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Anal. Chem. 1988, 6 0 , 549-552
Application of Temporal Optimization to Enzyme-Based Reaction Rate Methods R. Kay Calhoun and F. James Holler*
Department of Chemistry, University of Kentucky, Lexington, Kentucky 40506
The applkation of temporal optbnizationto enzyme-catalyzed reactions is explored. Theory Is presented that shows minlmization in the impreclslon in rate measurements for pseudo-firstorder enzyme-catalyzed reactions when the meaExperimental data on the surement tlme Is t = 7 = Kdv,. urease-catalyzed hydrolysis of urea show rates measured at 7 to be more precise than initial rates by a factor of 8. Simulated substrate vs tlme data are used to Illustrate the relationship between the time of mlnimizatlon of uR/R and the K,/[SIo ratio. Minimization in uR/R occurs for ratlos as low as 0.06, and for K,/[SJ,, > 4 the time of minimum error Is at 7. Experimental data are also presented that show mlnimizatlon in uR/R even when the reaction Is not strktly pseudo first order.
Many approaches have been taken by investigators in the effort to minimize imprecision in reaction rate methods of analysis. These approaches range from contiol of experimental variables such as pH and temperature (1-4) to numerical correction of data for variations in such variables (5,6). There have also been studies on the limitations of different methods of kinetic analysis (7-11). In a previous paper we explored the minimization of errors in reaction rate methods for first-order reactions through the use of temporal optimization (12). In this paper we extend temporal optimization to enzyme-based reactions. Both theory and experiment are presented that show that under pseudofirst-order conditions imprecision in reaction rate measurements minimizes at the time corresponding to t = Km/v,,. The effect of the K , / [ S ] , ratio on the time of minimization of error will be shown by the use of simulated substrate vs time data. Simulated and experimental data on the ureasecatalyzed decomposition of urea are presented that show error minimization even when the reaction is not strictly pseudo first order. THEORY For a first-order or pseudo-first-order reaction the rate depends on the initial concentration of reactant, [A],,the time of reaction, t , and the rate constant, k, as shown in eq 1. The
R = k [ A ] = [A]oke-kt
(1)
rate constant depends on variables such as temperature, pH, and ionic strength, however. Fluctuations in any of these variables therefore result in fluctuations in the rate. As we have shown (12) the imprecision in the rate due to changes in the rate constant in first-order processes can be expressed as QR _ 11 - ktl
Qk
R
where k and t are as defined for eq 1 and uR/R and Uk/k are the relative standard deviations of the rate and rate constant, respectively. Inherent in the development of eq 2 are the assumptions that it is possible to estimate the rate at any time 0003-2700/88/0360-0549$01.50/0
and that timing and instrumental errors are negligible. It can be seen from eq 2 that it is possible to optimize the rate measurement by a prudent choice of measurement time (12). In fact, at t = l / k = T imprecision in the rate constant has no effect on the run-to-run precision of the rate measurement at all. Furthermore, for measurements at any time less than 27, those determined at t = 0 exhibit the worst precision. To extend temporal optimization theory to enzymatic reactions it is necessary to consider the generalized mechanism for an enzyme-catalyzed reaction k
E+S&ES-P
k2
(3)
k-1
where E represents the enzyme, S the substrate, ES the intermediate complex, P the products, and k l , k+ and k2 are rate constants (13). The rate expression for this mechanism can be expressed as
R = (vmax[Sl)/(Km+ [ S I )
(4)
where u, is the maximum rate and K , is the Michaelis constant (14). If experimental conditions are chosen for which K , >> [SI, then eq 4 simplifies to
R = (uma,/Km)[S] = k ’ [ S ]
(5)
