Anal. Chem. 2006, 78, 3138-3143
Redox Enzymes Immobilized on Electrodes with Solution Cosubstrates. General Procedure for Simulation of Time-Resolved Catalytic Responses Claude P. Andrieux, Benoıˆt Limoges, Damien Marchal,* and Jean-Michel Save´ant*
Laboratoire d’Electrochimie Mole´ culaire, Universite´ de Paris 7-Denis Diderot, 2 place Jussieu, 75251 Paris Cedex 05, France
In view of the existing and potential applications of electrochemical enzymatic catalysis with redox enzymes immobilized on the electrode surface in biosensors, a numerical calculation procedure for simulating their cyclic voltammetric responses is presented. It is applicable to systems involving a redox cosubstrate in solution. The cosubstrates, substrates, products, and inhibitors are assumed to diffuse linearly (planar electrode) between the electrode and the solution. The reactions in which the various forms of the immobilized enzyme participate may be as numerous and intricate as required by the simulation with no other restriction than the computing time. They may, at will, follow or not follow Michaelis-Menten kinetics. Slow charge-transfer cosubstrates are treated in the framework of Butler-Volmer kinetic law. Biosensors constitute the main application of electrochemical enzymatic catalysis with redox enzymes immobilized on the electrode surface. They are of two types. In one of them, the target analyte is simply the enzyme substrate.1 A leading example in this area is the worldwide spread glucose oxidase electrode used to measure the blood glucose concentration.2 Another relates to affinity biosensing as, for example, in electrochemical enzyme immunosensors3 and DNA sensors.4 In these systems, the specific molecular recognition between a target analyte and an electroactive receptor immobilized on the electrode is measured by the current flowing through the electrode. In this framework, enzyme labeling is a powerful manner of amplifying the electrochemical response compared to simple redox labels.5 Theory and calculation procedures allowing the prediction of the voltammetric responses of such systems and the derivation of their kinetic characteristics provide useful information for their optimization and for the design of new devices. They have been developed both for systems in * To whom correspondence should be addressed. E-mail: marchal@ paris7.jussieu.fr;
[email protected]. (1) (a) Armstrong, F. A.; Wilson G. S. Electrochim. Acta 2000, 45, 2623. (b) Wilson G. S.; Hu Y. Chem. Rev. 2000, 100, 2693. (c) Ikeda T. Bull. Electrochem. 1992, 8, 145. (d) Brajter-Toth, A.; Chambers, J. Q. Electroanalytical Methods for Biological Materials; Marcel Dekker: New York, 2002. (2) (a) Cass, A. E. G.; Davis, G.; Francis, G. D.; Hill, H. A. O.; Aston, W. J.; Higgins, I. J.; Plotkin, E. V.; Scott, L. D. L.; Turner, A. P. F. Anal. Chem. 1984, 56, 667-671. (b) Matthews, D. R.; Holman, R. R.; Bown, E.; Steemson, J.; Watson, A.; Hughes, S.; Scott, D. Lancet 1987, 778. (c) HarnShen, C.; Benjamin, K. I.; Chii-Min, H.; Kuang-Chung, S.; Ching Fai, K.; Low-Tone, H. Diabetes Res. Clin. Pract. 1998, 42, 9. (d) McGarraugh, G.; Price, D.; Schwartz, S.; Weinstein, R. Diabetes Technol. Ther. 2001, 3, 367.
