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Ion Fluxes to Channel Arrays in Monolayers. Computing the Variable Permeability from Currents J. Monne´,† J. Galceran,*,† J. Puy,† and A. Nelson‡ Departament de Quı´mica, Universitat de Lleida, Rovira Roure 191, 25198 Lleida, Spain, and SOMS, Centre for Self-Organising Molecular Systems, University of Leeds, Leeds LS2 9JT, United Kingdom Received November 28, 2002. In Final Form: February 10, 2003 Chronoamperometric currents arising from Tl+ reduction at a Hg electrode covered with a phospholipid monolayer containing gramicidin monomer ion channels can be quantitatively understood as an interfacial permeation process with a variable local permeability. Any interfacial behavior (described by the permeability) can be combined with the properties of semi-infinite diffusion to yield a simple expression that allows: (a) the computation of the current corresponding to any given variation of the permeability with time and (b) the recovery of the permeability through a semi-integration of the currents. As a result, the diffusion process is accounted for, and one can focus on the obtained permeability, whose variation can, in a further step, be ascribed to phenomena such as a decaying number of active channels, a change in their translocation efficiency, etc. Analysis of the experimental data for the Tl+ permeation through gramicidin channels with the derived expression confirms the validity (as a good first approximation within the range of concentrations explored) of considering a simple first-order relaxation process for the gramicidin channel conversion between the conducting and the nonconducting forms coupled with firstorder heterogeneous kinetics for the channel crossing.
1. Introduction The transport of ions through ion channels and their conformational dynamics have great importance in many physiological processes. Since the work of Hodgkin and Huxley1 on the transmission of the nerve impulse, there have been numerous efforts to model ion-channel dynamics2 and ion-channel transport,3 and our understanding of these processes has improved enormously. Despite this, the combination of the three key processes has not been considered collectively in one model. The three processes are (i) the diffusion of an ion to a monolayer surface with an array of channels, (ii) the conformational dynamics of the individual channels, and (iii) the passage of the ion through the channels. Such a global approach has particular relevance because the transfer of ions from the solution media across the cell membrane to the cell interior is of particular importance to studies of the physiological uptake of ions, for example.4 Because of this, a program of work has been instigated to attempt to model the passage of ions to and across an array of ion channels, taking into account all the transport processes involved. To develop a suitable mathematical interpretative framework, a well-characterized experimental system is necessary for its validation. In the past decade, an electrochemical technique has been introduced consisting of a phospholipid monolayer on a mercury electrode with incorporated gramicidin monomolecular channels.5 The experimental evidence drawn from the potential step transients supported a description whereby the active conducting gramicidin was in equilibrium with a larger * Corresponding author. † Universitat de Lleida. ‡ University of Leeds. (1) Hodgkin, A. L.; Huxley, A. F. J. Physiol. 1952, 116, 449-472. (2) Colquhoun, D.; Hawkes, A. G. Proc. R. Soc. London, Ser. B 1981, 211, 205-235. (3) Jordan, P. C. Biophys. J. 1982, 39, 157-164. (4) Ritchie, R. J.; Larkum, A. W. D. Plant Cell Physiol. 1998, 39, 1156-1168. (5) Nelson, A. J. Electroanal. Chem. 1991, 303, 221-236.
