current fluctuation analysis
Earlier studies have shown that sugar binding reduces channel conductance in a concentration-dependent manner (Benz et al.,1986, 1987). Based on this observation, the pore conductance can be taken as a probe for the apparent equilibrium constants of sugar binding to maltoporin by means of linear regression analysis according to Eq. 3 a. Since we know that the three channels of a maltoporin molecule are equivalent and independent of each other (see below), we can so determine the apparent equilibrium binding constant based on an established simple binomial distribution of open and closed channels (Kullman et al., 2002) (stationary approach). More recently the power spectral densities of the sugar-induced current fluctuations were fitted to Lorentzian frequency functions (Nekolla et al., 1994; Bezrukov et al., 2000), suggesting a simple one-step model where the channels are either fully open (i.e., conducting) or completely blocked for ion current (i.e., nonconducting) when a sugar molecule happens to be inside and assuming that the three channels of a single protein are independent of each other (Verveen and De Felice, 1974; Lindemann, 1980; De Felice, 1981). A Lorentzian curve as a function of frequency f reads
 | (15) |
fc = 1/ on sugar concentration can thus be taken advantage of to determine the appropriate on- and off-rate constants according to Eqs. 6 and 12 (dynamic approach). |
RESULTS |
|---|
|
TOPABSTRACTINTRODUCTIONTHEORETICALEXPERIMENTALRESULTSDISCUSSIONCONCLUSIONSREFERENCES |
|---|
|
). In the case of -150 mV and +150 mV, we determined 91 pS and 84 pS, respectively, for one channel. For each transition this demonstrates the homotrimeric nature of the inserted maltoporin molecule as well as the asymmetry of the conductive state of the channel. The larger conductance at positive voltages has been observed for all insertions (more than 200) and was taken to probe the unidirectional insertion of maltoporin into the bilayer.
Stationary state
The apparent equilibrium constant of maltohexaose binding as it varies with the applied voltage is exhibited in Fig. 2. This includes measurements of stationary conductance changes versus concentration (Eq. 3 a) as well as findings obtained in the dynamic way based on a ratio of pertinent rate constants. Obviously both approaches yield results that agree rather well in view of a possible error margin that may extend up to 20–25%. Also we note a good conformity with relevant reports in the literature.
|
The peculiar course of the data indicates a weakening of the effective binding affinity. Such a situation actually occurs when a fast voltage gating of the channel interferes with the binding process so that an apparent binding constant according to Eq. 13 a would be measured using the present experimental approaches. Indeed this extended model does solve the problem at issue. The solid curve in Fig. 2 has been calculated accordingly with
 |
In the present context it should be noted that the curve fit under consideration involves the choice of two adjustable parameters (Ko, a in Eq. 10 a) for each of the two equilibrium constants. This implies a larger range of uncertainty affecting their quantitative evaluation. Actually the smaller values K = 1.7 · 104 · exp(0.01 · V/mV) M-1 and K• = 0.55 · exp(0.015 · V/mV) would also allow a still fairly satisfactory curve fit within the error margin of the measured data. However, there can be no doubt that the proposed basic approach does well describe the observed course of the apparent binding constants.
Kinetics
Rate constants have been determined based on our power spectral density data (see Eq. 15) They were evaluated applying linear regression analysis of 2fc = 1/ versus sugar concentration. Examples are presented in Fig. 3.
|
 |
|
±10%). A linear regression analysis results in rather well-fitting straight line described by
 |
±20–25%) owing to the substantially lower absolute values. Within such given limits they can nevertheless rather well be fitted using the same exponential factor as
 |
 |
|
|
DISCUSSION |
|---|
|
TOPABSTRACTINTRODUCTIONTHEORETICALEXPERIMENTALRESULTSDISCUSSIONCONCLUSIONSREFERENCES |
|---|
In view of the experimental facts reflected in the observed voltage dependence, it appears to be opportune to propose a model mechanism involving one binding site that gets lost upon a fast structural transition. This did indeed provide a successful means to describe the measured results. We note that the involved structural gating apparently does not very much affect the ionic channel current conductance. Thus only a minor change of the channel aperture is indicated that still allows a largely unrestricted passage of small ions. Nevertheless it may certainly abolish the binding and permeation of the bigger sugar molecules just by some minor but crucial alterations of the atomic structure.
