2896
Anal. Chem. 1984, 56, 2896-2904
platinum surface, then a potentiometric platinum-enzyme glucose sensor might become a reality. ACKNOWLEDGMENT We appreciate the skillful assistance of L. C. Cantin in carrying out the experimental part of this work. Registry No. Glucose, 50-99-7;hydrogen peroxide, 7722-84-1; platinum, 7440-06-4;glucose oxidase, 9001-37-0;platinic chloride, 16941-12-1. LITERATURE CITED Solsky, R. L. CRC Crlt. Rev. Anal. Chem. 1083, 14, 1-52. Ryan, M. D.; Wilson, G. S. Anal. Chem. 1082, 5 4 , 20R-27R. Wingard, L. B., Jr. Fed. Proc., Fed. Am. SOC. Exp. 1083, 42, 288-291. Wingard, L. B., Jr.; Ellls, D.; Yao, S. J.; Schiller, J. G.; Liu, C. C.; Wolfson, S. K., Jr.; Drash, A. L. J. Solid-Phase Blochem. 1070, 4, 253-262. Wlngard, L. B., Jr.; Schlller, J. G.; Wolfson, S. K., Jr.; Liu, C. C.; Drash, A. L.; Yao, S. J. J. Biomed. Mater. Res. 1070, 13, 921-935. Llu, C. C.; Wingard, L. B., Jr.; Wolfson, S.K., Jr.; Yao, S. J.; Drash, A. L.; Schiller, J. 0. J . Nectroanal. Chem. 1070, 104, 19-26. Wlngard, L. B., Jr.; Castner, J. F.; Yao, S. J.; Wolfson, S. K., Jr.; Drash, A. L.; Liu, C. C. Appl. Biochem. Biotechnol. 1084, 9, 95-104. Wingard, L. B., Jr.; Llu, C. C.; Wolfson, S. K., Jr.; Yao, S. J.; Drash, A. L. Diabetes Care 1082. 5 . 199-202. Hoare, J. P. I n "Encyclopedia of Electrochemistry of the Elements"; Bard, A. J., Ed.; Marcel Dekker: New York, 1974; Vol. 2, pp 2 10-238. Angersteln-Koziowska, H.; Conway, B. E.; Sharp, W. B. A. Electroanal. Chem. Interfacial Electrochem. 1073, 43, 9-36. Rasmussen, K. E.; Albrechtsen, J. Histochemistry 1074, 3 8 , 19-26.
(12) Gordon, A. J.; Ford, R. A. "The Chemist's Companion"; Wlley: New York. 1972: 438. -(13) Marincic, L.; Soeldner, J. S.; Coiton, C. K.; Giner, J.; Morris, $. J . Nectrochem Soc 1070, 726, 43-49. (14) Wingard, L. B., Jr.; Cantin, L. A.; Castner, J. F. Blochim. Blophys. Acta 1983, 748. 21-27. (15) Proctor, A.; Castner. J. F.; Wlngard, L. B., Jr.; Hercules, D. M. Anal. Chem., submitted. (16) Rowe, M., Johnson, Matthey Co., Oct 1983, personal communicatlon. ( 1 7) Zysk, E., Engelhard Industrles, Sept 1983, personal communication. (18) Castner, J. F.; Wlngard, L. E., Jr. Blochemistry 1084, 2 3 , 2203-2210. (19) Wingard, L. B., Jr.; Castner, J. F. I n "Enzyme Engineering"; Ghlbata, I., Fukui, S., Wingard, L. B., Jr., Eds.; Plenum Press: New York, 1982; VQI. 6,pp 415-416. (20) Giner, J.; Malachesky, P. Proceedings of the Artiflclal Heart Program CQnference, June 9-13, 1969, Washington, DC. (21) Dobos, D. "Electrochemical Data"; Elsevier: Amsterdam, 1975; pp 259-260. (22) Hoare, J. P. Nectrochim. Acta 1081, 2 6 , 225-232. (23) Heyns, K.; Paulsen, H. I n "Newer Methods of Preparative Organlc Chemistry"; Forest, W., Ed.; Academic Press: New York, 1963: Vol. 2, pp 303-335. (24) Rao, M. L. 8.; Drake, R. F. J. Nectrochem. SOC. 1060, 116,
. - - . ~ . .
