8532
J. Phys. Chem. B 1999, 103, 8532-8537
Analyzing Product Inhibition and pH Gradients in Immobilized Enzyme Films As Illustrated Experimentally by Immunologically Bound Glucose Oxidase Electrode Coatings Christian Bourdillon,1b Christophe Demaille,1a Jacques Moiroux,*,1a and Jean-Michel Save´ ant*,1a Laboratoire d’Electrochimie Mole´ culaire, Unite´ Mixte de Recherche UniVersite´ sCNRS No. 7591, UniVersite´ de Paris 7, Denis Diderot, 2 place Jussieu, 75251 Paris Cedex 05, France, and the Laboratoire de Technologie Enzymatique, UPRESA No. 6022, UniVersite´ de Technologie de Compie` gne, BP 20529, 60205 Compie` gne Cedex, France ReceiVed: April 8, 1999; In Final Form: June 30, 1999
Many enzymes may be inhibited by the products of the reaction they catalyze by means of a MichaelisMenten kinetic retro-action. Protons, which are involved as products or reactants in a number of cases, may also influence the enzymatic kinetics. The course of the reaction may therefore be altered by the attending production or depletion of protons. These effects are expected to become particularly important in multilayered films developing high catalytic efficiencies. In the case of redox enzymes, connecting the prosthetic group with an electrode by means of a redox cosubstrate allows a quantitative analysis of the kinetics of product inhibition by means of electrochemical techniques such as cyclic voltammetry. This approach is illustrated with the example glucose oxidase electrode coatings obtained by successive antigen-antibody attachment of a series of monomolecular layers. The kinetics of the inhibition by gluconolactone and the depiction of the pH gradients across the enzyme film could thus be derived, after the appropriate theory was established, from the hysteresis exhibited by the forward and reverse current traces. The theory, derived for the case of successive monomolecular layers, can be easily extended to more disordered enzyme assemblies. A general strategy is thus made available that allows a full description of the space-dependent dynamics of the system not only for the primary species of the enzymatic catalysis but also for secondary species that may be involved in feedback processes.
One way by which an enzymatic reaction may be inhibited by one of its product is related to the Michaelis-Menten kinetics of its release from the prosthetic group. Protons, being involved as products or reactants in a number of enzymatic reactions, may also influence, albeit in a different fashion, the enzyme kinetics. The course of the reaction may therefore be altered by the attending production or depletion of protons. Concerning immobilized enzyme systems,2 early studies have mentioned such phenomena as a possible cause of hysteresis (“memory effects”) observed in membranes containing pH-dependent enzymes.3 With redox enzymes, immobilization onto an electrode surface and “wiring”4a of the enzyme to the electrode by means of a redox mediator,4-8 has, besides applications in the field of biosensors, the advantage of converting the physical and chemical events taking place in the film into electrical responses, thus greatly facilitating their analysis. In this framework, systems based on glucose oxidase have received particular attention, not only because of possible applications in the sensing of glucose but also because they are convenient model systems allowing the illustration and analysis of phenomena of general interest. One aspect of these efforts has been the construction of spatially ordered enzyme assemblies bound by means of antigen-antibody or avidin-biotin interactions.8 Combined with the possibility of varying at will the number and activity of the successive monomolecular layers, treatment of the cyclic voltammetric responses by means of appropriate theoretical analysis thus allowed a precise description of the diffusion-reaction coupling and, hence, the demonstration and characterization of the spatial order of these assemblies.