By applying our pertinent theoretical relations, all the measured data could be very well fitted quantitatively with no inconsistency. So the voltage dependence of the various equilibrium and rate constants has been evaluated. Eventually the net translocation flux could then be calculated by means of Eq. 14. The actual flux for concentrations added only on one side is displayed for illustration in Fig. 6. We note that (due to K' < K''!) the cis to trans flux is generally somewhat larger than the reverse trans to cis flux, although this can hardly be distinguished any more at the lower concentration of 10 µM. Furthermore it turns out that a comparatively high rate of translocation occurs in the negative voltage range (about an order of magnitude below free diffusion through an unrestricted water-filled channel). It does, however, level down substantially at increasing positive voltages.
|
Turning to the rate constants, it proves to be somewhat unexpected to find the same activation dipole moments of +60 D (or alternatively +44 D) for each koff and -3 D for each kon, respectively. In other words, the electrical charge distribution changes do not differ for either one of the activation barriers. These barriers do nevertheless have different overall energy levels since the trans rate is about four times higher than the respective cis rate.
Looking at the sugar translocation process in terms of a basic one site/two barrier model appears to be a great simplification of a problem that comprises a whole sequence of interactions of the sugar molecule with various different parts of the channel (Dutzler et al., 2002). Crystal structures of maltodextrins bound to maltoporin reveal three glycosyl-binding subsites at the channel constriction (Dutzler et al., 1996). The results suggest the existence of a specific sugar translocation pathway with an aromatic amino acid "greasy slide" being aligned by "polar track" residues. Indeed it has recently been shown that aromatic residues at each extremities of the channel facilitate sugar transport (Hilty and Winterhalter, 2001). Nevertheless, the resulting "smooth" energy profile determined by Meyer and Schulz (1997) showed that no large energy barriers have to be overcome during the sugar movement from one subsite to the next. The sugar binding can then be described by a merge energy minimum, in agreement with our simplified reaction model that can quite well describe the data as they reflect all successive steps of sugar translocation.
It stands to reason that the apparent reaction dipole moments must arise from a difference in the structure of the involved molecular states. The crucial point is the concomitant change of the electrical charge distribution. So far the atomic structure of the open and ligated maltoporin channel, respectively, was seen to be the same at a resolution of 3.1 Å (Schirmer et al., 1995) and 2.4 Å (Meyer et al., 1997). Nevertheless, it is quite conceivable that a sufficient number of charge displacements below 2 Å may occur, resulting in dipolar changes of the present order of magnitude. In addition we emphasize the possible role of polar water domains. The constriction zone of unligated maltoporin was indeed found to be filled with strongly bound water molecules (Schirmer et al., 1995). For steric reasons the sugar transport implies the expulsion of bound water both from the sugar molecules as well as from the channel lumen. Also the ionic residues of the "polar tracks" provide a specific polar environment designed as solvation substitute and for favorable electrostatic interactions with the sugar. Reaction-induced changes of such structural factors may substantially contribute to the particular reaction dipole moments.
|
CONCLUSIONS |
|---|
|
TOPABSTRACTINTRODUCTIONTHEORETICALEXPERIMENTALRESULTSDISCUSSIONCONCLUSIONSREFERENCES |
|---|
Submitted on September 19, 2002; accepted for publication January 17, 2003.
|
REFERENCES |
|---|
|
TOPABSTRACTINTRODUCTIONTHEORETICALEXPERIMENTALRESULTSDISCUSSIONCONCLUSIONSREFERENCES |
|---|
Bezrukov, S. M., L. Kullman, and M. Winterhalter. 2000. Probing sugar translocation through maltoporin at the single channel level. FEBS Lett. 476:224–228.[Medline]
Boos, W., and H. Shuman. 1998. Maltose/maltodextrin system of Escherichia coli: transport, metabolism, and regulation. Microbiol. Mol. Biol. Rev. 62:204–229.
De Felice, L. J. 1981. Introduction to Membrane Noise. Plenum Press, New York.
Dumas, F., R. Koebnik, M. Winterhalter, and P. van Gelder. 2000. Sugar transport through maltoporin of Escherichia coli. J. Biol. Chem. 275:19747–19751.
Dutzler, R., T. Schirmer, M. Karplus, and S. Fischer. 2002. Translocation mechanism of long sugar chains across the maltoporin membrane channel. Structure. 10:1273–1284.[Medline]
Dutzler, R., Y.-F. Wang, P. J. Ritzkallah, J. P. Rosenbusch, and T. Schirmer. 1996. Crystal structures of various maltooligosaccharides bound to maltoporin reveal a specific sugar translocation pathway. Structure. 4:127–134.[Medline]
Hilty, C., and M. Winterhalter. 2001. Facilitated substrate transport through membrane proteins. Phys. Rev. Lett. 86:5624–5627.[Medline]
Kullman, L., M. Winterhalter, and S. M. Bezrukov. 2002. Transport of maltodextrins through maltoporin: a single channel study. Biophys. J. 82:803–812.
