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Optimized Wide-Interval Rate Measurements of Substrate J. E. Davis* and Brian Renoe' Division of Laboratory Medicine, Departments of Pathology and Medicine, Washington University and Barnes Hospital, Si. Louis. Missouri 63 1 10
Variation in the value of the rate constant of a first-order or pseudo-first-order reactlon, is a major contributor to error in the estimatlon of substrate from the rate of reaction. Rate measurements over any arbitrary but fixed time interval can be optimized to provide impressive reduction in errors from that source. Wide-interval measurements include a large amount of the available change in absorbance and reduce the need for high precision photometers.
Estimation of chemical substances by rate methods is becoming more common as a result of instrumentation which makes rate methods convenient to use. The simplest and most common methods are based on f i t - o r d e r or pseudo-first-order reactions. T h e particular advantage of first-order reactions is the fact that at any fixed time the velocity (rate of reaction) is proportional to the initial amount of substrate (1). A great many methods measure t h e rate as close to zero time as possible in an attempt to measure the initial rate. For first-order reactions, i t is also true that the average velocity between any two time points is proportional t o the initial amount of substrate (1). I t has been shown by Atwood and DiCesare (2) that the rate constant could be adjusted to give a maximum velocity over a relatively short interval which occurred at a fixed time after initiation of the reaction. T h a t an optimum rate constant exists, can be seen from the following argument. Suppose the rate constant were small, then the rate of reaction would be small. Increasing the rate constant would increase the rate of reaction. However, in contrast to the initial rate measurement, there comes a point at which the amount of substrate remaining at t h e time of measurement is so small t h a t the rate of reaction would be small. T h u s there is a compromise between a n increased rate due to an increased rate constant and decreased rate due to substrate utilization. I n addition to providing the maximum velocity, those same conditions provided resistance to the effects of variations in the rate constant (enzyme activity). Those authors were able t o show t h a t f 2 0 % variations in the rate constant resulted in a mere 2% error in the measured rate and, hence, in the estimated substrate concentration. The same variation would cause a 20% error in results measured by initial rate methods. Our purpose is to extend the optimization theory to the use of time intervals that are large in comparison to the time after initiation. Large time intervals will result in larger changes in absorbance which would permit the use of instruments that are neither so sensitive nor elegant as t h a t used by Atwood and DiCesare. Techniques for quickly and simply optimizing a n assay are presented and illustrated by application to the estimation of uric acid by the use of uricase.
THEORY A first-order reaction is characterized by a n exponential function of time: Present address: Department of Pathology, University of Virginia, Charlottesville, Va. 22901. 0003-2700/79/0351-0526$01 .OO/O
S = So exp(-kt)
(1)
where Sois the initial amount of substrate, S is the amount at time t , and k is the rate constant. For an enzymatic reaction with substrate levels well below K,, k = V,,/K, (2). The average rate of reaction (velocity) between times t l and t 2 is;
V," =
so
exp(-ktl)
-
exp(-ktz)
t2 - t l
(2)
In order to find the optimal value of k which minimizes the variation in V,, for variations in k , the derivative of Equation 2 is taken with respect to k ;
By setting this derivative equal to zero, t h e condition for optimization results; k t l exp(-ktl)
-
k t 2 exp(-ktz) = 0
(4)
For other than the trivial solution, t , = t,, the solutions t o this equation must be found by numerical methods. T h e physical situation justifies this condition as a minimum and obviates need to examine the second derivative. Furthermore, the natural variables are k t , and k t , and their use allows a general solution that does not depend on the particular time scale, i.e., minutes, seconds, hours, etc. For example, if k t , = 0.5, then Newton's method can be used to find that a value of k t , = 1.7565 will satisfy the equation within 0.01%. In a like fashion, numerous other values of k t , between 0 and 1 were chosen and the value of k t , was found. T h e relation between k t , and k t , is symmetrical so that values of k t l greater than 1 yield values of k t , less than 1. For that reason, k t , was limited to values less than 1. These data points are shown as a smooth function in Figure 1, where it is seen that t h e optimum value for k t , increases rapidly as k t , approaches zero. As an example in the use of Figure 1, suppose t h e rate constant, k , is 0.01 5-l and t l is chosen as 50 s. T h e value of h t , is 0.5 and the optimum k t , is 1.76; hence the second measurement is to be made a t 176 s. Also shown in Figure 1 is the time interval of the measurement, k A t , which of course can be calculated by subtracting k t , from kt,. Its purpose here is to illustrate that the rate