3138 Analytical Chemistry, Vol. 78, No. 9, May 1, 2006
which direct electron transfer between the electrode and the enzyme occurs6 and for systems in which a soluble mediator shuttles the electron transfer between the electrode and the enzyme, thus serving as artificial cosubstrate for the enzyme.7 Although the limiting situations or particular reaction schemes that have been dealt with so far are extremely useful in the deciphering of mechanisms and kinetic characterization, a general procedure able to address any type of reaction scheme is still lacking. Filling this gap was the objective of the work reported hereafter. We thus describe a procedure for the simulation of cyclic voltammetric catalytic responses of an immobilized enzyme on the electrode with substrates and cosubstrates that diffuse in the solution. Diffusion is assumed to be linear (planar electrode), and the cosubstrate is assumed to exchange electrons with the (3) (a) Safina, G. R.; Medyantseva, E. P.; Fomina, O. G.; Glushko, N. I.; Budnikov, G. K. J. Anal. Chem. 2005, 60, 546. (b) Micheli, L.; Grecco, R.; Badea, M.; Moscone, D.; Palleschi, G. Biosens. Bioelectron. 2005, 21, 588. (c) Wilson, M. S. Anal. Chem. 2005, 77, 1496. (d) Kreuzer, M. P.; Pravda, M.; O’Sullivan, C. K.; Guilbault, G. G. Toxicon 2002, 40, 1267. (e) Scheller, F. W.; Bauer, C. G.; Makower, A.; Wollenberger, U.; Warsinke, A.; Bier, F. F. Anal. Lett. 2001, 34, 1233. (f) Bagel, O.; Degrand, C.; Limoges, B.; Joannes, M.; Azek, F.; Brossier, P. Electroanalysis 2000, 12, 1447. (g) Vallat, C.; Limoges, B.; Huet, D.; Romette, J.-L. Anal. Chim. Acta 2000, 404, 187. (h) Campbell, C. N.; Lumley-Woodyear, T.; Heller, A. Fresenius J. Anal. Chem. 1999, 364, 165. (i) Del Carlo, M.; Mascini, M. Anal. Chim. Acta 1996, 336, 167. (j) Pritchard, D. J.; Morgan, H.; Cooper, J. M. Anal. Chim. Acta 1995, 310, 251. (k) Kalab, T.; Skladal, P. Anal. Chim. Acta 1995, 304, 361. (l) Rishpon, J.; Ivnitski, D. Biosens. Bioelectron. 1997, 12, 195. (m) Gyss, C.; Bourdillon, C. Anal. Chem. 1987, 59, 2350. (4) (a) Marchand, G.; Delattre, C.; Campagnolo, R.; Pouteau, P.; Ginot, F. Anal. Chem. 2005, 77, 5189. (b) Liu, D.; Perdue, R. K.; Sun, L.; Crooks, R. M. Langmuir 2004, 20, 5905. (c) Carpini, G.; Lucarelli, F.; Marrazza, G.; Mascini, M. Biosens. Bioelectron. 2004, 20, 167. (d) Zhang, Y.; Pothukuchy, A.; Shin, W.; Kim, Y.; Heller, A. Anal. Chem. 2004, 76, 4093. (e) Dequaire, M.; Heller, A. Anal. Chem. 2002, 74, 4370. (f) Azek, F.; Grossiord, C.; Joannes, M.; Limoges, B.; Brossier P. Anal. Biochem. 2000, 284, 107. (g) Caruana, D. J.; Heller A. J. Am. Chem. Soc. 1999, 121, 769. (5) (a) Azek, F.; Grossiord, C.; Joannes, M.; Limoges, B.; Brossier, P. Anal. Biochem. 2000, 284, 107. (b) Caruana, D. J.; Heller, A. J. Am. Chem. Soc. 1999, 121, 769. (c) Gyss, C.; Bourdillon, C. Anal. Chem. 1987, 59, 2350. (6) (a) Heering, H. A.; Hirst, J.; Armstrong, F. A. J. Phys. Chem. B 1998, 102, 6889. (b) Leger, C.; Elliott, S. J.; Hoke, K. R.; Jeuken, L. J. C.; Jones, A. K.; Armstrong, F. A. Biochemistry 2003, 42, 8653. (c) Honeychurch, M. J.; Bernhardt, P. V. J. Phys. Chem. 2004, 108, 15900. (7) (a) Bourdillon, C.; Demaille, C. ; Moiroux, J.; Save´ant, J.-M. Acc. Chem. Res. 1996, 29, 529. (b) Bourdillon, C.; Demaille, C.; Gue´ris, J.; Moiroux, J.; Save´ant, J.-M. In Protein Architecture: Interfacing Molecular assemblies and immobilization biotechnology; Mo ¨hwald, H., Lvov, V., Eds.; Marcel Dekker: New York, 1999; pp 311-335. (c) Limoges, B.; Moiroux, J.; Save´ant, J. M. J. Electroanal. Chem. 2002, 521, 8. (d) Limoges, B.; Save´ant, J. M. J. Electroanal. Chem. 2003, 549, 61. (e) Limoges, B.; Yazidi, D.; Save´ant, J.M. J. Am. Chem. Soc. 2003, 125, 9122. (f) Limoges, B.; Save´ant, J. M. J. Electroanal. Chem. 2004, 562, 43. 10.1021/ac052176v CCC: $33.50
© 2006 American Chemical Society Published on Web 03/24/2006
