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Optimization of the Coupled Enzymatic Measurement of Substrate J. E. Davis" and Jeff Pevnick' Division of Laboratory Medicine, Department of Pathology and Medicine, Washington University and Barnes Hospital, St. Louis, Missouri 63 110
I n the kinetic analysis of substrate, factors which affect the enzymatic activity will affect the estimate of the amount of substrate. Likewise variations in the activity of coupling enzymes will affect the estimate. By proper selection of the time at which to measure the rate, variations in enzymatic activities will cause only minor variations in the rate, contrary to the case where measurements of the initial rate are made. Optimization theory is developed for variations in the activity of the primary or the coupling or both enzymes. Four methods are presented for estimating the kinetic parameters used in the optimization.
Measurement of the initial rate of reaction in an enzyme catalyzed assay for substrate is not necessarily the best rate measurement. Atwood and DiCesare ( I ) present a technique for optimization which measures the rate a t a time when approximately 63% of the substrate has been consumed. That rate measurement exhibits a n impressive resistance to variations in activity of the enzyme with a resultant improvement in methodological precision, and obviation of the need for making early absorbance measurements. While their theory is developed for simple first-order reactions, some of the methods involved one or more coupled reactions, with lag phases t h a t were significant fractions of the time from triggering to measurement. Although they did not make a rigorous analysis of t h e effects of lag phase resulting from coupling reactions, they did offer an explanation which approximated the effect as a fixed time delay. Bergmeyer ( 2 ) has investigated the kinetics of coupled reactions. T h e case which is of interest for the present discussion is that of consecutive irreversible first-order reactions. When the end-product is measured and the rate constants are comparable, i.e., neither step is rate limiting, a lag phase is evident in the reaction curve. Following the lag phase is the straightest portion of the reaction curve which corresponds to the maximum velocity of the reaction. This maximum velocity, which occurs a t the inflection point of the velocity curve, could be substantially different from the initial rate of reaction of the first step. Nevertheless, the maximum velocity is proportional to the initial concentration of the reactant and the time at the inflection point is independent of the initial concentration. Furthermore, the product of the time a t the inflection point and either rate constant is equal to a respective term which depends only on the ratio of the rate constants. In other words, the relative shape of the curve depends on the ratio of the rate constants, not their individual value. Tiffany et al. ( 3 ) also studied the case of two irreversible first- or pseudo-first-order reactions occurring in series. They demonstrate the linear relationship between the concentration and the average rate measured between any two fixed times. 'Present address: Missouri Institute of Psychiatry, 5400 Arsenal Street, St. Louis, Mo. 63139. 0003-2700/79/0351-0529$01 OO/O
While making no claims about the desirable amount of coupling enzyme or the lag phase, they d o state that the concentration of substrate must be less than 0.1 K , for both enzymes. In order to provide a firm theoretical foundation for the optimization of coupled enzymatic measurements of substrate, a kinetic analysis needs to be made for the case of two irreversible first-order reactions occurring in a series. The resulting equation will be equivalent to t h a t derived by Bergmeyer ( 2 ) , although it will be factored differently for purposes of emphasis. Subsequently, the technique of Atwood needs to be expanded to the case of coupled reactions. Thus our purpose is severalfold: (1) to derive a theory of optimization which includes the effects due to coupling enzymes, (2) to present methods for estimation of the primary and coupling rate constants, and (3) to demonstrate the improved resistance of the optimized assay with respect to variation in enzymatic activity. EXPERIMENTAL Reagents. Urease type 111, glutamate dehydrogenase type 11, 2-oxoglutarate, and NADH were from Sigma (St. Louis, Mo. 63118). All other chemicals were reagent grade. Deionized water was used throughout. Instrumentation. A Gemsaec (ENI, Fairfield, N.J. 07006) was used t o gather spectrophotometric data. Procedure. The conditions of the assay were: 60 rmol/L urea; 40 mmol/L Tris; 4.0 mmol/L EDTA; 2.0 mmol/L 2-oxoglutarate; 1400 IU/L urease, nominal; 20000 IU/L glutamate dehydrogenase, nominal; 200 pmol/L NADH. The reagent was adjusted with HCl to a pH of 7.8 at 30 "C before the addition of enzymes or NADH. THEORY Assume the reactions are irreversible first order so that; ki
