Characterization and Mathematical Modeling of a Bienzyme Electrode

The use of L-malate dehydrogenase (decarboxylating,. EC 1.1.1.40) as a model enzyme resulted in a novel. L-malate sensor. It had improved characterist...
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Anal. Chem. 1999, 71, 4657-4662

Characterization and Mathematical Modeling of a Bienzyme Electrode for L-Malate with Cofactor Recycling Nenad Gajovic,*,† Axel Warsinke,† Tina Huang,‡ Thomas Schulmeister,§ and Frieder W. Scheller†

Institute of Biochemistry and Molecular Physiology, Potsdam University, c/o Biotechnologiepark Luckenwalde, Im Biotechnologiepark, 14943 Luckenwalde, Germany, ZEIK, Potsdam University, Am Neuen Palais 10, 14469 Potsdam, Germany, and Department of Chemistry, The University of Kansas, Lawrence, Kansas 66045 The coimmobilization of a NADP+-dependent dehydrogenase with p-hydroxybenzoate hydroxylase (PHBH, EC 1.14.13.2) in front of a Clark electrode yields a flexible design for highly selective, dehydrogenase-based biosensors. The use of L-malate dehydrogenase (decarboxylating, EC 1.1.1.40) as a model enzyme resulted in a novel L-malate sensor. It had improved characteristics compared with those of earlier sensor approaches: a strongly reduced NADP+ requirement (0.01 mmol L-1), an extended linear range from 0.005 to 1.1 mmol L-1 L-malate, and a working stability of more than 30 days. Only inexpensive chemicals (p-hydroxybenzoate, MgCl2) were needed in millimolar amounts. A linear mathematical model for the steady state helped to elucidate the sensor operation. Both experimental and simulation results indicated that the bienzyme sensor behaved like a quasi monoenzyme electrode with a hypothetical “L-malate hydroxylase”: The response was determined by the substrate concentration and diffusivity only, indicating the perfect coupling of both enzyme reactions by the intermediate NADPH. The presented scheme based on PHBH and the Clark electrode is a promising and reliable approach for other NADP+-dependent dehydrogenases. Together with citrate, L-malate is a major organic acid in most fruits and vegetables and contributes significantly to the flavor and taste of fruit juice and soft drinks. As an indicator of wine maturation during the fermentation process it is an interesting parameter in wine making. Currently, L-malate is determined offline by an enzymatic assay with photometric detection1san expensive and laborious method. In the first sensing approaches, enzyme reactors with malate dehydrogenase (MDH, EC 1.1.1.37) were used in flow injection systems.2-6 Enzyme electrodes using * Corresponding author: (tel.) ++49-3371-681-320; (fax) ++49-3371-681-312; (e-mail) [email protected]. † Institute of Biochemistry and Molecular Physiology, Potsdam University. ‡ The University of Kansas. § ZEIK, Potsdam University. (1) Mo ¨llering, H. In Methods of Enzymatic Analysis; Bergmeyer, H. U., Ed.; VCH: Weinheim, 1989; Vol. VII, pp 39-47. (2) Almuiabed, A. M.; Townshend, A. Anal. Chim. Acta 1989, 221, 337-340. 10.1021/ac9806355 CCC: $18.00 Published on Web 09/18/1999

