025
Anal. Chem. 1992, 64, 625-630
Model Analysis of Enzyme Monolayer- and Bilayer-Modif ied Electrodes: The Steady-State Response Tetsu Tatsuma* and Tadashi Watanabe*
Institute of Industrial Science, University of Tokyo, Roppongi, Minato-ku, Tokyo 106, Japan
The steady-date sensor respomes of enzyme monolayer- and hetwobilayermodWbd ektrodes were theoretlcayY anatyzed wtth a slmpla model. Parameten used here are the surface coverage and rate constants of the enzyme, the concentraUonr and dtfhdon mdkknts of the substrate and medlator, the rate constant of the electrode proteas, the thlcknew of the dlhrdon layer, and the dbtance between the enzyme and electrode. The sensor response was divided Into three facton; the effkJencles of substrate and mediator supply to the amyme, the enzyme/ekcbudechargetransfer dfichcy, and the potentlal rate of enzymatic reaction where the supply Mclencies are unity. Contrlbutlons of the parameters to them factors or the total sensor response are described by simple formulas. The experimental observations are succerrfully Interpreted in terms of these theoretical results.
INTRODUCTION We have been developing amperometric biosensors with monolayers or heterobilayers of such functional molecules as enzymes immobilized on a solid electrode.'" The monolayerand bilayer-modificationtechnique provides sensors with high sensitivity per functional molecule. This point is crucial in constructing biosensors or other devices functioning on the molecular scale.6 The functional efficiency of each enzyme molecule in an amperometric enzyme electrode is dictated by three factors: (1)the efficiencies of substrate and mediator supply, (2) the efficiency of enzyme/electrode charge transfer, and (3) the potential rate of enzymatic process which is expected in the case where the supply efficiencies are unity.6 We demonstrated experimentally that enzyme monolayer- and bilayermodified electrodes are advantageous with regard to the first factor'B2 and that the second factor could be improved by appropriate design of the s y ~ t e m . ~ * ~ * Analysis of enzyme electrode kinetics is of importance for designing a sensor or optimization of parameters. However, there have been no systematic studies on the steady-state kinetics of enzyme monolayer-modified electrodes which can corroborate the above-mentioned observations and predict the three factors mentioned above. Though many studies were reported on the response analysis of enzyme electrodes, they are mostly concerned with enzyme multilayer-modifiedelectrodes fabricated by enzyme cross-linking or polymer entrapment."' These analyses are complicated because the enzyme molecules are distributed homogeneously in a relatively thick layer, in which the local concentrations of substrat&) and product(s), the local rate of enzymatic reaction, and the local velocities of mass transfer are not uniform even a t the steady state. In most of such cases the differential equations had to be handled numerically and analytical solutions were not attained. In contrast, steady-state kinetics of an enzyme monolayer-modified electrode, where enzyme molecules are arranged on a plane, can be analyzed without recourse to differential equations, and the sensor response can be described by simple formulas. 0003-2700/92/0364-0625$03.00/0
In this paper, we present models for enzyme monolayerand heterobilayer-modifiedelectrodes and then analyze their steady-state kinetics and interpret the experimental observations collected by ourselves in terms of the theoretical results. The ultimate aim of the model analysis in this work and the subsequent paper12 is to rationalize the mechanism of an enzyme electrode and to give criteria for optimization of parameters and sensor performance; their quantitative justification is largely left for future investigations. MODEL Preconditions and Assumptions. Here we model an amperometric enzyme monolayer-modified electrode which senses a substrate on the basis of charge transfer from the substrate to the electrode via an enzyme and a dissolved electron mediator. The enzyme is a redox enzyme with the following elementary reactions:
E+S
-
E' + M
kl
+P E+N
E'
(1) (2)
where E and E' are native and charge-carrying enzymes, respectively. The charge(s) (positive or negative) of the latter is (are) supplied from the substrate S, which is thereby converted into the product P. M and N are mediators before and after the charge exchange with E'. The rate constants kl and kz are in units of M-' s-l, or mol cm-2 s-'/(M mol c m 3 . The charged mediator N diffuses toward the electrode surface and is discharged there:
N
k, -+
M
+ charge(s)
(3)
