Progress Curve Algorithm for Calculating Enzyme Activities from Kinetic Assay Spectrophotometric Measurements Jack W. London,*’ Leslie M. Shaw, and David Garfinkel Department of Computer and Information Science, University of Pennsylvania, and the Wm. Pepper Laboratory, Division of Laboratory Medicine, Hospital of the University of Pennsylvania, Philadelphia, Pennsylvania 19 104
An algorithm is presented for calculating enzyme activities from the absorbance measurements obtained with an enzyme analyzer. The algorithm is based on the integrated MichaeliiMenten equation, and can therefore accurately analyze the nonlinear data which frequently arise in kinetic enzyme assays. The actual computations are equivalent to the common linear least-squares regression analysis. The algorithm is used to calculate activltles of a series of dilutions of several human serum enzymes from data obtained with an automated enzyme analyzer. On the average, these calculated activity values correspond to 99.9 % of the expected dllutlon ratios, whereas actlvitles calculated with a commonly used linear algorithm correspond to only 95.4%.
Using an enzyme analyzer, enzyme activity is estimated in a kinetic assay by making a series of absorbance measurements over a time period of several minutes. These absorbance measurements correspond to the disappearance of a substrate or the appearance of a product. The measurements are then converted to enzyme activity by carrying out a mathematical procedure, or algorithm. For example, for those enzyme assays in which the absorbance values decrease linearly with time, the enzyme activity is equal to the time rate of change of the absorbance (Le., slope) multiplied by a constant conversion factor. The algorithm used in this instance would consist of the steps involved in performing a linear least-squares regression analysis. In any case, for a given set of absorbance values, the calculated enzyme activity will often be influenced by the algorithm used. In this paper we present an algorithm based on the integrated Michaelis-Menten equation. While this equation has long been used in the laboratory for evaluating enzyme kinetic constants ( I ) , it has not been applied to estimating enzyme activities from data obtained with an automated enzyme analyzer. It is suitable for all kinetic assays in which enzyme activity is estimated by measuring the enzymatic conversion or production of a single substance. Also, the assay enzyme-substrate system must be compatible with the assumptions of the simplest integrated form of the Michaelis-Menten equation. These assumptions are that any decrease in velocity with time must be the result only of decreasing enzyme saturation, and not of product inhibition or approach to equilibrium. Examples of such assays are those for alanine and aspartate aminotransferase (E.C. 2.6.1.2 and 2.6.1.1) and lactate dehydrogenase (E.C. 1.1.1.27), which are based on the conversion of NADH to NAD, y-glutamyltransferase (E.C. 2.3.2.2) which is based on the production of 4-nitoaniline, and creatine kinase (E.C. 2.7.3.2) in which NADPH is produced. Because the progressive depletion of substrate or accumulation of product is being measured in these assays, the concentration vs. time plots obtained are called progress curves. ‘Present Address, Department of Radiology, Hospital of the University of Pennsylvania, Philadelphia, Pa. 19104. 1716
ANALYTICAL CHEMISTRY, VOL. 49, NO. 12, OCTOBER 1977
This algorithm has an advantage over previously developed linear algorithms (2) in that it consists of a nonlinear regression analysis of all the absorbance measurements. Inherently nonlinear situations, as encountered with the lactate dehydrogenase kinetic assay ( 3 ) as well as nonlinearities arising from the depletion of substrate, thus pose no problem. Furthermore, this algorithm has an advantage over previously developed nonlinear algorithms ( 4 ) in that it is linear with respect to the regression coefficients themselves. No sophisticated nonlinear regression procedures, requiring large computers, are needed. This algorithm can be carried out with pencil and paper, pocket calculator, or small computer, in much the same manner as the familiar linear least-squares regression analysis. The Algorithm. This algorithm may be rationally developed from the biochemistry of these kinetic assays. Although more than one indicator enzyme substrate may be present, all substrates other than that whose absorbance is being monitored are present a t constant, if not saturating, levels. The conversion or production of this monitored substrate can be described by a single-substrate MichaelisMenten equation:
In the above equation, [SI is the substrate concentration, V, is the maximum velocity obtained at infinite [SI, and K , is an apparent Michaelis constant. If the measured absorbance is linearly related to the substrate concentration (Le., Beer’s law applies), Equation 1 may be rewritten as
dA
- ----
dt
A
CIA
+ C,
where the absorbance A replaces [SI,and V, and K , are replaced by the more general notation, C1 and C 2 . The integrated form of Equation 2 is
A
t = k , ( A o - A ) + k2 In Ao
(3)
In Equation 3, A,, is the absorbance when t = 0, k l = l / C 1 and k 2 = -C2/C1. This equation is, of course, the integrated Michaelis-Menten equation; it is this equation that is used for our regression analysis. I t should be noted that while Equation 3 is nonlinear with respect to the absorbance A (because of the In A term), it is linear with respect to the regression constants, k l and kp. This linearity results in a closed-form, non-iterative, analytical solution for the regression coefficients. A detailed procedure for the regression analysis, suitable for pocket calculator or small computer, consists of the following steps. (1) a t equal time intervals, N spectrophotometric absorbance measurements, A,, have been obtained. (2) form a set of normalized absorbances, P,:
$ = 1 A i - A , l , i = 1 , 2 , . . . N - l , j = 1 , 2 ,... rn (4)
subject to the condition that for assays in which the absorbing species is disappearing (e.g., the aminotransferase assays, in which NADH is converted to NAD),
reject any Ai
< A,
since these contain no information as to the enzyme activity: while for assays in which the absorbing species is being produced (e.g., the creatine phosphokinase assay),
reject any A i2 A,. Because some A , may be rejected by the above conditions,
constant corresponds to the slope of the linear component of the absorbance curve). Aside from just calculating an activity value, an algorithm should also give some indication as to how reliable that calculated value is, based on the data obtained. Exhaustion of assay reagent substrates or instrument malfunction can result in data which do not precisely define the enzyme activity. Fortunately, the variance of the estimated enzyme activity is easily calculated. (5) Calculate the statistical quantities relating to the reliability of the estimated activity:
mGN-1 (3) If m < 4, do not proceed any further, as there should be a t least four data points to properly define the two regression coefficients. (4)With the PI computed in step 2, perform the following regression analysis, outlined previously by Wilkinson ( 5 ) : (a) compute -
XIj =
and then the variance of the regression coefficient,
var(kl) = c1
(22)
The standard error of estimate can then be computed
q
xZj= In -
s
Finally, the coefficient of variation for the estimated activity may be computed:
CV = (100%) (SE/Activity) EXPERIMENTAL The E, are weighting factors necessary because we transformed the independent variable into the dependent variable and also necessary because the early data points usually contain the most information on the enzyme activity. These weights are only approximate, since to use the exact weights would require an iterative procedure. (b) Compute the regression coefficient k l of the function
+kz~2
Y = klx1
(9)
by calculating the following sums: m
a l l = C wjxlj2 j= 1
m
alz= C w j ~ l j ~ z j j=1
m j=
1
(24)
Apparatus. All enzyme activity data used to evaluate the proposed algorithm was obtained with a CentrifiChem analyzer and minicomputer (Union Carbide Corporation,Rye, N.Y.), except for the creatine kinase data. In the latter case a Gilford Stasar I11 (Gilford Instruments, Oberlin, Ohio) was used. Reagents. Reagent kits (Boehringer Mannheim, Indianapolis, Ind.) were used for each enzyme assay: alanine aminotransferase, kit No. 15925; aspartate aminotransferase, kit No. 15923; creatine kinase, kit No. 181188; lactate dehydrogenase, kit No. 15977. Procedure. The principles of operation of the CentrifiChem have been described (6). The procedure followed with the Gilford Stasar I11 was as described by the manufacturer’s protocol. The temperature was maintained at 30 “C for all assays with the exception of creatine kinase which was done at 37 “C. Final concentrations for the assay constituents were as stated in each reagent kit. Sample volume to total reaction mixture volume for the assays were: 0.090, alanine and aspartate aminotransferase; 0.0196, creatine kinase; and 0.024, lactate dehydrogenase. In all of these assays the reaction was initiated by the addition of enzyme (serum sample) t o the reagent mixture.