Lindemann, B. 1980. The beginning of fluctuation analysis of epithelial ion transport. J. Membr. Biol. 54:1–11.[Medline]
Meyer, J. E. W., M. Hofnung, and G. E. Schulz. 1997. Structure of maltoporin from Salmonella typhimurium ligated with a nitro-phenyl maltotrioside. J. Mol. Biol. 266:761–775.[Medline]
Meyer, J. E. W., and G. E. Schulz. 1997. Energy profile of maltooligosaccharide permeation through maltoporin as derived from the structure and from a statistical analysis of saccharide-protein interactions. Protein Sci. 6:1084–1091.[Abstract]
Montal, M., and P. Mueller. 1972. Formation of biomolecular membranes from lipid monolayers and a study of their electrical properties. Proc. Natl. Acad. Sci. USA. 69:3561–3566.
Nekolla, S., C. Anderson, and R. Benz. 1994. Noise analysis of ion current through the open and the sugar-induced closed state of the LamB channel of Escherichia coli outer membrane. Evaluation of the sugar binding kinetics of the channel interior. Biophys. J. 66:1388–1397.
Schirmer, T., T. A. Keller, Y. F. Wang, and J. Rosenbusch. 1995. Structural basis for sugar translocation through maltoporin channels at 3.1 Å resolution. Science. 267:512–514.
Schwarz, G. 1978. On the physicochemical basis of voltage-dependent molecular gating in biological membranes. J. Membr. Biol. 43:127–148.[Medline]
Van Gelder, P., F. Dumas, and M. Winterhalter. 2000. Understanding the function of bacterial outer membrane channels by reconstitution into black lipid membranes. Biophys. Chem. 85:153–167.[Medline]
Verveen, A. A., and L. J. De Felice. 1974. Membrane noise. Prog. Biophys. Mol. Biol. 28:189–265.[Medline]
Winterhalter, M. 1999. Sugar transport through channels reconstituted in planar lipid membranes. Colloids Surfaces A. 149:547–551.
Woodhall, A. M. 1973. Ionic blockade of sodium channels in nerve. J. Gen. Physiol. 61:687–708.
This article has been cited by other articles:
 |
|
 |
|
  W. R. Bauer and W. Nadler From the Cover: Molecular transport through channels and pores: Effects of in-channel interactions and blocking PNAS, August 1, 2006; 103(31): 11446 - 11451. [Abstract] [Full Text] [PDF]
|
|
|
|
|
|
J. J. Kasianowicz, T. L. Nguyen, and V. M. Stanford Enhancing molecular flux through nanopores by means of attractive interactions PNAS, August 1, 2006; 103(31): 11431 - 11432. [Full Text] [PDF]
|
|
|
|
 |
|
 G. Duret and A. H. Delcour Deoxycholic Acid Blocks Vibrio cholerae OmpT but Not OmpU Porin J. Biol. Chem., July 21, 2006; 281(29): 19899 - 19905. [Abstract] [Full Text] [PDF]
|
|
|
|
 |
|
 A. M. Berezhkovskii and S. M. Bezrukov Optimizing Transport of Metabolites through Large Channels: Molecular Sieves with and without Binding Biophys. J., March 1, 2005; 88(3): L17 - L19. [Abstract] [Full Text] [PDF]
|
|
|
|
|
|
C. Danelon, T. Brando, and M. Winterhalter Probing the Orientation of Reconstituted Maltoporin Channels at the Single-protein Level J. Biol. Chem., September 12, 2003; 278(37): 35542 - 35551. [Abstract] [Full Text] [PDF]
|
|
| ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| HOME | HELP | FEEDBACK | SUBSCRIPTIONS | ARCHIVE | SEARCH | TABLE OF CONTENTS |
) and dynamic approaches (
), respectively (see text). Some data taken from the literature are shown for comparison: (
) (Bezrukov et al., 2000
) (Hilty and Winterhalter, 2001
) (Kullman et al., 2002
) or trans (•) sides, respectively. Within the given error margins, either set can be well fitted by a single exponential relationship (see text). (B) Voltage dependence of the apparent flow-equilibrium constants. Averaged cis (