The subscript 0 indicates that the corresponding species is not involved in the reaction. For example, if ei does not react with sj, then the rate constant writes ki,0,*i,*j. All four first types of reactions are taken into account in these expressions of the kinetic terms. The distribution of the enzyme over its various forms at time zero provides the initial conditions for eq 1. In the preceding analysis, formation of an enzyme-substrate adduct and its successive reductive decomposition constitute two separate reactions, characterized each by their own rate constants:
Table 1. Enzymatic Reactions indexing reactions
forward
backward
ki,j,*i,*j
k*i,*j,i,j
ki,j,*i,0
k*i,0,i,j
ki,0,*i,*j
k*i,*j,i,0
ki,0,*i,0
k*i,0,i,0
kf
ei + sj y\ z e*i + s*j k b
kf
ei + sj y\ z e*i k b
kf
ei y\ z e*i + s*j k b
kf
ei y\ z e*i k b
ki,j,mm,0
Michaelis-Menten kinetics
mm,0,i,j
ki,j,*i,*j KMi,j,*i,*j
k
ei + sj 9 8 e*i + s*j K
kmm,0, *i,*j
z emm 98 e*i + s*j ei + sj y\ k
M
electrode according to the Butler-Volmer kinetic law. There is no limitation to the number of enzymatic reactions taking place at the electrode surface other than the calculation time, which increases with the complexity of the reaction mechanism. The procedure we developed in this purpose is the basis of a software package (SIMSENSOR) available on the Internet (http:// www.lemp7.cnrs.fr/). The software allows simulation of cyclic voltammograms and may also be used for single potential-step chronoamperometry experiments. This technique can be a useful alternative to cyclic voltammetry if one wants to avoid the interference of the electrode electron-transfer kinetics in the current response. RESULTS AND DISCUSSION Governing Equations, Initial and Boundary Conditions. Dealing with catalysis, we make the assumption, fulfilled in most practical cases, that there exists at least one irreversible step in the reaction sequence. It follows that all reactions may be considered as thermodynamically independent. The reactions involving the various forms of the immobilized enzyme, the substrates, cosubstrates, and products are of four types, as summarized in Table 1. They all fit in the same indexing system as defined in the table with ei for enzymes and sj for substrates, cosubstrates, and products. The differential equations governing the surface concentration of the various forms of enzymes, Γei, are thus given by eq 1 i i dΓei/dt ) T eprod - T econs
(1)
where the kinetic terms for the consumption and production of the enzyme form ei are given by eqs 2 and 3. i T econs )(
∑k
j,*i,*j
i,j,*i,*j[sj]x)0
+
∑k
i,j,*i,0[sj]x)0
∑k
i,0,*i,*j
+
*i,*j i T eprod )
∑k
*i,*j,j
+
j,*i
*i,*j,i,jΓe*i[s*j]x)0 +
∑k *i,j
∑k *i
∑k
*i,*j
i,0,*i,0))Γei
*i,*j,i,0Γe*i
*i,0,i,jΓe*i
+
∑k *i
(2)
+
*i,0,i,0Γe*i
(3)
In certain circumstances it can be useful to stipulate a priori that the Michaelis-Menten kinetics apply. The advantage of such a specification is that the two reactions are kinetically defined by two parameters instead of three. In this case, the enzymesubstrate adduct obeys the steady-state assumption:
Γemm ) Γei[sj]x)0/KM The Michaelis-Menten constant, KM, and the second-order rate constant, k, being defined as
KM )
kmm,0,i,j + kmm,0,*i,*j ki,j,mm,0
and
k)
kmm,0,*i,*j KM
We are thus led to introduce the sum ΓeMM ) Γei + Γemm to i replace Γei and Γemm. It follows that eq 1 applies to ΓeMM , leading i to eq 4. i iMM dΓeMM /dt ) T eprod - T econs i
(4)
with, after introduction of MM ki,j,*i,*j )
iMM T econs )(
∑
ki,j,*i,*j
ki,j,*i,* j[sj]x)0 +
j,* i,* j
+
∑k
*i,*j
i,0,*i,*j
(5)
1 + [sj]x ) 0/KM
∑k
i,j,*i,0[sj]x)0
j,*i
+
∑k *i
i,0,*i,0
+
∑k
j,*i,* j
MM i,j,*i,*j[sj]x ) 0)
ΓeMM (6) i