kz
A-B-C where k l = V,,,,,/K,, for the primary enzyme and k 2 = VmaX2/Km2 for the coupling enzyme. Differential equations representing the above can be solved to give the rate of change or velocity ( v ) of the final product, C;
T h e term A, is the amount of substrate at time zero and the term A&, exp(-klt) is equal to the rate of change in substrate, A. The additional terms represent effects due to the coupling enzyme and intermediate product. R. Thus the rate of appearance of product is different from the rate of disappearance of substrate by a steady-state term in brackets and a transient term in braces. T h e customary technique of measuring the initial rate is not feasible in this case because the initial rate is zero, Le., there is a lag phase while the intermediate product builds up. An alternate technique chooses a region of the absorbance vs. time plot that approximates a straight line. The slope (dA/dt) in this region is equivalent to measuring the peak velocity where the rate of change of velocity and, hence, curvature of .s' 1979 American Chemical Society
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plotted in Figure 1 where it is seen, for example, that the optimum TI for a ratio of kz to k l of 4 is 1.3. Thus, if kl = 0.01 s-l, then t I = 1.3/0.01 or 130 s. It is apparent from Figure 1 that the optimum time with respect to k l for k z / k , greater than 10 is only slightly different from Atwood’s limiting case with a result of 1.0. I t is appropriate to inquire whether t h e optimization depends on the nature of the change in the rate constant, since K , will be affected by competitive inhibitors and V, will be affected by noncompetitive inhibitors as well as by dilutional and temperature variations. T h e optimization with respect to k l is valid for small changes in K , or V,, or both. Proof of this assertion is seen from the partial derivatives of k l :
k2’kl
Figure 1. Optimal time (dimensionless) of rate measurement. T,, T!I, and T,,, provide maximum suppression of errors d u e to variations in k,,k,, and k, and k, respectively. Tpprovides the peak velocity. These dimensionless times are to be divided by the rate constant, k,,to obtain
(4)
and noting that
the actual time the absorbance vs. time plot is smallest. However, Equation 2 reveals that the velocity is proportional to the substrate for any time greater than zero. The question can now be asked; is there a n optimum time a t which to measure the velocity where the greatest precision will be obtained. Of course the answer will depend on the source of the variations. Of concern here are variations in t h e activity of the enzymes. In the following, the derivative of the velocity shall be set equal to zero and the resulting equation shall be solved for the time. Because there are two variables, hl and k,, the derivative can be taken with respect to one or the other or an infinite number of combinations of the two. Several conditions are particularly useful, that representing variations in kl, that representing variations in k2, and that representing equal percentage changes in k l and k 2 . The first situation occurs, for example, when a variable amount of inhibitor for the primary enzyme is present in the sample or when the primary enzyme is unstable and loses activity from one assay to the next. Optimization with respect to variations in h i shall be called Type I conditions. Similarly, when the coupling enzyme is subject to inhibition or is unstable, then optimization with respect to variations in k 2 is appropriate. This is a Type I1 condition. Finally, a Type I11 condition shall denote equal percentage changes in k l and ha. This situation would arise with dilutional errors, assuming that the ratio of enzyme activities remains constant. Also, changes in temperature would affect the enzyme activities in approximately equal proportion. Thus the optimum time will depend on which kinds of variation are most important in a particular assay. Later on, we shall be able to comment on desirable ratios of the primary and coupling enzyme. T y p e I-Variations i n kl. The derivative, set to zero, of the velocity with respect to k l is :
(3) where t~ is the optimum time a t which to measure the rate of reaction. In the event k , is very large, Le., an excess of coupling enzyme, the last two terms of Equation 3 can be neglected and the resultant equation is equivalent to that obtained by Atwood ( I ) . When h2 is not particularly large, then Equation 3 must be solved in its entirety. An analytical solution for tI could not be found, so resort to a numerical solution was made utilizing Newton‘s method. In order to obtain a solution for the optimum time which was not dependent on the particular values of h l and h,, the equation was scaled using a parametric time, TI = hit,. The results are