These studies have, however, neglected the possible effects of alteration of the course of the enzymatic reaction by the products formed, in particular by protons. The purpose of the work reported hereafter was to address these general questions, illustrating the theoretical predictions with the example of antigen-antibody bound glucose oxidase electrode coatings (Scheme 1). Results and Discussion As a possible reflection of product inhibition effects, our attention was attracted by hysteresis phenomena appearing in cyclic voltammograms such as that shown in Figure 1. Simulation of the cyclic voltammogram according to Scheme 2, without taking product inhibition into account (dotted line in Figure 1a) leads to an exact superimposition of the forward and backward trace unlike what is observed experimentally (the values of the various parameters used for the simulation depicted in Figure 2 were taken from ref 8d). There are two possible causes of hysteresis. One is inhibition by the product resulting from the transformation of glucose (G), namely, glucono-δ-lactone (GL). Although, qualitative studies in solution have indicated that product inhibition by GL is small,9 it cannot be excluded that it may play a significant role with electrode coatings containing several monolayers, such as those dealt with in Scheme 1 and Figure 1, where the catalytic currents are large, implying a high rate of GL generation. Another possible cause of hysteresis resides in the production of protons during the catalytic reaction (Scheme 2). The
10.1021/jp991186v CCC: $18.00 © 1999 American Chemical Society Published on Web 09/16/1999
Immobilized Enzyme Films
J. Phys. Chem. B, Vol. 103, No. 40, 1999 8533
SCHEME 1: Antigen-Antibody Immobilization of N Monolayers of Glucose Oxidase on the Glassy Carbon Electrodea
a A is an affinity-purified mouse IgG, C-GO, a glucose oxidase antimouse conjugate, the IgG moiety being produced in goat. Ago is a cocktail of two monoclonal antibodies to glucose oxidase produced in mouse.
Figure 2. Effect of the addition of glucono-δ-lactone on the cyclic voltammetry of the catalysis of glucose oxidation at a glassy carbon disk electrode coated with one glucose oxidase monolayer in the presence of glucose (0.4 M) and ferrocene methanol (0.2 mM) in a pH 7.0 phosphate buffer (ionic strength, 0.1 M). Temperature: 25 °C. Scan rate: 0.04 V/s. (a) Raw data. The dotted curve is the reversible voltammogram recorded in the absence of glucose. (b) Catalytic part of the current response (see text). (c) Primary plots (solid lines: experimental data; dotted lines: best linear fit). E0P/Q ) 0.19 V vs. SCE.8a (d) Secondary plot. The number on each curve in a, b, c, is the value of C0GL in mM. Figure 1. Cyclic voltammetry of the catalysis of glucose oxidation at a glassy carbon disk electrode coated with 10 glucose oxidase monolayers in the presence of glucose (0.5 M) and ferrocene methanol (0.2 mM) in pH 8.0 phosphate buffer (ionic strength, 0.1 M). The surface concentration of each monolayer in catalytically active enzyme o is ΓE,n ) 1.5 × 10-12 mol/cm2 (1 e n e 10). Temperature: 25 °C. Scan rate: 0.04 V/s. Recorded (solid lines) and simulated (dotted line) cyclic voltammograms. The effect of product inhibition is neglected in the simulation (parameters: k2 ) 700 s-1, k4 ) 1.1 × 107 M-1 s-1, kred ) 1.1 × 104 M-1 s-1, ks ) 0.19 cm/s, R ) 0.5, D ) 5.5 × 10-6 cm2/s, DP ) 3.3 × 10-6 cm2/s, and l ) 470 Å).
influence of pH on the enzymatic kinetics mainly concerns the rate constant k4 which increases with pH in a sigmoidal manner related to the pKa of the FADH2/FADH- couple (see eq 4 below).8a Since protons are produced during the reaction, the pH decreases within the enzyme film thus slowing down the reaction insofar as the diffusion of the acidic form, BH, and of the basic form, B-, of the buffer through the film and the adjacent solution, is not fast enough to maintain locally the pH at the value it has in the bulk of the solution. Inhibition by glucono-δ-lactone and protons have thus a similar effect. The separation of the two effects may be performed as follows. Inhibition by Glucono-δ-lactone at a Single Monlayer Electrode. At an electrode coated with a single monolayer of enzyme there is no significant hysteresis, indicating that inhibition by protons and glucono-δ-lactone produced inside the coating is negligible. We may therefore investigate separately the inhibition by glucono-δ-lactone by measuring the changes in the cyclic voltammetric response that take place upon addition of this compound to the solution. The experiments were carried out in a pH ) 7 buffer of 0.1 M ionic strength where gluconolactone hydrolysis is slow enough (1 h half-reaction time)10 to allow a reliable cyclic voltammetric analysis. It was observed that the catalytic cyclic voltammetric current decreases upon
SCHEME 2
SCHEME 3
addition of glucono-δ-lactone (Figure 2a) as expected from the Michaelis-Menten kinetics depicted in Scheme 3. The catalytic part of the current, icat, may be obtained as the difference of the cyclic voltammetric currents in the presence and absence of glucose respectively.8b icat is expected to obey eq 1 in which the parameters governing the interference of glucono-δ-lactone are those depicted in Scheme 3 where reaction 2′ and 3 replace 0 , the reaction 2 in Scheme 2. E is the electrode potential; EP/Q 0 standard potential of the mediator couple; CP, the bulk concentration of the mediator, [Q]0, surface concentration of the oxidized form of the mediator; CoGL, the concentration of glucono-δ-lactone in the solution; and ΓoE, the surface concentration of the enzyme in the monolayer.
