in 86c using the solution doping technique with 85b as a comparative standard. The average deviation was 12% but if the analysis for the Cu major constituent is ignored, the deviation drops to about 4 %. The real test of the technique is the ability to use the synthetic, matrix-free standards for the analysis of impurities from a variety of matrices. This experiment was performed by analyzing several elements in NBS lOle stainless steel alloy, NBS 462 low alloy steel, and NBS 86c aluminum alloy using the solution doped silver powder standards. Table VI indicates that quite reasonable analyses can be performed when a comparative standard is not available. The average deviation between the spark source analysis and the NBS certified values is approximately 10%. This average deviation is reasonable when compared to the results of Evans et al. (2) who found an average deviation of =t7% for several elements in iron alloys using a similar matrix standard. These results indicate that a rather simple chemical procedure can be used to obtain quantitative analyses even when a comparative matrix is not available. At first examination the limit of detection using solution doping should not differ from that generally obtained in the analysis of a compacted powder.
However, the presence of anionic species from the dissolution process can lead to the molecular ion interferences and a consequent reduction in detection limits. These interferences were not found serious in these studies. However, some difficulties were encountered during the analysis of trace species in sea water due to the formation of numerous molecular ions which precluded the determination of several elements of interest. Obviously this technique has several disadvantages (for example, sample loss or contamination) but it can certainly provide more quantitative results than when a relative sensitivity factor is assumed. ACKNOWLEDGMENT
The authors wish to thank F. D. Leipziger and B. N. Colby for their editorial comments during the preparation of this manuscript. RECEIVED for review February 25, 1972. Accepted May 18, 1972. The contributions of C. A. Evans, Jr., were supported by the Advanced Research Projects Agency under Contract HC 15-67-C-0221.
Kinetic Behavior of Enzymes Immobilized in Artificial Membranes W. J. Blaedel and T. R. Kissel Department of Chemisfry, University of Wisconsin, Madison, Wis. 53706
R. C . Boguslaski Ames Research Laboratories, Miles Laboratories, Inc., Elkhart, Ind. 46514
Steady state flux and distribution equations are presented to show how enzymes fixed in gels may be used for analysis or for the study of immobilized enzyme kinetics. Experimental support of the equations has been obtained with urease in three systems: a membrane-covered sensor, a membrane separating two solutions. and a membrane immersed in a solution. The relative merits of the three systems for analysis and for the determination of rate constants for the immobilized enzyme are examined.
IMMOBILIZED ENZYMES have assumed great importance in both theoretical and applied work. Interest has run high in their use as models for cellular enzyme systems. Recently, their potential for industrial processing and for therapy has become apparent. The properties of k e d enzymes that are of particular advantage to these applications are their stability and their insolubility, which permit easy removal and/or reuse after reaction. Weetall has summarized these advantages in a short review on the preparation, properties, and applications of enzymes immobilized on inorganic carriers ( I ) . The usefulness of immobilized enzymes in analytical devices has already been firmly established. Some typical analytical applications include : glucose assay with glucose oxidase entrapped in a polyacrylamide gel, used in conjunction with an oxygen electrode (2); peroxide assay through its oxidation of benzidine, catalyzed by peroxidase bound to (1) H. H. Weetall, Res. Develop., 22 (12), 18 (1971). (2) S. J. Updike and G . P. Hicks, Nature, 214,986 (1967). 2030
carboxymethylcellulose (3); assay of organophosphorus insecticides through their inhibition of the hydrolysis of a fluorogenic ester with cholinesterase immobilized in a starch gel (4); urea assay with urease in a polyacrylamide gel covering a cation selective electrode (5); L-amino acid assay with L-amino acid oxidase held in a Nylon net covering a cation selective electrode (6); amygdalin with P-glucosidase in a polyacrylamide membrane covering a cyanide ionselective electrode (7). For automated glucose determination in a flowing system, glucose oxidase has been bound to the inner surface of a chemically modified polystyrene tube (8). Urease has been similarly attached to Nylon tubes (9). All of this work justifies an experimentally supported theoretical treatment of the kinetic behavior of these systems in order to obtain fundamental information on the kinetic parameters of the immobilized enzyme. Lilly and coworkers found that the rate of hydrolysis of o-nitrophenyl galactoside solution passed through a porous sheet of DEAE-cellulose containing P-galactosidase was flow (3) H. H. Weetall and N. Weliky, Anal. Biochem., 14,160 (1966). (4) M. H. Sadar, S. S. Kuan, and G. G. Guilbault, ANAL.CHEM., 42,1770 (1970). (5) . . G . G . Guilbault and J. G. Montalvo, Jr., J. Amer. Chem. SOC., 92,2533 (1970). 16) 42, 1779 . , G . G. Guilbault and E. Hrabankova, ANAL.CHEM., (1970). (7) R. A. Llenado and G. A. Rechnitz, ibid.,43, 1457 (1971). (8) W. E. Hornby, H. Filippersson, and A. McDonald, Fed. Eur. Biochem. SOC.Lett. 9 ( l ) , 8 (1970). (9) P. V. Sundaram and W. E. Hornby, ibid.,10 ( 5 ) , 325 (1970).