and
Hence the derivatives are different only by a constant and would be equal to zero for the same value of time. This finding also applies to k,. T y p e 11-Variations i n k 2 . The derivative, set to zero, of the velocity with respect to k z is:
Newton’s method was again used to obtain the numerical solution for tII. The parametric time, TII= kltII is plotted in Figure 1. T y p e 111-Variations in k l a n d k 2(Dilutional). The derivative of the velocity in the most general form is:
for the particular case of dilutional errors, dk, = k , / k l dk,. Making this substitution into Equation 8 and setting d V = 0, gave an equation which was again solved by Newton’s method to find the optimum time. The parametric time in this case was TIII= kltIII. The result is plotted in Figure 1 where it is seen that TIIIis generally shorter than T I and for k 2 / k l > 5 , TIIIis approximately equal to Atwood’s limiting case with a result of 1.0. Error Surface. It is instructive to consider an error surface around some optimum time. For these purposes, the optimum ill be seen, this choice time was chosen as TI for k 2 / k l = 4. As w will amply illustrate the distinction between T I and 7’111 and further represents a not unlikely set of operating conditions, i.e., the observed velocity vs. time curve is characteristic of that for coupled reactions. The error surface is plotted in Figure 2 for selected levels of error. In the center of Figure 2 is the nominal operating point. I t can be seen that a &20% change in k l is required before the velocity measured a t time tI has changed (decreased) by 2 % . On the other hand only an 8% change in k 2 is required before the velocity is changed by 2 % . Note that a decrease in h2 results in an increased velocity. Because optimization was made with respect to variations in k l it is not surprising that the nominal operating conditions provide the greatest resistance to changes in h l , nevertheless the magnitude of insensitivity to such changes is impressive. Figure 2 also shows that a 7l/,% increase in k l and k , (un-
ANALYTICAL CHEMISTRY, VOL. 51, NO. 4,APRIL 1979
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Flgure 2. Error surface at T I for k , l k , = 4. See text for description
derdilution or temperature change) results in 2 % decrease in t h e velocity while a 32% decrement in k l and k 2 is required before a 2% change in velocity is incurred. This latter observation suggests t h a t kl and k 2 could be reduced approximately 15% so t h a t equivalent dilutional errors on the up side and t h e down side could be tolerated. Additionally, Figure 2 shows t h a t such a n operating point would be approximately parabolic with respect to dilutional errors, hence there would be a very small error in t h e velocity for small dilutional errors. However it is not really necessary to change t h e values of k l and k 2 , but merely t o measure the velocity at a different time, tIII. This fact can be appreciated by realizing that equal changes in k , and k2 are equivalent changes in t h e time scale. T h u s decreasing in k , and k 2 cause the velocity curve to evolve more slowly. Equivalent results would be found by decreasing the time where the velocity was measured on the velocity curve resulting from the original values of k l and kZ. This is consistent with the fact that tIII is less than tI. Measurement of k,and k,. In order to relate the theoretical curves of Figure 1 to a physical system, the rate constants k l and k Z or their equivalents must be determined by any of several approaches. Method A . Direct purposeful variation of the activity of t h e primary and/or coupling enzyme activity by 10 to 25% yields a straightforward estimate of the optimum time. In particular, the intersection of the time curves for f 2 5 and . the inter-25% variation in kl is virtually a t t ~ Similarly, section of the curves for variation in k , and k 2 together is virtually tIII. A distinction is to be made between these crossover times a n d the respective optimum times since the latter is based on infinitesimal variations in the rate constants. There would be no difference if the variation in the rate were a parabolic function of variation in the rate constants. The function is only approximately parabolic. However, in practice t h e difference is negligible. I n many cases it is not necessary to actually determine the values for k l or k,. However, if it is desirable to change conditions, for reasons of operator convenience or instrumental constraints, then some estimate is required. An estimation of k z / k l can be made from the measurement of time of crossover and the time of occurance of the peak velocity, t,. Because the center of the peak is difficult to locate, it is useful to use a least squares fit to the data points around the peak. For velocity data which were measured over equal intervals of time, the following simplified least squares formula is helpful,
