Anal. Chem. 1995,67, 312-317
Microfabrication and Characterization of Diaphorase-Patterned Surfaces by Scanning Electrochemical Microscopy Hitoshi Shiku, Toshihiro Takeda, Hiroshi Yamada, Tomokazu Matsue,* and lsamu Uchida Department of Molecular Chemistry and Engineering, Faculty of Engineering, Tohoku University, Sendai 980-77, Japan
Scanning electrochemical microscopy (SECM) was used to microfabricate and guan* diaphorase-patternedglass surfaces. Deactivated circular and linear micropatterns were produced at diaphorase-immobilizedsubstrates by a localized surface reaction. The oxidation of Br- and Clat a microelectrode generated a reactive species which deactivated the localized enzyme molecules at the substrate. The diaphorase-patterned surfaces were characterized by SECM on the basis of detection of catalytic current of ferrocenylmethanol coupled with oxidation of NADH. The concentration of the immobilized diaphorase was mapped fiom the quantitative analysis of the catalytic current. Recently, much attention has been focused on micropatterning with enzymes to fabricate miniaturized, integrated biochemical devices such as biosensors. Photolithography, currently the most popular technique for microfabrication, is basically unsuitable for biological materials because the photoresists, solvent, and atmosphere in these processes damage the materials. Although attempts at photolithographic microfabrication to meet the requirements for biological materials have been there is a need to develop novel methods for the microfabrication. We report here on the micropatterning of an enzyme-immobilized substrate based on a localized surface reaction initiated by an electrode reaction at a microelectrode. The micropatterned surface was characterized by scanning electrochemical microscopy (SECM).4,5SECM has been introduced as a new system6-s for viewing surfaces based on differences in electrochemicalproperties. Bard and c o - w ~ r k e r s have ~ - ~ ~developed SECM and applied it to image various surfaces with localized biological materials. SECM is particularly suitable for obtaining quantitative information on localized bioelectrochemical phenomena such as enzyme(1)Hanazato, Y.; Nakako, M.; Maeda, M.; Shiono, s.Anal. Chim. Acta 1987, 193,87-96. (2) Vopel, T.; Ladde, A; Muller, H. Anal. Chim. Acta 1991,251,117-120. (3) Bhatia, S. R; Hickman, J. J.; Ligler, F. S. J. Am. Chem. SOC.1992,114, 4432-4433. (4) Engstrom, R C.; Weber, M.; Wunder, D. J.; Burgess, R ; Winquisf S. Anal. Chem. 1986,58, 844-848. (5) Liu, H.-Y.; Fan, F.-R F.; Lin, C. W.; Bard, A. J. J. Am. Chem. SOC.1986, 108, 3838-3839. (6) Engstrom, R C.; Pharr, C. M. Anal. Chem. 1989,61, 1099A-1104A. (7) Emgstrom, R C.; Meaney, T.; Tople, R ; Wightman, R M. Anal. Chem. 1987, 59, 2005-2010. (8) Engstrom, R C.; Wightman, R M.; Kristensen, E. W. Anal. Chem. 1988, 60, 652-656. (9) Bard, A J.; Fan, F.-R F.; Kwak, J.; Lev, 0. Anal. Chem. 1989,61,132138. (10) Kwak, J.; Bard, A J. Anal. Chem. 1989,61, 1221-1227. (11) Kwak, J.; Bard, A J. Anal. Chem. 1989,61, 1794-1799.