© 1999 American Chemical Society

MDH,7 MDH together with diaphorase8 or NADH oxidase,9 or MDH (decarboxylating, EC 1.1.1.40) and pyruvate oxidase together with a H2O210 or O2 electrode11 were developed, eliminating the need for expensive flow-system instrumentation. Although the analytical task, detection of millimolar malate concentrations in food samples, was solved, these sensors, as do most other dehydrogenase based sensors, have some serious shortcomings: an often small linear range and the need for millimolar concentrations of the costly cofactors NAD+ and NADP+, raising the cost of L-malate determination to a prohibitive level. The incorporation of NAD(P)+ in a carbon paste12 or a solid binder carbon electrode13 was only partial remedy for this major problem, since both the sensitivity of these sensors and their operational stability were not satisfying, because of leaching of NAD+. Recently, we have developed a new dehydrogenase sensor utilizing salicylate hydroxylase (SHL, E. C. 1.14.13.1) and malate dehydrogenase (decarboxylating, EC 1.1.1.40) to improve the costeffectiveness and precision of L-malate detection.14 It was characterized by cofactor recycling and use of the Clark electrode and offered the highest sensitivity reported so far and a very low NADP+ demand. A linear mathematical model for this type of electrode is presented in this report as a first approach to elucidating its operation. Along with the model we report on some improvements, e.g., higher selectivity, from replacing SHL, which (3) Chemnitius, G. C.; Schmid, R. D. Anal. Lett. 1989, 22, 2897-2913. (4) Yoshioka, S.; Ukeda, H.; Matsumoto, K.; Osajima, Y. Electroanalysis 1992, 4, 545-548. (5) Prodromidis, M. I.; Tzouwarakarayanni, S. M.; Karayannis, M. I.; Vadgama, P.; Maines, A. Analyst (Cambridge, U.K.) 1996, 121 (4), 435-439. (6) Matsumoto, K.; Higuchi, S.; Tsukatani, T. Biosci. Biotechnol. Biochem. 1996, 60 (5), 847-851. (7) Silber, A.; Bra¨uchle, C.; Hampp, N. Sens. Actuators, B 1994, 18-19, 235239. (8) Gilis, M.; Durliat, H.; Comtat, M. Am. J. Enol. Vitic. 1996, 47(1), 11-15. (9) Mizutani, F.; Yakubi, S.; Asai, M. Anal. Chim. Acta 1991, 245, 145-150. (10) Messia, M. C.; Compagnone, D.; Esti, M.; Palleschi, G. Anal. Chem. 1996, 68, 360-365. (11) Gajovic, N.; Warsinke, A.; Scheller, F. J. Chem. Technol. Biotechnol. 1996, 68, 31-36. (12) Huan, Z.; Persson, B.; Gorton, L.; Sahni, S.; Skotheim, T.; Bartlett, P. Electroanalysis 1996, 8(6), 575-581. (13) Katrlik, J.; Pizzariello, A.; Mastihuba, V.; Svorc, J.; Stredansky, M.; Miertus, S. Anal. Chim. Acta 1999, 379, 193-200. (14) Gajovic, N.; Warsinke, A.; Scheller, F. J. Biotechnol. 1998, 61, 129-133.

Analytical Chemistry, Vol. 71, No. 20, October 15, 1999 4657

Figure 1. Schematic representation of the bienzyme electrode (MDH, malate dehydrogenase (decarboxylating); HBH, p-hydroxybenzoate hydroxylase).

oxidizes both NADH and NADPH,15 with the related, NADPHspecific p-hydroxybenzoate hydroxylase (PHBH, EC 1.14.13.2). The influence of increased hydroxylase loading on the sensor response was also examined with PHBH, since it offered a specific activity that was 20 times higher than that of SHL. The new sequence was as follows:

L-malate + NADP+s(MDH, EC 1.1.1.40) f pyruvate + CO2 + NADPH (1) NADPH + p-hydroxybenzoate + O2s(PHBH, EC 1.14.13.2) f 3,4-Dihydroxybenzoate + NADP+ + H2O (2)

