Digital Simulation of Scanning Electrochemical ... - ACS Publications

May 14, 2009 - using SECM has been performed in different measuring modes assuming a rate law according to Michaelis-Menten kinetics. Microscopic spot...
0 downloads 0 Views 456KB Size
Download PDF
Anal. Chem. 2009, 81, 4857–4863

Digital Simulation of Scanning Electrochemical Microscopy Approach Curves to Enzyme Films with Michaelis-Menten Kinetics Malte Burchardt,† Markus Tra¨uble, and Gunther Wittstock* Carl von Ossietzky University of Oldenburg, Faculty of Mathematics and Natural Sciences, Center of Interface Science, Institute of Pure and Applied Chemistry, D-26111 Oldenburg, Germany The formalism for simulating scanning electrochemical microscopy (SECM) experiments by boundary element methods in three space coordinates has been extended to allow consideration of nonlinear boundary conditions. This is achieved by iteratively refining the boundary conditions that are encoded in a boundary condition matrix. As an example, the simulations are compared to experimental approach curves in the SECM feedback mode toward samples modified with glucose oxidase (GOx). The GOx layer was prepared by the layer-by-layer assembly of polyelectrolytes using glucose oxidase as one of the polyelectrolytes. The comparison of the simulated and experimental curves showed that under a wide range of experimentally accessible conditions approximations of the kinetics at the sample by first order models yield misleading results. The approach curves differ also qualitatively from curves calculated with first order models. As a consequence, this may lead to severe deviations when such curves are fitted to first order kinetic models. The use of linear approximations to describe the enzymatic reaction in SECM feedback experiments is justified only if the ratio of the mediator and Michaelis-Menten constant is equal to or smaller than 0.1 (deviation less than 10%). The investigation of immobilized enzymes using scanning electrochemical microscopy (SECM) has been applied in numerous studies such as biosensor development, readout of bioanalytical assays, and investigation of new surfaces for enzyme immobilization.1-4 It can provide information about the lateral distribution of the enzyme5-11 and kinetic information about the * Corresponding author. Prof. Dr. Gunther Wittstock, phone: +49-441-798 3971. Fax: +49-441-798 3979. E-mail: [email protected]. † Present address: Fraunhofer Institute for Manufacturing Technology and Applied Materials Research, Wiener Str. 12, D-28359 Bremen, Germany. (1) Wittstock, G.; Burchardt, M.; Pust, S. E.; Shen, Y.; Zhao, C. Angew. Chem., Int. Ed. 2007, 46, 1584–1617. (2) Horrocks, B. R.; Wittstock, G. In Scanning Electrochemical Microscopy; Bard, A. J., Mirkin, M. V., Eds.; Marcel Dekker: New York, 2001; pp 445-519. (3) Sun, P.; Laforge, F. O.; Mirkin, M. V. Phys. Chem. Chem. Phys. 2007, 9, 802–823. (4) Szunerits, S. Handb. Biosens. Biochips 2007, 1, 269–289. (5) Wilhelm, T.; Wittstock, G. Angew. Chem., Int. Ed. 2003, 42, 2247–2250. (6) Nogala, W.; Burchardt, M.; Opallo, M.; Rogalski, J.; Wittstock, G. Bioelectrochemistry 2008, 72, 174–182. (7) Ciobanu, M.; Taylor, D. E.; Wilburn, J. P.; Cliffel, D. E. Anal. Chem. 2008, 80, 2717–2727. 10.1021/ac9004919 CCC: $40.75  2009 American Chemical Society Published on Web 05/14/2009