312 Analytical Chemistry, Vol. 67, No. 2,January 15, 1995
catalyzed redox reaction^,'^-'^ energy-coupled metabolic react i ~ n s , and ~ ~ Jpermeation ~ processes through bilayer lipid membranes.20-2zThe localized electrochemicalreactions occurring in SECM systems provide a way for microfabrication. Metal depo~ition,2~-~~ electrochemical etching,26-28and photochemical reaction^^^^^^ have been used to create microstructure at various surfaces. In the present paper, we describe for the first time the micropatterning of an enzyme-immobilized substrate using SECM technology. Electrochemicaloxidation of bromide or chloride at a microelectrode generates a highly reactive species31 which deactivates immobilized enzyme molecules in localized surfaces (Figure 1). Arbitrary patterns can be made by scanning the electrode tip along the surface. We have used diaphorase,a flavin enzyme, as the enzyme to be patterned. Diaphorase catalyzes the electrochemical oxidation of a nicotinamide coenzyme (NADH) by a redox mediator and has been used in NADH sensing device^.^^-^^ The diaphorase-patternedsubstrate was characterized by SECM on the basis of detection of the catalytic current for oxidation of NADH. (12) Wang, J.; Wu, L.-H.; Li, RJ. Electroanal. Chem. 1989,272,285-292. (13) Pierce, D. T.;Unwin, P. R ; Bard, A J. Anal. Chem. 1992,64,179551804, (14)Horrocks, B. R ; Mukin, M. V.; Pierce, D. T.; Bard, A J.; Nagy, G.; Toth, IC Anal. Chem. 1993,65,1213-1224. (15) Pierce, D. T.;Bard, A J. Anal. Chem. 1993,65,3598-3604. (16) Horrocks, B. R ; Schmidtke, D.; Heller, A; Bard, A. J. Anal. Chem. 1993, 65, 3605-3614. (17) Yamada, H.; Shiku, H.; Matsue, T.;Uchida, I. Bioelectrochem. Biome%. 1994,33, 91-93. (18) Lee, C.; Kwak, J.; Bard,A J. Proc. Natl.Acad. Sci. U.SA. 1990,87, 17401743. (19) Bard, A J.; Fan, F.-R F.; Pierce, D. T.; Unwin, P. R ; Wipf, D. 0.;Zhou, F. Science 1991,254, 68-74. (20) Antonenco, Yu. N.; Blychev, A A Biochim. Biophys. Acta 1991,1070,279282. (21) Antonenco, Yu. N.; Blychev, A A Biochim. Biophys. Acta 1991,1070,474480. (22) Yamada, H.; Matsue, T.; Uchida, I. Biochem. Biophys. Res. Commun. 1991, 180, 1330-1334. (23) Craston, D. H.; Lin, C. W.; Bard, A J.J. Electrochem. SOC.1988,135,785786. (24) Husser, 0. E.; Craston, D. H.; Bard, A J. Vac. Sci. Technol. 1988,B6,18731876. (25) Bard, A. J.; Dennuaulf G.; Lee, C.; Mandler, D.; Wipf, D. 0. Acc. Chem. Res. 1990,23,357-363. (26) Mandler, D.; Bard, A. J. J. Electrochem. SOC.1989,136,3143-3144. (27) Mandler, D.; Bard, A J. J. Electrochem. SOC.1990,137,2468-2472. (28) MacPherson, J. V.; Unwin, P. R J . Phys. Chem. 1994,98,1704-1713. (29) Lin, C. W.; Fan, F.-R F.: Bard, A J.J. Electrochem. SOC.1987,134,10381039. (30) Sugimura, H.; Uchida, T.; Shimo, N.; Kitamura, N.; Masuhara, H. UltramiCYOSCO~Y 1992,42,468-474. (31) Bailar, J. C.; Emeleus, H. J.; Nyholm, R; Trotman-Dickenson, A F. Comprehensive Inorganic Chemistry; Pergamon Press Ltd.: Oxford, 1973; Chapter 26.
