Anal. Chem. 1995, 67, 1669-1678
Transient Response of Multilayer Electroenzymatic Biosensors Saliha Bacha, Alain Bergel,* and Maurice Comtat Laboratoire de Genie Chimique et Electrochimie, Laboratoire de Genie Chimique, U.R.A. C.N.R.S. 192, Univetsite Paul Sabatier, 1 18 route de Nadmnne, F-31062 Toulouse, France
A numerical model is used to analyze the transient response of amperometric enzyme sensors. Different biosensor types are considered according to whether the enzyme is fixed on the inner or the outer surface or inside the membrane itself. The numerical simulation is performed with a tbite volume method which makes it possible to easily solve the transient mass balance equations in all the juxtaposed layers of the biosensors. Phenomena induced by the transient mass transfers in the different layers are emphasized particularly by analyzing the effects of the stirring strength and the assay protocol on the shape of the current-time curves. This makes it possible to suggest theoretically significant opportunities to improve the reliability and performance of biosensors. The peak current, which appears in some cases when stirring is decreased, offers a measure that is h t e r and less sensitive to the operating conditions than the steady-statevalue. After determination of a maximum number of physicochemical parameters by independent experiments, the model satisfactorily fits experimental results performed with free enzyme and immobilized enzyme sensors. The number of experimental studies dealing with amperometric biosensors is continually increasing and their field of application continually Modeling now allows the stationary behavior of such biosensors to be depicted correctly,l-1° but the models devoted to their transient response remain relatively scarce. The first such model was applied to an immobilized glucose oxidase sensor." It gave a good understanding of the different current controlling steps and solved the apparent discrepancy between some observed results and known kinetic data, A free enzyme sensor has also been modeled under transient conditions,12 and an analytical approach has been proposed for bienzyme sensors.13 (1) ScheUer, F.; Schubert, F.Biosenson; Elsevier: Amsterdam, 1992. (2) Tumer, A P. F.;Karube, I.; Wilson, G. S. Biosensors, Fundamentals and
Applications; Oxford University Press: Oxford, 1987. (3) Blum, L. J.; Coulet, P. R Biosenson, Principles and Applications; Marcel Dekker: New York, 1991. (4) Gough, D. A; Leypoldt, J. K Anal. Chem. 1979, 51, 439-444. (5) Leypoldt, J. IC; Gough, D. A Biotechnol. Bioeng. 1982, 24, 2705-2719. (6) Leypoldt, J. K; Gough, D. A Anal. Chem. 1 9 8 4 , 5 6 , 2896-2904. (7) Gough, D. A; Lucisano, J. Y.; Tse, P. H. S. Anal. Chem. 1 9 8 5 , 5 7 , 23512357. (8) Bartlen, P. N.; Whitaker, R G. J. Electroanal. Chem. 1987, 224, 27-35. (9) Tatsuma, T.; Watanabe, T. Anal. Chem. 1992,64, 625-630. (10) Albery, W. J.; Cass, A E. G.; Shu, 2. X. Biosens. Eioelectron. 1990,5,367378. (11) Mell, L. D.; Maloy, J. T. Anal. Chem. 1975, 47, 299-307. 0003-2700/95/0367-1669$9.00/0 Q 1995 American Chemical Society
The equations to be solved are mass balance partial derivative equations coupled with enzymatic nonlinear kinetics. The integration domain, defined by the geometry of the biosensor, generally consists of a set of layers of different physicochemical nature disposed side by side (electrode, electrolyte layer, one or more membrane($, diffusion layer, etc.) with a number and a space ordering that depend on the configuration of the sensor. The earlier models simplify the geometric representation by taking into account only one layer, which is assumed to be preponderant. The later transient models take into account the effects of a diffusion layer assuming that the steady state is always reached inside it.14-16 A previous paper17demonstrated that a finite volume method was particularly useful for solving the transient equations whatever the number of different juxtaposed layers. The purpose of the present work is to exploit the versatility of this integration method to predict the transient behavior of various kinds of biosensors without any simplication of the basic transient mass balance equations. Various kinds of biosensors are considered here. On the one hand, the enzymatic reaction was homogeneous, with the enzyme either immobilized inside a polymeric membrane or free in solution, confined in a reactional chamber bound by the electrode and a semipermeable membrane. On the other hand, the enzymatic reaction was heterogeneous, with an enzyme linked to one or two sides of a membrane mechanically held against the electrode. Improving the performances of such biosensors according to various operating conditions is emphasized. Validation of the model by comparison with experimental data is carried out. In order to fit the experimental curves by means of a minimum number of adjustable parameters, as many physicochemical parameters as possible were determined by independent experiments. THEORETICAL SECTION
Description of the Model. All the biosensors considered consist of a metallic electrode with a semipermeable membrane held against it. The enzyme is either free in solution in an electrolyte layer between the electode and the membrane or immobilized either on the surface of the membrane or inside it. In all cases, the geometrical domain can be depicted by the diagram in Figure 1. The sensor consists of the electrode surface, a layer of electrolyte, and the membrane. The electrolyte layer (12) Bergel, A; Comtat, M. Anal. Chem. 1 9 8 4 , 5 6 , 2904-2909. (13) Kulys, J. J.; Sorochmskii, V. V.; Vidziunaite, R A. Biosenson 1986,2,135146. (14) Tse, P. H. S.; Gough, D. A Anal. Chem. 1987, 59, 2339-2344. (15) Lucisano, I. Y.; Gough, D. A. Anal. Chem. 1988, 60, 1272-1281. (16) Battaghi, F.;Calvo, E. J. Anal. Chim. Acta 1992,258,151-160. (17) Bacha, S.; Bergel, A; Comtat, M. J. Electroanal. Chem. 1993,359,21-38.