enzymatic reaction.12-14 Quantitative analysis of enzyme activity using SECM has been performed in different measuring modes assuming a rate law according to Michaelis-Menten kinetics. Microscopic spots of immobilized enzyme can be investigated in the substrate-generation/tip-collection (SG/TC) mode.14-19 The flux of product of the enzymatic reaction can be calculated using a theory developed for diffusion through a disk shaped pore.20,21 Typically, the substrate(s) X of the enzyme is (are) provided in high concentration ([X]* . KM,X; [X]*, bulk concentration; KM,X, Michaelis-Menten constant of the enzyme with respect to X) so that the enzymatic reaction proceeds at apparent zeroth order with respect to X. The feedback (FB) mode can be applied for quantitative analysis of immobilized oxidoreductases also at macroscopic samples.12-14,22,23 The electron mediator (R) is typically provided in a very low concentration ([R]* , KM,O) and the substrate (S) of the enzyme in a very high concentration ([S]* . KM,S) so that the enzymatic reaction follows a pseudo first order rate law with respect to R. In such cases, analytical approximations of simulated approach curves are available and can be used to determine normalized effective heterogeneous rate constants keff.24-26 If the condition [R]* , KM,O cannot be fulfilled, the analysis of SECM approach curves (8) Schaefer, D.; Maciejewska, M.; Schuhmann, W. Biosens. Bioelectron. 2007, 22, 1887–1895. (9) Hussien, E. M.; Erichsen, T.; Schuhmann, W.; Maciejewska, M. Anal. Bioanal. Chem. 2008, 391, 1773–1782. (10) Luo, H. Q.; Shiku, H.; Kumagai, A.; Takahashi, Y.; Yasukawa, T.; Matsue, T. Electrochem. Commun. 2007, 9, 2703–2708. (11) Oliveira, E. M.; Beyer, S.; Heinze, J. Bioelectrochemistry 2007, 71, 186– 191. (12) Pierce, D. T.; Unwin, P. R.; Bard, A. J. Anal. Chem. 1992, 64, 1795–1804. (13) Burchardt, M.; Wittstock, G. Bioelectrochemistry 2008, 72, 66–76. (14) Zhao, C.; Wittstock, G. Anal. Chem. 2004, 76, 3145–3154. (15) Wittstock, G.; Yu, K.-j.; Halsall, H. B.; Ridgway, T. H.; Heineman, W. R. Anal. Chem. 1995, 67, 3578–3582. (16) Nunes Kirchner, C.; Szunerits, S.; Wittstock, G. Electroanalysis 2007, 19, 1258–1267. (17) Shiku, H.; Matsue, T.; Uchida, I. Anal. Chem. 1996, 68, 1276–1278. (18) Wittstock, G.; Schuhmann, W. Anal. Chem. 1997, 69, 5059–5066. (19) Wijayawardhana, C. A.; Ronkainen-Matsuno, N. J.; Farrel, S. M.; Wittstock, G.; Halsall, H. B.; Heineman, W. R. Anal. Sci. 2001, 17, 535–538. (20) Saito, Y. Rev. Polarogr. 1968, 15, 177–187. (21) Scott, E. R.; White, H. S.; Phipps, J. B. Anal. Chem. 1993, 65, 1537–1545. (22) Pierce, D. T.; Bard, A. J. Anal. Chem. 1993, 65, 3598–3604. (23) Pellissier, M.; Zigah, D.; Barriere, F.; Hapiot, P. Langmuir 2008, 24, 9089– 9095. (24) Bard, A. J.; Mirkin, M. V.; Unwin, P. R.; Wipf, D. O. J. Phys. Chem. 1992, 96, 1861–1868. (25) Wei, C.; Bard, A. J.; Mirkin, M. V. J. Phys. Chem. 1995, 99, 16033–16042. (26) Cornut, R.; Lefrou, C. J. Electroanal. Chem. 2008, 621, 178–184.

Analytical Chemistry, Vol. 81, No. 12, June 15, 2009

4857

is complicated,12 especially if the kinetic regime changes from pseudo first to zeroth order with decreasing working distance d within one approach curve. This situation is frequently encountered, because a roughly constant flux of mediator molecules formed at the ultramicroelectrode (UME) is distributed to a larger area under the sample at large d but has to be converted at small area at decreased d. Although deviations from zeroth order as well as from first order rate laws have already been observed by Pierce et al.,12 no digital simulations or more advanced theoretical description are available for this practically important situation. Other detection schemes such as redox competition mode27,28 and tip-generation/ substrate-collection mode29 have not been used for quantitative investigation of enzyme samples. However, in principle such treatment should be possible.30 Glucose oxidase (EC 1.1.3.4) is the most frequently used enzyme in biosensor research. Besides its commercial importance, it serves as a model enzyme for surface modifications. The published KM values for electron mediators lie in the range of a few tens of micromoles per liter to a few millimoles per liter,31,32 a concentration range typically used in SECM FB experiments. Thus, quantitative kinetic analysis of SECM approach curves by either pure first order or by zeroth order rate laws is difficult, because neither treatment applies in these cases. For this purpose, we extended our simulations with the boundary element method (BEM).33-36 While we have already demonstrated finite first order kinetics at the boundaries,37-39 the application to enzymatic reactions still relied on using linear first order or zeroth order approximations to describe the enzyme kinetics.40 Here, we present a new method for the digital simulation of SECM approach curves if the regeneration rate of the mediator at the sample cannot be linearized to a pseudo first order or pseudo zeroth order rate law, but true Michaelis-Menten kinetics have to be considered. The results are compared to experimental SECM approach curves to macroscopic GOx films prepared by the layer-by-layer technique and recorded at different mediator concentrations. (27) Eckhard, K.; Chen, X.; Turcu, F.; Schuhmann, W. Phys. Chem. Chem. Phys. 2006, 8, 5359–5365. (28) Eckhard, K.; Schuhmann, W. Electrochim. Acta 2007, 53, 1164–1169. (29) Fernandez, J. L.; Mano, N.; Heller, A.; Bard, A. J. Angew. Chem., Int. Ed. 2004, 43, 6355–6357. (30) Fernandez, J. L.; Bard, A. J. Anal. Chem. 2004, 76, 2281–2289. (31) http://www.brenda-enzymes.info/php/result_flat.php4?ecno)1.1.3.4, accessed February 17, 2009. (32) Bourdillon, C.; Demaille, C.; Moiroux, J.; Saveant, J. M. J. Am. Chem. Soc. 1993, 115, 2–10. (33) Fulian, Q.; Fisher, A. C. J. Phys. Chem. B 1998, 102, 9647–9652. (34) Fulian, Q.; Fisher, A. C.; Denuault, G. J. Phys. Chem. B 1999, 103, 4387– 4392. (35) Fulian, Q.; Fisher, A. C.; Denuault, G. J. Phys. Chem. B 1999, 103, 4393– 4398. (36) Sklyar, O.; Wittstock, G. J. Phys. Chem. B 2002, 106, 7499–7508. (37) Sklyar, O.; Kueng, A.; Kranz, C.; Mizaikoff, B.; Lugstein, A.; Bertagnolli, E.; Wittstock, G. Anal. Chem. 2005, 77, 764–771. (38) Sklyar, O.; Ufheil, J.; Heinze, J.; Wittstock, G. Electrochim. Acta 2003, 49, 117–128. (39) Sklyar, O.; Treutler, T. H.; Vlachopoulos, N.; Wittstock, G. Surf. Sci. 2005, 597, 181–195. (40) Sklyar, O.; Tra¨uble, M.; Zhao, C.; Wittstock, G. J. Phys. Chem. B 2006, 110, 15869–15877.