0003-2700/95/0367-0312$9.00/0 0 1995 American Chemical Society
aminopropyl)triethoxysilane/benzene solution for 30 min. After the remaining resist was removed, the substrate was successively dipped into a 1%(vlv) glutaraldehyde/water solution and a 0.1 mM diaphorase/phosphate buffer solution (PH 7.5). Between these steps, the glass substrate was washed thoroughly with water under supersonication. HOBr Br2 MeasurementSystem. The microelectrode for inactivation of the local enzyme and for SECM was fabricated as follows. A Pt wire (radius, 7.5 pm) was inserted into a soft glass capillary. The tip was fused carefully in a small furnace at 320 "C in vacuo to coat the Pt filament with the soft The tip was then polished on a turntable (Narishige, Model EG-4) and finally with 0.05 pm alumina powder to obtain a disk-shaped electrode. The tip radius including the insulating part was -50 pm. The microelectrode gave a steady-state voltammogram in 2.0 mM &Fe(CN)G/O.l M KC1when the scan rate was less than 50 mV/s. The Pt disk radius was determineda from the steady-state reduction current (4.3 f Figure 1. Microelectrochemicalpatterning at diaphorase-immobi0.1 nA) and found to be 7.5 f 0.2pm, which coincided with the k e d surfaces by electrogenerated HOBr. geometric size. The counter electrode was an Ag/AgCl immersed in saturated KCl. We also fabricated a smaller microelectrode (Pt disk radius, 2.3 pm; tip radius, -30 pm) by the above EXPERIMENTAL SECTION procedure using a fine Pt wire prepared by anodic polishing.42 Materials. Ferrocenylmethanol @MA) was synthesized by The smaller electrode was used for drawing the deactivated reduction of ferrocenecarboxyaldehyde (Aldrich) with N a B h and enzyme lines and also for SECM. recrystallized twice from hexane. NADH was purchased from The measurement was carried out in a two-electrode conliguraSigma and used as received. Glutaraldehyde was obtained from tion. The current was amplified with a current amplifier (Keithley, Wako Chemicals (Osaka, Japan). Diaphorase (NADH, acceptor Model 427). The amplified data were digitized and transferred oxidoreductase (EC 1.6.99.-)) purified from Bacillus sfearofherto a computer. Micromovement of the electrode was performed mophilus was donated by Uniticka Lid. (Kyoto, Japan). This by means of a motor-driven Xm stage (Chuo Seiki, M9103), enzyme has one flavin mononucleotide as an electroactive site controlled by a computer through a GP-IB interface. The tip per molecule, and its molar mass is ca. 30 OOO Da. The basic properties of diaphorase are described in the previous paper~.4O*~* surface of the microelectrode was carefully aligned to be parallel to the substrate surface under a microscope. For quantitative All the aqueous solutions were prepared with water purified by analysis to determine active diaphorase at the substrate, the Milli-Q I1 (Millipore Co.). current-distance profile was recorded by moving the tip normally Preparation of DiaphoraseImmobilizedGlass Substrates. to the surface at a speed of 2.4 p m l s (see below). No obvious Diaphoraseimmobilized glass was prepared by dipping a slide difference in the profile was observed when the moving speed glass successively