__
334-337
(25) deMeie, M. F. L.; Videia, H. A.; Arvia, A. J. Bloelectrochem. Bloenerg. 1082, 9 , 469-487. (26) Hsueh, K. L.; Gonzaiez, E. R.; Srinivasan, S. J. Electrochem. SOC. 1084, 131, 823-828.
RECEIVED for review January
3,1984. Accepted July 31,1984. This work was supported by Grant lROl AM26370 from the National Institute of Arthritis, Diabetes, and Digestive and Kidney Diseases of NIH.
Model of a Two-Substrate Enzyme Electrode for Glucose J o h n K. Leypoldt' a n d David A. Gough*
Department of Applied Mechanics and Engineering Sciences, Bioengineering Group, University of California, S a n Diego, La Jolla, Californiq 92093
A mathematlcai model of a two-substrate enzyme electrode Is presented. The electrode is a glucose-specific sensor In which enzymes are lmmobiilzed wlthln a membrane attached to an electrochernlcal sensor for oxygen! the cosubstrate of the enzyme reaction. The model presented here focuses on the reaction and dlffuslon phenomena that occur wlthln the enzyme membrane. I t Is shown that when the membrane has the approprlate permeability to both substrates and the enzyme reaction Is dlffuslon-llmited, the difference In the enzyme electrode current from that of an oxygen-sensitive reference electrode Is proportional to the bulk glucose concentration, irrespective of the ratlo of substrate concentrations In the bulk solutlon. Moreover, when the bulk oxygen concentration Is low, It may be advantageous to immobilize Ilmited amounts of enzyme In the membrane in order to control the range of glucose detectablllty. These predictions differ notably from those of a one-substrate model. The modellng results are summarlred concisely in three simple figures and verified by comparison with experimental observations. A method for identlfying the llmitlng substrate Is described and guldellnes are glven for sensor design.
"Enzyme electrodes" show promise for a variety of novel analytical applications (1-3). Broadly defined, these are Current address: Department of Medicine, V.A. Medical Center, San Diego, CA 92161.
chemical-specific sensors in which biological catalysts such as immobilized enzymes, immobilized cells, or layers of tissue are coupled to electrodes that are sensitive to a product or cosubstrate of the enzyme reaction. Early investigations with the type of enzyme electrode that requires only a single substrate ( 4 , 5 ) have demonstrated that this type of sensor can be used in some cases to analyze chemicals with little or no sample preparation, a feature that is not only convenient, but particularly advantageous for monitoring directly in living tissues. Many enzymes, however, require a cofactor or cosubstrate in order to be catalytically active. One approach to providing this reactant is to simply add it to the sample in sufficient amounts. Unfortunately, this eliminates the advantage of no sample preparation. In other situations where the cosubstrate is endogenous to the sample, an alternative is to design the sensor for aperation under existing cosubstrate concentrations. This may require, for example, specifying the immobilized enzyme content or tailoring the membrane permeability to the respective substrates. In order to proceed rationally with this approach, it is necessary to have a working model of the two-substrate sensor. Mathematical models of one-substrate enzyme electrodes proposed by a number of investigators (6-9) have been reviewed elsewhere ( I , IO) and have some relevance to twosubstrate enzyme electrodes. These models show that imposing diffusional limitations on the enzyme reaction can lead to improvements in sensor performance in various ways, such as, increasing the sensitivity to the substrate ( 6 ) ,extending the range of linearity (7,8), decreasing the sensitivity to en-
0003-2700/84/0356-2896$01.50/0 0 1984 American Chemical Society
P{NALYTICAL CHEMISTRY, VOL. 56, NO. 14, DECEMBER 1984
and the dimensiopless boundary conditions at the membrane-solution interface are
zyme inactivation (6),and reducing the time to reach steady state after a perturbation of the system (9). It has $90 been shown that such results are independent of whether the product of the enzyme reaction is detected potentiometrically or amperometrically (10). Such models are also successful in simulating calibration curves of two-substrate enzyme electrodes operating under certain conditions of fixed cosubstrate concentration (7)but are unable to predict the response under varying cosubstrate concentrations. Certain other models have been unrealistically formulated (11). Few attempts have been made a t modeling the two-substrate enzyme electrode. In one notable case (12),a model of a two-substrate enzyme electrode for glucose was developed based on numerical solutions of the governing equations for specific values of the properties of the enzyme membrane. Such an approach is a useful first step toward sensor design but unfortunately does not provide results that can be applied generally. The use of dimensionless variables and parameters would produce more general results and help guide further experimentation. More useful still would be a closed-form mathematical model expressed in terms of dimensionless parameters and variables, since iterative computations would not be necessary. Regardless of form however, the important advantage of a successful model is the ability to simulate the ideal sensor response and indicate the necessary sensor properties without the need for extensive experimentation, thus making possible a more directed developmental effort. In this paper we describe a general mathematical model for two-substrate enzyme electrodes, using as an example a sensor for glucose based on a membrane containing immobilized glucose oxidase and catalase. These enzymes catalyze the following reactions: glucose
