Molecular biologists and biochemists periodically want to determine the activity and kinetics of enzymes. In order to determine the activity, plots of the data are ...
Vol 6 no.2. 1990 Pages 6 3 - 6 5
Computer programs for the rapid determination of enzyme kinetics on MS-DOS compatible microcomputers R.Myers1, S.OIser)2 and S.Maloy*
Molecular biologists and biochemists periodically want to determine the activity and kinetics of enzymes. In order to determine the activity, plots of the data are usually drawn by hand and fitted by eye. In order to determine the kinetics of the enzymatic reaction, the activity data are subjected to various replots (also hand-drawn) and fitted by eye to estimate the kinetic parameters. These graphical approaches to study enzymes are arduous and prone to line-fitting errors. Professional enzyme kineticists (somewhat rare these days) infrequently draw plots, relying instead on powerful computer programs that combine numerical analysis of the data with various statistical routines. These programs often run on minicomputers and time-sharing machines which are not convenient for researchers that do not usually require large amounts of computation power. In contrast, microcomputers are found in most laboratories, and molecular biologists are becoming increasingly comfortable with them for word processing and data analysis. With this in mind, we wrote two FORTRAN progams on MS-DOS compatible microcomputers for quickly calculating enzyme kinetic parameters. In order to determine the specificity (V^JK^, maximum velocity (Kmax), and affinity (Km) of an enzyme for a given substrate, the kinetic parameters are calculated from the initial velocity (V,) of the enzymatic reaction at several substrate concentrations (5,) (Cornish-Bowden, 1979). The programs MEDIAN and HYPER are variations of previously published programs (Cornish-Bowden and Eisenthal, 1974; CornishBowden et al., Cleland, 1963) for determining enzyme kinetics from such data. We rewrote them in FORTRAN 77 using IBM/Ryan-McFarland Professional FORTRAN so they could be run on the now ubiquitous MS-DOS compatible microcomputers. Since we used this particular FORTRAN compiler, a math co-processor is required to run the programs. The program HYPER calculates Km, VmiX and their standard errors by a least-squares fit of enzyme activity data to a hyperbola defined by the Michaelis-Menten equation: Vm s i * S
as described (Cleland, 1963). This program fits (V,, S,, W,) values to the Michaelis-Menten equation through five iterations and then calculates the standard errors. The weights (W-t) may be omitted if they are all the same. If the weight value is omitted, Wt is set to 1.0. Otherwise, an appropriate weighting scheme can be inserted (eg. Wt = V} when errors are a fixed percent of V). HYPER also calculates the variance of the data from the fitted curve. A sample data file and output of HYPER is shown in Figure 1. The program MEDIAN fits enzyme activity data to the Michaelis-Menten equation according to the Median method (Cornish-Bowden and Eisenthal, 1974). Briefly, the program calculates the point of intersection of lines defined by V.
y=
—
*.V+
K,
where V, is the initial rate of the enzyme reaction at substrate concentration 5,. The program derives values for the Michaelis constant ( / i j and the maximum rate of the enzymatic reaction (Kmax) from the median point of intersection (x,y): Km corresponds to the median x value and V^ corresponds to the median y value. If the number of intersections is even, the median value is calculated as the mean of two median points. The program also calculates the points of intersection that define the upper and lower borders of the 68% confidence limits using the normal approximation to the distribution of Kendall's score (68% is taken as 1 standard deviation of Kendall's score, which corresponds to standard errors in least squares estimation to allow direct comparison of the two approaches). A sample data file and output of MEDIAN is shown in Figure 1. The programs complement each other in their approaches: HYPER determines enzyme kinetics via a least-squares fit of the data, while MEDIAN uses a non-parametric (distributionfree) approach. The HYPER method rests on a number of assumptions which can potentially mislead the researcher, MEDIAN relies on few assumptions. MEDIAN generally gives good estimates of the kinetics (assuming good experimental V = design) but when all of HYPER's assumptions are satisfied and Km + S an appropriate weighting scheme is selected, HYPER gives slightly better results. The relative strengths and weakness of Department of Microbiology, University of Illinois, Urbana, IL 61801, USA least-squares versus non-parametric methods of data analysis 1 Present address. Institute of Molecular Biology, University of Oregon, Eugene, have been exhaustively debated (e.g. Cornish-Bowden, 1979). OR 97403, USA 'Present address' International Biotechnologies, Inc., 275 Winchester Avenue, However, the kinetic parameters calculated by HYPER and New Haven, CT 06535, USA MEDIAN agree with each other and with the values estimated 63
R.Myers, S.OIsen and S.Maloy
C
A
I 6
DAT* SET
13
B FIT 10 HYPERBOLA V ' V M U S / I K t S I 68 PERCENT INTERVAL FRON
3 TO 11
DATA SET I 1 LONER BOUN1
BEDIM
1UPPER BOUND
1. •2N
I.I2II
I.I2II
» Pro Oiidiu Kmitici
3. I3N
3.1511
3.1311
157.49947
137.49994
137.49998
49, 999*
49.9999
3I.IHI
I/VM1
1. 3173
1.3173
1.3173
KR/VRAI
1. M63
I.IM3
I.IU3
VIMI VIWI/KH
K
•
I.I2II36 S.E.(K)
I.IMI42 I •
I.!3!69E»I9
V
•
3.151811 S.E.IVI
1.111631 I •
I.3U73E*«
MI6363 S.E.U/VI
l.llllll I •
I.7933*E»II
I/UN
K/V • 1/V .
