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Anal. Chem. 1980, 52, 977-980
CORRESPONDENCE Theoretical Evaluation of the Steady-State Response of Potentiometric Enzyme Electrodes Sir: The application of enzyme electrodes to analytical problems has seen steady growth over the past few years. Rechnitz et al. have recently developed electrodes based on the use of whole bacterial cells and tissue sections ( I , 2 ) . A general view of the construction and properties of enzyme electrodes has appeared recently ( 3 ) . Despite a n extensive literature concerning the preparation and the applications of enzyme electrodes, only a limited number of theoretical studies have been attempted. This is mainly due to the nonlinear nature of the partial differential equations which govern the behavior of these devices. Theoretical studies published thus far have derived closed-form solutions to the limiting kinetic cases ( 4 ) , digitally simulated the response of amperometric ( 5 ) and potentiometric (6) enzyme eleclrodes, and modeled t h e transient response of potentiometric enzyme electrodes under limiting kinetic conditions using Fourier analysis (7). T h e present study was motivated by the previous reports on the properties of potentiometric enzyme electrodes. Of particular interest to this work were statements that potentiometric electrodes displayed linear response to bulk substrate concentrations only up to approximately one tenth the Michaelis constant, KM, and that the range of linearity was essentially independent of the amount of enzyme in the layer surrounding the sensor. T h e requirement for linear response from a potentiometric probe is that the relative depletion of substrate a t the sensor surface be independent of bulk substrate concentration. This result should be contrasted with a detailed study of the response of amperometric enzyme electrodes which found linear response to bulk substrate concentrations in excess of KM( 5 ) . An amperometric probe will produce a current which is linearly related to bulk substrate concentration provided the current limiting process is first order. Thus, an amperometric probe will show linear response if diffusional mass transport of substrate, which is always first order, or some other first-order process, such as an enzymatic reaction operating at substrate concentrations below K M ,is current limiting. This appears to be a more rigorouis requirement than that for the potentiometric probe. On this basis one expects that the response of potentiometric probes should be linear to bulk substrate concentrations in excess of ICM, and that the upper limit of linearity should exceed that seen with the amperometric probe for the same enzymatic system. Detailed treatments of the mass transport equation that results from the enzyme electrode, assuming MichaelisMenton kinetics, have been frustrated by the inefficiency of the numerical methods suitable for the resulting nonlinear differential equation. Numerical solutions of complicated heat and mass transfer problems have been successfully and efficiently accomplished by the method of orthogonal collocation (8-13). Noting the power this method has shown in problems of a similar and more complex nature, a simulation of the steady-state response of the potentiometric enzyme electrode was attempted here in order to better establish the theoretical, steady-state behavior of enzyme electrodes.
of Fick’s second law of diffusion and a chemical reaction obeying Michaelis-Menton kinetics. The chemical system in question is
whose rate in homogeneous solution is assumed to be governed by the equation
where Cs is the substrate concentration (mole/cm3), k2[ElO is the enzyme activity (~mol/s-cm3), and KM is the Michaelis constant ((k, k - J / k J (mole/cm3). Incorporation of a diffusional mass transport term results in the net equation governing the rate of change of substrate concentration within any portion of the enzyme containing membrane surrounding the sensor. Thus
+
(3) where Dsis the substrate diffusion coefficient in the membrane (cm2/s). Under steady-state conditions, the time derivation is zero, leaving a nonlinear second-order ordinary differential equation. Solution of any second-order differential equation requires the imposition of two additional constraints. The boundary conditions used in this study were
(4)
CsIx=L = Cso (bulk substrate concentration)
cs
x
MODEL T h e model used in this study consisted of a superposition 0003-2700/80/0352-0977$01 .OO/O
(5)
For simplicity the diffusion coefficients of the product and substrate were assumed equal, and a bulk product concentration of zero was specified. The system as described is pictured in Figure 1. Equation 4 states that the sensor surface is impenetrable to both substrate and product, while Equation 5 maintains that the mass transfer of substrate to the outer surface of the membrane from the bulk solution is not rate limiting and the substrate does not undergo selective partitioning into the membrane phase. At this point it is wise to cast Equation 3 into a dimensionless form so the net factors which influence the system response become apparent. The dependent and independent variables are the substrate concentration, Cs, and distance, X,respectively. A natural choice for the reduced variables is to define distance in terms of the total membrane thickness and concentration in terms of bulk substrate concentration. This change of variables results in the dimensionless equations
C 1980 American Chemical Society
=
c,/c;
=X/L
(6)
(7)
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ANALYTICAL CHEMISTRY, VOL. 52, NO. 6, MAY 1980
T h e system response thus depends on two factors, cy and KIM. Note that a is a property of a given enzyme electrode membrane, while EMwill depend on the enzyme used and the bulk substrate concentration. Of course, one must recognize that a bound enzyme may not have the same numerical values of its properties (KM,k 2 , etc.) as the soluble system.