4858

Analytical Chemistry, Vol. 81, No. 12, June 15, 2009

EXPERIMENTAL SECTION Materials. The following chemicals were used as received: polystyrene sulfonate (PSS, MW ∼70 000 g mol-1, 30% in water, Aldrich, St. Louis, MO), poly(diallyl dimethyl ammonium chloride) (PDDA, MW 100 000-200 000 g mol-1, 20% in water, Aldrich, St. Louis, MO), glucose oxidase (GOx, from Aspergillus niger, EC 1.1.3.4; 40 300 units g-1, Sigma, St. Louis, MO), NaCl (Sigma), ferrocene methanol, (Fc, ABCR GmbH, Karlsruhe, Germany), Na2HPO4 · 2H2O (Scharlau, Barcelona, Spain), NaH2PO4 · 2H2O (Fluka, Buchs, Switzerland), and D-(+)-glucose (99.5%, Sigma). Solutions were prepared using deionized water obtained from a water purification system (resistivity g 18.2 Ω cm-1, Seralpur PRO 90 C, Seralpur, Ransbach, Germany). Aminated microscope glass slides were obtained from MenzelGla¨ser (Braunschweig, Germany). Sample Preparation. Samples were prepared by the layerby-layer technique according to a protocol described previously.13 Briefly, a precursor layer (PSS/PDDA)5 was formed on an aminated glass slide by alternating dipping of the slide in solutions of PDDA and PSS (1 mg mL-1, pH 4.5 ± 0.1) and intermediate excessive rinsing with deionized water. For enzyme films (GOx/PDDA)n, solutions of GOx and PDDA were used (1 mg mL-1, pH 7). Deionized water for rinsing was adjusted to pH 7. Scanning Electrochemical Microscopy. The SECM approach curves were recorded with a home-built instrument consisting of a stepper motor system (Scientific Precision Instruments, Oppenheim, Germany)41 and a monopotentiostat µ-P3 (M. Schramm, Heinrich Heine University, Du¨sseldorf, Germany). The UMEs were fabricated according to Kranz and co-workers42 by sealing Pt wires (25 µm diameter, Goodfellow, Bad Nauheim, Germany) into borosilicate glass capillaries (Hilgenberg GmbH, Malsfeld, Germany). The ratio RG of the radii of the insulating sheath rglass and the active UME rT was approximately 10. The electrochemical cell was completed by a Pt wire as the auxiliary electrode and an Ag|AgCl|3 M KCl reference electrode, to which all potentials are referred to. The working solution contained 0.045-0.9 mmol L-1 Fc as the redox mediator and 50 mmol L-1 glucose as the enzyme substrate in 0.1 mol L-1 phosphate buffer (pH 7.0) and 0.1 mol L-1 NaCl. Before the experiment, the solution was deaerated with inert gas (N2 or Ar) for 5-10 min, and a gentle stream of inert wetted gas was passed over the electrochemical cell during the measurement. The UME was held at a potential ET ) +300 mV and moved at a vertical translation rate vT ) 0.87 µm s-1. The BEM simulations were performed on a Linux openSUSE 10.3 platform with a GenuineIntel-Intel(R) Core(TM)2 Duo CPU 2.66 GHz and 4 GB of RAM. RESULTS AND DISCUSSION Extension of the Boundary Element Method for Nonlinear Boundary Conditions. The system to be described is a smooth sample with the immobilized enzyme GOx (Figure 2 in ref 13). It is investigated in a solution containing glucose and Fc. At the UME, the reaction (41) Wilhelm, T.; Wittstock, G. Microchim. Acta 2000, 133, 1–9. (42) Kranz, C.; Ludwig, M.; Gaub, H. E.; Schuhmann, W. Adv. Mater. 1995, 7, 38–40.

UME: 2Fc f Fc+ + e-

(3)

Quantitative analysis of approach curves to immobilized GOx with higher [Fc]* was enabled by a new simulation algorithm using nonlinear boundary conditions in the BEM. The reaction scheme (eqs 1-4) was implemented as the exterior formulation of the Laplace problem. The exterior formulation calculates the concentrations outside of the simulation objects (UME, cross section of insulating sheath 293 µm diameter, height of electrode shaft 250 µm; sample, 900 µm × 900 µm). It avoids arbitrary assumption of a simulation domain boundary in the solution.36 Boundary conditions are specified as fluxes at the surface of the UME and sample. We found that this is critical to achieve high accuracy in BEM simulations.36 The exterior formulation requires concentration variables whose values are initially zero in the solution bulk. For the experimental system (reactions in eqs 1-4), such a concentration c(Fc+, r) ) c(r) at the point r is obtained according to eq 9.