- + + + -
+ O2+ H 2 0
gluconic acid
H202
H2O
'/202
+ H202
deg
_ - Big(l - cg) dx
1/202 gluconic acid
(7)
da, dx
The boundary conditions at the membraneelectrode interface reflect the fact that oxygen is consumed rapidly at the electrode surface whereas glucose is not electrochemically active, a situation that is readily attainable experimentally. In dimensionless form these conditions are c, = 0 x = o (9) dcg/dx = 0
x = o
(10)
In these equations the following dimensionless variables and parameters were used:
(1)
R(eg,E,) =
(2) (3)
This overall process can be generalized (13) in the form glucose v02 products (4) where v is a stoichiometry coefficient that can have the value of either one or one-half, corresponding respectively to the situation in which catalase is either absent or coimmobilized in excess. The sensor operates by producing an oxygen (cosubstrate) difference current that is modulated by glucose, as proposed previously by others (14). Similar glucose sensors based on the detection of hydrogen peroxide or hydrogen ion, the products of the enzyme reaction, rather than of oxygen show identical behavior when the enzyme reaction is diffusion-limited and are qualitatively the same otherwise (IO).
THE MODEL The description given here is of a homogeneous enzyme membrane placed immediately adjacent to the electrode surface. Features of more complex membrane structures, if required in practice, can easily be incorporated into the model later. The steady state governing equations for the system are based on the principle of conservation of mass and have been previously formulated (IO). Equating reaction and diffusion in the membrane, the dimensionless governing equations are
x =1
- = Bio(Co*- eo) x = 1
to give the overall reaction glucose
2897
J
r(agCgBcg, VagDgCgBco/Do) r(agCgB, aocoB)
where D is the solute diffusivity within the membrane and c is the concentration of the subscripted solute. Mass transfer to the membrane-solution interface is described by the appropriate mass transfer coefficients hi, where i is respectively g or 0,and partitioning of the solutes in the membrane is allowed for by employing the respective partition coefficients cq. The variables cgB and c,B are the bulk concentrations of glucose and oxygen, respectively. The respective dimensionless solute concentrations ci are expressed in forms that are particularly convenient for this problem. The parameter co*, the effective substrate concentration ratio, gives the ratio of substrate concentrations in the membrane and can be useful to specify which substrate is limiting, as is discussed below. The parameter Bii is the mass transfer Biot number corresponding to the respective solute and represents the ratio of the permeability of the external solution to that of the membrane. The parameter 4: is the square of the Thiele modulus or the ratio of the potential rate of reaction to the potential rate of diffusion and is useful to indicate whether the process is under reaction control or internal diffusion control (15). It is advantageous to first simplify the above model equations by solving for Eo with eq 5-10. Substituting the result into eq 5 gives the following reduced description of this system:
d2cg --
dx2
4ZR(cg,cg + Bx
+ A) = 0
dcg _ -- Big(l - cg) dx
dc, _ -- 0
05x 5 1
(12)
x =1
x=o (14) dx where A and B are unknown integration constants dependent
2808
ANALYTICAL CHEMISTRY, VOL. 56, NO. 14, DECEMBER 1984
upon the boundary conditions and are given by A = F&O)
1.2
I
I
1 .o
2.0
I
I
3.0
4.0
(15)
Equations 12-16 describe a nonlinear boundary value problem that is not solvable generally in analytical form. However, accurate numerical solutions can easily be obtained by the orthogonal collocation method (16) provided that the value of +g is sufficiently small. A novel method for obtaining asymptotic solutions when the value of +g is relatively large is outlined in the Appendix. The form of the solution that is of interest here is the glucose-dependent difference current ( I 7), which is the difference between the oxygen-dependent current of the medium generated by an oxygen reference electrode and the glucosemodulated oxygen-dependent current of the enzyme electrode. This difference current is
"0
5.0
1 I&* Figure 1. The dimensionless diference current 7, normalized by effective oxygen concentration E," plotted as a function of the effective glucose concentration l / E , * , for various values of relative catalytic activity ug. Big, Bi, and K are held constant.
