Anal. Chem. 1993, 65, 3129-3133
Theoretical Evaluation of Mediation Efficiency in Enzyme-Incorporated Electrodes Tetsu Tatsuma**+ and Tadashi Watanabe’ Institute of Industrial Science, University of Tokyo, Roppongi, Minato-ku, Tokyo 106, Japan
A theoretical examination was made to determine whether the mediation efficiency is enhanced in enzyme-incorporatedelectrodesowing to the close proximity of the electrodeto all enzyme molecules. Two enzyme electrodes, an enzyme-incorporated conductingfilm electrodeon an insulating support and a solid electrode carrying an enzyme-incorporated insulator film (models A and B, respectively), were modeled, and the steady-state mediation efficiencies were evaluated chiefly by numericalcomputation. The results thus obtained show that a fast electrochemical reaction of the mediator leads to higher mediation efficiency in model A than in model B. The difference in the efficiency beween the two models increases with an increase in the rate of the enzymatic reaction. When the electrochemical reaction is slow and the enzymatic reaction is fast, the mediation efficiencies in the two models are comparable to each other. In the case where both electrochemicaland enzymatic reactions are slow, the efficiency in model B is higher than that in model A. These features were interpreted in terms of the concentration profiles of the charged mediator. Dependencies of the difference in the mediation efficiencies between the two models on the thickness of the film and of the diffusion layer were also examined. INTRODUCTION The sensitivity of an amperometricenzyme electrode under the condition where the response is proportional to substrate concentration is dictated by three factors:1.2 (1) the total enzyme activity in the enzyme layer, (2) the efficiencies of substrate supply toward the enzyme, and (3) the enzyme/ electrode charge-transfer efficiency or mediation efficiency in the case of mediated enzyme electrodes. From the viewpoint of the microfabricationof enzyme electrodes, every enzyme molecule is desired to work efficiently. This is expected to be achieved by enhancement of the latter two factors,the substrate supplyefficiency and the charge-transfer efficiency. We have studied enzyme monolayer- and bilayermodified electrodes in view of this .14 In these electrodes, tPreaent address: Department of Applied Chemistry, Faculty of Technology, Tokyo University of Agriculture and Technology, Koganei, Tokyo 184, Japan. (1) Tatsuma, T.; Watanabe, T. J. Electroanal. Chem. 1991,310,149167. (2) Tatsuma, T.; Watanabe, T. Anal. Chem. 1992,64,626-630. (3) Okawa, Y.;Tsuzuki, H.; Yoehida, 5.;Watanabe, T. Anal. Sci. 1989, 5,607-612. (4) Tatauma,T.;Okawa,Y.; Watanabe,T. Anal. Chem. 1989,61,23622366. (6)Tat”, T.; Tsuzuki, H.; Okawa, Y.; Yoshida, S.; Watanabe, T. Thin Solid Elmu 1991,202,146-160.
all the enzyme molecules are in contact with a solution containing an analyte, and the total amount of an enzyme is so low that the substrate supply efficiency is expected to be high: as has been verified experimentally9~4andtheoretically.2 Further, the close proximity of all enzyme molecules to the electrode surface in the monolayer electrode7 leads to high mediation efficiency when a mediator is sufficiently electroactive.112 Recently, enzyme electrodes in which an enzyme is incorporated in an electrode material such as a conducting polymer,a13redox polymer,14-16 carbon paste,17J8 or platinum black19 have been studied extensively. These enzymeincorporated electrodes are anticipated to exhibit a high enzyme/electrode charge-transfer efficiency, whether the transfer is mediated or direct, because of the close contact of all enzyme molecules with the electrode. Actually, it is clear that a solid electrode carrying an enzyme-incorporated electrically conducting film exhibita higher charge-transfer efficiency than that carrying enzyme-incorporated electrically insulating film. This is because both the solid support and the film work as an electrode in the former while the solid support alone works as an electrode in the latter. Hence, such a comparison cannot reveal whether the close proximity of the electrode to enzyme enhances the efficiency. Thus, we attempt to make a just comparison in the present work. To achieve this end, we should compare the