(4)

c ) [Fc]* - [Fc]

(1)

proceeds under a diffusion-controlled rate. Fc is provided in the bulk solution in submillimolar concentration. The generated ferrocenium methanol (Fc+) serves as an electron acceptor for immobilized GOx replacing the natural electron acceptor (O2) which was removed from the solution. A simplified reaction mechanism32,43 is used for quantitative analysis of the regeneration rate of Fc at the sample (eqs 2-4). k1

sample: GOxox + glucose {\} [GOxox· · · glucose]

(2)

k1

k2

sample: [GOxox · · · glucose] 98 GOxred + gluconolactone

k3

sample: GOxred + 2Fc+ 98 GOxox + 2Fc

The reaction mechanism can be transferred into a reaction rate law (eqs 5-7) following earlier publications:12,32,43 2k2ΓGOx d[Fc+]S ) dt KM,glc KM,Fc+ + +1 [glc]S [Fc+]

For efficient notation, we set c° ) [Fc]* as the bulk concentration of the reduced mediator. In order to solve the SECM problem, the Laplace eq 10 has to be solved.

(5)

∂c(r, t) ) 0 ) ∇2c(r), ∂t

S

with KM,glc )

k-1 + k2 k1

(6)

k2 k3

(7)

KM,Fc+ )

where [Fc+]S is the concentration of Fc+ at the sample surface, ΓGOx is the surface concentration of GOx, and KM,glc and KM,Fc+ are the Michaelis-Menten constants of GOx for glucose and Fc+, respectively. The glucose concentration of 50 mmol L-1 is higher than KM,glc ) 7-36 mmol L-1,44-46 and the reaction rate law can be simplified (eq 8). d[Fc+]S 2k2ΓGOx 2k2ΓGOx[Fc+]S ) ) dt KM,Fc+ KM,Fc+ + [Fc+]S +1 + [Fc ]S

(8)

In a previous paper,13 the analysis of SECM approach curves was performed by using an analytical approximation of digitally simulated approach curves proposed by Wei et al.25 for first order kinetics at the substrate. This was justified because of the low mediator bulk concentration ([Fc]* ) 0.045 mmol L-1) causing pseudo first order kinetics ([Fc]* , KM,Fc+). This method was also used by Pellissier et al.23 However, significant deviations from this theory were observed experimentally for higher [Fc]* by us13 as well as in previous reports by Pierce et al.12 (43) Bourdillon, C.; Demaille, C.; Gueris, J.; Moiroux, J.; Saveant, J. M. J. Am. Chem. Soc. 1993, 115, 12264–12269. (44) Castner, J. F.; Wingard, L. B., Jr. Biochemistry 1984, 23, 2203–2210. (45) Shu, F. R.; Wilson, G. S. Anal. Chem. 1976, 48, 1679–1686. (46) Gregg, B. A.; Heller, A. Anal. Chem. 1990, 62, 258–263.

(9)

r∈V

(10)

∇2 is the Laplace operator and V denotes the simulation domain. The BEM proceeds by transforming eq 10 into their integral representation at the domain boundary Γ according to Green’s second identity. The discretization of Γ into triangular boundary elements leads to eq 11. 1 c(r ) ) 2 i

N

∑ j)1

∂c(rj) ∂nj

N

A G(r , r )dΓ - ∑ c(r ) A Γj

j

i

j

j)1

Γj

∂G(rj, ri) dΓ ∂nj (11)

For a specific element i, the sum over all N boundary elements indexed by j is taken with values of concentrations c(r), the fluxes ∂c(r)/∂nj and the standard Green function G(ri,rj) ) (4π|ri - rj|)-1 for the Laplace equation calculated from the absolute distance |ri - rj| between the elements i and j. For the concentration variable c(r), a system of N equations with N concentration values and N flux values has to be solved, from which N concentration values are unknown and N flux values are given as boundary conditions. For the insulating parts of the UME sheath, the flux of all species perpendicular to the surface is zero (eq 12). ∂c(r) )0 ∂n

(12)

The diffusion-controlled reaction of Fc to Fc+ at the UME is modeled by a first order rate law with a high rate constant, i.e., kec ) 100. kec ∂c(r) ) (c° - c(r)) ∂n DFc+

(13)

The enzymatic reaction at the sample is assumed to produce a flux according to the Michaelis-Menten kinetics. Analytical Chemistry, Vol. 81, No. 12, June 15, 2009

4859

1 2k2ΓGOxc(r) ∂c(r) )∂n DFc+ KM,Fc+ + c(r)

(14)

Assuming [Fc+] , KM,Fc+ in eq 14 yields a linear boundary condition at the enzymatically active part of the sample (eq 15). This was used for calculating pseudo first order rate law approximations. 1 2k2ΓGOx ∂c(r) )c(r) ∂n DFc+ KM,Fc+

(15)

The new extension compared to previous BEM simulations of enzyme kinetics from this laboratory consists in the integration of the nonlinear boundary condition (eq 14) into the BEM formalism. Details of the implementation, especially the use of boundary condition matrices, are given in the Supporting Information, section S1. Comparison of Simulated Approach with First Order Kinetics and Michaelis-Menten Kinetics at the Sample Surface. In order to illustrate the influence of the rate law at the sample surface on SECM approach curves, approach curves were digitally simulated assuming either first order kinetics (eq 15) or Michaelis-Menten kinetics (eq 14) at the sample surface. The parameters kec, 2k2ΓGOx, and the ratio of [Fc]* and KM,Fc+ have been varied (Figure 1): Four different ratios [Fc]*/KM,Fc+ have been simulated ([Fc]*/KM,Fc+ ) 1, 2, 10, and 20, curves 1-4) assuming Michaelis-Menten kinetics and one for an assumed pseudo first order rate law. The normalized reaction rate constant is defined as κ)

keffrT 2k2ΓGOxrT ) DFc+ DFc+KM,Fc+

(16)

and was used with a value of 1.5. Simulations with other normalized rate constants are given in the Supporting Information. For comparison, approach curves for diffusion-controlled reaction at the sample surface and at an inert, insulating sample are included (Figure 1, curves 6 and 7). UME currents were normalized to the current iT,∞ in the bulk solution: IT ) iT /iT,∞