ANALYTICAL CHEMISTRY, VOL. 44, NO. 12, OCTOBER 1972
dependent (IO). The kinetic behavior of packed beds of ficin-CM-cellulose particles was also flow-dependent (11). These workers presented an equation accounting for diffusion and electrical potential gradients in a Nernst-type of diffusion layer, and showed it to be of a form that agreed with experimental behavior for ficin and ATP-creatine phosphotransferase attached to CM-cellulose (12). Katchalski and coworkers have published a n extensive mathematical treatment of substrate and product distribution in membranes containing enzymes (13). This treatment has been modified to account for diffusion through the unstirred liquid films adjacent to the membrane, and it was found experimentally that liquid film diffusion did indeed affect the apparent rate constants of the fixed enzyme in aikaline phosphatase-collodion membranes (14). Laidler et al. derived equations describing the kinetics of reaction in an enzyme-containing membrane immersed in a substrate solution, and took into consideration the partitioning of substrate between the liquid and membrane phases (15). SClCgny and coworkers studied the transport of substrate glucose from one solution to another through a glucose oxidase membrane, and concluded that the Michaelis constant for the fixed enzyme was about the same as for the soluble enzyme (16). Kasche and coworkers presented a model and equations describing steady-state catalysis by an enzyme immobilized in spherical gel particles, and showed that catalysis by the bound enzyme at low substrate concentrations differs markedly from catalysis by the unbound enzyme (17). The quantitative characterization of fixed enzyme systems is difficult because the enzyme-substrate reaction is complicated by interphase diffusion and mass transport of both reactants and products. Quantitative attempts to treat these multiphase systems rigorously or generally often led to equations that cannot be solved explicitly for the parameters of interest, or that cannot be easily interpreted or tested experimentally. The complexity has sometimes been evaded by the use of “apparent” rate constants, relating over-all reaction velocity to substrate concentration in the aqueous phase. Such apparent rate constants may include all of the above-mentioned chemical and geometrical parameters in an undefined way, and are of little use in characterization. In this paper, equations are derived for the steady state fluxes of substrate and product through a membrane in simple systems. Transport through the liquid and membrane phases is considered. Some of the equations are explicitly solvable for the fluxes, or for the substrate or product concentrations, or for the rate constants that characterize the immobilized enzyme. These equations are tested experimentally. THEORY
General flux equations of wide applicability are not sought. Instead, only simple, idealized systems are considered which (10) A. K. Sharp, G. Kay, and M. D Lilly, Biotechnol. Bioeng., 11, 363 (1969). (11) M . D . Lilly and W. E. Hornby, Biochem. J., 100,718 (1966). (12) W. E. Hornby, M. D. Lilly, and E. M. Crook, ibid., 107,669 (1968). (13) R. Goldman, 0. Kedem, and E. Katchalski, Biochem., 7 , 4518 (1968). (14) Zbid., 10, 165 (1971). (15) P. V. Sundaram, A. Tweedale, and K. J. Laidler, Can. J. Chem., 48, 1498 (1970). (16) E. SClCgny, G. Broun, J. Geffroy, and D. Thomas, J. Chim. Phys., 66, 391 (1969). (17) V. Kasche, H. Lundquist, R. Bergman, and R. Axen, Biochem. Biophys. Res. Commnn., 45,615 (1971).