RESULTS AND DISCUSSION
and
kl = cllql
+ c12q2
(18)
T h e estimated activity is then,
Activity
=
b/kl
(19)
where b is the conversion factor for converting absorbance/ time to U/L. The kl constant is related to the zero-order enzyme velocity, since the reciprocal of this constant corresponds to the saturated enzyme velocity, V, (Le., this
The proposed algorithm was tested by using it to estimate enzyme activities of human serum. The enzyme analyzers and procedures used to obtain spectrophotometric absorbance measurements are described in the Experimental section. In order to determine the accuracy of‘the algorithm, a series of dilutions were made of a human serum with a high activity of a given enzyme. Enzyme activities were estimated by a linear algorithm (2) and by our integrated Michaelis-Menten procedure. The cuvette with the lowest activity (which contained the most diluted serum, of course) was chosen as a reference, and the ratios of the other activities to this activity were calculated. Assuming an insignificant error in performing the serum dilutions, and also assuming no biochemical “dilution effect,” the ratios of activities should correspond to the dilution ratios. In Table I the results of these dilution studies are tabulated for the aspartate aminotransferase (AST),alanine aminotransferase (ALT), creatine kinase (CK), and lactate dehydrogenase (LDH) assays. In each instance, the percent of the expected ratio, based on the serum dilution, ANALYTICAL CHEMISTRY, VOL. 49, NO. 12, OCTOBER 1977
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Table I. Enzyme Activities and Ratios of Activities Determined with a Linear Algorithm and the Proposed Progress Curve Algorithm Linear algorithm Enzyme A ST
Expected ratio"
Activity, UIL
2.50 2.00 1.67 1.43 1.25 1.11
1.00b Mean % of expected ratio
ALT
i
5.00 2.50 1.67
20.00 10.00 6.67 5.00 4.00 2.00 1.00b
Expected ratio
2.38 1.93 1.61 1.38 1.21 1.08 - -.- - - -
95 96 96 96 97 97
?f
794 637 533 443 3 88 345 318
--__.-
2.20 1.90 1.56 1.37 1.21 1.07
2.50 2.00 1.68 1.39 1.22 1.08
__________
953 780 642 537 465 407 381
-.-_.94.3 = 3.3
4.90 2.56 1.69
--.__-_
98 102 101
-.__--
100.3 ? 7.86 6.07 4.68 3.93 1.86 - __ -.- -
i
_._.__
91.0
*
2.50 2.05 1.68 1.41 1.22 1.07
97 98 98
___._.
5.16 2.55 1.69
27 85 1375 912 540
21.51 10.35 6.51 4.74 4.01 1.83 --_-- - -
__
2
2.2
103 102 101
.__-._ ?
1.0
108 104 98 95 100 92 99.5
7.2
1.3
_____.
102.0 9101 4380 2755 2006 1696 776 423
i
100 102 101 98 98 96
99.2
2.1
? 79 91 94 98 93
100 100 100
98.8
88 95 94 96 97 96
..____
SDe
2941 2272 1752 1471 695 374 Mean % of expected ratio -r SDg
Progress curve algorithm Activity , % of Ui L iZatio Expected ratio
96.2 -r 0.8
SDd 2634 1376 911 538
i
Ratio
SDC 854 738 606 533 471 416 388
1.OOb Mean % of expected ratio
LDH
i
2.50 2.00 1.67 1.43 1.25 1.11 1.00b
Mean % of expected ratio CK
7 83 634 530 453 3 99 3 56 3 29
% of
i
5.8
a Expected ratio based on serum dilutions. This serum dilution chosen as reference. AST mean % expected ratios are significantly different, p < 0.01. ALT mean % expected ratios are significantly different, p < 0.02. e CK mean % expected ratios are not significantly different, p < 0.20. f Data too nonlinear for linear algorithm (substrate exhaustion). g LDH mean % expected ratios are significantly different, p < 0.10.