mediationefficiencies of two different enzyme electrodes, an enzyme-incorporated conducting film electrode on an insulator solid support (Figure 1A) and a solid electrode carrying an enzyme-incorporated insulator f i (Figure lB),in which enzyme-incorporatedfilms are permeable to the substrate and mediator. The film of the former and the latter electrodes may be polypyrrole and poly(vinyl chloride), respectively, and the solid support of the former and the latter electrodes may be glass and glassy carbon, respectively. However, it is almost impossible to fabricate these enzyme electrodes sharing the same value for all parameters. (6) Moody, G.J.; Sanghera, G. S.;Thomae, J. D. R. Analyst 1986,111, 1236-1238. (7) Bowdillon, C.; Bourgeois, J. P.; Thomas, D. J. Am. Chem. SOC. 1980,102,4231-4235. (8)Foulds, N. C.; Lowe, C. R. J. Chem. SOC.,Faraday Trone. 1 1986, 82,1259-1264. (9) Umana, M.; Waller, J. Anal. Chem. 1986,58,2979-2983. (10) Foulds, N. C.; Lowe, C. R. Anal. Chem. 1988,60, 2473-2478. (ll)Iwakwa, C.; Kajiya, Y.; Yoneyama, H. J. Chem. SOC.,Chem. Commun. 1988,1019-1020. (12) Yabuki, S.; Shinohara, H.; Aizawa, M. J. Chem. SOC., Chem. Common. 1989, 946-946. (13) Tatsuma, T.;Gondaira, M.; Watanabe, T. Anal. Chem. 1992,64, 1183-1187. (14) Gregg, B. A.; Heller, A. J. Phys. Chem. 1991,95,6976-6980. (15) Hale,P. D.; Boguelavsky, L. I.; Inagaki, T.; Karan, H.I.; Lee, H. 5.;Skotheim, T. A,; Okamoto, Y. Anal. Chem. 1991,63,677+82. (16) Pishko, M. V.; Michael, A. C.; Heller, A. Anal. Chem. 1991,63, 2268-2272. (17) Matuezewski, W.; Trojanowicz, M. Analyst 1988,113, 736. (18) Wang, J.; Wu, L.-H.;Lu, Z.; Li, R.; Sanchez, J. Anal. Chim. Acta 1990,228,261-257. (19) Ikariyama, Y.; Yamauchi, S.; Yukiashi, T.; Ushioda, H. J. Electrochem. SOC.1989,136, 702-706.
0003-2700/93/036~12~$04.00/0 (8 1993 American Chemical Society
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ANALYTICAL CHEMISTRY, VOL. 65, NO. 21, NOVEMBER 1, 1993
respectively. The charge(s) (positiveor negative) of the latter is(are) supplied from the substrate S, which is thereby converted into the product, P. M and N are mediators before and after the charge exchange with E’. The constants kl and k2 are the second-order enzymatic reaction rate constants. The charged mediator N is discharged in the conducting film (model A) or on the base electrode surface (model B):
B
A
-+ kA or ke
e, Y
9 VI 1 3 a Y
m
-E .“ f&
Insulator
‘
e .Y 4
Conductor
a
‘2 U
0 Enzyme
v)
Flgute 1. Schematic Illustration of the enzyme electrode models.
Here we treat this problem by theoretical modeling. The steady-state responses of the enzyme electrodes are simulated on the basis of the models for the above-mentioned two mediated enzyme electrodes. Here we call the model for the former electrode (Figure 1A) model A, and that for the latter electrode (Figure 1B) model B. In the present analysis, we used a common electrode reaction rate constant per geometrical surface area for the film in model A and for the solid support in model B, regardless of the film thickness and the reacting surface area. If the reaction rate constants are different between these models, the difference in simulated efficiency cannot be ascribed to conductivity of film and solid alone because it may be caused by the difference in the rate constants. Though several worker~6J,20-2~ have addressed themselves to model analysis of enzyme electrodes, analysis for enzymeincorporated electrodes has so far been presented only by Bartlett and Whitaker25to the best of our knowledge. They dealt with a special case alone where the electrochemical reaction of a mediator in the electrode material is infinitely fast. In such a case, all of the mediator molecules discharge on accepting a charge from the enzyme and cannot dissipate out the enzyme-incorporated electrode layer, and thereby, the mediation efficiency is necessarily unity. However, an electrode reaction is usually not sufficientlyfast and a portion of the mediator molecules may diffuse out of the membrane. Further, no work has been done from the viewpoint of chargetransfer efficiency. The aim of the present work is not to simulate the responses of actual enzyme electrodes, but to compare the steady-state characteristicsof modelsA and B to examine whether enzymeincorporated electrodes lead to higher mediation efficiency.