8534 J. Phys. Chem. B, Vol. 103, No. 40, 1999
(
1 1 1 1 1 ) + + icat 2FSΓo k2 k3 k Co E red G
Bourdillon et al.
CoGL K3 1 + 2FSk4ΓoE [Q]0 1+
)
(1)
with
C0P [Q]0 ) F 1 + exp - (E - E0P/Q) RT
[
(Nernst law)
]
kred )
k1k2 k-1 + k2
Primary plots (Figure 2c) obtained from the application of eq 1 are indeed linear. As expected, the intercept is independent of the concentration of glucono-δ-lactone. It corresponds to the same value of the rate constants as previously determined,8 when the possible inhibition by this product was ignored. The value of k2 (Scheme 2) thus obtained is now replaced by k2′k3(k2′ + k3). The slope of the primary plots, s, is a linear function of the glucono-δ-lactone concentration as seen in Figure 2d leading to the value of the equilibrium constant K3 ) k3/k-3 ) (1.6 ( 0.2) × 10-2 M, a result confirming that glucono-δ-lactone is not a strong inhibitor of glucose oxidase.9 However, as seen later on, inhibition by glucono-δ-lactone may become significant with the large catalytic currents obtained with thicker films. Product Inhibition (Protons and Glucono-δ-lactone) in Multilayered Films. Figure 3 shows typical examples of hysteresis exhibited by a 10-layer film as a function of the scan rate. Hysteresis increases upon decreasing the scan rate, suggesting that the diffusion of glucono-δ-lactone and/or the acid and base form of the buffer plays a crucial role in this phenomenon. This observation is confirmed, for protons, by the fact that inhibition decreases upon increasing the buffer concentration as discussed in more details later on. Together with hysteresis, inhibition manifests itself by the foward current trace becoming peak-shaped instead of sigmoidal and the peak current being smaller than the plateau current that would have been obtained in the absence of inhibition. The diffusion gradients of the various species throughoiut the film, the reduced and oxidized forms of the mediator, P and Q, glucono-δ-lactone, GL, and the acid and basic forms of the buffer, BH and B-, obey the following set of equations. For each monolayer within the film, i.e., for 1 e n e N - 1 (see Scheme 1):11a
{( ) ( ) } {( ) ( ) } {( ) ( ) } {( ) ( ) } {( ) ( ) }
∂[Q] ∂[P] ∂[P] ) DP ) + ∂x - n ∂x ∂x + n ∂[B-] ∂[B-] ∂[BH] )D D ∂x + ∂x - n ∂x ∂[BH] ∂[GL] ∂[GL] ) 2D ) ∂x + n ∂x + ∂x - n o 2FSk4,nΓE[Q]n
(
[GL]n 1 1 + [Q]n + K3 k2,n k Co red G
)
-
pHn ) pKBH/B + log([B-]n/[BH]n) a
(2)
x is the distance to the electrode surface, and the subscripts + and - mean at the right- and left-hand side of the monolayer, respectively. The pH in each monolayer is related to the local concentrations of BH and B- according to eq 3, where pKBH/B a designates the pKa of the buffer couple.
(3)
As shown earlier,8a the rate constant k4 (see Scheme 2) is pHdependent (eq 4).
k4,min + 10(pHn - pKa )k4,max E
k4,n )
(4)
1 + 10(pHn - pKa ) E
where k4,min ) 6.3 × 104 M-1 s-1, k4,max ) 1.2 × 107 M-1 s-1, and pKEa , the average pKa of the FADH2/FADH- and FADH•/ FAD•- couples is equal to 7.8a k2 also depends on pH (k2,n ) 1900 - 150pHn)7a whereas kred ()1.1 × 104 M-1 s-1) and K3 () 1.4 × 10-2 M as determined in the preceding section) may be considered as pH-independent. At the electrode surface (x ) 0), the kinetics of the mediator oxidation follows the Butler-Volmer law (eq 5).