The oxygen consumption was again monitored with the Clark electrode. The sensor scheme is depicted in Figure 1. EXPERIMENTAL SECTION Apparatus. A potentiostat EP 30 from Biometra (Go¨ttingen, Germany) and a chart recorder from Kipp and Zonen (Delft, Netherlands) were used. The electrodes (model SME 1/4) from Elbau (Berlin, Germany) were composed of a Pt working electrode (diameter 0.5 mm) and an Ag/AgCl reference/counter electrode. A stirred and temperature-controlled cell (30 °C) with a 5-mL volume was manufactured at our institute. Chemicals. L-malate dehydrogenase (oxaloacetate decarboxylating, EC 1.1.1.40, 21 U mg-1 protein) was from Fluka (Buchs, Switzerland), and p-hydroxybenzoate hydroxylase (PHBH, EC 1.14.13.2, 23.3 U mg-1, from Escherichia coli) was from Toyobo (Osaka, Japan). NADP+ (>98%) from Gerbu (Gaissheim, Germany) and L-malate (p.a.) from Roth (Neckarsulm, Germany) were used. Gelatin (for biochemistry) was from VEB Laborchemie (Apolda, Germany). All other chemicals were of reagent-grade. Dialysis membranes (MWCO 20 000 Da) were from ORWO (Wolfen, Germany) and Polyethylene (PET) membranes (d ) 10 µm) from Metra (Radebeul, Germany). Buffers. For all experiments except the pH study, Soerensen phosphate buffer (70 mmol L-1) pH 7.4, comprising 1 mmol L-1 MgCl2, 1 mmol L-1 HBA, and 10 µmol L-1 NADP+ was used. For the pH study, Soerensen phosphate buffer (70 mmol L-1 phosphate) and Tris/HCl buffer (50 mmol L-1 Tris) with all abovementioned additives were used. (15) White-Stevens, R. H.; Kamin, H. J. Biol. Chem. 1972, 247 (8), 2358-2370.

4658 Analytical Chemistry, Vol. 71, No. 20, October 15, 1999

Figure 2. NADP+ dependence of the response (sensor 2 U cm-2 MDH, 7 U cm-2 HBH; 70 mmol L-1 phosphate buffer pH 7.4, 1 mmol L-1 MgCl2, 1 mmol L-1 p-hydroxybenzoic acid, 1 mmol L-1 L-malate, 30 °C).

Procedure. The enzymes were immobilized and the biosensor prepared as previously reported.14 A schematic view of the sensor is shown in Figure 1. Measurements were performed by adding 1-25 µL of the sample to the cell which was filled with 2.5 mL of phosphate buffer. Current-time curves were recorded until the steady-state current was obtained. Optimization of the Working Conditions. For buffer optimization, 1 mmol L-1 of L-malate was measured with the standard sensor (2 U cm-2 MDH, 7 U cm-2 PHBH) under varying of buffer systems (phosphate and Tris/HCl buffer) and additives (0-1 mmol L-1 HBA, 0.001-1 mmol L-1 NADP+, 0-2 mmol L-1 Mg2+) and with various pH values (6-8) and temperatures (20-34 °C). For the optimization of enzyme loading, the amount of both enzymes in the sensor was varied separately and their response to 0.25 mM L-malate compared. PHBH was varied between 0.001 and 7 U cm-2, while MDH was kept constant at 2 U cm-2. MDH was varied between 0.05 and 5 U cm-2 with a constant PHBH loading of 7 U cm-2. Characterization of the Sensor Performance. The optimized L-malate biosensor was characterized with respect to the linear L-malate and NADPH detection ranges, response time, t(90%), operational stability, and cost per determination. For overnight storage, the sensor was left in buffer at room temperature. The enzyme and reagent costs were calculated assuming that 2.5 mL of buffer was consumed per measurement, and the sensor was routinely replaced after 200 samples. RESULTS AND DISCUSSION Optimization of the Working Conditions. All optimization steps were performed with the standard sensor comprising 2 U cm-2 MDH and 7 U cm-2 HBH. The NADP+ dependence of the sensor is shown in Figure 2: Only 10 µmol L-1 NADP+ was required in the buffer to achieve the maximum response with 1 mmol L-1 L-malate. Optimum concentrations for the other cosubstrates were determined to be 0.75 mmol L-1 HBA and 1 mmol L-1 MgCl2. The sensor response was not pH-dependent between pH 6 and 8. This was indicating a diffusion-limited sensor response, as pH 6 was far below the pH optima of both enzymes

Figure 3. Enzyme loading test with MDH at constant HBH, experimental and simulation (70 mmol L-1 phosphate buffer pH 7.4, 1 mmol L-1 MgCl2, 1 mmol L-1 p-hydroxybenzoic acid, 1 mmol L-1 L-malate, 1 mmol L-1 NADP+, 30 °C).