=-
Y-1-
531
Y+1
2 ~ - --I YO + ~ Y + I where yo is the peak velocity, y-, is the velocity in the previous interval, y+l is the velocity in the following interval, and AX is the distance to the estimated peak from the data point having the maximum velocity. T h e distance is in terms of the time interval, Le., if the velocity was measured every 30 s, and AX were -0.3, the estimated peak occurred 9 s before the data point with maximum velocity. Returning to the estimation of k 2 / k l , the ratio of t , to tI (for a crossover determined by variation of k , ) is equal to the ratio of Tpto TI. The latter ratio can be determined from Figure 1 and linear interpolation used if necessary to find the appropriate k 2 / k , . Once k 2 / k l is known, then the other optimal times can be determined, or individual adjustments to the activity of kl or k , can be made for improved resistance to change or to reduce the cost of the assay by using more of the inexpensive enzyme a n d less of the other. Method B. From Figure 1, the value of k 2 / k l can be determined if the value of T , is known. Since T p = k l t p , measurement of k , , in addition to t, as described in Method A above, is necessary. T h e value of k l can be measured from the terminal slope of a log velocity vs. time plot. This method is convenient because both measurements can be made on a single set of velocity data. Method C. T h e rate constants are determined from a nonlinear least-squares fit of Equation 10 to the data.
where X represents a measured variable such as absorbance, Y is a background level, and the other terms are the same as in Equation 2. The parameters to be determined are; Y , A,, kl, k 2 , and possibly t oin cases where there exists a significant dead time between mixing and initiation of the reaction timer. The Box Algorithm ( 4 ) is useful in dealing with three nonlinear terms, k,, k,, and to. T h e method requires a t least 10 good data points which include some measurements around t h e time where the velocity is a maximum. Method D. This method involves direct measurement of h , by measuring the slope of a log velocity vs. time plot resulting from a reaction mixture initially containing the intermediate substance B in Equation 1. In the example considered here, ammonium ion is the intermediate. From the terminal slope of a log velocity vs. time for a plot resulting from a reaction mixture initially containing the substrate, A, the rate constant, hl, can be determined also.
RESULTS In Figure 3 are plotted the velocity data (rate of change of absorbance) for a systematic variation in k l (Type I variation). T h e curve labeled 1 corresponds to the desired enzyme activity, curve 2 corresponds to an increase of 25%, and curve 3 corresponds to a 25% decrease. T h e peak velocity (corresponding to the straightest portion of the absorbance curve which is not shown here) occurs approximately a t the second velocity data point. At that time the velocities were 1370 greater and 17% less than the unperturbed velocity depending as k l was greater or less than the nominal. Such variation in results would be generally unacceptable. On the other hand, the crossover point labeled A represents the time a t which the various perturbations have very nearly the same velocity as the nominal, Le., minimal error. The time of crossover, A, is virtually the same as tI and illustrates the finding from the error surface t h a t increase or decrease in k , both result in a decrease in velocity a t that time. The measured decrease was
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ANALYTICAL CHEMISTRY, VOL. 51, NO. 4, APRIL 1979
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Flgure 3. Velocity curves resulting from systematic variation in k , . Conditions: 1 = nominal, 2 = +25% k , , 3 = -25% k ,
Figure 5. Semilog plot of velocity resulting from systematic variation in k,. Conditions: 1 = nominal, 2 = +25% in k,, 3 = -25% in k,. Curve 4 is due to the intermediate, ammonia, so that k , can be estimated directly from the slope
Table I. Estimation of Kinetic Constants method of estimationa measured AI AIII B C h , (s-’) 0.0061 0.0060 h, 0.0205 A (s) 223 183 t , (s) 78 78 I8 derived t~ ( s )
