EXPERIMENT 7
ALTERNATIVE DATA-PROCESSING OPTIONS FOR KINETIC DATA
(ENZYMATIC DETERMINATION OF GLUCOSE)
A. PURPOSES
The principal purpose of this experiment is to illustrate the relative merits
of two approaches to processing kinetic data, namely initial-rate and pseudoequilibrium options. On one hand, the initial-rate option is expected to be much
faster than the pseudo-equilibrium option. On the other hand, the pseudoequilibrium option is expected to be influenced less by changes in experimental
conditions than the initial-rate option.
B. OVERVIEW
Many different options have been developed for processing kinetic data.
The most commonly used options include initial-rate and fixed-time methods.
Principal advantages of these options include simplicity and short measurement
times. Principal disadvantages include low sensitivity, large dependencies on
signal noise and large dependencies on experimental variables such as
temperature, pH, and reagent concentrations.
Two less common options include a pseudo-equilibrium option and a tworate option. Principal advantages of these options include higher sensitivity and
less dependence on signal noise and experimental variables. Principal
disadvantages include more complex data-processing requirements and longer
measurement times.
This experiment is designed to illustrate implementation and performance
characteristics of two of these data-processing options, namely the initial-rate
option and the pseudo-equilibrium option.
Experimental data for the enzymatic determination of glucose using
glucose oxidase will be used to evaluate and compare these two data-processing
options. As with the experiment done earlier in the semester, changes in
absorbance, ∆A, are used to monitor the reaction as a function of time. Changes
in glucose oxidase activity will be used to evaluate the effect of variables on each
option.
C. RATIONALE
The rationale for variable dependencies is described below for the initial-rate
and pseudo-equilibrium options.
1. Initial-rate option
For low substrate concentrations, velocities (rates) of enzyme-catalyzed
reactions tend to be proportional to substrate concentration and enzyme activity.
For the glucose reaction, the initial rate of change in absorbance is approximated
by:
1
( dA
) = kγC
dt 0
GO
CG
0
(1a)
where k is a pseudo-first-order rate constant, γ is proportionality constant
between glucose concentration and absorbance, CGO is the activity of glucose
oxidase, and CG0 is the initial glucose concentration. Equation 1a is rearranged to
be explicit in glucose concentration.
0
CG =
( dA
)
dt 0
(1b)
kγCGO
It is apparent from Eq. 1b that undetected changes in glucose oxidase activity
are expected to cause errors in determined values of glucose concentrations.
2. Pseudo-equilibrium option
For a first-order process as illustrated in Eq. 1, it can be shown that the timedependent absorbance, At, is given by:
A = γCG [1 − exp(−kCGOt )]
t
0
(2a)
where all symbols are defined above. At long times (t → ∞), At → A∞, exp(kCGOt) → 0, and 1-exp(-kCGOt) approaches unity ((1-exp(-kCGOt)) → 1).
Accordingly, at long times (t → ∞), Eq. 2a simplifies to:
A = γCG
∞
0
(2b)
In other words, the maximum absorbance is expected to depend on glucose
concentration and to be independent of enzyme activity.
3. Summary
Glucose oxidase, like other enzymes, acts as a catalyst in these reactions.
The observations made above are consistent with the expectation that catalysts
influence rates of chemical reactions but do not influence the total extent of a
reaction at equilibrium.
A goal of this experiment is to determine how this basic principle impacts
quantitative results obtained using different data-processing options.
See Table E-8-1 and Figures E-8-1, 2 in EXP 8_SUPPLEMENT for graphical and
tabular illustrations of effects described here.
D. EXPERIMENTAL
Experimental data will be provided; your only responsibilities are to
process the data and to report your results.
The procedure used to obtain the data you will be processing is identical
to that used for Experiment 3, “Enzymatic Quantitation of Glucose.” The only
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difference between this exercise and procedures you used for Experiment 3 is
that kinetic data were obtained for three different enzyme activities rather than a
single activity. Enzyme activities identified as low, medium, and high correspond
to 34.5, 69, and 138 International Units (U), respectively, of enzyme in each
sample. (An International Unit, U, corresponds to an amount of enzyme that will
catalyze the conversion of one micromole of substrate per minute. µmol/min).
You are advised to review the description of Experiment 3 as you prepare
for this experiment.
E. EXPERIMENTAL DATA
1. Overview
Three data sets of absorbance vs. time for medium (69 U), low (34.5 U),
and high (138 U) enzyme activities are provided in Sheets 1, 2, and 3,
respectively, of an Excel file (EXP 8_DATA). You are required to process only
data for medium enzyme activity in Sheet 1.
Results for the data in Sheets 2 and 3 are tabulated below. The complete
data sets are included for any who might want to confirm results reported below.