measurement can be made over an interval which rapidly approaches the total time of reaction. In some situations, the rate constant must be modified (by adjustment of the amount of enzyme) to make the optimum times coincide with predetermined measurement times. The ratio, k t , / k t 2 shown in Figure 1, allows determination of the optimum rate constant. For example, if t l = 30 s and t z = 45 s, then the ratio is equivalent to t l / t 2 or 0.67 since the rate constant can be factored out of the ratio. From Figure 1 it is found that h t , = 0.82 and k = 0.027 s-'. In Figure 2, curve A shows the amount of material reacting within the measurement interval. Thus, in the example above where k t , = 0.82, about 15% of the original amount of material reacted to produce a difference in measurements at t l and tz. Curve B shows the amount of material reacted by the time of the final measurement, about 70% in the example above. 'C 1979 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 51, NO. 4, APRIL 1979
4,
527
I
k t 1 (Dimensionless) Figure 3. Variation of plus 25% (P) and minus 25% (M) in the rate constant and the resulting amount of error (decrease) in rate measurement over the optimum time interval which corresponds to an initial measurement at k t ,
1
160
' w
I
\\
1201
kt1
(Dimensionless)
Flgure 1. Optimum time relationship between measurement at the first time point, kf and the final time point, M,. The time interval between those two points is shown as k A t . Note that the rate constant has dimensions of inverse time so that kt is dimensionless
,,
1
0
I
I
50
,
I
60
I
90
I
120
150
I
SECOND5
Figure 4. Absorbance vs. time for two levels of urease: N = nominal, P = f23%
2 0.2 0.4 0.6 0.8 1.0
'0
kt
1
(Dimensionless)
Figure 2. (A) Amount of material reacted during the optimum time interval which corresponds to an initial measurement at k t , . (B) Total amount of material reacted by the end of the optimum time interval which corresponds to an initial measurement at k t ,
Notice t h a t at least 63% of t h e material will be reacted and that 100% of t h e material reacts in t h e interval when k t , = 0 and consequently k t , is infinite, i.e., the reaction has gone to completion. Finally, in Figure 3 is shown the error in velocity that results from a plus, P, or minus, M, 25% variation in the rate constant when the optimum time interval is chosen. It is seen that the error decreases as k t l decreases and correspondingly, k t , increases. Ultimately, t h e error due t o variations in rate constant goes t o zero, as k t , goes to zero, in which case t h e reaction has gone t o completion since the corresponding k t , is infinite. It is noteworthy that the error decreases rather slowly, even as t h e total amount of substrate converted surpasses 90% a t k t , = 0.3.
EXPERIMENTAL A CentrifiChem 400 (Union Carbide, Rye, N.Y. 10580) was used to collect spectrophotometric data at 292 nm at 37 "C. Reagents for uric acid analysis were from DuPont (Automatic Clinical Analysis Division, Wilmington, Del. 19898) and were chosen because of the large K , of the uricase obtained from Bacillus f a s t i d i o s u s ( 3 ) . Solutions of uric acid were prepared according to the method of Henry ( 4 ) . In an initial experiment wherein the reagents were reconstituted with 6.25 mL H 2 0 ,the rate constant was measured to be 0.033 s from a log velocity vs. time plot. For the purposes of illustration, it was desired to use a k value of 0.013 s-', therefore a new set of reagents was diluted to 15.9mL. No attempt was made to maintain the buffer capacity, ionic strength, or pH. In view of the results, this procedure was deemed adequate. Two time intervals were selected for the purposes of illustration. The first was ht, = 0.4 ( t l = 30 s) and k t , = 2.02 ( t z = 150 s). This condition permits a measurement of over 50% of the total absorbance available from the uric acid. A second interval was h t , = 0.8 ( t l = 60 s) and ht, = 1.23 ( t Z= 90 s). This condition was the smallest convenient time interval surrounding one reaction time constant (1,'k). The value of the rate constant (0.013 s-l) was chosen so that these convenient time points would approximately satisfy the optimization conditions. RESULTS In Figure 4 is plotted the absorbance readings from the experiment with h = 0.013 5-l. Inspection shows t h e initial rate (slope of the curves at t = 0) t o be greater for the curve
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ANALYTICAL CHEMISTRY, VOL. 51, NO. 4, APRIL 1979
where the rate constant was purposely increased 23% (increased amount of uricase). On the other hand, the average rates of reaction, as indicated by the slope of the long dashed lines between the absorbance a t 30 and 150 s, are nearly the same. Thus, any substrate concentrations derived from the slope of those lines would be nearly the same. In a similar way, the short dashed lines of Figure 4 have nearly the same slope and would give nearly the same results. Whether the narrow or wide measurement interval is chosen depends on the precision of the absorbance measurements and the time available for the measurement (or allowable cost for increased amounts of enzyme required for a short time scale). I t can be seen from Figure 4 that narrow time intervals will exhibit a steeper average slope than wide intervals.