oxygen at infinite concentrations of the other substrate, and V,, is the maximal velocity. In this case, the square of the Thiele modulus is The first term on the right side of this equation is the oxygen-dependent current of the oxygen reference electrode, whereas the second term is the oxygen-dependent current that is modulated by the enzyme reaction with glucose. This latter can be evaluated by use of the model and substituted back into eq 17 to give a convenient dimensionless expression for the glucose-dependent difference current
where 4 is the effectiveness factor (I5),defined here by the following: 7 = JIR(Cg, Eg
+ Bx + A ) dx
A similar expression was derived by us previously (IO)for a two-substrate enzyme electrode that detects a product of the enzyme reaction either potentiometrically or amperometrically. The difference here is that the expressions in eq 18 and 19 take into account the oxygen concentration gradient caused by the electrochemical consumption of oxygen. The model is formulated completely by incorporating the characteristic reaction rate expression for the soluble enzyme or an appropriate approximation in the dimensionless reaction rate expression. For membranes containing immobilized glucose oxidase and excess catalase, the following reaction rate expression is useful:
This expression is an approximation to the more general ping-pong rate expression determined for the soluble enzyme (18). The rationale for use of this expression is discussed below. For the assumed reaction rate model the kinetic parameters Vl,,, and K,' are related to the Ping-Pong kinetic parameters by Vl,, = Vm,/Kg and K,' = K,/K,. The values of Kg and KOare the Michaelis constants for glucose and
where K = D J J / u D g and is defined as the dimensionless kinetic constant, E,* is the effective substrate concentration ratio in the membrane defined in eq 11, and the parameter ,:u or the relative catalytic activity, is given by
The relative catalytic activity is convenient for comparing the potential rates of reaction and diffusion at specified substrate concentrations for different membranes, since it is a function of the properties of the membrane only. RESULTS Given the number of constants and variables involved in this system, it is advantageous to summarize results in terms of the dimensionless parameters defined in the model. These parameters are appropriate combinations of the constants and variables that indicate relative magnitudes of the physical phenomena. The advantage of such an approach is-apparent by noting that the dimensionlevs difference current i, can now be expressed a3 a function of only five dimensionless parameters, given mathematically by g;
= f(ug, K , Big, Bi,, F,*)
(23)
This represents a considerable reduction from the original 14 parameters from which these dimensionless quantities are derived. The Effects of Relative Catalytic Activity, ug. The effects of varying the relative catalytic activity on the sensor response to glucose can be seen in Figure 1. Here, the dimensionless glucose-dependent difference current, ig,is normalized by the effective substrate concentration ratio E,* and plotted as a function of l/E0* for various values of ug. This is equivalent to plotting the difference signal as a function of bulk glucose concentration, both divided by the bulk oxygen concentration, for specific values of relative catalytic activity. This method of plotting is convenient and particularly meaningful for sensor design since results for different oxygen
i-r
ANALYTICAL CHEMISTRY, VOL. 56, NO. 14, DECEMBER 1984
concentrations can be summarized in a single curve. Relative catalytic activity can be varied by, for example, fabricating membranes with different quantities of enzyme. The results are given for values of glucose and oxygen mass transfer parameters Big and Bi, and the kinetic constant K that correspond to realistic sensors. The results show that for membranes having low relative catalytic activity ( for example, ug = 0.5 and 1.0)) the normalized signal rises gradually with effective glucose concentration and is nonlinearly sensitive to glucose over a broad concentration range. As the relative catalytic activity is increased, the nature of the response changes. At higher relative catalytic activity (ug = 7.0, for example) the initial slope is steeper and more nearly linear, with a sharp break to the horizontal at a charaderistic effective glucose concentration, after which there is no sensitivity to glucose. Physically, the situation is as follows. For membranes of high relative catalytic activity, initially the glucose concentration in the membrane is low relative to the oxygen concentration, rendering the overall system limited by glucose and therefore potentially useful for glucose assay. In the region beyond the break in slope, however, glucose is present in relative excess and the system is limited by oxygen. Under these conditions, the system is not capable of glucose assay. For membranes of lower relative catalytic activity the reaction rate for a given effective glucose concentration is lower, leading to less oxygen consumption. The resulting oxygen reserve allows sensitivity to glucose at higher concentrations, although the corresponding signal is distinctly nonlinear. This suggests that it may be advantageous to employ membranes with low relative catalytic activity in two-substrate enzyme electrodes as a means of extending the range of sensitivity to glucose in the presence of otherwise