(17)

As expected, pre-examination of simulations for first order kinetics with different values for [Fc]* showed that the normalized curves overlap (dashed line, curve 5). Furthermore, they match fairly well with the analytical approximation proposed by Cornut and Lefrou (data not shown)47 indicating a reasonable agreement of the present simulations with the method used by Cornut and Lefrou for first order kinetics. Exemplarily, for [Fc]*/KM,Fc+ ) 2 and κ ) 1.5 as defined as in eq 16, the different behaviors of the simulated approach curves for first order kinetics (Figure 1, curve 5) and Michaelis-Menten kinetics (Figure 1, curve 2) are discussed. At distances L > 2, the approach curves with MM kinetics and first order kinetics do not differ significantly. First, at the given d, the reaction rate at the sample surface has only a small influence on the UME current iT. The diffusion of Fc from the bulk provides the dominating contribution to iT. This can be estimated by considering the (47) Cornut, R.; Lefrou, C. J. Electroanal. Chem. 2008, 604, 91–100. (48) Amphlett, J. L.; Denuault, G. J. Phys. Chem. B 1998, 102, 9946–9951.

4860

Analytical Chemistry, Vol. 81, No. 12, June 15, 2009

Figure 1. Simulated SECM approach curves with assuming Michaelis-Menten kinetics (solid lines) at the sample surface with different ratios [Fc]*/KM,Fc+. [Fc]*/KM,Fc+ ) 1 (curve 1), 2 (curve 2), 10 (curve 3), and 20 (curve 4) and assuming first order kinetics (curve 5, dashed line). The normalized heterogeneous rate constant is κ ) keffrT/DFc+ ) 2k2ΓGoxrT/(KM,Fc+ DFc+) ) 1.5 in all curves. For comparison, analytical approximations according to Amphlett and Denuault for diffusion-controlled reaction at the sample (curve 6) and an inert, insulating sample (curve 7) are given.48

currents for L ) 2, RG ) 10. The contribution of hindered diffusion is ITins ) 0.72 according to the analytical approximation by Amphlett and Denuault (Figure 1, curve 7).48 This is approximately 2/3 of the total current for [Fc]*/KM,Fc+ ) 2, κ ) 1.5, and L ) 2, IT ) 1.15 (Figure 1, curve 2). Second, because of the steep concentration gradient at the UME, the ferrocenium concentration at the sample surface [Fc+]S is smaller than KM,Fc+. This is analyzed in more detail in Figure 2a. Figure 2 shows the local concentration [Fc+]S at the sample surface along a line to the symmetry center. The upper panels show false color images of the local [Fc+](x,z) along a vertical plane through the center of the UME for different UME-tosample separations (L ) 2, 1, 0.5). For L ) 2 and [Fc]*/KM,Fc+ ) 2, the maximum concentration of Fc+ at the sample surface is [Fc+]S ) 0.27KM,Fc+ (Figure 2a). Figure 3 visualizes the relation between [Fc+]S/KM,Fc+ and conversion rate d[Fc+]S/dt at the sample for Michaelis-Menten kinetics according to eq 8 (curve 1) and for first order kinetics (d[Fc+]S/dt ) keff[Fc+]S, curve 2) with [Fc]*/KM,Fc+ ) 2, κ ) 1.5. For the given [Fc+]S at L ) 2, [Fc+] e 0.27 KM,Fc+, the reaction rate d[Fc+]S/dt for first order kinetics and for Michaelis-Menten kinetics are similar (Figure 3, deviation less than 21%). The current at the UME is thus described with acceptable accuracy by a first order rate law. Additionally, [Fc+]S calculated using a first order approximation and Michaelis-Menten kinetics overlap almost perfectly (Figure 2a). For decreasing distances (L ) 1 and L ) 0.5), maximum [Fc+]S increase to 0.7 KM,Fc+ and 1.3 KM,Fc+, respectively (Figure 2b,c). For Michaelis-Menten kinetics, the reaction rate does not increase linearly with increasing [Fc+]S anymore (Figure 3, curve 1). Therefore, the reaction rate for pseudo first order kinetics and Michaelis-Menten kinetics differ significantly at smaller L. The different reaction rates influences IT: For assumed first order kinetics, IT (L ) 1) ) 1.28 and IT (L ) 0.5) ) 1.40 with κ ) 1.5 (Figure 1, curve 5), whereas for Michaelis-Menten kinetics IT (L ) 1) ) 1.21 and IT (L ) 0.5) ) 1.15 with κ ) 1.5 and [Fc]*/ KM,Fc+ ) 2 (Figure 1, curve 2). Furthermore, the smaller tip-tosample separation implies a larger contribution of the mediator regenerated at the sample surface to the overall flux to the UME compared to L g 2 and thus IT being more sensitive to variations

Figure 2. Simulated surface concentrations of Fc+ for (a) L ) 2, (b) L ) 1, and (c) L ) 0.5 using [Fc]*/KM,Fc+ ) 2 and κ ) 1.5. The scales in parts a-c are identical. 9 represent simulated [Fc+]S using Michaelis-Menten kinetics, and 0 are simulated for 1st order kinetics. Upper panels are false color images (uniform scale for parts a-c) of local [Fc+] in the entire xz plane through the center of the UME.