concentration of nonconducting gramicidin in the layer.6 The distribution of active and inactive gramicidin varied with the applied potential; so that, following a voltage pulse, a relaxation transient corresponding to the reestablishment of the equilibrium between the conducting and nonconducting forms of gramicidin was seen. A semiquantitative model was then applied to this system relating the relevant rate constants to the current transient. Despite this, the system required a more rigorous mathematical definition of the problem so that the exact solutions regarding the values of the defining parameters could be obtained. It is timely, therefore, that the development of models describing ion-channel systems can be applied to the electrochemical ion-channel experiment. This paper describes an analytical approach to the diffusion and monolayer translocation of ions and applies it to the results derived from the mercury-supported phospholipid monolayer-gramicidin experimental system. 2. Experimental System The deposition of the phospholipid monolayer (dioleoyl phosphatidylcholine from Lipid Products, U.K.) on a mercury electrode and the incorporation of gramicidin channels in the monolayer by addition to the electrolyte of gramicidin D (Sigma chemicals) has been well-described elsewhere.5-8 Gramicidin D is a mixture of 72% gramicidin A, 9% gramicidin B, and 19% gramicidin C. The Tl+/Tl(Hg) redox reaction is used to probe the conducting activity of these channels. The main features of this system are illustrated in part a of Figure 1. It is important to note that the gramicidin channels diffuse in two dimensions and are presumed to have a mobility similar to that of the phospholipid with a diffusional coefficient of 10-10 m2s-1.9 Chronoamperometry with potential steps in the limiting current region of Tl+ reduction at an uncoated Hg electrode is used as the experimental technique.6 The potentials are quoted versus the Ag/AgCl 3.5 mol dm-3 reference electrode throughout. (6) Nelson, A. Biophys. J. 2001, 80, 2694-2703. (7) Nelson, A. Langmuir 1996, 12, 2058-2067. (8) Nelson, A.; Bizzotto, D. Langmuir 1999, 15, 7031-7039. (9) Stryer, L. Biochemistry, 3rd ed.; W. H. Freeman: New York, 1988.
10.1021/la0269210 CCC: $25.00 © 2003 American Chemical Society Published on Web 05/02/2003
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Figure 1. Schematic outline of the modeled system showing the lipid monolayer, including the ionic channels toward which the permeant (depicted as filled circles) diffuses. The solution-side surface of the monolipidic layer corresponds to the plane x ) 0 in the three-dimensional real system. In the model, x is the only relevant spatial coordinate. The model concentration cM(x1, t) represents the average of the concentrations at x ) x1 in the three-dimensional space. A typical concentration profile according to the model is also plotted.
3. Theoretical Approach to the Conduction 3.1. Mathematical Formulation for a Generic Permeability P(t). Let M represent the electroactive permeant species that reaches the electrode surface by diffusing from the bulk solution and crossing the monolayer through an array of channels. In planar geometry, the solution-diffusion process can be described by 2
∂cM(x, t) ∂ cM(x, t) ) DM ∂t ∂x2
c*M - cM(0, t) )
∫0
t
[
xDM
]
∂cM(x, t) ∂x
x)0
dτ
xπ(t - τ)
(2)
where the expression for the flux arises from
∂cM(x, t) -JM(x, t) ) DM ∂x
(3)
(1)
where DM stands for the diffusion coefficient of species M. The origin of the coordinates of the one-dimensional approach corresponds to the monolayer-solution interface (see Figure 1). The assumption of planar geometry is reasonable because the extension of the diffusion layer is extremely small (as a result of the short time scale of the experiment, 40 ms) in comparison with the drop radius (0.265 mm). More important is the fact that a mean field approach is used, which smears out or uniformly spreads (a) the permeant ion concentration at any x (including the monolayer surface at x ) 0) and (b) all the conducting sites on the monolayer surface, thus, rendering possible a macroscopic one-dimensional formulation. This approach averages the overlap of the diffusion spheres to the pores (which appears at very short time scales) and the lateral movement of the pores. This approach is supported by other workers studying similar systems.10 When semi-infinite diffusion (boundary condition limxf∞ cM ) c* M, which is reasonable given the volume of solution of 50 cm3 and the short time of the chronoamperometric experiment) and an initial homogeneous profile c* M are assumed, it can be shown11-14 that there is a relationship between the flux and concentration of the electroactive species at x ) 0 on the solution/monolayer surface: (10) Becucci, L.; Moncelli, M. R.; Guidelli, R. Biophys. J. 2002, 82, 852-864. (11) Oldham, K. B. Anal. Chem. 1986, 58, 2296-2300. (12) Oldham, K. B. J. Appl. Electrochem. 1991, 21, 1068-1072. (13) Oldham, K. B.; Myland, J. C. Fundamentals of Electrochemical Science; Academic Press: San Diego, 1994; Chapter 7, pp 219-262. (14) Mahon, P. J.; Oldham, K. B. Electrochim. Acta 2001, 46, 953965.
