Anal. Chem. 1994,66, 2163-2770
Model for an Immobilized Oxidase Enzyme Electrode in the Presence of Two Oxidants Nlcolas Martens and Elizabeth A. H. Hall’ Institute of Biotechnology, University of Cambridge, Tennis Court Road, Cambridge, UK CB2 1QT
In mediated oxidase electrodes,the artificialelectron acceptor competes with oxygen for the reoxidation of the enzyme. A steady-state model for a three-substrate electrode is employed to investigate the influence of oxygen on the current response of an amperometric enzymeelectrodeand concentrationprofiles predicted in the enzyme layer for the substrates. The model predicts the electrode response for different 0 2 levels, and the influence of model parameters on the electrode performance is tested and discussed with the view of improved sensor design. The general principle of enzyme electrodes was introduced about three decades ago by Clark and Lyons.’ Since then many biosensors based on electrochemical enzyme electrodes have been described, the majority of these devices operating in an amperometric mode. However, although the basic principle of these enzyme electrodes is well understood, the development of practical sensors has been somewhat limited. The fact that relatively few enzyme electrodes have been commercialized may be due to the technical problems associated with either the enzyme reaction or the underlying product/substrate-sensitiveelectrode, such as the availability of appropriate enzymes, their successful immobilization, and probably most importantly the achievement of an acceptable catalytic lifetime of the enzyme. A primary requirement for the optimization of the performance of an enzyme electrode is the detailed understanding of the mechanisms which govern the behavior of the electrode. The physiochemical phenomena involved are complex, including mass transfer of the reactants in the immobilization matrix, the enzyme kinetics, and the electrochemical process that links the enzyme reaction to the electrode. The development of models for enzyme electrodes provides a better understanding of the individual processes influencing the response of the device, and this information may be used as a guide for directions for improvement of the sensor design. There have been many reports on models for enzyme electrodes. Schulmeister has described models for multilayer and multienzyme electrodes; these models assumed operation of the electrode under diffusion control, such that the enzyme kinetics are linear with substrate. This allows the reaction and diffusion system to be described by a parabolic differential equation with linear inhom~geneities.~~~ A model for a twosubstrate enzyme electrode has been devised by Leypoldt and ( 1 ) Clark, L. C.; Lyons, L. Ann. N.Y.Acad. Sci. 1962, 102, 29-45. (2) Schulmeister,T. Selecrfve Electrode Reo. 1990, 12, 203-260. (3) Schulmeister,Th.; Pfeiffer, D. Eiosens. Eioelecrron. 1993, 8, 75-79.
0003-2700/94/03662?63~04.5~/0 0 1994 American Chemical Society
Gough where the nonlinear enzyme reaction was taken into account. This model was employed to describe the behavior of a glucose oxidase (GOx) e l e ~ t r o d e The . ~ transient response of a mediated amperometric enzyme electrode was studied by Bergel and Comtat, employing an implicit finite difference method.5 In the models mentioned above, either a one- or twosubstrate enzyme reaction was considered. However, many reports on enzyme electrodes deal with the replacement of the natural cosubstrate by an artificial redox mediator to improve the sensor performance. Nevertheless, it has been found that these mediators could not totally replace the natural cosubstrate when both were present in the assay solution, so that here, a three-substrate model would be required. In these cases, a complex calibration curve of the enzyme electrode was observed.”* In this paper, we describe a model for this three-substrate enzyme electrode and employ the model to investigate the influence of the oxygen on the current response of a mediated oxidase electrode.
THE MODEL We consider the following reaction scheme for the immobilized oxidase in the presence of two oxidants: ki
E,,
+ S 8 ES ki
E,,
+ Med,,
-
k,
k5
E,,
+P
(1)
+ Med,,
(3)
E,,
The following assumptions are made in the simulation of the kinetics of immobilized oxidase: (i) Steady state, Le., the concentration profiles of all reactants and all enzyme intermediates remain constant over time and the concentration of the total active enzyme [Et] remains constant in the matrix. (ii) The concentrations of the reactants in the bulk electrolyte [S]b, [02]br [Med,d]b, and [Med,,]b remain constant. (iii) Diffusion of the reactants can be described by Fick’s second law. (4) Leypoldt, J. K.; Gough, D. A. Anal. Chem. 1984,56, 28962904. ( 5 ) Bergel, A.; Comtat, M. Anal. Chem. 1984, 56, 2904-2909. (6) Martens, N.; Hindle, A.; Hall, E. A. H. Bimem. Eimlecrron., in press. (7) Pallachi, G.; Turner, A. P. F. Anal. Chim. Acta 1990, 234, 459-463. (8) Ohara, T. Y.;Rajagopalan, R.; Hellcr, A. Anal. Chem. 1993,65,3512-3517.