where kE is the heterogeneous rate constant in cm s-'. We assume the following for the model analysis: (1)The electrode is an ideal plane with infinite area (2) Active centers of enzyme molecules are homogeneously distributed on a plane a t a distance a from the electrode surface. (3) S and M are consumed and then N is generated, all on the plane of the enzyme active center. (4) N is consumed and M is regenerated on the electrode surface. ( 5 ) S is electrochemically inert. (6) The diffusion layer thickness b is constant. (7) No convection or migration occurs within the diffusion layer. (8) The concentrations of S and M are constant (Csand CM,respectively) and the concentration of N is 0 outside the diffusion layer. (9) The substrate, enzyme, and mediator carry the same number of charges, n. In all of the experiments described later, we did not stir the electrolyte solution Under such a condition, the diffusion layer must be, ideally, of infinitethickness. However, there inevitably exists convection due to local temperature gradient arising from the heat supply from the thermostat. Therefore, it is reasonable to assume a finite diffusion layer thickness. Figure 1schematically illustrates steady-state concentration profiles for S, M, and N in the vicinity of the electrode surface. Cas,Ca, and CaNare concentrations of S, M, and N on the ~ CONare concenplane of the enzyme active center. C Oand 0 1992 American Chemical Society
626
ANALYTICAL CHEMISTRY, VOL. 64, NO. 6, MARCH 15, 1992
to
Bilayer
CS (b - u ) / D+ ~ l/klI'
J=
Monolayer
(9)
The output current density i is denoted as follows:
i = nFecesJ,,
(10)
where ec and es are the enzyme/electrode charge-transfer efficiency and the substrate supply efficiency, respectively. Equations 4-8 afford ec and es:
ec = i / n F J = kECON/J
O
es = CBs/Cs = J / k l F C s 1 ( b - a)klI'/Ds + 1
b
a
-
Distance (cn)
Flgure 1. Schematlc drawing of concentration profiles of S, M, and N for a model enzyme monolayer-modified electrode at steady state.
trations of M and N on the electrode surface. In the steady state, the substrate concentration is constant from the reaction center plane to the electrode surface. In the case of an enzyme heterobilayer-modified electrode, the overlayer enzyme is considered to be placed on the active center plane and the underlayer enzyme is considered to be placed on the electrode surface (Figure 1). In what follows we analyze the following three cases in succession: linear region, saturated region, and nonlinear region. In the linear region, substrate concentration is so low that the sensor response is proportional to the concentration. In the saturated region, the substrate concentration is so high that the sensor response does not depend on the concentration any longer. The nonlinear region bridges the linear and saturated regions. Though all these regions behave essentially nonlinearly, approximations are applied to the linear and saturated regions since the analysis of the nonlinear region is somewhat complicated. I. Linear Region. First, we consider the kinetics where reaction 1 determines the total enzymatic reaction rate. A sensor responds linearly to the substrate concentration under such conditions, and the linear dependency is desirable for practical measurements. In this case,enzyme is mostly present as E, and the fraction of E' is negligible. Hence the rate of enzymatic reaction, J (in the dimension of flux), is given by where r is the total surface density of enzyme. The flux of the total substrate and mediator supply toward the active center plane (eqs 5 and 6, respectively) and the flux of the total charge-carrying mediator (N) generation from the plane (eq 7) are equal to J because of the mass balance, where D
(12)
The last factor in eq 10, Jwt, is the potential enzymatic reaction rate given by Jpot.
= k,rCs
(13)
Equations 10-13 can be summarized as nFCs 1 =
@N/kE
+ b ) ( l / D s + l/[kir(b - a ) ] )
(14)
Equations 10-13 or eq 14 expreaws the response chwacteristica of an enzyme monolayer- and bilayer-modified electrode in the linear region. 11. Saturated Region. Next we deal with the kinetics where reaction 2 determines the total enzymatic reaction rate. The output current i is independent of the substrate concentration. In this case, enzyme is mostly present as E', and the fraction of E is negligible. Hence the enzymatic reaction rate J is given by
J =k2rCa~
(15)
Equations 5-8 and 15 yield
..
(DN + b h @ M
+ -k J
The output current density i in this case is where eMis the mediator supply efficiency:
(5) Equations 6-8 yield CONproportional to J . Hence ec (= k&,N/J) is independent of J and is expressed by eq 11 in every region. The formula for Jpot,in the saturated region is (7) is a diffusion coefficient. The consumption rate of N on the electrode surface is equivalent to the flux of N supply toward the electrode surface and to the flux of M regeneration from the surface toward the active center plane.