It is assumed here that the monolayer does not accumulate significant amounts of M10,15 either within it (so its thickness is irrelevant for our models) or in some kind of adsorbed form on the monolayer-solution interface. This is consistent with the results of previous work5,7,8 where the cyclic voltammograms of the Tl+/Tl(Hg) couple at the gramicidin-modified phospholipid-coated electrode showed no evidence of the adsorption or accumulation of Tl+ on and in the monolayer, respectively. Thus, the absence of accumulation implies that the measured current (considered positive) follows from the incoming flux of M at x ) 0:
[
I(t) ) nFADM
]
∂cM(x, t) ∂x
x)0
(4)
where n is the number of electrons exchanged, F is the Faraday constant, and A is the electrode area. It is noticeable that, just as a result of the nature of process i (the transport of the permeant by diffusion toward the surface), by combining eqs 2 and 4, one can have experimental access to the (average) concentration of M at the surface from the measured currents. Figure 2 shows how cM(0, t) decreases with time; such a variation can be valuable information to take into account in any modeling of the remaining processes. A further key concept that will prove useful is the definition of the (local) permeability as the ratio between the flux and the concentration, both at the monolayersolution interface: (15) Hepel, M. J. Electroanal. Chem. 2001, 509, 90-106.
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Figure 2. Plot of the evolution of the permeant concentration at the lipid-solution interphase computed using eq 2 with the experimental currents obtained by stepping the potential at V ) 0.5 V, with a bulk concentration [Tl]* ) 10-4 M, and by taking DM ) 2.00 × 10-9 m2 s-1 and nFAc*M ) 8.49 × 10-3 C m-1.
Figure 3. Plots of experimental data of current versus time (fluctuating line) and the currents simulated with eq 18 (open circles) or eq 20 (×) using the parameters P∞ ) 7.71 × 10-5 m s-1, P∆ ) 8.92 × 10-5 m s-1, and kΣ ) 1141 s-1, with the other parameters the same as those in Figure 2.
[
P(t) ≡ DM
]
∂cM(x, t) ∂x
/cM(0, t)
x)0
(5)
The permeability so defined is an operational variable with no associated assumption and, as is shown in the following, P(t) is experimentally accessible whenever semiinfinite diffusion without accumulation is accepted. Indeed, the combination of eqs 2-5 leads to
[
P(t) nFAc*M -
1
I(τ) dτ
∫0txt - τ
xπDM
]
) I(t)
Figure 4. Plot of the experimental values of the permeability (using eq 6 and currents from Figure 3) versus time (fluctuating line). The open circles correspond to the fitted P given by eq 17 with the same parameters as those in Figure 3.
In general, once a series of permeability values at different times are computed, ordinary fitting procedures can be applied to find the parameters in an assumed functionality for P(t). 3.2. Simple Model for P(t) Based on the First-Order Interconversion of Nonconducting and Conducting Ionic Channels Coupled with First-Order Heterogeneous Passage Kinetics. 3.2.1. From the Physical Basis to the Mathematical Expression. We now proceed to introduce some additional hypotheses to construct a very simple model for P(t) compatible with the experimental data (such as the currents seen in Figure 3). Several previous works show that gramicidin exists in two forms: an active conducting form, which is the monomolecular β6.3 species,16,17 and an inactive nonconducting form (which could in reality be a collection of species such as Gramicidin aggregates,18,19 double stranded forms,20 or helixes17,21 in some orientation with the channel axis parallel to the surface). The active and inactive species are represented by the symbols Gr* and Gr, respectively. Let us suppose, then, that the model for process ii considers the conversion from one form to the other following a firstorder kinetics with constants k+ for activation and k- for deactivation:6 k+
Gr y\ z Gr* k -
For this process on the surface, we can write the following kinetic equation:
(6)
Apart from being computed from experimental currents, P(t) can be modeled, for instance, by combining a model for process ii (the number of active channels) and a model for process iii (the passage kinetics). The integral eq 6 can, therefore, be used in a twofold manner: (a) Given a known (or assumed from a model) functional dependence of P(t), one can compute the expected current at any time (see section 3.2). (b) Given the experimental current I(t), one can find the function P(t). When this approach is used, the dependence of P(t) on time can be obtained from a set of experimental current-time data via eq 6 (which can be seen as a semi-integration11 of the current data). For instance, applying eq 6 to the experimental currents seen in Figure 3, one obtains P(t) (see continuous line in Figure 4), which exhibits an exponential-like variation with time.