Analytical Chemistry, Vol. 67, No. lo, May 15, 1995 1669
free enzyme sensor
immobilizedenzyme sensor
electrode
KOX
IC,=-
(3)
membrane diffusion layer
diffusion layer
solution
solution
Using the superscripts m and s relative to the membrane and solution regions, respectively, the diffusivities are written as follows
1 Enzyme Dsm; Di,, = Dism= D,,
general diagram
DSS r; Di,, Oxm
= 1; Die, =
DO,
Oxm
(4)
The thicknesses of the inner and outer solution layers become electrolyte layer
membrane
diffusion layer
&=’.
6. 6”
6
=o
L O
(5)
6,
I Figure 1. General scheme of the geometry of a multilayer amperometric biosensor.
is either the reaction chamber containing the free enzyme or an unavoidable electrolyte layer when the membrane is mechanically held against the electrode. A diffusion layer whose thickness is controlled by hydrodynamics completes the model. In all cases, all the species except the enzyme are able to diffuse through the liquid layers and the membrane. The enzymatic system is
S+yOx--+$-Red where S is the substrate to be assayed, Ox and Red are the oxidized and reduced forms of the cosubstrate, respectively,and y is a stoichiometric coefficient. The substrate S and the oxidized form of the cosubstrate Ox are in the bulk of the solution, but there is no Red species. The area of the electrode is very small with respect to the volume of the solution to ensure that the concentrations do not vary in the bulk of the solution during the assay. The kinetics of the reaction is assumed to be a steadystate “ping-pong”type:
r=
1 + &/[SI
A
+ K,,/[OXl
where A is the enzyme activity, and KSand K& are the Michaelis constants for the substrate and the cosubstrate, respectively. Amperometric detection is based on the rapid oxidation of Red on the electrode:
Red
- Ox + ne
Dimensionless Parameters. All the concentration parameters are divided by the concentration of Ox in the bulk of the solution, [Ox]b: 1670 Analytical Chemistry, Vol. 67, No. 10, May 15, 1995
The dimensionless catalytic activities are expressed as
:4,
Ah0dm2 =
Do,[Oxlb
Ah“,
Damo=
Doxm[Oxlb
(7)
where Aho (mol m-3 s-l) is the activity of the free enzyme or the enzyme uniformly entrapped inside the membrane, and Ah‘ (mol m-2 s-l) is the activity of the enzyme linked to the membrane surface. q$,,02 is a modified expression of the square of the Thiele modulus 8,which expresses the ratio of the potential reaction rate without any mass transfer limitation to the maximum diffusion rate of the cosubstrate:
In the case where the enzyme is entrapped inside the membrane, c # J is ~ ~the ~ maximum value of the Thiele modulus, but that is no longer so when the enzyme is free in solution. Strictly speaking, DoXmwould have to be replaced by DoXs.A similar discussion could be held concerning the heterogeneous kinetics. The dimensionless activity Damo is a m o d ~ e dexpression of the Damkohler number. Damorepresents the maximum value of Da when the enzyme is linked to the inner side of the membrane, Le., when the species have to diffuse through the membrane before they react. When the enzyme is linked to the outer side, Dop would have to be replaced by Dw. For the sake of simplicity, the same expressions of #mo2 and Dam, are used in all cases. Mass Balance &tuitions. The mass balance equations relative to the substrate S and the two forms Ox and Red of the cosubstrate fully describe the system. For instance, for the case where the enzyme is entrapped inside the membrane, the
dimensionless governing equations are as follows: as-Di -aT
_ a2S _
2
4mo
+
2
aRed - $Red
aT
+ K,/OX
1 K,/S
S”g2
providing that either the electrolyte layer or the membrane is close in the vicinity of the electrode surface. When a reactive surface is directly applied against the electrode surface, one obtains
4mo +
y 1 + K,/S
+ Ic,/ox
assuming that the diffusivities of the two forms of the cosubstrate are equal and that the enzyme is homogeneously distributed. One then has within the electrolyte diffusion layers