Figure 3. Flux of ferrocene methanol f(Fc) generated at the sample surface at a given [Fc+]S. The curve is calculated according to eq 8. For comparison, the flux for 1st order kinetics with keff ) 2k2ΓGOx/ KM,Fc+ is shown (dashed line).

of the regeneration rate. A fit of experimental approach curves to an enzyme modified surface assuming first order kinetics would thus lead to an erroneous results at the given [Fc]*/ KM,Fc+ and κ (vide infra). The different reaction rates at the sample surface for first order kinetics and Michaelis-Menten kinetics are also evident from the comparison of [Fc+]S (Figure 2b,c). The lower reaction rate for Michaelis-Menten kinetics at the given L leads to an accumulation of Fc+ (Figure 2b,c, 9) whereas for an assumed first order reaction substantially lower [Fc+]S would result (Figure 2b,c, 0). It should be noted that the concentration at radial distances |x| > rT from the center of the active UME area are still smaller than KM,Fc+ and that the reaction proceeds according to a pseudo first order reaction rate law there. Consequently, a sufficiently precise description is not possible neither by first order nor zeroth order reaction rate law approximations for two reasons: (i) The change from pseudo first order rate law at the sample surface to zeroth order rate law occurs gradually with L. The transient range of the reaction rate for [Fc+]S ≈ KM,Fc+ cannot be described by any of the simplified models. (ii) Even if below the center of the active part of the UME, the reaction already proceeds at zeroth order at small L due to high [Fc+]S and the reaction still proceeds at pseudo first order kinetics or a transient rate law at larger radial distances from the UME center. For larger [Fc]*/KM,Fc+, [Fc+]S reaches KM,Fc+ at larger L. For example, for [Fc]*/KM,Fc+ ) 20, at L > 4, [Fc+]S is already bigger than KM,Fc+ and the reaction rate for Michaelis-Menten kinetics is significantly slower than they would be for first order kinetics (Figure 3). However, because of the large distance, the shape of the simulated approach curves with first order kinetics

and Michaelis-Menten kinetics do not differ significantly for L > 3, because the sample kinetics have only a minor influence on IT (Figure 1, curve 4). At small L, the deviation of Michaelis-Menten kinetics from first order kinetics is much more pronounced (Figure 1, curve 4) and the approach curves differ even qualitatively from those of smaller [Fc]*/KM,Fc+. As a further detail, it can be seen that the maximum of IT(L) occurs at larger L with Michaelis-Menten kinetics at the sample than for curves calculated with first order kinetics. We noted this effect repeatedly when trying to fit approach curves to GOx-modified surfaces to the analytical expressions for first order kinetics by Wei et al. and Cornut and Lefrou25,26 and our own simulations. When the considerations presented above are ignored, it is possible to find acceptable fits for the region around the maximum in the IT ) f(L) curves for a pseudo first order model by adjusting d0, rT, κ, and iT,∞. In this case, quite misleading parameters will be obtained. In order to use first order kinetic models and still avoid the described problem, one has to work with very low mediator concentration that may be inconvenient or impossible depending on the KM value of the enzyme for the mediator used. In order to determine reasonable limiting cases up to which first order rate law can be used safely to fit experimental SECM feedback approach curves to enzyme-modified samples with acceptable deviations, simulations were carried out for different ratios [Fc]*/KM,Fc+. The simulation considered the MichealisMenten kinetics and κ ) 1.6. Simulated approach curves were fitted to the analytical approximation for first order kinetics proposed by Cornut and Lefrou.26 A least-squares fitting was used varying κ and iT,inf. The closest point of approach was fixed to the one used in the simulation. Different sections of the simulated curve were used for the fitting procedure in order to explore the effect of limited approach to the samples. The minimum normalized tip-to-sample distance Lmin used for fitting was varied between 0.1 e Lmin e 0.5. A selection of simulated approach curves and the fitted curves is shown in Figure 4. As expected, it can be seen qualitatively that the fitting becomes worse with increasing [Fc]*/KM,Fc+. For comparison, a curve was simulated assuming a first order rate law with κ ) 1.6 and fitted assuming a first order rate law (Figure 4, curve 4). There is a fair agreement between simulation and fitting to the analytical approximation. This curves also overlaps with the data simulated with Michaelis-Menten kinetics and [Fc]*/KM,Fc+ ) 0.01 and its fitting with a first order rate law indicating that under these conditions fitting an approach curve to a surface Analytical Chemistry, Vol. 81, No. 12, June 15, 2009

4861

Figure 4. Simulated SECM approach curves (filled symbols) with Michaelis-Menten kinetics at the surface: [Fc]*/KM,Fc+ ) 10-2 (curve 1), 10-2/3 (curve 2), and 100 (curve 3) κ ) 2k2ΓGoxrT/(KM,Fc+ DFc+) ) 1.6 (curves 1-3). Fitted curves assuming 1st order kinetics at the surface are shown as lines. As a reference, a simulated approach curve with 1st order kinetics κ ) 1.6 at the surface is given (open symbols, curve 4, complete overlap with curve 1). Values within 0.1 < L < 10 were considered for fitting. Table 1. Fitted Normalized Rate Constants K Using Analytical Approximations for 1st Order Kinetics26 for Approach Curves Simulated with Michaelis-Menten (MM) Kinetics at the Sample Surface for Different Ratios [Fc]*/KM,Fc+ and for Different Minimum Normalized Distances Lmin Considered in the Fitting Procedurea κ for

simulation modelb MM MM MM MM MM MM MM 1st order

for

log([Fc]*/ KM,Fc+)