When products are obtained from a Michaelis-Menten kinetics reaction (for example, ei + sj f e*i + s*j), the contribution of this MM MM reaction to T e*i prod is ki,j,*i,*j Γei [sj]x)0, the other terms being the same as in eq 3. Calculations may thus be carried out with the sole consideration of ΓeMM with the time-dependent rate “coni stant” defined in eq 5. The substrates, cosubstrates, and products are all noted sj. They are transported by linear diffusion between the electrode, where their concentrations are noted [sj]x)0 (x is the distance to the electrode surface), and the bulk of the solution, where their concentrations are noted [sj]0. The time- and space-dependent concentration of sj thus obeys second Fick’s law: Analytical Chemistry, Vol. 78, No. 9, May 1, 2006
3139
validity of the Butler-Volmer law, eq 13 serves as boundary condition:9
∂[sj] ∂2[sj] ) D sj 2 ∂t ∂x
(Dsj is the diffusion coefficient of the subscript species) at time t ) 0, [sj]x)0 ) [sj]0. It follows that the concentration of sj and its gradients at the electrode surface are related by the following integral equation.8a
0
[sj]x)0 ) [sj]
xDs xπ
∫(
)
∂sj ∂x
t
j
0
dη
x)0xt
(7)
-η
RF i ) ks exp (E - E0) [sox]x)0 FS RT
[
]
ks exp
[
E° is the standard potential, kS, the standard rate constant, and R the transfer coefficient. The electrode potential, E is varied linearly with time according to
E ) Ei - vt
0 et etR: The flux of nonelectroactive substrates at the electrode surface is nil, leading to the following boundary conditions:
( )
D sj
∂[sj] ∂x
x)0
j j + T sprod - T scons )0
(8)
with j T scons )
∑k
i,*i,*j
+
i,j,*i,*jΓei[sj]x)0
∑k
i,j,*i,0Γei[sj]x)0
i,*i
∑k
i,*i,*j j T sprod )
∑k
*i,*j,i
*i,*j,i,jΓe*i[s*j]x)0 +
+
MM MM i,j,*i,*jΓei [sj]x)0
∑k
*i,*j
(9)
*i,*j,i,0Γe*i[s*j]x)0
]
(1 - R)F (E - E0) [sred]x)0 (13) RT
E ) Ef + v(t - tR) ) 2Ef - Ei + vt (14)
tR et e2tR:
v, the scan rate is positive for a reduction and negative for an oxidation. Ei and Ef are the starting and inversion potentials, respectively. Numerical Calculations. The successive steps of the simulation procedure are summarized in the flowchart (Chart 1). The potential scan interval is first divided into n elementary intervals of length ∆E, to which correspond n elementary time intervals ∆t ) ∆E/v (v is the scan rate). This first division is adapted to the numerical resolution of the substrate and cosubstrate diffusion equations. In most cases, it is too gross for the numerical resolution of eq 1 according to eq 15.
(10) i i Γei(t) ) Γei(t - ∆t) + (T eprod - T econs ) ∆t
(15)
and/or
∑k
MM *i,*j,i,j
ΓeMM [s*j]x)0 *i
*i,*j,i
terms for the reactions following Michaelis-Menten kinetics. For electroactive substrates, we call sj and s*j, sox, and sred, respectively. Their fluxes at the electrode surface are directly related to the current, i. Counting currents resulting from a reduction as positive
( )
∂[sox] i ) Dox FS ∂x
x)0
ox ox + T sprod - T scons
(11)
where the two kinetic terms defined by eqs 2, 3, and 6 contain the enzyme and substrate concentrations relative to time t - ∆t. The inappropriateness of the division manifests itself by the fact that some of the enzyme form concentrations become negative. As shown in the Chart 1, the interval ∆t is thus divided into finer and finer subintervals of length ∆t' until all enzyme form concentrations are positive. Using the Γei(t) values thus obtained, the substrate and cosubstrate concentrations at t are then derived from the numerical resolution of the diffusion equations according to the following procedure. The convolution integral in eq 7 may be expressed as follows.8b
1
xπ
or
( )
∂[sred] -i ) Dred FS ∂x
x)0
red red + T sprod - T scons
(12)
the kinetic terms being given by application of eqs 9 and 10 applied to sox and sred. S is the electrode surface area. For sox and sred, the boundary condition at the electrode surface is provided by the electron-transfer rate law. If we assume the (8) (a) Imbeaux, J. C.; Save´ant, J.-M. J. Electroanal. Chem. 1970, 28, 325. (b) Amatore, C.; Save´ant, J. M. J. Electroanal. Chem. 1977, 85, 27.