limiting oxygen concentrations. However, this strategy will be truly useful only if the enzyme remains catalytically active over the period of use, since small changes in the relative catalytic activity have a large effect on sensor calibration. These results illustrate that the need to develop a detailed model for a two-substrate enzyme electrode revolves around considerations of the limiting substrate. When the cosubstrate is present in excess, the more simple one-substrate model should be entirely adequate. But as shown above, when the cosubstrate is limiting, the design of a two-substrate enzyme electrode will be considerably more involved than that suggested by the one-substrate model. For example, increasing the relative catalytic activity may severely limit the detectable concentration range of the substrate of interest. This is unique to the two-substrate enzyme electrode, since in the one-substrate enzyme electrode the detectable range increases with increasing immobilized enzyme activity (6-8). Effects of the Dimensionless Kinetic Constant, K. The sensor response a t low relative catalytic activity is also dependent upon the value of the dimensionless kinetic constant. Figure 2 summarizes the results for membranes with different values of the dimensionless kinetic constant, where the normalized dimensionless difference current is plotted as a function of the effective glucose concentration as in Figure 1. In the lower curves corresponding to higher values of K , the normalized dimensionless current rises only slightly with effective glucose concentration and approaches an asymptotic limit. These large values of the kinetic constant imply a closer approximation to zero-order kinetics and therefore insensitivity to glucose. The upper curves correspond to lower values of K , where the reaction rate is better described by a first-order kinetic model. Here, the normalized dimensionless signal rises higher and remains sensitive to substantially higher glucose concentrations. This shows the dependence of the response of two-substrate sensors with low relative catalytic activity
1.01
ug Sig = 50.0 1.0
2899
0.8
[ / K - l
0.6
0.4
0.2 I
I
1.o
2.0
I
I
4.0
5.0
I
3.0 1/Eo*
Figure 2.
The normalized dimensionless dlfference current as a
function of effectlve glucose concentration, for various values of dimensionless kinetic constant K . Big, Bi, and ug are held constant. on the dimensionless kinetic constant, a parameter that may, in practice, be difficult to alter substantially. Effects of the Mass Transfer Parameters, Bigand Bi,. When the relative catalytic activity is low, variations in the external mass transfer conditions as specified by the respective mass transport parameters have little effect, since the system is dominated by reaction kinetics. However, when the relative catalytic activity is high a change in external mass transfer conditions can play a significant role. With relatively rapid reaction kinetics (i.e., when the Thiele modulus & >> I), an asymptotic overall effectiveness factor can be computed for arbitrary values of the mass transfer parameters (19). Under these conditions, the difference current is now given by the expressions lg
= nFAvcuP,cgE 6,(1
+ Bicl)
[
1-
n,a:(
&
-
&)I
(glucose-limited) (24) lg
- nFAcu~ocoE - 6,(1 + BiLl)
(oxygen-limited)
(25)
where qa is the asymptotic effectiveness factor described in the Appendix. The definitions of glucose- and oxygen-limited regimes are complex functions of the reaction and mass transfer parameters as described in the next section. Simplifying further, if the values for the mass transfer parameters of both substrates are the same (Le., Big = Bi,) the response of the sensor can be reduced to a particularly simple dimensional form and the conditions specifying the glucose- and oxygen-limited regimes given as follows: lg
=
nFAvcrgDgcgB
6,(1
+ Bill)
' %Dg
ffoDocoB
___
(26)
Equation 26 suggests a strictly linear response with glucose concentration independent of the value of relative catalytic activity. When the bulk glucose concentration is less than the characteristic value specified at the right in eq 26 or, in dimensionless form, when the effective substrate concentration ratio EO* L 1
(28)
2900
ANALYTICAL CHEMISTRY, VOL. 56, NO. 14, DECEMBER 1984
-"0
1.o
2.0
3.0
K = 1.0 Bio = 10.0
og = 8.0 B i g = 1.0
4.0
-
5.0
1/Eo*
sensor geometry, increases in the range of detectability are always accompanied by a decrease in the ability to resolve small differences in glucose concentration, suggesting that specific sensor designs should be developed for different applications. Limiting Substrate, These considerations indicate that identifying the limiting substrate is of key importance. The special case in which Big/Bi, = 1leads to a particularly simple determination of the limiting substrate, namely, when Eo* 1 1, glucose is limiting. However, it is necessary to develop a more general method that applies to other situations. We previously described such a method based on a similar model and a more complex reaction rate expression (20). With this approach, it can be shown that for dg >> 1, glucose is the limiting substrate for arbitrary values of Big/Bi, when any of the following conditions prevail: Big 1 I 7-< E,* arbitrarily high ug
B10
The normalized dimensionless difference current as a function of effective glucose concentration. I n the solid lines, up is held constant and the ratio Bl,/Bi, is varied. I n the broken lines, Bi,/BI, is held constant and uo is varied.