into a 10 mM (3-aminopropyl)triethoxysilane/ was less than 5 pm/s. For typical SECM imaging, the microbenzene solution, a 1%(v/v) glutaraldehyde/water solution, and electrode was scanned over the substrate with a constant tipa 0.1 mM diaphorase/phosphate buffer solution (PH 7.5),17 substrate distance in a solution containing FMA and excess followed by thorough washing with water under supersonication. NADH. The oxidation current of the FMA at the microelectrode A glass substrate patterned with diaphorase was fabricated by was monitored to obtain surface images. The tip-substrate conventional photolithography. A positive photoresist Vokyo distance was estimated from the decrease in the reduction current Oka, TSMR-V3 (15cP)) was spincoated on the glass substrate, of Fe(CN),j3- or the oxidation current of FMA, as compared to followed by prebaking at 80 "C for 30 min. A xenon lamp light the current observed when the tip was far from the substrate.17 source irradiated the photoresistcoated substrate through a photomask. The image was obtained in a developer rokyo Oka, Quantitative Analysis of the Catalytic Current. Digital simulation was used for the quantitative analysis of the oxidation NMD-W). The substrate was then dipped into a 10 mM (3current of FMA Diffusion of redox species into and from the (32)Cass, A E. G.; Davis, G.; Green, M. J.; Hill. H. A 0.1.Electmanal. Chem. microelectrode is nonlinear; therefore, two-dimensional diffusion 1985,1W,117-127. should be considered. We have used digital simulation with an (33)Miki. K;Ikeda, T.;Todoriki, S.; Senda, M. Anal. Sci. 1989.5, 269-274. (34) Matsue, T.;Kasai, N.; Naruni, M.; Nishizawa. M.; Yamada, H.; Uchida, I.]. explicit finite difference (EFD) method10.bl~45 incorporating the Electmanal. Chem. 1991,300,111-118. heterogeneous enzyme reaction at the substrate. As described (35)Chang, H. C.; Ueno, A; Yamada, H.; Matsue, T.; Uchida, I. Analyst 1991, in the previous paper,% in the presence of excess NADH, the 116,793-796. (36)Matsue, T.;Nishizawa, M.; Sawaguchi, T.;Uchida, I. J. Chem. Soc., Chem. Michaelis treatment of the enzyme reaction at the surface affords
E
+
3
Commun. 1991,1029-1031. (37)Tatsuma, T.;Watanabe, T.J. Electmanul. Chem. 1991,310,149-157. (38)Sawaguchi,T.;Matsue. T.; Uchida. I. Bioelectrochem. Bioeneq. 1992,29, 127-133. (39)Yamada, H.; S h i h , H.; Matsue, T.;Uchida, I. Bioelectrochem. Bioeneq. 1993.29.337-346. (40)Matsue, T.;Yamada, H.; Chang, H.C.; Uchida, I.; Nagata, K; Tomita, K Biochim. Biophys. Acta 1990,1038.29-38. (41)Matsue, T.; Yamada, H.; Chang, H.4.; Uchida, I. Bioelectrochem. Bioeneq. 1990.24.347-354.
= 2(KorDpcFMA++)/(K +mCf+) (1) where ko is the molecular activity, D rp is the surface concentration
ff
(42)Matsue, T.; Koike, S.; Uchida, I. Biochem. Biophys. Res. Commun. 1993, 197,1283-1287. (43)Wight", R M. A w l . Chm. 1981.53,1125A-l133A. (44)Feldberg, S.W.J. Electroanal. Chem. 1981.127.1-10. (45)Feldberg, S.W.J. Electroanal. Chem. 1990,290,49-65. Analytical Chemistry, Vol. 67, No. 2, January 15, 1995
313
-= 1.1
c/ 0
l**r 2
rX,I,
4 012/mol cm-*
6
Figure 2. Theoretical the dependence of the relative maximum current (IR") on r D p . Concentration of FMA, 0.5 mM.