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Figure 4. Velocity curves resutting from systematic variation in k , and k , together (dilutional or temperature type variations). Conditions: 1 = nominal, 2 = +25% k , and k,, 3 = -25% k , and k ,
2.5?% which compares well with the predicted 2% and is a substantially smaller error than that a t the peak velocity. In Figure 4 are plotted the velocity data for a systematic variation in k l and k , together (Type 111 variation). The results are entirely analogous to those for perturbations in hl above. At crossover A, corresponding to t111,the 25% perturbations resulted in a decrease in velocity of 4 % . Data for systematic variation of k 2 (Type I1 variation) are plotted in Figure 5. A semilogarithmic scale was chosen to illustrate estimation of k l from the terminal slope. At crossover A, corresponding to tII, 25% perturbations in k 2 resulted in error too small to be measurable from the figure. As expected, Figure 5 demonstrates that the three curves exhibit the same terminal slope. Also this illustrates the point that the velocity is different from that of a simple exponential decay (first-order reaction) by a steady-state term. The transient term has negligible effect on the terminal portion of the curve. For a value of h z / k l of 4,a 25% increase in h2 results in a 6.25% decrease in the value of the steady-state term. T h e observed decrease was about 6%. Table I summarizes the kinetic constants estimated by the several methods. T h e agreement with the experimental finding is generally good. For instance, from Figure 5 the type I1 crossover occurred at 124 s. The estimates of t , ranged from 127 to 145 s. In the latter case, it will be seen that Method C generally overestimated the various times. One would have expected Method C, the nonlinear least squares, to provide
0.0226
3.4 3.1 227 218 t11 ( s ) 133 145 134 t111 ( s ) 1 9 4 197 189 t, ( S I 85 79 a The method of estimation denotes the particular experimental manipulations. In Method A I , one measures the time at which the rate of reaction is a maximum and also the time at which the reaction curves intersect for variations in h , (see point A of Figure 3). Method AIIIis like AI except that both h , and 12, are varied keeping their ratio constant (see point A of Figure 4). In Method B, one measures the time at which the rate of reaction is a maximum and also estimates k , from the terminal slope of a semilog plot of the reaction data (see Figure 5). I n Method C , one fits the absorbance data by a nonlinear least-squares routine. In Method D, one estimates k , as in Method B and also estimates h , by observing the reaction of the intermediate substance (see curve 4 of Figure 5). See text for full description of methods. kJk,
4.1
D 0.0061
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3.9 3.8 210 216 127 131 187 -
better agreement since the maximum deviation between the measured and fitted absorbance was only 1.6 milliabsorbance. It may be that the method was adversely affected by early measurements before the reaction mixture was fully temperature equilibrated. DISCUSSION In the design of an assay for a substance by kinetic methods, a great many factors have a bearing on the selection of conditions. The optimization, presented here, presumes that variations in the rate constants (enzymatic activities) are a dominant source of error. Depending on the rate constant which is most subject to variation, conditions can be selected that substantially minimize the effects of such variation. In the selection of those conditions, the optimization theory provided the time at which the rate measurement should be made. If that time were, for example, 10 min and it was desired to complete the analysis in 2 min, then both rate constants (amount of enzyme) would simply be increased by a factor of 5. It may be observed from Figure 4 or 5 that, for
ANALYTICAL CHEMISTRY, VOL 51, NO. 4, APRIL 1979
a fixed analysis time, the amount of enzyme required for the optimized method is greater than that for the maximal velocity criteria (approximating an initial rate measurement) yet less than that for the end-point criteria (99% complete a t 5 time constants). In the one-step kinetic analysis described by Atwood, the rate of reaction at the optimum time was only 37% of the initial rate. Even so, t h a t rate was the greatest rate which could be achieved a t a nonzero measurement time. More or less enzyme would have given a smaller rate. By way of contrast, the coupled kinetic analysis exhibits a peak rate at some time after initiation of the reaction and the rate of reaction a t the optimum time can range from 37% to 100% of t h a t peak rate. For instance, when k z / k l is greater than 5 t o 10, the optimum time of type I or 