( )
( ) }{
∂[P] ∂[Q] i ) -D ) ) DP FS ∂x x)0 ∂x x)0 0 RF(E - EP/Q) F(E - E0P/Q) ks exp [P]x)0 - [Q]x)0 exp RT RT (5)
{
∂[Q] D ∂x
1+
Figure 3. Cyclic voltammetry of the oxidation of glucose (0.4 M) at a 10-layer electrode (surface concentration of enzyme in each monolayer: 2.65 × 10-12 mol/cm2) in the presence of ferrocene methanol (0.4 mM) in a solution containing 1.48 mM NaH2PO4, 16.5 mM Na2HPO4, and 16.4 mM Na2SO4 (pH ) 8, total ionic strength: 0.1 M). Temperature 25 °C. The number in each diagram is the value of the scan rate in V/s. Full lines: experimental data. Dotted lines: simulation of inhibition by protons and glucono-δ-lactone production. Dashed lines: no inhibition.
[
]}
(the standard rate constant for ferrocene methanol is kS ) 0.19 cm s-1, the transfer coefficient R ) 0.5, and the standard 0 potential EP/Q ) 0.19 V vs SCE.8d) The other species are electrochemically inactive and thus their flux at the electrode surface is equal to zero (eq 6).
( ) ( ) ( ) ∂[GL] ∂x
)
x)0
∂[BH] ∂x
)
x)0
∂[B-] ∂x
x)0
)0
(6)
At the film solution interface, i.e., for x ) L ) (N - 1/6)l (l, the distance between two successive monolayers, is equal to 470 Å11b), the boundary conditions express the time (t)-dependent relationship between the concentration of the various
Immobilized Enzyme Films
J. Phys. Chem. B, Vol. 103, No. 40, 1999 8535
Figure 5. Variation of the peak current with the scan rate for the same conditions as in Figures 3 and 4. Open circles: experimental data; full line: simulation of the combined effect of proton and glucono-δ-lactone production; dotted line: simulation of proton production alone; dashed line: simulation of inhibition by glucono-δ-lactone alone; dashed-dotted line: no inhibition. Figure 4. Simulation of the cyclic voltammograms of the oxidation of glucose (0.4 M) at a 10-layer electrode (surface concentration of enzyme in each monolayer: 2.65 × 10-12 mol/cm2) in the presence of ferrocene methanol (0.4 mM) in a solution containing 1.48 mM H2PO4-, 16.5 mM HPO42-, and 16.4 mM SO42- with Na+ as counterion (pH ) 8, total ionic strength: 0.1 M). Temperature 25 °C. The number in each diagram is the value of the scan rate in V/s. Full lines: simulation of the effect of proton production alone. Dotted lines: simulation of glucono-δ-lactone inhibition alone. Dashed lines: no inhibition.
species and their diffusion flux to or from the solution under the form of integral equations (7).
x ∫( ) x ∫( ) x ∫( ) DP π
[P]x)L ) C0P [Q]x)L ) -
D π
t
0
t
0
∂[P] ∂x
∂[Q] ∂x
D π
[B-]x)L ) CB0 - -
x (
[GL]x)L ) -
t
0
∫0t
D π
xDπ∫ ( t
0
-η dη
x)L+xt
)
∂[B-] ∂x
)
∂[GL] ∂x
-η
dη
x)L+xt
∂[BH] ∂x
[BH]x)L ) C0BH -
dη
x)L+xt
-η
dη
x)L+xt
-η
dη
x)L+xt
-η
(7)
C0BH and CB0 - are the bulk concentration of the acid and base form of the buffer, respectively. The resolution of the above system is eased by the fact that the concentration profiles of any diffusing species, X, can be regarded as linear between two successive monolayers.8d,g Thus, eqs 2 may be simplified by substitution of eqs 8.