The reaction is then pseudo first order in substrate, and the quantity u, f K , is the pseudo-first-order rate constant k’. Under these conditions, temporal optimization theory predicts that at time t = l / k ’ = 7 fluctuations in K m and urn, have minimal effect on the imprecision of rate measurements. In K , f u, = 7, QR f R 0. other words, as t The general effect of fluctuations in K , and u, on the imprecision of rate measurements can be studied by simulating reaction data. The Michaelis-Menten equation, eq 4,can be integrated as shown by Henri (15), and the integrated form is shown in eq 6 (16). This equation cannot be solved ana-
-
-
Urn&
= K m In
([sl~/[sl) + ( [ S I , - [SI)
(6)
lytically for [SI,but it is possible to use it to numerically obtain simulated [SI vs time data, which contain fluctuations in apparent substrate concentration due to random changes in K , and.,u These data are then used in the calculation of rates according to eq 4 and the values of gRf R resulting from the fluctuations in these rates are calculated. These simulated data are useful predictors of trends in aRf R values for a wide range of K,/[S],ratios and noise levels on K , and .,,v EXPERIMENTAL SECTION Instrumentation. A bipolar pulse conductance instrument was used to follow the appearance of ionic reaction products. This conductance technique and instrument, BICON, have been discussed in detail elsewhere (17,18). The instrument was interfaced to an Intel SBC204 single-board computer, which was used to control the instrumental parameters, timing, and data acquisition. The conductance cell used is depicted in Figure 1. The cell consists of three parts: a ‘/,,-in.-thick piece of Teflon, which acts as a spacer, and two pieces of Plexiglas each of which has a 1/16-in. stainless steel rod sealed into its center and an opening near one end to allow connection to the sample stream. The Plexiglas pieces 0 1988 American Chemical Society
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ANALYTICAL CHEMISTRY, VOL. 60, NO. 6, MARCH 15, 1988
Time, s
F w e 3. plot of uR/Rvs time for pseudo-first-order enzymecatalyzed reaction with K , / [ S ] , = 10. Rates obtained from simulated data containing random fluctuations in K, and v,, equivalent to relative standard deviations in each of 0.01.
TEFLON SPACER
Figure 1. Diagram of conductance flow cell.
STOP SYRINGE
& WASTE SLIDER VALVE
CONDUCTANCE
MANUAL DRIVE
Flgure 2. Schematic diagram of stopped-flow apparatus.
were polished so that the stainless steel rods are smooth and flush with the surface of the Plexiglas. The Teflon spacer has a slot milled in its center so that when the cell is assembled, a channel for solution flow is created and the two stainless steel electrodes are in direct opposition. This orientation of electrodes has been shown to give large, consistent signals even for low-volume cells (19). A two-port tangential jet mixer (20) was located directly adjacent to the conductance cell. A manual stopped-flow syringe drive system was fabricated locally and used to deliver the solutions to the mixer. Figure 2 shows a schematic of the stopped-flow instrument. Solutions and Procedure. The enzyme solutions were approximately 100 units of urease/mL (Sigma Chemical Co., Type VI). One unit of urease liberates 1.0 pmol of NH3from urea/min at pH 7.0 and 25 OC (21). The solutions were kept refrigerated at 4 "C while not in use and were used within 4 days of preparation. There was no noticeable deterioration of the enzyme solution within this time period. The urea (Aldrich Chemical Co., Inc., 99+%) solutions were also used within 4 days of preparation. Two buffer systems were used: (1)pH 7.3, 0.05 M Tris buffer was prepared from desiccated Trizma hydrochloride and Trizma base (Sigma Co.) according to directions provided by the manufacturer; (2) pH 7.0 phosphate buffers were prepared in various concentrations from potassium hydrogen phosphate (J.T. Baker Chemical Co.) and potassium dihydrogen phosphate (Fischer Scientific Co.) All chemicals were reagent grade unless otherwise noted and were used without further purification. Deionized distilled water was used for all solutions.