Figure 4. Enzyme loading test with HBH at constant MDH, experimental and simulation (70 mmol L-1 phosphate buffer pH 7.4, 1 mmol L-1 MgCl2, 1 mmol L-1 p-hydroxybenzoic acid, 1 mmol L-1 malate, 1 mmol L-1 NADP+, 30 °C).

(HBH, pH 8.0; MDH, pH 7.75). The optimization of enzyme loading revealed that ∼5 U cm-2 of both enzymes was sufficient to get maximum response. If MDH was varied, 80% of the maximum response were found at 0.25 U cm-2, and further increasing MDH loading by a factor of 20 only led to 25% additional gain in response (Figure 3). When HBH was varied, 80% of the maximum response were reached at 0.05 U cm-2, and a 100-fold increase in HBH loading resulted only in a response only 25% higher (Figure 4). The thicknesses of the enzyme membranes in these tests were 0.1-0.15 mm. The optimized working buffer with saturating concentrations of all reagents was pH 7.4 phosphate (16) Mell, L. D.; Maloy, J. T. Anal. Chem. 1975, 47(2), 299-307. (17) Gough, D. A.; Leypoldt, J. K. Appl. Biochem. Bioeng. 1981, 3, 175-206. (18) Leypoldt, J. K.; Gough, D. A. Biotechnol. Bioeng. 1982, 24, 2705-2719. (19) Leypoldt, J. K.; Gough, D. A. Anal. Chem. 1984, 56, 2896-2904. (20) Bergel, A.; Comtat, M. Anal. Chem. 1984, 56, 2904-2909. (21) Bartlett, P. N.; Whitaker, R. G. J. Electroanal. Chem. 1987, 224, 27-35. (22) Tatsuma, T.; Watanabe, T. Anal. Chem. 1992, 64, 626-630. (23) Bacha, S.; Bergel, A.; Comtat, M. Anal. Chem. 1995, 67, 1669-1678. (24) Schulmeister, T.; Scheller, F. W. Anal. Chim. Acta 1985, 170, 279-284. (25) Schulmeister, T. Anal. Chim. Acta 1987, 201, 305-310. (26) Horvath, C., Engasser, J. M. Biotechnol. Bioeng. 1974, 16, 909-923. (27) Rutter, W. J.; Lardy, H. A. J. Biol. Chem. 1958, 233, 374.

Figure 5. Calibration curves for L-malate (experimental and simulation) and NADPH (sensor: 2 U cm-2 MDH, 7 U cm-2 HBH; 70 mmol L-1 phosphate buffer pH 7.4, 1 mmol L-1 MgCl2, 1 mmol L-1 p-hydroxybenzoic acid, 0.01 mmol L-1 NADP+, 30 °C).

buffer with 10 µmol L-1 NADP+, 1 mmol L-1 MgCl2, and 1 mmol L-1 HBA. Characterization of the Sensor Response. The sensor covered a linear detection range from 5 µmol L-1 to 1.1 mmol L-1 L-malate (Figure 5, slope 25 µA cm-2 mmol-1 L, R ) 0.99997, N ) 8). The sharp break at 1.1 mmol L-1 was due to oxygen depletion by the enzymatic reaction, as indicated by a net current of zero at the Clark electrode. The sensitivity for L-malate was more than 5 times higher than that for NADPH (Figure 5, slope 4.8 µA cm-2 mmol-1 L). An explanation for such behavior will be given in the mathematical treatment section. With a 90% response time of 30 s and a total analysis time of 90 s, a sampling frequency of 40 tests per hour could be realized. The long-term stability at room temperature was very high: The response did not fall below 90% of the initial value even after 33 days. In contrast to H2O2, the byproduct of oxidase-based sensors, the reaction product 3,4dihydroxybenzoate, did not seem to harm the immobilized enzymes. The reagent and enzyme cost was only 0.003 US $ per test, in comparison with 0.2-0.5 $ for a homemade enzyme assay and 2.4 $ for the Boehringer F-Kit. Linear Modeling of the Steady-State Response. The general diffusion/reaction problem in enzyme sensors can only be solved numerically. While there are detailed mathematical models with numerical and analytical solutions for one-enzyme/one-substrate,16 one-enzyme/two-substrate problems,17-23 there are much fewer examples for more involved two-enzyme/two-substrate or even more complicated schemes. As there was no exact solution for our two-enzyme/five-substrate problem in the literature, we have adopted a linear model24,25 similar to our problem and extended it by calculating the solution for the Clark electrode. What are the consequences of using a linear model, i.e., approximating the Michaelis-Menten equation by first-order rate terms? The first and most important consequence is a limited application range. The model is exact only at low substrate concentrations (,0.1Km), at which the approximation error is small. In immobilized enzyme systems, however, the first-order region can be extended to higher concentrations, as indicated by the apparent Michaelis-Menten constant, Km,eff. As Km,eff can be calculated approximately from the intrinsic Michaelis constant, Km, and the Thiele-modulus Φ,26 it can be used to estimate the range of application of a linear model. Here, d is the thickness of the Analytical Chemistry, Vol. 71, No. 20, October 15, 1999