of diaphorase, K m is the Michaelis constant for FMA+, and C w + is the concentration of FMA+ at the surface. The values for ko M, re~pectively.~~ These K m and and K m are 110 s - l and 1 x ko values are for the immobilized enzyme and cannot be compared directly to those for a homogeneous system because of the difference in the dimensions of the collision However, the mechanisms of the enzyme reactions itself should be basically the same. We calculated the steady-state current at 0.4 V vs Ag/AgCl in voltammograms at various tip-substrate distances by digital simulation incorporating eq 1 (see Appendix). The steady-state current was plotted against distance. The maximum value ( Z R ~ ) in the steady-state current vs distance profile depends on the surface concentration of diaphorase (rDp).17 Figure 2 shows the dependence of the ZR- on rb calculated by the digital simulation. we used this working curve for determining rDp. RESULTS AND DISCUSSION Characterization of Diaphorase-Patterned Substrate Prepared by Photolithographic Method. A substrate with stripe
patterns (diaphoraseimmobilized bandwidth, 100pm; gap width, 200 pm) was dipped into a 0.5 mM FMA, 5.0 mM NADH/O.l M KCl, 0.1 M phosphate buffer solution (PH 7.5). The position of the microelectrode tip was changed stepwise by 20 pm along the surface (x axis). At each point, we recorded the oxidation current vs distance (z axis) curves to obtain the ZR- values. Figure 3 shows the profile of IRmaX dong the photolithographically pattemed substrate. Peaks in the profile appear periodically every 300 pm, in accordance with the enzymeimmobilized pattern. The surface concentration of active diaphorase can be determined from the maximum current using the working curve shown in Figure 2. At the diaphoraseimmobiliied band, the enzyme concentration was found to be -2 x 10-l2 mol cm-2. Some active diaphorase mol cm-2) were also observed, even in molecules (-4 x the gap region, probably due to physical adsorption at the glass surface. Electrochemical Micropatterningby ElectrogeneratedActive Bromine Species. Enzymepatterned surfaces are also fabricated using surface reactions initiated by microelectrochemical conversion. We have used bromine species generated at a microelectrode to deactivate localized enzyme molecules immobilized at the surface. The enzyme-immobiliied substrate was immersed into a 25 mM KBr, 2 mM &Fe(CN)6/0.1 M KCl, 0.1 M phosphate buffer solution (PH 7.5). The electrochemical (46) Andrieux, C. P.; Dumas-Bouchiat, J. M.; Saveant,J. M.]. Efedmunuf.Chem. 1981, 171-187.
314 Analytical Chemistry, Vol. 67, No. 2, January 15, 1995
1
I
Figure 3. Bottom, profile of /Rmax along the diaphorase-micropattemed surface. Top, an illustration of the micropattemedsurface. Enzyme-immobilizedbandwidth, 100 pm; gap width, 200 pm. The catalytic current was measured in a 0.5 mM FMA, 5.0 mM NADH/ 0.1 M KCI, 0.1 M phosphate buffer solution (pH 7.5).
oxidation of Br- yields Br2, which subsequently reacts with HzO to generate HOBr, a strong oxidant as well as a bromination reagent3* We placed the microelectrode tip -7 pm from the diaphorase-immobilized substrate and applied a potential pulse of 1.7 V vs Ag/AgCl with a 5,20, or 80 s period to deactivate the localized enzyme molecules nearby. The substrate was then transferred into a solution containing 0.5 mM FMA and 5.0 mM NADH to obtain SECM images based on detection of catalytic current for oxidation of NADH. The catalytic activity of the immobilized diaphorase was determined from the SECM images. Figure 4 shows the SECM images of the diaphorase-immobiliied substrates with deactivated enzyme areas. Circular areas with low enzyme activities are clearly seen in the images. The diameter of the deactivated circles is roughly proportional to the square root of the pulse period for deactivation. Since we fixed the position of the microelectrode in the deactivation process, the electrogenerated species diffused spherically from the electrode tip and reacted with the immobilized enzyme. The size of the deactivated circle reflects the diffusion length of the electrogenerated species. At the center of the circles, the enzyme activity is almost lost, indicating that the present procedure is an effective way for deactivating localized enzyme molecules. If the tip for deactivation moves along the enzymeimmobilized substrate, one can obtain micropatterned enzyme surfaces. Figure 5 shows an SECM image of a diaphoraseimmobilized substrate with a deactivated line. In this case, we used a smaller microelectrode (Pt disk radius, 2.3 pm; whole tip radius, 30 pm). The microelectrode for deactivation was held at 1.7 V vs Ag/AgCl and scanned linearly (73 pmls) along the diaphorase-immobilized substrate (tip-substrate distance, less than 5 pm). After the deactivation, 5.0 mM NADH was added to the solution, and the electrode potential was changed to 0.4 V vs AgIAgCl to monitor the oxidation current of FMA. The microelectrode was then scanned (2.4 pm/s) over the substrate with a deactivated enzyme line (tip-substrate distance, 8pm) . About 10 min was necessary to obtain the SECM image shown in Figure 5. The image indicates that the width of the deactivated line is -20 pm. It is also possible to quantify the surface concentration of the active diaphorase from the image. The r D p values at the center of the
(a)arbrx’o
12
-2 nA
4
2 Figure 6. SECM image of diaphorase-immobilizedsubstrates with a deactivated circle in a 0.5 mM FMA, 5.0 mM NADH/1.O M KCI, 0.1 M phosphate buffer solution (pH 7.5). Tip potential, 0.4 V vs Ag/ ASCI. Deactivationwas carried out in a 0.5 mM FMN1.O M KCI, 0.1 M phosphate buffer solution (pH 7.5). Tip potential for deactivation, 1.9 V vs Ag/AgCI, 5 min.