(7)
d[Gr*]/dt ) k+[Gr] - k-[Gr*]
(8)
The solution of this kinetic scheme leads to
[Gr*] ) [Gr*]∞ + ([Gr*]0 - [Gr*]∞)e-kΣt
(9)
k Σ ≡ k+ + k-
(10)
where
can be seen as a total decay rate (equal to the sum of the (16) Killian, J. A. Biochim. Biophys. Acta 1992, 1113, 391-425. (17) Ulrich, W. P.; Vogel, H. Biophys. J. 1999, 76, 1639-1647. (18) Naydenova, S.; Petrov, A. G.; Yarwood, J. Langmuir 1995, 11, 3435-3437. (19) Ogoshi, S.; Mita, T. Bull. Chem. Soc. Jpn. 1997, 70, 841-846. (20) Dhathathreyan, A.; Baumann, U.; Muller, A.; Mobius, D. Biochim. Biophys. Acta 1988, 944, 265-272. (21) Biron, E.; Voyer, N.; Meillon, J. C.; Cormier, M. E.; Auger, M. Biopolymers 2000, 55, 364-372.
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individual constants) of gramicidin from an initial (surface) concentration [Gr*]0 to an equilibrium value [Gr*]∞ (i.e., when time tends toward infinity). Prior to the application of a potential pulse at t ) 0, the conducting gramicidin exists as a fraction of the total gramicidin in the monolayer and is in equilibrium with the nonconducting species of gramicidin.21 The physical picture in eq 9 is that, following the potential step, a fresh equilibrium between the gramicidin species is established. The attainment of this equilibrium takes a finite relaxation time during which the number of conducting gramicidin channels changes. Despite the overall neutrality of the gramicidin molecule, there are localized partial charges22 responsible for the change in the permeability with the applied potential. This physical picture is consistent with the preliminary experiments (not shown here) in which a gramicidin-modified monolayer is pulsed in the absence of Tl+. The resulting current transient, which is solely due to the charging of the capacitor, exhibits a “tail” compared to a current transient resulting from the same experiment carried out on a pure lipid layer. This charging current “tail” suggests some conformational change of gramicidin. Now, we assume first-order heterogeneous kinetics for the passage across the channel (process iii), corresponding to the scheme k2
Gr* + Msolution 98 Gr* + Minternalized
(11)
where Msolution stands for M at the surface-lipid solution, Minternalized corresponds to the internalized (or amalgamated) permeant species, and k2 is the heterogeneous rate constant. This particular model for process iii can be seen as an extension of the usual approach7,23-25 or a specific case of other more complex proposals.6,8,10,26 The rate of internalization (measured as a current) is
I(t) ) nFAk2[Gr*]cM(0, t)
(12)
Combination of this expression with the definition of permeability in eq 5 yields
P(t) ) I(t)/[nFAcM(0, t)] ) k2[Gr*]
(13)
which allows the interpretation (within this model for process iii) of the experimentally accessible P(t) as a measure proportional to the number of active channels. In this specific combination of models for processes ii and iii, P(t) is derived from the kinetic eq 12 together with the prescribed number of active gramicidin channels provided by eq 9: -kΣt
P(t) ) k2[Gr*] ) k2[Gr*]∞ + (k2[Gr*]0 - k2[Gr*]∞)e (14) We define
P∞ ≡ k2[Gr*]∞
(15)
(permeability at infinite time) and (22) Hollerbach, U.; Eisenberg, R. S. Langmuir 2002, 18, 3626-3631. (23) Nelson, A. J. Chem. Soc., Faraday Trans. 1993, 89, 2799-2805. (24) Nelson, A. Langmuir 1997, 13, 5644-5651. (25) Mauzeroll, J.; Buda, M.; Bard, A. J.; Prieto, F.; Rueda, M. Langmuir 2002, 18, 9453-9461. (26) Rueda, M.; Navarro, I.; Ramirez, G.; Prieto, F.; Prado, C.; Nelson, A. Langmuir 1999, 15, 3672-3678.