AnalyticalChemistty, Vol. 06, No. 17, September 1, 1994 2763
(iv) The enzyme is uniformly distributed throughout the matrix and enzyme activity is not a function of position. (v) No product or substrate inhibition occurs. (vi) The partitioning coefficients for oxygen, substrate, and mediator are 1. Using these conditions eqs 1-3 give [E,] = [E,,] + PSI + [Eredl -= d[ES1
dt
kl[Eo,][S] - [ES](k2+ k3) = 0
(4)
(5)
Theconcentrations of [ES], [Er4],and [E,] can bedescribed as follows using the nomenclature of Atkin~on:~
= [O,]
[=I [Med,,]
-+Po
(7)
Bm
4,Dm, and Do are the respective diffusion coefficients of substrate, mediator, and oxygen. The boundary conditions at the matrix/electrolyte and matrix/electrode interfaces are aty=d
aty=0 At steady state, the diffusion of a substrate into the enzyme layer is equal to the reaction rate of that substrate within this matrix. We consider a planar matrix of the thickness y = d, and diffusion is considered to be in they direction only (edge effects are neglected). The material balance of substrate, oxygen, and mediator, within a differential thickness in the matrix, gives
(9) Atkinson, E.;Lester, D.E. Blotcchnol. Bioeng. 1V4,16, 1299.
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AnalytlcalChemktty, Vol. 86, No. 17, September 1, 1994
w 2 1 --d[SI --0
dY dY The first boundary condition states that the substrates are in equilibrium with the surrounding fluid at the matrix/ electrolyte interface. The boundary condition at the electrode/ matrix interface states that no flux of oxygen or substrate occursat thisboundaryand that theredudmediator israpidly oxidized (diffusion controlled) at the electrode. KO,Ks,and Kmare the partitioning coefficients for oxygen, substrate, and mediator, respectively. These equations and the accompanyingboundary conditions can be normalized employing the following parameters:
Equation 21 demonstrates that the normalized substrate concentration, within a polymer matrix containing an immobilized oxidase, is linearly dependent on the normalized concentrations of mediator and oxygen. Since we are interested in the electrode response, due to the oxidation of the mediator at the electrode, and the influence of oxygen on this response, only the concentration profiles for oxygen and mediator are considered hereafter. Substitution of eq 21 into eqs 16 and 17 results in
Parameter group A Equations 12-14 become d2F, dx2 =
@:['F s 0+ FmBm FaB, +L]-l
+
(15)
with the boundary conditions atx= 1
F, = Fo = F, = 1 atx=O F m = l , dFo -=-= dF, dx dx
0
It is advantageous to simplify the above set of second-order differential equations by solving for Fs.Adding eqs 16 and 17 and substitution in eq 15 yields d2F, -- - Po d2Fo -+ Pm-
d2F, dx2
dx2
dx2
(18)
The boundary conditions stated for the eqs 16 and 17 remain unchanged for eqs 23 and 24. These equations govern the reaction/diffusion system in the polymer matrix, and all parameters are contained in the normalized groups a2,p,, and B,. The model above describes a mediator which exists in the oxidized form in the bulk solution. However, some of the commonly employed redox mediators for enzyme electrodes (e.g., ferrocenes) are present in the electrolyte in their reduced form. To account for this situation, we now consider the concentration profiles of the reduced mediator in the enzyme matrix. d2Fd --d2Fm dx2
with IL,
Do[oZlb
= - =-
Ds[slb
(19)
dx2
The governing normalized equations now become dzFs dx2
-- - "[ ' -IFOBo - +'F,Bm Integrating eq 18 and applying the appropriate boundary conditions gives F,
P$'~
+ rmFm + (1 - w0 - pm + p,,,JOa(1 - x))
with Joa
=
(z),
+-!]-I
FBB,
(26)
dx2
(21)
with the boundary conditions given by AnaWc8l Chembtry, Vol. 66, No. 17, September 1, IS94
2765
atx= 1
Tablo 1. Ddautl Pararmtom Empbyod for tho “ o r l c a l Solutbn ot tho Qovomkrg Dmorontlrl Equatio”
Fmd = Fo= F, = 1
oxygen ([021b), mediator ([Med&,, [Meddlb), mmol/L enzyme layer thickness (4, cm enzyme mncn ([&I), mmol/L kl, L mol-’ s-l k2, S-’ kj, S-’ kd. L mol-’ s-l k5, L mol-’ s-l D,, cm2 s-I Do, cm2 s-* D,, cm2 s-I
atx=O
The normalized parameters are now defined as a
Parameter group C Again this set of differential equations can be simplified by solving for F,, This now results in F,
P Z O - P ~+ F 1~- P~O
+ ~ m - ~ m J r d (-x) l
(29)
with
(32)
(33)
Parameter Group D 2766
AmlytIcaIChemistry, Vol. 66, No. 17, September 1, 1994
1.2 1 .o
1.0 x 1 v 3
0.26 14 OOO 0
lo00 2.0 x 2.0 x 1.0 x 1.0 x 1.0 x
106 105 104 10-5 104
Unless otherwise stated in the respective figure legends.