(∂[X] ∂x )
+n
)
[X]n+1 - [X]n , l
(∂[X] ∂x )
-n
[X]n - [X]n-1 l for 1 < n e N-1 (8)
)
Dimensionless transformation and calculation procedures to be used for simulating the cyclic voltammograms for any set of values of the parameters are given in the Supporting Information section. As seen by comparing the experimental voltammograms in Figure 3 to the simulated voltammograms in Figure 4, inhibition
by glucono-δ-lactone alone, although significant, especially at low scan rates, is not sufficient to account for the observed hysteresis. The effect of proton production is much larger (Figure 4). Combining the two effects leads to a satisfactory agreement between experiment and simulation as can be seen in Figure 3 and a good reproduction of the effect of the scan rate on the extent of inhibition, at least for what concerns the heights of the peak currents. These conclusions are confirmed by a more extended comparison between simulation and experiment as a function of the scan rate as shown in Figure 5, which focuses attention on the variation of the peak current. We may now examine what happens when the buffer concentration increases. As expected, inhibition by proton production decreases, resulting in an overall decrease of inhibition which results, all other parameters being the same, in an increase of the peak current and a decrease of hysteresis. Out of an extended series of experiments, Figure 6 provides a typical example of this behavior. It is interesting to observe the attending increase of pH through the film. The lowest value is reached at the electrode surface, and we see that this may be as much as 2 pH units lower than the bulk pH (right-hand side of Figure 6). As expected, the variation is less when the concentration of the buffer is larger (left-hand side of Figure 6). We also note that, in the latter case, the ionic strength at the electrode surface undergoes a large variation along the catalytic wave, significantly larger than when the buffer is diluted by Na2SO4 so as to maintain the bulk ionic strength equal to 0.1 M. This difference parallels the observation that the simulation of inhibition by protons and glucono-δ-lactone production fits the experimental data better in the latter case than in the former, indicating that the ionic strength exerts a distinct influence on the enzymatic kinetics. In a separate experiment, we found that the catalytic current obtained at a monolayer electrode decreases by ca. 30% when the ionic strength of the solution is increased from 0.1 to 0.15 M. This observation, together with previous findings concerning the role of the charge borne by the oxidized form of the mediator,8 suggests the following qualitative interpretation of the influence of ionic strength. As this factor increases, the activity of the FADH- negatively charged group decreases and so does the rate of the enzymatic reaction. We may thus envision that the rate of the enzymatic reaction decreases in a sigmoidal manner as the ionic strength increases. In the experiment reported in the right-hand side of Figure 6, there is a large variation of the pH which decreases to a value as low as 5.8, triggering a strong inhibition. The ionic strength
8536 J. Phys. Chem. B, Vol. 103, No. 40, 1999
Bourdillon et al. is formed at the electrode surface that has no time to diffuse out during the time of the experiment. The fact that more glucono-δ-lactone is formed with the undiluted buffer than with the diluted buffer is simply a consequence of the catalytic current being larger in the former case than in the latter because there is less inhibition by proton production. This observation underlines the fact that the role of protons and glucono-δ-lactone are not simply additive as can also be seen in Figure 5. The magnitude of the catalytic current may also be varied by changing the concentration of mediator and/or the surface concentration of enzyme in each monolayer. The effect of these factors on inhibition is seen upon comparison of Figures 1 and 6 (left-hand side). In the first case the concentration of mediator is 0.2 mM and the surface concentration of enzyme in each monolayer is 1.5 × 10-12 M cm-2 resulting in a peak current of 90 µA, whereas these factors are 0.4 mM, 2.65 × 10-12 M cm-2 and 150 µA, respectively, in the second case. While inhibition is quite severe in the second case, it is practically negligible in terms of peak (or plateau) currents in the first. This is the reason that inhibition by protons and glucono-δlactone production could be neglected in previous studies where the experimental conditions were identical or similar to those depicted in the caption of Figure 1. Conclusions
Figure 6. Cyclic voltammetric oxidation of 0.4 M glucose by glucose oxidase at a 10-layer electrode (2.65 × 10-12 mol/cm2 enzyme per monolayer) in the presence of 0.4 mM ferrocene methanol. Scan rate: 0.05 V/s. Temperature: 25 °C. pH ) 8.0. Ionic strength: 0.1 M. Right: 1.48 mM NaH2PO4, 16.5 mM Na2HPO4, and 16.4 mM Na2SO4. Left: 32.4 mM NaH2PO4, 26.4 mM Na2HPO4. From top to bottom: cyclic voltammograms (solid lines: experimental data; dashed lines: no inhibition; dotted lines: simulation of inhibition by proton and glucono-δ-lactone production), pH, ionic strength, and gluconoδ-lactone concentration at the electrode surface. (b) Start of the potential scan, (O) inversion of the potential scan.