Lmin ) 0.101

Lmin ) 0.202

for Lmin ) 0.303

for Lmin ) 0.505

-2.00 -1.67 -1.33 -1.00 -0.67 -0.33 0.00

1.60 1.59 1.57 1.53 1.44 1.29 1.05

1.59 1.58 1.56 1.52 1.45 1.31 1.08

1.59 1.59 1.57 1.54 1.47 1.35 1.13

1.62 1.61 1.60 1.58 1.53 1.42 1.24

1.61

1.59

1.60

1.60

a

A fit to a curve simulated using 1st order kinetics and fitted to analytical approximations for 1st order kinetics26 is given for comparison in the last row. b MM, reaction at sample simulated as MichaelisMenten kinetics; 1st order, reaction was modeled as true 1st order reaction.

with Michaelis-Menten kinetics by an analytical approximation assuming first order rate laws is indeed appropriate. The κ values obtained by the fitting are shown in Table 1. For comparison, the last row in Table 1 is the fit of a first order model to the curves simulated with a true first order model (Figure 4, curve 4). Fitted κ with deviations to the fitted κ obtained for the true first order reaction larger than 10% are printed in bold. If simulated values with 0.1 < L < 10 are fitted and the accepted deviation is 10%, Michaelis-Menten kinetics can be described by first order kinetics up to [Fc]*/KM,Fc+ ) 0.1. For 0.5 < L < 10, first order kinetics and Michaelis-Menten kinetics do not show more than 10% deviation in κ up to [Fc]*/KM,Fc+ ) 10-2/3 (Table 1). If 20% deviation in κ is acceptable, fitting of SECM approach curves with Michaelis-Menten kinetics at the sample surface is possible up to [Fc]*/KM,Fc+ ) 10-1/3. Please, note that the agreement between simulated approach curves and fitted analytical approximations for first order kinetics 4862

Analytical Chemistry, Vol. 81, No. 12, June 15, 2009

may still be as good as in Figure 4, curve 2, even if the obtained parameters deviate more than 10% from the real values. Obviously bad fitting results are obtained only for large [Fc]*/KM,Fc+ ratios indicating that the first order assumption is clearly not valid (Figure 4, curve 3). In this fitting, the closest point d0 of approach was kept constant at the value used in the simulation. This would correspond to experiments, where d0 is determined by independent experiments and there is no uncertainty in this value. Such independent distance information may come from optical observation with an inverted (confocal) microscope or shear force measurements. For cases in which additional equipment is not available or not applicable, we have suggested to measure with exactly the same UME mounting an approach to a microscope slide, identify the last data point before the UME sheath touches the substrate, fit all points of the curve before the mechanical touch between the UME sheath and sample to the theory for hindered diffusion by varying rT, iT,∞, and the distance d0 of the last valid point to the sample surface. Afterward, the UME is retracted but kept in the same mounting and the approach to the sample with finite kinetic is carried out. The last point before the mechanical touch is identified and assumed to have the same distance d0 from the sample surface as in the case of the approach to glass.49 This procedure avoids the use of the absolute coordinate of the positioning system and can be used even for different thicknesses of the glass slide and the sample to be characterized. However, experimental determined values for d0 have always an uncertainty range. The uncertainty for the SECM-based procedure49 is connected to the step size used in the approach (typically 0.5-1 µm). Making step sizes smaller enhances the uncertainty of identifying the touching points. With such realistic uncertainty range of d0 or if d0 is varied in the fitting procedure, apparent agreement of first order fittings for samples with Michaelis-Menten kinetics is obtained for even larger [Fc]*/ KM,Fc+ ratios, although κ values would then deviate by almost 30% (data not shown). Simulations were also carried out for κ ) 0.12, 0.55, 1.1, 1.6, 3, 5 (see Supporting Information), and similar guidelines were found detailed here for κ ) 1.6. Analysis of Experimental Approach Curves. Polyelectrolyte multilayer films of GOx and PDDA have been prepared by the layer-by-layer technique. Characterization of such films by UV-vis spectrometry and scanning force microscopy have been described in a previous paper.13 Simulated approach curves were compared to experimental SECM approach curves to (GOx/PDDA)3 films. Input parameters for the simulations were KM,Fc+, 2k2ΓGOx, rT, RG, D, and d0. The parameters rT, RG, D, and d0 were determined from independent experiments. The radius of the active part of the UME rT ) 15 µm. The scan rate vT ) 0.87 µm s-1 was selected in order to maintain steady-state conditions in all situations as the steady state is obtained by BEM simulation. The smaller rT and the larger κ, the faster an approach curve can be carried out in steady state.50 From approaches to glass with UME of the used rT and RG, we found that vT as low as 1 µm s-1 are required in order to obtain a (49) Nunes Kirchner, C.; Wittstock, G. In Electrochemical Sensor Analysis; Alegret, S., Merkoci, A., Eds.; Elsevier: Amsterdam, The Netherlands, 2007; Vol. 49, pp e363-e370. (50) Bard, A. J.; Denuault, G.; Friesner, R. A.; Dornblaser, B. C.; Tuckerman, L. S. Anal. Chem. 1991, 63, 1282–1288.