4659

Φ ≡ dxVmax/KmD′

(3)

Km,eff ≈ KmΦ

(4)

enzyme layer, Vmax is the enzyme reaction rate with substrate saturation, and D′ is the apparent substrate diffusivity inside the layer. The calculated value for L-malate with the standard sensor (2 U cm-2 MDH, 7 U cm-2 PHBH) was Km,eff(malate) ≈ 7.5 mmol L-1 and, thus, more than 5 times higher than the upper detection limit. The maximum deviation from the model would be ∼10% at the highest detectable concentration (∼1 mmol L-1). There is, however, an even simpler way: As Leypoldt and Gough have demonstrated, the linear portion of a calibration graph can be described by a linear, one-substrate model.18 Since the calibration graph of the real sensor was linear up to the upper detection limit, this meant that our simple model was adequate for the entire concentration range. At low enzyme loading, however, as the linear range of the calibration graph will be smaller, the model will represent only this linear portion. Rationale of the Applied Model. Using an approach by Schulmeister,24,25 the bienzyme electrode was assumed to be planar and infinitely extended, with a catalytic layer of thickness d, in which both enzymes were distributed homogeneously, and no outer (protective) or inner (electrode-covering) membranes were present. Both enzyme reactions were treated as irreversible and first-order with respect to the limiting substrate, the product of the first reaction (NADPH) being the rate-limiting substrate of the second, signal-generating reaction. O2 consumption by the second reaction was monitored with a PET membrane-covered Clark type electrode. The electrode current was proportional to the O2 flux through the PET membrane. It is important to note that the rate of O2 consumtion by the enzyme was assumed to depend only on the limiting substrate NADPH. Looking at the real enzyme reactions 1 and 2, it is obvious that this model only fits if all cosubstrates are present in excess, making the two reactions pseudo first order with respect to malate (1) and NADPH (2). In reality, this was achieved by adding saturating concentrations of NADP+, HBA, and MgCl2 to the buffer. The O2 concentration cannot be manipulated so easily, but experimental and theoretical studies of Clark electrode-based sensors revealed that O2 is not limiting unless a critical analyte concentration (ccrit, Analyte ≈ (DO2/DAnalyte)[O2]),17,18 indicated by a sharp break to the horizontal in the calibration graph, is reached. Below this critical concentration, O2 has no impact on the response (and the enzyme reaction) whatsoever. So (1) and (2) could be approximated as follows:

M + A1sK1 f NADPH + A2+ A3

(I)

NADPH + A4 + A5sK2 f A1 + A6 + A7

(II)

Here, M ) L-malate and Ai ) intermediates with no impact on reaction rates, for instance, A1 ) NADP+, A2 ) pyruvate, A3 ) CO2, A4 ) O2, A5 ) HBA, A6 ) 3,4-dihydroxybenzoic acid, and A7 ) H2O. 4660

Analytical Chemistry, Vol. 71, No. 20, October 15, 1999

The Conservation Equations. For the rate-limiting substrates M (L-malate) and NADPH:

DM(d2M(x)/dx2) - K1M(x) ) 0

(5)