At the diffusion layer-bulk interface, the concentrations are supposed to be constant: Red=O; O x = 1; S = S b
(15)
Discretization of the integration domain was performed by systematicallyselecting a smaller mesh near the interfaces where the concentration gradients could be steeper, particularly when a surface reaction occurs. A great advantage of the finite volume integration method is that there is no problem if the space grid is enlarged or reduced in any part of the integration domain. The resolution algorithm and the numerical finite volume method have been described in a previous paper.17 Balance of the mass flux densities at a membrane-diffusion layer interface should involve the enzymatic surface kinetics. No partition effect is taken into account:
EXPERIMENTAL SECTION General. The experiments were made with free enzyme and immobilized enzyme sensors. In all cases, the enzymatic reaction was the oxidation of the pyridinic coenzyme, NADH, by potassium hexacyanoferrate(IID, &Fe(CN), catalyzed by a diaphorase @):
NADH + 2Fe(CN)?- -!?.NAD’
+ 2Fe(CN);- + Ht
This reaction served as a basis for the NADH sensor, which used oxidation of hexacyanoferrate (ID, Boundary Conditions. If the substrate is not electroactive, its mass flux density is thus nil at the electrode:
as
=0
The mass flux densities of the two forms of the electroactive cosubstrate are opposite:
aox ax 1x4 = - xaRed Ix=o The electrochemical reaction is assumed to be fast enough to ensure a nil concentration of Red at the electrode: Red[,,
=0
Current density due to the mass flux of Red at the electrode surface is provided by
Fe(CN);-
-
Fe(CN)?-
+e
on a platinum electrode maintained at a potential of 0.35 V vs saturated calomel electrode (SCE) . Diaphorase (EC 1.8.1.4.) from Clostridium kluyveri, potassium hexacyanoferrate(III), and reduced nicotinamide adenine dinucleotide (NADH)were provided by Sigma. All the other compounds were purchased from Merck. A 0.1 M phosphate solution, pH 7.5, was used for the measurements with the free enzyme sensor, and a 0.1 M carbonate solution, pH 9.1, was used for the immobilized enzyme experiments. The immobilized enzyme sensors were built with a preactivated Immunodyne membrane Ball Industries, BIA065HC5,0.65pm porous middlesize, 150pm dry-state thickness) which permits a very easy covalent linking of some enzymes such as diaphorase on its surface. In free enzyme sensors, diaphorase was confined in the electrolyte layer with a cellophane semipermeable membrane 40 pm thick. All the sensors used a 2 mm2 platinum working electrode and a platinum wire as auxiliary electrode. A potential of 0.35 V/SCE was maintained by means of a Tacussel potentiostat, and stirring was achieved by a magnetic barrel. Enzyme confining methods Analytical Chemisty, Vol. 67,No. 10, May 15, 1995
1671
Table 1. Parameter Values Used in the Slmulation
solution
KNADH (mol m-3) KF~(cN) 3(mop m-3) diaphorase ' activity geometrical parameters
(4
a
freea
enzyme immobilized*
4.50 x
0.24 x
0.65 x
11.83 x
0.47 x
1.85 x
0.5
0.5
0.4
0.4
Ah0 = 1
he = 7.5 x 10-4
mol s-1 m-3 do, numerically 6, = 45 x adjusted 4,numerically adjusted
mol m-2 6, = 160 x 6i = 0
were sufficientlyhigh for the denominator of the enzymatic rate expression to be neglected. The Michaelis constants were measured with variable concentrations of substrate or cosubstrate, by means of a classic Lineweaver-Burk representation of the results. These measurements were made only with the free enzyme, assuming that the Michaelis constants kept the same value for immobilized diaphorase. The values are reported in Table 1. RESULTS AND DISCUSSION
Effect of Stirring on the Current-Time Curves. The theoretical study is carried out around a standard case with the following constant values of the dimensionless parameters:
Di, = Di,,, = 1
K s = Kox= 0.5
(17)
*
Cellophane membranes. Pall Immunodyne membrane.