MODEL ANALYSIS Here we model a mediated enzyme electrode which responds to a substrate on the basis of charge transfer from the substrate to the electrode via an enzyme and a dissolved electron mediator. The enzyme is a redox enzyme with the following elementary reactions:
E+S
kl +
E’+ P
k2
+
E’ + M E N where E and E’ are native and charge-carrying enzymes, +
N
M
charge(& (3) where kA is the first-order rate constant in reciprocal seconds, and kB is the heterogeneous rate constant in centimeters per second. As mentioned above, we used a common electrode reaction rate constant per surface area in both models regardless of the thickness of the conducting film, that is k,I = k B (4) where I is the thickness of enzyme-incorporated films. We assume the following for the model analysis: (1)Every interface involved is an ideal plane with infinite area. (2) Enzyme molecules are homogeneously distributed in the film with a concentration of CE. (3) No convection or migration but diffusion alone contributes to mass transfer within the film and within the diffusion layer in the solution. (4) The concentrations of S and M are constant (CS* and CM*, respectively) and the concentration of N is 0 outside the diffusion layer. ( 5 ) Mass transfer does not occur in the solid support. (6) S and P are electrochemically inert. (7) The substrate, enzyme, and mediator exchange the same number of charges, n. (8) The substrate concentration in the film is so low that reaction 1limits the enzymatic reaction rate and that the depletion of mediator in the film is negligible. Indeed, a sensor should work under such a concentration for proportionality between the concentration and the sensor response. (9) The enzymatic reaction is so fast that it can be treated as in steady state. The film thickness and the diffusion layer thickness are I and d, respectively. Diffusion coefficients and partition coefficients are represented by D and K, respectively. The rate of concentration change owing to diffusion UD is u&)
a2C(x)
=D ax2
(5)
where x is the distance from the electrode surface (0 d x d I). Since it is assumed that the enzymatic reaction is in a steady state and that the rate of it is determined by reaction 1as mentioned above, the enzymatic reaction rate U E is given by uE(x) = klC&s(x)
(6)
The electrode reaction rates in models A and B, uc and j c (flux), respectively, are given by uC(X)
= kACN(x)
(7)
Thus, mass balances at steady state can be summarized as follows: (9)
(20) Mell, L. D.; Maloy, J. T. Anal. Chem. 1976,47, 299-307. (21) Gough, D. A.; Leypoldt, J. K,J. Electrochem. SOC.1980, 127, 1278-1286. (22) Leypoldt, J. K.; Gough, D. A. Anal. Chem. 1984,56,2896-2904. (23) Bergel, A,; Comtat, M. Anal. Chem. 1984,56, 2904-2909. (24) Tse, P. H. S.; Gough, D. A. Anal. Chem. 1987,59, 2339-2344. (25) Bartlett, P. N.; Whitaker, R. G. J. Electroanal. Chem. 1987,224, 27-35. (26) Tatsuma, T.; Watanabe, T.; Okawa, Y. Anal. Chem. 1992, 64, 630-636.
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0
-1 01
00
3 The differential eqs 9 and 13 are solved for the case where d = 0 as follows:
c,
= c,
-2
*em + eea' e*'
+
-3
(model A)
-5
-4 Log (kAl or k
-3
-2
cm s-')
Flgurr 2. Dependencies of the medlatlon efflciency, e, In models A (0,A, V) and B (0,A,V)on the electrode reaction rate constants
and the enzymatlc reactlon rate constants. Values of parameters: C, = (0, O), 10" (A,A),or 10-6(V,V)mol cm4; & = 10-a mol cm-? C, = 104 mol cm4; k, = k2 = 108 cms mol-' s-1; 4 = Q, = lo-' cm2s-l;Ks = KM = KN = 1.0; I = lo4 cm; and d = 0 cm.