P∆ ≡ k2([Gr*]0 - [Gr*]∞)
(16)
so that P∆ can be physically interpreted as the change in P(t) from t ) 0 to ∞. Then, one finds that the combination of the gramicidin interconversion model and first-order heterogeneous kinetics model corresponds to the functionality
P(t) ) P∞ + P∆e-kΣt
(17)
which is consistent with the observed exponential-like decay in Figure 4. From now on, we will refer to this combination of models associated with eq 17 as the “Exponentially Decaying Permeability” (EDP) model. Moreover, Figure 4 also shows that P(t) tends toward a constant value after the first few milliseconds. This is interesting and shows that after a short time a constant permeability P∞ describes the translocation of Tl+ through the gramicidin-modified monolayer. This is entirely consistent with previous findings, for example,7 that the entry of the Tl+ ion into the gramicidin channel is the rate-controlling step in its reduction, or Guidelli et al.’s model,10 which identified this step with the partial dehydration of the ion at the pore mouth. The combination of P(t) corresponding to EDP model, eq 17, with the generic integral eq 6, leads to
[
(P∞ + P∆e-kΣt) nFAc*M -
1
I(τ) dτ
∫0t xt - τ
xπDM
]
) I(t) (18)
The numerical solution of the integral eq 18 allows us to obtain the current at any time once some values for the parameters P∞, P∆, and kΣ of the EDP model are given. This is done by a Fortran program, which uses a discretization of the integral in small intervals where the unknown variable is linearized (sometimes referred to as Huber’s method).27 The program has been validated by comparing the results of a case having an analytical solution. When P∆ ) 0, P(t) ) P∞ (see eq 16). Physically, this means that there is no change in the conducting sites. The dependence with time of the current is well-known:28
( ) (
)
P∞2 P∞ xt t erfc I(t) ) nFAc*MP∞ exp DM xDM
(19)
Figure 5 shows the good agreement between the current obtained analytically and the one given by the program. Equation 19 suggests the trial of
[ ] [
I(t) ) nFAc*MP(t) exp
]
P(t)2 P(t) xt t erfc DM xDM
(20)
as a candidate for solution of the EDP model (i.e., the solution of eq 18). It can be checked that eq 20 is not the rigorous solution of this problem, but numerical assays have shown that it can be applied in the preliminary examinations of the data with good accuracy. For instance, discrepancies between the crosses and the open circles in Figure 3 are only significant for very short times. However, the approximation is less accurate for larger values of P∆ (see markers in Figure 6). (27) Feldberg, S. W. Electroanal. Chem. 1969, 3, 199-296. (28) Bard, A. J.; Faulkner, L. R. Electrochemical Methods, Fundamentals and Applications; Wiley: New York, 1980.
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Figure 5. Plot of the current obtained with the simulation program (continuous line) solving eq 18 compared to the analytical solution (eq 19; open circles) versus time when P∆ ) 0. The parameters are P∞ ) 7.6 × 10-5 m s-1, kΣ ) 1080 s-1, DM ) 1.00 × 10-9 m2 s-1, and nFAc*M ) 8.49 × 10-3 C m-1.
Monne´ et al.
Figure 7. Impact of changing P∞ on the current evolution. The other parameters are the same as those in Figure 3.
Figure 8. Plots of the currents for different values of k+ with the other parameters the same as those in Figure 3. Figure 6. Plots of the currents for different P∆ values. The plot with P∆ ) 0 corresponds to the case with an analytical solution (eq 19), and the line labeled as “Cottrell” corresponds to the diffusion-controlled current. The markers stand for the approximation given by eq 20 (crosses for P∆ ) 0.5 m s-1 and open circles for P∆ ) 10-2 m s-1). The other parameters are the same as those in Figure 3.