DISCUSSION Measurements of analytes by rate methods are becoming increasingly important compared to end-point measurements. This is because rate methods inherently correct for blanks (interfering absorbance unrelated to the analyte). Generally, they take less time, since it is not necessary for the reaction to go to completion. Furthermore, the availability of instrumentation which permits convenient rate measurements has made the use of rate methods more popular. However, variation in the rate constant is one of the major sources of error. T h e rate constant in enzyme catalyzed methods is particularly sensitive to variations in temperature (often 7-8%/degree) and in amount of inhibitors and activators from t h e sample. Furthermore, if calibration is to be performed occasionally, then loss of enzymatic activity on storage becomes a factor. Finally, the rate constant can \ ary because of dilutional errors as well as variations in the amount of the second substrate in pseudo-first-order reactions. These are compelling reasons for optimizing the rate measurement to reduce the effects of such variations. T h e optimization presented by Atwood and DiCesare (2) provides many advantages, not the least of which is aLoidance of measurements near to. In addition to those advantages, the wide-interval optimization presented here permits the measurement of larger changes in absorbance, therebv reducing the need for a high precision photometer. Furthermore. to a lesser extent the optimized wide-inten a1 measurement is even less affected by variation in the rate constant. When it is not practical to adjust the rate constant, then t l must be chosen to be less than the reciprocal of the rate
constant. With that selection, t 2 is automatically set and will be greater than the reciprocal of the rate constant. Conversely, t 2 may be selected and t , determined by the optimization condition. If no conditions predispose a particular selection of t l or t 2 ,then they may be selected to achieve a particular absorbance change. On the other hand, both t, and t 2 may be fixed by instrumental considerations in which case the rate constant must be adjusted. All of these situations can be handled by referral to the appropriate figures. In order to apply the methods of this paper, it has been necessary to know or to measure the rate constant. In some circumstances, it is inconvenient to measure the rate constant. The wide-interval optimization is still valid in those circumstances, in which case the average rate of reaction can be measured for a number of rate constants (or simply, amount of enzyme) and the value which maximizes the average rate of reactions selected as the optimum condition. This approach is valid since the derivative of the average rate of reaction is zero a t the optimum conditions, indicating a maximal value in this case. However, measurement of the rate constant is preferred because the optimal conditions can be achieved with fewer measurements and with greater precision. Because the peak in a graph of rate vs. rate constant is broad, it is difficult to select the best value for the peak. I t is anticipated that the theory of optimization for wide-interval rate measurements can be extended to include coupled reactions as well as measurements a t multiple time points in order to improve the precision. Nevertheless, the theory developed thus far is applicable to any first- or pseudo-first-order reaction, even though enzyme catalyzed reactions were treated explicitly. The optimized wide-interval rate measurement of substrate is easy to use and offers substantial advantages.
LITERATURE CITED (1) T. 0. Tiffany, J. M. Jansen, C. A. Burtis. J. B. Overton, and C. D. Scott, Clin. Chem. ( Winston-Salem, N . C . ) , 18,829-840 (1972). ( 2 ) J . G. Atwood and J. L. DiCesare, Clin. Chem. ( Winston-Salem, N . C.), 21, 1263-1269 (197.5). (3) G. Lum and S. R. Gambino, Clin. Chem. ( Winston-Salem. N . C . ) . 19, 1184-1186 (1973). (4) J. DiGiorgio, "Clinical Chemistry Principles and Techniques", 2nd. ed.. R. J. Henry, D. C. Cannon and J. W. Winkelman, Ed., Harper and Row Publications, Inc., Hagerstown. Md., 1974.
RECEI~XED for review September 18, 1978. Accepted December 21. 1978.