yield explicit solutions, which can be tested experimentally, and which show analytical promise. Nature of the Enzyme Reaction. It is assumed that the reaction follows simple Michaelis-Menten kinetics:
vs
v=-
K + S
In Equation 1, v is the reaction rate as measured by the rate of disappearance of substrate, S. V is the maximum rate when the enzyme is saturated with respect to substrate, and K is the Michaelis constant, equal to that substrate concentration for which the rate is equal to V/2. When the over-all rate of reaction is limited by the first order rate of decomposition of the enzyme-substrate complex (ES),and when this complex is formed rapidly and reversibly from the enzyme ( E ) and substrate (S), the theoretical over-all rate expression has the same form as Equation l :
E
+Se ES k? k1
- dS dt
1 dP
n dt
-
ka ----f
E -t nP
kaES
ka
+ k2 + S
(3)
ki
E, S, and P represent enzyme, substrate, and product, respectively, when they appear in chemical equations, and total molar concentrations when they appear in mathematical equations. Activity effects are disregarded. Lower case k’s are rate constants. For most enzyme catalyzed reactions, the over-all rates are accurately described by Equations 1 and 3, even though the detailed rate mechanisms are usually more complicated than that of Equation 2. Despite their simplicity, the over-all rate constants are of great practical use in describing and characterizing enzymes. For finding rate constants, or for determining concentrations from reaction rate or flux measurements, the limiting cases are of most interest :
- dS - -
1 dP - V
-. .-
dt
n dt
dS dt
1 dP n dt
V
ks
- - = - c =
(high substrate, S >> K )
(4)
(low substrate, S R. Fluxes of 3 and P are caused by diffusion and by the occurrence of the enzyme-catalyzed reaction. Material balances on 8 and p yield simple and easily-solved differential equations:
where 62 =
P
=
7 RD, ~
F1, -
(7)
To find the fluxes ( J , and J,) of S and P in the membrane, gradients of S and P are formed by differentiating Equations 19 and 21 with respect to 2 . Evaluating the gradients at the surfaces (2 = 0 and 3 = 2)gives the fluxes at the surfaces: (9)
To find the fluxes ( J , and J,) of
8 and P in the membrane
film,gradients of 8 and P are found by differentiating Equations 8 and 9 with respect to 3. Evaluating the gradients at the surfaces ( 2 = 0 and = 2) gives the fluxes at the film surfaces.
J,,,
=
-D,
(g)
i=O
2032
-
-
E&2,
-
PI,
+ ----=--
nBaz) 20,
ANALYTICAL CHEMISTRY, VOL. 44, NO. 12, OCTOBER 1972
As in the high substrate case, the fluxes of 8 and P are not independent: at each point within the membrane, they are related through the stoichiometry of the enzyme-catalyzed reaction, the relationship being expressed in Equation 16. Effect of Mass Transport through the Liquid Films. The membrane surface concentrations in the flux equations (12-15 and 24-27) are influenced by transport through the adjacent liquid films. The liquid film conditions are highly different for the sensor and the two-solution systems, and it is necessary to define them carefully. For measurement purposes, in the sensor system, we must solve the flux equations for and PZm,which the sensor measures. In the two-solution system, we must solve the flux equations for Si, P I , SZ,and Pz, which can be measured by analysis of the solutions. Additional relationships are needed to solve the flux equations for these concentrations. If it is assumed that equilibrium exists at the surfaces at steady state, then there can be no net accumulation or transport of S and P at the surfaces :
Steady State Concentrations in the Sensor System. For high substrate (8 >> E), Equations 12-15 and 28-30 give
B,,
= 6,Sl
&VX 2 -- 2
(t.
+6
3
For low substrate (8 > 8. The eight unmeasurable surface concentrations may be eliminated from Equations 12-15 and 28-30, giving the fluxes at the surfaces in terms of composition and other fundamental parameters which describe the system:
(35)
(37)
-+tP
I
To describe transport through liquid films, the popular Nernst boundary layer theory assumes a laminar film with diffusion through a linear concentration gradient. When solutions are stirred, however, at least a part of the boundary layer involves a transition through turbulence to the laminar condition that exists close to the surface. In such cases, it is preferable to express the transport in terms of an empirical mass transport coefficient, t (18, p 50). For the two-solution system, if the geometries and stirring regimes are the same for both solutions, and if both solutions have the same composition with respect to supporting electrolyte, a particular solute species would have the same mass transport coefficient in both solutions. In terms of transport coefficients, substrate and product fluxes through the solution boundary layers become, for the two-solution system:
- Si) J , zm = - ts(Sz - SZ,) Jp i m = -tp(Pim - Pi) J p 2m = - tp(P2 - Pzm) Js
im =
6PDP
In the two-solution system, the experimental method of measuring steady-state fluxes is to pass Solutions 1 and 2 by the membrane surfaces at constant flow rates, F1 and Fz, respectively. The only source of substrate is Solution 1, containing substrate at an entering concentration Si. The steady-state fluxes may therefore be calculated from the steady-state concentrations and substituted into Equations 35-38, giving (St
Fi
Si - Sz
- Si)AT = 2 - + ts
78
x + l w,
(39)
-t,s(Sim
(Zero for sensor system) (36) (Zero for sensor system)
(18) R. E. Treybal, “Mass Transfer Operations,” 2nd ed., McGrawHill, New York, N.Y., 1968. ANALYTICAL CHEMISTRY, VOL. 44, NO. 12, OCTOBER 1972
2033
Steady State Fluxes and Concentrations in the Two-Solution System. Low Substrate: 8