3.2.2. Impact of the Parameters in the EDP Model. With the tool to compute currents, one can determine the impact of the different parameters involved in a given temporal dependence of P(t). The current in the EDP model (found from eq 18) depends on the following relevant parameters: DM, P∞, P∆, kΣ, c* M, and t. We now discuss the effects of some of them. (a) Effects of P∞ and P∆. The knowledge of the impact of P∞ and P∆ on the currents may add valuable theoretical information that can help us to validate or discard a model. The effect of P∞ is reflected in Figure 7. The larger the values of P∞, the larger the current. However, the differences between the curves decrease with increasing time as a result of the common tendency of all the curves to approach 0. Figure 6 shows the evolution of the current for systems with varying P∆. Larger values of P∆ produce steeper curves at short times. The least-steep curve arises from the value P∆ ) 0. As is seen in Figure 6, the curves with higher values of P∆ give larger currents at shorter times and smaller currents at longer times. This can be interpreted as being due to a very large ionic flux crossing the channels at the initial times for larger values of P∆, giving rise to a depletion of M close to the phospholipidic layer and less-steep gradients of M for subsequent times. It is noted that all the currents are less than those in the diffusion-limited case (Cottrell current) and that for large values of P∆ the current goes through a minimum.
The mechanism behind the I(t) versus the minimum t can be understood as follows. If P∆ is large, at short t, the permeability is high leading to high currents and a significant depletion of M close to the lipid layer. At an intermediate time (e.g., around 10 ms), the depletion of cM(0, t) is still significant (diffusion is slow to restore the reduced material), while the permeability reaches some kind of stable value and the minimum in I(t) is reached. The current recovers because the permeability stays constant, and some replenishment due to diffusion from the bulk takes place. Finally, at longer times the current goes through a relative maximum before reaching (at infinite time) the steady-state value of 0. (b) Effects of k+ and k-. As can be seen in Figure 8, an increase in k+, with no change in k- , leads to an increase of the current, as is expected from a larger ratio of the conducting ionic channels in the phospholipid monolayer and, of course, an increase of P∞ (the limiting value at infinite time). But for the values of k+ that render P∆ negative (so the ratio of active channels is greater at infinite time than that at the beginning), a maximum in the plot of the current versus time is observed. This maximum can be seen as a switch in the predominance from an increasing permeability to the decreasing availability of M in the neighborhood of the monolayer. Further computations, not shown here, indicate that the current and P(t) decrease when k- increases. Because the ratio of k+/k- is the equilibrium constant of the interconversion scheme (eq 7), P∞ could be kept constant when increasing both k+ and k- by the same factor, but then larger values of kΣ ) k+ + k- would lead to a faster approach of the permeability to P∞ and currents would converge sooner to values predicted by eq 19. The currents are very sensitive to the parameters k+ and k-
Channel Arrays in Monolayers
Figure 9. Plots of the ratio of the measured currents over the Tl+ concentration versus time. The markers correspond to different thallium(I) bulk concentrations: 10-5 M (solid line), 4 × 10-5 M (open circles), 10-4 M (crosses), and 4 × 10-4 M (dotted line). All the currents are determined at a potential of -0.5 V. Each point in the plot is the average of 16 experimental data points within 0.4 ms of the represented time.
Figure 10. Linear dependence of the current on the thallium bulk concentration for the series recorded at -0.5 V. Each point corresponds to the average (between 10 and 40 ms) of the currents measured within a repetition.
at short t, but the differences decrease for larger t when the changes in the ratio k+/k- are less important (see Figure 8). 4. Analysis of the Experimental Results 4.1. Testing of the EDP Model. 4.1.1. Linearity with Tl+ Bulk Concentration. The master equation in the EDP model, eq 18, prescribes linearity between the current and the bulk concentration because the interconversion model for process ii assumes that the parameters P∞, P∆, and kΣ are independent of c* M according to eqs 10, 15, and 16. To check for this linearity property, we have plotted in Figure 9 the ratio of the measured currents (where the corresponding blank has been subtracted) over the metal bulk concentration versus time, for four different values of c*M. When the estimated uncertainty is taken into account, the agreement is quite satisfactory, although at very short times (t < 1 ms), there is some deviation from the expected superposition of the curves. The linearity of the long time currents with c* M is compatible with the data represented in Figure 10, where the average of the currents between 10 and 40 ms is plotted against the bulk Tl concentration (R2 of the linear fitting is 0.9991). Although the EDP model is not fully demonstrated by the current experiments because nonlinear relationships could apply, especially for large bulk concentrations or short times, it can be seen as a good first approximation. Further refinements on processes ii and iii leading to better fittings of the data (using the suggested methodology of computing