SIMULATION Application of orthogonal collocation to Equation 10, with boundary conditions specified by Equations 4 and 5 , will generate a set of nonlinear algebraic equations which may be solved for substrate (or product) concentrations a t the collocation points. Solution of such systems of equations is generally attempted using Newton’s method (14) or one of the quasi-Newton methods (15, 16). Rapid convergence of the iterative scheme makes Newton’s method quite attractive. Unfortunately, the method is only locaUy convergent. The requirement for a good initial guess, and t h e belief that the Jacobian matrix of the system of equations might become computationally singular as the solution is approached, led to consideration of possible alternative methods. Results of the study of two-dimensional problems indicated the two-point secant model (14) would be the most promising numerical method. Implementation of the secant method in more than two dimensions requires a cautious approach. Orthogonal collocation is a global method, that is, all function values (concentrations) are explicitly dependent on all the other function values. This is a major advantage of the technique in being able to generate an accurate numerical solution to a given problem. However, since the equations generated are tightly coupled to one another, the iteration must be pursued in a rather conservative manner. This is especially true when the equations are nonlinear. It was nonetheless possible to write rather efficient programs which solved Equations 3-5. Details are available upon request. Solution of Equation 10 was carried out for enzyme loading factors (CY) of 0.5 to 100 and bulk substrate concentrations of to 100 KM. Convergence to a final solution was found to be independent of the initial guess used for t,he cases tested. In all cases, the final values of concentrations and their distance dependence conformed with our physical intuition of what should occur. Some representative product concentration profiles are shown in Figure 2. Those profiles obtained when first-order kinetics held within the entire membrane layer (Cos > KM)displayed analogous quantitative agreement with the relevant analytic solution ( 4 ) . Thus, the limiting potential at Cos >> KMwas found to be the same from both the numerical solution developed here and from the analytic solution to the zero-order kinetics differential equation. Knowledge of the concentration of product a t the sensor surface allows one to construct response calibration curves for the sensor. The previously known analytic solutions to the two limiting cases allow for calculation of the two extremes of the calibration curve. In light of this, the ultimate purpose of the present work was to precisely define the transition portion of the response curves. More specifically, the intent was to determine the concentration a t which nonlinear response starts, the concentration where the response becomes independent of the bulk substrate concentration, as well as the range over which the transition from Nernstian response
SENSOR
K MEMBRANE
x: 0
cp:0
x: L
Figure 1. Schematic of enzyme electrode model. Enzyme containing layer extends from X = 0 to X = L . Bulk solution begins at X = L . Product and substrate profiles drawn to reflect boundary conditions used in the simulation
rce .4
E
\\ -
X Figure 2. Product concentration profiles. Numbers shown denote K, value for the particular curve. All curves were computed with LY = 20
0-
- 59 -10 0
s
-30
-25
-w
-Is
-to
-5
LOG
e
0
5
Io
u
w
Figure 3. Calibration curve calculated from simulated concentration profiles. N values shown with respective curves. Concentrations are in units of K, to saturation occurs, and how these three characteristics were affected by the enzyme loading factor and the Michaelis constant. The simulated concentration profiles were used to generate potentiometric calibration curves, i.e. plots of the logarithm of the surface product concentration vs. the logarithm of the bulk substrate concentration for the various enzyme loading factors (see Figure 3). It is evident that even at low enzyme loading factors (CY < 1.0) the electrode response will be Nernstian, provided that the bulk substrate concentration is significantly less than KM. Indeed, the curves in Figure 3 indicate that even when the surface product concentration is substantially less than the bulk substrate concentration,
ANALYTICAL CHEMISTRY, VOL. 52, NO. 6, MAY 1980
979
"r
65
-
30
-
t.