(model B)
RESULTS AND DISCUSSION
2 em where constants a and j3 are given by
ea' -
+ e-e*'- + 11 +
a = (klCE/Ds)1f2
(19)
(20)
0 = (kA/DN)lf2= (k$DNZ)lf2
(21) The substrate concentration profiles are common for both models, since the mediator concentration is sufficiently high so that the mediator concentration in the film is equal to that in the solution bulk, namely, C d x ) = CM*.Equations 18 and 19 are for models A and B, respectively. It must be noted that eq 18 does not hold in the case where a = 8. The steadystate mediation efficiencies in model A, eA, and that in model B, eB, are evaluated by the following equations:
Concentration profiles and the mediation efficiencies are evaluated not only from the above equations but also by computation on the basis of the difference equations derived from the differential equations (eqs 9 and 13). The enzymeincorporated f i b s are divided into lo00 layers with thickness Ax (=Z/lo00). A concentration at x = 0 is assumed and a concentration at x = Ax is evaluated from this value by eq 10 or 14. A concentration at x = 2Ax is obtained by substituting these values into the difference equation. A concentration at x = 1 obtained by repetition of such an operation is fitted to the value given by eq 11 or 15 by changing the concentration at x = 0. Computation was carried out on a PC-9801DA computer (NEC, Japan) with a coprocessor, 80387DX-20 (Intel). Results obtained by analytical and numerical methods were in excellent agreement.
Dependencies of the Mediation Efficiency on the Parameters. Figure 2 shows how the mediation efficiency in models A and B depends on the electrode reaction rate constants and the enzyme concentration in the film. The parameters are given in the caption. Larger rate constants give higher mediation efficienciesin both models, as expected. The dependence of the mediation efficiencyon ~ / D is Nsimilar to the dependence on the electrode reaction rate constants. Thus, the mediation efficiency is high when the electrode reaction is controlled by diffusion of the charged mediator (N); the charged mediator is difficult to dissipate out of the membrane under diffusion-controlled conditions. Lower enzyme concentration in the film makes the mediation efficiency higher. It was found that the mediation efficiency depends on 1/Ds in a manner similar to its dependence on the enzyme concentration. The mediation efficiency is low when the enzymatic reaction is limited by diffusion of the substrate. In such a case, the enzymatic reaction occurs only in the vicinity of the film/solution interface, and hence the charged mediator generated as a result of the enzymatic reaction dissipates out of the f i b easily so that the mediation efficiencyis low. On the contrary, when the enzymaticprocess is reaction-controlled,the charged mediator is generated in the film bulk so that the mediation efficiency is higher. Next we compare the mediation efficiencies in models A and B. When the electrode reaction is controlled by diffusion of the charged mediator (namely, j3 is large), the efficiency is higher in model A than in model B. The difference between eA and e B is larger when the enzymatic reaction is diffusionlimited (namely, a is large). When the electrode reaction is not limited by diffusion of the charged mediator, e A and eB are closer to each other. In the case where the enzymatic reaction is not controlledby diffusion, the mediation efficiency is higher in model B than in model A. These observations can be explained in terms of the concentration profiles of the charged mediator. Figure 3 shows the concentration profiles of substrate (S)and charged mediator (N). When a is large enough, substrate concentration is low at the film bulk (curve 1 in Figure 3B) so that the charged mediator is generated in the vicinity of the film/ solution interface. When j3 is large, a large portion of the
ANALYTICAL CHEMISTRY, VOL. 65, NO. 21,NOVEMBER 1, 1993
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1.0
?
-5
z
0.5
r(
X
u
n
-*
I O
0 X
-4
1 X
Flguro5. Slmuiated concentratkm profiles of the substrate (1)and the charged mediator (2-5)In the fllm. Values of parameters are the same as In Figure 2 except for C, [ = 10-e(A) or lo-' (B) mol cm4] and kA/= kB [=IO-' (2,3) or 10-2 (4, 5) cm 8-11. Solid and broken lines are for models A and B, respectively.