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Analytical Chemistry, Vol. 78, No. 9, May 1, 2006
∫
t
0
() ∂sj ∂x
dη
xt - η
≈
x)0
1
l)n-1
xπ
∑ ∫
l)0
(l + 1)∆t
l∆t
() ∂sj ∂x
dη
xn∆t - η
x)0
Linearizing the variation of (∂sj/∂x)x)0 between l∆t and (l + 1)∆t
() ∂sj ∂x
(η) ≈
x)0
() {( ) ∂sj ∂x
x)0
(l∆t) +
∂sj ∂x
[(l + 1)∆t] -
x)0
() ∂sj ∂x
(l∆t)
x)0
}
η - l∆t ∆t
(9) (a) Butler, J. A. V. Trans. Faraday Soc. 1924, 19 , 729. (b) Erdey-Gru´z, T.; Volmer, M. Z. Phyz. Chem. 1930, 150A, 45. (c) Delahay, P. Double Layer and Electrode Kinetics; Wiley: New York, 1955; Chapter 7.
Therefore
1
∫
t
xπ
0
() ∂sj ∂x
+
() ∂sj ∂x
dη
xt - η
≈
x)0
[(l + 1)∆t] -
x)0
xπ
l)0
() ∂sj ∂x
∑∫
(l + 1)∆t
l∆t
(l∆t)
{
(n - l)∆t
x)0
xn∆t - η
∆t
() ∂sj ∂x
1
l)n-1
[(l + 1)∆t] -
x)0
() ∂sj ∂x
(l∆t)
x)0
∆t
Chart 1
() ∂sj ∂x
(l∆t)
x)0
xn∆t - η -
}
n∆t - η dη xn∆t - η
It follows that
1
xπ
∫
[ [
t
0
() ∂sj ∂x
dη
xt - η
≈
x)0
x
4
∆t l ) n - 1
3
π
∑
×
l)0
]( ) )]( )
∂sj 3 (l∆t) + (n - l)1/2 - (n - l)3/2 + (n - l - 1)3/2 2 ∂x x)0 1 ∂sj (n - l)3/2 - (n - l - 1)1/2 n - l + [(l + 1)∆t] 2 ∂x x)0
(
Finally, eq 7 becomes
[( ) ∂[sj] ∂x
[sj]x)0(n) ) [sj]0 - Aj
(n) + Bj
]
(16)
(l)R(n - l)
(17)
x)0
with
Aj )
Bj )
x
4
x
4 3
Dsj∆t π
( )
Dsj∆t l)n-1 ∂[sj]
3
π
∑ l)0
∂x
x)0
R(n - l) ) (n - l - 1)3/2 - 2(n - l)3/2 + (n - l + 1)3/2 (18) At each time, the substrate and cosubstrate concentrations at the electrode surface can thus be calculated from their past values and those of their gradients according to eqs 8 and 16-18.
( )
Ds j
∂[sj] ∂x
x)0
j j + T sprod - T scons )0
(19)
This equation is applied to all the nonelectroactive species in solution sj leading to a system of linear equations that is solved according to the substitution method, thus leading to the values of all [sj]x)0. For the electroactive species two linear equations are obtained from eqs 11-13, for leading to the determination of [sox]x)0 and [sred]x)0.
The current is finally derived from eq 11 or 12. The two test steps m < 10 000 and n < 10 000 are introduced to avoid lengthy useless calculations in case of inadvertent dysfunction. The test of accuracy involves the comparison between the values of the current, inold and innew, obtained in two successive simulations with nnew ) 2nold. The test is repeated until |(innew - inold)|/inold e 0.02. For the simulation of a chronoamperogram, the test is performed for two values of the current corresponding to 10 and 20% of the total time excursion. For cyclic voltammetry, simulation, the test is performed at Ec - 10∆E and Ec - 20∆E, with Ec as the potential when the sign of the current derivative toward time changes for the first time. Analytical Chemistry, Vol. 78, No. 9, May 1, 2006
3141
Figure 2. Simulation of the potential step responses for the catalysis of the electrochemical reduction of hydrogen peroxide by immobilized horseradish peroxidase. Surface enzyme concentration, 2.8 × 10-12 M cm-2; [P]bulk ) 21 µM; [H2O2]bulk ) 0, 0.1, 0.5, 1, and 4 mM. Same simulation parameters as in Figure 1.