Figure4. SECM images of diaphorase-immobilizedsubstrates with circularly deactivated areas in a 0.5 mM FMA, 5.0 mM NADH/O.l M KCI, 0.1 M phosphatebuffer solution (pH 7.5). Deactivation reaction was initiated by electrogeneratedbromine species at a microelectrode located, near the substrate by applying a potential pulse (1.7V vs Ag/AgCI). Pulse period for inactivation: (a) 5, (b) 20, and (c) 80 s.
A nA -\-
an SECM image of the diaphorase-immobilized substrate with a deactivated circle induced by electrooxidation of the chloride ion.3l In this case, a microelectrode (Pt disk radius, 7.5 pm; tip radius, 50 pm) for the deactivated circle was placed less than 5 pm apart from the substrate. A potential pulse of 1.9 V vs Ag/AgCl with a 5 min period was applied to the microelectrode in a 0.5 mM FMA/ 1.0 M KCl, 0.1 M phosphate buffer solution (PH7.5). The area of the deactivated circle is small compared to those in Figure 4, even though the deactivation time is much longer. Voltammetric investigation suggested that most of the oxidation current at 1.9 V vs Ag/AgCl was consumed by decomposition of water. In addition, the electrogenerated chlorine species may react with watefll to lose the reactivity before diffusing into the enzymeimmobilized substrate surface. Although the detailed mechanism for the deactivation of an immobilized enzyme by an electrogenerated halogen species is unknown at the present stage, the present procedure is widely applicable to fabrication of enzyme-patterned surfaces. The resolution of the patterns will be improved using a smaller microelectrode, as has been vigorously studied by Bard and coworker~!~ ACKNOWLEDGMENT This research has been partly supported by Grants-in-Aid for
Scientific Research (Nos. 05235102 and 06558117) from the Ministry of Education, Science and Culture, Japan. Figure 5. SECM image of a diaphorase-immobilizedsubstratewith a deactivated line in a 0.5 mM FMA, 5.0 mM NADH/25 mM KBr, 0.1 M phosphate buffer solution (pH 7.5). Tip potential, 0.4 V vs Ag/ ASCI. The deactivated line was created by moving the electrode tip (1.7 V vs Ag/AgCI) for deactivation at 73 pm/s along the surface in a 0.5 mM FMA/25 mM KBr, 0.1 M phosphate buffer solution (pH 7.5).
deactivated line and at the untreated surface were found to be -9 x and -2 x 10-l2 mol cm-2, respectively. Electrochemical Micropatteming by Electrogenerated Active Chlorine Species. Electrogenerated chlorine species were also used to deactivate the immobilized enzyme. Figure 6 shows
APPENDIX
DigitalSimulationfor Microelectrode. The catalytic current was calculated by digital simulation with the explicit finite difference (Em)methodMor the fast quasiexplicitfinite d ~ e r e n c e (FQEFD) method&based on the axisymmetric grid model shown in Figure 7. In this model, the volume elements are exponentially expanded for both the r direction and the z direction. For the radial (r) direction, the grid was divided from the edge of the electrode to the center of the electrode and to infinity on the electrode. For the z direction, the grids are expanded both from (47) Lee,C.;Miller, C.J.; Bard, A