1
-Lo
-.5
0
5
LOG
l),the limiting bulk substrate concentration is governed by the relationship
which is obtained as a t least squares fit to t h e data from ci = 3 to a = 100. Combining Equation 3 and the definition of enzyme activity (It?), one obtains an equation for the minimum required enzyme activity per unit volume of membrane a t a specified limiting bulk substrate concentration
where CLSis the limiting substrate concentration in molarity and the other symbols are as previously defined. T h e expression shows a positive fifth root dependence on KM,a result which is superficially surprising when one considers that it
OO
I x) .25
I
I
.os
.lo
15
Z Figure 5. Functional dependence of the limiting Michaelis constant on the extent of deviation from linearity. The data are the slopes of the linear sections of the data presented in Figure 4 for the indicated deviation from linearity. The plot resutts from a generalization of Equation 12 to E = AKMYwith Ziog,, units deviation from linearity defining this relation. The value of A depends on the Zvalue chosen, D s , and L 2
Table I. Required Enzyme Concentrations for Prescribed Limiting Substrate Concentrations'
Cis, mM 100. 10. 1.0 0.1
1.0
KM, mM 0.1
1035 (667)b
164 (66.7)b 26 (6.67)b 4.1 (0.67)b
6 53 103
16.4 2.6
0.01
164 65.3 10.3
1.6
a These are based on Equation 12. A substrate diffusion coefficient of cm*/s,L = 100 pm, 0.05 log,, units deviation from linearity. The amount of enzyme is in fimol/min.cm3. Value in parentheses corresponds to deviation of 0.25 log,, units and is independent of K M .
states that a t a given enzyme activity the enzyme with the lower K Mwill display a slightly higher limiting bulk substrate concentration. However, one should realize that when soluble enzymes are used the amount of enzyme or the time required to convert a given fraction of sample under pure first-order conditions is proportional to the KM. In the case of zero-order enzyme kinetics, the amount of enzyme (or the time) needed to achieve any desired fractional conversion is independent of KM. Thus, in this case, we should expect some positive dependence of the amount of required enzyme activity upon the K M . The important specification to note here is that of equivalent activity. The dependence of Equation 12 on K M with the constraint defining deviation from linearity was checked by determining the upper limit of substrate concentration at varying amounts of deviation from Nernstian behavior. A plot of this dependence against the deviation allowed is shown in Figure 5. Figure 5 shows that the relationship between the required enzyme activity and K M depends on how the limit of linearity is defined, and that a KM-independentequation results when a deviation from linearity of 0.25 log,, units or greater is allowed. These results are consistent with the behavior of a homogeneous enzyme system. Equation 12 was used to estimate the required enzyme loading to achieve an upper limit of linearity. Results obtained are shown in Table I. T h e loadings indicated are in many cases experimentally achievable and indicate that extension
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Anal. Chem. 1980, 52, 980-982
of the range of enzyme electrodes to substrate concentrations in excess of K M is possible. Table I indicates that to achieve deviation from linearity of less than 0.05 loglo units a t bulk substrate concentrations equal to 100 times the Michaelis constant, over 1000 enzyme units/cm3 are required for Khl < 1 mM. Such a n electrode should be linear to nearly 100 mM. Enzyme electrodes with activities of 50-100 units/cm3 are more realistic. We expect that linearity to 10 mM for KM equal to 1 m M can be achieved. One should note that the required amount of enzyme decreases as the diffusivity of the substrate decreases. This implies t h a t use of a highly cross-linked gel to support the enzyme might be desirable. But, as we have shown previously (7), electrode response time will increase in proportion to the decrease in diffusivity. Thus there is a trade-off between linearity and response time. The same is true of the thickness of the enzyme layer, i.e., a thick layer promotes linearity and increases the response time. In summary, i t has been shown that the linear dynamic range of potentiometric enzyme electrodes can be extended beyond KM when the enzyme immobilized in the membrane surrounding the sensor is of sufficiently high activity. In a practical sense, this illustrates the point that only enzymes capable of being immobilized while retaining a high specific activity hold promise for use in routine applications. Furthermore, since stabilization of the immobilized enzyme is still a problem, significant deterioration of the linear dynamic range of the sensor with time is implied. This suggests the need for frequent recalibration and assessment of the time dependent of the sensor's dynamic range.