is listed. For the most part, the activities estimated by the integrated Michaelis-Menten algorithm are different than those estimated by the linear method (except for CK, mean 7'0 expected ratios are significantly different, p < 0.101, and more accurately reflect the dilutions. Furthermore, the proposed algorithm gives more accurate results at higher activities than the linear method, increasing the centrifugal analyzer "through-put" by decreasing the need for sample dilution and reprocessing. This is a definite benefit of the new algorithm. The greater accuracy of the integrated Michaelis-Menten algorithm is attributable to its ability to properly handle the nonlinear measurements which may occur in these assays. The most common biochemical cause of such nonlinearities is exhaustion of reagent substrate. This occurs when high enzyme activities are present, resulting in the conversion of a substantial portion of the reagent substrate. For example, a serum sample with a very high alanine aminotransferase activity will result in the depletion of NADH during the course of an ALT assay. As the substrate concentration falls, the enzyme velocity is decreased, and the absorbance change per time interval correspondingly decreases. This results in a curved absorbance vs. time plot. Linear algorithms yield erroneous estimates in these situations, since they are attempting to fit a straight line to nonlinear data (see Figure 1). The integrated Michaelis-Menten algorithm will usually yield a reliable estimate. This procedure fails only when the enzyme activity is sufficiently high that almost all the substrate is converted early in the assay period in these instances, there are not enough acceptable data points (i.e., m < 4) to make a reliable estimate of the zero-order enzyme velocity. 1718
ANALYTICAL
CHEMISTRY,
VOL. 49, NO. 12, OCTOBER 1977
Q
a\ \
\
1 6
i o
__"_
-
:1
I O
@ e 4
Figure 1. Spectrophotometric absorbances obtained for an ALT assay during which the reagent substrate NADH was exhausted. Linear
algorithm regression line is dashed. Progress curve algorrthm regression line is solid It should also be noted that for some assays, such as LDH, the absorbance change with time is inherently curved (3);while linear algorithms have problems with these nonlinear situations, the integrated Michaelis-Menten procedure performs well (see Table I). In addition to accuracy, a measure of the worth of a n enzyme activity algorithm is its ability to distinguish those estimated enzyme activities which are unreliable. The source
above 3% should be considered suspect. In these instances, the absorbance data obtained should be examined, and the sample reprocessed (with dilution, if substrate exhaustion was the source of lack of precision). I t should be noted that the rejection of absorbance values A , C Ah. (for measurement of substrate) or A , > A,v (for measurement of product) could result in biased results if AN were in error (see Equation 4 above). An erroneous final absorbance measurement could cause valid points to be rejected. However, significant bias would result only from a substantial error in Ah‘. In such an instance, the reliability of the estimated activity would be questionable since a large coefficient of variation would probably result from error in the other measurements and/or a small number of accepted data points. Finally, it should be noted that an apparent K , (having absorbance units) may be obtained from a simple extension of the algorithm. Using the nomenclature presented earlier, Figure 2. Spectrophotometric absorbances obtained for an ALT assay during which instrumental noise occurred. The solid line is the progress curve algorithm regression line
of error may be “biochemical” or instrumental in nature. “Biochemical” error refers to extreme instances of the nonlinearities discussed above. In these situations, the substrate concentration is very low (much less than the indicator enzyme apparent Michaelis constant) during almost all of the assay. The enzyme velocity equation, Equation 1, then becomes:
From Equation 4, it can be seen that while the data may define well the ratio V,/K,, the V , constant is ill-defined. Since the enzyme activity is directly related to the V,, there is little information in such data as to the enzyme activity. Instrumental error can have a variety of sources, such as a failing power supply or an improperly washed cuvette. If such a problem uniformly affects the measured absorbances, it may go undetected; as most frequently happens, only a few absorbance readings per enzyme assay are affected, giving rise to scatter in the data (see Figure 2). Both scatter in the data and the amount of zero-order kinetic information in the data w ill be reflected in the standard error of estimate of the regression coefficient, k l . The coefficient of variation, calculated by dividing the standard error by the value for the activity and then multiplying by l o o % , is a useful criterion for judging the reliability of an activity estimate. By processing a number of samples, we have found that an estimated activity with a coefficient of variation
K
= - - k2
hi The constant k l may be calculated by Equation 18 and
k2 = clzql + where
c22q2
(27
cZ2= a l l i d
Since K , is a characteristic of the enzyme system, its calculation could conceivably be used as a check on the validity of a calculated activity. However, certain pitfalls must be considered before such use is made of the apparent K,. First, when the assay is proceeding properly. the enzyme is saturated throughout the assay and the measurements contain little information about the K,. Second, if varying mixtures of isoenzymes are being assayed (as with some serum assays), then apparent K , determinations may rightly vary from sample to sample.
LITERATURE CITED (1) L. Michaelis and M. L. Menten, Biochem. Z., 49, 333 (1913). (2) . , G. P. Hicks. R. A. Ziesemer. and N. W. Tietz. Ciin. Chem. ( Winsfon-Salem. N . C . ) , 19, 27 (1973). (3) H. Gumeund, R. Cantwell, C. H. McMurray, R. S. Criddle, and G. Hathaway, Biochem J . , 108, 683 (1968). (4) B. E. Stathnd and A. L. Louderback, Ciin. Ghern. ( Winston-Salem. N.C.), 18. 845 (1972). (5) G. N. Wickinson, Biochem. J., 80, 324 (1961). (6) D. L. Fabiny and G. Ertingshausen. Ciin. Chem. ( Winston-Saiem, N.C.), 17, 696 (1971).
RECEIVED for review March 7, 1977. Accepted July 11, 1977. This research was supported by Grants GM 16501, RR15, and GM 17522 from the NIH, USPHS.
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