111, will give approximately 37% of the peak velocity, whereas a Type I1 condition will approach 100%. When t h e rate constants are approximately equal, then the velocity at the optimum time will be approximately 75% of the peak velocity in all types of optimization. I t is observed from Figure 1that the various optimum times converge when t h e rate constants are equal. Under those conditions, there are minimum errors resulting from variation of k , or k 2 or any combination of them. This is in contrast t o other conditions where, for example, selection of a Type I11 optimum time sacrifices the suppression of errors due to variation in kl alone. However, that sacrifice will be small for k z / k l greater than 10 or less than 2. While the selection of equal rate constants is a generally desirable condition, other conditions might be dictated by cost consideration. Thus, less of the expensive enzyme and more of the other enzyme would be used. This strategy generally leads to amounts of enzyme proportioned for equal cost and the ratio, k z / h l ,is thus fixed. The actual amounts of enzyme are set by the choice of analysis time. It has been customary to make h2 greater than hl, and the theory so far has only considered k z equal to or greater than k l . However, the symmetry of the differential equations is such t h a t t h e same curve would be generated for k 2 / k i = 4 as for h z / h l = 1/4. For this case ( h z / h l= 0.25), the optimum time would be obtained from Figure 1 by choosing a value of 4 for the ratio of rate constants and using the corresponding value of TIif error from h2 is to be minimized, TIIif error from k 2 is to be minimized, and TI11 for simultaneous variations. T h e actual time will be calculated as from the relations; TI = hztI, TI1 - k2tI1, and TI11 = h2tIII. Of the various methods for determining the optimal time, Method A offers particular advantage because it is not necessary that the upside and downside variations be equal. For example, if it were expected that an increase in k l could
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be expected because of endogenous enzyme in some samples, then the time of crossover between the nominal and exceptional sample could be chosen. I t is customary to calibrate enzymatic measurements of substrate. Even though the actual time of measurement may be far from optimum, the results will be without error if the conditions do not vary. T h e consequence of non-optimum conditions is only to make t h e analysis more sensitive to variation in conditions. It is fortunate that the optimum time need not be known with great precision since the results of Table I exhibit a considerable range of estimates for t h e various optimal times. T h e measurement of urea under the conditions presented here was taken as an example that would be a serviceable assay. Nevertheless the conditions could be modified for the sake of cost and convenience without sacrifice in quality. The anticipated changes would reduce the amount of glutamate dehydrogenase by a factor of two and increase the amount of urease by a factor of two. This would make the rate constants approximately equal. Another set of experiments could be performed to verify that the expected conditions had been achieved. Then, possibly, the amount of both enzymes might be increased to move the optimum time to about 2 min, which time is long (to reduce the required amount of enzymes) but not affecting throughput of the centrifugal analyzer (whose sample loader limits the throughput to a t least a 3-min cycle). Extensions of the theory presented here could consider additional coupling steps. Also, wide time intervals for the measurement of the rate deserves consideration. It is not necessary that the reaction be enzyme catalyzed, but only that it be first- or pseudo-first-order in each step. Furthermore, the coupling step is mathematically equivalent to an RC filter used in analog rate measurements; hence the optimum time for the measurement of a first-order reaction by a n analog ratemeter can be determined. Nevertheless, the point made here is that coupled kinetic assays for substrate can be made less subject to errors in a straightforward manner.
ACKNOWLEDGMENT We thank a reviewer for bringing the work of H. U. Bergmeyer to our attention. LITERATURE CITED ( 1 ) J. G.Atwood and J. L. DiCesare, Ciin. Chem. (Winston-Salem, N . C . ) , 21. 1263-1269 (1975). (2) H. U. Bergmeyer, Biochem. Z . , 324, 408-432 (1953). (3) T. 0 . Tiffany. J. M. Jansen, C. A. Burtis. J. 8.Overton. and C. D. Scott, Clin. Chem. ( Winston-Salem, N . C . ) , 18, 829-840 (1972). (4) G. E. P. Box, Ann. N . Y . Acad. Sci.. 86, 792-816 (1960).
RECEI\.ZD
22, 1978.
for review September 18, 1978. Accepted December