There is no electrolyte layer between the electrode and the membrane: and measurement protocols have already been described elsewhere.18 Determination of the Dfisivities. The diffusivity of NADH in solution was measured by means of a platinum rotating disk electrode. In order to obtain a very fast oxidation reaction of NADH as compared to its mass transfer to the electrode surface, the potential was maintained at 0.9 V/SCE and the rotation speed kept at low values. The same experiment, carried out with hexacyanoferrate in solution, was used to calibrate the method. The ratio of the slopes of the current variation as a function of NADH concentration was expressed according to Levich's equation:
)
ratio = 2( DNADH DFe(CN),3-
L,=O
The following protocol was chosen. Prior to measurement, the sensor was washed in a solution that contained only the cosub strate. At the outset it was plunged into a beaker containing the cosubstrate and the substrate to be assayed. Initial conditions corresponding to this protocol are the following. (i) The cosubstrate is present within the membrane:
O5X51+L0, ox=1
(18)MontagnC, M.; Durliat, H.;Comtat, M. Anal. Chim. Acta 1993,278,2533. (19)Weast, R C.; Astle, M. J.; Beyer, W. H. Handbook of Chemistry and Physics; CRC Press: Boca Rato, FL, 1987-1988.
1672 Analytical Chemistty, Vol. 67,No. 70,May 15, 7995
(19)
(ii) The substrate concentration is equal to the concentration in solution up to the membrane surface and nil within the membrane:
05x51, where the coefficient 2 takes into account the difference of the number of electrons transferred by NADH and F~(CN)G~-. This ratio method avoided several sources of error which might be possible when using Levich's equation, such as errors in determining the density, viscosity, rotation speed, or possible effect of surface roughness. NADH diffusivity was obtained by taking the value of hexacyanoferrate diffusivity as reference.lg Diffusivities of NADH and hexacyanoferrate in the membranes were measured with a membrane covered working electrode, according to the method described by Gough and Leypoldt.4 All the measured values are reported in Table 1. Determination of the Kinetic Parameters. The values of the free (Aho) and the immobilized (Ahe) diaphorase activities and of the Michaelis constants relative to the substrate (Ks) and cosubstrate (KOAwere measured spectrophotometrically. One diaphorase unit corresponds to the oxidation of 1pmol of NADH/ min at 25 "C. The activity was deduced from the measured initial rate, when an aliquot of free enzyme solution or a known area of diaphoraselinked membrane was introduced into the spectrophotometric cell. The substrate and cosubstrate concentrations
(18)
s=o; 1 < X 5 1 + L o , s=sb
(20)
The other values are varied according to the purpose of the study. Figure 2 shows the current-time curves obtained for different values of the diffusion layer thickness, Lo, when the enzyme is linked on the inner side of the membrane, i.e., the side against the electrode. The parameter values are Sb= 0.5, Da, = 1,and DSS= 1. For the highest stirring strengths, the values of L, are the lowest, and the current density continuously increases until a constant steady-state value is reached. With L, = 2, a poorly defined maximum is visible. For greater values the maximum is clearer. The curves are identical over longer and longer periods of time as L, increases. The common part of the curves describes the response curve which would be obtained with no stirring.The appearance of a steady state for the lowest Lo values has no physical meaning; it is due only to the use of a very thick diffusion layer in the model, which would very difficult to maintain experimentally over a period sufficiently long to reach a steady state. The most important conclusion to be drawn from Figure 2 is that a maximum appears when the stimng strength is sufficiently low. When the current peak does occur, stirring does not affect the height or the time of the peak, while the steadystate current always depends on stirring strength. Consequently, replacing the steady-statevalue, commonly used as the measure-
1
x
.-U
v)
5
20
-0
r C
E 3 0 v) v)
-aJ
.10
C
.-0
v1 C
.-E
U
I 0.
2.
4.
Dimensionless time Figure 2. Effect of the diffusion layer thickness, Lo, on the response curves. The sensor is dipped into the solution to be assayed after being washed in a solution containing the cosubstrate. The enzyme is linked to the inner side of the membrane. Li = 0; KS= Kox= 0.5; Dis~=1;Dio~.=1;Sb=0.5;Damo=1;DiS.=1. (l)L,=O.O1;(2) L, = 1; (3) L, = 2; (4) L, = 3;(5)L, = 4; (6)L, = 5;(7)L, = 7;(8) L, = 10. 1 .o
.-U21 v)
C
0
U
5
A
U
C
e L
a
U
.O )
.2
.4
Dimensionless time Figure 3. Effect of the diffusion layer thickness, b,on the response curves. The sensor is dipped into the solution to be assayed after being washed in a solution containing the cosubstrate. The enzyme is linked to the outer side of the membrane. Sb = 0.5; Dam,= 100; Dism = 10. The other parameter values are the same as in Figure 2. (1) L, = 0.01; (2) L, = 1; (3)L, = 3;(4) L, = 4; (5)L, = 5;(6)L, = 7;(7)L, = 10.