Thus, eqs 17-19 and 11 show the dependence of i on the parameter values. 111. Nonlinear Region. In the nonlinear region, where the rates of reactions 1 and 2 are comparable, the enzymatic reaction rate is no longer given by eq 4 or 15 but as follows:
Equations 4 and 5 yield
From eqs 5-8 and 20, J can be reformulated into the equation
ANALYTICAL CHEMISTRY, VOL. 64, NO. 6, MARCH 15, 1992
E
'
""
"
'
""""
'
""""
Cnk d
''?"
' """'
627
''7
kI
cs (mol c i 3 )
In this case i takes the form
Flgure 2. Correlations between the potential enzymatlc reaction rate J, and the substrate concentration CsIn the linear (l), saturated (2), and nonlinear (3) reglons.
where es and eM are given by
b-a es=l--J CSDS (DN + akE)(b - a) eM=1J
(DN+ ~ ~ E ) C M D M
(26) (27)
The formula for ec is again eq 11. Thus the behavior of i is governed by eqs 20-27 and 11. Incidentally, JWt. in the nonlinear region is 10-9
10-8
cS (mol
The Michaelis constant, KM, is equivalent to CMk2/kl here. EXPERIMENTAL SECTION Materials. An Sn02-coatedglass plate from Nippon Sheet Glass (Japan) was rendered hydrophilicwith sulfuric acid which was heated by dilution (1:l)for a few minutes prior to sensor fabrication. (3-Aminopropy1)triethoxysilane (APTES) and a glutaraldehyde solution were obtained from Tokyo Kasei Co. (Japan) and Sigma Chemical Co., respectively. Glucose oxidase (E.C. 1.1.3.4,from Aspergillus niger) solution (ca5OOO unita mL-', Boehringer Mannheim) was used as received. Electrode Derivatization. The hydrated SnOzelectrode was treated successively at room temperature with a 10% aqueous solution of APTES for 30 min, then with a 2.5% glutaraldehyde aqueous solution for 30 min, and finally with a glucose oxidase solution for 30 min to yield a glucose oxidase monolayer-modiied electrode. Electrochemical Measurements. Measurements were performed with a conventional three-electrode system without continuous stirring. Air-saturated 1/15 M phosphate buffer (pH 6.4) kept at 30 O C with a water bath was used as electrolyte. Ag/Agcl and F% black were employed as reference and counter electrodes, respectively. The electrode potential was controlled with a potentiostat 2020 (TohoTechnical Research, Japan) and a function generator HB-111 (Hokuto Denko, Japan). RESULTS AND DISCUSSION I. Linear Region. Enzymatic Process. As is seen in eq 10, the factors eo eS,and the potential enzymatic reaction rate Je define the output current in the linear region. Equation 13 indicates that Jwt. increases with increasing I?. Substrate Supply Efficiency. With (b - a ) k l r / D sand es denoted by x and y, respectively, eq 12 can be rewritten as (29) y = l / ( x + 1) The magnitude of DS/klr determines whether the enzymatic process is under kinetic or diffusion control. h e r r results in diffusion control and lower es, and smaller r results in kinetic control and higher e% Smaller b as well as larger a makes es higher. Charge-Transfer Efficiency. With DN/kEband ecb/(b - a ) (=ec if b >> a ) denoted by z and y, respectively, eq 11
10-6
10-5
cm-?
Flgure 9. Dependencies of es, eM,and J/J,
on Csin the nonlinear regbn. Parameter vaiues are k , = k , = 10 cm3mop' s-l, kE = lo5 cm s-l, I' = lo-" mol cm-,, b = lo-* cm, a = lo-' cm, Os = 0, = om2 s-'. and CM = mol om3. is rewritten as eq 29. The magnitude of D N /kE determines whether the electrode process is under kinetic or diffusion ControL A s d e r DN as well as a larger k E renders the process diffusion controlled and affords higher ec. Larger b and smaller a is desirable for enhancing ec, in opposition to the case for es. Influence of the Parameters on the Output Current Density i. The contributions of I', b, and a to Jpt.,es, and ec are now clear. Their contributions to the final response i, however, still need to be elucidated. This could be done by examining eq 14 which contains all of the three factors, JWt., est and ec. Greater r yields greater i, and sufficiently large r affords r-independent i. Smaller a is desirable to enhance i, though a does not control i when klr or b is eufficiently large. The contribution of b to i depends on the magnitudes of DNfkE and Ds/klr. If DN/kE and D s / k l r are much larger than b, i is proportional to (b - a). DN/kE >> b and D s / k l r (b - a) yield i proportional to ( b - a ) / b , and D N / k E