good fit to the theory of hindered diffusion.48 The closest point of approach d0 ≈ 3.5 µm was determined from curves to the sample without glucose in the working solution. In this situation the enzyme cannot convert Fc+, and the sample behaves like an inert, insulating sample. The procedure is detailed in section IV of the Supporting Information of a previous paper.51 In addition, these experiments serve as control experiments. The data of the control experiments were fitted to analytical approximation of simulated approach curves proposed by Amphlett and Denuault48 as outlined in section IV of the Supporting Information of a previous paper.51 The ratio RG ≈ 10 of the radii of the insulating sheath rglass and rT was derived from optical micrographs of the UME recorded by confocal laser scanning microscope in reflection mode. The diffusion coefficient of D ) 6 × 10-6 cm2 s-1 was measured by chronoamperometry at a macroscopic electrode and is in good agreement with previously reported values.52 Experimental approach curves with glucose were recorded for four different [Fc]* in deaerated solutions. Under our conditions, a strict exclusion of O2 is not possible. However, no oxygen reduction current could be measured with the Pt electrode in the cell. Small traces of O2 which may have remained in the cell would be consumed continuously by the GOx-containing layer on the sample. If the UME approaches the sample, the O2 diffusion to the sample regions underneath the UME is further blocked by the presence of the UME with its insulating shield, which further decreases a possible competition between the electron acceptors Fc+ (high flux from UME) and O2. Therefore, the limited exclusion of O2 will not significantly affect the obtained results. The regeneration of Fc by the enzymatic reaction at the sample surface is detected at the UME as an enhanced Fc oxidation current. The obtained results were compared to BEM simulations using one set of adjustable parameters (KM,Fc+ and 2k2ΓGox) that describes the set of experimental approach curves for all four [Fc]* satisfactorily. Only the parameters KM,Fc+ ) 1.3 × 10-7 mol cm-3 and 2k2ΓGOx ) 7.54 × 10-10 mol cm-2 s-1 (κ ) 1.45) provide a sufficient fit simultaneously for all tested bulk concentrations of Fc (Figure 5). As it has been described earlier, the experimental approach curves cannot be described well by using simulations with approximations using first order kinetic models for the reaction at the sample surface. Attempts to describe the approach curves with first order kinetics at the sample surface using keff ) 2k2ΓGOx/KM,Fc+ are shown for comparison (Figure 5, curve 5). The Michaelis-Menten constant obtained in the fitting procedure, KM,Fc+ ) 1.3 × 10-7 mol cm-3, is reasonable and lies within the range of KM,Fc+ published in literature, e.g., approximately 5 × 10-8 mol cm-3 by Bourdillon et al.32,43 and 1.8 × 10-6 mol cm-3 by Pellissier et al.23 CONCLUSIONS A new extension of the three-dimensional boundary element formalism allows the incorporation of nonlinear boundary condi(51) Rianasari, I.; Walder, L.; Burchardt, M.; Zawisza, I.; Wittstock, G. Langmuir 2008, 24, 9110–9117. (52) Pust, S. E.; Scharnweber, D.; Baunack, S.; Wittstock, G. J. Electrochem. Soc. 2007, 154, C508–C514.

Figure 5. Experimental and simulated SECM approach curves to (GOx/PDDA)3 at different concentrations of Fc. [Fc]* ) 0.045 mmol cm-3 (curve 1), [Fc]* ) 0.09 mmol cm-3 (curve 2), [Fc]* ) 0.45 mmol cm-3 (curve 3), [Fc]* ) 0.9 mmol cm-3 (curve 4). Symbols represent experimental curves; solid lines correspond to simulated data using the BEM algorithm with Michaelis-Menten kinetics (2k2ΓGOx) 7.54 × 10-10 mol cm-2 s-1, KM,Fc+ ) 1.3 × 10-7 mol cm-3, κ ) 1.45); dashed line (curve 5) shows simulated data for assumed pseudo first order kinetics at the substrate with keff ) 2k2ΓGOx/KM,Fc+ ) 5.8 × 10-3 cm s-1 (κ ) 1.45). See text for details.

tions. Such nonlinear behavior can result from catalytic and enzymatic reactions, corrosion, or adsorption at metal surfaces. The routine will work if the nonlinear boundary conditions are either monotonically increasing or decreasing. In these cases an iterative Newton algorithm can be used to find the solution from any start value. These conditions are fulfilled for SECM approach curves in the feedback mode to enzyme-modified surfaces. Experimental approach curves to glucose oxidase-modified surfaces over a range of mediator concentrations could be described simultaneously by one set of kinetic parameters. At the same time it became evident that large discrepancies may result from using first order approximations. Moreover, the shape of the approach curves changes also qualitatively if Michaelis-Menten kinetics are dominant. This is confirmed by the simulation and corresponding experiments. The use of first order approximations seems only justified under the condition that the mediator bulk concentration is equal or less than 10% of the corresponding KM value. In many cases such low mediator concentrations will be out of a range conveniently accessible in experiments so that a description by models using Michaelis-Menten kinetics should be used. ACKNOWLEDGMENT The project was partially supported by Deutsche Forschungsgemeinschaft (Grant Wi 1617/7). SUPPORTING INFORMATION AVAILABLE Details of the implementation of the BEM with nonlinear boundary conditions and simulated approach curves with Michaelis-Menten kinetics at the sample surface with different normalized heterogeneous rate constants κ. This material is available free of charge via the Internet at http://pubs.acs.org.

Received for review March 6, 2009. Accepted April 29, 2009. AC9004919

Analytical Chemistry, Vol. 81, No. 12, June 15, 2009

4863