LITERATURE CITED Rechnitz, G. A.; Kobos, R. K.:Riechei, S . J.; Gebauer. C. R. Anal. Chim. Acta 1977, 9 4 , 357. D'Orazio, P.; Meyerhoff, M. E.; Rechnitz, 6. A. Anal. Chem. 1978, 50, 1531. Guilbautt, G. G. I n "Comprehensive Analytical Chemistry", Vol. 8, Svehla, G., Ed.; Elsevier: Amsterdam, 1977. Blaedel, W. J.: Kissei, T. R.; Boguslaski, R. C. Anal. Chem. 1972, 4 4 , 2030. Mell, L. D.: Maloy, J. T. Anal. Chem. 1975, 47, 299. Tran-Minh, C.: Brown, G. Anal. Chem. 1975, 4 7 , 1359 Carr, P. W. Anal. Chem. 1977, 49, 799. Villadsen, J. V.; Stewart, W. E. Chem. Eng. Sci. 1967, 22, 1483. Karanth, N. G.; Hughes, R. Chem. Eng. Sci. 1974, 2 9 , 197. Ferouson. N. B.: Finlavson. B. A. Chem. €no. J . 1970. 1 . 327. Serih, R.'W. Int. J . Num. Math. Eng. 1975: 9 , 691. Whiting, L. F.; Carr, P. W. Anal. Chem. 1978. 50, 1997. Whiting, L. F.; Carr. P. W. J . €lectroana/. Chem. 1977, 8 1 , 1. Johnson, L. W.; Ries, R . D. "Numerical Analysis"; Addison-Wesley: Reading, Mass., 1977. Dennis, J. E. P Mor& J. E. SIAM Rev. 1977, 19, 46. Brent, R. P. SIAM J . Num. Anal, 1973, 10, 327. Bowers, L. D.; Carr, P. W. Anal. Chem. 1976, 48, 544A. Guilbault. G. G. "Handbook of Enzymatic Methods of Analysis"; Marcel Dekker: New York, 1977.
James E. Brady P e t e r W. Cam*
Department of Chemistry University of Minnesota Minneapolis, Minnesota 55455
RECEIVED for review September 17,1979. Accepted February 4,1980. This project was partially supported by the University of Minnesota Computer Center and a grant from the National Science Foundation (CHE 78-17321).
Test for Dehydrogenation in Gas Chromatography-Mass Spectrometry Systems Sir: During a recent synthesis of 2-hexyl-5-pentylpyrrolidine (a), the pheromone of the fire ants Solenopsis molesta and S. texanas ( I ) , we had occasion to compare the gas chromatograph/mass spectrum of our product (Figure 1) with that of the natural product run earlier (Figure 2). Gross differences are apparent that were traced to the presence in t h e former of overlapping spectra of the related pyrrolines b and c. Homologues of b and c had been encountered earlier
.
a
c
b
Hw R
11
111
as natural products in S. punctaticeps ( 2 ) ;their spectra are characterized by an abundant m / z 82 ion (iii) arising from &-cleavageof the rearrangement ion (ii). However, there were no such impurities in the synthetic sample of a. Under the 0003-2700/80/0352-0980$01.00/0
gas chromatographic conditions employed here [2-m, 3% SP-1000 (stabilized Carbowax) packed column, programmed 10"/min, see Refs. 1 and 2 for details], the pyrrolines separate easily from the pyrrolidines and even trace quantities would be easily detected. Clearly, dehydrogenation of a was occurring either in the ion chamber, jet separator, or the intervening valve of our system (LKB-9000). As expected, the spectra were not highly reproducible, more serious degradation being encountered a t low levels of samples. A sample admitted by direct insertion probe produced an excellent spectrum (Figure 3), absolving the ion chamber as the source of trouble. On the other hand, the jet separator and following valve had been replaced recently; earlier spectra were run on a system that had been in continuous operation for many years. Indeed, the spectrum of a improved considerably after a few weeks of further use of the mass spectrometer (Figure 4 ) ; several months later i t was indistinguishable from that obtained earlier with the natural product (Figure 2). We have observed that once this stage is reached, the extent of dehydrogenation no longer increases as the sample size is diminished to the nanogram level and the spectra become entirely reproducible. Sherman has observed a related loss of sample at low levels after installing a valve that prevents column bleed from entering the jet separator when the system is not in use (3). He suggests, and we agree, that a continual flow of column bleed (usually siloxane polymer) appears to be important in mainC 1980 American Chemical Society