The boundary conditions stated for eqs 26-28 are, of course, still valid for eqs 32 and 33. Numerical Solution. In both cases described above, the nonlinear two boundary value problem does not have a general analytical solution. However, accurate numerical solutions can be obtained, using, for example, a shooting algorithm described by Press.lo In this method, values for all dependent variables are “randomly” guessed at one boundary, satisfying the boundary conditions at that point. The differential equations are then integrated to arrive at the other boundary. The discrepancy from the desired boundary values is then minimized by adjustment of the initial guesses. The routine for the integration of the differential equations was based on the fourth-order RungeKutta method with adaptive stepsize control. Employing this technique, parameter values examined gave numerical results which were in agreement with the boundary conditions of the model and did not show sensitivity to the initial estimates.
RESULTS AND DISCUSSION The default parameters employed in this study are given in Table 1. The kinetic parameters, for thereaction of enzyme, used in this paper, were those determined for glucose oxidase by Weibel and Bright.” Model with Oxidized Mediator in Bulk Solution. Variations in the Thiele Moduli (ai; ai). The Thiele modulus can be varied by changing either the thickness of the enzyme layer or the amount of enzyme immobilized in the matrix (see parameter group A). This parameter describes the relative importance of diffusion and reaction in the enzyme layer. When a2is small, the kinetics are the dominant resistance; the overall uptake of substrate, oxygen, and mediator in the enzyme matrix is kinetically controlled. Under these conditions, the substrateconcentration profile across the membrane is essentially uniform. The overall kinetics are determined by the total amount of active enzyme. In contrast, when the Thiele modulus is large, diffusion limitations are the principal determining factor. Concentration profiles for oxygen and mediator are shown in Figure 1 for various values of Thiele moduli and constant concentrations of substrate, mediator, and oxygen in the bulk. Oxygen is depleted within the matrix as it is consumed by the (IO) Press,W. H.; Flannny,B. P.;Teukolsliy,S.A.;Vetterling, W.T. InNumerfcd Recipes in Pascal; Cambridge University Press: Cambridge, UK, 1989. (11) Wcibcl, M. K.; Bright, H. J . Biochem. 1971,121,801.
Table 2. Olo#clry
Symbols B
normalized concentration at the polymer/ electrolyte interface diffusion coefficient thickness of the polymer film enzyme enzyme/substrate complex normalized concentration within the polymer matrix partitioning coefficient mediator product normalized distance from the electrode distance from the electrode normalized reaction constants Thiele modulus Subscripts mediator oxygen oxidized species reduced species substrate total bulk solution
D d
E Es F K Med P X
Y
B
a* m 0
ox red S
t OD
al ci
0.8 c Q)
B
0
0.2
0.0
0.4
0.6
0.8
1.0
Normalised Distance
b) 0 C
0
=-
z
0
0.0 0.0
.
.