DN(d2NADPH(x)/dx2) - K2NADPH(x) + K1M(x) ) 0 (6) For the cosubstrates A1 (NADP+), A4 (O2), A5 (HBA):

DN(d2NADP(x)/dx2) + K2NADPH(x) - K1M(x) ) 0

(7)

DO2(d2O2(x)/dx2) - K2NADPH(x) ) 0

(8)

DHBA(d2HBA(x)/dx2) - K2NADPH(x) ) 0

(9)

Here, Di (i ) M, N, O2, HBA) are apparent diffusivities of malate, NADP(H), O2, p-hydroxybenzoic acid and K1 and K2 are apparent first-order rate constants of reactions I and II (Ki ) Vmax,i/Km,i,eff). The left boundary conditions (x ) 0) were trivial:

M(0) ) M0, NADPH(0) ) 0, O2(0) ) O20, NADP(0) ) NADP0, HBA(0) ) HBA0 (10, 11, 12, 13) Here, M0, NADP0, O20, HBA0 are bulk concentrations of the respective compounds. At the right boundary (x ) d) there were no-flux conditions for all compounds but O2:

dM(d)/dx ) 0; dNADPH(d)/dx ) 0; dNADP(d)/dx ) 0; dHBA(d)/dx ) 0 (14, 15, 16) The O2 flux at the right boundary was proportional to the local O2 concentration at x ) d, multiplied by a constant (Kmemb) reflecting thickness (δ) and an O2-permeability coefficient (P) of the hydrophobic membrane in front of the Clark electrode:

dO2(d)/dx ) - KmembO2(d)

(17)

Here, Kmemb ) 44 cm-1 ) (RDO2,memb)/(δDO2) ) P/(δDO2) where δ ) 10 µm, R ) O2 partitioning coefficient at the enzyme layer/ PET interface, DO2,memb ) O2 diffusivity inside PET membrane. Without knowing P, R and DO2,memb, Kmemb were simply calculated from the measured current density of a PET-covered Clark electrode (without an enzyme layer) in air-saturated water. In a mathematical sense, Kmemb is an adjustable parameter to cover all possible conditions for the Clark electrode boundary between the two extremes of no flux () very thick membrane, Kmembf 0) and O2(d) ) 0 (no hydrophobic membrane, Kmembf∞). For eqs 5-7 there were existing solutions. With NADPH(x) known, eq 8 was an inhomogeneous second-order differential equation with constant coefficients and was solved by integration. Equation 9 was equivalent to eq 8 with a no-flux right boundary condition. Its solution was derived from the solution for eq 8. Solutions of the Boundary Value Problem (Eqs 5-17). Solutions for the eqs 5-7 and the related boundary conditions were already known from the literature:24,25

M(x) ) M0 NADPH(x) ) M0

cosh(Q1(d - x))

[

cosh(Q1(d - x))

K1 DN(Q22

(18)

cosh(Q1d)

-

Q21)

cosh(Q1d)

-

]

cosh(Q2(d - x)) cosh(Q2d) NADP(x) ) NADP0 - NADPH(x)

(19) (20)

Equation 8 was solved by substitution and integration. The solution is:

(

O2(x) ) 1 -

(

x d+

cosh(Q2d) - 1 cosh(Q2d)Q22

(

1 Kmemb -

)[

M0K1K2

O02 +

DNDO2(Q22 - Q21)

)]

cosh(Q1d) - 1 cosh(Q1d)Q21

cosh(Q2(d - x)) - 1 cosh(Q2d)Q22

-

-

M0K1K2 DNDO2(Q22 - Q21)

)

(21)

)

(22)

cosh(Q1(d - x)) - 1 cosh(Q1d)Q21

The solution for eq 9 was derived from eq 21:

(

M0K1K2

1 1 DNDHBA(Q22 - Q21) Q22 Q21

HBA(x) ) HBA0 +

cosh(Q2(d - x)) cosh(Q2d)Q22

+

cosh(Q1(d - x)) cosh(Q1d)Q21

Here, Q1 ≡ xK1/DM; Q2 ≡ xK2/DN; Ki ) Vmax,i/Km,i; Kmemb ) 44 cm-1, DM ≈ 5 × 10-6 cm2 s-1, DN ≈ 2 × 10-6 cm2 s-1,21 DO2 ≈ 2 × 10-5 cm2 s-1, DHBA≈ 8 × 10-6 cm2 s-1, O20 ) 0.24 mmol L-1, HBA0 ) 1 mmol L-1, NADP0 ) 0.02 mmol L-1, d ≈ 0.2 mm. Apparent diffusion coefficients were estimated from literature data in aqueous solution.23,29 The O2 reduction current density was proportional to the oxygen flux at x ) d and was obtained by derivation of eq 21 at x ) d.

dO2(d) I ) FnDO2 ) FnDO2 A dx

-

(

d+

1

Q21)

)[

O02 +

Kmemb

cosh(Q2d) - 1

M0K1K2 DNDO2(Q22

(

-1

cosh(Q2d)Q22

-

)]

cosh(Q1d) - 1 cosh(Q1d)Q21

(23)

Here, I ) oxygen reduction current (µA), A ) electrode area (0.196 mm-2), F ) 9.648 × 104 C mol-1, n ) 2. It is worthy of note that eqs 21 and 23 can be adjusted to all possible right boundary assumptions for Clark electrodes by proper choice of Kmemb. A high value (Kmemb f∞) stands for “no(28) Entsch, B.; Van Berkel, W. J. H. FASEB J. 1995, 9, 476-483. (29) Lide, D. R., Ed. CRC Handbook of Chemistry and Physics; CRC Press: Boca Raton, 1994; Chapter 6, p 253.

Figure 6. Typical simulated concentration profiles of all reactants (2 U cm-2 MDH (K1 ) 0.23 s-1), 7 U cm-2 HBH (K2 ) 1.13 s-1); DN ) 2 × 10-6 cm2 s-1, DM ) 7 × 10-6 cm2 s-1, DO2 ) 2 × 10-5 cm2 s-1, d ) 0.2 mm, HBA0 ) 1 mmol L-1, M0 ) 0,4 mmol L-1, NADP0 ) 0.05 mmol L-1, O20 ) 0.24 mmol L-1).

membrane” (used, among others, by Leypoldt and Gough18,19). All formulas were programmed on a computer (with Turbo Pascal 7.0 compiler and 64 Bit precision) and eq 23 was used to simulate the L-malate response of a typical sensor (2 U cm-2 MDH, 7 U cm-2 HBH, Km(malate) ) 0.39 mmol L-1,27 Km(NADPH) ) 0.023 mmol L-1,28 w K1 ) 0.23 s-1, K2 ) 1.13 s-1, Kmemb ) 44 cm-1, d ) 0.2 mm), the influence of sensor thickness (d), and enzyme loading (variation of K1 and K2). Typical computed concentration profiles are depicted in Figure 6. The simulated response showed a very good correlation with the measured curve (Figure 5, “response” ) Imax - Imeas; with Imax ) resting O2 reduction current, Imeas ) steady-state current after addition of analyte). Both curves were linear to 1.1 mmol L-malate. At this concentration, the computed value for Imeas became zero (i.e., O2(d) ) 0), and consequently, this point marked the maximum response and the end of the applicability of formulas 23 and 21. The simulated response depended on d and showed a broad optimum between 0.15 and 0.2 mm (with 2 U cm-2 MDH, 7 U cm-2 HBH). Because there was no way to vary this parameter precisely, this trend could not be checked experimentally. The reaction layer thickness of a representative real sensor was determined as ∼0.2 mm (in the wet state). After variation of the apparent diffusivity of NADPH (DN) in eq 23, we found that the response was not affected at all, as long as DN was lower than 10-5 cm2 s-1. The real value for DN is clearly lower than this even in aqueous solution,23 so the sensor response was independent of DN, and the estimation error for DN did not contribute to the simulation. The simulated influence of MDH loading (variation of K1) showed a satisfying correlation with the experiment (Figure 3). Both simulation and experiment revealed a saturation of the response at approximately 5 U cm-2. At less than ∼0.2 U cm-2 MDH, however, the real responses were considerably lower than predicted by the simulation. This behavior can be explained by the linear model assumptions, as with decreasing enzyme loading the first-order approximation deviated the more from the real Michaelis-Menten term. The simulated variation of HBH (variation of K2) showed less of a good correlation with the experiment (Figure 4). According to the simulation, saturation of the response should have been reached at ∼0.5-1 U cm-2, but in the experiment saturation did not occur before ∼7 U cm-2. One reason for this difference may have been that the estimated value of DN Analytical Chemistry, Vol. 71, No. 20, October 15, 1999