decreases as the pH decreases because the doubly charged phosphate ion is replaced by the singly charged phosphate. This decrease of the ionic strength counteracts the effect of pH of the kinetic of the enzymatic reaction as can be seen from the fact that the actual cyclic voltammogram exhibits less inhibition than predicted by the simulation where the effect of ionic strength is ignored. This counteracting effect of the ionic strength is, however, modest as compared to the strong pH effect. In the experiment reported in the left-hand side of Figure 6, the variation of pH is less because the buffer is more concentrated. The counteracting ionic strength effect is thus more pronounced in relative value. The result is that inhibition is less on total and the difference between the actual cyclic voltammogram and the simulation ignoring the ionic strength effect is larger than in the right-hand case. It is also worth noting that a similar difference between actual and simulated cyclic voltammograms is also found for the same diluted solution as in the right-hand side of Figure 6 but at a higher scan rate, 0.2 instead of 0.05 V/s (Figure 3). Due to the higher scan rate, the decrease of the pH is less dramatic (simulation shows that it does not decrease below 6.5) and, again, the counteracting ionic strength effect is more pronounced in relative value. We also see that, although inhibition is mainly caused by proton production, a significant amount of glucono-δ-lactone
Films containing large concentrations of enzymes, thus giving rise to large catalytic currents when wired to an electrode surface, may exhibit quite significant inhibition by products. Even with an enzyme such as glucose oxidase where the Michaelis constant for glucono-δ-lactone is as large as 1.4 × 10-2 M, inhibition by glucono-δ-lactone interferes quite significantly with film containing a total surface concentration of enzyme of the order of 3 × 10-11 mol/cm2, giving rise to current densities of the order of 2 mA/cm2. Under the same conditions, the inhibiting effect of proton production is much larger than the effect of glucono-δ-lactone, even with buffer concentrations in the commonly used range. These inhibiting effects can be treated theoretically by means of a diffusion-reaction model which uses as key parameters the product Michaelis constant and the pH dependence of the enzymes kinetics which can be derived from homogeneous experiments. It thus allows one to rationalize and to predict the microenvironmental effects and the feedback processes that accompany the enzymatic reaction. Although the simulation procedures were established for films made of spatially ordered successive monolayers, they can be easily extended to the case of more disordered enzyme assemblies.11 Experimental Section Glucono-δ-lactone was purchased from Aldrich and used as received. All other chemicals as well as instrument and procedures were as described earlier.8 The working electrode was a 0.07 cm2 glassy carbon disk. Supporting Information Available: The changes of variables and introduction of characteristic parameters leading to a dimensionless formulation of the diffusion/reaction problem. This material is available free of charge via the Internet at http:// pubs.acs.org. References and Notes (1) (a) Universite´ de Paris 7sDenis Diderot. (b) Universite´ de Technologie de Compie`gne.