Langmuir, Vol. 19, No. 11, 2003 4699
the permeability) would involve a larger number of parameters than the simple EDP model developed here. 4.1.2. Goodness of Fit. A set of experimental currents obtained from the phospholipid-gramicidin system (see slightly fluctuating line in Figure 3) have been “semiintegrated” to obtain P(t) drawn as a continuous line in Figure 4. The fitting of P(t) to the proposed EDP model (eq 17) yields the model parameters P∞, P∆, and kΣ. The resulting parameters from the best fit of these particular data using a nonlinear regression algorithm29 are P∞ ) 7.71 × 10-5 m s-1, P∆ ) 8.92 × 10-5 m s-1, and kΣ ) 1141 s-1. The expected permeability from these values has been plotted as open circles in Figure 4, where the agreement with the recovered P(t) via semi-integration is good, with the theoretical permeability practically reaching a constant value slightly before the experimentally retrieved one. Further, one sees that the currents obtained (see the open circles in Figure 3) reproduce the experimental ones reasonably well. 4.2. Recovered Parameters. The experimental currents arising at three different potentials and four different bulk concentrations of thallium have been analyzed following the procedure explained in section 4.1. For each combination of potential and concentration, we have used three repetitions to allow for the estimation of the uncertainty of the fitted parameters. The fitted values for P∞ are gathered in Table 1, values for P∆ are in Table 2, and values for kΣ are in Table 3. Two relevant observations from the tables are as follows. (i) P∆ and kΣ, physically related to the initial milliseconds of the experiment, are less independent of the Tl bulk concentration (recovered P∆ and kΣ values in Tables 2 and 3, show some diminishing trend with increasing c* M). The estimated values of P∆ (Table 2) especially exhibit a large scattering, as is expected from this parameter being very sensitive to the values of the short time currents. Especially divergent is the P∆ value for the smallest concentration, which can be due to the relative importance of the fluctuations for low-concentration currents compared to those of the low measured current (see oscillating line in Figure 9). This variability of P∆ and kΣ with the permeant bulk concentration could be due to either a lack of accuracy of the current measurements at extremely low values of t or a slight deviation from the suggested EDP model (see the observed loss of linearity of the short time currents in Figure 9). However, the deviation does not seem prominent enough to discard the EDP model from the current data as a simple first approximation. Further refinements in the models of processes ii (interconversion) and iii (passage) could better fit the data, but more than three parameters would then be required. In contrast to the variability of the P∆ and kΣ values, P∞ is essentially independent of c*M and can be considered the most robust recovered parameter. (ii) When the uncertainties are taken into account, the effect of the assayed final potential seems to not be significant for any parameter. This could be physically interpreted as the interconversion not depending on the potential (in this region of Tl+ reduction), because a certain threshold has been overcome.30,31 Conclusions For a solution containing an electroactive permeant metal ion, the current at a channel-modified monolayer(29) Wolfram, S. Mathematica. A System for Doing Mathematics by Computer; Addison-Wesley Publishing Company: Redwood City, CA, 1988. (30) Andersen, O. S. Biophys. J. 1983, 41, 135-146. (31) Kurnikova, M. G.; Coalson, R. D.; Graf, P.; Nitzan, A. Biophys. J. 1999, 76, 642-656.
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Table 1. Values of P∞ (m s-1) Obtained by Fitting the Permeability P(t) (arising from the Semi-Integration of the Currents) to Eq 17 for Three Different Potential Steps and Four Different Thallium Concentrationsa [Tl+] ) 10-5 M [Tl+] ) 4 × 10-5 M [Tl+] ) 10-4 M [Tl+] ) 4 × 10-4 M
-0.5 V
-0.6 V
-0.7 V
9.37 × 10-5 ( 1.79 × 10-5 9.13 × 10-5 ( 4.2 × 10-6 9.01 × 10-5 ( 9.7 × 10-6 6.89 × 10-5 ( 2.6 × 10-6
8.40 × 10-5 ( 5.3 × 10-6 8.79 × 10-5 ( 4.2 × 10-6 8.16 × 10-5 ( 8.9 × 10-6 6.55 × 10-5 ( 3.2 × 10-6
9.01 × 10-5 ( 2.5 × 10-6 8.81 × 10-5 ( 5.6 × 10-6 7.44 × 10-5 ( 7.5 × 10-6 6.03 × 10-5 ( 1.5 × 10-6 b
a Three repetitions (unless otherwise stated) of each experiment were considered to obtain the variability range of each parameter. The fixed parameters are DM ) 2.00 × 10-9 m2 s-1, n ) 1, and A ) 8.825 × 10-7 m2. b Only two repetitions were available.