Big 1 I E,* I Bi,
glucose is the limiting substrate, and the response of the sensor is linear with glucose concentration and independent of oxygen concentration. Under these conditions, eq 26 is applicable and the response is similar to that of a one-substrate sensor discussed elsewhere (10). However, when the bulk glucose concentration exceeds this value, the response is then given by eq 27 and is independent of glucose concentration but directly dependent on oxygen concentration. It is notable that the range of sensitivity to bulk glucose concentration is not only a function of the bulk concentration ratio of glucose and oxygen and the stoichiometry of the reaction but of the overall permeability to both substrates as well. Importantly, this result is independent of the particular form of the reaction rate expression used, as might be expected under diffusionlimited conditions. The results of these asymptotic solutions that apply when the system is dominated by mass transfer limitations are shown Figure 3. The normalized dimensianless difference current is plotted as a function of effective glucose concentration for various values of relative catalytic activity and mass transfer parameters. In the solid lines the relative catalytic activity is large and the glucose mass transfer parameter is varied, whereas in the broken lines, the glucose mass transfer parameter is small and the relative catalytic activity is varied. In all curves, the dimensionless kinetic constant and oxygen mass transfer parameter are held constant. In the solid lines, the normalized dimensionless signal rises quasi-linearly with the effective glucose concentration when the mass transfer paradeters are different, or linearly when the same, until reaching unity at which point the system becomes limited by oxygen and insensitive to glucose. Thus, the sensitivity to glucose can be adjusted by specifying the glucose mass transfer conditions. In practice, this might be achieved by placing a second membrane with the appropriate substrate permeabilities adjacent to the enzyme membrane as described later. The broken lines in Figure 3 indicate that when the glucose mass transfer parameter is low, the effect of increasing relative catalytic activity is to extend the range of sensitivity to glucose. This is a result of excess oxygen in the membrane due to mass transfer differences. This effect shows a distinctly different trend with increasing values of relative catalytic activity from that shown in Figure 1, or from predictions based on a onesubstrate enzyme electrode model (6-8, 10). This behavior is obtained only when Bi, > Big and the relative catalytic activity is very large. It is important to note that for a given
- I E,*
Figure 3.
ag
I gcr
gg
2
(29)
Big
Bi,
I 1
gcr
with the critical relative catalytic activity expression Bi,(l gcr
=
gcr
given by the
+ K ) ~ / ~ ( E , -* 1) Bi,/Bi, - 1
(30)
The dependence of the critical relative catalytic activity on the enzyme kinetic and mass transfer parameters has been presented graphically elsewhere (20). The conditions in eq 29 illustrate that a determination of the limiting substrate is more difficult when the mass transfer parameters for the two substrates differ. When E,* 2 Big/Bio, the response of the two-substrate enzyme electrode is very similar to that when the mass transfer parameters are equal: glucose is the limiting substrate independent of the value of the relative catalytic activity, as long as this latter parameter remains arbitrarily high. When either of the other two conditions in eq 29 are met however, the identification of the limiting substrate must take into account the value of the relative catalytic activity. For example, when E*, IBig/Bio, glucose is the limiting substrate provided that the value of the relative catalytic activity is less than the critical value. Above this critical value, oxygen is the limiting substrate. This behavior is illustrated in Figure 1,where the oxygen-limited plateau current is found a t values of E,* 1 1 for ug = 4.0 and 7.0. Nevertheless, glucose can still be the limiting substrate even though E,* 5 1, provided that the value of the relative catalytic activity is sufficiently high and Bi,/Bi, IEo*, as suggested by the third condition in eq 29. Increasing the value of the relative catalytic activity extends the range over which glucose is the limiting substrate. This is illustrated by the broken lines of Figure 3, where the range of sensitivity to glucose is extended to values of l/E,* that are much greater than unity by increasing the relative catalytic activity. These results suggest that both internal and external mass transfer conditions, in addition to enzyme kinetics, may be important in defining the limiting substrate. The practical significance of the critical relative catalytic activity is in indicating the point a t which the change from glucose- to oxygen-limitation occurs and a t which the difference current becomes independent of glucose concentration. To facilitate determining this point, eq 30 is rearranged as follows:
ANALYTICAL CHEMISTRY, VOL.