0.2
0.4
0.6
0.8
1.0
Normalised Distance F h r o 1. Concentration profiles for (a) the normalized oxygen concentration and (b) the normalized oxidized mediator concentration in the poiymer matrix for increasing Thiele rnoduil: ai (A)5, (A)IO, (M) 20, and (0)5 0 (A)50, (A)100, (B) 200, and ( 0 )500. The model parameters used for the calculations are given in Table 1.
enzyme reaction. The slope of this decrease in oxygen increases with the increase in the Thiele moduli (Figure la). The concentration of the oxidized mediator decreases within the enzyme matrix from both interfaces (x = 0 and x = l), reaching
a minimum at a distance within the membrane which is determined by the kinetics of the enzyme reaction and the diffusion properties.of the reactants. From x = 1 to x < 1, the concentration decreases due to consumption by the enzyme reaction, whereas the concentration of oxidized mediator also increases from x > 0 to x = 0 due to the reoxidation at the electrode. The minimum value of the concentration profile drops toward zero with increasing values of and (Figure lb). Variations in Substrate Concentration. Similar profiles are obtained when the Thiele moduli are held constant and the substrate concentration is varied in the bulk solution (Figure 2a,b). The concentration of oxygen in the vicinity of the electrode surface drops significantly even at relatively low substrate concentrations (Figure 2a), indicating that the reoxidation of the enzyme proceeds primarily through its reaction with oxygen. This competition is highlighted in Figure 2c, where the concentration profile of the mediator is plotted for a particular substrate concentration in the presence and absence of oxygen. The minimum in the profile becomes more pronounced, for a given substrate concentration, in the absence of oxygen. This is not surprising, as first, the reaction constant, for the reoxidation of the enzyme by oxygen, was assumed to be 10times higher than thevalue for the mediator reaction (Table 1) (which is appropriate if one compares the second-order reaction constants, determined for various ferrocene derivatives,I2 with the value found for oxygen13 ). Second, the mediator was recycled via the oxidation at the electrode, whereas neither oxygen nor the product was assumed to be recycled, nor even electroactive at the working potential. Oxidation Current Response due to Mediator at the Electrode ( x = 0 ) . The parameter of greatest interest in an amperometric biosensor is the current. In mediated amperometric enzyme electrodes, this is directly proportional to the first differential of the mediator concentration at the electrode surface. Simulated calibration curves for increasing oxygen concentrations are shown in Figure 3, where the normalized current is plotted against the glucose concentration as a function of oxygen concentration. It is immediately apparent that the shape of the calibration curve changes from the typical Michaelis-Menten-type behavior, observed for one oxidant (Le., mediator in the absence of oxygen), to the sigmoidal dependence of the normalized current on the substrate concentration as the concentration of oxygen increases in the bulk electrolyte. This influence on the current response was also observed experimentally for mediated enzyme electrodes employing glucose oxidase and other oxidases.68 By consideration of the concentraion profiles (Figures 1 and 2), it can be seen that the inflection in the calibration curve for mediator in the presence of oxygen is the result of the depletion of oxygen in the enzyme layer. At concentrations below this inflection, reoxidation of the enzyme occurs mainly via oxygen, while above this concentration, oxygen is depleted in the layer adjacent to the electrode and the reoxidation proceeds mainly via the mediator route, leading to a "takeoff" in the current response of the enzyme electrode. (12) Caw, A. E. G.;Davis,G.; Francis, G . D.;Hill, H. A. 0.;Aston, W. J.; Higgins, I. J.; Plotkin, E. V.; Scott, L. D. L; Turner, A. P. F. Anal. Chem. 1984, 56, 667-67 1, (13) Wcibel, M. K.; Bright, H. J. J . Biol. Chem. 1971, 246. 2734.
Ana!vticalChemistty, Vol. 66, No. 17, September 1, 1994
2767
c
Q
Y!
6
1
OxyQen (mmolll)
Substrate (mmolll)
0.0
0.2
0.4
0.6
0.8
1.0
Normalised Distance
Flgwe 3. Simulatednormakedcurrent((F,,,/dxlr0)plattedas a funotkn of glucose and oxygen concentration.The model parametersempbyed are given In Table 1.
1.00
U
Q
cn
kbIk4
m
z
0.0 0.0
0.2
0.4
0.6
0.8
1.0
Normalised Distance 100 80
ri
60 Current (W)
E
0"ri
40
20
b
1
0
.-m
L
0.7
1
P
kblkl Substrate (mmol/l)
26
Flgure 4. Current obtained in (a) oxygen-satwated solution and (b) air-saturated solution given as a percentage of the current calculated in the absence of oxygen, as a function of glucose and mediator competltion efficiency (k5/k4).The model parameters employed are given in Table 1.