2. Data for standards and unknown
Data included in Sheet 1 are for five standard solutions and one
“unknown” solution. Procedures for treating these data are described in the
following sections.
3. Background correction
Data in Sheet 1 (Medium enzyme activity) and Sheet 2 (Low enzyme
activity) have been background-corrected for your convenience by subtracting
the absorbance at t = 0 from absorbances at all times. Data in Sheet 3 (High
enzyme activity) have been included without background correction so that you
can perform your own background correction if you wish.
F. DATA PROCESSING
Two data-processing options, the initial-rate and pseudo-equilibrium
options, will be applied to the data for absorbance vs. time in Sheet 1. Results
obtained by applying these options to data for low and high enzyme activity in
Sheets 2 and 3 of the Excel file are summarized in tables below.
Your first task is to complete Tables 1 and 2 by applying the two dataprocessing options to data in Sheet 1. Your second task is to use graphical and
mathematical procedures to interpret the results. You are free to do these tasks
any way you wish; procedures using Origin are summarized below.
Notes regarding figures: Your report will include several figures. You are
required to label all axes, to identify all plots on each figure, and to include figure
legends that identify the figures. You are not required to fully format figures or to
replot data so that data points overlay fitted lines.
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1. Initial-rate option
The object here is to obtain slopes of plots of data for absorbance vs. time
near the beginning of each response.
a. Calculate least-squares slopes. Highlight (Select) and copy data for
time and absorbance from t = 0 to t = 40 s and paste these data into an Origin
worksheet. Highlight columns containing absorbance data and Click Plot/Scatter.
Then calculate least-squares fits of the six plots (five standards and an unknown)
using the Tools/Linear fit sequence. Include the figure of absorbance vs. time
with best-fit lines in your report as Figure 1.
Record the best-fit values of slopes of the six plots (five standards and an
unknown) in the center column of Table 1 below. Values for the lowest and
highest concentrations are included in the table to help you test whether you are
doing the process correctly. Include this table in your report as Table 1.
b. Calibration plots. After you have completed Table 1, prepare a single
graph containing calibration plots of Rate vs. Concentration for the three enzyme
activities, do least-squares fits of the three plots. Include this figure in your report
as Figure 2. Record the least-squares values of slopes and intercepts with
standard deviations in a table similar to Table 2 below and include this table in
your report as Table 2.
c. Unknown concentrations. To simulate a situation where undetected
changes occur in enzyme activity, use the slope and intercept for the medium
enzyme activity and the rates for the unknown for all three sets of conditions to
calculate three values of the unknown concentration.
d. Concentration error. Using the concentration calculated using medium
enzyme activity as the “true” value, calculate percentage errors for
concentrations calculated using rates for the low and high enzyme activities.
e. Ratios of slopes of calibration plots. Divide the slopes of the three
calibration plots by the slope of the plot for the lowest enzyme activity and record
the ratios in Table 4 given below. These ratios will be used later.
Table 1. Initial-rate data calculated using method of least-squares.
Rate (∆A/∆t (10-4 au/s))
Glucose Concn.
Low Enzymea
Medium Enzymea
High enzymea
-5
(10 mol/L)
(5)
(10)
(20)
5
2.57
5.45
7.84
10
4.89
14.4
14.4
7.57
20.9
19.4
10.0
27.3
24.9
13.3
25.2
35.2
Unknown
7.70
21.5
a
Enzyme activities: 34.5, 69, 138 U.
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Table 2. Slopes and intercepts for calibration plots and concentrations calculated
using these results.
Initial-rate option
Pseudo-equilibrium option
Enzyme
Slope (sd)
Intercept (sd)
Slope (sd)
Intercept (sd)
activity
Low (34.5 U)
Med. (69 U)
High (138 U)
Calculated
Concentration
Calculated
Concentration
concentration
error (%)a
concentration
error (%)a
Low (34.5 U)
Med. (69 U)
0.0
0.0
High (138 U)
a
Percentage difference between concentration calculated for medium enzyme
activity and concentrations calculated for low and high enzyme activities.
2. Pseudo-equilibrium option
For reasons to be explained in lecture, responses of absorbance vs. time
follow combined zero-order/first-order kinetic behavior. Objects of this part of the
experiment are to use a curve-fitting method to resolve the first-order component
from the zero-order component and to relate the maximum value of the first-order
component of the response to concentration.
a. Plotting/fitting data. Highlight and copy all the time and absorbance data
in Sheet 1 of the Excel file and paste these data into an Origin worksheet.
Highlight the six columns of absorbance data and Click Plot/Scatter to generate a
plot containing all six data sets.
Follow procedures for fitting nonlinear curves in the Origin tutorial to fit a
model for simultaneous zero-order/first-order processes to each of the data sets.