+ ,)lI2 ucr(Big/Bio - 1) + Bi,(l +
C,B/
Bi,(l
l/C,*
=
56, NO. 14, DECEMBER 1984
1.o
K)'/'
Knowing the dimensionless kinetic constant and relative catalytic activity of a given membrane, this expression can be employed by equating the relative catalytic activity for a given membrane to the critical relative catalytic activity allowing calculation of a corresponding effective substrate concentration ratio. This simple calculation can be a powerful tool in designing practical sensors since the effective substrate concentration ratio corresponding to the characteristic critical relative catalytic activity, expressed in terms of dimensionless glucose concentration l/co*,defines the range of detectability of glucose.
DISCUSSION Model Verification. A difficulty in verifying the model is experimentally determining the values of all necessary parameters. An approach we suggested previously (17) is to first determine the mass transfer properties for each substrate separately in the absence of the enzyme reaction and then, with both substrates present simultaneously, evaluate the kinetic parameters of the immobilized enzyme. We reported the use of a membrane-covered rotated disk electrode to quantitatively characterize diffusion and reaction properties of collagen membranes containing immobilized glucose oxidase and catalase (17). The complete mass transfer properties for glucose and oxygen were determined by techniques based on the membrane-covered rotated disk electrode (21-24). The only remaining parameters, the intrinsic kinetic parameters of the immobilized enzyme V&=and Klg, were estimated by using the model and a nonlinear least-squares fitting of experimental results (13). This approach to the estimation of kinetic parameters of immobilized enzymes eliminates the need to employ so-called apparent kinetic parameters (25), in which unresolved mass transfer resistances are included in the kinetic terms and is a direct extension of nonlinear regression techniques used for soluble enzymes (26)to these differential equation models. More details of membrane fabrication and parameter estimation have been given previously
COB
12.0
0
(31)
24.0
I
n
-
36.0 20.0
ap= I 79
/-
- 12.0
0.6 -
ln
\
.3
0.4 -
c , ~ = 0 . 2 4rnM
-
0"
8.0
2
-.-
0 c0~=0.16mM
A cO~=O09mM
= 0.076 Kh = 0,005 Big = 65.5 B,,= 19.2
m
K
0.2 -
OO2
X g
2-
'0 1 0
,
0.4
1.2
0.8
1.6
- 4.0
2.0 0
1lEo' Flgure 4. Experimental results. Normalized dimensionless difference current plotted on the left axis as a function of effective glucose concentration on the lower axis or current density on the right axis as a function of the glucose concentration on the upper axis, both normalized by oxygen concentration. Current plotted as a function of effective glucose concentration for two membranes of different enzyme loading.
measurements made with a given membrane at different oxygen concentrations to be displayed on the same curve, thus focusing attention on the response to glucose. This approach also suggests convenient algorithmic methods for determining glucose concentration from the observed signals at different oxygen concentrations. Estimation of all three kinetic parameters of the Ping-Pong reaction rate expression for the immobilized enzyme was not possible from these data (13), although analogous kinetic parameters have been determined for the soluble enzyme (18). The full Ping-Pong reaction rate expression can be written in the following form:
(13).