Q)
charged mediator is consumed in the film for model A, while a large portion of it is dissipated out of the film according to the concentration gradient (curve 5 in Figure 3B) for model B. Thus e A is higher than e B in this case. For a small a,the enzymatic reaction is not controlled by the substrate transport (curve 1 in Figure 3A), hence the charged mediator is generated in the bulk of the film. Therefore, when B is large enough, almost all of the charged mediator is consumed in the film for model A and about the half of it diffuses toward the solid electrode surface and the other portion diffuses out of the film according to the concentration gradient (curve 5 in Figure 3A). Thus, e A is about 2-fold higher than e B . For a large a and a small 0, the concentration profile of the charged mediator for model A (curve 2 in Figure 3B) is similar to that for model B (curve 3 in Figure 3B) and the average concentration in the film is nearly equal to that on the solid surface ( x = 0). Since the electrochemical reaction rate is proportional to the average concentration (model A) or the concentration at the solid electrode surface (model B), e A and e B are close to each other. The concentrationprofiles of the charged mediator for both models (curves 2 and 3 in Figure 3A) are similar to each other when both a and B are small enough. In this case, however, the average concentration of the charged mediator is lower than the concentration on the solid surface so that eA is lower than e B . Examination of Limiting Cases. Four limiting cases arising from the conditions that the enzymatic reaction is limited by the substrate transport or not, and that the electrode reaction is limited by the charged-mediatortransport or not, are examined in this section. We assume that d = 0 and K = 1, and that 1 has a finite value. The mediation efficiencies in the limiting cases are evaluated from eqs 22 and 23. In the case where a = and B = =, that is, both the enzymatic reaction and the electrode reaction are limited by mass transfer, the mediation efficiencies are obtained as follows: eA =
+ B)
(24)
e B = l/ffl= 0 (25) The efficiency for model A dependson the a/@ratio. However, e A >> e B even if a >> 8, where t?A = 0. When a = 0 and @ = are obtained. In the case where a = 0, e A = 1 and f?B =
-2
-3
Log (1, cm) Figuro 4. Dependencies of the medlatlon efficiency, e, In models A (0)and B (0)on the film thickness. Values of parameters are the same as In Figure 2 except for C,(=1od mol cma), kA/= kB(=lo-' cm e-'), and 1.
and @ = 0, the mediation efficiencies are
= es a @'1/a When a = 0 and B = 0, the efficiencies are given by eA = @'1'/3 = 0 e B a 8'1'/2
0
(26) (27)
(28) Therefore, 1 . 5 = ~ These results agree completely with results obtained by numerical simulation. Effect of Film Thickness. Figure 4 depicta the dependencies of the simulated mediationefficiency in the two models on the film thickness. In thick f i s , mass transfer tends to be limited and hence the concentration profiles approach those for large a and B (curves 4 and 5 in Figure 3B) so that the e d e B ratio is high. In contrast, in thinner films, mass transfer tends to be facile and hence the concentrationprofiles approach those for small a! and /3 (curves 2 and 3 in Figure 3A) so that the e d e B ratio is low and may be lower than unity. Here we note again that the first-order electrochemical rate constant in model A, k ~is, inversely proportional to the film thickness 1 in the present model (eq 4). Effect of Diffusion Layer Thickness. Next we consider the case where the diffusion layer thickness is not negligible. When the diffusion layer is thick, the charged mediator accumulates in the layer. A thicker diffusion layer makes the concentration gradient of the charged mediator smaller and thereby the mediation efficiency higher because the charged mediator is difficult to diffuse out of the film. Furthermore, the difference between e A and e B decreases with increasingthickness of the diffusion layer. When the diffusion layer is sufficiently thick and B is sufficiently large, e B = 1.0 as well as e A becauae the concentrationgradient of the charged mediator is steeper in the film than in the diffusion layer. The e J e B ratio is nearly equal to unity also with small a, since the concentration gradient of the charged mediator in the film is nearly zero so that the average concentration in the film is nearly equal to that on the solid surface ( x = 0). Thus, the comparative study presented in this paper has no significance when the diffusion layer is sufficientlythick, that is, IIDK