The model for the absorbance change for combined zero-order/first-order
processes is:
At = A0 + A∞ (1 − e − k1t ) + k0t
(3a)
where At is absorbance at time t, A0 is the absorbance at t = 0, A∞ is the
maximum value of the first-order component of absorbance, k1 is a pseudo-firstorder rate constant, k0 is a zero-order rate constant, and t is time. The maximum
value of the first-order component of the response, A∞, is expected to vary
linearly with glucose concentration.
The relationship in Eq. 3a implemented in Origin syntax as follows:
P1 + P 2 * (1 − EXP(− P3 * X )) + P 4 * X
(3b)
where P2 is the maximum value of the first-order component of absorbance (P2
= A∞).
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Suitable initial estimates for the four fitting parameters are: P1 = 0.01, P2
= 0.1, P3 = 0.008, and P4 = 0.00004. Assuming that you proceed with all the fits
at one time, it should be necessary to enter these initial estimates only once
because best-fit parameters for each data set should be satisfactory for each
subsequent data set. Start by clicking 1 Iter once and then Click 10 Iter
repeatedly for each plot until fitting parameters remain constant.
See Figure E-8-3 in EXP 8_SUPPLEMENT for a graphical illustration of a typical
fit with resolved zero-order and first-order components included.
Enter best-fit values of P2 into Table 3 for each of the solutions. The
expected values for the lowest and highest concentrations are included to help
you confirm that you are implementing the process correctly. Include Table 3 in
your report as well as a graph containing the six response curves with best-fit
lines as Figure 3.
b. Processing data. After you have completed Table 3, use procedures
similar to those described in Steps F-1-b to e for the initial-rate option to process
these data. Include results in the two columns headed “Pseudo-equilibrium
option” in Tables 2 and 4.
Table 3. Maximum values of first-order component of absorbance vs. time
response calculated using deconvolution method.
Absorbance change (P2 = A∞)
Glucose Concn.
Low Enzymea
Medium Enzymea
High enzymea
(10-5 mil/L)
(34.5 U)
(69 U)
(138 U)
5
0.0756
0.0779
0.0791
10
0.134
0.142
14.4
0.191
0.207
19.4
0.253
0.258
24.9
0.333
0.352
0.339
Unknown
0.1932
0.1937
a
Enzyme activities: 34.5, 69, 138 U
4. Comparison of data-processing options
The purpose of this section is to compare the ruggedness of the two dataprocessing options described above.
a. Tabulated ratios of calibration slopes. You will have tabulated (Table 4)
ratios of calibration slopes for the initial-rate and pseudo-equilibrium options.
Include Table 4 in your report.
b. Plot of ratios of calibration slopes. Plot the ratios of calibration slopes
(Table 4) vs. enzyme activity (34.5, 69, 138 U) on a single graph, and do a leastsquares fit of each plot. Include this graph with the least-squares lines and the
slopes of these lines in your report as Figure 4. Include the slopes of these plots
in the last row of Table 4. These slopes are identified in the remainder of this
document as Relative Error Coefficients, REC’s .
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c. Interpretation. It can be shown that the slopes of the plots described in
the previous paragraph represent relative concentration errors (RCE’s) per unit of
error in enzyme activity. When multiplied by 100, the slopes have units of (%)/U.
Accordingly, these slopes are called Relative Error Coefficients, REC’s. For
an error, ∆U, in enzyme activity, the relative concentration error, RCE (%), is
given by:
RCE (%) = 100
∆C
(%) = 100 x REC (% / U ) x ∆U
C
(2)
in which REC ≡ slopes of ratio plots. Use this equation and values of slopes
(REC’s) of plots of in the last row of Table 4 to calculate concentration errors
corresponding to an enzyme-activity error of ∆U = 3 U. Do the calculation for both
data-processing options. Include calculated values of relative concentration
errors for the two data-processing options in your report. Use plots in Part F-4-b
to rationalize differences in concentration errors caused by a fixed error in
enzyme activity.
Table 4. Slopes and ratios of slopes for calibration plots at low, medium and high
enzyme activity.a
Initial-rate option
Pseudo-equilibrium option
Enzyme
Slopes of
Ratio of
Slopes of
Ratio of
b
activity
calibration
slopes
calibration
slopes
plots
plots
Low (34.5 U)
1.00
1.00
Med. (69 U)
High (138 U)
Relative Error Coefficients (REC’s).
Slopesc
a
Enzyme activities: 34.5, 69, 138 U
b
Ratios of slopes for medium and high enzyme activities to the slopes at low
enzyme activity.
c
Slopes of plots of slope ratio vs. relative enzyme activity.
G. REPORT
In addition to the usual information, your report should contain all the
figures and tables described above, excluding those in the SUPPLEMENT.
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