4661

Figure 7. Simulation of NADPH peak concentration with varying MDH loading () variation of K1) (K2 ) 1.13 s-1, DN ) 2 × 10-6 cm2 s-1, DM ) 7 × 10-6 cm2 s-1, DO2 ) 2 × 10-5 cm2 s-1, d ) 0.2 mm, HBA0 ) 1 mmol L-1, M0 ) 1 mmol L-1, NADP0 ) 1 mmol L-1, O20 ) 0.24 mmol L-1).

(DN ) 2 × 10-6 cm2 s-1) was too high, resulting in false high K2 values. It was also possible that the apparent enzyme loading was lower than estimated, because HBH, a homodimer, is reported to associate to oligomeric chains of as many as 40 enzyme molecules.28 This could lead to an inhomogeneous enzyme distribution and diminish the apparent enzyme loading. An important question was the computed NADPH peak concentration, when enzyme loading was varied, as this was related to the required NADP+ concentration in the buffer (NADPH(x) + NADP+(x) ) NADP+0). NADPH peak concentration increased with decreasing DN and increasing K1 (Figure 7) and decreased with decreasing DM and increasing K2 (Figure 8). The key factor required for low NADP+ was a high ratio of the first-order kinetic constants K2/K1. In the standard sensor this ratio was 5.0 (K1 ) 0.23 s-1, K2 ) 1.13 s-1), and the computed saturating NADP+ concentration was 4.5% of the L-malate concentration to be measured. This was in good agreement with the experimentally observed NADP+ requirement (1% of L-malate) taking into account estimation errors (DN, DM) and the disregarded outer dialysis membrane. CONCLUSIONS Compared with former L-malate sensor approaches, two crucial problems were solved by effective recycling of the cofactor: The

4662 Analytical Chemistry, Vol. 71, No. 20, October 15, 1999

Figure 8. Simulation of NADPH peak concentration with varying HBH loading () variation of K2) (K1 ) 0.23 s-1, DN ) 2 × 10-6 cm2 s-1, DM ) 7 × 10-6 cm2 s-1, DO2 ) 2 × 10-5 cm2 s-1, d ) 0.2 mm, HBA0 ) 1 mmol L-1, M0 ) 1 mmol L-1, O20 ) 0.24 mmol L-1, NADP0 ) 1 mmol L-1).

need for NADP+ was reduced from ∼2 mmol L-1 10,11 to 10 µmol L-1, and the linear range was extended, because the sensor operated under NADP+ saturation. At saturating enzyme loading the coupling of the two enzyme reactions was very efficient and resulted in an operation mode which can be described as “quasi monoenzyme” behavior with a hypothetical L-malate hydroxylase enzyme. Simulation and experiments revealed that the response in this case was determined by the L-malate concentration and its diffusivity only. This could explain why L-malate was indicated to have a sensitivity 5 times higher than NADPH (because of the higher diffusivity). The side effect was an improved selectivity, because with the Clark electrode based sensor NADPH was the only “interference”. An interesting question in the future will be whether the favorable analytical characteristics of the MDH/ PHBH system can be reproduced with other dehydrogenases. In principle, all NADP+-dependent dehydrogenases can be used with this simple scheme. The presented linear model should be helpful in optimizing such systems. Furthermore, the analytical solution can be used to check a yet to be developed, more general, numerical model for this type of bienzyme sensor. Received for review June 10, 1998. Accepted July 31, 1999. AC9806355