J. Awl. Chem. 1991, 63.78-83.
Analytical Chemistry, Vol. 67, No. 2, January 15, 1995
315
i=l 2 5 i 5 i, i = i, i=i,+l i,+25i
-
1
i=i,o’5 -1)
a
Where D* = DAt/(Ar)2.* Same as above for i,
+ 2 5 i.
Same as above for j = 1 (1 5 i 5 ia).
and kb is the backward electron transfer rate. We describe here the FQEFD method. From the diffusion equation, we can calculate new concentrations (At later) using the concentrations in the previous iteration (At earlier):
t‘
I
I
I
I l l
I
I
I
I
I
I
I
I
I
I
I
1
r i 1 2
...
l
I,
l
.....
i,,
..
2 i
.
rQ Figure 7. Spatial grid modeled for SECM simulation and notations.
the surface of the electrode (z = 0) and from that of the substrate (z = d) to the center of the tip-substrate distance (d/2). The diffusion equation was solved digitally under the following initial and boundary conditions:
t = o , o l r s a , z = o C F ~ = CbulkF, ~C F ~ + = O(AI) 0 < t, 0 5
?‘
5 a, = 0
DFm(acFm/az) = -&m+ (acFm+/az) = kEFhlA - kbCFM.4- (-42)
a < r, z = 0 aCFm/az = aCFm+/az = 0
0
5
r, Z = d DFm(aCF-/az)
(A3)
=
-&m+ (acFm+/aZ) = f F m (A4) where C Fis the~bulk concentration ~ ~ ~for FMA, fFmis the flux for FMA at the substrate, kf is the forward electron transfer rate, 316 Analytical Chemistly, Vol. 67, No. 2,Janualy 75,7995
Subscript x signifies FMA or FMA+. Single, double, and triple prime on c, denote the previous, present, and new concentrations separated by At. The variables Doufi, Din,*,D,j, and Ddomj are functions of geometry and are related to the diffusion coefficient. The values are listed in Table 1. For j = jd, 1 5 i, we must consider the enzyme reaction:
+
Ac,(ijJ = [Dout,i{~/(i 1jJ - ~ , ’ ( i j J }- Din,i{~,’(ijJc,”(i - IjJ} - Ddomjd{C,’(ijJ- c,”(ijd - 1)) mzFMA,Enzl/(l
+
+ Dout,i + Djn,i + D d o m j )
(A7)
Forx = FMA, m = 1;forx = FMA’, m = -1 The term Z reaction 1:
Z F ~ E is ~the
flux of FMA caused by the enzyme
F o r j = 1, 1 5 i 5 i,,
where j3 characterizes the degree of exponential expansion of the spatial grid.” We calculated the cyclic voltammograms in the potential range, 0-0.4 V vs Ag/AgCl, and found that the current at 0.4 V was practically independent of the scan rate when the scan rate was less than 5 mV/s. For a typical case (tip-substrate distance, 15 pm), the decrease in the scan rate from 5 to 0.5 mV/s changes the current at 0.4 V only by 0.01%. The programs were written in Turbo Pascal and executed in double precision on an Intel 486 (50 MHz)-based IBM-compatible computer. The CPU time was -60 s to simulate a cyclic voltammogram at 5 mV/s (io= 8, irg= 20,jd = 16,/3 = 0.5,4600 iterations). Further division of the spatial grid gave no obvious difference in the calculation results.
Received for review June 28, 1994. Accepted October 18, 1994.@ AC940653W ~~
@Abstractpublished in Advance ACS Absfructs, December 1, 1994.
Analytical Chemistry, Vol. 67, No. 2, January 15, 1995
317