Table 2. Values of P∆ (m s-1) Obtained in the Same Process as That for Table 1 -0.5 V [Tl+]
10-5
) M [Tl+] ) 4 × 10-5 M + -4 [Tl ] ) 10 M [Tl+] ) 4 × 10-4 M a
10-4
-0.6 V 10-4
7.69 × ( 1.78 × 1.74 × 10-4 ( 5.0 × 10-5 -4 1.10 × 10 ( 1.6 × 10-5 7.91 × 10-5( 6.0 × 10-6
10-4
-0.7 V 10-4
8.67 × ( 2.07 × 2.29 × 10-4 ( 6.9 × 10-5 -4 1.31 × 10 ( 2.5 × 10-5 6.04 × 10-5 ( 3.1 × 10-6
10-3
1.17 × ( 1.4 × 10-4 2.34 × 10-4 ( 4.7 × 10-5 7.54 × 10-5 ( 2.53 × 10-5 5.95 × 10-5 ( 3.26 × 10-5 a
Only two repetitions were available. Table 3. Values of kΣ (s-1) Obtained in the Same Process as That for Table 1 [Tl+] ) 10-5 M [Tl+] ) 4 × 10-5 M [Tl+] ) 10-4 M [Tl+] ) 4 × 10-4 M
a
-0.5 V
-0.6 V
-0.7 V
2.28 × 103 ( 7 × 101 1.63 × 103 ( 3.2 × 102 1.23 × 103 ( 1.3 × 102 9.64 × 102 ( 2.15 × 102
2.32 × 103 ( 1.6 × 102 1.93 × 103 ( 2.7 × 102 1.52 × 103 ( 1.6 × 102 8.75 × 102 ( 2.39 × 102
2.70 × 103 ( 4 × 101 2.05 × 103 ( 1.5 × 102 1.42 × 103 ( 7 × 101 1.56 × 103 ( 1.09 × 103 a
Only two repetitions were available.
covered electrode can be modeled as a simple diffusion process from the bulk followed by transport across an array of channels incorporated in the monolayer. A fundamental tenet of this approach is the well-known relationship (eq 2) that holds between the concentration of permeant ion and the gradient at the monolayer-solution surface (disregarding the nature of the operating interfacial process). Using the concept of local permeability [P(t), see eq 5], we obtain the general eq 6, which relates the measured currents to the local permeability (we assume that there is no accumulation of the permeant). The treatment can be used either to predict currents from a given expression for P(t) or, conversely, to determine P(t) from the current using a semi-integration technique. The experimental data of Tl+ reduction at the gramicidinmodified monolayer-coated electrode show an exponentiallike decay of P(t). This behavior of P(t) supports a specific model for the permeability (the EDP model), where the conduction is controlled by the interconversion of the conducting and nonconducting forms of gramicidin followed by a first-order heterogeneous kinetics process. The reasonable linearity of the experimental currents (in particular for times that are not too short) with the Tl+ concentration supports the validity of the EDP model as a good first approximation. The resulting expression of
the permeability (given by eq 17) involves three parameters: P∞, P∆, and kΣ, with P∞ being the most robust of them. In summary, the general expression shown in eq 6 is applicable to any system of ion transport through a conformationally labile ion-channel array in an impermeable monolayer where the conditions of diffusion are welldefined because it enables the unambiguous identification of the permeability of the system. The specific combination of a first-order model for the interconversion of the conducting channels and a first-order heterogeneous kinetics model for the channel passage leads to the permeability given by eq 17 and reasonably agrees with the experimental data. Acknowledgment. A.N. was supported by the MoD-NERC JGS grant program. A.N. thanks the University of Lleida for a visiting grant. Lleida researchers acknowledge the support of this research by the Spanish Ministry of Education and Science (DGICYT: Project No. BQU2000-0642). J.M. thanks the Catalan Government for the grant to stay in Leeds. LA0269210