Normalised Distance Flgure 2. Concentration profiles for (a) the normalized oxygen concentration and (b) the normalizedoxidized mediator concentration in the polymer matrix for increasing concentration of substrate ((A)5, (A) 10, (H) 20, and (0) 40 mmoi/L) in the bulk electrolyte; (c) concentration profiles of oxidized mediator (2 mmoi/L glucose, = 25, = 250) in the absence (A)and presence (A)of oxygen. Other model parameters employed are given in Table 1.
Different parameters in the model can influence the sigmoidal behavior of the enzyme electrode in the presence of the two oxidants. Although it is known that the immobilization of enzymes may influence the kinetics of the enzyme reaction itself, this is difficult to predict, and it is not obvious how the values of kinetic constants for the enzyme reaction (kl, k2, k3) could be deliberately influenced in the design of a sensor. However, the reaction constant for the reoxidation of the mediator may be increased (in theory!) by 2768
AnalVticaIChemistw, Vol. 66, No. 17, September 1, 1994
the employment of more efficient redox couples. In Figure 4a, the mediator current in oxygen, displayed as a percentage of that obtained in the absence of oxygen, is shown as a function of substrate concentration and mediator competition efficiency (Le., ks/k4). As one would expect, the difference in the response becomes less pronounced with the increase of ks, as the oxidation of the enzyme via the mediator route becomes more favorable, but oxygen dependency remains significant even when ks = k4. It is also further apparent from Figure 4a, that the influence of oxygen on the current is greatest at low substrate concentrations for all ratios of kslk4, but that the current response at higher concentrations of substrate is almost independent of the ks/k4 ratio. However, the latter independency offers little consolation as far as a sensor is concerned, since the normal operational range is at lower concentrations. Even the more relevant comparison for a device operating in air (20.9% oxygen partial pressure) shows
4
a>
*I
c E
1 oum
2
0 4um
0 6pm
0 8pm IOpn
0
10
20
30
40
50
0
Glucose (mmol/l)
10
20
30
40
50
Glucose (mmolii)
b)
b)
1.0 pm 4-
C L
3
04um
60 50
E €
5E
z, x 83-
0
10
20
30
40
50
Glucose (mmol/l) Flgure 5. Calibration curves calculated for different enzyme layer thicknesses inthe presenceof (a) mediator onlyand(b)the two oxidants, medlator and oxygen.
that calibration of the signal due to the mediator is difficult (Figure 4b) and susceptible to considerable error, particularly at low substrate concentrations, where the sensor is most likely to be employed. Obviously, the influence of the parameters described in this model must be addressed if calibration and/or oxygen dependency errors are to be overcome. The most accessible parameters in the design of a sensor are the thickness of the membrane and the actual loading of active enzyme in the matrix. Both values influence the Thiele moduli in the model described above. The thickness of the membrane appears as a squared term, and thus, small changes could have a pronounced effect on the response of the enzyme electrode. In Figure 5 , the simulated responses are plotted for different thicknesses of the enzyme layer in the presence of only one oxidant (mediator) (Figure sa) and of both mediator and oxygen (Figure 5b). The maximum current decreases significantly with the decline of the membrane thickness, due to the decrease in the total amount of enzyme present in the system. Furthermore, the sigmoidal shape of the calibration curve observed in the presence of two oxidants (mediator and oxygen) becomes less pronounced as the thickness of the enzyme membrane is diminished. A convenient way of identifying this anomalous behavior more readily is the representation of the data in a Hanes plot, as shown in Figure 6. A response showing normal MichaelisMenten kinetics gives a parabolic curve, and Figure 6a shows
20
0.6pm
10
0.8prn 1.Opm
z
0
10
20 30 40 Glucose (mmol/l)
50
Figure 6. Hanesplotsfor the data obtained from the simulations shown in Figure 5: (a) madlator only; (b) oxygen and medlator present In the bulk electrolyte.