ment value, by the peak height could permit a significant improvement in the reliability of the sensor. Such behavior has already been remarked in a theoretical simulation of mono- and bilayer modified electrodes.20 Enzymatic reaction occurred in this case only on the electrode surface, which significantly simplifies the set of mass balance equations and permits an easy solution of the transient equations in the diffusive layer. By means of the numerical method used here, solving of all transient mass balance equations in all the layers of complex biosensors allows identification, explanation, and exploitation of these phenomena with classic multilayer devices. Figure 3 shows a wider variety of response curve shapes. They are plotted with a higher dimensionless enzyme activity, Dam, = 100, assuming that the substrate diffuses faster in solution @is. (20) Tatsuma, T.; Watanabe, T.; Okawa, Y. A d . Chem. 1992, 64, 630-635.
.oo
a.
16.
Dimensionless time Figure 4. Effect of the assay protocol on the response curve. Sb = 1; Dam, = 1; 4m02= 1; Disa = 1; L, = 3. The other parameter values are the same as in Figure 2. (1) The sensor is dipped into the solution to be assayed after being washed in a solution containing the cosubstrate; (2) the sensor is dipped into the solution to be assayed after being washed in a buffer; (3) the substrate is injected into the solution containingthe cosubstrate; (4) the substrate and the cosubstrate are injected into the solution.
= 10, Sb = 0.5) and for a sensor with the enzyme linked to the outer side of the membrane. In Figure 2, the current normally decreases as Loincreases, because the mass transfer resistance increases in solution for low stirring strengths. On the contrary, Figure 3 depicts a significant current increase when L,increases from 0.01 to 1and also a weak enhancement for L, = 3, followed by a normal decrease for the higher values. In this case, two antagonisticphenomena occur. The enzymatic catalysis produces the Red species at the membrane-solution interface. When stirring strengthens, a large proportion of Red diffuses back to the bulk of the solution and is lost for amperometric detection. Lowering the stirring strength first results in an increase in the amount of Red reaching the electrode surface. For weaker stirring, the influence of the mass transfer resistance in solution becomes predominant. A similar phenomenon has been noted relative to the influence of the polymer thickness on the behavior of an enzyme sensor where enzyme was immobilized inside the polymer.8 As in Figure 2, a maximum appeared for high values of L,. Once again, the height and the time of the peak were not affected by stirring strength, while here the steady-state current varied greatly. For this sensor, a high stirring strength could be detrimental to its performance. A proper choice of L, could enhance the response, but a slight variation in stirring could drastically bias the result. As before, the best solution could be to choose a stirring strength for which the peak appeared on the response curve and to use its height as the measure. Moreover, Figure 3 shows that the peak value ensures a response not very far from the higher steady-state value, which would be accessible only after a tedious search for the optimal stirring strength. Influence of the Assay Protocol on the Current-Tie C w e s . Occurrence of a maximum also depends on the choice of initial conditions. Current-time curves are plotted in Figure 4 for a sensor with enzyme linked to the inner side of the membrane with Sb= 1,Dam, = 1,DSS= 1,and L, = 3. Before an assay, the sensor is washed in a solution containing only the Analytical Chemistty, Vol. 67, No. 10, May 15, 1995
1673
cosubstrate. At the initial time, the sensor is dipped into the solution to be assayed. As a result, there will be no diffusion layer at the initial time. The current density increases very quickly. Depletion of the substrate in the diffusion layer then causes the diffusional transfer rates to decrease within the membrane. When these rates become too low to compensate for the consumption of the substrate, the intensity decreases (curve 1). The sensor response does not differ greatly if the device is washed before the assay in an electrolyte solution without the cosubstrate (curve 2). The initial conditions are then
ox=o;
0 5 x 5 1, 0 5 x 5 1,
1< X I l + L o ,
ox=l
s=o; 1 < x s l + L o , s=sb
(21)
I
0.
i
2. Diffusion
On the other hand, the maximum no longer occurs if it is assumed that no substrate is initially present within the diffusion layer. Such a case would be experimentally realized when the substrate is introduced last, after stirring has been well established in a beaker containing only the electrolyte solution with the cosubstrate (curve 3),
O5X51+L0, 05X