Prior to a given series of experiments, the flow system was flushed with deionized water and then with buffer solution. The volume of the stopped-flow syringes was such that one flow cycle was sufficient to rinse the previous solution from the conductance cell; therefore, replicate experiments were done on sequential pushes. For experiments on solutions of different concentration of analyte, three additional pushes were done between solutions. No attempt was made to control the temperature, and the temperature range is reported for the individual experiments. A switch actuated by the stop syringe of the stopped-flow apparatus signaled the computer to start data acquisition. The time between bipolar pulses and the number of pulses per point were determined by the analyst for each run based on the time required for each reaction. At the end of a run the data were stored for analysis. Several computer programs were used in the data analysis and rate measurement. These programs include KINFITI, a nonlinear curvefitting program (22) used to estimate enzyme kinetics paand K,, and a program to generate rate vs time rameters u, curves with a software implementation of the integrating ratemeter (23-25). The simulations presented in this work were performed with a PDP 11/03 laboratory computer in FORTRAN IV. RESULTS AND DISCUSSION Simulated reaction data generated from eq 6 were used to predict the effect of noise in K , and ,v on the minimization of error in rate measurements for enzyme-catalyzed reactions. Figure 3 shows a plot of uR/R vs time obtained for rates calculated from simulated data. For this set of data the ratio Km/ [SI, was 10 and the relative standard deviations of K , and u, were both 1%. At this ratio of K,/[S], the approximation shown in eq 5 is valid and the reaction is pseudo first order in substrate. Therefore T = Km/u,, = 100 s. The curve in Figure 3 has the expected V-shape and a minimum a t approximately 100 s, i.e. a t T . The effect of the K,/[S], ratio on the minimum in uR/R was studied by simulating substrate vs time data that had low levels of random noise in K , and u,, and calculating the corresponding uR/R vs time curves. The minima for these curves were calculated as the intersection of linear leastsquares lines drawn through each arm of the V. Figure 4 shows the results of this study for K,/[SI0 ratios from 0.2 to 20. As in Figure 3, = K J u , = 100 s and the times of minimization were normalized to this time. This plot exhibits two significant features: (1) for K,/[S], > 4 the minimum in imprecision occurs approximately at r; (2) there are minima in (FR/R vs time curves even a t low K,/[S], ratios. In fact, there is minimization a t ratios as low as 0.06. However, as K,/[SI0 decreases in the range 0 < K,/[S], < 4 the time of minimization becomes increasingly greater than T . The simulated data show that when the pseudo-first-order approximation expressed in eq 5 is valid, it is possible to optimize the rate measurement by choosing the measurement
ANALYTICAL CHEMISTRY, VOL. 60, NO. 6, MARCH 15, 1988
551
2 001
,
0 0
4
0
12
16
20 [Urealo, rnM
Figure 4. Plot of the time of minimization in u,/R vs K,I[SIo obtained from simulated data containing random fluctuations in K, and vmx.
time to be T. These data, however, also show that optimization is obtainable even for non-pseudo-first-order cases and that the optimum time is well-removed from time zero. The reaction chosen for experimental study is the ureasecatalyzed hydrolysis of urea. As a result of the clinical importance of urea, much research has been done on improving methods for its determination (26-37). Since the hydrolysis products are ionic, as is illustrated in eq 7, conductometry has NHzCONHz
+ 2H20
urease
2NH4'
+ COS2-
Figure 5. Working curves of rate vs initial urea concentration. (1) Rates calculated from eq 4 for times (0)5.5 s, (A)50.5 s, and (0) 104.5 s. (2) Rates estlmated with double-integrating ratemeter for times (X) 50.5 s and (+) 104.5 s.
1201
I,
96
(7)
often been used as the detection technique in the determination of urea (26-28, 37). Thus, this chemical system is well-suited for testing the application of temporal optimization to enzymatic reactions by using the BICON instrument. As suggested previously, the time at which uR/Rapproaches a minimum is directly related to K,. However, K , varies greatly with reaction conditions (33, 38-46). Values of K , under the reaction conditions chosen for these studies were estimated by using KINFIT4 to analyze conductance versus time data from the experiments. We chose reaction conditions that resulted in two K, values roughly an order of magnitude apart. First, K , was estimated to be 1.48 f 0.91 mM under conditions of 0.05 M Tris buffer, pH 7.3, =50 units/mL urease, and urea concentrations ranging from 1 to 13 mM. Second, it was possible to increase K , to 17.6 f 9.9 mM by using 33 units/mL urease and 0.2 M phosphate buffer at pH 7.0 (39). The approximation of pseudo-first-order behavior as expressed in eq 5 is assumed to be valid when K,/[S] 3 10 (47). By using urea concentrations ranging from 1 to 15 mM under both sets of reaction conditions, we were able to carry out reactions in which the K,/[S], ratios ranged from 0.1 to 18. Thus, it was possible to determine the imprecision in rate measurements for pseudo-first-order and non-pseudo-first-order cases. Sets of replicate reactions were carried out and the reaction rates were estimated in two ways: (1) by application of a software version of the double-integrating ratemeter (25);and (2) by calculation from eq 4, using estimates of K , and u,, generated by KINFIT4 (22). These parameters were also used in the estimation of 7. Five replicate kinetic runs were collected for each concentration of urea. Rates were estimated by both methods and uR/Rvalues were calculated for each concentration. Working curves of rate vs initial urea concentration are shown in Figure 5 . Curves are shown for three different measurement times: 5.5, 50.5, and 104.5 s. These are time zero, the earliest time allowed by the ratemeter algorithm, and the average T for all runs, respectively. For two of the measurement times shown in Figure 5, 50.5 and 104.5 s, it was possible to obtain rates both by the application of the ratemeter and by calculation from eq 4. As Figure 5 shows, a t these times both methods give consistent estimates of the rate. The concentration of a pseudounknown urea solution was determined with an error
r5 N
P X
\
b"
(L \
b" 0
50
100
150
200
250
Time , s Flgure 6. Plot of relative standard deviation of rate vs time for the hydrolysis of a 3 mM urea solution in pH 7.0, 0.2 M phosphate buffer, which contains -33 units/mL urease at =25 OC; K , = 20 mM and r N 105 s: (lower) rates calculated with eq 4; (upper) rates estimated with double-integrating ratemeter.