the predicted influence of layer thickness on the response in the absence of 02.However, in the two-oxidant case (Figure 6b), the 1-pm-thick enzyme layer exhibits a clear minimum, which corresponds to the point of inflection in the current calibration curve. With the decrease in the thickness, the minimum becomes less apparent, and for an enzyme layer of 0.4-pm thickness, the typical parabolic curve is observed but with a high intercept. This clearly indicates the importance of the thickness of the enzyme layer on the current response, as only a 2.5-fold decrease in the thickness has changed not only the maximum current significantly but also altered the general shapeof thecalibration curve obtained in the presence of two oxidants. A comparison of parts a and b of Figure 6 identifies the potential problems of calibration of this electrode in a varying oxygen environment. Made1with Reduced Mediatorin Bulk Solution. Variations in Substrate Concentration. In Figure 7, the concentration profiles of reduced mediator (Figure 7a) and oxygen (Figure 7b) are shown for different substrate concentrations in the bulk electrolyte. In the absence of substrate, a straight line is observed for the mediator profile. With increasing substrate concentration in the bulkelectrolyte, the profile of the reduced mediator "bends" upward as the generation of reduced mediator by the enzyme reaction increases. The profiles for oxygen are, as expected, similar to those obtained for the simulation, assuming oxidized mediator in the bulk solution. Analytical Chemistry, Voi. 66,No. 17, September 1, 1994
2768
5
c
.-m
0
&
'
0
'0
U
al
4m E
.. &
0.6-
0.4. * 0.2-
. o
c
.
0
&
&
M
0
u
a
A
t
&
i A
O
W 0
.
1
20
40
60
80
Glucose (mmolll)
1.0>
------
--
..- ; ; . - 8
.-v)
m
E
0.2
0
z 0.0 0.0
0
.
0.2
e
*
0.4
0.6
0.8
1 .O
Normalised Distance Figure 7. concentratlon profiles calculated with reduced mediator in the bulk electrolyte for (a) the normalized mediator concentrationand (b) the normalized oxygen concentration in the polymer matrix for 16, increasing concentration of substrate ((A)0.1, (A)4, )(. 8, (0) and (0)40 mmollL) in the bulk electrolyte.The model parametersused for the calculatlons are given in Table 1.
Oxidation Current Reponse due to Mediator at the Electrode ( x = 0 ) . Figure 8 compares the normalized twooxidant calibration curves obtained from simulations with either oxidized or reduced mediator in the bulk solution. For both cases the same maximum current was reached. However, the normalized background current in the case of the reduced mediator was higher compared with the case of oxidized mediator, as the mediator has to be oxidized at the electrode before it can react with the enzyme. Both graphs exhibit a sigmoidal relationship between current and substrate concentration, due to the depletion of oxygen in the matrix. The inflection is more pronounced when the mediator is in its reduced state in the bulk electrolyte. From the point of the sensor design, it is therefore advantageousto employ mediators which exist as the oxidized species in aqueous solutions. (14) Albery, W. J.; Craston, D. H. In Biosensors: fundamentals and Applications; Turner, A. P. F., Karube, I., Wilson, G. S., Eds.; Oxford University Press: Oxford, UK, 1989.
2770
Ana!vticalChemistty, Voi. 66, No. 17, September 1, 1994
Flgure 8. Figure 8: Calibration curve obtained from simulations when the mediator is present In its oxMized (El) or reduced (+) state In the bulk electrolyte In the presence of both mediator and oxygen.
CONCLUSIONS A model for a three-substrate electrode has been described and applied for the simulation of a mediated enzyme electrode in the presence of oxygen. Simulations employing this model confirm the hypothesis that the sigmoidal behavior of the calibration curve, observed experimentally when mediator and oxygen are competing for the reoxidation of the enzyme, is due to depletion of oxygen in the enzyme layer. Investigation of the sensitivity of the model to changes in the Thiele moduli showed that only relatively small variations in the thickness of the enzyme membrane caused a significant change in both the magnitude of the current response and the general behavior of the system. The inflection observed in the presence of oxygen disappears when the value of the Thiele moduli is decreased (e.g., decrease in the layer thickness, d C 0.5 pm). Albery and Carston had previously described a simple model for a membrane enzyme electrode. They assumed that if the electrolyte layer behind a membrane is sufficiently thin (a few micrometers), the concentration polarization in the enzyme layer can be omitted.14 However, our model suggests that not only the thickness of the enzyme layer alone, but the Thiele moduli, defined by the layer thickness and the enzyme concentration, have to be considered, and even below 1 pm, the concentration polarization of the reactants can be significant. The model described here can be used to obtain the parameters required to improve the design of sensors where the mediator is to replace a natural cosubstrate, and it can be seen that considerable improvement in mediator kinetics is required if oxygen competition is to be eliminated from the signal. ACKNOWLEDGMENT We thank the Gottlieb Daimler- und Karl Benz- Stiftung for the financial support. Received for revlew February 9, 1994. Accepted May 9, 1994.' Abstract published in Advance ACS Absfracfs,june 15, 1994.