Data reported elsewhere (17)for membranes of somewhat high immobilized enzyme loadings, along with previously unreported results for lower immobilized enzyme loadings are plotted in Figure 4. The left and lower axes are given in dimensionless parameters, according to the previous convention of normalized dimensionless current vs. effective glucose concentration, whereas the right and upper axes are given in terms of the corresponding directly determinable experimental parameters and variables for this specific experiment. The symbols correspond to solutions of different oxygen concentrations in the physiologic range prepared by equilibrating buffer with atmospheric, 67% atmospheric, and 38% atmospheric oxygen, respectively. The solid lines are fitted by using the nonlinear least-squares technique, from which the kinetic parameters were estimated. In the upper curve corresponding to the higher value of u,, the normalized dimensionless signal rises quasi-linearly with effective glucose concentration and turns to the horizontal. Although sensitive to glucose over only a limited range, the signal in this range can be conveniently approximated as linear with concentration. The lower curve is for a membrane with identical mass transfer properties that differs only in enzyme loading. The signal rises gradually and distinctly nonlinearly with effective glucose concentration but is sensitive to glucose concentration over a much broader range. The data are slightly less scattered in the upper curve, as might be expected for a system that is affected to a greater extent by diffusion. This figure demonstrates the utility of the normalization, which allows glucose
2901
VmaxCgCo
r(cg, co) =
c,(cg
+ K,) + KoCg
(32)
The value of Kgreported for the soluble enzyme at 38 "C is 0.125 M (18) and similar values have been described over a wide range of temperatures (18)and pH values (27,28). The data shown in Figure 4 were, however, taken over a bulk glucose concentration range of 0 to 5 mM. The use of eq 20 to adequately explain the data is justified by noting that over this concentration range Kg + cg = Kg (33) and therefore the product c,cg in the denominator of eq 32 can be neglected, leading to eq 20. The important glucose concentration here is the concentration in the vicinity of the enzyme, which in general is less than the bulk concentration and may be lower still when the sensor is operating in the diffusion-limited regime. It is important to note that others have shown that eq 20 can adequately describe the kinetics of soluble glucose oxidase when studied over a similar concentration range of oxygen and glucose used in these data (29). A simple test of the validity of this assumption can be made by examination of the dimensionless parameters. If valid, values of ig/cOBvs. cgB/cOB for various oxygen and glucose bulk concentrations lie on a single curve for a given membrane. This feature should also apply to two-substrate enzyme electrodes designed to detect the product of the enzymatic reaction potentiometrically or amperometrically. Such a response is, however, unique to the reaction rate expression
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assumed in eq 20 and is not expected to be observed when more general reaction rate expressions are used. The use of eq 29-31 to identify the limiting substrate in diffusion-limited membranes is also demonstrated in Figure 4. The effective substrate concentration ratio that equals Big/Bi, and for which ug = urnin the higher loaded membrane is indicated. These values can be useful in the following way. The effective substrate concentration ratio where up = u,, for this membrane, as calculated by eq 31, is indicated in Figure 4. Good agreement between the prediction of eq 31 and the experimentally determined range of detectability for this membrane is seen. This is mildly surprising since the value of ug is not very large, a prerequisite for the application of eq 31. This result suggests that this membrane is operating in an essentially diffusion-limited mode. Increasing the value of ug by incorporating higher enzyme loading would move the response curve to the left reducing the range of sensitivity to glucose, asymptotically reaching the point at which Bi,/Bi, = Eo*, where further increases would have little effect. Sensor Design. The necessity to account for enzyme decay in sensor design suggests that diffusion-limited operation is most desirable. If it is assumed that the mass transfer parameters are identical for both substrates (Big = Bi,), diffugion-limited operation can be simply expressed by eq 26 and 27. Under these conditions the dimensionless difference current is a function of a single parameter E,*. Changes in the range of glucose detectability can be made only by changes in the values of the quantities that make up this dimensionless parameter. This analysis thus suggests several potential design strategies for extending the range of detectability. For example, immobilization of excess catalase gives a stoichiometry coefficient v of one-half;the difference between excess catalase and no catalase corresponds to a 2-fold difference in the detectable glucose range. An increase in the membrane permeability ratio CYJI,/CY,D,also increases the detectable range in direct proportion. Specification of this parameter may be achieved by employing novel materials for immobilization of the enzymes with the appropriate differential permeability to the substrates, that is, a reduction in glucose permeability, while retaining oxygen permeability. Altering membrane thickness, however, is not useful in determining the limiting substrate. When significant differences in mass transfer parameters exist, appending a noncatalytic membrane of preferentially restricted glucose permeability adjacent to the enzyme membrane may also extend the range of glucose detectability. Incorporation of this additional membrane into the present model simply changes the values of the mass transport parameters hi. Equation 29 gives an indication of the desired properties of the additional membrane that would result in glucose limitation even when E,* < 1. The requirements are that hg
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