of 0.20% when the calibration curves for rates taken at T were used. The errors were 5-8 times greater when the other curves were used. Figure 6 shows uR/Rvs time curves for a set of 3 mM urea solutions. At this urea concentration the reaction is pseudo first order so estimation of the rate by the ratemeter is as valid as its calculation from eq 4, and the two uR/R vs time curves are in agreement as expected. Both curves show larger imprecisions in the rate a t times close to zero than are observed at times approaching 7. Thus, for enzyme-catalyzed reactions run under pseudo-first-order conditions, a judicious choice of measurement time will result in minimal error in the rate measurement. Many of our experiments were not run under pseudofirst-order conditions. Figure 7 shows uR/R vs time curves for a set of 1 mM urea solutions taken under conditions such that K , N 2 mM, i.e. for these reactions K , is not much greater than [SI. Hence, uR/Ris not expected to strictly obey eq 2. But, as was indicated by the simulated data of Figure 4,it is not necessary that the reaction be strictly pseudo first order for minimization in the error curve to occur. The curves shown in Figure 7 lend experimental support to these findings. Both curves in Figure 7 not only exhibit large values of uR/R early in the reaction but also reach a minimum near T = 7.3 S.
CONCLUSION The results presented in this paper show that the choice of measurement time is an important consideration in the optimization of enzyme-based reaction rate methods. The
552
ANALYTICAL CHEMISTRY, VOL. 60, NO. 6, MARCH 15, 1988 (2) (3) (4) (5)
0
4
I//(
8
I2
I6
23
lime,s Figure 7. Plot of relative standard deviation of rate vs time for the hydrolysis of 1 mM urea solution in pH 7.3,0.05 M Tris buffer which contains 4 0 units/mL urease at =24 OC;K, E 2 mM and T N 7.3 s: (-) rates calculated with eq 4; rates estimated with dou(-e)
ble-integrating ratemeter. range of urea concentrations studied encompasses the normal concentrations of 2.5-6.3 mM found in blood plasma or serum (a), so the information presented here provides a solid basis for the extension of this work into actual clinical analysis of urea. Generally, procedures for applying temporal optimization to enzyme-based analyses should first include preliminary experiments to determine the reaction conditions under which pseudo-first-order behavior exists. But, it should be noted that, as we have demonstrated, strict pseudo-first-order behavior is not necessary for temporal optimization to be beneficial. If the reaction curve exhibits some exponential character, then minimization of uR/R is observable a t times well removed from time zero. Estimates of T for the reaction conditions chosen must then be made. In cases where repetitive analyses are performed under the same controlled experimental conditions, the value of T simply becomes another operational parameter. Rates determined at T can be treated by the usual fixed-time procedures; e.g., they can be used in single-point calibrations or in working curves of rate vs concentration (12). However, by choosing the measurement time to be 7,the effects of fluctuations in variables affecting K , and urnax,and thus imprecision in the measurement of rates of enzymatic reactions can be substantially decreased. Registry No. Urease, 9002-13-5; urea, 57-13-6. LITERATURE CITED (1) Holler, F. J : Enke, C. G.: Crouch, S. R. Anal. Cbim. Acta 1980, 117, 99.
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RECEIVED for review June 23, 1987. Accepted November 7, 1987. We are grateful to the National Science Foundation for supporting this work under Grant CHE-8217348.