1.317379 8.E.I1/VI •
I.IM166 I •
l.3il«4E*N
V/K •
157.114233 S.E.IV/M •
1.276733 « •
I.I3I34M2
VARIANCE •
V
I.4I233E-I3 SI6RA •
S
BEST V
POINT
S
DBS V
CALC V
ERROR
REL ERROR
1
Mil
1.131
1.131
I.IMM
1. HIM
2
1.121
1.371
1.573
-I.II3II
3
1.141
2.IM
2.IM
I.IIIM
1. HIM
4
I.M
2.321
2.521
I.IMM
1. Mill
3
1.141
2. I N
2.BN
I.IIIM
1. HIM
4
I.IM
3.188
3.188
I.IMM
1. Mill
I.II2N63
1/V
I/S
BEST 1/V
S/V
BEST S/V
K1BHT
1.I3IM
I.IIIH
t.l«H
1.73231 1 M . I I M
I.93J9I
I.11932
I.IRM
LIMN
I.S7IN
I.I2IM
1.37322
1.63*94
1.63344
1.11274
1.11271
l.INN
3I.NNI
2.HIM
I.I4IN
2.I9M
1.47619 2 3 . I N N
1.47451
I.II9K
I.II9I6
l.INN
2.32IN
I.I8IN
2.31923
1.39*13
12.3NN
1.39694
1.13173
1.13176
l.INN
2.BNN
l.lilN
2.79984
1.33714
6.25NI
1.35716
MS7I4
1.13713
MINI
3.I8B24
l.INN
3.1888*
1.32381
l.INN
1.32374
1.32311
1.32374
LIMN
PROSRM COMPLETED FOR (ATA SET
PMffiAH COMPLETED FOR
-1. •1318
I
1 OUT* SET
Kig. 1. Sample data file and output from the enzyme kinetics programs HYPER and MEDIAN. Simply type HYPER to call up the least-squares program or MEDIAN to call up the non-parametric program. A screen prompt requests the name of the data file. The data file should contain the initiaj velocity of the enzyme reaction for each substrate concentration tested. The initial velocity can be calculated on a spreadsheet, or estimated from a graph of product formation versus reaction time. Details about the format of data files are provided. (A) Sample data file with the initial velocities of proline oxidase activity at six proline concentrations. (B) Output from HYPER using the sample data file. HYPER calculates Km, ymal, three transformations of Km and l^.,,, the standard errors (S.E.) of each calculation, and W(S.E.2). VARIANCE is the variance of the data from the fitted hyperbola and SIGMA = (VARIANCE)2. S = the substrate concentration, V = the measured velocity of the enzyme reaction for a given S, BEST V = the expected initial velocity of the enzyme reaction at S as calculated from the least-squares fit of the data to the Michaelis-Menten equation (see text), and WEIGHT = the weighting values for the data (see text for details). (C) Output from MEDIAN using the sample data file. MEDIAN calculates the median Km, Kmax and four transformations of Km and Vmax. The lower and upper bound values indicate the 68% confidence interval (see text for details). S = the substrate concentration, Obs V = the measured initial velocity of the enzymatic reaction for a given S, and Calc V = the expected initial velocity of the enzyme reaction at S as calculated from the Michaelis-Menten equation (see text) using the median values of Km and l/max. Error is the difference of Obs V and Calc V, and Rel error is the percent deviation of Obs V from Calc V as defined by: Rel error = (Obs V/Calc V) - 1
64
Enzyme kinetics for microcomputers
from linear transformations of the Michaelis —Menten equation (e.g. Lineweaver —Burk, Hanes, Eadie — Hofstee) for welldesigned enzyme kinetics experiments (R.Myers, D.Townsend and S.Maloy, submitted). These programs and documentation are available on request from the authors for the cost of supplies and postage. Included on the diskette is the source code, object file, and executable file for each program, and a sample data file (MOCK) to run through the programs. Acknowledgements This work was supported by NIH grant GM39096 to S.M.
References Cleland.W. (1963) Computer programmes for processing enzyme kinetic data. Nature, 198, 463. Cornish-Bowden.A. (1979) Fundamentals of Enzyme Kinetics. Butterworth, London. Cornish-Bowden.A. and Eisenthal.R. (1974) Statistical considerations in the estimation of enzyme kinetic parameters by the direct linear plot and other methods. Biochem. J.. 139, 721-730. Cornish-Bowden.A., Porter.W. and Trager,W. (1978) Evaluation of distributionfree confidence limits for enzyme kinetic parameters. J. Theor. Biol., 74, 163-175. Received on September 21. 1989. accepted on September 25. 1989
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