Scheme 1a
a E, E , E , and E are the various enzyme forms. P/Q is the 1 2 3 redox cosubstrate couple.
Figure 1. Simulation of the cyclic voltammetric responses for the catalysis of the electrochemical reduction of hydrogen peroxide by immobilized horseradish peroxidase. [P]bulk ) 21 µM. Temperature, 293 K; glassy carbon electrode surface area, 0.1 cm2. (a) Surface enzyme concentration, 2.8 × 10-12 M cm-2; [H2O2]bulk ) 0.515 mM, scan rate (V/s), 0.01 (magenta), 0.02 (cyan), 0.05 (red), 0.1 (green), and 0.2 (blue). (b) Surface enzyme concentration, 4 × 10-12 M cm-2; [H2O2]bulk ) 0.2 mM; scan rate, 0.01 V/s. (c) Surface enzyme concentration, 4 × 10-12 M cm-2; [H2O2]bulk ) 0.01 mM (blue line), [H2O2]bulk ) 0 (green line); scan rate (V/s), 0.01. Simulation parameters: E0 ) 0.2 V vs SCE, kS ) 0.02 cm s-1, R ) 0.5, k1 ) k1,1k1,2/(k1,-1 + k1,2) ) 1.7 × 107 M-1 s-1, K1,M ) (k1,-1 + k1,2)/k1,1 ) 128 µM, k2 ) 109 M-1 s-1, k3 ) k3,1k3,2/ (k3,-1 + k3,2) ) 7.85 × 106 M-1 s-1, K3,M ) (k3,-1 + k3,2)/k3,1 ) 37 µM, k4 ) 30 M-1 s-1, k5 ) 2080 M-1 s-1, k4 ) 0.01 s-1. DH2O2 ) 1.2 × 10-5 cm2 s-1, DP ) DQ ) 4 × 10-6 cm2 s-1. 3142
Analytical Chemistry, Vol. 78, No. 9, May 1, 2006
Example of Application from the Catalysis of the Electrochemical Reduction of Hydrogen Peroxide by Immobilized Horseradish Peroxidase. Previous studies of the title reaction with the enzyme in solution10a or immobilized on the electrode surface10b have shown that the catalytic mechanism consists of the set of reactions shown in Scheme 1. Besides the main cycle, made of two rate-determining reactions obeying Michaelis-Menten kinetics and of a fast irreversible reaction, three other reactions are related to inhibition by the substrate, namely, the inhibition reaction itself (k4) and two regeneration pathways. One is a slow spontaneous reaction (k6) while the other is a faster reaction involving the cosubstrate (k5). The rate of these reactions is such that their interference falls within the time window of cyclic voltammetry, making the characteristics of the cyclic voltammetric responses depend on the scan rate as illustrated in Figure 1a. Figure 1b gives a second example where the time dependence results in trace-crossing features, while Figure 1c illustrates the case where the catalytic current is governed by substrate diffusion. Figure 2 shows an example of potential step chronoamperometric responses obtained with the same system. Simulation of these various curves requires 2-10 s with a standard PC (1.8-Ghz processor, 256-MB memory). (10) (a) Dequaire, M.; Limoges, B.; Moiroux, J.; Save´ant, J.-M. J. Am. Chem. Soc. 2002, 124, 240. (b)Limoges, B.; Save´ant, J.-M.; Yazidi, D. J. Am. Chem. Soc. 2003, 125, 9192.
CONCLUDING REMARKS An efficient numerical calculation procedure has been presented and tested for systems involving a solution redox cosubstrate and an immobilized enzyme. It may involve a theoretically unlimited number of enzyme forms and enzymatic reactions. In practice, the computer time is the only limit in this connection. As shown with the example of catalysis of the electrochemical reduction of hydrogen peroxide by immobilized horseradish peroxidase, the computation time remains quite reasonable for
complex mechanisms. This system also illustrates the time resolution capability of the calculation procedure. In other words, Michaelis-Menten assumptions can be lifted and the time evolution of each enzyme form can be taken into account.
Received for review December 9, 2005. Accepted January 30, 2006. AC052176V
Analytical Chemistry, Vol. 78, No. 9, May 1, 2006
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