Immobilized Enzyme Films (2) (a) Katchalsky, A.; Silman, I.; Goldman, R. AdV. Enzymol. 1971, 34, 445. (b) Goldman, R.; Goldstein, L.; Katchalsky, A. Water-Insoluble Enzyme Derivatives and Artificial Enzyme Membranes. In Biochemical Aspects of Reactions on Solid Supports; Stark, G. R., Ed.; Academic Press: New York, 1971; pp 1-78. (c) Mattiasson, B.; Mosbach, K. Biochim. Biophys. Acta. 1971, 235, 253. (d) Broun, G.; Thomas, D.; Selegny, E. J. Membr. Biol. 1972, 8, 313. (e) Srere, P.; Mattiasson, B.; Mosbach, K. Proc. Natl. Acad. Sci. U.S.A. 1973, 70, 2534. (f) Mosbach, K. FEBS Lett. (Suppl.) 1976, 62, E80. (3) (a) Naparstek, A.; Romette, J. L.; Kernevez, J. P.; Thomas, D. Nature 1974, 249, 490. (b) Thomas, D.; Barbotin, J. N.; David, A.; Hervagault, J. F.; Romette, J. L. Proc. Natl. Acad. Sci. U.S.A. 1977, 74, 5314. (4) (a) Heller, A. Acc. Chem. Res. 1990, 23, 128. (b) Ohara, T. J.; Vreeke, M. S.; Battaglini, F.; Heller, A. Electroanalysis 1993, 5, 825. (c) Cosnier, S. Electroanalysis 1997, 9, 894. (5) (a) Turner, A. P. F.; Karube, I.; Wilson, G. S. Biosensors; Oxford University Press: Oxford, UK, 1987. (b) Lomen, C. E.; de Alvis, W. U.; Wilson, G. S. J. Chem. Soc., Faraday Trans. 1 1986, 82, 1265. (c) de Alvis, W. U.; Hill, B. S.; Maklejohn, B. I.; Wilson, G. S. Anal. Chem. 1987, 59, 2688. (d) Jung, S. K.; Wilson, G. S. Anal. Chem. 1996, 68, 591. (6) (a) Muller, W.; Ringsdorf, H.; Rump, E.; Wilburg, G.; Zhang, X.; Angermaier, L.; Knoll, W.; Liley, M.; Spinke, J. Science 1993, 262, 1706. (b) Spinke, J.; Liley, M.; Guder, H.-J.; Angermaier, L.; Knoll, W. Langmuir 1993, 9, 1821. (c) Lvov, Y.; Ariga, K.; Ichinose, I.; Kunitake, T. J. Am. Chem. Soc. 1995, 117, 6117. (7) (a) Bartlett, P. N.; Birkin, P. R. Anal. Chem. 1993, 65, 1118. (b) Bartlett, P. N.; Birkin, P. R. Anal. Chem. 1994, 66, 1552. (c) Hodak, J.; Etchenique, R.; Calvo, E.; Singhal, K.; Bartlett, P. N. Langmuir 1997, 13, 2708.
J. Phys. Chem. B, Vol. 103, No. 40, 1999 8537 (8) (a) Bourdillon, C.; Demaille, C.; Moiroux, J.; Save´ant, J-M. J. Am. Chem. Soc. 1993, 115, 2. (b) Bourdillon, C.; Demaille, C.; Gue´ris, J.; Moiroux, J.; Save´ant, J-M. J. Am. Chem. Soc. 1993, 115, 12264. (c) Bourdillon, C.; Demaille, C.; Moiroux, J.; Save´ant, J-M. J. Am. Chem. Soc. 1994, 116, 10328. (d) Bourdillon, C.; Demaille, C.; Moiroux, J.; Save´ant, J-M. J. Am. Chem. Soc. 1995, 117, 11499. (e) Bourdillon, C.; Demaille, C.; Moiroux, J.; Save´ant, J-M. Acc. Chem. Res. 1996, 29, 529. (f) Anicet, N.; Bourdillon, C.; Moiroux, J.; Save´ant, J-M. J. Am. Chem. Soc. 1998, 120, 7115. (g) Anicet, N.; Anne A.; Moiroux, J.; Save´ant, J-M. J. Phys. Chem. B 1998, 102, 9844. (9) (a) Nakamura, T.; Ogura, Y. J. Biochem. (Tokyo) 1962, 52, 214. (b) Gibson, Q. H.; Swoboda, B. E. P.; Massey, V. J. Biol. Chem. 1964, 239, 3927. (c) Duke, F. R.; Weibel, M.; Page, D. S.; Bulgrin, V. G.; Luthy, J. J. Am. Chem. Soc. 1969, 91, 3904. (10) Pocker, Y.; Green, E. J. Am. Chem. Soc. 1973, 75, 113. (11) (a) All diffusion coefficients are set equal (D ) 5.5 × 10-6 cm2 -1 s ) with the exception of P (DP ) 3.3 × 10-6 cm2 s-1).8d (b) The distance l was derived, in a previous work,8d,g from experiments in which the electrode coating was made of one active monolayer deposited on top of 10 inactivated monolayers. Under these conditions, the catalytic currents are low enough for product inhibition to be negligible. (12) (a) For a film made of N successive monolayers, one obtains (see Supporting Information) a series of equations which are the same as the finite difference equations that appear in the approximate resolution of continuous systems obeying Fick’s law. Thus, for continuous systems, one may divide the film thickness into a finite number of intervals and apply the same procedures as used here to the resulting finite difference equations. (b) Crank, J. Mathematics of Diffusion; Oxford University Press; London, 1964; pp 789-790. (c) Smith, G. D. Numerical Solutions of Partial Differential Equations; Oxford Mathematical Handbook: London, 1971, pp. 923.
